Principles of Magnetic Resonance (3rd Enlarged and Updated Edition)

(4.140)

The notation f(k, s) means f(E), of course, where E is the energy of an electron with wave vector k and spin coordinate s. Typically, E = Ek+Espin, where Espin is the energy associated with the spin orientation and Ek is the sum of the kinetic and potential energies of an electron of wave vector k. (We shall call Ek the

f(E)f---_~

(aj

f(E)

'---=<-]l

(b)

i\ £'1,.

E

E j ,'

E

Fig. 4.7. (a) Fermi rundion feE) at absolute zero. (b) Fermi rundion at a temperature T above absolute zero 118

(4.141) As we can see, the quantity in the brackets is (apart from a minus sign, since IJ- e = ~l'enS) the average contribution of state k to the z-component of electron magnetization of the sample. We shaH denote it by I'z k. The towl zmagnetization of the electrons, /-lz, is then '

~=L/-lzk

(4.142)

.

k

If we define the total spin susceptibility of the electrons,

Ilz = X~Ho

x~,

by

(4.143)

and define a quantity

xi:

by

-/-lzk=Xk 'H0

81r 2" 2 31'c1'nf, Izj L.!Uk,s(O)! msf(k,s) k,s

f(E) =

translational energy of the electron). For example, Espin is the Zeeman energy of the electron spin in the static field Ho. There may be other contributions to Espin, however, from the electrostatic coupling between electrons, which depends on their'"'Telative spin orientation. If we consider a typical tenn in the sum of (4.139) corresponding to a single value of k, there are two values of m s , giving us

(4.144)

then (4.142) is equivalent to

X~ =

Lxi:

(4.145)

k

We can therefore write (4.141) as

- 8 1r l'n nIzj !u,.:eO) 12xi: H o 3 so that the total effective interaction for spin j is

8, H zj '" -31'n L. 1 Uk(O) I" XkHO

(4.146)

(4.147)

k

Our problem now is to evaluate the summation. It would be simple to make the assumption of completely free electrons, but it is actually no harder to treat the case of a material with a more complicated band structure. We shall do the laller, since the resulting expression will enable us to make some important comparisons between experiment and theory. We assume, therefore, that the energy of the eleclrons, apart from spin effects, is detennined by its k vector. For free electrons, the dependence would be

(4.148) so that in k-space, all electrons on a sphere of given k would have the same energy. That is, the points of constant energy fonn a surface, in this case a sphere 119

"

/

"

for any slates k having the same value of translational energy E1.:' Even when we allow the electrostatic coupling between the electrons to affect the energy to tum over a spin, it may be reasonable to assume Ihm this modification depends at most on Ek- We therefore assume that is a function only of the energy Ek: Xl(E k ) (4.151)

"

xl

'.

'. "(b)

(0)

xl::o

"/

We

Fig.4.8a,b. Intersection of two surfaces of constant energy wilh the k. = 0 plane. (:1) Circular section of a free elCdron. (b) A less syrnmetric seetion for a hypothetical "real" substance

(Fig. 4.8). In general the effecl of the lattice potential is to distort the surfaces 2 varies slowly as one from spheres. We shall assume that lhe function IUk(O)1 moves the point k in k-space from one allowed k-value to the next, so that we can define a density function to describe the number of allowed k-values in any region. Let us define 9(EI.:' A)dEkdA as the number of allowed k-values lying within a certain region of k-space, defined as follows: It is a small cylinder lying between the energy surfaces Ek and Ek + dEk (Fig. 4.9). Its surface area on the top or bottom surface is an element of dA of the constant energy surface. We denote the particular coordinates on the surface also by the symbol A in 9(E, A). The total number of states dN between Ek and Ek + dEk is found by summing the contributions over the entire surface:

therefore rewrite (4.150) as

Clln

L IllkCO)!2 Xt = JIUk(O)12Xs(E,,)g(Ek,A)dAdEk

•

If we have any funclion F of E, we define ils average value over the surface of constant translational energy Ek' (F(k)}e • as

xt

(4.149) We can use these functions to evaluate the summation by replacing it with an integral:

L

•

1",(0)1' xl, =

J

1".(0)I'xk9(E•• A)dE.dA

(4.150)

Now xl: depends on the Fermi functions f(k,~) and f(k,-~) :ll1d thus on the energy E1.:, and on the difference in energy of a spin in Slate k with spin up versus that with spin down ('YeftHo for free electrons). Therefore would be the same

xl

Fig.4.9. Volume in k-space associated with dEdA

JJ

g(E•• A)dA

J

F(k)y(E•• Jl)dA

J

(4.153)

1".(O)I'y(E•• A)dA = 9(E.)(I".(0)I') E,

(4.154)

J(1".(0)1')

(4.155)

giving

L

•

1".(O)I'xk =

E,X'(E.)g(E.)dE.

Now XS(E k ) is zero for all values of E k that are not rather ncar to the Fenni energy, since for small values of E k the two spin stales are 100 percent populated, whereas for large Ek' neither spin state is occupied. XS(EIJ must look much like Fig. 4.10. XS(Ek) will be nonzero for a region of width about kT around the Fenni energy EF. We are therefore justified in assuming (Itt~:
L

•

",,(O)I'xk = (1".(0)1') E,·

J

X'(E.)g(E,)dE.

(4.156)

The integral remaining in (4.156) is readily evaluated in teons of (4.145), since =

.'CE,JIL

LXt =

• J

J

xl:y(Ek' A)dEk dA

(4.157,)

x'(E.)9(E•• A)dE.dA

A

L+.''---_--:OEJ!

120

1 = -(E) 9.

ll1en we can set the integral ovcr dA in (4.152) equal to

=

.,

-,

F(k)9(E•• A)dA

(F(k» E, =

x~

"

(4.152)

Fig.4.10 Function XS(E",) versus E",

E k 121

which, performing the integral over A, becomes

x::::: JXS(Ek)e{E,,:>dEk

(4.157b)

Therefore, using (4.147), (4.156), and (4.157), we may say that the interaction with jth nuclear spin is

8.

,.]

(4.158)

-'Yn lllzj [ 3(1l1k(0)1 ) EpXeHo

This is entirely equivalent to the interaction with an extra magnetic field aH, which aids the applied field Ho, and is given in magnitude by the equation

aH

8'11"

2

(4.159)

We see that this formula has all the correct propenies to explain the exper· imental results. I) It predicts that a higher frequency is needed for the metal than in the diamagnetic reference. 2) The fractional shift is independent of w. 3) Since both (luk(0)12) Ep and are independent of temperature, all/Ho is as well. Since the larger·Z atoms will have a larger value of (llIk(0)1 2) Ep

r

corresponding to the pulling in of their wave function by the larger nuclear charge, the in<:rease of Knight shift with Z is explained. The Knight shift fonnula can be checked if one can measure independently aH/Ho, (Itlk(O)12)Ep and.t:. There is only one case for which all three quan· tilies are known (Li metal). The spin susceptibility has been measured by Schumacher J4.7] by a method we will describe shonly. Ryler [4.8} has measured (l"k(O)1 ) E by measuring the shift of the electron resonance by the nuclear moments. This shift,LlHe, is given by

8. (I Uk (0)1') EpXn • Ho :::: 3

(4 . 160)

IJ.H,

where X~ is the nuclear susceptibility of the U 7 nuclei. Denoling the number of nuclei per unit volume by N, we have ::::

N"(~1t2 I(I + I) 3kT

(4.161)

)L..

Thus, since X~ is known, mcasurement of LlHe gives (11l/.:(O)1 2 In order to enhance the size of the shift, RYler polarized the nuclei, using the so·callcd Overhauser effect (see Chap.7). It is necessary then to modify the formulas slighdy, but the principle remains the same. For comparison with experiment it is convenient to compute (lu/.:(O)1 2) E. using wave functions normalized to the atomic volume, which we shall cafl 122

X:.

Table 4.1 5

Ho = T(I"k(O)1 )E,X,

s Xn

PF. We shall use PA to denote the wave function density at the nucleus for a free atom. It is then convenient to discuss the ratio PF/PA for Li and Na. A comparison of RYler's values, theoretical values, and values deduced from combining SchwlIacher's measurement of X~ with measured Knight shifts is given in Table 4.1. There is excellent agreement among all three values. For Na, Kolm and Kjefdaas [4.9,10] find PF/PA = O.80±O.03. We can combine this value with the measured Knight shift to obtain a quasi-experimental value of ~. Before presenting these results, we shall describe Schumacher's direct measurement of

l\,/PA in Li 0.'19 ± 0.05 0.45 ±0.03 0.442±0.015

Kohn and KjetdafU lheordical Experimenlal ptus Knigh~ shift.) Ryter (experimenlal)

ex:

x:

The fundamental problem in measuring is how to distinguish it from the other contributions to the total susceptibility, which (in a melal) are quite comparable in size. The method employed by SCluwlOcher is to isolate the spin contribution by use of magnetic resonance. From the Kramers·Kronig relations we have for the electron spin susceptibility .t:: - 2 JX~dw x:-'. w

=

(4.162)

o

where X~ is the imaginary pan of the conduction electron spin susceptibility. For a sufficiently narrow resonance, we may neglect the variation in w across the absorption line, taking it out of the integrnl. We can then change the integration from one over frequency to one over field, using w:::: "(H: X• :::: -2 - I e lI" W o

= J "dw Xe

o

'J= "IH

1

= -2 -

'll"WO

Xe (

(4.163)

0

An absolute measurement of the area under the resonance curve will therefore enable one to determine X~. (Actually, the approximation of a narrow resonance is not well fulfilled in Schumacher's case; however, the final formula can be shown still to be correct. For a discussion of these points the reader is referred to Schumacher's paper [4.7]. We may also note that the resonant absorption of energy at the cyclotron frequency occurs degenerate with that of the spin. However, the rapid electron collisions broaden it so much as to render it unobservable. One may be confident thai it is only X~ that is being measured.) Absolute measurements of absorption are always very difficult. Schumacher circumvented them by making use of the nuclear resonance of {he Li7 or Na 2J 123

nuclei in the same sample for which he measured the conduction electron resonance. For the nuclei, onc has a spin susceptibility X~I' given by (4.161) lind for which X

s=~ n

00

"Ynj X"dH

1T WO

(4.164)

n

Ho = O. as with ferromagnets or anliferromagnets. lei us discuss the case of ferromagnetism briefly.~ Once again the electron-nuclear interaction will consist of the sum of the convellfional dipolar coupling 1td and the 8-state interaction 1ts : (4.166)

o

By choosing wo/2-;r So' 10 Me/s, Sc/llullacher could observe either the electron or nuclear resonances simply by changing HOI the remainder of the apparatus being left unchanged. The nuclear resonance occurred at abollt 10,0000au55, whereas the electron resonance was al only a few Gauss. Then, if we denmc the area under the electron or nuclear resonances by A c or An, respectively, we have

We shall average this over the electron wave function .,pe, as with the Knight shift, to get an effeclive nuclear Hamiltonian 1t~n' which will contain the nuclear spins as operators:

1t~n =

j .,p~(1td + 1ts).,pedre

X~ = 'Ye A c X~l 1'n An

(4.165)

Since X~I can be computed, we are thus able to determine X~. Note that we can measure the "area" in any units we wish (such as square centimeters on the face of an oscilloscope) as long as they are the same for bOlh resonances. We do not even need to know how much sample we have, since it is lhe same for both electrons and nuclei. The experimental values obtained are listed in Table 4.2 IOgether with various theoretical values. The first column of experimental and theoretical numbers [4.5] gives theoretical values based on non interacting electrons, blll an effective mass has been inttoduced to take account of the lattice potential. The effective masses are computed by Harvey Brooks, using the quantum defect method. The second column shows the theoretical values obtained by Sampson and Seitz, who took into account the electron-electron coupling by means of an interpolation formula of Wigner. The next column shows theoretical values due to Pines, based upon the Bohm-Pines collective description. Next we give values obtained by using the Knight shift and the Kohn-Kje/daas theoretical values of PFIPA . The last column shows Schumacher's results. (See [4.11] for subsequent resulls.)

where dre stands for integration over all electron coordinates (spin and spatial), and where the symbol G signifies that the spatial integration goes over the entire physical volume of the sample. By breaking G into the atomic cells G I, G2, ... GN of the N atoms of the crystal, we may fonnal1y interpret the contribution of 1td to (4.167) as arising from the summation of the dipolar fields of the electrons on the various atoms. The wave function .,pe will provide a detailed picture of the spatial distribution of the electron magnetization in each atomic cell. Evaluation of this tenn is identical to computing the local field due to a volume distribution of electron magnetization. Suppose the magnetization is unifonn throughout the sample. If the lattice has cubic symmetry, 1td will contribute an effective field given by the Lorentz local field:

4, -M -

3

_ a ·M

I,i Na

Sampson and Seitz

1.17

2.92 1.21

0.04

-

."j' WeSjo(r/- Rj)1/JedTe

8<, e h L H&j = - 3

Bohm-Pines

Knight shift. and theoretical Pp/PA

Schumacher

1.87 0.85

1.85±0.20 0.83 ± 0.03

2.08±O.1O 0.95± 0.10

If we took the wave function to be a product of one electron stales of quantum numbers fl, we would have, then,

8,

We point out that if Ryter' s value of PFIPA is used, the Knight shift value is raised, providing excellent agreement wilh that of Schumacher. The Knight shift calculation is closely related to the problem of the nuclear resonance in samples in which the electron magnetiz..1tion does not vanish when

(4.169)

I

H,j =

124

(4.168)

where M is the magnetic dipole moment per unit volume and 0' is a demagnetizing factor (in general a tensor) that expresses the effect of the "magnetic poles" on Ihe outer surface of the sample. For example ;; = 471i3(ii + jj + kk) for a sphere. The s-state lenn may be interpreted as contributing a magnetic field H&j at the jth nucleus.

Tabte 4.1 x: (all values are 106 cgs volume unit..s) Free eledrons

(4.167)

G

-310"

L:

(PIS,(r - Rj)IP)

111) of II

set

(4.170)

fJoccupied

~ See the references lisled under "Nuclear Hesonance in Ferromagnet.s and Antifcrromagncts" in the Bibliography.

125

"

where "occupied" means that we include in the sum only those states 1m containing an electron and we have omiued lhe subscript 1 from Sand r. By using the fact thai Ile = -"YenS, we have

8. H,j = 3"

'" L

(PI".'(r - Rj)IP)

(4.171)

lJoccupicd

Since the matrix element involves coordinates of only one electron, the various values of I now appear as Ihe values of p Ihal 3rc occupied. In a substance such as iron, we may think of some values of {3 as corresponding to closed shells, some to the 3d band, and some to the 48 band. We shall discuss these contriblllions shortly. The contribution of the tenn 7-{d is somewhat different for a ferromag-

net than for a paramagnet. For the latter, the magnetization is tlllifonn in both magnitude and direction for ellipsoidal samples, nnd the simple dcmugllclizing arguments follow. For a ferromagnet the magnetization within a domain is uniform, but the various domains have differing magnetization vectors. Thus, for a soft ferromagnet in zero applied field, the magnetization averaged over a volume large compared with the domain size is zero. The density of magnetic poles on the outer surface therefore vanishes. Within the body of the ferromagnet, div M "" 0 even at domain boundaries. If, then, we calculate the dipolar contribution to the magnetic field al a nucleus, we may proceed as follows. We draw a small sphere about the nucleus, of radius small enough 10 lie within one domain. We compute the field due to magnetization on atoms within the sphere by a direct sum. The atoms outside the sphere are treated in the continuum approximalion. For cubic symmetry, the atoms within the sphere give zero tOial contribution. The atoms outside the sphere contribute as a result of the surface pole density on the inner sphere and the Ollter sample surface. The former is the contribution 411" MI3, where M is the magnetization within the domain containing the nucleus. The latter contributes - ;; . M ' , where l\t! is the magnetization averaged over a volume large compared with ;\ domain size. TIle total field seen by the jth nucleus HTj is therefore given by

4.

HTj""Ho+

3

_ M-

0'

'M+Hsj

(4.172)

where lIo is an externally applied field. Although lIo and M ' vanish in zero applied field, HTj does nol. Therefore we have a "zero field" resonance. Such a resonance was first observed by Gos.tard and Portis [4.12] in the face-centered cubic fonn of cobalt. Using the Co59 resonance, the measured Hsj = 213,400Gauss. In iron, Hsj is 330,000 Gauss. Hsj has also been observed by means of the Mossbauer effect. It was discovered there that application of a static field H o lowered the resonance frequency, showing the fIsj points opposed to the magnetization M. The contribution from the 3d and 48 shells in iron is expected by Marshall [4.13] to give field of 100,000 (Q 200,OOOGauss parallel to the local magneti126

zation. Therefore the inner electrons must give a field of about 400,OOOGauss opposed to the local magnetization [4.14]. This phenomenon, called core polarization, was actually already known from e.lectron magnetic rcsonan.::e of paramagnetic ions for which the 48 electrons are missing. In principle the 3d electrons are incapable of giving an isotropic hyperfine coupling, since d-states vanish at the nucleus. However, the d-electrons are coupled to inner shell electrons electrostatically, the coupling for an inner electron of spin parallel to the d-electron spin being different from that of an electron whose spin is opposed 10 that of the d-electron. Consequently the spatial part of two wave functions such as the 38 are different for the two spin states. The spin magnetization of the two electrons does not add to zero at all points of the electron cloud. We can see from (4.171) that if the 3s electron densities at the nucleus differ, there will be a nonzero contribution from the 3s electrons to Hsj, even though their spins are opposed.

4.8 Single Crystal Spectra

"

We saw in Sect. 4.5 that the chemical shift for a p-state such as xf(1") depended on the orientation of the static field lIo, with respect to the bond axes. If Ho was either parallel or perpendicular to the bond direction, an extra field was induced, acting on the nucleus, which was parallel to Ho. For an arbitrary orientation, one can resolve Ho into X-, y-, and z-components. Ho z , Hoy, Ho z , in general gClling an interaction of extcrnal field with nuclcar spin involving bilinear products such as Hozly etx. Blocmbergen and Row/alld [4.15] showed theoretically and experimentally that the same thing is (nle for a Knight shift if one includes the conventional dipolar coupling (4.98) of the nucleus with the spins of the conduction electrons in addition to the Femli contact tcrm. Although the conventional dipolar tenn does not give rise to a net shift of the resonance if one averages over all orientations of Ho with respect to the crystal axes, as for a powder sample or as in a liquid metal, it nevertheless gives an orientation dependence to the Knight shift. To deal with these anisotropies, one expresses the bilinear form with coupling coefficients which are components of a tensor interaction. Then, writing the Knight shift J( and the chemical shift CT as tensors using a dyadic notation where

-

CT = iCTui + iuzyj + iCTzzk + jCTyzi + jCTyyj + jCTy:k

+ kCT;zzi + kCTzyj + kCTzzk and similarly for

J( ,

we get a Hamiltonian

-

'H = -'YliHo'( I - ';; + where

(4.173)

J(

)·1

(4.174)

I is the identity dyadic 127

2 2 ]1/2

=ii+ 11 + kk

+(1-uzz+Kzz) az

(4.175)

Explicit evaluation of u er /3 and I<':X/3 (a = x, y, z; both tensors are symmetric

fJ '"

x, y, z) shows that

Since the shift parameters ux X, Kxx, etc. are in general very small compared to I ,_and since 0').- +O'} +a~ = I, weean rewrite (4.183) to get

(4.176)

W = 'YHo[l + (Kxx - uxx)ak + (l{yy - Oyy )a~ + (Kz z - uzz)a~]

etc.

so that one can find principal axes. Denoting these axes by X, Y, Z with unit vectors i p , 1p ' and k p , we have

'; =ipuxxip+jpoyyjp+kpuzzkp

(4.177)

-

~

and similarly for K. In general, K and '; do not have to have the same principal axes, but we will assume for simplicity that they do. Then if Ho lies along a principal ax.is the resonance frequency is

= wo +waak +wba~ +wca~

'lip H = Ho +Hp

(4.186)

Then with Ho = kHo, where k is a unit vector along the laboratory z-direction, we get

(4.178)

Wz ='YHo(l-dzz+I
(4.185)

The approximation that the shifts are small could in fact be introduced in the original Hamiltonian. 1l1en we write 11. as a large part 1lo plus a penurbation

(4. 187a)

Ho = -jnHolz

wX='YHo(l-oxx+Kxx) :::::Wa+WO Wy = 'YHo(1 - oyy +Kyy) ::::: Wb +Wo

H p = +-yftHok· ( '(;

1<) ·f

(4. 187b)

But (4.187a) shows that I z commutes with the large pan of the Hamiltonian, so we keep only those tenns in H p which are diagonal in I z . Thus

where WO='YHo

(4.179)

If Ho lies in a more general direction we can express (4.174) by defining an effective field Heff

'H = -'YnHcrr' I

(4.184)

H p = 'YtIHo k · (;; Utilizing (4.183) we have k.ip=ax

(4.180)

where

(4.188)

(4.189)

giving us Herrx = (I - uxx + Kxx)Hox H effy = (1 - oyy + K)'y)Hoy

H p = jnHolz[ai-(ux

(4.181)

Herrz =(1- o z z +Kzz)Hoz

+ a~(J{zz - uzz)]Iz

(4.182a)

where

(4. I82b)

aX =sin Bcos tP ay = sin B sin tP

ax = HoxlHo

crz

(4.183)

ay = HoylHo az = HozlHo

= cos

(4.192)

B

The" 2 w=wo+wa sin Bcos 2 tP+Wb sin 2 B sin 2 ¢J+wccos 2 9

we get from (4.180) and (4.181)

126

(4.191)

It is often convenient to express the orientation of Flo in terms of spherical coordinates (B, tP), so that

Defining the three direction cosines aX, ay, and az by

[(I - ox X + K xx )2a~ + (1 -

(4.190)

H p = -jnHo[t + a~(J{xx - ux x) + a'~(l{yy - on')

Herr = JHJrrx +HJrry +HJrrz

W = 'YHo

KX x) + a}(uyy - I
+ cr~(uzz - I
0'

Then the resonance frequency w is given by W = 'YHeff

X -

O"y)'

+ Ky),)2a~

(4.193)

a"d

129

1i = -')'llHoP + (J(XX -
q, (4.194)

This resuh can be expressed in another manner which is useful when discussing the techniques of magic angle spinning which we take up in Chapter 8. Utilizing the trigonometric idenlities cos 2

q, = !(I + cos 21$)

2

4(1 - cos 21$)

sin 1$ =

(4.195)

,

(4.196) Defining (j

2
(4.197) = (
(LO for "longitudinal")
=:
(TR for "transverse") and correspondingly for the components of J(, we get iT) + (l( La -
- 11. = - '111Ho [ I + (/(

0 - 1)

+ ( I(TR ;
(4.198)

The two angular functions are linear combinations of the spherical hannonics Y2.no Therefore, they average to zero over a sphere. Since in a liquid there is generally rapid tumbling motion, such a spherical average is appropriate, giving W =wO+!(Wo+Wb+WC>

= '1 H o [

(I +

I<XX

4.9 Second-Order Spin Effects - Indirect Nuclear Coupling We have discussed the role of electron spin coupling to nuclei in paramagnetic or ferromagnetic materials. Since, in a diamagnetic substance, the tOlal spin of the electrons vanishes, Ihe nuclei experience zero coupling to the electron spins in first order. Effects are found if one considers the coupling in second oroer,e however. The coupling is manifested through an apparenl coupling of nuclei among themselves. the so.-called indirect coupling. The indirect coupling was discovered independently by Nahn and Maxwell (4.16] and by GUlOWSIey el al. {4.17]. The phenomena they observed are illustfUted In liquid PFJ , lhe by the case of PF3 , a molecule in which all nuclei have spin rapid tumbling narrows the line. It is found that both the pJI and F19 resonances consist of several lines, as illustrated in Fig. 4.11. Since all the fluorine nuclei are chemically equivalenl, the splittings cannot be due to chemical shifts. (FurthemlOre there is only one phosphorus atom per molecule but four phosphorus frequencies.) The fact that Ihe individual lines themselves lire narrow shows that the motion is sufficiently rapid to narrow the direct dipolar coupling. Moreover, the splittings are found to be independent both of temperature and static field. The number and relative intensity of lines are as though each nuclear species experienced a magnetic field proportional to the z-compooent of the tOlal spin of the other species. It was found that OWF = owp (see Fig.4.11), where OWf.' and owp are the f-requency separalions of adjacent lines in the phosphorus and fluorine spectra, respeclively. These facts indicated the coupling was somehow related to th~ nuclear magnelic momenls. The original explanation proposed was that one nucleus induced curreniS in the electron cloud, which then coupled to Ihe other nucleus. In a simple picture. the induced currents are represented by an induced electron magnetic moment.

4.

= !(

Often in working with solids one has a sample consisling of many small crystallites in the form of a powder. Then the orientation of the crystallites is random with respect to Ho. The anisotropic nature of the chemical and Knight shifts then gives rise to a line broadening and to specrra which have quite distinct sha~ depending on the relative values of woo Wh. and We' Such absorption spectra are called "powder patterns" and are discussed in Appendix I.

+ I<;y + I
+
--I

(4.199) For a system such as a bond xf(r) which possesses axial symmetry, it is convenient to take We to lie along the axis, so Ilial Wo = Wh. giving w=wo+ 130

'(

22 )+ (2wo-w.-w,)(3COS 0-1) ' (4.200) 3

3 w o+ Wb+ Wc

_

'w, J-

U

(b) L-

Fig. 4.11. (a) The p31 resonance in 1'10'3. The lines are equlIlly SPllCOO all amount &..,., and the intensilies are 1: J: J: 1. (b) The FU n'SOmmce in PF3 &

See rererfflccs to -Il"h Couplins" in the Bibliography. 131

If this moment were isotropic (as one changes the orientation of the molecule with respect 10 the nuclear moment), the coupling to a second nucleus would average to zero in a liquid, owing to the rapid random tumbling of the molecules. However, as we observed in connection with the chemical shifts, the induced moment is in general not isotropic. We can estimate the size of the coupling between the two nuclei from second-order perturbation theory. The first nucleus exerts a magnelic field I' l h(l/r 3 ) where (1/1'3) is the average of the inverse cube of the distance between the electron and the first nucleus that partially unquenches the orbital angular momentum, producing a fractional admixture of excited state of Il/e1i2(I/r3)/LJE, where LJE is the energy to the excited state. A complete unquenching would produce a magnetic field at the second nucleus of leMR3, where R is the distance between the nuclei (we treat the electron orbital magnetization as equivalent to a magnetic dipole). ll1erefore the order of magnitude of the nuclear-nuclear interaction energy EI2 on this model is El2 ~

I'IIe1i2(i"1r3) leh LJE

RJ 12 h

(4.201)

This formula fails by an order of magnitude or more in accounting for the facts. However, it has the virtue of making it seem reasonable that the splittings in PF J were an order of magnitude larger than those in PH3, since clearly this mechanism is closely related to chemical shifts that are always smaller for hydrogen than for fluorine. As was pointed out by Hahn and Maxwell, and GUlowsky and McCall, any mechanism such as we have described, and which will lead to a result that is bilinear in the two nuclear moments, must take a very simple fonn. Since the interaction is averaged over all molecular orientations, it can depend only on the relative orientation of the nuclei; hence it must be of the form (4.202) where Al2 is independent of temperature and field. These workers also pointed out that this panicular form would also explain the puzzling fact that, for example, there were apparently no splittings of f1uorines by fluorines in PF3 . We shall not give the proof here but physically the explanation is based on the idea that the interaction energy, (4.202), which depends on the relative orientation of spins, is unchanged if bOlh spins are rotated through the same angle. For equivalent nuclei such as the three f1uorines in PF J , one cannot rotate one fluorine spin without rotating the others by an equal amount, since the alternating and static fields are identical at all three f1uorines. Therefore the coupling between equivalent nuclei does not affect the resonance frequency. Ramsey and Purcell [4.18] proposed another mechanism utilizing the electron spins, which was substantially larger because, as we shall explain, it allowed the two nuclei to interact with nearby electrons, in contrast with the orbital mechanism where only one nucleus is on the same atom as the electron that is polarized. We may schematize their mechanism as shown in Fig. 4. 12. 132

AtomB

Atom A

I

I

Stat,r

t

Sta'" II

--------------

.I

Fig. 4.12. Bc<:ause of the bonds betwc<:n atoms A and B, the electron wave function is formed by an equal mixture of the state I, in which the electron Elloment on atom A points up (that 011 B, down), and state II, in which the spin orientation is reversed

In the absence of a nuclear moment the electron bond will consist of an equal mixture of the states I and II, shown in Fig.4.12. If now we put a nucleus on atom A with its magnetic moment pointing up, state I will be slightly favored over state II. The electronic spin magnetic moment of atom A will have a slight polarization up; that on atom B, down. Therefore a nucleus on atom B will find a nonzero field owing to its own electron. Since this field would reverse if the nucleus on atom A were reversed, an effective nuclear-nuclear coupling results. We can easily estimate the size of the coupling. The fractional excess of state I over state II will be

¥llIeh2JU(O)I~ = hyperflne energy LJE

electrostatic energy

(4.203)

where lu(O)I~ is the wave-function density of the electron lit atom A, and LJE is the energy to an appropriate excited state. The coupling of the electrons on atom B to nucleus 2 are thus given by the proouct of the electron spin coupling if the electron spin is in one orientation only, (81r/3ht1'eIi2Iu(O)11, times the excess fraction of the time the electron is in the favored orientation. Thus the coupling is

(~llle1t2Iu(0)1~) (¥llle1t2Iu(O)11)

(4.204) c,E This coupling turns out to have the correct order of magnitude. If the electron funclions do not contain an s-part, we should instead use the ordinary dipolar coupling between the electron and nuclear spins. The extension of these ideas to solids was made independently by Bloembergen and Rowland [4.19] and by Ruderman and Kittel [4.20]. We shall discuss the situation for metals, confining our attention to the coupling via the s-state hyperfine coupling. Since the metal is not diamagnetic, we should concern ourselves with the possibility of 11 first-order effect - related, therefore, to the Knight shift. This mechanism of coupling was originally proposed by Frohlich and Nabarro [4.21]. As Yosida [4.22] has explained, however, the Frohlich-Nabarro effect is actually included in the second-order calculation. We shall discuss the physical reason shortly, but for the present, we shall simply ignore any first-order effect. Van Vleck [4.23] also discussed this effect. 133

1be effect of the magnetic moment of a nucleus at any lattice site is to make that site a region favorable for an electron of parallel magnetic moment but unfavorable for an electron of anliparallel moment. In order to take advanlage of the magnetic interaction, an electron of parallel moment will distort its wave function to be larger in the vicinity of the nucleus. The distortion is brought about by mixing in other states k of the same spin orientation. As we shall see, the result is as though only states above the Fermi surface were added. TIle wave functions of the Bloch states are added so as to be in phase with the unperturbed function (Fig. 4.13) at the nucleus in order to interfere constnlctively at that point, but because of the spread in wavelengths, they rapidly get out of step as one moves away from the nucleus.

,,-,,

(a)

,

,

~

(b)

"'-----./

x-o

"'-----./

•x

Fig.4.13a,b. The UnllerturbM fun<;lion and two of the higher states mixed in. The nucleull is at z = 0, which gUIlrAntees th"t the WIIYes be in ph"se lit z = O. Nole thl\llhc IIdmixed Waycs beal wilh one Allother. (a) Two of lhe WllVes mixed in by lhe perturbation. (b) Unperturbed function

As a result of the beats between the unperturbed and perturbed functions, the original uniform distribution of spin-up charge density (we neglect the variations due to the lanice charge) is changed to have an oscillatory behavior, which dies out as one goes away from the nucleus. The characteristic length describing the atlenuation is the wavelength of electrons at the Fermi surface. The resulting charge density of electrons whose momenl is parallel to the nucleus is shown in Fig. 4.14.

,

We now tum to the actual calculmion of these effects. For simplicity we shall calculate the interaction between the two nuclei directly rather than compute the changes in the spatial distribution of electron spin. However, the oscillatory natun; of the charge will be apparent from the answer. We consider, therefore, an electron-nuclear coupling 7ten involving only two nuclei of spins II and 12 and for simplicity, treat only the effect of the s-state coupling. We have, then,

7ten = 81\" ['Yl'Yeh211 • 3

L Slb(rl -

R I ) + "I2'Yeh2 12 .

I

L: Blb(r/ -

R2)]

I

= 'Ii, + 'Ii,

(4.205)

where we have allowed the nuclei to be different by using two values of gyromagnetic ratio "II and /2 and spins II and 12. We must lake into account the exclusion principle for the electrons. Two methods are available. We could use (4.205) to find penurbed one-elecrron functions and then fill these functions in accordance with the exclusion principle. Or, we could do penurbation theory in which we used the many electron functions as the unperturbed states. It is this laller procedure that we shall utilize, since it emphasizes that bas.ical1y we are dealing with a many-electron problem and that the states we use are but one approximation. Let us therefore consider a many-electron Slate 10) with energy Eo and excited states In) with energy En. and compute the second-order energy shift due to ?ten. We shall as usual assume the total wave function of the system to be a product of the electron and nuc1e:lr functions. Denoting the latter by 1/;0 with energy Ecn where 0 represents the nuclear spin quantum numbers, we shall wish to compute the second-order energy shift of states 10)"'0'. Therefore, for the second-order shift, we have .6£(2) = 00'

L ",0

.6E~~ of the

(Oal'h'enlna')(na'I'HenIOa) + EO') - (En + Eo-')

slate 10)"'0': (4.206)

,(Eo

Since the electronic energy differences are generally much greater than the differences in nuclear energy, we can neglect Eo and Ecr , in the denominator. 7 Writing 7ten as 'Ht + 7t2, we find

.6E~~ = L

11,0.'

E

I E [(Ool7illua')(ua'I1tJ 100') 0 -

n

+ (Oal7tzlna/)(na'l7izIOa) + (00"11-£ II Ha/)(na'I1-£2100") + (Oo:l7i2Ino:/)(na'[7iIIOa)j (4.207)

o

x

.1g.4.14. The charse delll;ity of eleclrons whose magnetic InOlnClI1$ aN: paral1el to the nuclear moment. The nucleus is located al z o. Po is the charse denllity in the absence or a nuclear moment. At z = z\, the dedron eharse is deficient, lIQ that lhe net electron moment there is oppoosoOO to the nuclear moment

=

134

7 One can think of lhis as being, thel'dore., basically :a calcul:alion of the cOlll'lin.g betW(lCn nuelei in zero external field. However,.t turns olllthat the fidd dependence IS very small. This resull$ from the flld thal the statcs E .. have a continuous distributioll in Cflergy slarting from Ea. Neglecting E.. - t:;. docs not. ~rKnlSly JX'rtur~ lhe.s~tra of excited stales or the malrix elemenl$. A further discuSI510n or the;e f'Omu 15 gIven immediately following (4.223).

135

The first two terms in the brackets represent the changes in energy we should have were onc or the other of the nuclei the only one. The last two terms represent the extra energy when bOlh are simuhaneously present, and are thus an interaction energy. Since it is only the interaction thai we wish 10 calculate, we shall consider the last two terms only. We have then .1E(2) "" ' " (Oal1tdno')(no'j1i21 0o ) +complex conjugate "'"

(4.208)

Eo - £ n

L..n,oJ

Now. fTOm the form of 'HI and 112. we can write them as

L

HI '" I) . Gl;:;

I\fJG11J

L

.jjiif

.Jiif

00'

x

",,(OIGjpln)(nIG,p,IO) ,,( II I ')( 'II I) L E _ J;' L 0' 1/3 0' ex 2{j' 0' +C.C. fJ,P' II 0 LJJj

"" L

"

,,(OIG,pln)(nIG,p' 10) E ~ (alhpI,p'la)+c.c..

/J.P'

n

(4.210)

(nWIO) =

(1,p'

[ " (OIG'P1n)(nIG,p,IO)] E _ J;' + C.c.

11(112(1' L..

..

0

(4.211)

.

LJJJ

(01

1felf =

L 5,6(r, -

L.. fI

Rj )Io)(nl

L 5,6(r, -

, _ E,)

(E 0

I

(-I)('P+P")

=

J

P" P[A'(I)B'(2) . . .j'

VP(A(I)B(2)C(3) ... )dT

-2,

L

(_1)1'"

N. P,P"

= L(-I)I'"

JP"

P[A'(1)11'(2) ... j' V P[A(I)B(2) ... )dT

J

P"(A'(I)B'(2) ... j'V(A(I)B(2) ... jdT

(4.215)

'"'

where the last step follows because V is unchanged by relabelling the electrons. lei us now consider V to be a sum of one-electron operators: V = LV
(4.216)

,

R,)I0)

·h +C.c.

(nWilO) = L(-I)I'"

64.'

C = -9-;11"21';11'· We shall now take the states 10) to be prcxiucts of Bloch functions. Denoting the product of a Bloch function and a spin function by letters A, and so on,

J

P"[.4'(I)B'(2) .. .j' V(1)[A(l)B(2)C(3) . . ·ldT

pit

(4.212)

where

136

(4.214)

where V(f) depends only on the coordinates of the lth particle. Consider, for example, the contribution from l = I:

By expressing the couplings G] and G2 explicitly, we obtain 1 ]. , C"

P'[A'(I)B'(2) .. .j'VP[A(l)B(2)C(3) ... ]dT

,pit

X

" 1felf = L.J

J

-2, L N. P

O.un

To evaluate the energy of (4.210), we should need to specify the nuclear states 10), lust what they would be would depend on the total nuclear Hamiltonian, which would include such things as the coupling of the nuclei to the external static field Ho, the dipolar coupling between nuclei, and so forth. It is convenient to note that whatever the states 10) may be, the energy ~E:;;! is just what we should find as the first-order penurbation contribution of an extra tenn in the nuclear Hamiltonian 1felf given by

(-I)P+I"

Since V is symmetric, it is only the relative ordering of the states that counts. We can express this fact by defining the pennutation pll. pll is the pennutation that, following p, gives the same ordering of electrons as pi alone. That is, pO p = p'. By making this substitution into (4.214), we get

<)',

= L..- L..-

L

;.,.

vN! pp' ,

I,pG,p

where Gj and G2 do not involve the nuclear spin coordinales. By using these

A£(')

P

Of course the pennutation causes any function to vanish if any two functions such as A and B are identical. Consider the matrix element of a perturbation V, which is symmetric among all the electrOns; that is, it is unchanged by interchanging the elecrron numbering:

(4.209)

relations,

(4.213)

jn) = _1_ L(-I)P P[A'(l)B'(2)C'(3) ... )

(J=z,l/,z

.u

p

(nWIO) =

P=Z,II,Z

'Ii, = I, . G, =

10) = _1_ L(-I)P P[A(I)B(2)C(3) ... ]

This will vanish unless the state In) contains B(2)C(3). write such an excited state as

In) = _1_ D-I)P P[A'(I)B(2)G(3) ... 1

.jjiif

.

(4.217)

. Let us therefore

(4.218)

p

This makes 137

(nlVtlO) , I:<-I)P"

J

P"IA'(I)D(2)C(3) ... ]' p" x V(I)IA(I)B(2)C(3) ... ]dT

=0

irA' is idenlicalto any of the funclions othe.,.;,. .

n,e,D.

etc. (4.219)

, (A'IV(I)IA)

It is clear thai the various values of I in (4.216) will simply pick out the various states A. B. C in 10), and the sum over ex:cited stales In) will pick out the states A', n', and so on, which are not occupied in 10). We can therefore write (4.212)

as

'Ii," '

L

C

Now. the Fermi funclions as well as the energies E kll depend on the energy associated with the electron spin quantum number. For example. the electron spin Zeeman energy changes with 3. However, the Fermi levels of the spin-up and spin-down distributions coincide. Thus. at absolute zero. there is a continuous range of Ekll up to the Fenni energy and a cominuous mnge of E k ,.. from the Fermi energy on up. The matrix elements of the 6·functions vary slowly with energy. Therefore the variation of 'Herr with the electron spin energy is very small, and we may forget about the energy of the electron spin coordinate, writing to a good approximation (4.223) in the fonn it would have in zero magnetic field:

'Ii,"' C L

II

k,.s occupied k', s' unoccupied

X

(ksIS6(r - R,)lk's')(k'iIS6(r - R2)ks)

Ek. - £1.:'"

where we have replaced

Eo - En

x

k -

k'

(4.224)

We can now perform the sums over" and s':

by Ek. - £1.:'/," since lhe stales

Eo

if k, s is occupied in State 10)

°

=

·1,

(klo(r - RIl!k')(k'lo(r - R,llk)f(k)[1 - f(e)] E E +C.c.

·12 + c.c. (4.220)

and En differ in energy solely by Ihe rransfer of one electron from the stale lk,,) 10 Ik', S'), The tenns "k, s occupied", "k'. i unoccupied", refer to whether or nOi these siales are occupied by eleclrons in the wave funClion 10). If we define the functions P(k.s) by

p(k, s) = I

II' ('15Ii)(,'151')

kk'; 11,11'

LII·(,151,')(iI5I,)·I" .,11'

L

.,.'

IIp(,15pliXi I5p'I')[w

11"":'""

,L

(4.225)

ltp[,p' T, {5p5p'}

PoP' But, as we saw in Chap. 3,

if k,3 is unoccupied in state 10)

(4.221)

1 , }5(5 + 1)(25 + I)opp'

we can easily remove the restrictions on k,3 from the summation:

0IJIJ'

,--

'Ii,"' C L

since

2

k,lI;k',.'

(4.226)

1 S=-

2

which gives, finally.

P(k,,)(1 - p(k',il] E E ·/z+c.c. (4.222) k. - k'" In order to express the variation of 'Herr with the tempemture of the electrons, we must average 'Herr over an ensemble. This will simply replace IJ(k, 3) by f(k. 3), x

'Heff = I

,C {" (kl<5(r - Rtllk')(k'I<5(r - R,)lk)f(k)(1 - f(e)]

t· .1'2-

2

L. kk'

E k - Ek ,

+ c.c.}

(4.227)

the Fermi function. We have then,

'Herr = C

"

L. k,II;k',If'

= A I 2II'/z

II' (kslS6(r - R J )J'/s')(k's'IS6(r - R:z)lks)

E kll

f(k,,)[1 - f(e,i)] X

-

where AI2 is a constant independent of spin. We now evaluate the matrix elements in terms of the Bloch functions:

E k 'II'

, ·..I2+c.c.

!f;k = 11k ( r ) e ik· r

Eklf - Ek'If'

, C

L

,

where. as before, lIk(r) has the periodicity of the lanice. We have. then,

II' ('15Ii)(iI5I') ·1,

(k'16(1' - R2)lk) = U~,(R2)Uk(R2)ei(k-k')' R]

(kl<5(r - Rtlk')(k'I<5(r - R,)lk)f(k")11 - f(k',i)] X

E

E

kif -

138

+c.c.

k'.'

(4.228)

(4.229)

so that.

(4.223) 139

I: co, I(k -

(klo(r - RI)Ik')(k'lo(r - R,)lk) 2 -R I ) , (R)'(R) (R)i(k-k')'(R = Uk'(R2)Uk' I uk 1 "k 2 e

(4230) .

k,k'

If we assume that R, and R 2 are equivalent sites (as, for example, in a simple mctal), and define

=

(4.231)

k')· RI,]f(k)[1 - f(k')] k2 k,2 -

(-,-)6 2'11'

j'J [COS (kR COS 0) cos (k'R COS 0') k2 _ k,2

+ ~'i:::n-'.(kc:'R::..:c::o,,-';:;0,,):::,':::n.,(;;.k..:'R.:..:c:::o,:.:.O'.!..!)] k2 _ k,2

we have =

HeW

C1112 '. '" 1".,(0)I'I".(0)I'2co, [(k 2

=

E -E,

L-

k,k'

k

64 2 2

I"[r9'Tr 'Yc/1'Y2 h X

k') . R12J f(k)[1 _ f(k')]

E1,;

I.:,k'

Ekt

",

(4.232)

2

j'J sin kR sin k'R kk'f(k)[1 -

4 1 -- (2'11')4 R2

k')· RI,]f(k)[1 - f(k')]

It is not possible 10 evaluate the summation without either some further approximations or some explicit infonnation on the k dependence of wave functions and energy. If one assumes spherical energy surfaces, an effective mass m*, and that !uk/(O)12 and lu/.:(0)1 2 may be replaced by a v;\lue appropri:ltc to k and k' near the Fenni energy, one can evaluate the sums. In these terms,

Ek = 2m*k

(4.236)

f(k)[1 - f(k')jk'k"dfldfl' dkdk'

The integrals over dn = -2'11'd(cos 0) and dn' = -2'11'd(cos ( 1 ) are readily performed to give, for the summution of (4.236),

k

1

I: 1".,(O)l'I".(O)I'co, [(k -

X

k2 _ k,2

f(k')Jdk dk'

This integral may be evaluated at absolute zero by noting that the limits on k' can go from 0 to 00, nO( just kr to 00, since if k' < kp for each k = k J , k' = 1.:2, there is a k = k2, k' = k J which has the opposite sign for the integrand. The range of k l is then extended 10 cover -00 to +00, and the integral is evaluated by a contour integral. The infinity at k = k' is avoided by taking a principal parI. The final result is 2 'Heff = - 9 '11' ,;,1I2 1i2m *!Uk F(0)j4

(4.233)

(sin 2k p R - 2krR cos 2kFR) I

I

x I ' 2

R'

This gives us

'Herr = I, . h fl 2

I: co, [(k I

k,k

k')· R12] f(k)[1 ~ f(k')]

}.;2 _ }.;/2

(4.234)

The number dN of states in k-space within a solid angle ef[} and between two spherical shells of radii k and dk is dn k 2dk dN = 41f 21f 2 Denoting the angle between k and R l 2 as 0, that between Fig.4.15), and using R for IR 121, we have

(4.235)

e and R12 as 0' (see

Fig.4.IS. Relalive orienlations of k, k', and R I 1

140

(4.238)

We note that this expression indeed corresponds to an oscillatory behavior as one varies R. For large distances the coupling goes as

~ 1r 21';'YI1'2 h1 Iu kF(O)]"

X 2m'

(4.237)

(4.239) We see that the dependence on lu(0)1 4 will cause the coupling to be large for large-Z atoms. In fact the coupling for the heavier elements is substantially larger than the direct dipolar coupling. Had we used the conventional dipolar fonn of coupling between the nuclear and electron spins appropriate to "non" s-states, we should have obtained

~ = neff

[r • r _ 3(l1 . R J2)(l2' R I2)] B 1

2

R2

12

(4.240)

12

where B12 is a complicated function. It vanishes if there is no "non" s-character to the wave function as seen by a single nucleus; for large distances, B 12 typically falls off as 1/R12' For a discussion of B 12 the reader is referred to the paper by B/oembergen and Row/and [4.19]. In the case of molecules such as PF3, Gutowsky et al. [4.17] show that the assumption of p-type wave functions on both P and F together with the "non" s-state coupling gives a coupling of the lonn 141

(4.241)

when averaged over all orientations of the molecule in the external field. The coupling does nol vanish when averaged over molecular orientations, since the induced spin magnetization itself depends on the orientation of the molecule with respect to the nuclear spins. Couplings such as (4.240) have the same spin dependence as thai of the direct dipolar coupling. To emphasize this similarity, the coupling is often referred to as the "pseudo-dipolar" coupling. On the other hand, a coupling such as AI2II • h has the same fonn as the electrostatic exchange coupling. Since in our case the physical origin is not an exchange integral, the tenn is referred to as the "pseudo-exchange" coupling. The effect of pseudo-exchange or pseUdo-dipolar coupling on the width and shape of resonance lines can be analyzed by simply adding these tenns to the dipolar tenns. The number of situations that arise are very npmerous. In liquids. the pseudo-dipolar coupling averages 10 zero, but the pseudo-exchange does not, giving rise to resolved splittings. For solids, both tenns have effects. The pseudo-exchange lenn, since it commutes with [~ = [1% + h%, has 110 effect on the second moment of the resonance but does increase the fourth moment. As Van Vleck discusses, this must mean that the central portion of the resonance is narrowed, but the wings are enhanced (see Fig. 4.16), since the fourth moment is more affected by the wings than is the second moment. The fact that the central portion appears sharper gives rise to the tenns exchange narrowing or pseudo-exchange narrowing. Where a real exchange interaction exists. as in electron resonances, the exchange narrowing may be very dramatic.

\

--

Fig.4.16. Solid curve, the resonance shape wilh vanishing pseudo-exchange coupling. The dashed curve shows the shape when pseudo-exchange i" 111· c1uded

analysis of the electron-nuclear interaction. Denoting the wave vector of a Fourier component by q, Yosida points out Ihut the Frohlich-Nabarro effect is the q = 0 leon (infinite wavelength). In the second-order perturbation, a component q '" 0 joins--two electron slates k and k' that satisfy the relation

k' = k + q

Since in a proper second-order calculation the excited and ground states mllst differ, we see from (4.242) that we must exclude q = O. Yosida treats q = 0 by first-order theory, but q f. 0 by second order, adding the results. The answer he obtains in this way is identical to that of Bloembergen-Rowland and Ruderman· Kittel, taking a principal part as mentioned on p. 141. The simplest way to see that the answers will be the same is to consider instead the Knight shifts. There are two ways one can calculate the Knight shift. The first method is to assume a unifonn static field andJO compute the first-order nuclear-electron coupling of the polarized electron state. The second method is more complicated. We assume a static field but one that is oscillating spatially with wave vector q. For such a spatial oscillatory field there is no net spin polarization, since the field points up in some regions of space as much as it points down in others. The usual first-order interaction vanishes. If one now goes to second-order perturbation in which one matrix element is the electron-nuclear coupling, the other, the electron-applied field interaction, a nonzero result is found. That is. the static field induces a spatially varying spin polarization. Let us choose the maximum of the static field 10 be at the position of the nucleus. When q gets very small (long wavelength), we expect that the result must be the same as if the field were strictly unifonn. Therefore the limit of the second-order answer as q ...... 0 must be the usual Knight shift. This result can in fact be verified. In evaluating (4.238) using principal parts. one takes the limit as k ' --+ k. This procedure, as with the Knight shift, includes the first-order perturbation repopulation contribution. or the Frohlich-Nabarro effect. Before concluding this chapter, we should consider the role of electron spin in the chemical shift of diamagnetic substances. In the absence of an applied magnetic field, diamagnetic substances are characterized by 11 total electron-spin quantum number S of zero. The application of a field Ho in, say, the z·dircction adds a tenn Hsz. the spin Zeeman interaction, to the Hamiltonian:

If the two nuclei are not identical, one can approximate the pseudo-exchange ~o~pling as AI2[lz[2z' Since this interaction does not commute with 11%

+ h%.

11 ltlcreases the second moment. The resonance curve then appears broadened, and one speaks of "exchange broadening". If there is a quadrupole interaction that makes the various m-states unequally spaced, the exchange coupling can lead to a broadening. even when the nuclei are identical. We conclude by remarking on the Frohlich-Nabarro effect that the presence of one nucleus causes the electrons to repopulate their spin states, producing a magnetic field at other nuclei in much the same way as the static field produces a Knight shift. Yosidtl has analyzed the problem by perfonning a spatial Fourier 142

(4.242)

Hsz

= 'YehHo

E" $zj = 'YehHoSz

(4.243)

j=1

where j labels the electrons, and where N

S,"

L: S,;

(4.244)

j=1

Since the ground-state wave function 10) is a spin zero function. we have that

S,IO) =

°

(4.245) 143

so that all matrix elements of 'H.sz to excited states In) vanish:

°.

(4.246) (nl1iszIO) = 7.hHo(nIS, 10) = The ground state is therefore strictly decoupled from all other states as far as the spin Zeeman coupling is concerned. 1be applied field is therefore unable to induce any net spin, and there is no phenomenon analogous to the unquenching of the orbital angular momentum. The fact that the spins actually couple to a magnetic field makes this result seem strange. Intuitively we except that, given a strong cnough magnetic field, the spins must be polarized. The parndox is resolved by considering an example, the hydrogen molecule. The ground state is the singlet bonding state, but there is a triplet antibonding state. In the presence of an applied field, the states split, as shown in Fig. 4. 17. Energy

Triplet

Singlet

S-l,M._O

S.O,M.-O

Ag. 4.17. Erred of lhe applied field /10 on the singld. And triplet-spin stales of a hydrogen mole<:ule. If 110 were large enough, a triplet slate would be lowesl Ilnd the ground state would pOSSCliS & magnetic moment

H,

As we can see, for large enough Ho the S = I, /tI/s = -I state crosses the S = 0, Ms = 0 state. The ground state is then a triplet state, corresponding to a spin polarization. However, since the singlet-triplet splitting in zero field is several electron volts, the crossing of levels could never be produced by an auainable laboratory field. Further insight is obtained by considering the effect of a hypothetical mixing of a triplet state into the ground state. If the result is to induce a net spin polarization in the positive z-direction on one atom, it induces an equal and opposite spin polarization in the negmive z-direction on the other atom. Clearly there is zero net spin Zeeman interaction with an applied field in the z-direction. Since such a spin polarization gives no net lowering of energy, it is not in fact induced. Note, however, that if the two atoms are dissimilar, there may be a different induced orbital moment which, through the spin-orbit coupling, could then induce such a spin polarization. Thus, in a molecule such as H I, the iodine orbital magnetization could induce a spin polarization into the bond, giving a spin contribution to the chemical shift on the hydrogen as well as on Ihe iodine. 144

5. Spin-Lattice Relaxation and Motional Narrowing of Resonance Lines

5.1 Introduction We turn now to a discussion of how the nuclei arrive at their thermal equilibrium magnetization via the process of spin-lanice relaxation. We shall find it convenient to cliscuss two techniques for computing TI· The first method is appropriate when the coupling of the nuclei with one another is much stronger than with the lanice. In this case an allempt to compute the population changes of an individual nucleus due to th~ coupling to th~ lattice is complicated by the presence of a much stronger coupling of the nuclei among themselves. The first method makes the assumptio.n that the strong c~u. piing simply establishes a common temperature for t.he SpillS and that the .laHlce coupling causes this temperature to change. There IS a close analogy wllh the process of heat transfer between a gas and the walls of its contain~~, i~ which the role of the collisions within the gas is to maintain a thermal eqmhbnum among the gas molecules. In the collision of molecules with the wall, we con~i~e~ the molecules to have the velocity distribution appropriatc to themlal eqUlhbnum. As we shall see the first method leads to a fomlUla for TI that is particularly convenient when the lattice is readily described in quantum mechanical tenus. For example, relaxation in a metal involves the transfer of energy to.the conduction electrons, which are readily thought of in terms of Bloch funcuons and the . ' exclusion principle. The second method is that of the so-ealled density matnx. Although thiS is a completely general method, it finds its greatest utility for systems in w~ich the lallice is nalurally described classically and in which the resonance WIdth is substantially narrowed by the motion of the nuclei. Moreover, when motio~ takes place, the relaxalion time T2, which describes coupling between the nUcl~I, becomes long, and it may be a very poor approximation to assume that a SplO temperature is achieved rapidly as compared to T l . lllUS the second ~ethOO is useful when the first one fails. Because of their large mass, the motion of the nuclei is often given very well in classical tenns. In fact an attempt to describe the motion of molecules in a liquid quantum mechllOically would be quite cumbersome. Consequently the density matrix. method is well suited t.o discussing cases in which motional narrowing takes place. An added feature IS that both TI and T2 processes (relaxation of I z and I z or III) can be treated by lhe density matrix method, provided there is motional narrowing. 145

The density matrix method is very closely related to the conventional timedependent perturbation theory. Actually the two are entirely equivalent. The density matrix method, however, gives results in a particularly useful form. It is ideal for treating problems in which phase coherence is important, and in fact the density matrix or a mathematical equivalent is necessary to treat such problems. In any event, much of the fonnalism of this chapter applies equally well to systems other than spins. For example, dielectric relaxation can be treated by these methods. As we see, the two approaches complement one another, one applying to the broad resonances of a rigid lattice and the other being most useful when the resonance has been narrowed by nuclear motion.

would enter only when we tried to express the wave function of the total system in terms of the wave functions of individual spins. We shall say that any system whose population obeys (5.1) is described by a temperature T, even when the system is nOt in equilibrium with a reservoir. Equation (5.3) enables us to make a simple plot to schematize the populations. We illustrate in Fig.5.1 a case based on our first interpretation: we consider the population of the various energy states of a single spin of I = acted on by a static magnetic field.

t

,

I

I

5.2 Relaxation of a System Described by a Spin Temperature I

P(Ea )

e-E~/kT

P(EI1) - e E.lkT

so that, since

(5.1)

E.

we have P(Ea )::=.

e-E.. IJcT

L: e ,

Ec/kT =

e- E.. IJcT Z where

(a)

(b)

.'ig. 5.1. (a) Energy levels ar II spin 3/2 nucleus. (b) Bllr grlll>h af POpullllion vel'$US energy.

The lengths or the bal'S are determined by the exponential em'e1ope

Energy

(5.2)

LP(E.) = I

,

+% +% - - -

I

Exponential I envelopo ".i=-~

-'h

A system with a set of energies E u , Ell' and so on, which is in thermal equilibrium with a reservoir of temperature T, occupies the levels with probabilities p(Ea ), p(E/I), and so forth, which arc given by

I

I

- 'It -1ft, (5.3)

'I, 'I,

Population

Ca)

Cb)

(5.4)

is the partition function or "sum of states". These equations may actually have two interpretations, which we might illustrate by considering N identical spins. The first interpretation considers the spins as isolated from one another. The "system" consists then of a single spin, and the energies E a represent the possible energies of the single spin. The second interpretation considers the system to be formed by all N spins. In this case, E a represents the total energy of all N spins. We shall find it convenient to use both interpretations. We should perhaps note that the first interpretation is correct only if the spins can be considered to obey Maxwell-Boltzmann statistics, . but the second interpretation holds true whether or not the individual particles obey Maxwell-Boltzmann, Fermi-Dirac, or Bose-Einstein statistics. The statistics

1<1g. 5.2. (a) Population distribution not describable by a temperature. (b) A po56ible lransition or a pair or spins rrom the st...tes designaled by CI"OMes to lhose designated by circles

A system such as shown in Fig. 5.2 clearly does not correspond to thennal equilibrium, since the bar graph envelope is not an exponential. In Fig.5.2b we indicate a transition that could take place, conserving the total energy of the spins. Two spins designated by crosses couple, inducing transitions to the states designated by circles, one spin going up in energy and the other down. (Such a transition would be induced by the Ii tenns of the dipolar coupling.) The number of transitions per second from cross to circle, dNldt)r ..... O. will be the product of the probabilities of finding two spins in the initial state times Ihe probability of transition W., ..... 0 if the spins are in the initial state. Thus,

It

I See rererences to "Spill TemperMure- in the Bibliography

146

147

~)

._0

= P_I/2P_I/2W,;_O

(5.5)

The inverse reaction from the circle 10 the cross will have a rale dNldOo .... ~, given by

dN)

dt

0-.

=:

(5.6)

P_J/2PI!2 WO_J:

If we equate these rates, we guarantee equilibrium. This is the assumption thai equilibrium is obtained by detailed balance. Since WO_:t: "" W:>: ..... o, we find P-I/2P-I/2 = P-3/2PI/2

P-3/2

P-I/2

P-I/2

1'+1/2

or - - = - -

(5.8) n

The average energy of the system, E, is then (5.9)

n

We shall assume further that the energies E u nre measured from a reference such that

L;En =Tr1i=O

(5.10)

,

n

a condition that is fulfilled for both the Zeeman and dipolar energies. To compute the relaxation, we shall consider changes in the average energy. If we define fJ = l/kT to represent the spin temperature, we have that

dE dE dP di"=dpdi"

dpni ::::: "" d LJ(Pm W mn - pu W nm )

(5.11)

'"

dE dt

m,n 1 = '2 L(PmWmn - PnWnm)(En - Em)

(5.14)

'",n where the second form is introduced because it treats the labels m and 11 more symmetrically. By equating (5.11) and (5.14), we obtain a diITerenlial equation for the changes in spin temperature. We have two problems: (I) finding dE/dfJ and (2) seeing what becomes of (5.14) when we introduce the requiremenl that at all times a spin temperature apply. We tum first to evaluating dE/dP: e-fjE.

Pn = - -

Z

dE

and

(5.15)

d

dP = dP ~Pn(P)En

(5.16)

We first seek an approximate expression for Z. Once again, assuming the temperature to be high enough so that PEn <:: I for the majority of states, we expand exp( -PEn) in a power series and keep only the leading terms:

P2E~)

Z= ~ ( l-fJEn+--:v:- + ...

(5.17)

Approximating such a power series expansion by the leading terms clearly has validity if IEnl <:: kT for the significant energies. However, the approximation proves legitimate under less stringent conditions using an argument similar to that in Appendix E both here and in (5.19-26). When we utilize (5.10), the second lenn on the right of (5.17) vanishes. We then neglect the f32 term, and Z becomes equal to the total number of states. Since this is also Z for infinite temperature Zoo, we may say that

Z = Zoo

But since E::::: EnPnEn, we have also thai

(5.13)

This equation is frequently called the "master" equation. By substituting into (5.12) we have, then, that

(5.7)

But this is just the condition of thermal equilibrium among the states, since they are equally spaced in energy. We see, therefore. thai thennal equilibrium is reached by processes such as we have indicated in Fig.5.2b. The typical rate for such a process is of the order of the inverse of the rigid lattice line breadth, or between 10 10 lOO/tS for typical nuclei. Therefore, if Tl is milliseconds to seconds, we should consider the nuclear populations 10 be given by a Boltzmann distribution. We shall now proceed to consider the relaxalion of a system of nuclear spins whose Hamiltonian 1t has eigenvalues En, and in which the fractional occupation of state n is Po. (Thus n designates a state of the total system, rather thall the energy of a single spill). Nonnalization requires thal

E=LPrIEn

We shall assume that the Po's obey simple linear rate equations. Introducing Wmn as the probability per second that the lattice induces a transition of the system from m to n if the system is in state m, the rate equation is

(5.18)

When we utilize this fact and (5.15), (5.16) becomes

de : : : dt '48

d "" dPn dt LJPoEo ::::: "" LJEndi n

(5.12)

n

149

where

L =

Wmn(E m - E n )2

1 m,1I

2

(5.28)

L:E~ n

-;::t

(5.19)

I " 2 -ZLJE" n

00

again in the high-temperature limit. Thus

" E' dj3~n

dE

(5.20)

dt<::-dt~

We now tum to evaluation of (5.14). Since the system is always describable by a temperature, we have PII

_ P e(Em~En){J

(5.21)

m

-

We shall furthennore assume that when the system is in thennal equilibrium with the lattice, the transitions between every pair of levels are in equilibrium. This is the so-called principle of detailed balance. Denoting by p~ the value of Pn when the spins are in thennal equilibrium with the lattice. the principle of detailed balance says thai

LWmn = PIIlOWnm Pm I.

P w:mn-- W nfllL ---!:!... Pm

(5.22)

or that

5.3 Relaxation of Nuclei in a Metal

= W nm e(Em-En).B"

(5.23)

where PI. = I/(kTL)' By subsliluting (5.21) and (5.23) into (5.14), we find

dE dt

=.!. L: pm W mn [1

- e(Em-En)({J-fJ!.l](E" - Em}

(5.24)

2 m,n

Now, expanding the exponential, we find dE dt

(5.25)

Now pm=

e- f3Em Zoo

~

1 - (3E m + (/32 E~/2!) ....., =

Zoo

Zoo

(5.26)

Thus, combining (5.26) with (5.25) and equating the resultant dEldt to that of (5.20), we find

dP dt

L: Wmn(Em -

=

En)']

(P _ P) ~ ,"m""n_~::;;;- _ _ L

[2

Equation (5.28) was first derived by Gorter on the assumption that 1(3 - (31, 1« (3 [5.1]. As we can see, this restriction is not necessary. The great advantage of (5.28) is that, by postulating a temperature, it has taken into account the spin-spin couplings. The rate equations, (5.14), would by themselves imply that there are multiple time constanlS that describe the spinlattice relaxation, but the assumption of a temperature forces the whole system to relax with a single exponential. We may get added insight by viewing our states n as being nearly exact solutions of the nuclear spin Hamiltonian, between which (since they are not exact states) transitions take place rapidly to guarantee a thermal equilibrium, but between which the lanice also makes much slower transitions. After each lattice transition, which disturbs the nuclear distriblllion, the nuclei readjust among their approximate levels so that the lattice once again finds the spins distribllled according to a temperature when it induces the next spin transition. Our formalism implies that treating the states n as being exact makes a negligible difference in the answer.

LE~

(5.27)

We now turn to an example of the application of (5.28). We shall consider the relaxation of nuclei in a metal by their coupling to the spin magnetic moments of the conduction electrons. This is the dominant relaxation mechanism. In a T, process, the nucleus undergoes a transition in which it either absorbs or gives up energy. In order to conserve energy, the lanice must undergo a compensating change. For coupling to the conduction electrons, we may think of the nuclear transition as involving a simultaneous electron transition from some state of wave vector k and spin orientation .5, to a state k l , l. We may think of this as a scattering problem. Denoting the initial and final nuclear quanlum numbers as m and n, respectively, we have that the number of cransitions per second from the initial state of nucleus and electron Imks) to the final state Ink'i), W mkB,lIk' B" is W mkB,nk'B' =

2;

2

l(mkslVlnk' i)1 o(Em + E kB - En - Ek'B')

(5.29)

where V is the interaction that provides the scattering, and where (5.29) assumes that there is an electron in Iks) and there is /lone in Ikll). The total probability per second of nuclear transitions is obtained by adding up the IVmks nk'B' 's for all initial and final electron states. We have '

n

150

151

L

W mn '"

(5.30)

Wmks,Ilk's'

x

ksoccupicd 1,,'6' unoccupied

L

(5.31)

vVmb ,llk's'Pks(l- Pk'6')

ks;k's'

By averaging (5.31) over an ensemble of electron systems, we simply replace Pb by the Fermi function f(Eb), which we abbreviate as f(k,s):

L

W"'>I '"

(5.38)

and, assuming E ks '" E", + E s • we get for W mn

We must now express W mb nl/6' explicitly. To do so, we must specify the interaction V. For metals with a substantial s-character to the wave function at the Fenni surface, the dominant contribution to V comes from the s-state coupling between the nuclear and electron spins:

8. , V'" "3,e,nh 1·50(1')

Imks) '" Im)ls)uk(r)e ik • T

(5.34)

It is a simple matter to compute the matrix element of (5.29):

(mkslVlnk's') '" 81r Iclnh2(mII[n). (sISls')uk(O)uk'(O)

(5.35)

3

which gives us

2:

(mII.ln)(nllo,lm)

(5.36) x o(E", + Eb - En - E k '6') We can substitute this expression into (5.32) CO compute W mn . We are once again faced with a summation over k and k ' of a slowly varying function. As before, we replace the summation by an integral, using the density of stales g(Ek , A) introduced in Sect. 4.7. This gives us

r; -9-'C Inti

'"'

L.

0I,OI'6,S'

(mIIOI[n)(nllo,[m)(sISOI[s')(s'[So'ls)

01,01'6,6'

x

J<1"k(O)I')

X

e(Ek)e(Ek ,)dEk

E

J:

0"",(0)1') E f(E b '"

Il

)1I -

f(Ek", + Em - E,,)] (5.39)

~,~~+~-~,+~-~

(5~

Since E", - En. the nuclear energy change, is very small compared with kT, the Fenni function f(Ek6 + Em - Ell) may be replaced by f(Eb)' Actually, doing so makes W mn = W nm . It is, in fact, at this point that the slight difference between W mn and Wmn arises. It is the slight difference, we recall, that gives rise to the establishment of the thennal equilibrium nuclear population. We are allowed to neglect the difference here in computing W"'>I' since we have already included the effect in (5.23). Moreover, since both u(Ek ,) and ([uk,(0)1 2 ) Ell are slowly varying functions of E k" we may set them equal to their values when Ek '" Ek" In fact we shall evaluate U(Ek ,), and so on at E ks ' This gives us for the integral in (5.39):

J
x (,15.1,')(/150,1,)I"k(O)I'I".,(O)I'

2 2 4

6~2 1;,~h'l L

=

OI,OI'=Z,Y,z

27l' 647l'2

Wmn '" 2;

(5.33)

where we have chosen the nucleus I to be at the onglO. For the electron wave function, we shall take a product of a spin function and a Bloch function uk(r)exp(ik· r). Therefore the initial wave function is

I

I

(mIIOIln)(nIIOI,[m)(s[SOIls )(s 1501 , Is)

f(E)JdE

(5.41)

o where we have set the lower limits as zero, since the only contributions to the integral come from the region near the Fenni surface. E'" E F . Since (5.41) is independent of the spin quantum numbers sand S', we may now evaluate the spin sum of (5.39):

2:('15.1,')(/150,1,) = 2:(,150 5.,1,) S,6'

6 = Tr {SOISOI' } = 00101' !S(S + 1)(2S + I)

'" 00101,/2 152

(5.37)

We integrate first over dA and dA ' , utilizing the relations of (4.153, 154) to perform the integrals, and introducing again the average of [uk(0)1 2 over the energy surface, Ek' (IUk(0)[2) EJ:' We also assume that the energy Eks appearing in the Fenni functions remains constant on a surface of constant Ek' an assumption that would be fulfilled unless the spin energy depended on the location on the surface Ek . The integralion over dEI,,' is then easy because of the delta function. We have that

(5.32)

Wmks,nk's,f(k,s)[l - f(k',s/)]

k6;k's'

W"'>I '"

f(k, ')11 - f(k', ,')Jg(Ek • A)

x g(Ek " A')o(Em - En + Eks - Ek'6,)dEJ.,dAdEk'dA'.

The sum over "ks occupied" is, of course, equivalent to summing over electrons. We can remove the restrictions on ks and k's' by introducing the quantity Pks' which is defined to be unity if ks is occupied; zero, otherwise. This gives us W mu '"

Jl"k(O)I'I"k,(O)I'

since

5 '"

!

(5.42) 153

This gives us

W mn

::::

W mn ::::

~ 1l'3h3'Y~'Y~ L(mIIQC[n)(n[Iolm)

•

x

J(IUk(O)I')~.'(E)f(E)[1

- f(E)JdE

(5.43)

Now f(E)[1 - f(E)} = -kT :~

(5.44)

which follows directly from the fact that

feE) =

~4 1l'31l3'Y;'Y~([uJ.:(0)12)h,£l(EF)kT L

•

L ](mIIQC[n)1 2

W mn =

1

•

L.;j L(mll;.ln)(nllj.lm) iJ

Fig. 5.3. f'mdions j(E), t -j(E), and j(E){I - j(E)I. The $Olid line .how. I(E)(I -/(E)I

Since /(0) = 1 and /(00) :::: 0, and since /(£)[1 - /(E)] peaks up only within a width kT, it is also closely related 10 a 5-function when in an inlegral of other functions that vary slowly over kT.

(see

where the coefficient i = j is aoo, and where aij for i 1: j falls off rapidly with the distance apart of the nuclei i and j. 11lese terms arise because the elecrron wave function extends over many nuclei, so that more than one nucleus may scatter the electron from a given initial 10 a given final state. By returning to (5.47) and employing our fonnula for T I , we have

I TI =

SF

L

_

",pC

I

J(f.,-E~,)j(1-J)dE+dE7

dG 00 , +dE SF 0

154

(ml[?t.I.J1n)(nl[?t. l.Jlm) __=-=::_

a~ cm~,~n~,.,---

+ Ep

00

.

j(E-Ep )2jCt-ndE ...

S 0

EE;n 00

J G(E)j(E)[1 -j(E)]dE = G(E~') J j(E)/1 - jCE)ldE

I

EE;n

(5.45)

7 Equation (5:4.5) can be derived simply. Let C(E) be a slowly varying function of energy. Then, utlh1.lIIg Lhe fact that j(E)[l-j(E)J is nonvanishing within only kT about Ep, we c~n expand C(E) in a power series "bolll EF:

Thus 00

En)'

00

= _

2 2 +(E-Ed d CI 2! dE2

-

~=------

aoo Z-

aoo

"

(mll.lnXnll.lmXEm

'00 •• ~,n~,~.

By utilizing this fact, we have, finally,

G(E)=GCEFl+CE-EddC[ dE

(5.48)

0

L

footnole 7 ).

(5.47)

,

defining the quantity aoo, which is independent of the nuclear states nand m. If we have more than one nucleus, it can be shown [5.2] that Hlmn is given by a sum over the N nuclei, labeled by j or j, (i,i:::: 1 to N):

and f(E)[1 - f(E») peaks up very strongly (Fig. 5.3) when E " E•.

f(E)[1 - feE)] = kT6(E - E.)

(5.46)

We note that Wmn is proportional to the temperature T. This fact has a simple-physical interpretation. When the nuclei undergo a transition, they give an energy to the electrons that is very small compared with kT. Most of the electrons are unable to take pan in the relaxation because they have no empty states nearby in energy into which they can make a transition. 11 is only those electrons in the tail of the distribution that are important. Their number is proportional to kT. We can write (5.46) as W mn :::: aDO

I e(E-Ep)lkT + I

[(m[Io [n)[2

=-2

o=::e,y,::

Tr {?t'}

(5.49)

A similar expression is found by using (5.48). The important point is that we do not need to solve for the explicit eigenstales and eigenvalues, but only evaluale the traces in a convenient represent:ltion.

The first term, using C5.44), gives G(EF )kT. The second term vanishes, since the integrand is an odd function of E - EF, and the third term gives a contribution proportional to (kT)3, as seen by changing the integrand from E lo ::e := ElkT. If we neglect the third and highcr terms, the answer is just what we should have if we replaced j(1 - n by kT6(E - Ep). The correction!! are generally of order (kTIEd 7 smaller, the exact form depending on the functional dependence of C on the energy E.

155

For our problem of a single spin, the quantum numbers m and label the 2I + 1 eigenstates of I r • Then, using the fact that

11

would (5.59) (5.50)

[Ir , I z ]

:::

iIy

(5.51)

etc.

we find

L Tr {['H, IO']2} ::: _1'~h2 H6 Tr {(I; + I;)} • E Tr {'H 2 } ::: 1~'12 H6 Tr {I;} •

(5.52)

Thbl~

so that, since Tr{I;} ::: Tr {I;} :::Tr{I;J,

L a

Tr {[?i.E.I'

:::-2

Tr (11')

(5.53)

~:::: aoo:::: 64 'lf3h3;:;~(luI(O)I)}Fe2(Ef)kT 9

TI

(5.54)

The quantity (lul;:(O)l)}; appearing in the expression also occurred in the expression for the Knight shift, LJ.HIH :

LJ.H

H·

h

2

S

T(lu.(O)I> E,X.

.

(5.55)

We can therefore use (5.55) to evaluate (Iuk(O)!fh., giving

TI

(LJ.:Y : : [£>{~F)r 1f~T 1'~~ih3

The T appearing in the Korringa relation represents only one contribution to the relaxation time - that due to the coupling of nuclei to the magnetic moment of ,!I-state electrons. One expects that the experimental TI should, if anything, be shoner. It is therefore interesting to examine a table given by Pines {SA]. In it he lists e;Jlperimental Tj's, those computed from (5.58) (the Korringa relation) and those computed from (5.59), using Pines' theoretical values of relx& and I!O(E.)/e(E.).

(5.56)

5.1. Experimental and Lheoretical T I '. (all tillW$ in ms)

T 1 (Experimental)

T. (Korrillga) T. (Pines)

t5O±5 15.9±0.3 2.;5±0.2 3.0±0.6 6.3±0.1

88 10.3 '.1 '.3 '.1

232

18.1 2.94 4.'

•••

We nOle that the Korringa TI'S are all sllOrur than the e;Jlperimental ones. The discrepancy cannot be removed by appealing to other relaxation processes, since if we included them, the theoretical T, would be even shorter than those computed by the Korringa relation, and the discrepancy would be still greater. On the other hand, the Pines' values, based on inclusion of the electron-electron couplings, make the predicted values longer than the experimental. The discrepancy between the Pines' values and the experimental is perhaps a measure of the imponance of relaxation processes we have not computed.

For a Fenni gas of noninteracting spins, one can show that X~ given by

5.4 Density Matrix - General Equations

2h2

1 Xo:::: Teo(Er)

(5.57)

where we have put subscripts "0" on XS and e(E F ) 10 label them as appropriate to noninteracting electrons. In this approximation one has

C>H)2

TI ( - H

h

"(2

• - - --"

41fkT "(~

(5.58)

Equation (5.58) is commonly called the "Korringa relation", after Dr. J. Korringa who first published it [5.3]. It provides a very convenient way to use measured Knight shifts to predict spin-lattice relaxation times. A more accurate expression is obtained from (5.56) and (5.57):

156

As we have remarked, the concept of a spin temperature is nOl always valid. We lum now to discussion of a mel hod of attack that is very useful when the spin temperature concept breaks down - the technique of the density matrix. An exceptionally good discussion of lhe densily matrix is that of Tolman [5.5}. The method has the further advantage of giving one a discussion of both T l and T2 processes in a natural way. It is ideally suited 10 treating problems in which the resonance is narrowed by lhe bodily motion of the nuclei. It is also applicable to broad line spectra, whcre it can in fact be used for an alternate derivation of our equation for TI of Scct.5.3. As we shall see, the method is simply a variant of the usual time-dependent perturbation theory, but one that is in a panicularly useful fonn. 157

We begin by considering a system described by a wave function t/J, at some instant of time. and ask for the expectation value (M~) of some operator such as the x-component of magnetization. M~. We have. then, ~

(M.)

Therefore

PMztln =

L PUm(mIM~ln) m

(5.60)

(,p, M.,p)

SupJXlse we now expand in a complete set of orthononnal functions Un, which are independent of time:

(5.69) m,n' so that (5.70)

(5.61) If

t/J

"

m

By using (5.62), we have that

varies in time, so must the cn's. In tenns of the functions

tin,

we have

(M.) ~ I:(nlPlm)(mIM.ln)

(5.62)

m,"

lI,m

If we change the wave function, (M~) will differ because the coefficients C~.CII will differ, but the matrix elements (mlMz In) will remain the same. Correspondingly, for a given ¢, the effect of calculating expectation values of different operators is found in the different matrix elements, but the coefficients c;,lc" remain the same. We can conveniently arrange the coefficients cnc;" to fonn a matrix. We note that, to compute any observable, we can specify either all the c" 's or all the products c"c:n. However, since we always wish the c's in the fonn of products. to calculate observable properties of the system, we find knowledge of the products more useful than knowledge of the individual c's. It is convenient to think of the matrix cncin as being the representation of an operator P, the operator being defined by its matrix elements: (5.63)

(nIPlm) = Cnc:n In tenns of (5.63) we have. then,

(M.) ~ I:(nlPlm)(mIM.ln)

(5.64)

lI,m

The result of the operator P acting on a function

tim

may be written as (5.65)

~DnIPM.ln)

"

(5.71)

Also we note that P is an Hennitian operator. This we prove by noting that the definition of an Hennitian operator P is

JU~PUmdT ==

J(Pun)*UmdT =

(J u~IPu

.. dT

r

(5.72)

0'

(nIPlm)

~

(mIPln)'

(5.73)

But (nIPlm) = CIIC:"

(mIPln) = cmc~

(5.74)

so that (5.73) is satisfied. Often we shall be concerned with problems in which we wish to compute the average expectation value of an ensemble of systems. The matrix elements c"c:n will then vary from system to system to the extent that they have differing wave functions. but the matrix elemems (mIM:r:]n) will be the same. If we use a bar to denote an ensemble average, we have, then,

(Mz ) =

L Cnc~l(mIMz]n)

(5.75)

",m

since the un's fonn a complete set. As usual, we find the an's by multiplying both sides from the left by Uil and imegrating:

an

J

= U~P1LmdT = (nIPlm)

so that

(5.66)

The quantities C"C;" fonn a matrix. and it is this matrix that we call the "density matrix". We shall consider it to be the matrix of an operator e, defined by the equation

(n]elm) = cnei'll - (nIPlm) (5.67) n

Likewise we have M~ltll = LUm(mIM~ln)

(5.68)

(5.76)

Since P is an Hennitian operator, it is clear that e is as well. Equation (5.64) becomes, then,

(M.) ~ I:(nlelm)(mIM.ln)~T'{eM.}~Tc{M.e}

(5.77)

lI,m

m

158

159

For the future, we shall omit the bar indicating an ensemble average to simplify the notation, but of course we realize lhat whenever the symbol (J is used, an ensemble average is intended. Of course the wave function t/J, describing whatever system we are considering, will develop in time. Since the 11 11 's are independent of time, the coefficients C n mUSI carry the lime dependence. It is straightforward to find the differential equation they obey in terms of Hamiltonian 1i of the system, since

_'!. a~ , at

=

'H~

(5.18)

The density matrix is the quantum mechanical equivalent of the classical density (J of points in phase space, and (5.82) is the quantum mechanical fonn of Liouville's theorem describing the time rate of change of density at a fixed point in phase space. In the event that 1i is independent of time, we may obtain a formal solution of (5.82), (](t):: e-(i/l)'ltI{](O)e(i!l)UI

In terms of functions have, for example,

J :: J

which gives, using (5.61),

11

(k1{](t)lm) ::

dCn

--;-1 Ln -dUn:: Lcn'Hu .. t .. We can pick out the equation for one panicular coefficient, both sides by uk and integrating: h dCk -7 -

I

dt

=

2:Cn(kl1tln)

CI;,

by multiplying

U'I>

(5.83)

which are eigenjimcrions of Ihe Hamiltonian 'H, we

uke-(i/')'H'{](O)e(i/l)'ltfumdr (e(i/l)'HI Uk ) •{](O)eWA)'HIUmdr

(5.84)

By utilizing the fact that HUm:: EmU"" and using the power series ex.pansion of the exponential operator, we get

(5.19)

(5.85)

n

This equation is the well-known starting point for time-dependent perturbation theory. We can use (5.79) to find a differential equation for the matrix elements of the operator P, since

."

= -,i

E [ckc;.(nIHlm) -

(5.80)

(kl'Hln)cnc:"l

for the time-dependent matrix element in lerms of the matrix element of t! at t :: O.

So far we have talked about the density matrix without ever exhibiting explicitly an operator for e. For the sake of concreteness, we shall do so now. We shall take an example of a spin system in thermal equilibrium at a temperalUre T. We shall take as our basis states, Uri, the eigenstates of lhe Hamiltonian of the problem, 'Ho. The populations of the eigenstates are then given by the Boltzmann factors, giving for the diagonal elements of (!:

e- E ... / kT

CmC~ :: =--'0--

Z

i

(5.86)

where, as usual,

= h(kl P1t -1tPlm)

Z = 'Le-E./kT

n

where we have used (5.70) for the last step. We can write (5.80) in operator form

as

dP

ill =

i

hlP, 1t1

If we wrile

(5.81)

This equation looks very similar to that of (2.31) for the time derivative of an observable, except for the sign change. If we perfonn an ensemble average of the various steps of (5.80), assuming 'H to be identical for all members of the ensemble, we find a differential equation for the density matrix. (J. Since the averaging simply replaces P by {!, the equation for (J is (5.82)

160

Cn ::

lcllieia-n

we have that CmC~:: ICmIICnlci(Q'... an)

(5.87)

It is customary in statistical mechanics to assume that the phases all arc statistically independent of the amplitudes Ie" I and that, moreover, 0'", or On have all values with equal probability. This hypothesis, called the "hypothesis of random phases", causes all the off-diagonal elements of (5.87) to vanish. If, for example, we were to compute the average magnetization perpendicular to the static field 161

for a group of noninteracting spins, as we did in (2.88), the vanishing of the off-diagonal elements of !! would make the transverse components of magnetization vanish, as they must, of course, for a system to be in thennal equilibrium. More generally, we see from (5.85) Ihal the off-diagon..l elements of fI oscillate harmonically in time. If they do not vanish, we expect that there will be some observable property of the system which will oscillate in time according to (5.75). BUI we should then not have a true thennal equilibrium, since for thennal equilibrium we mean that all properties are independent of time. Therefore we must assume that all the off-diagonal elements vanish. Note. however, from (5.85) (which applies to the situation in which the basis functions are eigenfunctions of the Hamiltonian) that if the off-diagonal elements vanish at anyone lime, they vanish for all time. We have, therefore, (nlelm) = (omll/ Z )e-En /k'l"

(5.88)

It is worth noting that the operator for f! is on a different fOOling from most other operators such as that for momentum. In the absence of a magnetic field, the latter is always li.\J/i. For a given representation the density matrix may, however, be specified quite arbitrarily, subject only to the conditions that it be Hennitian, thai its diagonal elements be greater than or equal to zero. and lhat they sum to unity. There is therefore no operator known a priori. However, in certain instances the matrix elements (nJelm) can be obtained very simply from a specific operator for fl. When this is possible, we can use operator methods to calculate properties of the system. We now ask what operator will give the matrix elements of (5.88) bearing in mind that the u,,'s, and so 011, are eigenfunctions of 'HO. Using the fact that

e

-"Ho/kT

um=e

-Em/kT

Urn

(5.89)

(which can be proved from the expansion of the exponentials), we can see readily that the explicit fonn of e is

e = ~e-"Ho/kT

(5.90)

Z

We can use this expression now to compute the average value of any physical property. Thus suppose we have an ensemble of single spins with spin I, acted on by a static external field. Then ?to is the Hamiltonian of a single spin:

In the high-temperature approximation we can expand the exponenlial, keeping only the first tenns. By utilizing the fact that Tr {M z } = 0, we have

(M,)

=

~ T'{M,(,- ~+)}

~ ~ Tr{ (~~h::oI;) } Now, in the high-lemperanlre limit, Z = 2I + I. Since Tr {4} = we get

(M z ) =

~~fI2 I(l + 1) H 3kT

(5.93)

j I(I + 1)(21 + 1), (5.94)

0

which we recognize as Curie's law for the magnetization. The density matrix gives, therefore, a convenient and compaCI way of computing thennal equilibrium properties of a syslem. One situation commonly encountered is that of a Hamiltonian consisling of a large time-independent interaction 'Ho, and a much smaller but time-dependent tem H,(t). The equation of motion of the density matrix is then

de i dt = !i[e, 'Ho + Hd

(5.95)

If HI were zero, the SolUlion of (5.95) would be e(t) = e -(i/" )1io/ e(O)e(i/h)1i oi

(5.96)

Let us then define a quantity e* (the siar does not mean complex conjugate) by the equation e(t) = e -(i/")1iot e* (t)e(ijh )"Ho/.

(5.97)

If, in fact, HI were zero, comparison of (5.96) and (5.97) would show that e* would be a constant. (Note, moreover, Ihat at t = 0, e* and e are identical.) For small?tl, then, we should expect e* to change slowly in time. Substituting (5.97) into the left side of (5.95) gives us the differential equation obeyed by e*:

-~[?to, el + e- O/")1iotd: * e(ifh)1iot = ~(e, Ho + ?til t

(5.98)

We note thai the commutator of e with ?to can now be removed from both sides. Then, multiplying from the left by exp (il1l)'Hot) and from the right by exp (-(i/h)Hot), and defining

(5.91) We shall illustrate th~of the density matrix 10 compute the average value of the z-magnetization (1I1 z ). It is

(M z ) =Tr{Mze} =

162

~ Tr{M ze-"Ho/k1'}

(5.92)

Hi = ei1iot/h?tle-(i/h)"Ho/

(5.99)

we get, from (5.98),

de* i * * dt = hie, 1£,(')]

(5.100)

163

Equation (5.100) shows us, as we have already remarked, thai Ihe operator r/ would be constant in time if the perturbation ?i1 were sel equal to zero. The transformation of the operalor?il given by (5.99) is a canonicallTansformation, and Ihe new representation is termed the interaction representation. The relationship of U and (J. is illustrated by considering the expansion of the wave function tP in a form

tP = L:ane-(i/l)E.l un

We can make a closer approximation by an iteralion procedure, using (5.109) to get a bener value of U·(t l ), to put into the integrand of (5.108). Thus we find

·e(l) = ,'(0) +

,'(0) +

.

(5.101)

= ,'(0) +

o

(e(O), "H;(,")Jd," }. "H;(")] d"

0

(,(0). "H;(")]d"

o

n

instead of

2 t tl

+

(5.102)

tP=L:Cnu n

(nll?·lm) = e{i/Ii)(En -E".)t(nll?lm)

((,'(0). "H;(''')J. "Hi(")]d"

d'"

(5.110)

(5.103)

We could continue this iteration procedure. Since each iteration adds a term one power higher in the perturbation 'Hi, Ihe successive iterations are seen to consist of higher and higher perturbalion expansions in the interaction ?i I, For our purposes, we shall not go higher than the second. Actually, we shall find it most convenient to calculate the derivative of (J*. Taking Ihe derivative of (5.110) gives

Since (5.101) and (5.102) give the same tP, we must have

d,;,(1) = *["(0). "H;(I)] +

(5.104) (5.105) Comparison with (5.103) shows that ana~ :::::

(D JJ o 0

n

where the un's and En's are the eigenfunctions and eigenvalues of the Hamiltonian ?io. In the absence of ?iI, Ihe an's would then be constant in time. We shall show Ihat Ihe matrix ana~ is simply (nlu·lm). We note first that replacing I?(O) by U· in (5.83-85) gives that

(nil 1m)

Q.E.D.

(5.106)

There is likewise a simple relalionship between (nl1ii 1m) and (nl1i 1 lm). By an argumem quite identical to that of (5.83-85) we have (nj1iiJm) =

Ju~e(i/l)1tot1ite-(i/l)1totumdT

::::: e(ifA)(E"-E,,,)t(nl1i 1Im)

,'(I) = ,'(0)

+

(5.107)

u"

(5.100). By inte-

,

*J .

[,(I'), "H;(I')]dl'

(5.108)

o

This has not as yet produced a solution, since e·(t') in the integral is unknown. We can make an approximate solmion by replacing e"(t l ) by e"(O), its value at t = O. This gives us

.

'(') = ,'(0) +

,

i J(,'(0). "H;(,')Jdl' o

US J

((,'(0). "Hi(I')]. "Hj(l)]dl'

. (5.111)

o

It is important to note that (5.111) is entirely equivalent 10 ordinary timedependent perturbation theory carried to second order. However, instead of solving for the behaviors of a" and am, we are solving for the behavior of the products ana:"n which are more directly useful for calculating expectation val-

"es.

5.5 The Rotating Coordinate Transformation

.

Now we proceed to solve the equation of motion for grating from t = 0, we have

164

*J[{, *J" , *J

(5.109)

We saw in Chap. 2 that it is often convenient 10 go 10 the rotating frame when a system is acted on by alternating magnetic fields. We explored both a classical treatment and a quanlum mechanical treatment. In the laller we ITansformed the Schrtldinger equation. We now examine how to transform Ihe differential equation for the density matrix.. Let us consider first the case of an isolated spin acted on only by the static field Ho and a rotating field HI(t) given by (5.112) where we have already recognized the sense of rotation for a positive "'( in the negative sign in front of j. In the laboratory reference frame de i - = -(,"H - "H,)

d'

(5.113)

h

165

11. = -")'h[HoIz + HI (Iz cos wt - I y sin wt>]

(5.114)

Utilizing (2.55) we express this as

11. = -")'h[HoIz + Hle+ iwU , Ize- iwU ,]

where 17k expresses the chemical or Knight shifts. We omit a chemical shift correction from H I since HI <: Ho always. Defining

(5.115)

Defining an operator R as

(5.125) we now express 11. as a sum

R =: eiwll •

(5.116)

'H = 'Hz + 'H p + 'HI

with

(5.126)

we have

11. = -")'h(HoIz + HIRIzR- I )

so that

(5.111)

~: = itrr(HoIz + HI RlzR- I ) -

i")'(Holz + HI RIzR-1)e

(5.118)

Following the same reasoning as in Chap. 2. we now seek to eliminate the operators by defining a new variable eR

en

==

R-leR or

"a'

(5.119)

e=RenR-1

(5.120)

'H p =

k

. deR _I l(wIze- f!W1z) + R-;[lR

'HI = _")'hHleiwll'Ize-itNlI. Following the reasoning of Chap. 3, we keep only that portion of the spinspin coupling which commutes with 1iz. Thus. we express 'H il• as a sum of teons like the A, B. C• ...• F of (3.7), keeping only 1f!!k' the teons which commute with I z : J

(5.121)

11 =

Multiplying from the left by R- I , from the right by R. utilizing the fact that I z and R commute, and employing (5.119) and (5.120), we get

i

= h(eJ~11.eff -'Heffeu)

where

(5.122)

(5.123) is our old friend, the Hamiltonian of (2.63), which expresses the interaction of the spin with the effective magnetic field in the rotating reference frame. Let us now allow there to be more than one spin. allow the spins to interact with each other by lhe dipole-dipole coupling or by indirect spin-spin coupling through bonding electrons, and include Knight shifts and chemical shifts as well. Then, labeling individual spins by k or j, we have the Hamiltonian in the laboratory frame as

(5.124)

166

(5.128)

Thus we take as our total Hamiltonian 'H' in this perturbation sense

= -i,,),Ho«(!Iz - I z (!) -i")'H I (eRlzR-I - RIzR- I e)

deR

(5.127)

kJ

11f;,.I.)=0 .

Substituting (5.120) into (5.118). we get

dt

I

L")'h1ioUk I zk + '2 E'Hjk

,,),hHoIz + E ")'hHoUkIzk

,

-

+~

E1CJk - ")'hH,RIzR-'

(5.129)

j,k

We can now repeat the derivation of (5.122) above, noting that our new Hamiltonian differs from that of (5.114) by addition of the teons I

L; ,hHou,I., + 2 L;1I1j k

(5.130)

jk

all of which commute with R. (It IS Interesting to note that this expression through its 17k seems to require that we know the absolute chemical shift relative to a bare nucleus, i.e. a nucleus stripped of all its electrons. That viewpoint is wrong. All one needs do is pick a convenient reference compound for which one defines the chemical shift to be zero. ")' is then defined as being the ratio of the angular frequency to the static field for resonance for that reference compound. We discuss this topic further lit the end of this section). Thus we readily find

den &

i

I

I

= h(en'Jterr - 'Heff(!R)

where

(5.131)

167

'H~ff '" - ,MHo - w/-y)I, - 1hH I I r

I

+ "E1 IiH0t1 IJ,k+"2

L'H2 j

(5.139) (5.132)

k,j

Ir;

This new effective Hamiltonian reminds us by the presence of the tenns 'Hqlr; that if spins interact with one another in the laboratory frame, they do also i/the rotating frame. It also reminds us from the tenns involving UIr; that when there is more than one chemical shift, one cannot be simultaneously exactly on resonance for all nuclei. Since 'H~tr is independent of time, we can solve formally for en(t) in terms of its value at an earlier time (t '" 0) by eR(t) '" e -i'H~ffL/. e R(O)ei'H~ffl/lI

(5.133)

It is imponam to keep in mind that in general the interaction representation differs from the rotating coordinate transformation. Both may be used with a large time-independent Hamiltonian 'Ho and a time-dependem ?it(t). e- (e expressed in the interaction representation) is related to e by

e- (t) '" ei'Hol/h efJ)e -i11{Jtlh whereas

eR

is related to

(5.134)

and a new zero of chemical shifts

'171, '" Uk -

(j

(5.140)

1(z =

--I hHoI,

(5.141,)

1tp

L 7 'IHOUkI,1r;

(5.14Ib)

which makes

=

1

k

where in getting the second equation we drop terms such as 7iult. Clearly the fonn of (5.141) is the same as that of (5.127). We have simply redefined 7 to COlTeSpond to a different zero of chemical shifts. While (5.138) guarantees that the Zeeman term falls in the midst of the resonance lines, it seems to imply through (5.137) that we know the individual uk's, and go through some evaluation to get 7i. What is imponant to realize is that there is some arbitrariness in the division between 'Hz and 1tp which really corresponds to the arbitrariness in the choice of a zero of chemical shift.

e by

eR(t) '" e-K..>tI. e
(5.135)

5.6 Spin Echoes Using the Density Matrix

If we have the Hamiltonian of (5.114),

l

(t) '" e -K..>oll. e
(5.136)

These do become the same if we are exactly at resonance (wo '" w), but othelWise would differ. If the only time dependence in the problem is from HI, the rotating coordinate transformation eliminates it. The division of the Hamiltonian of (5.127) into ?i z and ?ip is nOt the only division one might think to make. For example, if one defines an average chemical shift for the system of N nuclei, 7i, by the equation 1 7i'" NLt1k

(5.137)

k

then one might prefer to have the Zeeman energy be the average value including the chemical shifts. We can do this by making a division into an 'HZ and ?il> defined as (5.138') ' 1 hH o(Uk - -U)I,k 'H' '" 'LJ p' k

, +"2I ",--?tij

(5.138b)

j,k

Note, however, that (5.138a) can be converted to (5.127) if we define a new 1, called

i,

168

Now that we have obtained the differential equation for the density matrix in the laboratory frame, in the interaction representation, and in the rotating frame, we tum to methods of solution. The rest of this chapter is devoted to that task. We will first discuss several problems involving pulse techniques, then we tum to use of the density matrix to discuss relaxation. As topics involving pulse techniques we shall first examine how to use the density matrix to derive the spin echo, giving us a chance to use this formalism to examine a problem whose physics we have now seen both classically and quantum mechanically (using wave functions). We then turn to some new physics which underlies one of the major techniques employed in NMR today, Fourier transform medlOds. To do this we first set the stage by deriving a result important to linear response theory, the theory underlying Fourier transform NMR. This result is the response of a system to a 8-function excitation. Anned with that result, we next go on to explain what is meant by Fourier transfonn NMR, and then prove the fundamental theorem which specifies what it measures. We then further develop a concrete appreciation of the density matrix by exploring what happens to the actual matrix elements under the action of an alternating magnetic field. The most general approach to solving the equation is to pick an appropriate family of wave funclions as a basis set, then conven the operator equation to a set of coupled differential equations between the matrix elements of e and ?i. Thus if we pick a basis set 10), where 0 stands for the set of eigenvalues which distinguish the states, we convert the equation 169

d(!

i

- = -«(!H - ?i(!) dt h

to

j;

d (aleld) = D(aleIP)(P1Hla') - (aIHIP>(Plela'» • p d,

H,

(5.142)

(5.143) I

In many cases there are then straightforward melhods of solution. Once one has

r------ T ----l

I

iI

obtained the (alula'rs, one can then calculate the expectation values of observ~ abies using (5.77),

I

I

"

t1

t2

t]

(M.> =T,{M.e) = L:(a'IM.la)(alela')

(5.144)

o

Fig. 5.4. III vs time, giving the definition of the times 0, tl, t2, and 13 which spc<:ify the beginning and end of the tr/2 pulse (I = O,td and of the tr pulse (t = t2,13)

0',0"

The weakness of Ihis appro,\lch is that it may fail to reveal important physical insights available from other methods. The "other methods" are based on manipulating operators, and are analogous to the quantum mechanical derivation of the echo carried out in Chap. 2. We now tum to several illustrations of the use of the operator approach. Later we return to use of (5.143). We begin with a derivation of the spin echo of a group of non interacting spins using a (rr!2,7r) pulse sequence. We deal with this problem by first transfonning to the rotating frame so that our basic equation is given by (5.123)

d(!R

--;It

As in Chap. 2, we approximate this as (5.146)

1i"err = -1'hH I T;I; Then, utilizing (5.83), we write I? n(t I) = e iWI tl f" I? n(O+)e -iWl 11 I"

= eiW1 tl f z I?n(O-)e-iW1 tIl"

(5.148)

WI = 1'HI

i

= "h«(! llHeff - 'Herru n)

(5.147)

where

Likewise, during the interval tl to t2, where HI is zero, we have

As in Chap. 2, we assume that the static field is inhomogeneous with a distribution function p(ho)dho for the N spins, where dN, the number of spins whose resonance lies between ho and ho + dho, is given by (2.119):

fm(t2) = e i ,.Il o(t2 -( 1) I, I? n(t l)e -i,.lt o(t2 -11 )1.

Defining [as in (2.122)] X(8) =

dN = Np(}to)dho

(5.149)

e ifJl"

(5.150,)

and where ho is given by (2.118):

(5. 150b)

ho = H o -wh We take the distribution function to be symmetrical about a center value (lto = 0) and we assume w is to be at resonance when ho = O. Since we have already treated this problem in Chap. 2 using wave functions, we will employ the same notation (Fig. 5.4). At t = 0-, the density matrix has its initial value un(O-) which represents thennal equilibrium in the laboratory frame. At t = 0, HI is turned on. Although there is a discontinuity in 1i"eff at t = 0, the right-hand side of (5.123) remains finite. Therefore, integrating (5.123) with respect to time from t = 0- to t = 0+, we find

selecting WI tl = 7r/2 and WI (tJ - t2) = 7r for the rr/2 and rr pulses, and choosing times t later than t3, we get (!n(t, ho) for those spins off resonance by ho as I?n(t, 110) = T(t - 7, ho)X(rr)T(T, h o)X(rr/2)l?n(O-) x X-I(rr!2)T-I(7,ho)X-I(rr)T-I(t - r,ho)

(5.151)

Following our procedure from Chap. 2, we insert X(rr)X- I (7r) = I to the left of T(t - T, ho) and utilize (2.134) to get X(7r)X- t (1l")T(t -

T,

ho)X(rr) = X(rr)T-1(i - 7, ho)

(5.152)

(5.145) A similar argument holds of the other discontinuities. During the time interval from t = 0 to t = t I or from t2 to tJ, the Hamiltonian is given by (2.117)

1teff = -1'1i(h oI z + H IIz) 170

I?n(t, ho) = X(7r)T- I (t - T, ho)T(T, ho)X(rr/2)l?n(O-) x X- 1(7r/2)T- 1(T, ho)T(t - T, ho)X-I(rr)

Thus, when t -

T

(5.153)

= 7, or t = 27, 171

(l/l(2r, 110) '" X(7r)X( 7r(l.)e n(O-)X- 1(7r(l.)X- 1(7r)

(5.154<1)

= X(3./2)eR(0-)X-'(3./2l

(5.154b)

= X(-'/2leR(0-)X- 1(-./2)

(5.154c)

where the last step utilizes the fact that X(2Jf) '" X(O) when X operates on fl [contrast this with (2.78)]. Equation (5.154) shows that when t '" 2r, fl is independent of Ito, hence we have an echo. Moreover, flR(2r) is the same as we would have if instead of applying two pulses spaced apan by T, we had applied a single pulse which produced a -1f/2 rotation. We now proceed to calculate the magnetization signal at the echo peak. Since (;lR(O-) corresponds to thennal equilibrium in the laboratory frame, we have, using (5. tl9) and (5.90),

Tr{G} ",Tr{I+X(-7r!2)(1 + -Yli:'~lz)X-I(_7r/2)}

'" ..!.-e-x....II. e -'Ho/kTeilollf• Z Since 'Ho '" -'YhHolz, we get

~eY"l1o/./kT

(5.155)

T,(I+X(-./2lX-'(-./2») =T,(r+) =0

_ I ('YhHolz ) (;lR(O )"'(21+1) 1+ kT + ...

{ h)'N H

,H, {e/l(t)I.) ,H, {e/l(t)Iy )

(5.158)

Combining we get

(M.(t) + i(My(t) = (M+(t» =

,I, Y, {e/lI+)

(5.159)

Since both (M~(t» and (My(t» are real, we can get them both readily by taking the real and the imaginary pans of (5.159). We utilize (5.154), (5.156), and (5.157) to calculate (.M~(2r» and (My(2r»

Jp(l'o)dh"h (r+ 1I<2T, ho») ,hN J + P(ho)d11o {G}

172

T,

I)

1

Tr

(5.162)

Since Tr {l>;Ig }

'"

~;;~~

:;y,

(r+(-ly))

(5.163)

0 and Tr{l;} '" (21 + 1)1([ + 1)13, we get

N-y 2 h 2 I([ + 1) . . 3kT Ho' = -IXoHo

.

(5.164)

(5.157)

"0.

'" (21

(5.161)

Now X(-1(/2) '" X-I (1f!2). Hence, using (2.124) (X- t ('lf!2)lzX(Jr!2) '" -llIl we get

(M+(2T» = -

(Note that since the spins do not interact, we can work with the Hamiltonian of one spin. However, we eventually include the effect of all N spins when we This situation is to be contrasted with that of SecI.S.7 where integrate over the Hamiltonian is that of N coupled spins). We have operators M>;(", -ylJi>;) and My corresponding to magnetization along the x·axis (the HI direction) and the y-axis in the rotating frame given by

(M+(2T» = N

.

The physical significance of this result is that at infinite temperature, where only the a term remains, the thermal equilibrium magnetization vanishes. Therefore, no pulse sequence can produce a signal. We will encounter a similar situation in all pulse sequences stnning from thermal equilibrium. The b term gives

In the high temperature approximation, Z '" 21 + I, giving

(M.(t) = (My(t» =

,

"

In the trace, the term labeled a is readily shown to be zero:

(M+(2T» = (5.156)

Z

(5.160b)

'--v-'

(M+(2T» = (;I + I) k;Tr{r+ X(-./2lI.X- (-./2))

(;lR(O-) '" e- iloltl • e(O-)eilo'll.

(;lR(O-) '"

where

(5.160<1)

The -i shows that at t '" 2T the magnetization has zero component along the x-directjon (i.e. HI), that its magnitude is the same as the thermal equilibrium magnetization, xoHo, and Ihat it points along the negative y-direction. The approach we have described can be extended readily to deal Wilh arbitrary sequences of pulses. Let us consider that at t '" 0- the system is in thennal equilibrium with density matrix (;In(O-) '" e(0-), as in (5.152). Stnning at t '" 0 the Hamiltonian in the rotating frame takes on a value 'HA for a time interval tAo then jumps suddenly to 'Hn for a time lB' and so on. In the example above tA '" t]

11.A '" --y1i(holz + H l l%)

tB",t2- t l

1{B '" -"'f1ihOlz

tc'"

He'" --yli(IIoIz + J[ll~) 'HI) '" -,liholz

t3 - t2

tD"'t -tJ

(5.165)

Corresponding to t A and 'H A , there is a time development operator Til given by (5.166)

Til '" exp(-i7tAtA/li) Similar equations hold for the succeeding inlervals. Thus, we get en(tz) '" TZTy ... TBT"{;l(O-)T; I Tjj I ... TylT I

Z

(5.167) 173

We chose to neglect all tenns in the Hamihonian but the tenn involving HI when HI is on. That approximation is often useful (as illustrated above) but would not be allowable if one were trying to understand what happens when HI is comparable to the line width. The general fonnula (5.167) does nOI necessarily include this approximation, but the approximation is readily introduced if desired.

Now lhat we have looked at a familiar problem with the density matrix fonnalism. we tum to a problem we have not treated quantum mechanically before. the response of the system to a o-function. This topic is imponam in pan because, as we saw in Chap. 2. if we know this response we can use it to calculate both I and I'. But it is also important because it underlies the theory of Fourier transfonn NMR which we take up in the next section. We therefore stan by recalling (2.156): ~

J,

~

J,

mer) cos (wr)dr

x" =

mer) sin (wr)dr

(2.156)

where mer) is the response at time r to a delta-function applied at r = O. We now tum to the calculation of mer). To do this we must first senle on the Hamiltonian. We take a Hamiltonian which has substantial practical utility. that of a group of N identical spins. coupled to each other with dipolar. pseudooipolar, and pseudoexchange interactions, experiencing chemical and Knight shifts. A more general Hamiltonian would include different nuclear species. quadrupole coupling. and relaxatiOll phenomena to a thennal reservoir. With the approximation above we will then have a Hamiltonian

'H = -

2:: ,hH,I,k(l k

1

Uk) + 1: 2::'H;j i,i

2:: ,hC'(t)Iz

(5.168)

k

which is similar to that of (5.123) except for the replacement of the HI driving teno with a magnetic field CE(t). The parameter C is used to define the size of the driving tenn. This is a Hamiltonian in the laboratory frame. In (5.168), the 8·futlction field has been applied along the x-direction in the laboratory. It will produce magnetization in general with components along the laooratory X-, y., and z-directions. Thus, in general such a drive produces a magnetization in the a'·direction for a o-function in the Q-direction. This fact requires that we label m(r) by subscripts a and 0'1 just as we label the X by X(I"(l' on p.58. Thus, let 111(1"(1'(1") be the magnetization in the a'·direction pf<xluced by a 6-function field applied in the a-direction. Equation (2.93a) gives X~", and X~",· The meaning of the constant C can be seen by considering the imegral of the driving field CE(t): 174

JC'(t)d'

,-

= C

(5.169)

But, 'by definition, m(t) is the response when this integral is unity, i.e. when C = I. or. alternatively, if M(r) is the response when C 4- I, meT) = M(T)/C

5.7 The Response to a 6-Function

X' =

,+

(5.170)

.

While in principle one can set C = I, in the process one conceals some units. and thus complicates the use of dimensional arguments to check the correctness of fonnulas. We will therefore keep C, realizing that eventually we will utilize (5.170). Indeed. to keep the units explicit. we will replace C by a product H TfJ:

Hro == C

(5.171)

where one can think of H as the height and 1"0 the duration of a rectangular pulse which. in the limit that rO goes to zero, represents the 6-function. To find mu(r) we need to solve the density matrix with the Hamiltonian of (5.168). with the initial condition that e(O-) represents thennal equilibrium. At t = 0, the driving teon dominates. since the 6-function is the biggest teon. At all ocher times. only the Zeeman and spin-spin teons are present Thus, if t > O. we have a time-independent Hamiltonian. We will treat it in the same spirit as the Hamiltonian of (5.126) in which we recognize that the spin-spin coupling lenns are perturbations, SO (hat it is appropriate to drop the teons in the spin-spin coupling which do not commute with the Zeeman coupling. We thus have as our Hamiltonian 'H.

=

-ihHol~ + L'Yl~Hou~JJ~1e + ~ L,'H.11c -

'YhlzHroo(t)

, (5.172)

i,k

Ie

where 'H1k is the ponion of 'H.jk which commutes with the mean Zeeman energy

II.. 'H1j l = 0

(5.173)

It is convenient, as in (5.126), to define three quantities

'Hz == 'H.p ==

'YhHoI~

L'YhHou~kI~k +~ "L'HJk

(5.174)

i,k

Ie

H6 = -ilil",HrOO(t) Note from (5.174) that 'H.z and 'Hp commute

['HI'. 'Hz] = 0

.

(5.175)

'Hz is the Zeeman energy, apart from chemical and Knight shift differences. Hp is a much smaller tenn ("P" for perturbation) which gives rise to splittings and 175

line widths. For t > 0, since we have then 'H6 = 0, we have a lime-independenl Hamiltonian 'Hz + 'Hr. Then, using (5.95), for t > 0,

e(t) = .. p (-i(Hz + Hp)t/IiJe(O+)exp(i(Hz + Hp)t/Ii)

.

therefore

(5.185)

(5.176)

To find e(O+)we must solve for the time development under the action of the C-function. Since we seek the linear response, it is appropriate to treat 'H6 as a perturbation, keeping only the fim-order lenn. We therefore turn to (5.11 I), using 'H6 for 'HI> keeping only the first lenn on the right. In this equation

e'(.) = exp (-i(Hz + Hp)t/Ii)e(t)exp (;(Hz + Hp)t/Ii) 'H (t) = exp( -i('Hz + 'Hp)t/h)'H6(t)exP (i('Hz + 'Hp)tlh)

(5.177)

de'(t) = .'.[e'(O-)H;(t) - H;(l)e'(O-)J

(5.178)

6

Then

dt

(5.186) The first tenn, linear in I z , corresponds to a thennal equilibrium magnetization. Therefore, when put in (5.176), it will not lead to any transverse magnetization. To get mn(t), therefore, we keep only the tenn involving I y , utilizing (5.170) (H Tl)mxx(t) = (Mx(t»

h

=,/oT,{Ixe('»

Integrating from t = 0- to t = 0+, we gel

,/oHo

= ,liT, { IxT(1) Z(oo)kT,(Hro)IyT

0+

e'(O+) - e'(O-)

=.'./0 J

[T(I)e(0-)H,(t)T- 1(.)

0'

0-

(5.179)

- T(t)H,(t)e(0-)T-1(t)]dt with

(5.180.)

T := exp (-i('Hz + 1ip)tlh)

and (for future use) Tp := exp (-i'Hpt/li)

.

(5. I 80b)

But over the zero time interval we can neglect the time dependence of everything except the c-function giving

.(0+) = e'(O-) - *,/oHTo[e(O-)Ix - Ixe(O-)J

(5.181)

and indeed

(5.182) Now e(O-) corresponds to Ihennal equilibrium, hence is given by I

1

e(0-) = Z ..p[-(Hz+Hp)/kT] '" Zexp(-1iz/kT)

(5.183)

In the high temperature approximation this gives

(I

+ ,/oHoI,) kT

(5.184)

Following the argument after (5.158), we drop the first tenn in the parenthesis since it contributes zero magnetization, getting 176

(I)

.

}

(5.187) (5.188)

A useful variant on (5.188) can be obtained utilizing the fact that 'Hz and 'Hp commute, and expressing Tz as exp (iwotlz), with wo = "tHo. Then

Tr{IzTIIIT- 1 }

Tz := exp (-i'Hztlh)

e(0-) = _1_ Z(oo)

"'I 2h2 Ho -1 mn(t) = Z(oo)kT "'I Tr {IzT(t)IyT (t)}

-I

= Tr {Til IzTZTpIyTpl} = Tr {e- lwotl• IzeifNOLf'TpIyTpl} = Tr {IzTpJyTp I} cos wot + Tr {Jy T p l yT p t} sin wot

This equation is closely related

10

(5.189)

(B.18) in Appendix B. At t = 0+, Tp =

Tj;l = 1 so that the coefficient of the cos wot tenn vanishes (Tr {IzIy } = 0), and the coefficient of the sin wot term is Tr {J;}, which is nonzero. For times shon compared to h/11ipl, where by l1-lpl we refer to the magnitude of a typical nonzero matri", element of 'Hr, we can slill appro"'imate Tp as 1. Consequently, the magnetization nlzz(t) will start as sin wot, a signal oscillating at the Lannor frequency. Since 'Hp gives rise to spectral width, over times of the order of the inverse of the spectral width, Tp will differ significantly from I, and we may e"'pect both traces to contribute. In Appendix B, Eq. (B.18), we show that if 'Hp contains only spin-spin terms and 7 is chosen such that the Uk'S are all zero, the first trace always vanishes. The effect of the operators Tp is to modulate the coefficients of the oscillations at Wo over times of the order of the inverse of the spectral frequency width, L1w. 177

Thus, for times short compared to the inverse of the spectral frequency width, ~, we may write mzz(t) = 7

721i2 Tr {12} Z(oo)kT Ho sin wot

with

t

<: l/L1w

•

It is straightforward to show that the term a for a many-spin system is the static susceptibility Xo. Hence

for

t
(5.191)

This result has a simple classical significance (Fig. 5.5).

[cos (woOTr {Io;Trl yT p1}

=

J

mu(T) cos (wT)dT

o

X~:I:"(w) =

y

,

....ig.55. A magnetization, Alo = xoJ1" initially along the :--direction is rotated through II. small angle dB = "'I J1 TO by a field J1, of duration TO, along the laboralory :I:"-direction. The rC:$Ulting magnetization has .. component Mod6 along the laboratory ",-direction. At later times, it precesses about z, producing .. nonzero component along the :I:"-axis which oscillates all sin Io'(Jt

(5.192)

giving My = XoJIO(7HTo) = HToxowo

(5.193)

This magnetization precesses at Wo, in the left-handed sense, giving a sin wot dependence to M:I:"' lllUs, utilizing (5.170), mo;o;(t) will be given as

(5.194)

for times short compared to the dephasing effects arising from chemical and Knight shift differences and spin-spin couplings. We see, therefore, that (5.188) and (5.189) have simple physical significance. In fact, we can usefully rewrite them as 178

=

J

(2.156)

mu(T) sin (wT)dT

o

Considering the 6-function to be a field in the x-direction of strength H>- Ho, on for a time 7"0 < I/wo, we see that the field tilts the thermal equilibrium magnetization xoHo to have a component in the positive laboratory y-direction. Since we are seeking a response (Le. an x-y magnetization plane) linear in the driving tenn, H, we keep the angle of rotation, L18, small

mu(O = xowo sin wot

(5.195)

While (5.195) looks relatively simple, involving just a few traces, its actual evaluation may be enonnously difficult since the equation contains all the content of the theory of line shapes! Nevenheless, the equation gives a compact statement for attack by the various fonnal methods which have been developed to deal with the line-shape problem. In principle, this mu(t) will yield both the absorption and dispersion spectra utilizing (2.156)

x~z(w) =

z

X{w~}

Tr l y

+sin (woOTr {IyTplyTj;I}]

(5.190)

~

mu(t)=xowo sinwot

mzz(t) =

5.8 The Response to a ,,/2 Pulse: Fourier Transform NMR We tum now to the topic of Fourier transfonn NMR, a method of doing magnetic resonance which is today all pervasive in the field of high resolution liquid spectra, and which fonns the basis for one of the most important other developments in NMR, so-called two-dimensional Fourier transfonn NMR (which we take up in Chap. 7). In 1957, Lowe and Norberg [5.6] discovered an imponant theoretical result which was confinned experimentally by them in conjunction with Bruce [5.7]. Lowe and Norberg showed theoretically that in solids with dipolar broadening, the Fourier transfonn of the free induction deCa'yfoliowing a -rr/2 pulsegave the shape of the absorption line. They verified their result experimentally with a single crystal of CaF2 by comparing their experimenlal free induction decays with the Fourier transfonn of Bruce's steady-state absorption data taken on the same crystal. In 1966, Ernst and Amfer.fon [5.8] pointed out that there were substantial experimental advantages to the use of pulses over steady-state meth<X1s when dealing with complex spectra. (We discuss advllntagcs later in this section.) Their method, which consists of first recording the free-induction signal, then laking ils Fourier transfonn, is referred to as Fourier transfonn NMR. The method allows on~ to obtain both the absorption spectrum and the dispe-;;ion spectrum'"":", ~how ttlatthe cosine transfonn of the free induction decay gives one whereas the sine transfonn gives the other. (See also the book by Ernst et al. [5.9].)

TheY

179

[ These remarks are reminiscenl of linear response theory discussed in Sect. 2.11 where we found thut X' and i l are given by cosine and sine transforms of the function m(T), (2.156):

J

~

x' =

~

m(T) cos (wi)dT

X" =

o

J

m(T) sin (wT)dT

(5.196)

~n

_1_(1

(0-)= (0-)= U Z(oo)

1C = 'Hz +'Hp +1-{1

with

(5.197)

(5.198)

(5.199) (5.200) expressing the fact that the spin is acted on by a rotating magnetic field which is to generate a 7f(l pulse. During the pulse, we go to the rotating frame where UJl is described by (5.131):

i i ,

= "h(un'H eff - 'HeffUIl)

(5.201)

with 'H~rr given by (5.132) and UIl by (5.116) and (5.119). While the H t is on, we take it to be so large that we can neglect everything else in 'HI giving

'H' = -"(fi.HtI~

.

hH 1 o ,) kT

(5.204)

1

-I)

un(t,) = Z(oo) (lhHO 1+ kT X(wltt>I~X (Wltl)

(5.205)

where X(D)

== e+ i8Jz

(5.206)

Now, ulilizing (2.55),

X(8)I z X-'(8) = I y sin 8+ I~ cos 8

so that UR(tl) =

Z(~) (I + 1::0 [Iv sin Witt + I z cos Wtttl)

(5.207)

(5.208)

The first term on the right will nOt contribute 10 the magnetization, following the argument of (5.160) and (5.161) (i.e. it is allihal remains as T-+oo). Moreover, the term involving I z is, apart from a constant of proponionality, the same as the temperature-dependent contribution to a (! R corresponding to thermal equilibrium. Thus, it contributes nolhing to transverse magnetization. (The validity of neglecting the tenns involving 1 and I z can be verified by direct calculation of (Mz(t)) and (M!I(t)).] We are left, then, with I 1liHo. UR(tl) = Z(oo) Sill (wltdIv

---;;r

(5.209)

Therefore, utilizing (5.120),

e(tl) = RUR(tI)R- I I l'IHo. -I = Z(oo) kT ,m (w",)[R(t,)I,R ('1))

(5.210)

where R is given by (5.200): ROI) = ei""lt J•

(5.211)

To gel (!(t) for t> tl' we introduce the operators T, Tz, and Tp defined as in (5.180),

(5.202)

If the HI is turned on suddenly at t = 0 and turned off at t = tit we have, then [os in (5.147)1. UR(tI) = e+iw]It r. un(O+)e-iwtltJ" = e+iW ]lt 1"un(O-)e-i""t 1tl"

+7

whence, as in (5.150a) and (5.151)

•

where m(i) is the response of Ihe syslem 10 a S-function which we treated in Sect. 5.6. We therefore demonstrale that in the linear response regime al high temperatures, a 1f(l pulse (in faCI, any pulse) is equivalent to a S-funclion excitation, hence Ihat the transforms of the free induction signal give ;( and X". Let us consider, then, Ihe Hamiltonian of (5.124) describing a group of N coupled spins, allowing there 10 be both chemical and Knight shifts. We then have a Hamiltonian 'HI given by keeping only Ihat part of the spin-spin coupling which commutes with the Zeeman coupling as in (5.172)

dUll dt

I

with WI = "(HI. Taking UR(O-) to represenl thermal equilibrium, we have, utilizing (5.155-157) generalized for the Nobody Hamiltonian as in (5.184),

(5.203)

Tz(t) = exp (-i'H.ztl1l) = exp (iwotIz) Tp(t) = exp (-i'H.pt/h)

T(t) = exp (-i(1ip + 1iz)tlh) = TzTp '" TpTz

(5.212)

(Do not confuse the operator T(t) with the temperature T!) 18.

181

Thus, since 1tz + 7-ip arc independent of time, we get from (5.83)

Utilizing (5.196) and (5.219) with Hro = I, we have then

=

e(l) = T(t - tt)e(tl)T-I(t - til =

-yIiHo

Z(oo)kT

sin (Wltl)T(t _ tl)R(t\)lyR-I(tl)T-IU - t\)

T(t - tl)R(tl)

= exp (iwotI:) exp (-i1tpt/fi) exp (iHptdh)exp [i(w - WO)t1 1:] (5.214)

If, now, we neglect the effect of Hp and (w-wo)t I during t\ we can approximate Trl(tl)ei(""-~)tl/. ~ I

(5.223b)

o

= exp [iwo(t - t 1)1:] exp (-i7-ip(t - t I )/1i.) exp (iwt I [:)

p

.'0 J(Mz(r»o sin (wr)dr sm

=

Xl/(w) =

= T(t)T I(tl) exp (j(w - WO)tl 1:)

(5.223.)

Sin 0 0

..

(5.213)

Now

(5.215)

getting

Equation (5.223b) includes the result of Lowe and Norberg (they did not include chemical or Knight shifts). Equation (5.223) includes the result of Ernst and Anderson that the cosine and sine transfonns yield respectively the dispersion and the absorption. [It is useful to keep in mind that the tenns "cosine" and "sine" imply a particular phase convcntion. Ours is defined by (5.221).] The validity of (5.218) rests on the approximation that during the pulse we can neglect the effect of Hp and (w - woH I. Since HI' leads to a spectral width, Llw, this approximation is equivalent to

(5.216) Therefore, the expectation value of the transverse magnetization (Mz ), following (5.158), is

(M.(t)

=

(5.217)

This result is identical to the result of (5.187) except that sin WI tl replaces -yHro. We introduce the notation

(Mz(t»o = -y

2,.2H

t 0 sin (O)Tr{IzTlyT-I} Z(oo)kT

(5.218)

the magnetization following a pulse with wltl

(5.219)

== 0

Then we can write sinO

(Mz(t))e = 7n zz (t)-H ,

TO

(5.220)

Now, (Mz(t»e is just the signal we would observe from a pulse in which -yHltl = 0, produced by a rotating field which at t = 0 lay along Ihe x-axis

according to the convention of (5.114). This is also the rotating field generated by a linearly polarized field along the x-axis given by Hz(t) = Hzo cos wt

(5.221)

where, following (2.83),

Hzo 182

= 2H I

"YHI »Llw

and

(w - WO)tl «I

and

"'fHI» Iw - wol

or

(5.224.) (5.224b)

The classical derivation of the b-function response given in (5.192-194) involved assuming that a magnetic field H satisfy

,1m {g(t)I.}

2h2H O Sin . (Wltl )Tr (I z TIy T- 1) = -y Z(oo)kT

j

(Mz(r»o cos (wr)dr

i(w) = ."1

(5.222)

H»Ho

(5.225)

where the field H lasts for a time ro. This condition signifies that the magnetization is rotated during a time less than a Lannor period in H o. This appears to be a much stricter condition than (5.224b). The reason one can use an HI «Ho is that HI is rolating at a frequency w which is nearly the same frequency as the free precession frequency WOo Thus, while HI is on, the total phase angle about the Ho axis which the spin advances is wtl. If a l5-function had kicked the spin at t = 0, the subsequent precession angle in the same time tl would have been wotl' But since I(wo - w)tll and Llwtl «I we can take wOtl = wtl' Thus, the HI builds up the rransverse magnetization over the time it> but does not in the process inrroduce a phase error relative to a "zero time", or 6-function, creation of transverse magnetization followed by free precession for t I. We note also that this argument explains why there is nothing magical about exciting the free induction decay wilh a 0 = 7T/2 pulse. Equation (5.221) shows that any angle of 0 will do. [The fonnula (5.221) appears to blow up for sin 0 = 0, which simply reflects the vanishing of (Mz(t»o for that case]. There are several very imponant advantages to the Fourier [Tansfonn approach. The first is that since one is recording the NMR signal when the HI is zero, one eliminates noise from the oscillator which drives the HI. For steadystate apparatus, the oscillator noise is reduced by bridge balance, but typically such an approach reduces noise which arises from amplitude modulations of HI but does not reduce noise which arises from frequency modulation. 183

If one is dealing with a complex spectrum which has many absorption lines well separated from one another, as in many high resolution spectra of liquids, there is an additional advantage to the Fourier transfonn method. A steady-state search necessarily spends much of its time looking between the resonance lines of such a spectrum. The search procedure is thus inefficient. By pulse excitation, all the spins are induced to broadcast their presence at their characteristic frequencies, thus one immediately gains in dam collection by eliminating the dead time of a steady-state search. Of course, one must use a broader band system, since the steady-state system need only have the band width necessary to examine a single line whereas the pulse system needs a band width large enough to include the whole spectrum. However, since the noise voltage goes as the square root of the band width, one still is better off with the pulsed approach. An important point is that the Fourier transfonn method is the basis for the enonnously powerful two-dimensional Fourier transfonn techniques which have revolutionized NMR. II is useful at this point to make several remarks about instrumentalion. The signal M:c(r) consists in general of a sum of oscillating signals (corresponding to each resonance frequency which has been excited) which decay as a result of line broadening and relaxation phenomena. In general, all of the spread in oscillations is small compared [Q the Larmor frequency. To record these frequencies takes very high frequency response. It is customary to use a technique called "mixing", which in essence translates the fTequency of oscillation while preserving phase relationships. We will not go into the details of how mixers work, but essentially they work as shown in Fig. 5.6. The mixer has three pairs of terminals, one at which the signal is applied, one to which a reference voltage is applied, and one which is the output, the "mixed" signal.

Mixer Signal voltage input o---+~

Mixer signal output

If Vsig(t) = Vs

cos (wst

+,p)

= Vs(cos ¢ cos wst - sin r/J sin wst)

(5.226,) (5.226b)

Vrcr
Then, Vout = Vs cos [(ws - wn)t + ,p

- 0]

(5.227)

In expressing the reference, we have intnxluced a phase angle 0 to represent the fact that the phase of the reference can be set by the experimenter. Suppose 0 = 0, then (5.228) At firs! sight, this expression appears to be equivalem 10 (5.226a). However, there is an ambiguity since the cosine is unchanged when the sign of its argument is changed. Thus, one does not know whether wn < Ws or ws < Wit from recording (5.227) alone. To resolve the ambiguity, we record a second signal with (} = -1r/2. If 0 = -1r/2 Vout = Vs cos [(ws - wn)t + (
+,pl

(5.229') (5.229b)

Together, (5.228) and (5.229) enable us to determine both the sign of (wS - wn), and the phase angle ,p. Recording both such signals is called quadrature detection. It requires two mixers, whose reference signals differ by rr/2 in phase, and the sels of recording apparatus to record the output signals from both mixers. In the example above we took the reference phases to be "0" and "-1rI2". To apply (5.196) we must recognize that there is an absolme meaning to the phase since we took the rf driving magnetic field to vary as cos wt in our definition of Xl and XII [see, for example, (2.92), (2.95) and (5.198)]. I{ is best, then, to consider thal the two reference phases are (5.230)

Reference voltage input "'ig.5.6. Schematic diagram of a mixer. The radiofrequency magnetic resonance signal is fed inlo the signal voltage input terminats; a referenee voltage whose phase can be adjusted is fed into the reference volLage input. The voltage at the mixer signal output goes to recor
184

where 0 1 is unknown until one goes through some procedure to set it. Using 01> we then have from the first mixer Voul = Vs

cos [(ws - wn)t + ,p

-

Oil

(5.231')

+,p - Oil

(5.23Ib)

and from the second mixer Vout = Vs sin [(WS - wn)t

185

These two outputs together enable one to detennine Ws and ; - 8 t . If one has a procedure to detennine 81 in absolute tenns so that one can set it to zero, the cos output gives the dispersion signal and the sin output gives the absorption signal.

E p = (pl1tzIJI) = --yhHoJt = -1""'OIJ

We use these as basis states to express

e,

.

(5.241)

(5.235), in matrix fonn:

A(I I dt pen I') P = '"i .... L;:«P Ien III Ii )(/1 /I j'Hefflp) 5.9 The Density Matrix of a Two·Level System

"

- (/tl'Herrlpll)(Jt"lenll/» We have seen examples of the use of operator methods in conjunction with the density matrix to calculate the results of a sequence of pulses. These methods are extremely powerful and help us see what is happening in physical tenns. However, there is also great insight to be found in actually following what is happening to the individual matrix elements of the density matrix for an example which we already understand thoroughly, the effect of an altemating field on a spin! system. [The spirit is that of (5.143).] The Hamiltonian in the laboratory system for this problem is then

'H = -"(hHoI: - "'(1tH I (Iz cos wt - I,I sin wt)

(5.232)

where we assume a positive "( and a rotating field. As in (5.115), we write 'H. = -"(h(HoI: + e'loItl z Ize-iloltlz) Defining, as in (5.116),

R = e ilolU•

(5.234)

we then transfonn to the rotating coordinme frame to have (5.122)

deR

"""dt

i = ",(eR'H.err -'HerreR)

(5.235)

with

'Herr = -"(1t[(Ho - wl"'()I: + HtIzl

and

(5.236.) (5.236b)

We take Ho = wh, exact resonance, giving 'Heir = -1twl I z

with

,

'86

p=±!

(5.242)

and so forth for the other matrix elements. Recognizing that

(pll,I"') = 0

(5.243)

unless p 1= p', we then get four equations such as

d"++ . [ --:it = -lw\ g+_<-Il,I+) - (+ll,l-lg-+]

de++

dt de__

iWI

= -T(e+- - e-+)

de+_

iWI

de_+

iw)

--:it = -2(g++ --:it = -2(g-- -

e+-)

(5.245b)

g--)

(5.245c)

g++)

(5.245<1)

We note immediately that if HI = 0, all elements of eR become independent of time. They change with time only while the pulse is on. Adding (5.245a) and (5.245b) we get

e++(t) + e--(t) = const = "++(0) + g__ (O)

(5.240)

(5.245.)

iWI

d t = -T(e-+ -

(5.238)

(5.239)

(5.244)

t, so that

de++ + de __ = 0 dt dt '

hence are given by eigenfunctions and eigenvalues of I::

1,lp) = pip)

(1Ignl- ,) = g+-

(5.237)

The energy levels and eigenstates in the laboratory system in the absence of HI are those of the Zeeman Hamiltonian 'H.z = -"(!tHo I:

A somewhat more compact natalion is useful, defined as

But (+IIzl-) =

(5.233)

.

or

(5.246)

(5.247)

a stalement of the conservation of the tOlal probability of finding the spin in either the up or down spin states. In a similar way, adding (5.245c) and (5.245d) gives

e+-(t) + e-+(t) = const = "+-(0) + g_+(O)

.

(5.248) '87

If we take the differences instead of the sums of the two pairs of equations, we get

~(E?++ -

l?--) = -iwt(l?+_ -

d'

(5.255b) Expressing these in matrix form, we get

l?~+)

(M+(t)) = 1"

(5.249)

L:(,.len(')I,/)(P'Ir+I,.)

(5.256)

(M,(I)} = 1/' L:("len(I)I,I)(/II,I") These equations state that the off-diagonal marrix elements are "fed" by the action of HI on the population difference l?++ - l?-_. By taking the lime derivatives of (5.249), we get

Jl

2

Jl

2

dtZ(£,++-l?--)+wl(l?++-l?--)=O

/,,/,' which give, on utilizing the explicit matrix elements of 1+ and I:.

(M+('» = 1"e-+(t)

(5.257a)

(5.250)

(5.257b)

(5.251)

These equations show immedialely that the existence of transverse magnetization depends on the exislence of nonzero off-diagonal elements of l?R' Moreover, the z-magnetization is proportional to the difference in the diagonal elements of l?R. hence it will vanish unless £,++(0 j. l?--(t). The significance of (5.252) and (5.253) is clarified by considering e(O) to correspond 10 thermal equilibrium, in which case the off-diagonal terms l?+-(O) and l?_+(0) vanish. Note., incidentally that (5.73) requires that£,+_(t) and £,-+(t) are complex conjugates of one another. TIlen we get

Therefore,

l?++(t) - l?--(t) = A cos wIt + B sin wIt

C cos wit + D sin wit

A and C are obtained by evaluating the left-hand sides at t = O. Band D are found by evaluating the derivatives of the left-hand sides at t = O. utilizing (5.249). The result is

"++(') - e--(I) = ["++(0) - e--(O)J cos

•

and

dt 2 (l?+- - e-+) + wI (e+- - l?-+) = 0

l?+-(O - l?-+(O =

and

/,,/,'

e++(') - e--(') = ["++(0) - e--(O)] cos WI'

WI'

- i[e+_(O) - l?-+(O») sin

and

WI t

(5.258b)

(5.252)

e+-(I) - e-+(I) = ["+-(0) - e-+(O)J cos

WI'

- i[e++(O) - E?-- (0» sin

If we let the HI stay on for a time tl. we gel, then, for all later limes t that

WI t

. (M +(» t = I.,,,[ l!++( 0) - l!-_(O) ] Sill

Combining (5.252) with (5.247) and (5.248) gives

E?++(t):

I

and

1

'2 + '2 [e++(O) -

1

~[l?+-(O) -

Z

and

+~[£>++(O) -£>__ (0)] sin

wit

11ms following a 7r/2 pulse (5.254)

Suppose. now, we wish to calculate the expectation values of the x-. y-, and z-components of magnetization in the rotating frame. We utilize (5.159) (5.255') I ••

1"

= (M:(O» cos Witt WI I

(5.259a)

= Z[E?++(O) - l! __ (O)] cos WI tI

(5.253)

E?-+(O») sin wit

e-+(I) = 2["+-(0) + e-+(O)J + 2[e_+(O) - "+-(0)] cos

wlt l

= i(M:(O» sin Wltl

e __ (O)] cos wit

1

(5.258a)

(WI

(5.259b)

t I = "lf/2)

(M,(t» = 0

('»

(My = (M,(O» (M,('» = 0 •

(5.260)

which agrees with the classical picture of rotation about HI by rr/2 in the lefthanded sense.

'.9

Following a

"II"

pulse

(Wit)

="11")

(nle'(O)lm) = (nle
(M.('» = (My(l)) = 0 (M,(<) = -(M,(O» .

(5.261)

Again, the result agrees wilh Ihe classical result. Equation (5.257a) Iclts us useful facts about Ihe off-diagonal elemenls of

=0

unless

n=m=k

(5.265)

(kle'(O)lk) = (kle
ell: fH- must be nonzero for Ihere to be transverse magnetization.

a) b)

u+- must be pure real for the transvcrse magnetizations to lie along the

c)

u+- must be pure imaginary for the transversc magnetitation to lie along

d (mI' d' e (t)lm) = d'd (mlel m )

x-axis.

=

*~ !(ml"(O)I,:)(nll£jlm~

Ihe y-axis.

u-+ = Ae+ itJI (and hence u+- = Ae- itJI ) (M+(t» = l'flA(cos

q, + i sin tP)

SO that

(M.('» = 1hA cos ~

Thus ¢ gives the phase angle of the

in the

-

0'"

!ii(")e'(O)1ti('~ .....

C

D

- l£j(<)e'(O)l£i(l') + l£i(<)l£W)e'(O)llm)d,'. (5.266) ,

'"

.....

"

B

'

F

(5.263)

(ml,' (0)1t; (t')l£i (')Im) = ~)mle' (0)1 m')(m' 11£;(,')1£; (<)1 m) Since m'

(5.264b)

x-v magnetization

G)' j !mll"(O)1ti(t')1ti('~

(5.262)

(5.264.)

~

+

~e .consider first the lenus A and B. Since m i= k (Ihe rransilion is between two dislmCI stales), A and B both vanish according to (5.265). To treal tenns such as C, we write

where A is real,

('» = 1M sin

B

A

Since use of an HI which lies along Ihe x-axis will produce components of magnetization perpendicular to Ihe x-axis (assuming thennal equilibrium al t = 0), it will necessarily produce a purely imaginary u+- no matter what angle, Wltl, is chosen. To produce magnetization along the x-axis, HI mllst have a component nonnal to the x-axis. Such a magnetic field will produce a u+- with a nonvanishing real component. In general, if e-+ is given by

(My

- !mll£iln)(nle'(O)lm)!

x-v plane.

(mle'(O)lm') = 0

(m" k)

•

Ihe tenus C and F vanish, leaving

Jo

d 1 ' d' (mlelm) = h' [(mll£;(t')lk)(kll£;(<)lm)

+ (mll£;('JIkXk/l£j(,')lm)jdt'

S.lO Density Matrix - An Introductory Example

(5.267)

We convert Ihe malrix elements in the integrand by (5.107); Since the fonnal equations of the density matrix get rather involved, it is a good idea al this point to consider an example that will make ~he notation more concrete. We shall calculate the probability per second of transitions from a slate k to a state m. We consider thaI only state J.: is occupied at t = O. This assumption is not necessary, but it has the advantage, Ihat, with it, (dldtXmlulm) is directly Ihe probability per second of a lransition. Thus inilially all en's are zero except Ct;. As a result, at t = 0, the only nonvanishing element of the density matrix is (klulk), which is equal to I. By using (5.103), we see that 190

(mll£; (')1 n) = c(;!')(Em - E. )'(mll£ I (<)In)

.

(5.268)

Wc shall also adopt a convenient abbreviation. We shall define the quantum nU~bers m, n, and so on, in tenns of the corresponding energies, measured in radians per second: Em

T

sm

Eft

T

sk

~.

B.~

Thcn we have, using (5.268) and (5.269),

191

:'(mlelm)"

~2

t

J

The funclion Gmdr) is called the "correlation function" of "HI(t), since it tells how "HI at one time is correlated to its value at a later time. For a typical perturbation we have

(ml1i,(t')lk)(kl1i,(t)lm)e'(m-k)(t'-t)

o + (ml1i, (t)lk)(kl1i, (t') Im)e'(m-k)(t-t')Jdt'

(5.270) (5.277)

So far. our equations have been quile general as 10 the nature of the perturbation 1i 1(f). For example. it could vary sinusoidally in time. We shall assume for OUT example, however,that1f\(t) varies randomly in lime. By this we mean thai we shall consider a number of ensembles of systems of identical 110 and identical e(O). However, we shaU consider that1iI(t) varies from ensemble 10 ensemble. with properties described below. We shall therefore perfonn an average of the ensembles. We denote Ihis average by a bar. Then

+ (ml1i, (t) Ik)(k 11i, (t') Im)e'(m-k)( t-t')ldt'

(5.271)

As an example, suppose 'Ji\(t) were the dipolar coupling betwccn nuclear moments in a liquid. It might vary in lime. owing to the thennal motion of nuclei in the liquid. These motions would in general be different among various parts of the fluid, all al Ihe same temperature. We shall assume, moreover, that the ensemble average, such as

(ml1i, (t')lk)(kl1i, (t)lm)

depends on t and

Thus, if "HI (t) and "HI (t + r) were unrelated, we could average the two teons in the product separately, gelling (5.278)

=- 0,

However, we have, for r

(5.279) For typical physical systems, the penurbation "HI (t) varies in time, owing to some physical movement. For times less than some critical lime rc;, called the "correlation time", the motion may be considered negligible, so that hl(t) ';::' "HI(t +1'). For r > Te , the values of "Hl(t+r) become progressively less correlated (Q "HI(t) as l' is lengthened, so that Gkm goes to zero. Thus Gmk(r) has a maximum at r = 0, and falls off for 11'1 > TC;, as in Fig. 5.7. A function "HI(t) with the above propenies will be called a "stationary random function of time".

(5.272)

e only through their difference 1', defined by

Fig.S.7. Function G... t(T) (or lypical physie,,1 s)'slem

(5.273)

t-t'=T

II

T

That is, we assume (5.274) is independent of t but is a function both of T and of the pair of levels, m and k, with which we are concerned. The fact that (5.274) is independent of t is expressed by the statement that the perturbation is stationary. (A more complex case could be handled, but it would make everything much more complicated.) TIle dependence on 1', m, and k leads us to define a quantity GmkCr) by the equation (5.275) Since 11.1 (t) is a stationary penurbarion, we have that Gmk(r) = (ml"H1 (t)lk)(kl"H1 (t + T)lm)

" (k11i,(t + T)lm)(ml1i, (t)lk)

=- Gkm(-T) 192

(5.276)

Bearing in mind these properties, we now rewrite (5.271) in terms of Gmk(r):

d dt(mlelm) = -;.

I

J

[Gmk(r)e-i(m-k)T + Gmk(_r)ei(m-k)Tjdr

h 0 t

".!...2 JG mk ()e-l(m-k)'d T T h

-,

(5.280)

This equation tells us the rate of change of (mlelm) Te, the limits of integration may be taken as ± =, so thaI (d/dt)(mlelm) becomes independent of time. There is a range of 0 < t < Te; for which the transition rate is not constant. And, of course, if (mll?lm) becomes comparable to unity, we do not expect our pelUrbation approximalion to hold, since Ihe initial population (k1elk) would necessarily be strongly depleted. We shall now confine our anemion to times longer than Te;, assuming that for them (mlelm) has nOI grown unduly. Then we have 193

~ (mlelm) =

1 h

2

+=

J

GlIlk(T)e-i(m-k)r = Wkm

(5.281)

-=

where Wkm is the probability per second of a lransition from stale k to m. Equation (5.281) is closely related to the well-known resuh from timedependent perturbation theory:

W'm =

2: !(klVlm)I'e<Er)

l(ml'lt,(t)lk)I'· Gm,(O)

where (kIVlm) is the matrix element of the interaction between states k and m, and P(Er) is the density of final states in energy. In fact. by utilizing (5.275), we see that Wkm of (5.275) involves a product of twO matrix elements of the perturbation. In dlC present case the energy levels are sharp, but the perturbation is spread in frequency, whereas the usual lime-dependent perturbation is monochrommic but the energy levels are smeared. In view of the similarity to the usual time-dependent theory, we are not surprised that the tenns A and B of (5.266), which involve only one matrix elemen! of the penurbation, vanish. The integral of (5.281) is reminiscent of a Fourier transfonn. Let us then define a quantity Jmk(w) by the equation

+= Jmk(W)::

J

Gmk(T)e-il
(5.282)

-=

with the inverse relalion Gmk(T) ::

2~

kill

is independent of

TC'

1 Gmk(O) :: 2'1f

J

Jmk(w)eiwr dw

+=

J

h2

-=

which tells us that the area of the spectral density curve remains fixed as T c vanes. A sel of curves of Jmdw) for three different TC'S is shown in Fig. 5.9. A simple consequence of the fact that the area remains fixed as T c varies is found by considering Fig. 5.9. We note that if the frequency difference m - k were equal to the value WI. the spectral density Jmk(WI) of the curve of medium T c would be the greatest of the three at WI. Consequently, as T c is varied, the transition probability Wkm has a maximum. The maximum will occur when (m - k)Tc '::! I, since it is for this value of T c thai the spectrum extends up to (m ~ k) without extending so far as 10 be diminished. J m,

(5.283)

..-T,1ong

I

, J

J

\

o

(5.28la)

Now. in a typical case, the interaction matrix element (ml1t 1(Olk) runs over a set of values as time goes on. As one vanes something such as the temperature. the rate at which (ml1t\(t)!k) changes may speed upor slow down (Tc changes),

(5.284.)

Jmk(w)dw

-=

:: Jmk(m - k)

(5.284)

But. from (5.283)

+=

Jmk(W) may be thought of as the spectral density of the interaction matrix G mk . We expect that Jmk will therefore contain frequencies up to the order of IIT c (see Fig. 5.8). In tenns of Jmk' we have

W

but the set of values covered remains unchanged. As a physical example, the dipole-dipole coupling of a pair of nuclei depends on their relative positions. If the nuclei diffuse relative 10 each other. the coupling takes on different values. The possiple values are independent of the rate of diffusion, since they depend only on the radius vector from one nucleus to the other and on the spatial orientation of the moments. However. the duration of any magnitude of interaction will depend on the rate of diffusion. Then we have that

Wl

~

T"

short W

Fig.S.9. Jml(l
In the event thai Tc «::: I/(m - k). J",k(w) extends far above the frequency of the transilion. It is then often a good approximation to say that Jmk(m - k) '::! Jmk(O)

lei us define a quantity

"'it:. 5.8. Typical spectral density plOl

194

2(~)Jm'(O) =

0'

(5.285)

such that

+=

J Jm,(w)dw

(5.286)

-=

195

where 2(a/Tel is lhe width of a rectangular Jmk(W) whose height is Jmk(O) and whose area is the same as that of the real Jml:(w), From our remarks, a ~ I. Then we have, combining (5.284), (5.284a), and (5.286), T, h I(m I'H1(')lk)1 ' Jm.(O) = 2a

(5.287)

By utilizing (5.281a) and (5.285), then, we have

w

_ l(ml'HI(t)lk)I'

km -

h2

~ o'c

(5.288)

which holds for Tc < I/(m - k). Since one can often estimate the mean squared interaction as well as the correlation time, (5.288) provides a very simple method of computing the transition probability in the limit of shan correlation time (rapid motion). As a side remark, we note that for a flat spectrum Jmk(w). (5.288) will hold approximately for all Tc ~ 1/(m - k). We can use it to estimate roughly the maximwn rate of transition Ihat'H\(t) is capable of producing under the most favorable circumstances. by noting that this occurs when Tc ';;t I/(m - k). We get the approximate relation, then.

~ l(ml'HI(t)lk)!'

W)

km max -

"1

~

(k) am.

h~e-H/Tb

+ T)

(5.294)

where TO is lime defined by (5.295) in teons of the probability per second, that Hq(T) willjurnp from +h q to -It'!: I

W,

(5.295)

'lI

We assume for our example that this time is the same for all three components of the field. Oearly TO may be taken as the correlation time. 3 Substituting this form into (5.292) we find

., , '! (k Ilq Im) !" h q 1 2T," Jm.l.:(w) = "(nh +w '0

(5.296)

'"" 2 2

2

2TO 1l+(m-k) 2 2 TO

(5.297)

It is interesting to apply this formula to compute the T} for the case of spin For this case, as we saw in Chapter I, (5.291)

q,q'

where we have recognized that it is Hq(t) that varies from one member of the ensemble to another. For simplicity we shall suppose that the x-. y., and z-components of field fluctuate independendy. This means in effect that knowledge of Hz at some time is not sufficient to predict H y at that time. With this assumption. we get in (1.291) only terms for which q = q. Let us define +~

"H',("')"H'."(''''+'"'T'')e·-ilolTdT

(5.293)

The transition probability Wkm is then

+ T)lm)

=,~r.' L(mII,lk)(k1I,. Im)H,(')H,.(' + T)

196

k)

W.l.: m = [L. I'n hq l(mI 1q lk)1

Gm.(T) = (m!'H,(t)lkXkl'H,(t

J

Hq(t)Hq{t

(5.290)

The"

j~k(w) = l'~h21(mllqlk)12

;2 I:i:a,.,(m ,

Evaluation of j~k(w) requires information on the physical basis of the field fluctuation. Moreover, even when one possesses thal information, the malhemat~ ical problem of finding the correlation function may still be too difficult to solve. One can then often assume a function that, on general physical grounds, has about the correct behavior. For certain simple cases one can aClUally compute the correlation function. One such case arises when the field Hq(r) lakes on either of two values and makes transitions from one value to the other at a rate that is independent of the time since the preceding transitioo. Assuming the field takes on values of ± h q , we show in Appendix C that

(5.289)

HI (t) = -"(nh[H::t:(t)l::t: + Hy(t)Iy + H z(t)IzJ

-'or. q=r,y,z L H,(.)I,

W km =

-=2W

Our remarks so far apply to many forms of interaction 'H1(t). In order to be more coocrete, we now specialize. to consider a panicular fonn for the interaction 'HI (t). We shall assume that it consists of a fluctuating magnetic field with %-, y-. and z-eomponents that couple to the components of the nuclear moment:

=

Then we have

(5.292)

,

i.

(5.298) = 2Wt /, -1/' T, ' Assuming a strong static field in the z-direction, the matrix elements between the levels are -

I(W.I- ~)I' =

t

I(W,I- ~)I' = i I(~II,I

(5.299)

_ ~)I' = 0

3 Note thllt although we have not gi",:n II Pl'"ecise definition or the correlation time. once the correlation runction is specified, there is always a P'"ecisely definf!d parameter (in this case TO) thal enlers the problem and ehllfllCtC':n:res the time scale.

197

5.11 Bloch-Wangsness-Rcdfield Theory

We shall assume h z = h y = hz, so that

h; = !hij where hij =

hi + h~ + h;. Then, since m -

...!- = 2')'2 hij Tl

(5.300)

n 3 1

k = wo' the Larmor frequency (5.301)

TO

+wfiT6

This function is ploned in Fig. 5.10. We see that it indeed has the sort of variation we had predicted earlier from our general arguments based on the "constant area" under the curve of Jmk(W). We note thal the minimum comes when WOTO = 1.

We tum now to a more general treatment of the density matrix, following the ideas of Redfield [5.10], which are closely related to a treatment of relaxation due to Wangsness and Bloch [5.11]. All of the basic ideas are anticipated physically in the basic work of Bloembergen et al. [5.12]. The development of the basic equation of Redfield's theory is a generalization of the treatment given in the previous section for computing transition probabilities. Redfield shows that the elements of the density matrix obey a set of linear differential equations of the following fonn: 4

~I!:a' =

L

Ra(X',{3{3,ei(a-a'-{3+{3')tl!~{J'

(5.303)

{3,{3'

In T,

where R aa , ,{3{J' is constant in time. In this equation the time-dependent exponential has the property of making any tenns unimportant unless 0' - 0/ = {J - {JI. Therefore we could write (5.303) as Fig.5.10. Variation of TI with the correlation time "0

By utilizing this fact, we can calculate the minimum value of T 1 from (5.301) and obtain, in agreement with (5.289)

_I) __I ')'~h~

(5.302)

T} min - 3 WO

Now, when TO gets very long, the field Hz simply gives rise to a static line broadening (in this case, since only two discrete values can occur, the stalic line consists of a pair of lines at ± h z ) and is simply related to Ihis static width. Of course, wo is known for a typical experiment. Equation (5.302) therefore tells us the remarkable fact that knowledge of the rigid lattice line breadth, plus the resonant frequency, enables us to compute the most effective relaxation possible from allowing the line broadening interaction to fluctuate. In general the correlation time is changed by changing the temperature of the sample. Although (5.302) enables us to predict the minimum value of Tl' it cannot tell us at what temperature the minimum will occur unless, of course, we know the dependence of TO on temperature. We note in addition that the measurement of T} as a function of temperature can provide infonnation about the temperature variation of whatever physical process produces the fluctuations.

"&

~e:at = L'Raa',{3{3te~f1'

where the prime on the summation indicates that we keep only those tenns for whieh 0' - 0" = {J - {JI. The diagonal pan of this equation (that is, the part we should have if we kept 0' = ai, {J = {JI only) has the same fonn as the "master" equation, (5.13). The conditions under which (5.303) or (5.304) hold are given in tenns of R OtOt '(j{3" T c , and a time interval .1t. .1t defines a sort of "coarse graining". That is, we shall assume that we Ilever try to follow the details of e· for time intervals less than .1t. We must be able to choose such a time, subject to the simultaneous conditions that

.1t:» T c and

(5.305)

1 -;;--'---:» LIt

(5.306)

R aa ',{3{3J

Equation (5.305) will permit us to set the limits of certain integrals as ± 00, as we did in (5.281). Equation (5.306) will guarantee that, during .1t, the density matrix does not change too drastically, so thai our perturbation expansion has validity. Since the R aa ',{3{3"s are comparable to the inverse of the relaxation limes, these conditions are equivalent to saying that Tj and T2 are much longer than T e . These are also the conditions of motional narrowing. The reader will pehaps note that these are the very conditions stated in Section 2.11, which must hold if simple time-independent transition probabilities are to hold. What we are doing here is to generalize Ihe usual time-dependent perturbation theory so that it includes the coherence effects associated with phase factors in the wave function. 4 For compactllCS5 in writing, we use the notation eo,,"

(aiel a'). 198

(5.304)

for the matrix clement

199

The beauty of (5.304) is that it provides us with a simple set of linear differential equations among the elements of the density matrix, which in principle we can always solve. They wiJIlead to a set of "nonnal modes". We note that there is a great deal of similarity to the rate equation describing population changes. Moreover, expressions for the Raot'(J(J"s are given by Redfield's theory, so that we have the relaxation times given in tenns of the atomic properties. Before outlining the derivation of (5.303) and (5.304), we remark that there are twO ways of utilizing (5.303) or (5.304). In the first method, one solves for the behavior of each separate element of the density matrix and then computes the time dependence of the physical variables of interest (such as the x-components of magnetization M z ) by the fundamental equation

(Mz ) =

L: e••,(a'IMzla)

(5.307)

a,o' The second method involves seeking a differential equation for (Mz ). One does this by writing

d-d '" ' dt (Mz) = dt L.J {laQ',(a IMzla) a,a' =

.,.L:

,

d:" = *[,(0), :>lj(t)] +

G)' Ja

[[,(0), :>lj(t')], :>lj(')]dt'

d'

(5.308)

We then use (5.303) to express the time derivative (dldt){laQ'" In fact, since (5.309) we have

+ ei(a-a')! d{l(.ta' dt

(5.313)

We compute the 0'0'1 matrix element. There will be contributions from bOlh tenns on the right. We consider first the contribution of [{I·(O),1ij(t)]:

(alle'(O), :>li(t)lIa') = Dale'(O)IP)(PI:>lj(tl!a') p

- (al:>lj(t)IPJ(Plg'(O)la')

(dg·"')(a'IMz1a)

d{l~a' = i[a _ 0"].

these equations are fewer in number than are the equations of the elements of the density matrix, their solution may be considerably simpler than the original set. This trick will work when the relaxation mechanism and operators are such thai the expectation values of the operators pick out only a small number of the possible nonnal modes. We shall illustrate this use of (5.312) shortly. First, however, we tum to a description of the derivation of Redfield's fundamental equation. Our starring point is the basic equation for the time derivative of fl·, (5.111):

(5.314)

We shall now introduce the idea of an ensemble of ensembles whose density matrix coincides at t = 0 but whose penurbations 1i1(t) are different. (We are therefore flot allowing an applied alternating field to be present. That is, we are describing relaxation in the absence of an alternating field. One could add the effect of the alternating field readily. We discuss it in the next section.) We shall assume that the ensemble average of 1i 1(t) vanishes. This amounts to our assuming that1iI(t) does not produce an average frequency shift.~ Let us discuss this point. In general we expect

(5.310)

(5.315)

This relation enables us to transfoml (5.303). By substituting into (5.303) and utilizing (5.309), we find

where J(q is a function of the spin coordinates, and Hq(t) is independent of spin. For example, we saw that hi (t) had this fonn if it consisted of the coupling of a fluctuating magnetic field with the X-, y-, or z-components of spin. In that case q took on three values corresponding to the three components x, y, and z. If hI (t) represented a dipole-dipole interaction of two spins. there would be six values of q corresponding to the tenns, A, B, ... , F into which we broke the dipolar coupling in Chapter 3. Since we are dealing with stationary penurbations, the ensemble average of 1ij(t) is equivalent to a time average. In general we assume that the time average of Hq(t) vanishes, causing 1ii(t) to have a vanishing ensemble average. As a consequence we shall set

dt

d{lQ'a'

d'

fl aa'

= i(a' - a){laa'

+ L RaQ"

,{3{3' {I{3(J'

P,P'

,

= -,. [I?, 7-lo]Q'Q" +

L

Roo',pp'I?PP'

(5.311)

P,P'

l

We then substitute this expression into (5.308), obtaining

d(Mz ) d'

---=

L:

a,Q",pjJ'

(*[e, :>lo] ••' + n••"pp,gpw)(a'IMz1a)

(5.312)

Although it is not obvious from (5.312), under some circumstances the righthand side is proponional to a linear combination of (Mz(t», (My(t», and (AdAt», giving us a set of differential equations similar to those of Bloch. If 200

(5.316) ~ If there is zero shifl.

a

shifl,

we

<;ao

include

the lwerage shifl in ho, redefining 1-£1 (t) lo give a

201

where the bar indicates an ensemble average. This means, we have remarked, that we cannot let 1-£1 (t) be a time-dependent driving field such as that applied to observe a resonance. On the basis of (5.316) the first teon on the right of (5.313) vanishes when averaged over an ensemble. We proceed in a similar way to compute the aa' matrix element of the second teon on the right of (5.313). By utilizing the fact that


=~"...."...,. . . . Hq(t)Hq,(t + T)e-i... r dT

J

Lqq,(w):=

(5.324)

o By utilizing the fact that Hq(t)Hq,(t + r) is real' and is an even function of T, it is convenient 10 define the real and imaginary parts Lqq,(w):

Re {Lqq,(w)} =

,+=

2"

J·H·,"(t~)H~,-,(~t-+-T')

cos (wT)dT := kqq'

-=

(5.317)

(5.325)

J

=~~~~~ 1m {Lqq,(w)} = Hq(t)Hq,{t + T) sin (wT)dT

and defining T=t-t'

o

(5.318)

Since the only imponant contributions to (5.319) come from teons that satisfy the condition a - a' = fJ - If, we can combine the first twO teons on the right of (5.319) as

-!, (L: L: {(all<'I,8)CP'I[(" la')lk,.,(a - ,8) + k,~(a' h

/l')J

fJ,flq,q'

x e'(a-ft+p'-<>,), e~w

j)

(5.326)

The last two teons of (5.319) are

-!, L:

h fJ.fJ',q,q'

{e~ft 1
+ep..,I
(5.321)

is independent of t and goes to zero when T exceeds some critical value T e. In this case we can consider times t grealer than T c, peonitting us to extend the upper limits of integration to T = +00. We now define the correlation function G(l/(3(l/'(3'(T) as G,pa'W(T) "

(al?i, (t)IP)(P'I?i, (t +T)la')

(5.322)

GaPa'W(T) = L:(all<'IP)
We then define the spectral density Lqq,(w) of the interaction as

202

(5.327)

The imaginary pan of Lqq' can be shown to give rise to a frequency shift corresponding to the second-order frequency shirt of a staLic interaction. We shall neglect this effect, keeping only teons proportional to Re {L qq ,} because they give the relaxation. Therefore we replace the Lqq,'s by the kqq,'s. In analogy to our earlier discussion we now define the spectral densities J(l/(l/'{1IP(w) as (5.328)

-= Then, combining (5.326), (5.327), and (5.328), we have de:o:' _ "

~

By utilizing (5.315) we have that

/l'»)ei(a-P')'j .

- LJ

R

O'o-'IJ(3'

e i(o--O"-/H(3')L".

<:/J{1'

(0)

(5.329)

(3,{1'

(5.323)

where R(l/O",/J/3' is given as 6 As long as the 1<'s arc taken as Ilermitil\ll 0l)erl\tors, the U,'s arc real. If one ehooses the 1<, 's lIS non-Ilerrnitian, lhe /f, 's become complex, but the only f'ft' 's that are nonzero then involve q and q"s which make U,(t)II,,(L + r) real.

203

(5.334) (5.330) Equation (5.329) relates de"ldt at time t > T( to eO al t ::: O. II is the first leon in a power-series expansion. In order for the convergence to be good. it must imply that f!ppl at time t nOI be vastly.different from this value al t = ~. This implies thai we can find a range of tImes t such that t:> Tel but sull 1!~1J,(t) ':l!' Upp,(O). This laner condition implies that 1

~--'---

R".. ,~W

:» ,

(5.331)

The important trick now is 10 nOle thai if (5.331) holds true, we can replace l'flIJ'(O) by i!P13,(t) on the right side of (5.329). By this step, we convert (5.329) into a differential equation for [/, which will enable us 10 find [/ by "integration" at times so much later than t <:: 0 that l?PIJ,(t) will no longer be nearly its value at t = O. The resulting equation is (5.303). The physical significance of our conditions is now seen to be that we never ask for infonnation over time intervals comparable 10 Te, and that in this time interval the density matrix must not change too much. In practice. this implies that

TI. T2

>Tc

(5.332)

As we shall see in greater detail, the condition Tc < T2 is just that for which the resonance lines are "narrowed" by the "mOlion" that produces the fluctuations in 'HI (t). Because ~oJJfJ = R/J/J,OOl

(5.333)

(that is, the transition probability from 0' to fJ is equal to that from fJ to 0'), the solution of the Redfield equation is an equal distribution among all states. This situation corresponds to an infinite temperature. Clearly the equalions do not describe the approach to an equilibrium at a finite temperature. The reason is immediately apparent - our equation involves the spin variables only, making no mention of a thennal bath. The bath coordinates are needed to enable the spins to "know" the temperature. A rigorous method of correcting for the bath is to consider that the density matrix of (5.313) is for the total system of bath and spins. Since in the absence of HI the spins and lattice are decoupled, we may take the density matrix to consist of a product of that for the spins, a, and that for the lattice, ~L. We take for our basic Hamiltonian '110 the sum of the lattice and the spin Hamiltonians (which, of course, commute). 'HI commutes with neither and induces simultaneous transitions in the lattice and the spin system. Then we have 204

Introducing spin quantum numbers sand !l, and lanice quantum numbers / and /', we replace 0' by sf, and so on. Then we assume that the lattice remains in thermal equilibrium despite the spin relaxation:

ll}1' = Off'

e-"/IkT "/"/kT

Ee

(5.335)

I" We then find the differential equation for

d

L• •

I..d.

r

d/ ll /I'uu ') = o/f'llfl' dtan'

(5.336)

and sum over f. The result, in the high temperature limit, is simply to give a modified version of Redfield's equation, with density matrix for spin a replaced by the difference between u and its value for thennal equilibrium at the lattice temperature q(T). We therefore simply assert that for an intcraction in which the lattice couples to the spins via an intcraction 'HI (which, to the spins, is time dependent),7 the role of the lanice is to modify the Redfield equation to be

du;o-' ~ +i(Ol-Ol'-.B+.8')t.. - d - = L. Roo-',fJ.B,e [ufJ.B' - u.BfJ,(T)]

t

(5.337)

.oft'

where a,a',fJ,' stand for spin quantum numbers, and where u.8.8,(T) is the thennal equilibrium value of ufJ.8':

C"/J/ kT

"~~,(T) = 6~W Le WIlT

(5.338)

~"

That (5.337) should hold true is not surprising in view of our remarks in Chapter I concerning the approach to thennal equilibrium. We note here. however, that our remarks apply not only to the level populalions (the diagonal elements of u) but also to the off-diagonal elements.

7 "HI involves both spin Rnd lattice coordinRlcs. If we treat the lattice quantum mechanically. the lattice variables are operators, and 'HI dOClS not involve the lillle explicitly. If we treal the lattice classically, 'H\ ;n\'oln'$ the time c:o:plicity. Thill this must be so is evidcnt, since the coupling must be time dependent to induce spin transitions between spin stales of different energy. However, it is time indcll·cndent when the lattice ml'lkcs a simultaneous transitiOIl that just absorbs the spin energy. 205

5.12 Example of Redfield Theory

where q = x, y, z, and

We tum now to an example to illustrate both the method of Redfield and some simple physical consequences. The example we choose is Ihal of an ensemble of spins which do nOI couple to one another but which couple to an external fluctuating field, different al each spin. The external field possesses X-, y-, and z-componenls. This example possesses many of the features of a system of spins with dipolar coupling. However, it is subslanlially simpler to lTeal; moreover, it can be solved exactly in the limit of very short correlation time. For the case of dipolar coupling, Ihe nUClUations of the dipole field arise from bodily motion of the nuclei, as when self-diffusion occurs. The correlation time corresponds 10 the mean time a given pair of nuclei are near each other before diffusing away_ OUT simple model gives the main qualitative features of the dipolar coupling if we simply consider the correlation time 10 correspond to that for diffusion. In panicular, then. our model will exhibit the imponant phenomenon of motional narrowing, which has been so beautifully explained in the original work of BJoembug~n et al. Before plunging into the analysis. we can remark on cenain simple features that will emerge. At the end of this section we develop these simple arguments funher. showing how to use them for more quantitative results. We may distinguish between the effects of the Z·, y-. and z-eomponents of the fluctuating field. A component Hz causes the precession rate to be faster or slower. It. so to speak. causes a spread in precessions. It will evidently nor contribute to the spin-Ianice relaxation because that requires changes in the com· ponent of magnetization parallel to Ho. but it will contribute to the decay of the transverse magnetization even if the fluctuations are so slow as to be effectively static. In fact, as we shall see, it is Hz that contributes to the rigid-lattice Line breadth. The phenomenon of motional narrowing corresponds to a son of averaging out of the Hz effect when the fluctuations become sufficiently rapid. The z- and y-components of fluctuating field are most simply viewed from the reference frame rotating with the precession. Components fluctuating at the precession frequency in the laboratory frame can proouce quasi-static components in the rotating frame perpendicular to the static field. They can cause changes in componenls of magnetization, either parallel or perpendicular to the static field. The former is a T 1 process; the latter, a T2 process. Clearly the two processes are intimately related, since the magnetization vector of an individual spin is of fixed length. The transverse componentS of fluctuating magnetic fields will be most effective when their Fourier spectmm is rich at the Larmor frequency. For either very slow or very rapid motion, the spectral density at the Lannor frequency is low, but for motions whose correlations time, is of order I/wo, the density is at a maximum. The contribution of H~ and H y to the longitudinal and transverse relaxation rates therefore has a maximum as , is changed. Let us consider, then, an interaction 1t1(t) given by

'H\(t) =

-,"h L:Hq(t)Iq

(5.339)

1to = -"(n/tHol: = -liwolz

(5.340)

where. wo is the Larmor frequency. We characterize the eigenstates by a, the eigenvalues of (5.340). These are wo times the usual m-values of the operator I z . (Herem = I, I-I, ... -I). We shall continue to use thenotalion a,d,fJ,fJ', however, rather than m, in order to keep the equations similar to those we have just developed. The matrix elements (al1tI(t)la') are

(al'H,(')la') =

-,"h L:Hq(')(allqla')

.

(5.341)

q

Then the specrral density functions JOtpo'P'(w) are

We now use the symbol kqq.(w) introduced in the preceding section as +~

kq¢(w) =

4J 'H'q"(")H"q-,T.('--:+-:T">e- iw.. d,

.

(5.343)

-~

Clearly the fluctuation effects, correlation time, and so on are all associated with the kqq"s. For simplicity let us assume that the fluctuations of the three components of field are independent. That is, we shall assume

Hq(t)Hq,(t + r) = 0

if q "" q'

(5.344)

For example, (5.344) will hold true if, for any value of the component H q , the values of H q• occur with equal probability as IRq'1 or -IHq,l. We note that kqq(w) gives the spectral density at frequency w of the q-component of the fluctuating field. With the assumption of (5.344) we have, then, (5.345) We now seek to find the effeci of relaxation on the X-, y-, and z-components of the spins. To do this, we utilize the second technique described in the preceding section, that of finding a differential equation for the expectation value of the spin components. Let us therefore ask for (dldt)(Ir ), r = x,y, or z. By using (5.312), we find

d(I,) - = " L. i h[ ll,1to Joo,,(a' I.. 0'l l+) " L. Roa',f3fJ'llf3J3'(U ' IIrla) dt 0,0' o,o',P,P' (5.346)

q

206

207

"

+1'~

The first tenn on the right, involving 'Ha, can be handled readily:

L , 2.r.Q. 'H'o)Cf(f,(a'IIr 10) h

= _,i Tr {(g1iO

,

0,.

-1toe)Ir}

=

Ii Tr {eHoI, -

=

Ii Tr {e[Ho,!,]}

e I , Ho}

= -i'YnlIo Tr {e[I~, I r ]}

(5.347)

If r = z, this term vanishes. If l' = x, we have

-i1'nHO Tr {ily!?}

= +1'n Ho(Iy )

(5.348)

If r = y, we get --YnHo(l:t'). Thus we have

L 0',0"

*[e, 'h'o]O'O',(c/llrlo)

= 1'n«1) x HO)r

(5.349)

RO'O",(3(3'!?(3/3,(O"IIr la)

.

(5.350)

0',0" ,/3,(3' As we have seen, RCl'O" ,(3(3' is itself the sum of four tenns [see (5.330)]. We shall discuss the first teon. JO'(3O"(3'(O" ~ 13'). By using (5.345), we find

L

~

1; I: L

RO'O', ,(3(3' e(3(3' (ollIr 10)

0',0" ,(3,/3'

1; I: (PIIqlo)(oII,Iq -

= i1'~ku(wo)Tr {I:t'Iye - I:t'ely} =

i-y;ku(wo)Tr {U:t'I y - IyI:t')e}

= --y;k:t':t'(wo)Tr{IzU} = -1'~k:t':t'(wo)(lz)

(5.352a)

The tenn q = y gives, in a similar manner,

-1';kyy (wo)(Iz)

.

(5.353)

RO'O', '/3/3'!?/3/3'('/ 1I z 10) = --y;( k:t':t'(wo) + kyy(wo) J(I z )

. (5.354)

By combining (5.346), (5.349), and (5.354), we have

({3'jlq]O")(o'jlrlq!?I,O')kqq(o' - {1)

(5.351)

where the last step follows from the basic properties of orthogonality and completeness of the wave functions 10), and so on. We are able to "collapse" the indices 0 and .0. but we cannot do the same trick for 0' and p' because they occur not only in the matrix elements but also in the kqq's. In a similar way one can obtain expressions for the other three [enns in RO!O!',(3(3" getting finally

=

DW, 10)(011[/" I,lellPlk,,(o - P) ',p = ,,~[ I:
L

O",(J' ,q

L:

,~

0',0" ,(3,(3'

(oIIqIP)(P'IIqlo')(P1dP')(o'II,lo)kqq(o' - P')

O',O",/3,(3',q

"" -y;

where, in the last step, we have utilized the faci that kqq(w) is an even function of w. To proceed further, we must now specify whether r is x, y, or z. First let us consider r = z. Then, since lr will cornmUie with [z. we get nothing from q = z in the last line of (5.352). Since matrix elements of ['" vanish except for L1m = ± I, the only states (0' and (3) that are joined by I q for q = x have 1,6 - al = wo, the Larmor frequency. Since [Iz.I z ] = iIy and [Iz,I y] = ilz , we have, then,

All told, then,

JO'(3O"/3'(o/ - ,O')e(3(3' (0"1 Ir 10')

2h 0',0",(3,/3' =

(5.352)

0)

',p

which is the driving tenn of [he Bloch equations describing the torque due to the external field. The second tenn on the right of (5.346) involves the relaxation terms;

L

I: (Wqlo)(ol[[I" IqJ, eIIP)kqq(P -

=.,~

j

=

Irlq}IP)kqq(fJ - a)

O',fJ,q

j

-i1'nHO Tr felIz, I:t'J}

:L ({3IIq]a)(aleUqlr cr,fJ,q

IqI,)eIPlkqq(o - P)

z d(I ) "" -y«1) x Ho): dt

-1'~[k:t':t'(wo)+ kyy(wo)](Iz)

(5.355)

This equation relaxes toward (Iz ) "" 0 rather than the thennal equilibrium value 10· To remedy the silUation, we should replace U by e- f!{T), as discussed in rhe preceding section. This result simply makes (Iz) relax toward the thennal equilibrium value 10:

d~,)

= 1((1)

x Flo), -

1~[k,,(wo) + kyy(wo)I«I,) -

10)

(5.356)

This is clearly one of the Bloch equations, with T 1 given by the expression (5.357)

O!,(3,q 208

209

We can proceed in a similar way to find the relaxation of the x-component. For it, the value q = x contributes nOlhing, but q = y or z does. The situation for q = y is similar to the one we have just discussed, connecting states 0' and {J which differ by woo On the other hand, when q = z, the states 0' and {J are the same (Iz is diagonal), so that 0' ~ (J = O. The spectral density of Hz at zero frequency enters. Therefore we find

L

R ua' ,PP'(JPfJ,(O"II:l: 10') = -'Y~[kyy(wo) + ku(O)](I:l:)

dMx

&

Mz

M; MJ

,

We now introduce the precessional motion by replacing M:l: and My (5.359)

For this equation there is no effect of replacing (J by (J - (J(T), since in thennal equilibrium (I:l:) = O. Equmion (5.359) and a similar one for (Iy ) look very much like the transverse Bloch equation describing a T2 process, except that the T2 for (I:l:) differs from that for (Iy ). Labeling these as T2:l: and T2y we get

1

= 'Y~[kyy(wo)+ k:z(O)]

(5.360a)

-1' = 'Y~[k:r::l:(wo) + kzz(O)]

(5.360b)

1',. 1

"

If we did not have the driving tenn from Ho, these equations would cause (I:l:) and (I y ) to decay with the rates T2z and T2y respectively. However, in general the precession rate wo is much faster than I/T2 :r: or I/T2y. Thus, we must average the T2 effects over the precessional motion. That is, the amplitude of the transverse magnetization will decay slowly compared to the precession frequency. h is straightforward to detennine this rate. We define the transverse magnetization M1.. by the equation

M1.. = iMx + jM y

(5.364)

= -T2x - T2y

NO):l: - 'Yn[kyy(wo) + ku(O)](I:l:)

so that

(5.361)

Theo dMl=2M dM1.. dt 1.. dt =2M dM:l:+2M dM y Zdt Ydt From (5.359) and (5.360), we get

(5.363)

dM1.. M2 M2 M ~ -dt = lvfxwoMy - _z - MywoM:l: - - " T2z T2y

(5.358)

which gives for the derivative of (I:l:):

-

My -woM:l: - T2y

Using (5.363) and (5.362b), we get

u,u',fJ,P'

d(l.) ----;u= 1'n«1) x

dM y

ili =

= woM:l: - T2x

(5.362.)

M x = M1.. cos (wot + 4»

(5.365)

My = -lvh sin (wot + 1') in (5.364), getting

lvh dM1.. = -Ml

(COS 2 (wot + r/J) + sin 2 (wot + r/J»)

~

~x

~y

(5.366)

Averaging this equation over one period, we replace cos 2 (wot+r/J) and sin 2 (wot+ r/J) by getting

!

dM1.. dt

=

_1_)

-M1.. ~(_I_ + 2 T2z T2y

= _

M1.. T2

-!... _ ~ (_1_ + _1_) T2 - 2 T2z

(5.367)

(5.368)

T2y

Combining (5.360) with (5.368), we see that 1 1 T2 = 2Tl + 'Y;kzz(O)

(5.369)

Our relaxation mechanism therefore leads to the Bloch equations. Of course one cannot expect that in general the Bloch equations follow from an arbitrary 7-tl(t), and one would have to study each case to see whether or not the Bloch equations resulted. To proceed further, we need to know something about the spectral densities of the x-, y-, and z-components of the fluctuating field. We shall once again assume a simple exponential correlation function, with the same correlation time 7"0, for q = x, y, and z: Hq(t)Hq(t +7") = H~exp(-I7"I/To)

(5.370)

which gives (5.362b)

-

kqq(w) = Hi

TO

I +W

2 2

(5.371)

7"0

In tenns of (5.371) we have, then, 210

211

'1

f

relative to its neighbors by diffusion. In Ihe time phase angle 6¢ over its nonnal precession:

6¢

=

T,

a spin will precess an extra (5.315)

±l'nlH.:IT

(5.372) After n such intervals, Ihe mean square dephasing tJ.¢2 will be

1 '(2

;;:;- '" '111 ..I2y

tJ.¢2 = n6¢2 = n1'~H;T2

T() ) BlTO + "2" Hz I +W2T2

The number of intervals n in a lime t is simply

0 0

from which we gel

t

'n

(5.311)

n=-

1 1 =..,2H2 ro + T2 In l 2T) ;;; ....2

(5.316)

T

(H2z TO + ~(H2 + 82) TO ) 2 z Yl+w2r2 o

(5.373)

If we take as T2 the time for a group of spins in phase al t = 0 radian out of step, we find

0

10

get about one

(5.318)

We nOle first of all that Tl goes through a minimum as a function of TO when woTQ '"

1. The nuclualing fields Ihal determine T 1 are the z- and y-components

at the Larmor frequency. If we view the problem from the rotating frame (that is, one rotating at the Larmor frequency), these results are reasonable. since the T, corresponds to a change in the z·magnetization. Such a change is brought about

by "static" fields in either the z- or y-directions in the rotating frame. since in the rotating frame the effective field is zero (HI. of course, is zero). But "static" x- or y-fields in the meating frame oscillate at WQ in the laboratory frame. On the orner hand, lhe decay of the x-magnetization must arise from the "static" y- or z-fields in the rOiating frame. Since the z-axes of the laboralory and rotaling frames coincide, for the z-field it is the static laboratory component that counts, but for the y-field, il is the laboratory component at the Lannor frequency Ihal is important. We note that in the limit of very rapid motion (wOrD <:: 1), and assuming an isotropic fluctuating field

H'• • H'y = H'•

(5.314)

Tl and T2 are equal. Physically, for our model, this result signifies that for a very short correlation time, the spectral density of the fluctuating field is "white" to frequencies far above the L1.rmor frequency, so that the x', yo, and z-directions in the rotating reference frame see equivalent fluctuating fields. The two terms in the expression for T z have simple physical meanings. One tenn depends on H~. It represenls the dephasing of the spins due to the spread in precession rates arising from the fact that H ~ can aid or oppose Ho. This term can be readily derived by a simple argument, which we give below. The second term, as we shall explain, results from broadening pf the energy levels due to the finite Iifelime a spin is in a given energy state. Let us now tum to a simple derivation of the first term of the equation for T2. We assume the field as a value lH~1 for a time r; then il jumps randomly to ± IH~I. Such a change in field in practice arises because a nucleus moves

'I'

...!... T2

= ",,2 H 2 T In

r

(5.379)

We note that the shorter T (Ihal is, the more rapid the mOlio~), the narrower Ihe resonance. This phenomenon is therefore called motional narrowing. We see that Ihe motion narrows Ihe resonance because it allows a given spin 10 sample many fields H~, some of which cause ilia advance in phase; olllers, 10 be retarded. The dephasing takes place, lIlen, by a random walk of small sleps, each one much less Ihan a radian. In conttast, when there is no motion, a given spin experiences a constant local field. II precesses either faster or slower than the average, and Ihe dephasing of a group of spins arises from Ihe inexorable accumulation of positive or negative phase. The contraSI with "collision broadening" of spectral lines is greal. In thai case Ihe phase of the oscillation is changed by each collision. Since the frequency is unperturbed belween collisions, Ihere is no loss in phase memory except during a collision. Since each collision gives a loss in phase memory, a more rapid collision rate produces a shorter phase memory and a broader line. With motional narrowing, there is no phase change when H r is changing from one value to another because the change is very rapid, but lIlere is a phase change during the time a given value of H r persists. More rapid motion diminishes the loss in phase memory in each inlerval. We have considered just one (enn of our expression for T2. The other term, clearly has the same dependence on TO as does the spinwhich involves lattice relaxation. We interprel it as the bro,1.dening of the line due to the finite life of a spin in any eigenstale as a result of the spin-Ialtice relaxation. The lifetime

H;,

8 Of course the word ~motioll~ here refers to tn.nstation of the position of the nudeWl, not to the change in spin orientation. 213

is finite because a field in the V-direction can change the .z-magnetization. We should estimate the order of magnilUde of the lifetime broadening to be

.cl.E =!!... T,

or

(5.380)

c,E I L!w=-=-

h T, Assuming isotropic fluctuating fields, we see that our example actually gives _I = _I + _ 1 (5.381) T2 T 2, 2T1 where T2, is the broadening due to the spread in the .z-field. The quantity 1fl'2', is often called the sUlllar broadening; the term lilT) is called the nonsecular or lifetime broadening. More generally, we replaced lilT) by IIT I" where T I, (which gives the nonsecular broadening) is related to T). As we have remarked, if we consider the secular broadening, we nOte that as TO decreases, T2 increases, oc the line narrows. Of course, we have seen that as one increases TO (slows the motion), the validity of the Redfield equations ceases when T2 ";:! TO. For longer TO'S, we cannot apply the Redfield equation. The longest TO for which the Redfield theory can apply, then, is TO = T2 , or , 82TfJ _1 = _1 = "'( TO T2 nz

or

7n!H,lrO = 1

..!....

(5.383)

So far we have excluded applied alternating fields from the time-dependent coupling 'H)(t). Let us now assume that such fields are present, giving an extra tenn, 'H2(t), in the Hamiltonian. We can include its effect in a straightforward way, as has been shown by Bloch. Introduction of 1t2(t) simply replaces 1tj(t) by Hj(t) + Hi(t) in (5.313):

0'

t 1" T',

Secular (1~) and non$OCular (Tn broadening versus TO. For

Fig. 5.11.

Toe!l/wo

I

where TOO is the value of TO for infinite temperature. The temperature variation of T t or T2 gives one a convenient measure of E and TOO. The narrowing smdies of Andrew and Eades [5.13) performed on molecular crystals, provide one such example. Another one is the work of Holcomb and Norberg [5.14) on self-diffusion in the alkali metals, and subsequently, Seymour [5.15] and SpoJcas (5.16) on aluminium. Here one has the interesting fact that, using resonance, these workers could measure the self-diffusion rate of both Li and AI (for which there is no radioactive isotope for use in the conventional tracer technique).

5.13 Effect of Applied Alternating Fields

T',

T',

(5.384)

(5. 382)

As we can see from our simple model, Ihis is just the value of TO at which a Iypical spin gets one radian out of phase before there is a jump. For longer TO'S, the spins are dephased before there is a chance for a jump. That is, they do not dephase by a random walk. The line breadth is then independent of the jump rate, giving one the temperature-independent, rigid-lattice line breadth. The twO contributions to the line breadth (secular and nonsecular) are plotted in Fig. 5.11. If one analyzes the relaxation via other mechanisms, the same general features are found. The fact that more than one transition may be induced will make important the spectral densities at frequencies other than 0 and wOo Often, 2w0 /11

comes in. For example, when the relaxation arises from the coupling of one nucleus with another by means of their magnetic dipole moments, the E and F terms of Chapter 3, which involve the product of two raising or IWO lowering operators, connect stales differing in energy by 2hwo. Our formulas show us that the measurement of T 1 and T2 will enable us to detennine TO. When the fluctuations in the interaction 'Ht(t) arise from bodily motion that varies with temperature, we can use resonance to study the temperature variation of TO. Often one has a "barrier" to motion and an activation energy E such that

the eXilmple in the led, T[

=2Tl

d'

.

:, = *le'(O),Hj(t) + H;
+

2'

(*) Jo 11.(0), (Hj(t')

+ H;
The effect of Hi in the first tenn on the right vanished when averaged over an ensemble. For that reason we were forced to consider the second tem. In general the contribution of 'H 2(t) to the first term does 1I0t vanish, since 'H 2(t) is identical for all members of the ensemble. If 1t2(t) is not too strong, we expect Ihat the first-order tenn is all that is necessary, and therefore we neglect the role of 'H 2 in the integral. Physically, this approximation amounts to our saying thllt

dt/

de·)

dt = dt 1f.

de')

1

+ ill n'!lax

(5.386)

where de*/dt)1f.2 is the rate of change of e· due solely to 'H2, and de*/dt)relax is

Toe!I/'r"IH.1 214

215

the rale of change of r/ we should have if 1t2 were zero. What we are neglecting are, therefore, nonlinear effects in the interactions.

Under what circumstances do we expect this approximation to work? The answer is that neither perturbation must change

'l

t (which is the upper limit of lhe integral), for the

'Hi

too much during the time

'Hi

[enns in the integral

is acting on a (/. which is not in fact e-(O) bUI must be corrected for the driving by 'Hi. Since we wish to choose t as about Te. this represent the fact that

requirement means that

1("I?t,I,,')IT<

<0:'

(5.387)

"

If 1t2 is too large to salisfy (5.387), we should then attempt 10 solve first for the combined effect of 'Ho and 'H2. using perturbation theory for 'HI- For example,

we may use a rotating coordinate transfonnation rather than a transfonnation 10 the intemcrion representation, thereby converting 11i(t) to a static interaction, which could then be removed by a second transformation to the interaction representation of the effective field. It is interesting to note that there is in fact a very close similarity between the interaction representation and the usual rotating coordinate transformation that renders H I a static field. Both are transformations to rotating coordinate systems. The interaction representation is a transformation to a system rotating at the Lannor frequency, whereas the usual transformation goes to a coordinate system rotating with H t . For simplicity we shall assume that (5.387) is satisfied. Note that since T, and T2 are much longer than T c for all our equations to be valid, (5.387) can still be easily satisfied evcn under conditions of saturation. There is one further consequel'lCC of the addition of a tenn 112(t). We have remarked that when we treat the latticc classically, the density matrix relaxes to its value at infinite temperature rather than to the thennal equilibrium value

e!:T)o f!{T) =

(5.388)

Z(T)

where Z(T) is the sum of states. When the change of 1i2 is small during the time T e , which characterizes the "lattice" motion, we expect that 1i2 looks like a "static" coupling to the lanice and that we should assume that the system relaxes at each instant toward the instantaneous density matrix:

f!{T. t) = Z(T. t) exp (-(?to + ?t,(t»)/kT)

.

(5.389)

This equation can be verified by treating the lattice quantum mechanically. When T e is long compared with the period of 1i2' (5.388) applies. Under circumstances where the Bloch equations hold, the shon T e leads oflen to T l = T2 , and the Bloch equations become

216

M x H() t+ Mo-M Tj

where

(5.390)

Mo = xoH(t)

(5.391)

and H(t) is Ihe instantaneous applied field. Explicit solution shows that the solution of (5.390) differs significantly from the nonnal Bloch equations (assum+ ing T 1 = T2, but Mo = xoMo) only when the line width is comparable to the resonance frequency. We can combine (5.386) with (5.311) to obtain the complete differential equation for the density matrix, including an applied alternating field: dUoot' i d - = -,,(Eo-' - Erx)l!rxtY t

' + -,i '" L.. (l!rxtY'(o "I 112(t) I0) l 0"

- (,,(?t,(')I,,")eo""'] +

L

Ro"'JJ~'[e~p' - e~p'(T)J

(5.392)

~JJ'

where, for l!fJIP(T), we use either (5.388) or (5.389), depending on the circumstances. In order to make (5.392) more concrete, let us suppose we have a two-level system. Thus we may have a spin particle quantized by a static field in the z-direcrion. We label the states 1 and 2 and find that the density matrix has four elements Ull> 022, un, U'2t· In the relaxation tenns Ra~ ,/Jll', the only tenns of imponance (as we have seen) involve cr - crl = fJ - If. The only tenns Ihal count are therefore

!

R Il ,22 =

R22,1I

R12,12 = R21,21

== I/TI == - 1/1"2

(5.393)

By assuming 1i2 joins states 1 and 2 only, and denoting (l11i2(t)12) by 1i12(t), we find dUl1

_'_e-1{,/"·

,

dM dt

--=/

d.

dU22

=-dt" =

t!22 - ",1 -

[m(T) -

ell (T)] + ii; ( UI2 ~ (t) 1L21

TI

del2

Ul2

-- = -~

&

i

l

i

+ -(£2 - £1)UI2 + -:-(U11 - U22)1i 12(O h h

~ (t)n )

- Iq2

c:21

(5.394)

(5.395)

If E2 is larger than £1 and '}-{12 oscillates at frequency w, we can solve (5.394) and (5.395) in steady state by assuming that

="21e-i~t

l!12 = r'12ei~t,

.!?21

el1 = rll ,

l!22 = r22

(5.396)

where the roo, 's are complex constants. The details of the solution are left as a problem. but the form of the answer is identical to that of the Bloch equations.

'17

If we write

'H2(t) = V cos wt

(5.397)

6. Spin Temperatnre in Magnetism and in Magnetic Resonance

where V is an operator, and define Wo by the relation E 2 - El == hwo, we have in the limit of small V (that is, no saturation) e-El/ kT

1 -El/kT 1'" = ell(T) = e EI/kT + e E21kT ';::' -e 2

'"

_ _ V"T2 [(T) - 2'n 1·( ) ell +IW WOT2 S! IT2 V I 2W O I + i(w WO)T2 4kT

e22

(1')J

6.1 Introduction (5.398)

We note that 1'12 differs from zero only near to resonance and that T2 characterizes the width of frequency over which 1'12 is nonzero. If the states 1 and 2 are the two Zeeman states of a spin ~ nucleus and a static magnetic field parallel to the z-direction, then the transverse magnetization Alz has matrix elements only between states I and 2, the diagonal elements being zero. Therefore (M:r(t»

=

1'12eiwt(2IMzll) + 1'2Ie-iwt(lIMzI2)

= 2 Re {TI2eiwt(2IMzlI)}

(5.399)

Taking (5.400)

V = -MzHzo and recalling that X is defined as

(5.401)

(Mz(t)} = Re {xHzoeiwt} we see that

wo!(IIM.12)I'

'T2

X(w) = 1 + i(w _

2kT

WO)T2

(5.402)

Now, using the fact that I =~. we have X(w) =

iT2

1 + i(w -

WO)T2

wO'Y 2 1i. 2 ](l+I) -

2

3kT

(5.403)

which agrees with the expression for the Bloch equation derived in Chapter 2. We note that we could detennine both 71 and T2 from first principles by computing R",22 and R I 2,12' Alternatively we could simply treat Tl and 72 as phenomenological constants to be given by experiment. If there are more than two levels to a system, the solution may be carried out analogously by simply setting all off-diagonal elements of gaol equal to zero, except those near resonance with the alternating frequency (Err - Eo, S! liw).

218

In Chapter 5 we employed the concept of spin temperature to discuss relaxation. The idea of spin temperature was introduced by Casimir and du Pre [6.1] to give a thermodynamic treatment of the experiments of Gorter and his students on paramagnetic relaxation. It was Vall Vleck [6.2] who first employed the concept for a detailed statistical mechanical calculation of the relaxation times of paramagnetic ions. 80th in this case, and also in his general statistical mechanical treatment of static propenies of paramagnetic atoms [6.3], he recognized and emphasized the fact that expansion of the partition function Z in powers of lIT enabled one to calculate Z without the necessity of solving for the energies and eigenfunctions of the Hamiltonian. Waller evidently was the first person 10 use this property [6.4]. From the partition function, one can compute all the static properties of the system, such as the specific heat. the entropy. the magnetization, and the energy. For example, the average energy of a system. E, at a temperature T is given by

E = kT' :T In Z

(6.1)

In 1955. Redfield [6.5] showed that the conventional theory of saturation did not properly account for the experimental facts of nuclear resonance in solids. In one of the most important papers ever written on magnetic resonance. he showed that the conventional approach in essence violated the second law of thennodynamics. He went on 10 show that saturation in solids can be described simply by applying the concept of spin temperature 10 the reference frame that rotates in step with the alternating field H \. To understand his ideas, one needs to understand certain concepts which predate the discovery of magnetic resonance - ideas such as adiabatic demagnetization. In this chapter we begin by describing a simple experiment which displays the failing of the pre-Redfield theory of magnetic resonance. Then we tum to a discussion of the use of spin temperature in nonresonance cases to build background for the application of these same ideas in the rotating reference frame. We then discuss the Redfield theory of saturution in solids.

219

6.2 A Prediction from the Bloch Equations Let us consider a simple resonance experiment with a rotating magnelic field of angular frequency w. lI"ansverse to the stalic field B o• tuned exactly 10 resonance w: ,Ho

(6.2)

We discuss it by means of the Bloch equations. It is convenient 10 transfonn to a reference frame rotating at w with HI along the x-axis as is done in Sect 2.8. Exactly at resonance. the Bloch equations become dM~

----;It "" -..,MyH I + dM~

Mo - M~

(6.3a)

Tl

Jkrz

(6.3b)

-;U""-T2

dMy "" -vM HI _

dt

,:

~

(6.3c)

T2

Suppose we now orient M along HI sO that at t "" 0 Mz "" Mo. Jvly "" M: "" O. From (6.3b) we see that M z will decay 10 zero in a time T2 • The low HI steadystate solution of the Bloch equations shows Ihat they describe a Lorenlzian line with a frequency widlh I &:-

.~

T,

,

For solids typically Llt..,)

';!'

'1Hnei,hbor ~ '11l ~ '1.-../1(/ + I)/Z J (tJ

a

6.3 The Concept of Spin Temperature in the Laboratory Frame in the Absence of Alternating Magnetic Fields Let us now tum to a discussion of the application of the concept of spin temperature to magnetic experiments not involving resonance. A typical system we might consider is a group of N spins of spin I, gyromagnetic ratio '1, acted on by an external field Ho, and coupled together by a magnetic dipolar interaction represented by a dipolar Hamihonian Hd' We denote the Zeeman Hamiltonian by Hz. The solutions of the Schmdinger equation are then wave functions tPn of energy Ell of the total system.

'li",,: ('liz + 'lid)"" : E"""

(6.5)

(6.6)

Unfonunately, (6.6) is exceedingly difficult to solve, depending as it does on the coordinates of 10'22 spins. One can assume, however, that if the spin system is in thennal equilibrium with a reservoir of temperature 8, the various states n of the total system would be occupied with fractional probabilities PII given by the Boltzmann factor I'll ""

where H ~gl . I bo r is the nuclear magnetic dipole .field due to neighbors, and a is . . the distance to the Z nearest neighbor. For typical solids Llt..,) IS of order of a few tens of kilocycles (e.g. .t1w S! 211'" X lO kc for AI metal). Funher examination of (6.3a) and (6.3c) shows that they do not involve M~ and that if M and M z are initially zero they would remain zero were it not for the tenn invofving T t . If T} ::> T2, therefore, and for times up to about T I after M has been orienled along H I, we still have My "" M: "" O. Therefore these eqlw.tions predict that wilen M is aligned along HI, it wUl decay to zero in a time T2, typically of order 10- 4 to lO-5 sec. Experimentally this prediction (decay in T2 when M is along H) is found to be correct for liquids, but it is not correct for solids. Rather, for solids it is found that as long as HI is turned on and is sufficiently strong, the decay rate of M~ is much more like T, than a time T2 which characterizes the line width. Redfield first stated this fact on the basis of his steady-state experiments, but he did not actually perfonn lhe experiment we have described. That experiment was first performed by Holton et a!' [6.6]. Why do the Bloch equations fail for 220

solids but not for liquids? Redfield has given an explanation. We will discuss the conditions for validity of the Bloch equations later in this chapter, after we have discussed some important background material on spin temperature in cases where there is no alternating field HI present.

E,,/k8 .!..eZ

(6.7)

where Z is the panition function (6.8)

" Quantities such as the average energy E and magnetization M ~ would then be given by

E:E~&

~~

"

(6.9b)

"

As we have remarked, Van Vleck recognized thai expressions such as (6.9) could be evaluated wilhout solving the Schmdinger equation because they could be expressed as traces. For example we can express the panition function as a lrace as follows. (6.10)

"

"

Since the trace is independent of lhe panicular representation used

10

evaluate it. 221

we can use a convenient representation. For example. we could evaluate (6.10)

in principle by using as basis functions the eigenfunctions of the z-component of spin Izk of all the individual nuclei. However, to do so, we need to lake one more step, expansion of Z in a power series. It is often valid for nuclei and for eleclTons to use the high temperature approximation. We expand the exponential in a power series, keeping only the leading lenns. Then the sums are easy to do. 2 Hl1i N', } Z=Tr { 1-/.;8+2/.;28 2 -'" =(21+1) +2k26 2Tr {'Jt}+··(6.11) where we have used the faci that Tr {1t} = 0, as can be readily verified for both

Hz and 'HdUsing these methods one finds C(H2 + lJ2) 0 L

E=

where

8

N"f2fI 2I(I + 1)

C=

(6.12)

(6.13)

3k

is the Curie conslant, and HL is a quantity we call the local field, which is of the order of the field one nucleus produces at a neighbor (several Gauss) and is defined by (6.14) 1 T< {H'} L- k(2I+l)N d Since the trace in (6.14) can be computed, HL may be considered to be precisely known. One finds H[:: -y 2r!2[(I + I) ~)1/rjk)6 (6.15)

CH' -

j

One can compute the magnetization M and finds it obeys Curie's law

M= CHo 8

q=

E+kBlnZ

(6.17)

8

Evaluating Z and E we get C (H 2 +H 2) ~::Nkln(2I+I)-0 L

2

8'

(6.18)

6.4 Adiabatic and Sudden Changes The significance of these results is more fully realized by considering the behavior of the spin system when the applied field Ho is made a function of time. For simplicity, let us assume the spin system is thennally isolated from the outside world and that it mayor may not be in [hennal equilibrium. The first of these assumptions is satisfied if the experiments we conduct are perfonned on a rime scale short compared to the spin-lauice relaxation time. The assumption implies that the Hamiltonian of the system includes only variables intemal to the system since spin-lanice relaxation results from tenns involving variables both intemal and external 10 the system as illustrated by (5.33). If the system is in internal themlal equilibrium, a spin temperature applies, so that (6.18) holds. We can then consider three cases.

(6.16)

Note that this is a vector equation, so that M and Ho are parallel. A moment's reflection shows that (6.16) is truly remarkable. It states that the vectors M and Ho are parallel no matter what the size of Ho as long as the high temperawre approximation is valid. Suppose Ho is small, comparable to the local field a nucleus experiences from its neighbors. One might then suppose that nuclei would tend to line up along the direction of the local field, not along Ho. It would seem reasonable to suppose that the degree of polarization one could achieve per unit of applied field would be less when Ho «..H L than when Ho:» H L . Equation (6.16) shows Ihat this intuitive argument is incorrect - the degree of polarization per unit of applied field is independent of the size of Ho relative to the local field. Not only is this true of the magnitude of M, but also of the direction as well. 222

Another useful property of (6.16) to note is that when Ho :: 0, M :: O. Thus, suppose Ho were tumed 10 zero so suddenly that M does not have time to change. Immediately after Ho :: 0 we have a case where M f. 0, Ho :: O. But if there..were a temperature, (6.16) shows M must be zero. Therefore we can use (6.16) to conclude that at this instant of time the system is not describable by a temperature. Another quantity of great utility is the entropy ~. We know from statistical mechanics that Ihe entropy measures the degree of order in a system. In a reversible process in which there is no heat flow into or out of a system, the entropy of that system remains constant.

a) The Hamiltonian is independent of time (the applied field is static). In this case, the average energy is constant in time, whether or not the system is describable by a spin temperalure. If the system has many parts which are strongly coupled together, but is not initially in a state of internal equilibrium, we expect that irreversible processes within the system will eventually bring the system into an internal equilibrium describable by a spin temperature B. During that process, the energy is conserved since a time-independent Hamiltonian corresponds 10 a system on which Ihere are no applied forces. Proof of the constancy of the energy is left as a homework problem (Problem 6.5). b) The Hamiltonjan changes slowly jn time. The criterion of slowness is that at all times internal transfers of energy shall be fast enough so that the system is always describable by a single temperature B. Under this circumstance, Ihe changes are reversible, and the entropy of the system remains constant. 223

system. The condition will be satisfied if we make the changes on a rime scale rapid compared to the spin-lattice relaxation time T!. Frequently one has T1's of seconds, and by cooling may achieve Tl'S of hours. Such long limes are practically infinite. The second condition we must satisfy is that after each small change in Ho we must allow a new temperature to be reached before making another small change. This condition is typically that we must change Ho slowly on a time scale defined by the precession period of nuclei in the local field of neighbors (I/-yHL)' This time scale is a few tenths of a millisecond. Between these time intervals there is a readily achievable range for which Ho can be changed adiabatically. For an adiabatic change we have a constant entropy. Thus from (6.18) we get that (HJ+HVI02 remains constant. If we stan in an initial field Hi at temperature 0i and change the field adiabatically to a final value Hr, the temperature Or is given by the relation H.I2 +HI.2 _ H2r +H2). (6.23)

c) The Hamiltonian changes discontinuously in time. Such a change could occur if Ho can be changed quickly. The term "discontinuous" means that the change is so fast that the various spins making up the system do not change direction during the change. Let us now investigate these various cases more fully. I) Time-Independent Hamiltonian

Consider a system not in thermal equilibrium initially. Let us suppose that the parts of the system are coupled together. We then expect that eventually the system will achieve an internal equilibrium described by a "final" temperature If we know the energy of the system at t = 0 (call it Eo), we can compute I~e temperature Or as follows, making use of the fact that for a time-independent Hamiltonian the energy is conserved. _ . Utilizing (6.12) which relates the energy E to the temperature, and applymg conservation of energy, we get that

o.

E=Eo

or

8r -

(6.\9)

C(H6 +Hl) Sf = Eo

A famous result, cooling by adiabatic demagnetization, can be seen in (6.23) by taking and <: corresponding 10 what happens when a sample initially in a strong magnetic field has that field turned to zero. Then we

H;:> Hl,

(6.20)

eRo

8r H L -=-<:\ 8i Hi

(6.2\)

where we take Of from (6.20). During the process of establishing internal thermal equilibrium, the entropy is not constant since irreversible processes are taking place. But eventually the entropy O'r is given by the thermal equilibrium value for a temperature Or C(H 2 +H2) ~02 I.

(6.24)

Thus, the final temperature of the spins is much colder than the initial tempcrnture. If initially the spins were in thennal equilibrium with a thermal bath (such as the lauice cooled to liquid helium temperatures), the final temperature would be a good deal less than the bath temperature. Note that the bigger Hi and the smaller H L, the greater the cooling. To reduce HL' it is common to dilute the magnetic atoms. Figure 6.1 shows Or versus Hr for an initial field Hi much larger than the local field, and for an arbitrary initial temperature 0i' Note that as one lowers the

Sf

O'r= Nkln(2I+ l)-

Hi Ht

find

We can compute the magnetization Mr which finally results after internal thermal equilibrium is reached by means of Curie's law

Mr=--

Of

(6.22)

f

If the Hamiltonian can be divided into two pans which commute, the energy of each pan is separately conserved. The system then cannot be expected to reach an equilibrium described by a single temperature, but rather is expected to reach an equilibrium in which each commuting pan is described by a temperature. We encounter this situation when we apply these ideas to the rotating frame later in the chapter. In some cases the subsystem may not have enough complexity to make us confident it will eventually be describable by a temperature, but even so we ohen blithely proceed to assume that a temperature is achieved.

0,

2) Slow or Adiabatic Clwnges

We keep in mind that to be adiabatic, a change in Ho must satisfy two conditions. The first is that there should be no heat flow into or out of the spin 224

",

Fig.6.1. The final temperalure 6 f reached al a final magnetic field I1f ror adiabatic changes in the applied field rrom an initial value ilL at tcml>erature 8;. Noll' lhat 8r does not chllnge much ...·ith IIr for IIr < II.. 225

field from its initial value. the temperature drops until the applied field becomes comparable to the local field. Funher decreases in applied field do not then produce much lowering of the temperature. The magnetization can be computed from Curie's law if the temperature is obtained from (6.23). For the case that there is an initial magnetizmion Mj in an initial field Hi much larger than HL. the final magnetization M r is then easily shown 10 be Hr Mr = Mi (6.25) 2 V/H r +H2L

This result is shown in Fig. 6.2. NOIe that during me cooling process, M r remains at the initial value Mj until Hr gets comparable [0 the local field HL. because for large values of H. 9r is propor1ional to Hr.

thennal equilibrium between the spins and lattice in the field Bo. If Yo:» H L• the magnetization is small, but nevenheless it is not zero. One of the remarkable features of (6.25) or Fig. 6.2 is that when H f = 0, Mr = O. As we have remarked, this is a very general consequence of Curie's law no matter what the temperature. (Note, however, that the derivation of Curie's law utilizes the high temperature approximation. At low enough temperatures this approximation breaks down, allowing spontaneous magnetization to be a possible zero field state). What is remarkable is that the degree of order of the spin system is just the same when M = 0 as when M = Mi. Although it is clear that a system with net magnetization is ordered, how can there be order when AI = 01 The answer to this paradox is that even when H 0 = 0, spins still experience magnetic fields owing to the presence of their neighbors. A typical spin will point either with or against the local field it experiences. For a highly ordered system, there will be a substantial excess pointing with the local field rather than against it Since the local fields at different nuclear sites have ran· dom orientation (we here rule out highly ordered spin arrangements such as in a ferromagnet), there is no resultant muaoscopic magnetization resulting from the alignment along the microscopic local fields.

3) Sudden Switching

Fig. 6.2. MlIgno:ti:tation AI, versus "pplicd fidd IIf for adiabatic dl"ngcs in IIf. It is &lI5lImed that III > II~. Noto: thllt M, is independent of 'II{ for II~ > and goes to zero when IIr goes to :tetO

We have considered a process which is reversible. Now we tum to one where things happCn suddenly, resulting in irreversibility. Suppose we describe the system by a wave function t/J. TIle time-dependent Schrtxl.inger equation is

h

-i

a~ {It = 7i(l)~

11£,

II,

This equation is ploued in Fig.6.2. Note that if we lower the field until it is zero, M f = O. However, since the field changes are reversible (and in fact at all times keep the entropy constant), we can recover Mi by raising Hr from zero back up to its initial value Hi. The recovery of Mr in such a process d~s 1I0t involve spin-lattice relaxation since, as we have postulated, everything is done on a time scale shon compared to Tt! This curve is often a surprise to those of us who first learned magnetism by studying magnetic resonance, since we learned that one needs TJ to produce magnetization from an unmagnetized sample. Actually, if one uses Curie's law and (6.23), one sees that if one stans in zero field with spins which are in thennal equilibrium with the lauice (8 = ()/), the mere act of adiabatically turning on the static field to H o ,. HI. will produce a magnetization. The size is (6.26) where Mo is the magnetization one gets when the spin-lattice relaxation produces 226

= -M·H(l)~+7id'"

(6.27)

where M is the tOlal magnetic moment operator and 1id the dipolar coupling. The time dependence of 1i arises because the applied field H is time dependent. For the case of sudden switching, we take H as independent of time except for t = 0, at which time it jumps discontinuously from one value to another. Denoting by 0- and 0+ times just before and just after t = 0, we then have

~(O+) -

0+

"'(0-) = -

J*7i(l)~(l)dl

=0

or

(6.28)

0-

(6.29) since 1i(t), though discontinuous, is nevenheless never infinite. Thus we have that the wave function just before the switch is identical to its value juS! after. We can utilize (6.29) to see that a sudden change in H produces a change in the expectation value of the energy E. The expectation value of energy at any time t, E(t), is E(l) = (~(l). 7i(l)~(l))

(6.30) 227

Thus

whence we gel

E(O+) = -C H(O-) . H(O+) _ C H~

E(O-) = (,,(0-). 'II(O-),,(O-» = -(M(O-» . H(O-) + ('IId(O-»

8.

(6.32)

(6.33)

But

is conserved. 1Oerefore.

Ukewise

(6.41)

The fact that "'(0-) ::: "'(0+), however, means that the expectation values of both magnelization and dipolar energy are the same at t ::: 0+ as at t ::: 0-. This resuh expresses Ihe physical fact Ihal all spins point the same direction al t ::: 0+ as they did at t ::: 0-. Thus we can write

(6.34) This equation is very useful since in general the system just before Ihe sudden change is assumed 10 be in internal thennal equilibrium with temperature OJ. Thus we can compute the expectation values by the methods of Sect. 6.3. So far we have nor employed the density matrix notation, simply to make Ihis chapler accessible to those readers who have not yet become familiar with it. We can write our results compactly by recognizing that (6.29) is equivalent to saying that lhe density matrix f! obeys the relation

(6.35) Thcn

(6.36) Assuming thermal equilibrium at t ::: 0- at a lemperalUre Bi we get

_

riCO ) =

cxp(-'II(O-YkO,) Z(O )

(6.37)

Therefore

('IId(O-» = Tr {e(O-)'/{d} Tr {'lid cxp (-'II(O-)lkO,») Z(O-)

(6.38)

Manipulation, using the high temperature approximalion plus the definition (6.14), gives

('IId(O-» = 228

(6.40)

As discussed earlier, immediately after switching the field. the spin system is in general not in internal thermal equilibrium even though it was in thennal equilibrium at t = 0-. If we wait a long enough time, we expect a temperature to be achieved. Call that time if and the temperature 8r- We can compute 8 r by recognizing that fOT t > 0 the Hamiltonian is lime independent, hence the energy

where (M(O-» and (1{d(O-» are the expectation values of magnetization and dipolar energy at t = O~. defined as

(M(O-» " (,,(0-). M,,(O-» ('IId(O-» " (,,(0-). 'IId"(O-»

8j

(6.31)

CH'L 0,

(6.39)

E(tr) = _CIH'(O+) + HlJ

0, and E(O+) is given by (6.39). Thus

H(O+)' + H' 8r - (J. " - 'H(O ).H(O+)+Hl .

(6 42) .

The significance of (6.42) can be seen by considering a particular example in which the applied magnetic field is turned suddenly from its initial value. Ho, to zero at t = O. Then H(O-) ::: Ho. H(O+) ::: O. giving

Or = 0,

.

(6.43)

We contrast this with the result of turning the field slowly 10 zero

HL

Or::: OJ Ho

(6.44)

The sudden tum-off leaves lhe temperatu~ unchanged, the slow tum-off produces cooling. The results of sudden changes are summarized in Fig. 6.3. for Ihe case described above in which Ho is lurned suddenly to zero at t ::: O. Adiabatic and sudden changes in Ho have been very useful in magnetic resonance. One of the first uses of adiabalic changes was to measure the magnetic field dependence of the spin-lattice relaxation time. In 1948, Turner el al. [6.7] were studying [he spin-lattice relaxation time TI of protons in insulators. The relaxation times were, in some inslances, many minutes long. To observe Ihe dependence of T 1 on stalic field, they used a field cycling lechnique in which they observed the resonance at a high field. bUI allowed the spins to relax. in lower fields to which they cycled belween their observations. Pourui [6.8] discovered that the nuclear relaxation times in a crystal of LiF were so long thai he could take his sample out of the apparatus into Ihe earth's magnetic field and then return it with only a small loss in magnetization. With Ramsty [6.9] he demonstrated that if this sample were removed from the strong field 10 a small static field (40 Gauss

'29

III

i

/ P 'V!v ~

£1

Fig. 6.3. Magnetic field (11), wave function (l{!), energy (E), spin temperature (0), and rnagnc~ization (M) as functions of lime for the case of a magnelic field turned suddenly to zero

i - - - - - !-

,

, ,,

~N;;O~ _ _ {%
,

I

,

"I

-----'--"-----

or less), subjected to an audio-frequency magnetic field, and then returned to the original magnetic field, he could detect aI the strong field the resonant absorption of energy by the spin system when in the weak field. The experiment we have just described is closely related to the famous experiment on negative temperarures by Pound and Purcell [6.10]. They 100 started with the LiF sample in a strong field, but removed {he sample to a solenoid whose field they suddenly reversed, producing a situation in which the magnetization is anliparallel to the static field. They then returned the sample to the strong field where they observed the relaxation of the magnetiz.1tion component along the static field. They point out that such a circumstance, with the upper energy Zeeman levels more populated than the lower, corresponds to a negative Zeeman temperature. Any system with an upper bound on its energy levels can in principle have a negative temperature. In addition to the original paper by Pound and Purcell, we call the reader's attention to the wonderful account of this experiment and discussion of the negative temperature concept to be found in Van Vleck's [6.11] lecture on the concept of temperature in magnetism. (Professor Van Vleck cautions that there are some incorrect numbers in his paper.) This imponant experiment in which a population inversion was produced is the forerunner of the maser and the laser. Demagnetizing to zero field is the basis for the experimenls of Hebel and Slichter [6.12] and Redfield and Anderson [6.13,14] to measure the nuclear relaxation in superconductors. The problem here is that since superconduclOrs 230

exclude the magnetic field, it is hard to observe a magnetic resonance in a metal in the superconducting state. One starts the field cycle with a magnetic field of sufficient strength to suppress the superconductivity. When one turns the field to zero, one achieves two effects: (1) the nuclear spins are cooled and (2) the sample becomes superconducting. With the field zero, the nuclear spin temperature relaxed towards the lattice temperature so that when the magnet was turned on again after a variable time T, the metal returned to the nOTlllal state and the spins warmed to a holler temperature than they had before the cycle began. The increase in temperature is measured by a swift pass though resonance. By varying T, one could deduce the zero-field spin-lattice relaxation time. The NMR results showed that as one lowered the sample temperature below the superconducting transition temperature, Te , the nuclear relaxation rate was at first faster than it would have been if the metal were normal, then at lower temperature it became slower. The temperature dependence of the nuclear relaxation rate is dramatically different from the temperature dependence of ultrasonic absorption, as was shown by Morse and Boltm [6.15]. They found that ultrasonic absorption dropped rapidly relative to its value in the normal metal on cooling below Te · The contrast between the behavior of two low energy scattering processes, nuclear relaxation and sound absorption, is difficult to understand in a one-electron theory of metals, but finds a natural explanation in the Bardeen, Cooper, and Schrieffer (BCS) theory of superconductivity. These experiments constitute a direct verification of the concept of eleclTon pairs, which is the basis of the BCS theory. (See, for example, Leon Cooper's Nobel Prize lecture [6.16].) Anderson [6.17] working with Redfield, combined field cycling with the application of an audio-frequency magnetic field applied while the static field was zero to heat the spins to plot Ollt the zero-field absorption characteristics of a spin system. Thus they used field cycling to give them the sensitivity of resonance in a strong field to monitor the effects they produced in zero field. Abragam and Proctor [6.18] did further studies establishing the validity of the spin temperature concept, again using LiF. An important result of their experiments was their observation of the transfer of energy between the twO spin systems (Li 7 and F19) even in a static field which greatly exceeded the dipolar fields exerted by the nuclei on their neighbors. We discuss these experiments further in Sect. 7.10.

6,5 Magnetic Resonance and Saturation The analysis of magnetic resonance by Bloembergen et lIl. using standard perturbation theory is given rather compactly in (1.32) - the differential equation for the population difference, n, between the two energy levels of a system of spin particles,

4

dn

no-n

- '" -2W(w)n + - dt T1

(6.45) 231

W(w) is the probability/second that a spin will be flipped by the radio-frequency field H t . Standard perturbation theory shows W(w) = i-y2 Hrg(w)

(6.46)

where g(w) is a funclion nonnalized to unit area having the same dependence on frequency as the absorption line -that is, it expresses the fact that the frequency of HI must be close to resonance for H, to induce transitions. no is the thennal equilibrium population difference, and TI the spin-lattice relaxation time. It is always JX)ssible, at least conceptually, to consider TI infinite, in which case (6.45) is especially simple to solve. n = Ae- 2Wf (6.47) where A is a constant of imegration. It is well to recall the conditions for the validity of (6.46). 1bey are two: i) ii)

The perturbation matrix elements inducing transitions must be small compared to the width of the final state energy levels. This means HI < H L. The wave function must not change much. We note, however, that (6.47) predicts that fl --f 0 as t _ 00.

To satisfy condition (ii) we expect that we must consider times less than 1/1V. Thus, though it is always easy to meet condition (i) by making H, small, no matter how weak HI, if we wait long enough we violate (ii). [Note we are requiring here also that I/W < TI, otherwise the Tl tenn would rescue condition (ii) even for times long compared to I/Wj. We have the interesting problem, therefore, that we do not know how to integrate the equations of motion beyond a time for which n is almost its value at t = O. (See Fig. 6.4). The solution of this problem was found by Redfield [6.19] in a truly remarkable paper, the more so when one recognizes that it was his first work on magnetic resonance. In it he shows that the Bloch equations, when applied to a solid, violate the second law of thennodynamics. The essence of his approach is to note that a resonant time-dependent perturbation, no matter how weak,

".

of l"lliitiity of ~Rrgiolt com'elJliOIl(lI/H:1(llrlJo/io/f /lleor)'

will eventually produce large effects. Whenever a perturbation of small size can produce a big effect, it is dangerous to treat it lightly. He therefore eliminates the time dependence by transfonning to a reference system in which the Hamiltonian..is essentially time independent. The residual time dependence is not of the dangerous variety. For such a transfonned system, the energy is conserved. Moreover, the system is highly complex, consisting of many interacting spins. One can thus predict that after a sufficiently long time the system will be found in a state of internal equilibrium. That is, it will be in one of its most probable states. Phrased alternatively, the energy states will be occupied according to a Boltzmann distribution at some temperature 8. We consider the Hamiltonian 1t, given by

1/ = 1/z(I) + 1/d

where llz(t) is the Zeeman interaction with the static field Ho and the rotating field of amplitude HI and angular frequency w rotating in the sense of the nuclear precession. "The TOtating field makes 'Hz time dependent. We are, of course, considering T I infinite. Utiliz.ing the methods of Sect.2.6, we transfonn to the rotating reference frame, gelling a transfonned Hamiltonian J(:

'It = -~h(Ho-wh)I.+HII.J+1f.: + term oscillating at ± w, ± 2w -Iz sin? + I y cos; = e- i/ .? Iyei !.?

and

(6.50) (6.51)

to transform products such as ei!,w.l IzIye -iI

.w. t

(6.52)

to expressions involving sin Wit and cos Wit. The term 'H~ is that part of the dipolar coupling which commutes with Ii' Physically, it is the part of 1id which is unchanged by rotation about the z-axis. (The two statements are of course equivalent since one can consider Ii as generating rotations). II is the sum of the tenns A and B discussed in Sect. 3.2. The fonn of 1t~ is

o

"·ig.6.4. Conventional 6aturation theory predicts that the I>opuiation JI goes to zero exponentially with lime t. However, the assumptions on which il is based are valid only for the initial PlITt of the curve where n s:' no

(6.49)

In deriving (6.49) we have used relations such as

"(2/i,2

1id = - 4

"

(6.48)

L

j,k

(1-3cos 2 8j1.) :I r jk

(3I:jI z k - Ij'

h)

(6.53)

where 9jk and rjk are coordinates of nucleus j with respect to nucleus k. The term in the square brackets in (6.49) may be considered as the coupling of Ihe spins to an effective static field He as we noted in Chap. 2.

He

=

k(Ho -w/-y)+iH I

(6.54)

2/W 232

233

In the absence of the time-dependent tenns, the energy levels of 'HI would be split by He·and the dipolar couplings, so that we expect typical splittings to be IJE ~ iliJHi + (6.55)

M=C

HJ

(6.56)

Now, in the absence of HJ, 'Hz and 'H3 commute, since the fact that

H3

[Iz , 11:3] = 0 was the definition of1f.3. Under this circumstance 'Hz and would separately be constants of the motion. However, if HI f. 0, ['Hz, H~] f. 0, and the Zeeman and dipolar systems can then exchange energy. Since H is independent of time, the total energy is conserved. Moreover, the system is very complex. Redfield therefore postulates that no matter what the state of the system at t = 0, a long time later it will be in a state of internal equilibriulll described by a Boltzmann distribution. In other words, there will eventually be a temperature 6 which can be assigned to the spins. We can thus say that the density matrix g is

e=

..

e-'H/k9

(6.57) Z with Z = Tr {exp(-H/k8)}, where 'H is the effective Hamiltonian of (6.56). Of course, we expect that after a long enough time Redfield's hypothesis would be fulfilled (unless there were some hidden selection rule which we have overlooked, are perfectly isolated from one llnother if HI such as the fact that 1f.z and is zero). But the really important question becomes, how long does it take to reach equilibrium? The answer to this question clearly depends on the size of HI, since HI is needed to prevent isolation of the dipolar and Zeeman systems. We return to the question later, but for the present consider that the time is short enough to make the establishment of a temperature practical.

H?t

6.6 Redfield Theory Neglecting Lattice Coupling

2

CH,2 _

I T, «'lj0)2) L- k (2I+I)N d

E = _C(H; + H'I> 8 234

(6.58)

(6.60)

(6.61)

Evaluation of the trace of (6.61) gives, for a system with only one species, (6.62)

where {IJH 2 } is the second moment of the resonance line. Following our earlier convemion of omitting primes. we shall now use H L for H~, using the prime only when we wish to distinguish between the local field in the laboratory reference frame and the rotating reference frame. It is important to notice that the Redfield assumption leads to Curie's law and that the vector nature of the law shows that the nuclear magnetization is always parallel to the effective field when the system is describable by a temperature in the rotating frame. Thus, if one is exactly at resonance, where He = iHI' the magnetization is perpendicular to the static field. Since the fonn of (6.58), (6.59) and (6.60) is identical to that of the corresponding equations in the laboratory frame, most of the equations of Sect. 6.4 can be immediately applied to the rotating frame by repl:lcing Ho with He and with

HE

Ht[.

6.6.1 Adiabatic Demagnetization in the Rotating Frame An adiabatic demagnetization in the rotating frame can be perfonned readily, as was demonstrated by Holron [6.20]. Suppose that initially HI = 0, and the sample is magnetized 10 Its thermal equilibrium value kMo. Let us shift Ho far above the resonance value wI'"'(, and tum on HI' C'Ne assume Ho - wI'"'( to be much bigger than HL and HI.) We now have an effective field which is virtually parallel to M. Next we change Ho to approach resonance at 11 rate sufficiently slow to satisfy the criterion for a reversible change. C'Ne are here assuming T l to be infinite, which we achieve in practice by perfonning all experiments in a time shorter than T j ). We then have that

M = Mo The significance of (6.56) can be appreciated by calculating again the energy E, entropy a, and magnetization M. We find easily that

(P

where C is the Curie constant, and where

HE H;,

1f. = -'"'(hI· He + 1f.3 = 1f. z + 1f.~

(6.59)

a=Nkln(2I+I)- C (H;+HII)

HE

where the square root is a convenient way of including the two limiting cases of He>HL and He«H L . The time-dependent tenns will connect states of order iliHo apan in energy. Unless ~ + a very low resonance field indeed, the time-dependent tenns are far from resonance and can be neglected since they are unable to produce transitions. They are not dangerous. We thus obtain a Hamiltonian which we call 1f., omining primes for simplicity of notation.

He 8

He JHi+ H [

(6.63)

Notice that M is parallel to He' as is required by Curie's law. Thus, as one approaches resonance, M changes direction, always pointing along He. In general, the magnitude of M also changes, unless Hi ~ HE. We can experimentally measure M by suddenly turning off HI, leaving M}o precess freely about 235

Ho. The induced voltage immediately after turn-off is proportional 10 M',t.. (Use of a phase sensitive detector enables one to measure Al'1: and 1L1y , but for these experiments My = 0). One can measure M z by noting Ihm though M'I: decays to zero within a time of order I/-yH L, after tuming off HI, M l does not change. One can thus wait till M',t. has decayed, and then apply a 7r/2 pulse which rotates M z into the x-y plane for inspection. The theoretical values of M'I: and M l are HI

HI

MQ

------------ ----- -----------------_.-

(6.64)

M. = M - = MOr~==;; He ./H2 +H2

vel,

ho ho M, = M - = MO'f~~=,;' He I H 2+H2 V el.

(6.65)

where ho is the component of the effective field in the z-direction:

110

== (Ho - w/-y)

(6.66)

Notice that if one does an adiabatic demagnetization exactly to resonance, the value of magnetization ~=~

HI /H2 +H2

V

I

0~

L

will persist indefinitely as long as HI remains on (actually, when relaxation to the lattice is included, it decays, but on a time scale typically of order T I ). We contrast this prediction with our earlier conclusion from the Bloch equations that M:z; decays to zero in T2 , where T2 ~ I/-yH1•. The factlhat M does not shrink as long as He is kept constant, and that M precesses in step with HI is often is comparable 10 M:z; described by the graphic tenn "spin locking". If will be less than Mo. However, the "loss" of magnetization is recoverable. Were one 10 go back off resonance, M would grow back to M o when H;:» Figure 6.5 shows (6.67). When Redfield proposed his theory. the fact Ihat the spins were locked to He was one of the most surprising results. It is, of course, nOlhing but the rotating frame equivalent of the statement that the magnetization in the usual laboratory frame adiabatic demagnelization has a one-to-one correspondence with Ho, as expressed in (6.25). Note that if one pulses on Ho when (Ho -w/-y):»HI and H t • and then changes H o so thai one passes through resonance, continuing until one is far on the other side of the resonance (Ho - w/-y negative), one will have turned M antiparallel to Ho. Moreover, although near to resonance, one might have M < Mo (if HI < Hd; by the time one is far from resonance one would have M '";;£ Mo· This experiment provides a simple means of lllrning over the mngnetization. One can see from (6.64) that if one demagnetizes exactly to resonance, the magnetization will be the full Mo provided 1I I :» HI.. TIle resonance signal is then as big as can be achieved using a 1r/2 pulse. If HI ~ HI,. one will not

lIf

El,

Hl.

236

fl i . fI, )'ig.6.S. The transverse nmgnetization At" produccJ in a demllgnetization experiment in the rotating frame, as a runction of the strength or alternating field If, employed. Ih is assumed to be turned on when If a is we11 off resonance, with the slImple initially at its thermal equilibrium magnetization {\'fa

achieve the full magnetization at resonance. Were one to reduce HI slowly to zero after arriving at resonance, M would shrink to zero. In this manner all of the order represented by Mo prior to demagnetization would have been put into the dipolar system, thnt is, into alignment of spins along their local fields. One could, at a later time, slowly turn H t back on again, thereby recovering the magnetization. 6.6.2 Sudden Pulsing A situation frequently encountered in experiment and interesting to contrast with adiabatic demagnetization is the effect of a sudden change in He. We treat an especially simple case, that of suddenly turning on HI at time t = O. We assume that before we tum on HI, we are off resonance by an amount ho, with the system initially magnetized to Mo along Ho. We utilize the ideas of sudden changes discussed in Sect. 6.4. The sudden tum-on of HI is so fast that the system has the same wave function or density matrix just after turn-on as it had before. The dipolar energy Ed which depends on the relative orientation of spins is thus the same at t = 0+ as at t = 0-. Moreover, since the dipolar Hamiltonian 1t~ is the same in both laboratory and rotating frame.

_ C1I2 H2 Ed = -~ = -Mo.::.L. 6] Ho

(6.68)

where 61 is the lattice temperature. The Zeeman energy is

Ez = -M·He = -Moho

(6.69) 237

The tolal energy. E = Ez + Ed, is then

E = -Mo{ho + HVHo) ~ - Moho

6.7 The Approach to Equilibrium for Weak HI (6.70)

A long lime later a spin temperature will be established together with a magnetizalion M parallel 10 He giving

E=_C(Hi;Hl) =_M(HJ;eHl )

(6.71)

But the lolal energy is conSClVed once HI has been turned on, so that, equating (6.70) and (6.71), we get

M = Mo H~O:~2

(6.72a)

L

o

Hillo

M z = NIo H2

H2

(6.72b)

M: = Mo H2 : H2

(6.72c)

0+

L

,,2 o

L

This equation shows that exactly at resonance, M would vanish. The null is vel)' sharp, M:t varying linearly with ho, so that observation of the null provides a precise method of observing exact resonance. II is interesting to contrast these results with conventional saturation theory. For a spin! system, M: is related to the population difference n by the equation /'fm

~-~~ 2 Saturation theory says that the equilibrium population difference, assuming infinite Tl' is n = O. hence M z = O. Equation (6.72c), on the other hand, says M z will be zero only when saturation is perfonned exactly at resonance (ho = 0). Conventional saturation theory assumes the transverse magnetization vanishes as well as M z . Equation (6.72b) shows that in general, M:I; is large, and may in fact be larger than M z . If (6.72) has a simple geomelTical meaning. After H] is pulsed on. M precesses about He. The component of M parallel to He cannot decay without energy exchange to the lattice. but Ihe componenl perpendicular can decay since Ihe local field gives a spread in precession frequencies. Thus, after several times l/-yHL' M will be parallel to lIe. and will have a magnitude given by the projeclion of the il'lilial Mo on He.

Hi:» Hl.

We saw in Sect. 6.5 that standard perturbation theory predicted that following the turn-on of a weak HI the population difference 11 would go to zero for long times, although we recognized that we could not rigorously apply perturbation theory to times greater than l/W. The requirement of a weak HI was necessary in order that perturbation theory be valid for at least short times. Of course, since M z is proportional to n. this implies M z would go to zero. In Sect. 6.6, however, we saw that M z would, under thesc conditions, go to an equilibrium valuc Mz)equil = Mo , 2

10

il 2 °H 2

+

(6.74)

L

fIr

where we have assumed Hr« 116 and «: HE (although we note that the equilibrium expressions in Sect. 6.6 were not limited to weak H J). It is therefore clear, as we suspected, that for fong limes, perturbation theory does flat give correct predictions. For short time intervals, however, it must be correct. Recalling the proportionality between M. and n, we know that

d~~,

_ -2W(w)M,

(6.75)

for times short compared to I/W(w). How can we describe M z for longer times? The solution to this problem was worked out by Provotorov [6.21] in an elegant paper utilizing powerful techniques. Rather than outlining his analysis, we will give an altemative derivation of his result. We note that in the absence of HI, thc Zeeman interaction in the rotating frame is just

1iz = -Iflhol z

(6.76)

Let us make the assumption that we can assign a tempcrature 0z to this Zeeman Hamiltonian, and 0d to the dipolar This assumption may not be rigorously correct, bUI it is al least simple, and corresponds to the facts at the time HJ is turned on. Immediatcly before turning on H] the dipolar system is at the lauice is the samc in thc rotating or laboratory frame. In temperature 01, since rhe laboratory frame we have

'Hl

1t3

CHo

Mo=-8,

(6.77)

But in the rotating frame

Clio Mo=-8z

(6.78)

Inasmuch as Ho ~ h o , Oz «0,. Thus the Zeeman temperature in the rotating frame, 0z, is very cold compared to the dipolar temperature tJ d . Turning on II] couples the two reservoirs and they approach the final equilibrium value given by the analysis of Sect. 6.6. The coupling of HI produces lTansitions between the

°

238

239

energy levels of the Zeeman and dipolar systems which we assume are governed by simple rate equations for the population of the various slates. (This assumption is similar to our postulates of Sect. 5.2. It is quite common in all cross-relaxation calculations. Provotorov makes it implicit in his work when he evaluates the relaxation times.) Since there are many states, a large number of coupled rate equations similar to (5.13) result. As has been shown by Schumacher [6.22], when the two systems are char· acterized by temperatures, the many equations reduce to twO coupled linear rate equations, one for (1/8z), the other for (I/8d) much as the many equations represented by (5.13) reduce to a single rate equation (5.27). But the conservation of energy gives a relationship between 8z and 8J :

Ch 2

CH2

8z

8d

___ 0 _ ~ ::::: const

(6.79)

Equation (6.79) is a first integral of the coupled equations, so that one of the resultant time constants is infinite, and a single exponential results. Since M, oc 118z, this means M~ relaxes according to a single exponential towards its equilibrium value. Using the fact that M, ::::: Mo initially, we get an equation for M, as a function of time,

M" - M,,)equil::::: (Mo - M~)equil)e-t/T

(6.80)

The only unknown in this equation is 1". We can, however, easily calculate il as follows. Taking the derivative of (6.80), evaluating it at t : : : 0, and comparing with (6.75) (which must be valid initially where perturbation theory is correcl), we get

~ T

Mo

:::::2W(w)

Ala

(6.81)

Mz)equil

Using (6.72c) for M')equil we get

2

2

2

_I ::::: 2W(w) h 0 +HL::::: 7(72 H2 (h 0 +H2) II 9(W) 1" H2 IH2 L

(6.82)

L

This result is the same as that firs! found by Provotorov, as indeed it must be since we have made the same approximations as he. The complete time development of the magnetization is therefore

M,::::: 2Mo 2 ho+HLexp -1r"f HI2(h~+Hf) 2 y(w)t )] ho+H L HL

[2

2 (

2

I

Slope b,OW" I,om f'tnuroutioll tlltory

". Filial rolue gi~fm by Uedfield O,eo,y

, Fig.6.6. M. "eUIl! t during saturation ~ given by the argument in ~he te,,~. The equlltion ror M. 115 Il runction or lime is round by joining lhe initilll region, where dM./dt is known rrom perturba~ion theory, to ~he equilibrium "alue, give" by Redfield theory, using Sc1wlllochu's observation lhat ~he approa<:h lO equilibrium involves a single exponential

6.8 Conditions for Validity of the Redfield Hypothesis We have noted that lhe concep! of spin temperalure in the rOlating frame has as a basic requirement neglect of certain lime-dependent !erms in the Hamil!onian

J

transformed to the rotating frame. This means, in essence H a,. Hi + Ht. On the basis of the previous seclion we can now add a second requirement. We saw that in the absence of HI> the equality of dipolar and Zeeman temperalures in the laboratory reference frame implied inequality in the rotating frame. Spinlattice relaxation will allempl 10 bring the two temperalUres together at the lanice temperature in laboratory frame in a rime TI. On the other hand, the presence of H I attempts to equalize them in rotating frame. The two tendencies are in conflict. The slrength of the tendency is indicated by the corresponding relaxation times (a short relaxation lime means a correspondingly strong tendency for the associated equilibrium). Thus a spin temperature will be established in the rotating frame only if 1" is much less Ihan Tt. Thus we find that a spin temperature is established in the rotating frame and that we muSt employ Redfield's approach provided

T, 1 -:» T

(6.83)

This expression is remarkable since it involves the successful integralion of the equations of motion well beyond the time (l/W) for which perturbation lheory is usually valid (see Fig. 6.6).

240

".

1r"'(2 Hr

or

(h~

;r

l )9(W)T1 ,. I

(6.84)

This is almost exactly the conventional condition for saturation. Note that the longer T I , the smaller the HI which will satisfy (6.84). In particular, frequently (6.84) is satisfied even when HI <. HI..

241

6.9 Spin-Lattice Effects So far we have considered the case of magnetic resonance on a time scale shon compared 10 the spin-lattice relaxation time. In many instances one performs

l

transient experiments which satisfy this condition. However, an equally important

I

case arises when one does experiments on a lime scale long compared (0 T. as when one employs steady-state apparatus. One can still salisfy the criterion for validity of the Redfield theory given by (6.84), but the spin temperature 8 in the rotating frame is now determined by the coupling to the lattice. As we shall see, this slatement by no means implies thai () = 8,. bUI only that it is a function of 8/_ Fortunmely it is very simple to generalize oue previous treatment 10 include the lattice. As a maHer of fact. in Redfield's famous paper he considered just this case, but we have put off consideration of the lattice relaxation until this Stage to simplify the discussion. In general, when the spin system exchanges energy with the lattice, the internal equilibrium of the spin systcm in the rotaling frame is momentarily disturbed. The basic assumption we make is that the cross·relax;lIion bctween the dipolar and Zeeman systems is so rapid compared to T I that, following an exchange of energy of the spins with the lattice, a new spin temperature is rapidly established. Thus, the lattice always finds the spin system described by a temperature in the rotating frame. We considcr that thcre are three basic relaxation equations for the classical magnetizations lI{zl and Mz: and for the expectation value of the dipolar energy (1t~), which we write down phenomenologically in a fonn chosen to assure that in the absence of HI the spin system would reach thennal equilibrim with the lattice.

·l

DMz:

7ft =

1110 -Mz: Tg

'Dt=

(1i?,), - (1f.\) Te

-E = -M,ho -

°

M.H, + (1id )

(6.88)

we can find the rate at which the lattice coupling changes E by taking Ihe time derivative of (6.88):

I

I

(6.85)

dE

DM~

a

0

(6.89)

1be only conttibutions to the time derivatives we need consider are those given by (6.85). (6.86), and (6.87) since other contributions do not change the tOlal energy E. Employing (6.85-87). assuming that M always lies along He' and using (6.58) and (6.59). we readily find an equation for the magnitude of M

dM

••

DMz:

. dt = -h0 /5t - H'/5t + at (1id )·

-

(6.86)

o(1i?,)

statistical mechanics for which there are a number of collision terms, each of which changes the distribution function f and for each one of which one could compute af/Dt. Thus. if the Zeeman energy -M· He and the dipolar energy (1t3) ..w ere not in thennal equilibrium with one another. the magnetization would grow (or shrink) unlil equilibrium was reached. ThaI process clearly produces a contribution to the three time derivatives (6.85-87) which we have not included in the equations. The lattice coupling. represented by Ta • T/>, and Te , must be expected to push the spin system out of thermal equilibrium in the rotating frame, but the internal couplings of the spin system counteract that trend by producing energy exchanges within the spin system. Changes within the spin system must conserve the total energy of the spins. If then we consider the rate of change of the expectation value of energy, E, we can ignore any changes which simply redistribute energy within the spin system. and keep only changes associated with the lattice. Using the fact that

dt

I = -(M", - M)

(6.9Oa)

Ttl

and for the spin temperature 8

(6.9Ob) (6.87)

where To and T/) and T e are relaxation times corresponding to exchange of energy with the lattice. and where (1t3)/ is the value of (1t3) when the spin temperature is equal to the lattice tcmperature. We have used partial derivative signs in these equations 10 emphasize that they represent the changes in these quantities induced by lauice coupling only. Thus. although Ta is the usual TI. T/) is lIot the usual T2 (::::: Il-yH t ,}. Tb is gcnerally of onler TI, a much longer time. A good analogy to these equations is to think of a Boltzmann equation in.

1 We do nol worry !Iboul Ai, since its relAxlllion does nOl c:hange lhe el1crgy in lhe rol.llling frame (in essenc:e we llSSume At, = 0).

where we have introduced a notation Tie and where

M

_ cq -

MoHerr{hoITa ) (hyTo ) + (H?ITb) + (HlJTc )

(6.910)

"

2 2 I (h H2 +:.:J. H ) _I = .::2. + _, Tl e h~+H?+Hl To Tb Tc

(6.9Ib)

[We have here neglected the term (1t~)1 of (6.87)). Note in particular that Mcq := 0 exactly at resonance, is poslUve when Ito";> 0 (i.e., M is parallel to He). but is negative for 110 < 0 (that is, M is antiparallel to He)' The last case corresponds to a negative spin temperature in I

242

243

the rotating frame. Equation (6.91a) shows that the equilibrium 8 is far from the lattice temperature 8/ and may even be of the opposite sign. Since Mo is inversely proponional 108/, it is still true Ihat 8f detennines 8. even Ihough they are quite different. The negative temperature one sometimes finds is a simple manifestation of the fact that M;: always tends to be positive whether ho is positive or negative. In fact. one can say that the equilibrium is reached as follows: The strong internal coupling of the spin system (which guarantees a spin temperature) keeps M along He, since Curie's law is a vector law. The lattice is anempting (a) to make Ihe z·component of M be Mo. but (b) the xcomponent be zero. (a) would make M bigger than MO so that its projection on the z-axis is Mo, whereas (b) would make M be zero. The lauice is thus fighting itself since (a) and (b) are inconsistent. The equilibrium value of (6.91a) results.

6.10 Spin Locking, T 1g, and Slow Motion As discussed above, and shown in Fig. 6.7, a spin temperature in the rotating frame is established in a rime T from an arbitrary initial condition wilhoUl exchange of energy with the lauice. But this is only a quasi-equilibrium value since. over the subsequent time TIL" the spin temperature changes as energy is exchanged with the lanice to drive M to M equil of (6.91 a) (see Fig. 6.7). During this process M lies along HeW' Thus if one starts at resonance having oriented M along HI, M will not decay in a time characterized by the inverse of the

M

line width, but rather with the time Tt, which requires energy exchange with the lattice. In principle. by going to low enough temperalUres one should be able to make T t , as long as one pleases. If one does this, magnetization along HeW in the..rotating frame. following at time T to establish a spin temperature, will remain without decay for as long as one has chosen to make TIL" This time may be seconds in metals or even hours in insulators. Even though we can lock the magnetization along Herr for a time T, when H] is on, if we remove H, suddenly, M wUl decay 10 zero ill a lime oi order of the inverse line width. That is, the Bloch equations with the usual meaning of T2 give a rough qualitative description of what happens. In contemplating the Redfield theory, it is helpfUl to go back to the SilUalion of the laboratory frame without alternating fields. There we see that smning with a magnetized system we can tum Ho to zero slowly and later tum it back up to its original value. When Ho = O. M = 0, but the full M is recovered when Ho is lUmed back on, all without exchange of energy with the lattice. In zero field the order is manifested by the preferential alignment of nuclear moments along the direction of the local fields of their neighbors. The ex.istence of order in the local fields is the basis of a technique [6.13] for observing motions which are much too slow to be seen as a T, minimum or a line·narrowing. Consider a nucleus #1 with a neighbor nucleus #2. Suppose now that #2 makes a sudden jump. as in the process of diffusion. The duration of the jump is perhaps 10- 12 to 10- 13 , very fast compared to nuclear precession frequencies. Thus the local field at #1 arising from #2 changes suddenly in both magnitude and direction. The orientation of the spin of nucleus #1 is thus somewhat randomized relative to the local field. If the mean time a given nucleus sits between jumps is T m , the alignment of nuclei in the local fields of neighbors, that is the ordered state, can persist only a time of T m . Thus to carry out a full demagnetiz..1tion and remagnetization cycle with full recovery of the initial magnetization, we must remain in zero field for a time less than T m. Of course even were there no jumping, any T] process would change the entropy of the spin system. so that in any event we must complete the cycle in a time less than T,. We can conclude i)

Vatu .. glvclI by (6.12a)

~

ii) Votu .. glv,," by (6.9Ia)

T~,1

Fig.6.7. The hierarehy of times followin& sudden tum-on of If l

• During time T, a spin temperature is established in the rolating frame, the value of which is determined by the expectation value of the ener&y, E. immediately after tUMIing on 11\. Over a longer time T 1I. the spin temperature in the rotatin& frame challSes, changing /If correspondingly in response to ener-gy transfer with the lattice. Both processes require Redfuftr s spin temperature hypothesis to analyze sina: we have M5umed T<:T'I

244

The demagnetization-remagnetization cycle can be used 10 monitor the zero field TI. We can detect jumping when T m < T •.

We can apply these same concepts to adiabatic demagnetization ex.periments perfonned in the rotating reference frame, as was done by Slichtcr and Ai/ion [6.23,24] and by Look and Lowe [6.25]. The analysis is quite straight forward. Since jumping causes sudden changes in the dipolar energy, it gives a contribution to IlTc which is

2

;;;--'--- = -(1 - p) Tc jumping T m

(6.92)

245

where p is a quantity expressing the fact that the local field after a jump is not completely random relative to its value before the jump. It can be calculated. Its value depends on the jump mechanism (vacancy, interstitial, etc.). The 2 arises because the local field is a coupling between pairs of nuclei, either of which can jump. Thus, simple measurements of T1(J enable one to measure a10mic motion or molecular reorientation when T m is comparable to T1(J arising from other causes.

7. Double Resonance

7.1 What Is Double Resonance and Why Do It? One of the most important developments beyond the original concept of magnetic resonance is so-called double resonance in which, as the name suggests, one excites one resonant transition of a system while simultaneously monitoring a different transition. There are many reasons for doing double resonance. The goals include polarizing nuclei, enhancing sensitivity, simplifying spectra, unravelling complex spectra, and generating coherent radiation (e.g. masers and lasers). The field has been characterized by a rich inventiveness which seems to continue unabated to the present day, and which makes the task of understanding all that has been done rather overwhelming. It is useful, therefore. to find some means of classifying the work into broad areas which involve related concepts. We shall employ three broad categories. The first category of double resonance makes use of spin-Iauice relaxation mechanisms. We call it the Pound-Overhauser double resonance after two of the important pioneers. The method involves a family of energy levels whose populations are ordinarily held in thennal equilibrium by thennal relaxation processes. If one saturates one of the transitions (1.34) one so-to-speak clamps their populations together (Le. forces them to be equal). The thennal relaxation processes then repopulate all the levels. producing unusual population differences which may possess useful properties (for example, the upper of two energy levels may have a larger population than the lower, or a population difference which is nonnally small may become large). Included in this category are methods of dynamic polarizaliotl of nuclei (the Overhauser effect and solid effec!), electronnuclear double resonance (ENDOR). and masers and lasers. A second category depends on cross-relaxation phenomena, hence we call it cross-relaxation double resonance. The fundamental concept is that if two spin systems can exchange energy (i.e. cross-relax), one can detect the absorption of energy by a resonant altemating field tllned to one spin system by lise of a second alternating field to monitor the temperature of a second spin system. Various experiments involving cycling the "static" magnetic fall in this category, as does the Hartmann-Hahn method which is the basis for the technique referred to today as "CP" (meaning "cross polarization") introduced by Pines, Gibby, and Waugh for enhancing the sensitivity of CI3 spectroscopy. [n general these techniques 246

247

require that some appropriate cross-relaxation process exist and that it be much faster than certain spin-Ianice relaxation processes which would otherwise destroy the effect. The third category depends in general on the existence of spin-spin couplings which in many cases must not be unduly obscured by either spin-lattice relaxation or cross-relaxation. We shall therefore call it spin coherence double resonance because it depends on the ability of spins to precess coherently for a sufficient time to reveal the spin-spin splittings. Typically, one here makes use of the faCI Ihat when two nuclei are coupled, changing the spin orientation of one nucleus changes the precession frequency of the nuclei to which it is coupled, so that the second nucleus can reveal in this way when Ihe first nucleus is being subjected to a resonant alternating magnetic field. Among the examples of this category are topics known as spin decoupling, spin tickling, spin echo double resonance (SEOOR), coherence transfer, and two-dimensional Fourier transform NMR (2D-Fr NMR). 1be last of these is fundamental 10 many important areas of resonance ranging from NMR imaging to delennining the struClure of complex biomolecules. Our purpose in introducing three classifications is pedagogical. Other scientists might pick different classifications. Of course, most schemes of classification are not perfect. For example, the nuclear Overhauser effect leads 10 effecls in 20FT NMR which are exceedingly useful and important. Moreover, 2D-Fr NMR involving only one nuclear species is not a double resonance experimenl (only one oscillator is used) but it can be conceptually viewed as one in which lhe ability of a large HI 10 excite all the nuclei in a spectrum obviates Ihe necessity of having a separale oscillator for each NMR line. It is not our intention in picking three classifications to imply that there is only one idea involved in each category. Indeed, within the groupings we employ there are numerous important innovations. For example, it was many ye~ after lhe firsl spin decoupling experiments were perfonned that 20 Fourier transfonn spectroscopy was invented.

7.2 Basic Elements of the Overhauser-Pound Family of Double Resonance The first double resonance experiment was carried out by Pound p.l] on the Na 23 nuclear resonance in NaN03. His aim was to prove that the spin lattice relax:ltion mechanism was via time-dependent electric field gradients. He computed the thermally induced transition probabilities between the various spin states based on this mechanism. In the NaN03 crystal, the Na resonance is split by an axially symmetrical electric field gradient into three distinct absorption lines corresponding to transitions between the m values of ~ to to and to He predicted and observed that saturating any Olle transition (e.g., the ~ to would produce level populations of all the states which differ from

-4 -l 4) 248

!' ! -4,

4

thermal equilibrium. Thus when he salurated Ihe j to transition, the intensity of the to transition increased by a factor of ~ times its normal intensity. We will discuss below why such intensity changes occur in connection with the Over.IJauser effeci. By means of a series of experiments saturating the various transitions while observing the OIhers, he verified that the spin-Iallice relaxation was via electric quadrupole coupling. The second double resonance experiment perfomled was by Carver (7.2). Ollerhauser, while still a graduate student at Berkeley, had predicted (7.3J that if one salurated the condUClion electron spin resonance in a metal, the nuclear spins would be polarized 10000foid more strongly than their normal polarization in the absence of electron saturation. (For an excellent summary of dynamic nuclear polarization by one of the important pioneers, see [7.4].) Crudely speaking, Overhauser predicted a polarization of the nuclei which they would have if the electron spin BollZmann factor were used in place of the nuclear spin Boltzmann factor. It may be hard for readers today to appreciate the deep scepticism with which Ollerha"ser' s proposal was greeted by the resonance community, because today there are many schemes for dynamic polarization, and the principle forms the basis for many other important techniques. However, in 1953 Ihere was a widespread (Ihough short-lived) belief Ihal Ollerhauser's scheme must violate the second law of thennodynamics. Carver's experiment was not only Ihe first demonstration of the dynamic polarization of nuclei, but perhaps more importantly showed that Ollerhauser's revolutionary Ihoughl was correct. His concept reoriented the thinking of resonators. While his concept stimulated the subsequent invention of other schemes of dynamic polarization, perhaps more imponant was the stimulus it provided to exploration of other novel effects of pumping transitions and doing double resonance. Figure 7.1 shows the first dynamic polarizalion of nuclei, the Li nuclei in lithium metal. To perform this experiment it is necessary to have an HI of 510 10 Gauss to salurate the condUClion electron spin resonance. One also wants a size of metal particle small compared to the skin depth at the electron frequency so Ihat aU the nuclei are polarized by the electron saturation. Accordingly, Carver used a solenoid 10 generate a static field of about 30Gauss, which put Ihe electron spin resonance at 84 MHz, and the lithium nuclear resonance at a frequency of 50kHz. At that, low frequency, the nuclear resonance was too weak to be observed directly (Fig.7.1a), but popped up from the noise instantly when the electron saturating oscillator was turned on.. Carver calibrated the degree of polarization by observing the proton resonance in mineral oil (Fig. 7.lc). He went on to show Ihal dynamic pol:uization did not require a melal by working with the proton resonance of liquid ammonia in which sodium had been dissolved. The sodium atoms in such solutions ionize, giving free electrons whose resonance is readily saturated. The principles underlying the Overhauser effect are in fact identical to those on which POUlld based his experiment. The Overhauser effect provided a strong impetus to the development of double resonance melhods, in part because Over-

4 -4

249

........."p.p.................

~

a

c

.

~:,

Fig.7.1. Demonstration of the Overhauscr nudear polarization effed on Li 7 nuclei in metallic lithium by Carver. The osdlloscope picture shows nuclear absorption plotted vertically versus magnetic field. The magnelic field excursion is about 0.2 Gauss. The top line shows the normal Li 7 nuclear resOnanCe (lost in noi~ at the 50kHz frequency of the NMR apparatus). The middle line shows the Li 7 nuclear resonance enhanced by saturating the electron spin resonance. The experimental conditions of (3) and (b) differ only in turning on the electron spin saturating oscillator. The bottom line shows the proton resonance from a glycerine sample containing eight times l'lS many protons under the same expt'rimenlal conditions, from which one concludes the J.i7 nuclear polarization was increased by a factor of 100

m,.

Hence we take the eigenvalues of I energy eigenvalues are then

E = "fefi.Homs

ms

=±~

z.as another good. quantum number. The

+Amlms - ",(n!lHom, m, = ±~

(7.3)

It is convenient to label states and wave functions by a convention

(7.4)

where c = 2mS, p. = 2m,. A state with ms = +~, m, = -~ is written as 1+ -). The selection rule for transitions induced by an applied alternating field is Llms = ± 1, Llm, = 0, or LlmS = 0, Llm, = ± 1. The first corresponds to an electron spin resonance, the second to nuclear resonance. (See Sect. 11.3 for a detailed discussion). Their resonance frequencies, We and Wn, are (7.5.)

hauser had made an ingenious and daring prediction which many talented physi-

(7.5b)

cists judged to be wrong, and in pan because he addressed a topic of interest to a community of scienlists much broader than resonators. Although Pound did not recognize that if his concept were applied to other systems it could lead to large nuclear polarizations, there can be lillie doubt that even without Overhauser's contribution, Pound's invention of double resonance would have caused a burgeoning of the field.

There are thus four allowed transitions. They are shown in Fig. 7.2. The relative size of IAI and ("fnhH) influences the appearance of the energy level diagram slightly as is shown in Fig. 7.3. If the nucleus and electron under consideration are far apart, typically IAI < l"fnfi.Hol (Fig.7.3a). rf the nucleus and electron are close together, usually IAI > hu rlHo I (Fig. 7.3b). The fonner case is encountered for a typical nucleus in a solid which has a low concentration of paramagnetic centers. The latter case is encountered for nuclei of paramagnetic atoms, or for nuclei which are very close neighbors of a pararragnetic center. For the rest of our discussion we adopt the convention of drawing figures which look like Fig. 7.2a, though we do not mean to imply thereby anything about the relative size of IAI and hn!lHol.

7.3 Energy Levels and Transitions of a Model System To understand the principles of the many applications, one can consider a very simple system consisting of a nucleus with spin 1= coupled to an electron of spin S = acted on by an external static magnetic field Ho. The Hamiltonian for this system is

!'

11. ='YehHoSz+AI.S-'YnhHo1z

!

where we have used subscripts e and n to denote electrons and nuclei, and where we have taken the form of electron-nuclear coupling appropriate for s-states. We assume that IcrlHo» A (the strong field approximation). Of course ,e» l"fnl. These assumptions make Sz nearly commute with 11.; hence mS, the eigenvalue of Sz, is a good. quantum number. Only the term AlzSz of the electron-nuclear coupling gives diagonal terms. so the Hamiltonian is effectively (7.2) 250

++1

(7.1)

-+~+i~1 (iI) Nudl'lIr r
I

w .. '" 'Y.. 11 0 +

.!!.I 211

->

(b) I;·/,.,.trlill V!ill "'SOllllllce 1rl1!l)'ilirJIIs

Fig.7.2a,b. The energy level diagram and allowed transitions of a system consisting of an electron of spin S = coupled to 3 nuclcus of spin I acted ou by an externa.t static magnetic field flo. The figure assumes 'Yn is negative

!

=t

251

++--

++--

The resul/ of using the more general form offunction of (7.8), whether as a result of solving the simpler HamUlonian of (7.1) more exac/ly, or as a result of solving the more general (7.7), is thai, on application of an applied alternating magnetic field, transi/iolls other t!lan those shown in Fig.7.2 become possible. We adopt the convention of calling transitions other than the four in Fig. 7.2 "forbidden transitions" .

--+-

--+-

-+--

In the absence of applied alternating fields, populations of the energy levels of the combined spin system are given by the Boltzmann factors when the system is in thennal equilibrium. As discussed in Chapter I, the achievement of thermal equilibrium can be thought of as resulting from lransitions induced by the coupling to a thennal reservoir in which the thermally induced transition probability We'l,c,y from state 1!"'1) to state Ie'l') is related to the rate Wc"'"e.,

-+-(a) DisrulI/

(h) N"urb)' 'IIICIt-,ts

IIlIdt'('$

Fig. 7 Ja b. The cITed of lhe size of the electron-nucle;o.f coupling II/h relative to the nucle'at ~nallce frequency Col on Lhe appearance of the energy level dillgn"us. For nuclei far from the dectron, lJ$ually iAI < h'.h 110 For nuclei ne~r. to the electron, .rrcquen~ly IAI > b.h/fol. The figu~ ll$IIumes a n~"tive 1'n and A positIVe. What would It look hke for positive "fa?

I·

by

~w~<~,~,<,'-'L'" = _P<'_._'

We have assumed a particularly simple Conn of electton spin-nuclear spin coupling, that which arises from the Fenni COnlact hyperfine expression. The most general Conn of spin-spin coupling would be obtained by adding also the dipolar coupling of Sect. 3.2 between the electron spin and the nuclear spin. Then, as is discussed in Chapter II, the Hamillonian would be

Wc'l'"e'l

where Pe'l is the thennal equilibrium probability of occupation of state le'l). Or, using the Boltzmann relation, We."c''l' =exp[(E ., - EC'l,)/k TJ c IV ,.,. "e.,

'H. = 'YellHOS~ + Az'z,Sz,Iz ' + AII'y,[IIS" + Azzdz'Sz' - 7n1lHo1n (7.6)

jmsm/)

(1.12)

e

where the axes Z', y', z' are a set of principal axes. Solution of the more general Hamiltonian of (7.6) still gives an energy level diagram which looks much like either Fig.7.2a or 7.2b as long as 'Ye1,Ho:> IAz'Z'I, IAy'y'I, and IAz'z.l. In this approximation. ms, the eigenvalue of Sz, is still a good number, but m" the eigenvalue of It, is not necessarily so. Thus the lowest order wave functions tPi (where i distinguishes the four states) may not be

tPi =

(1.11)

Pe'l

It becomes convenient at this point to switch to a more compact notation. Since there are only four states, we label them 1,2,3, or 4 as shown in Fig. 7.4. The notation lVij then corresponds to the thennally induced transition rate from state i to state j. Transitions Wij in which the electron spin is flipped but not the nuclear spin are shown in Fig. 7.5. As an aid to memory, transitions between I and 2 or between 3 and 4 are electron transitions, lransitions between I and 3 or 2 and 4 are nuclear transitions. Also, level 2 lies below level I, level 4 lies below level 3 in Fig. 7.4.

(1.1)

but may instead be linear combinations of such states.

tPi =

L

cimSnI/lmSm,)

(1.8)

1/1,=1++)

"'S,ml

If we keep the Hamiltonian (7.2) but solve it more exactly, we find that the states are of the fonn of (7.8) rather than of (7.7). Since there are still only four energy levels, the notation left) [i.e. 1+ +), I - +), 1+ -), 1- -)] still has validity since it suffices to distinguish four levels. However, while

1/t 1

'"

1-+)

I/t. '" 1--)

Fig. 7.4. Definition of the four states 1,2,3, and <\ in lerms of the earlier llotalioll le./)

(1.9)

in general Izleft) '" !Jllell) 252

(1.10)

,

253

Fig. 7.5. Some thermally induced mp' ping rates which invotve ~he transilion of the electron spin. The direction of the lIrrow indicates the direction of the trallsition corresponding to the rates 1V21 , 1V12 , ete.

(7. 13c)

(7.13<1)

Since the probabilities of occupation must add to one, we have that PI +P2+P3+P4 = I

'h=)--)

We seek a steady-state solution, so we set the left side of (7. I 3a-
7.4 The Overhauser Effect

PI

Though Overhauser' s original proposal pertained to polarization of nuclei in a metal. the principle can be illustrated by considering the model system discussed in the previous section. We take the simplest Hamiltonian, (7.1). with the solutions of (7.3). We assume that the principal relaxation mechanisms are those shown in Fig.7.6 which involve electron spin relaxation (WI 2, W21> W34, W 43) and a combined nucleus-electron spin flip (W23. W32) such as one obtains from the Fermi contact interaction in a metal as explained in Sect.5.3. An applied alter· nating field induces transitions of the electrons between levels I
(7.14)

Pt W12 + (pz - PI)We

(7.13.)

(7. 13b)

= 1>2

(7.15)

Equation (7.l3d) then gives us W

4J va =P4--

W"

which is the normal thennal equilibrium population ratio for this pair of states. Use of (7.I3d) in (7.l3c) gives W2J

P3=P2W32

(7.17)

which is the normal thennal equilibrium population of states 2 and 3. Thus for the family of transitions shown, while the saturation will change all the populations, it only affects the population ratio of the pair of states between which We acts. This result is a special property of the assumption that there are no W;/s which also couple stale 1 to states 3 or 4. Since the ratios PJ!P4 and P2/va are both thennal equilibrium, it follows that the ratio of p., to 1>2 is also thennal equilibrium. r"Or a pair of levels in thennal equilibrium Pj = I)ie(Ei-B;)/kT

::::: PiBij 1/1,-1+-)

(7.16)

(7.18.) (7.ISb)

which defines the Quantity Bij, the Boltznumn ratio (whence the symbol B). Note t.h~ convention on the order of the symbols E;, Ej of (7.18a) and the subscripts t,) on B;j. We therefore have PI = P2

(7.19)

Whence

Fig. 7.6. The Overhauscr effcet. Thermally induced transitions lVi; shown attempt to Illaintain thermal equilibrium. An applied al~ernaling field induces elcetron spin transitions at 8 ra~e IV., between the 8ta~es I and 2

254

(7.20)

255

We have stated that the Overhauser effect produces a nuclear polarization. Let us compute the nuclear polarization. In a state le11), the nuclear spin expectation value is '//2 (7]:< +1 or -t). Therefore. the average expectation value of nuclear spin I z • (Iz) is (7.21.)

(I,) ~ L::p;(iII,li)

This expression is the same as would be found for the nucleus if acted upon by the external field only.1 Comparing (7.23b) with (7.26), we see that electron saturation has increased (Iz)·.by the ratio

~ "

(I,) (lz)th,mn

"" ~(Pl +P2 - P3 - P4)

(7.21b)

l2-B23- B 24

(7.21c)

"" 2 2+B23+B24

To appreciatc the significance of this expression, we evaluate it in the high temperature approximation:

E-E I ) kT

1+

I ) ~ ~ (Eo - E,) + (E, - E,) ( z 2 4kT

A

2" + inhHo

so that

(E, -E1)+(Eo - E,) +(E, - E,) 4kT

Since E2 - E 1 "" -icliHo -

(Iz )lIICTlll =

256

lin hHo

2

2kT

Equation (7.28) is the result Overhauser originally predictcd for a metal. In a metal. one electron couplcs to many nuclei, so there is only a single electron spin resonance, not a resolved pair as in our example.

(7.25)

which, in the high temperature approximation, gives ~ ([z) therm - 2

(7.28)

lOne is at first surpriscd that the coupling to the dedron docs nol come into {/'\h <m since one can view lhe encrgy levels as though there were an effective magnetic field /fe/T acting on the nucleus givcn by

(7.23b) 2 4J.:T If the electron spin resonance were not being saturated, it is easy to show that the thermal equilibrium (Iz)

_

in

(7.24)

.!. iehHo

I B21 + 1 - Bn ~ B24

(Izhherm

The essential feature of the Overhauser polarization of nuclei is that there be a dominant nuolear relaxation process which requires a simultaneous nuclear spin flip and electron spin flip. In mctals, the strong role of the Fenni contact tcrm leads to relaxation processes in which the nuclear and the electron spins flip in opposite directions through terms such as I+ S- or [- S+. Even with a conventional dipole-dipole coupling there are terms [the tenns B, E, and F of (3.7)] which have this propeny of correlated flips if they can lead to relaxation. In liquids, the translational and molecular rotational degrees of freedom introduce a time dependence to the dipolar coupling which enables these terms to produce relaxation. Therefore, paramagnetic ions in liquids should lead to an Overhauser effect. As wc have remarked. Carver and Slichter [7.2] demonstrated this result using solutions of Na dissolved in liquid ammonia. The Na ions give up their valence electron, producing isolated electron spins which relax the H nuclei of

z

2 B 21 + I +B23+B2'1

ie

(7.22b)

Using the approximation that iehHo is much the largest term, we get

(Iz)therm ""

(Iz)

~'-'-~­

7.5 The Overhauser Effect in Liquids: The Nuclear Overhauser Effect

(7.23.)

(I ) ~

which means the nucleus is polarized as though its magnetic moment were comparable to the much larger moment of an electron! If both electron transitions arc saturated simultaneously

(7.22.)

Now E3 - E2 "" iehHo +inhHo

E4 - E2 ""

(7.27)

2in

Jl eff = Jl o

A/2. this gives (7.26)

-

AlliS --

'Ynll

+t

-t,

Since IllS occurs with virtunlly e(111
257

the ammonia. A similar effect should arise with lWO nuclei. as was recognized by Nolcomb (7.5]. Bloch [7.6J, and Solomon [7.7], and was demonstrated by Solomon in a pioneering double resonance experiment in HF utilizing the HI and FI9 resonances. In Sect. 3.4 we saw that the dipolar coupling between nuclei in a solid could be used to obtain infonnalion about struclUre of molecules or solids. In liquids the dipolar coupling is averaged out owing to the rapid translational and rotational motion. Nevertheless. the dipolar coupling still plays an important role in detennining the relaxation times. It has tumed out that the nuclear Overhauser effeci (both stalic and transient) gives important information about which resonance lines conespond to nuclei that are physically close to one another in a molecule. It is thus playing an important role in struclUre detenninalions of complex molecules. especially when combined with two-dimensional Fourier transfoml methods. We therefore seek to understand the Overhauser effect in liquids in greater detail. In order to see more clearly how the liquid degrees of freedom make an Overhauser effect possible, we examine a concrete example, the nuclear Overhauser effect of a molecule containing two spin! nuclei. An example is HF, the system studied by Solomon. We shall demonstrate that saturating the resonance of one nucleus (the S species) produces a polarization of the Other species (the [ spins) and then show that if one disturbs one species from thennal equilibrium with a pulse, the effect is revealed in the transient behavior of the other species. It is, then. convenient to write the Hamiltonian as (7.29) where 'H 12(t) is the dipolar coupling between the two nuclei, augmented. perhaps, by any pseudoexchange or pseudodipolar coupling (Sect. 4.9). and where the explicit inclusion of t emphasizes that in liquids the dipolar coupling is in general time dependent since the radius vector from one nucleus to its neighbor within the molecule is continually changing direction. There is also coupling of the nuclei in one molecule with those in another. We omit these effects since we are concerned with demonstrating principles. Our calculation would therefore apply rigorously only to a case in which the molecules are present in low concentration in a liquid which does not otherwise contain nuclear spins. Since the Hamiltonian commutes with both It and St, we can label its eigenslates by mJ and ms, the eigenvalues of [z and St. The energy levels then become

E = -"YJ'iHomJ -,sliHoms + Amrms

(7.30)

where A is the pseudoexchange coefficient. the only tenn of 'H'12 which does not average to zero under the molecular tumbling. If A .,; O. the resonances of the I-nuclei or of the S-nuclei are doublets. We shall for simplicity assume that A is zero, SO that both the I resonance and the S resonance consist of single resonance lines. Figure 7.7 shows the energy levels. labeled by mrms, and also 258

4--

w w

FIg. 7.7. The energy levels ImsmJ> of II p"ir of nuclei (posilive .,'5). An IIpplied allernalins field induces lhe transition 1 to 2 and 3 to 4 in which rnJ remains fixed, but ms chanses. The lransition rllte IV ill defined in lhe text

l---.·

numbered 1. 2. 3. and 4. We assume that we are irradiating the S transition. producing a transition probability per second of W between states 1 and 2 and between 3 and 4. To find what this does 10 the [-spin polarization, we calculate (It) given by

,

(It) =

L

(il[:li)')i = ~(Pl + P2 - P3 - P4)

(7.3Ia)

.=1

In a similar way (S:) = !(PI + P3 - P2 - P4)

(7.3lb)

To find the Pi'S. we must solve the four rate equations for the populations Pi (i = I to 4)

dp,

dt = PI WH + P2 W 24

+ P3W34 - P4(W41 + W,n + W 43)

+
(7.32<1)

dp3

dt = PI W 13 + P2 W23 + 1'4 W43 - P3(W3t + W32 + W34)

+(P'1-P3)W

•

(7.32b)

etc. Adding these two equations. gives d(P3

+ p<j)

dt

= PI (W14 + W13) + P2(W24 + W23) - P3(W31 + W32) -J,,,(W
(7.33)

In the steady state (where all dp./dt's vanish). (7.32) shows that as W is increased. 1'3 and Pot become progressively more nearly equal. In a similar manner. PI and 1'2 become progressively more nearly equal. We therefore set PI = 1>2 and P3 = 1'4, obtaining from (7.33) ~=~

+ W13 + WZ4 + W23) F =1'1W.U + W31 + W"2 + W32 G

W14 (

(7.34)

259

which defines F and G. Now. utilizing (7.12) and (7.18)

To calculate these transition rates we need several equations. From (5.281) 1

(7.35)

and so on for W13. W2". and W23. Since EPi P" :::: PI (FIG), we get

= I.

PI

= P2.

P3

= p".

J

Gmk(T)eXp[ - i(m - k)TdT)

willi

-~

and

(7.300)

hence

+~

Wkm = 2" _ h

Gm.(T) " (ml1i l (l)!k)(kl1i , (t + T)lm)

Taking Gmk(T) = Gmk(O)eXP(-jTI/Td

(7.36b)

PI = 2(1 + FIG)

W

Therefore

km

I (I-(FIG») PI - Poi = '2 1 + (FIG)

(7.37)

Utilizing PI = 1>2 and])3 = P'I> (7.31) becomes (7.38)

Utilizing (7.34) and (7.35) we can get an exact expression for F and G. However, in the case of a liquid. the high temperature approximation is valid even for electrons (perhaps liquid He is an exception!) so that Bij = I + (Ej - Ej)/kT

giving =

_I_{ 2kT

(7.39)

(7.40)

1- -) to

1+ -)

(7.41)

As we shall see, W42 = W31, so that .

(7.43)

and recalling that

B = -t(S- I+ + S+ r)(1 - 3 cos 2 8)

E = -i(S+ J+) sin 2

ee- 2i6

~~ (/6) ((1 -

3 cos' 8)'),.(+ -1[+5-1 -

+X- + Ir5+1 + -) (7.44)

to 1+ +) in which the I-spin flips up, W 42 is the transition from 1- -) to 1- +). also a transition in which the I*spin flips up. Finally W32 is the transition from 1+ -) to 1 - +) in which the S-spin flips down and the I-spin flips up. We shall define a new notation which makes these processes more explicit by defining U/l1 where M is the totnl chnnge in mr + ms in the tmnsition:

Ul(W) = W31(W)

k)2 T ;

For dipolar coupling, the 'H.1(t)'S represent the various tenns A. B, C, etc. from (3.7). Defining

Uo =

Examination of Fig.7.7 shows that W41 is llie transition from

+) in which both spins flip up. W31 is the uansition from

2Tc

+ (m

we get that

E

W"I(E4 I ) + W31(EJ - E I ) 2(W41 + W31 + W 42 + W32)

+ W 42(£4 - £2) + W32(£3 - E2)} 2(W"1 + W31 + W 4 2 + W 3 2)

1+

= Gmk(O) h2 I

Ao = "'IJ1sh 2/r 3

IG-F (I,) = (P, - p,) = '2 G + F

(I,)

we get from (5.297)

where «(1 - 3 cos 2 8)2)" .. means the average over the 411" solid angle of (I 3 cos 2 8)2. Thus «1_3cos2 8)2)"

•

=~J(I-3COS28)2sin8d8d¢=4/5 4.

(7.45)

Likewise (sin 2 8 cos 2 (J),,'/I" = 2/15

(7.46)

(sin" (J)o\?r:::: 8/15. In this manner we get Uo =

A~ -.!....

U2 =

h2 5"

(7.42)

Tc

11.2 10 1 +(ws -wr)2rt

A33

Tc

I +(wr +wS)2 Tt

(7.47) 261

The resulting polarization of the I-spins is thus, using (7.40), (7.41), and (7.47),

Now, in thennal equilibrium we have from Pj(T) '" PiO')B ij thai (7.55)

Pj(T) - ]>;(1') = Pi(T)(Bij - I) = Pi(Tkij

(I,)

Therefore, in (7.54) we recognize that since Pi (7.48)

';:!

Pi(T) S!'

t

W 4 1P\e14 S!' W 4 1[P4(T) - Pl(T>]

(7.56)

Substituting into (7.54) we gel Suppose

Iwsl »wl (as for S being an electron spin, in which case ws is negative)

and thai W~7";

«: I. Then

(I ) = ''"'5 (3/5 - 1/10) = ~ "W5 z 41.:1' (3/5 + 1110) 7 41.:1'

(7.50)

which for negative Ws is posltlve. Therefore, dipolar relaxation produces an Overhauser effect of the opposite sign to the conventional Overhauser effect since for dipolar coupling U2 »Uo. We have carried out the solution for the steady-state polarization of the 1· spins. If one is doing pulsed experiments, as is frequently the case for double resonance experiments or for two-dimensional Fourier transform experiments, the pulses disturb the 1-S spin system from thermal equilibrium, following which the thennal processes bring the system back to thermal equilibrium. One then speaks of a transient nuclear Overhauser effect. Thus, in general, one wishes to find the time dependence of the observables (I~(t) or (S~(t) for some sort of initial conditions. Solomon, in [7.7J, calculates the transient response and demonstrates what happens in a set of classic experiments. To do the calculation, one needs to solve for the time dependence of the Pi'S, starting with equations such as (7.32) with W = O. Thus we have

dp4

dt =PJWI4 +P2 W 24 +]J3 W 34 -P'I(W41 + W"2+ W"3)

(7.51)

- Pol -

[PI (T) - 1'4(T)j}

+ W",{)" - P4 - [p,(T) - p4(T)j} + W",{p3 -Pol - U'3(T) - 1',1 (T)]}

(7.49)

which is negative for electrons. On the other hand, if there were a strong Fenni contact tenn, so that Uo »UI or U2, as with the conventional Overhauscr effect,

( I ) = _ lIws z 4k1'

dP4 dt = W ll {PI

with similar equations for dpddt for i = I, 2, and 3. We saw in Chap. I that the fact that Wij oF Wj; is important in producing a thermal equilibrium population. That is the physical significance of (7.56). Clearly the family of equations represented by (7.54) describes a system which will relax to the thermal equilibrium populations p/T). We can then replace the Wij'S by the Um's, taking

Uo

W32 s:' W23 =

W42 s:' W2.1 = VI = W3l S! W I 3 W31 s:' W l 3 = U2

and introducing

Ui

(7.58a)

by Jhe relations

WI2 s:' W21 ==

ui = W34

s:' W"3

EI

cl" =

where

~E4

kT

relationships, one can now show Ihal d dt (Pl +!J2 -1'3 -1',,) = 2UI [(pol -]>2) + (P3 - PI)]

+ 2UO(p3 - 1") + 2U,(P4 - P'> - 2U I [PI (T) - P2(1') + P3(T) - PI (T)] - 2Uo[P,(T) - p,(T)]- 2U I [P,,(T) - p,(T)]

(7.52)

From (7.31) one can show that

Therefore (7.59) can be rewritten utilizing 10 and So to represent the thermal equilibrium values of (I~) and (S~) as

(7.53) = [10 _ (1,)](U0 + 2UI + U,) + [So - (5,)](U, - Uo) dt In a similar manner, we get

d(I,)

dp4 dt = W 41 (PI - P4) + W 42U>2 - p,,) + W 43 (1'3 - P4)

,,,

(7.59)

(7,60)

we get

+ W41PIC14 + W42P2C24 + W43P3C34

(7.58b)

U: differs from UI (7.47) by the subslitution of Ws for WI. Making use of these

Writing Wu = W<\lB I4 and taking BI4 = I +cH

(7.57)

(7,54)

(7,61,)

(7.61b)

'63

These two coupled linear differential equations have solutions consisling of real exponentials. They may be viewed as a nonnal modes problem with imaginary frequencies, i.e. real exponentials. There are thus two time constants. These equations show that as long as (5,) = 50 and (I,) = 10, both d(5,)/dt and d(I,)/dt remain zero. However, if one disturbs either spin system from thennal equilibrium, both spin systems will respond. Thus, if one makes ($,:) = 0 by applying a 7fn pulse to the 5-spins, not only will d(S,)/dt be differem from zero, but also d(l,)/dt will be nonzero. In general, then, when one applies pulses to a system of coupled spins, all spin populations will be found to respond as a result of the relaxation. We pose the detailed solutions of these rate equations as a homework problem.

~,

=1++)

I,

,, , WI':, ,,, , ~1=1-+)

Woo

,,, , ,,, IV4.J: : 11'.,. ,, ,, ,, t

, ,,, w,. ,,

~J=t+-)

~.=r--)

Fig. 7.3. The use or • rorbidden tnonsilion which flips oolh lile dN:tron and nucleus simultaneously, !Vea , to produce nuclear polilfiZlltioll. h is I181SUl'll«I thallransilions which ilwolve an e1edfOn sl>in-nil> only are lhe only sip;nilicallt lhermlll PI"OCes6e$

7.6 Polarization by Forbidden Transitions: The Solid Effect In order for the Overhauser effect to work, the nuclear relaxation process (such as W13, W24) cannot be allowed to short circuit relaxation in which both an electron and a nucleus flip, such as W23 or W!1. It is not always possible to meet those conditions. The condition which one can, in general, be quite sure will hold is that pure electron spin-flip processes (such as W 12 or W34) are much the fastest W;j's because electrons couple more strongly to tile lanice than do nuclei. Jeffries [7.8] and independently Abragam et al. (7.9) recognized that the so-called forbidden transitions were not strictly forbidden in many useful cases, and that one could use them to good advantage in achieving polarization. In fact, Erb et al. (7.10) independently discovered the effect experimemally. Using this scheme, Jeffries and his colleagues at Berkeley, and experimentalists at Saclay collaborating with Abragam. have obtained proton polarizations of over 70%, and have made a rich variety of applications. Invention of this technique was another major step forward. The phenomenon is often referred to as the solid effect. There are two possible forbidden transitions. They are shown in Figs. 7.8 and 7.9. The transition of Fig. 7.8 can be induced by an alternating field parallel to the static field when the simple isotrOpic electron-nuclear coupling of 0.1) is solved to the next higher order to include the effect of the tenn AUz 5 z + IySy ) in admixing 1+ -) with 1- +). The transitions of both Figs. 7.8 and 7.9 can be induced by alternating fields perpendicular to ]fo when the more general Hamiltonian, (7.6), is solved to adequate precision. For example, the dipole-dipole coupling between a nucleus a distrance r from the electron makes the transition matrix elements of Fig. 7.8 and 7.9 of order 'Yefilr 3 Ho times the matrix element of Fig.7.6 in which only an electron is flipped. The ratio also depends on the angle the static field makes with the axis connecting the nucleus and the electron. (The effect of the dipolar coupling is expressible more precisely in terms of the contribution of the dipoledipole coupling components AZ'Zl, Ay'y" and A,,'z')' 264

FIg. 7.'). A rorbidden transition 1Iltemalive to that or Fig. 7.8, which producl'3 Iloclear polarization or the opposite sign

Even though the transition probability Wen may be small, it is frequently possible to achieve strong enough alternating magnetic fields to make it larger than the thennal transition rates Wij in which a nucleus flips, so that the transition connected by Wen produces effective population equalization. Lei us analyze the case of Fig. 7.9 for which the transition between states I (-IPt = 1+ +)] and 4(IP4 = 1- -)) is saturated. Since we assume Ihe only thermal transitions of consequence are those shown, we immediately can write down 1'1 = P4

(7.62,) (7.62b) (7.62c) (7.63,)

(7.63b)

265

P3 :

~~B::,4",-J~_

(7.630)

2+8 12+ B 43

relative to the donor atom. The solution is a straightforward generaljzation of that of Sect. 7.3, using the N quantum numbers mi, the eigenvalues of fZi' ..E = '1ehHOmS +

Using (7.2Ib),

) 1 812 - B
(7 64) .

To appreciate the meaning of this expression we look .at l~e high temperature limit (evaluation of the expression for all temperatures IS given as a homework

Electron resonance occurs when Ams ""

± I, Ami"" 0 for all i. 1'he frequency

We

= "feHo

A;

+ ~ -;;m j

•

I 7enHO

2

(765)

2kT

.

so that the enhancement over the normal polarization is (7.66)

the full Overhauser effect.

7.7 Electron-Nuclear Double Resonance (ENDOR) A double resonance experiment of great historical importance was performed by George Feher [7.11]. He named it electron-nuclear double resonance (ENDOR). The purpose of this technique is to resolve otherwise unresolvable resonance lines. The concept involves in essence observing a nuclear resonance through its effect on an electron spin resonance. Fehu was studying the electron spin resonance of electrons bound to donor atoms in silicon. 5 % of the silicon nuclei, the isotope Si 29 with spin ~, possess a nuclear magnetic mome~1. As a result of the large radius of the orbit, the electron spin couples magneucally to many Si29 nuclei. Since the Si 29 nucleus can have two spin orientations, and since the Si29 hyperfine fields depend on where in the donor orbit the nucleus sits, there are many different local fields Si 29 nuclei can produce at the electron. The essential situation is shown by considering a Hamiltonian of the one electron interacting with N Si 29 spins. (For simplicity, to avoid the needed discussion of the statistical effect of an isotopic abundance, let us suppose all Si sites had Si 29 atoms.) We take it to be of the foml N

11. "" "fchHoSz + S·

L

A;Ii - "fnhHolz

(7.67)

i=1

where Ai is the hyperfine coupling of the ith nucleus (for simplicity we assume a simple Fermi contact form of coupling rather than the more general tensor fonn). The Ai'S are determined by the crystallographic location of the ith nucleus

(7.68)

We IS

problem).

(1,) :

I: AjmjmS - I: "fnhHOmj

(7.69)

+!

Since there are many possible values of Ai, and since each mi can be or (7.69) describes many frequencies, in fact virtually a continuum, rather than a set of resolved lines such as in the case of an electron interacting with only a single nucleus. For a sample consisting of many donors, there are 2 N various ways of assigning the mj's. The problem of resolving the hyperfine lines may be likened to that of a man with several telephones on his desk, all of which ring at the same time. If he tries to answer them all, he hears a jumble of conversations as all the callers speak to him at once. Of course his callers have no problem - they hear only one voice, though he hears several. Feher recognized lhal each Si29 nucleus experiences the hyperfine field of only one electron. Thus the nuclear resonances are sharp. Each Si 29 site gives rise to two resonance transitions, corresponding to whether the electron hyperfine field aids or opposes the applied magnetic field:

-i,

Wnj ""

"fnHo - AjmS

(7.70)

The nuclear transition frequencies of Si 29 nuclei located at each distinct crystallographic location relative to the donor atom give rise to a distinct pair of lines. For many sites, the Aj is sufficiently large that these lines are well resolved from the other transitions corresponding to other sites. Recognition that nuclear resonance would give resolved lines even though electron spin resonance did not was a great insight on Feher's part. But, however, the nuclear resonance signal would be weak since the donor concentration (_10 17 per cm 3) is typically so low. Feher conceived of the clever method of detecting the nuclear resonance 1;ly its effect on the electron spin resonance. Consider the situation in Fig. 7.10. We show 4 levels only, which represent the four energies we get if we hold all mi's fixed except for those of the jth nucleus:

L

E '" CrehHo +

Ajmj)ms + Apnpns - "fn1iJIOmj

(7.71)

i • j

Clearly we can define an effective static field acting on the electron of Heff = Ho +

A-kmj ji-j"fe

L

(7.72)

for this case. 266

267

'It,-I++l

Tn, , :,,'/', ill , '

..~ I

.,1' I... I ' I I

'¥l-l-+)

7.8 Bloembergen's Three-Level Maser

.;...

,"" ",""

,

, I

,I· I',

I"

I I t

I" I

'

p,

p,

'v,

'e)

..Ll..

(,)

1J>l '" (x/!

E' -E') (-;;:r-

J'1g. 7.10a-c. Ft~,'s scheme for observing nuclear rcsomlll(e by nlOniloring il.$ effcct on the electron spin resonance. flg\lrc 7.IOa shows lhe stales of (7.71) for the electron and the jlh nucleus, with the eleclron transition being shown between stales I llnd 2, and thermal relaxation pro«sscs such as in the Overhauser efrect. Figure 7.IOb shows the populations produced by saturating the electron transition. Figure 7.1Oe: shoWlii the effcct of an Ildiah2.tie pIlIJ!:IlI.ge through the resonant frequency of states 1 and 3, with lV~ turned off. When 'V~ is turned back on, it finds the electron transition bct"'een stales I and 2 momentarily ul1!laturated

We assume for concreteness that we have the same thennal relaxalion processes as with the Overhauser effect (Sect. 7.4), and saturate the same transition, The resulting populations are shown in Fig.7.10b. Suppose we momentarily switched off We' switched on the nuclear oscillator (Wn ) close to the transition frequency between states 1 and 3, and did an adiabatic passage with W n through the nuclear resonance line. The passage will exchange the populations between states 1 and 3, producing the populations shown in Fig. 7.IOc. If one then turns on We, initially the electron resonance is no longer fully saturated, so the elec· tron spin resonance has a changed signal height which decays as the transition saturates. Thus, the nuclear resonance produces an effect on the electron spin resonance. We have described a panicular method of doing ENDOR. There are many variations. The technique has been exceedingly important in mapping wave functions of a variety of paramagnetic centers, and thus is one of the principal ways one knows the structure of many important point imperfections in solids. (For an excellent review of the use of ESR and ENDOR to study structure of point imperfections, see [7.12].) It is of great importance for structural detenninations of biological molecules. For his discovery of ENDOR and applications, Feher was awarded the Buckley Prize in Solid State Physics of the American Physical Society.

A maser (microwave amplification by stimulated emission of radiation) works on me principle that irradiation of an absorption line associated with transilion between two energy levels will lead to a net emission of energy when the upper ~evel is more highly populated than the lower one. Townes (7.13]. and [7.14J, and Independently Prokhorov and Baso'll [7.15J, recognized that if such a situation could be achieved, it could be used as the basis of a new type of oscillmor. Townes achieved population inversion with ammonia gas by physically separating lhe molecules in the upper energy level from Ihe lower energy level by means of properly shaped electric fields, in essence a molecular beam apparatus. The frequency of the oscillator fell in the microwave region. Shonly Ihereafler, Bloembergen recognized Ihat one could produce the energy level inversion by use of the Overhauser-Pound principle of level repopulation through saturation. It required a system with three or more energy levels (Fig. 7.11).

,, , ,,

,pV

,, ,,] 11/ ,

J1

,J hIll

21

FIg. 7.1t. Bl«mbugtfl'll three-level maser. The thermal relaxation is shown by the dashed arrows, the induced transition by the twoheaded solid arrow. If the populations PI and PJ arc cquali:.:cd by saluration, Pl reaches a value which ClI.LJses lin inverted population either of 1)1 rell\live to P' or of p, relative to p:!, depending on lhe relative strenglh of lhermal rates bclwe<:n stales I lLnd 2 versus 2 and 3

If, for simplicity, we can neglect the thennal relaxation between states 2 and 3, then (7.73a)

PI = P3 P2=P I B I 2

PI = P' = P2=

so that 1

:;-;-''n2+ B12

Du 2+B 12

(7.73b) (7.74.)

(7.74b)

so that P2/P3 = B 12. Since

B I2 = e(E,-E,)/kT > 1 268

(7.75) 269

we have that

">1 P3

a~

making the upper level more highly populated than the lower, the condition for maser operation at the frequency (E 1 - ~)lf1. B/oembergen pointed out that the theoretical situation he described could be achieved with paramagnetic ions, as was shonly demonstrated [7.16]. Bfoembergetl's concept has been of enonnous value. Level inversion by pumping another transilion is the principle on which all lasers operate. The pumping may be done optically, as when a light source is used, or by the equivalent of pumping (which generates nonequilibrium populations between a pair of levels) in chemical reactions (chemical lasers) or electric discharges (electric discharge laser). Prokhorov and Basov [7.15] independently developed many of the same ideas.

7.9 The Problem of Sensitivity Sooner or later all resonators want to observe a resonance wh.ich is too weak to be seen. Resonances may be weak because the number of spins is low, for example since the isotopic species is rare, or the nuclei occupy special positions such as being surface atoms on a crystal or neighbors of an imperfection. The resonances may be weak because the nuclear ""( is small. The development of superconducting solenoids with their high magnetic fields has helped enormously in increasing sensitivity of NMR equipment because the quanta absorbed in resonance are thereby increased. Reduction of noise by narrow banding also helps. The practical problem used to be that a narrow bandwidth was achieved by using long integrating time constants, and those in tum meant it took a long time to sweep across the resonance line. By the time one was near the end of the sweep, the apparatus was likely to have drifted, giving distorted line shapes or spurious signals from drift in the apparatus baseline. The advent of the multichannel digital signal averager has dramatically changed all that by permitting cumulative addition of a number of rapid sweeps. If the appamtus drifts, one stops sweeping, readjusts the equipment, and continues sweeping. Moreover one sweeps only as long as is needed to achicve the desired signal-to-noise ratio. TIle signal accumulated is proportional to the num~ ber of sweeps N. Since the noise is random and both positive and negative, its amplitude increascs as .JFi, giving an overall increase in signal-to-noise ratio proportional to NI.JFi = .JFi. As a practical maller, this means one must increase the averaging time fourfold to gain a mere factor of 2 in signal-to-noise. This increase may not matter if it means going from I min to 4 min, but if one is already averaging for 270

I h, going to 4 h is a hanl way to improve signal·to-noise! These facts illustrate that when one is using averaging there is a big premium on setting variables such as HI> modulation amplitude, sweep excursion, etc., to the optimum. A corollary is when looking for an unknown, be willing to set the equipment to give ~ maximum signal even though you may thereby distort the resonance line (for example, from using too large a modulation, or from panially saturating). After the resonance has becn found you can focus attention on it with parameters adjusted to avoid distonion.

7.10 Cross-Relaxation Double Resonance No mailer what one does to improve one's apparatus, one eventually reaches the limit of the current state of experimental an. What then if the signal is still too weak to see? We tum now to the use of double resonance, assuming there are two resonances which can be excited, one the "weak" one too difficult to see directly, the other a "strong" one observable by conventional means. For example, the weak one might result from a low abundance species with spin S, whereas the strong one might result from an abundant species with spin I. (For simplicity, we use the term "rare" to imply the species whose resonance is weak. Since a low ""( could also make the resonance weak, "low ""(" can be substituted for "rare" in most cases.) If an H I is applied at resonance to the rare species, those nuclei absorb energy and their spin temperature rises. If the rare and abundant spins could exchange energy, the abundant spins would thereby get holter, and their signal amplitude would diminish. If the abundant species is thermally isolated from the outside world (i.e., has a long TI), this temperature rise can be made quite large and thus readily observable merely by silling on the weak resonance long enough. By going to low temperatures, TI of many systems can be made exceedingly long. Thus the crucial question becomes how well can the two species exchange energy. The mixing of the Iwo spin systems was sludied by Abragam and Proctor [7.17]. They employed the Li 7 and FI9 resonances in LiF. Their experiments were part of their studies of the fundamentals of spin temperature and of adiabatic demagnetization. Working in a field of several thousand gauss to observe the resonances, they prepared the system in some nonequilibrium state in the strong field (for example, they inverted the F I9 magnctiz..'ltion), removed the sample to a lower static field, allowed the Lj7 and FI9 spins to mix, then returned the sample to the strong field for inspection of the Li 7 and F19 resonances. In this way they found that in a field of 75 Gauss the spins come to a common spin temperature in a mixing time of 6s, that the time was unobservably shon at 30Gauss, and longer than the TI 's (several minutes) above lOOGauss.

271

A detailed study of cross-relaxation was made by Bloembergen et al. [7.18] and by Pershall [7.19] for both electron spin systems and nuclear spin systems. They show that the crucial problem is the failure of the Zeeman energy to match when two different nuclear species undergo mutual spin Aips. The mismatch in Zeeman energy must be made up by the dipolar coupling between the spins. The simplest sort of process arises when two nuclei have nearly the same 'Y. For example, HI and F I9 have 'Y's which differ by 5%. Owing to the existence of tenns such as S+ I- or S- 1+ in the dipolar coupling [the B tenn of the dipolar Hamiltonian of (3.7)], the dipolar coupling couples states in which the proton spin flips up (down) and the fluorine spin flips down (up). If we could consider the 'Y's as being identical, we would then have a situation such as shown in Fig. 7.12. Because the individual energy levels match, the initial state of the two spins, indicated by the two x's, has the same energy as the final state indicated by the two o's. A system started in the x state will undergo a transition to the o state by means of the S+ I- part of the B term in the dipolar coupling. For HI and F t9 , the fact that the 'Y's differ by 5 % means that in a strong magnetic field such as 10 kGauss, the Zeeman energy difference between the x and the o states would correspond to the energy of FI9 or HI in a field of 500Gauss. Such an energy mismatch would prevent the transition unless there were some other energy reservoir whose energy could change to make up the difference. A possible candidate is the dipolar energy reservoir. For a pair of spins, it has a typical value of 'Yl'YSh2/r3, which, expressed in units of magnetic field, is only a few gauss for reasonable values of r. Thus, it cannot make up for an energy mismatch of 500Gauss. If, however, the static field were much lower, the mismatch would be correspondingly reduced, and the mutual flips might become possible! Returning to LiF in a field of IOkGauss, the Li 7 resonance occurs at 16.547 MHz and the Ft9 resonance occurs at 40.055 MHz, a ratio of 2.420. Clearly the mismatch here is even worse than for H t and F t 9. What Bloembergen and his colleagues recognized [7.18,19] was thai 2.4 is close to 2.0, hence a process in which two U 7 nuclei flip up and one F I9 nucleus flips down comes much closer to satisfying energy conservation than a process in which only one Li7 flips. Now, Li 7 has a spin of ~' but for the sake of argument we are going to pretend it has a spin since the explanation is then simpler. To estimate the rate, one must utilize a fonnula for transition probabilities such as (2.182) (with

!'

Ms

=

M l = -1/2

-1/2

M s = +1/2 Spccies 5

272

Specic" J

]<'ig.7.12. A mulual spin-IIiI' from thc (x) lo lhe (D) slale cxchangcs energy belween the [-spins and the S-spins. Shown verlically are lhe allowed energy levels. For lhis cxample thc cnergy spacing is assumed to l>c lhe same for lhe two spc<.:ies, and bolh nudei have spin

t

0). One therefore looks for a matrix element which connects the initial state (two Li 7 spins up, one FI9 spin down) to the final state (two Li7 spins down, one FI9 spin up). If we denote the Li 7 by S and the FI9 by I, and use subscripts i, j, k to distinguish individual nuclei, a matrix element of the dipolar !enn B. (aIS; Ib), between states a and b flips only olle U 7 spin. To describe two spin flips we must go to second-order perturbation to get the effective matrix element (al?ierrlb) joining the initial and final states:

W =

t:

(al1iofflb) =

L: (al1i~I'~'~1idlb) °

b

°

where one matrix element might involve

st I":

(7.77)

st IZj and the other st Ii. There

are other possibilities, for example combined with SziSt The energy difference Eb - E c depends on what the initial and intennediate states are, but is of the order of magnitude of the Zeeman energies involved. In general, if one considers a process Wilh I1.F fluorine spin flips down and n Li Li spin flips up, with !lwLi or flWF energy change per Li or per F spin, the energy mismatch L!E between initial and final states is (7.78) (7.79)

nLi = 2

dE nwF =0.17

but for llr =

2

Hl.i =

5

11E = 0.066 flWF

Thus, the more spins we allow to flip simultaneously, the closer we can make the energies of initial and final states match. However, more spins flipping requires more dipolar terms acting simultaneously and thus requires third- or higher-order perturbation expressions. The first-order matrix elements are of order HI., where HI. is some sort of dipolar field of one nucleus at a neighbor. The secondorder expression (7.77) is then of order Hl/Ho, where Ho is the static field. Each increase in order of perturbation contributes a factor HLlHo. Thus, since H L
.'ig.7.0. Dah. or Pershiln for the cr0S5relaxation Lime 121 ror l,i7_1~19 in Li(~ versus static magnetic field for three crystal orientations (sec 17.191)

T1\ YS. magnetic rield in LiF

l /

10

When one has a single crystal in which some nuclei have a quadrupole splilling and others do not, there are magnetic fields other than zero in which the energy level splittings of the two systems become equaL Cross-relaxation is then r.apid. Edmonds {7.26] has reviewed nuclear quadrupole double resonance.

/ J 7.11 The Bloembergen-Sorokin Experiment

••< B ••

l,,

I

0.1

/ ,t

,I /

o.Oll-~-l('-7:-~--~ o

SO

Ma9"etil: field IGallssl

'"

Clearly the use of cross-relaxation as part of double resonance leads in a natural way to cycling the static field between low values for cross-relaxation and high values for observation. Anderson [7.20). working with Redfield, combined field cycling with Ihe application of an audio-frequency magnetic field applied while the static field was zero to heal the spins to plot out the zero-field absorption characteristics of a spin system. Thus they used field cycling to give them the sensitivity of resonance in a strong field to monitor the effects they produced in zero field. Redfield [7.21], Fernelius [7.22], Slusher and Hahn [7.23], Minier [7.24], and others utilized field cycling to observe quadrupole splittings of nuclei near to foreign atoms. If one has a sample which is a powder (e.g. a metal sample where one employs a powder to overcome problems of the skin effect), an advantage of demagnetizing to zero field is that the quadrupole splittings are then identical in all the individual crystallites. There is therefore no longer any powder broadening of the NMR transitions [7.21-24]. To observe very large quadrupole splittings in powders, there is a problem of coupling the energy absorbed by the quadruple transition into the rest of the spin system. It is necessary then to adjust the amplitude of the "audio" frequency alternating magnetic field according to principles of Hahn and Hartmann discussed later in this chapter. Weitekamp et al. [7.25} have employed field cycling to zero field together with Fourier transfom NMR to simplify dipolar spectra of powdered samples. 274

An important advance in cross-relaxation double resonance was discovered by Bloembergen aod Sorokin in their studies of the cesium halides (7.27J, especially CsBr. In fact, they found vinually all the essential elements of what has come to be known as the Hartmann-Hahn method which we discuss later, though it seems evident that the full significance of the results of Bloembergen and Sorokin was not appreciated owing to the fact that they discussed the problem entirely in the Csl33 rotating frame instead of using the doubly JOtating frame which we take up in the next seclion. We give a brief account here of what they found, then return in the next section to discuss their results a bit further. using the concepts of Hannumn and Hahn [7.28] and of Lurie and S/ie/lter [7.29] concerning spin temperature in the doubly rotating frame. We shall discuss two important discoveries made by Bloembergen and Sorokin. 1be first has to do with spin-locking (Sect.6.6.1). We saw in Chap.6 that if the Ht is sufficiently strong (6.84) one must use Redfield's ideas of spin temperature in the rotating reference frame. Then, if one puts the magnetization along the HI in the rotating frame, it will persist for rimes as long as those for which the spin system may be considered to be thermally isolated from the surroundings. In their sample of CsBr, Bloembergen. and Sorokin found that the Cs t33 spin-Ianice relaxation time was lo-2Dmin at room temperature, ten thousand times longer than the Dr Tt 's. Thus they expected to be able to maintain (i.e. "lock") the Cs lJJ magnetization lined up along the HI)es for many minutes. Yet, when they auempted to observe the Cs l33 while sweeping through the line using an HI for which Redfield theory should apply. they found no signal even when they took only 15s to go through the line. They found that the explanation was that the short Br T t causes the Cs·Sr dipolar coupling to fluctuate in time, thereby providing a relaxation mechanism for Tie Eq. (6.9Ib). Since TIe specifies how long the spins can be locked. the Br T I thereby limits the Cs 133 spin-locking. A simple view of their discovery is that the Br T} contributes to the relaxation rates Tb and Tc of (6.87) and (6.9tb). The short relaxation time of the Br results from the modulation of the Br electric quadrupole interaction by means of lallice vibrations. Thus. we see that if one nuclear species has a dipole coupling to a second species with a fast spin-lattice relaxation time, Tie of the first species will be shortened, and one can detect the existence of the second species by comparing, for the first species, Tl with T( e' 275

Their second discovery was thaI if they utilized two oscillators. one tuned to the es l33 resonance, the other tuned near the 8r79 or BrS l resonance. they could utilize the rapid spin-lattice relaxation time of the bromine to polarize the es 133 . This effect they labeled the transverse Overhauser effect. In their experiment, they applied a strong rf field H de. tuned exactly to the Cs resonance. and a rather weak rf field HI )Br tuned to a frequency'; which differed from the Br resonant frequency VBr by an amount given by (7.80) As we explain rigorously in the next section, this condition can be understood by considering the effective fields Heff)Br and Heff)es in their respective rolating frames. In the bromine frame. HeffBr=' ) k

21r(v - VBr) 11l,

'H) IBr

+~

(7.81)

';;t

k2'IT(v - vBr)/rBr

The problem of matching the Zeeman splittings of two different species could be solvro if by magic one could apply one magnetic field to the I-spins, a second magnetic field to the S-spins. How can one do this to spins which are neighbors on the atomic scale? The magical solution was found by the Wizard of Resonance. Erwin Hahn and demonstrated by the Wizard and his Sorcerer's Apprentice Sven HartmiJnll [7.28]. Hahn recognized that alternating magnetic fields have a negligible effect on nuclei unless the frequency of altemntion is close to the precession frequency. Thus if he applied two allernating fields HdJ and Hds at frequencies WJ and ws, respectively, the frequencies being chosen to satisfy the respective resonance conditions, ~=n~

Since HdBr is small, Heff)Br

7.12 Hahn's Ingenious Concept

(7.82)

Likewise.

since the Cs is tuned exactly to resonance. The Bloembergen-Sorokin condition is a special case of the equation (7.83) which has come to be known as the Hartmann-Hahn condition. As we shall see, when this condition is satisfied, the two spin systems can come to a common spin temperature in the doubly rotaling frame. Equation (7.83) is a condition for rapid cross-relaxation in the doubly I'OIating frame. Since Bloembergen and Sorokin analyzed their experiment by means of the single rotating frame of the es 133 , they did not have spin systems which came to a common temperature in that frame of reference. They discussed the problem in tenos of an Ovemauser effect (indeed they called it the transverse Ovemauser effect). They showed that when (7.80) was satisfied. the es magnelizarion Me. and the Br magnetization M Br satisfied the relation

~=U~

(7~)

HIlJ would have negligible effect on the S-spins and Ht)s would have negligible effect on the I-spins. Each species could then be viewed in its own rotating reference frame. Figure 7.14 shows the two rotating reference frames. Note that the z-axis is the same in both frames. It is in fact the direction of Ho in the laboratory. In Fig. 7.14 the magnetization vectors MJ and M s are also shown. Note that in a general situation they do not lie along either the z-axis or the respective HI'S, though they might if special ways of preparing the syslem were used. For the case where an M is not parallel 10 its HI, it will precess about the HI in the rotating frame with precession frequency [}J or [}s given by ~=nHI)J

%=~HI)s

(7.8~

These equations suggesl immediately that we can make the precession frequencies match by adjusting the ratio of the H)'s to satisfy the relation HI), is ~~=W

n~

(7.84) In Sect. 7.16 we shall see how we can get their result in a simple manner by combining Redfield's results on the calculation of Meq, (6.9Ia), with the idea of spin temperature in the doubly rotating frame.

y,

,/' --",n/ , M,

\

...

__/

,

I

),"/

/l,p

flg. 7.14. The I'Olaling reference frames ahow lhe axes r, and rs along lhe respeelive rf fields Ill), and III )$' Note lhill the z-axis is in common

276 277

This condition is known as the Hahn condition. It corresponds to making the Zeeman splittings of the I-spins quantized along Hdr in the I-spin rotating frame equal to the Zeeman splitting of the S-spins quantized along H1)s in the S-spin rotating frame. Satisfying the Hahn condilion matches the Zeeman spJiuings, but in reference systems which are exotic to say the least Our real objective in producing the matching is to permit the two spin systems 10 couple. How can we use the exotic matching 10 couple the systems? We know that the spin systems are coupled via the dipolar interaction. For two spins of unlike species that portion which broadens the line is the A term of the dipolar coupling 1idA 1idA = 'Yr,}S h2IzSz(l - 3 cos 2 8rs ) T rs

(7.88)

Since the z-axis In the rotating frame is the same as in the laboratory, this coupling is unaffected by the transformation of spin variables to the rotating frame. (Note: Sometimes confusion arises as to what variables are transformed. In quantum mechanics one can formally introduce operators which transform only I, only 5, or only r, or any combination. Since r rs and 8r s are unaffected by rotations of spatial coordinates about the z-axis. (7.88) is unaffected whether or not r is transformed. As we show below, the usual approach transforms only I and 5 in order to remove the time dependence of the HI couplings to the respective spins.) Taking the classical view, if Ms is precessing around HI)S in the S·spin rotating frame, M~s will be oscillating sinusoidally at ils = 'YsHt)s. This produces, via 'J-idA (7.88), a time-dependent coupling to M:r. But M:r is the component of M r transverse to HI)r, hence the coupling produces the same effect as applying an alternating field along the z-direction of frequency ils. When the driving frequency as matches the I-spin resonance frequency il r, the S-spins cause the I-spins to absorb energy, and Mr to nutate away from the direction of HI) [. We have described two spin systems which are coupled. We know with coupled systems that when the natural frequencies coincide, resonant transfer of energy results and we need to worry about the back reaction. Thus for two coupled pendulums, one al rest initially, the other set in motion, after a while the first reaches a maximum amplitude, with the second one at rest. Then the energy exchange reverses, the pendulum which was driven now drives, the pendulum which drove now is driven. If we had only a pair of spins, a similar situation would occur. However, typically there are many coupled I-spins, many S-spins, perhaps coupled, perhaps not, but in different neighbor environments in any case. For such a case it becomes useful to assign a temperature 8/ to the I -spins in their rotating frame and a temperature 85 to the S-spins in their rotating frame. Under these circumstances, M r always poinls along HI)r and Ms always points along Ht)s, but 278

as energy is exchanged, the sizes of M r and Ms change. In equilibrium 8r = 85 , Causing 85 to increase produces a heating of 8r. We see, therefore, that there is a way of coupling two different spin systems so that they can exchange energy. We need now 10 do three things: l. 2. 3.

set the qualitative arguments on a firm quantum mechanical basis; describe the experimental steps involved in putting Hahn's idea to use; analyze the expected results.

7.13 The Quantum Description We star! by writing the Hamiltonian of the system in the laboratory frame. 'J-i = 'J-izr(t) + 'HZS(t) + (1{d)r / + ('Hd)SS + ('J-iJ)rs

(7.89)

where 1{zr(t) is the Zeeman energy of the I-spins. It includes both a static interaction with field Hok and a time·dependent interaction with the two alternating fields. It is most convenient to consider that rotating fields have been applied, instead of linearly polarized alternating fields. We have, then, 1{zr(t) = - "fr'il· [kHo + i(HI)r cos w:rt + j(Hlh sin W:1t

+ i(Hl)S cos w:st + j(Hlls sin w:st]

(7.90)

where W:1 and Wzs may be positive or negative, 10 represent either sense of rotation, and where

1= I:Ij

(7.90,)

j

is the total spin vector of the I-spins. The terms ('J-id)rs, etc., represent the magnetic dipolar coupling of the Ispins with the S-spins, and so forth. We now wish to transform to a rotating reference system. In so doing, we are following Redfield as discussed in Chapter 6. However, our problem is somewhat different from his since we have two rotating fields. We therefore wish to transfonn in such a way that we view the I-spins and S-spins in Iheir respective reference frames. The transformation is readily accomplished by introducing the unitary operator T defined as T = exp (iw~rIzt)exp (iw zs5zt)

I, = I:I,j ;'

where

(7.91)

S, = I:S,k

(7.92)

k

are the total z-components of angular momentum of the two spin species. We define a new wave function 1/;/ by the equation

(7.93,) 279

then substituting T-1ljJ' for

,po SchrOdillger's equation becomes

_!!. atjJ' ::; 'H'.// i

at

or

(7.93b)

where H' is a lransfonned Hamihonian. Explicit evaluation of 'H' using the techniques of Chapter 2 gives

'H!

= - 1'11/[(Ho + W:lhl)lz + (H ')1 IzJ

-1's1J[(Ho +w,shs)Sz + (H,)sSzl + ~Il + ~lS +~ss + lime-dependem terms we ignore . (7.94) The lenns 1t~11' elC., represent thai pan of the dipolar coupling 1tdl1 that commutes with the Zeeman inleraclion between the spins and the static laboratory field Ho. These terms are usually called the "secular part" of the dipolar interaclion. We write them OUI explicitly below. The lime-dependent terms are of two sons. One variety arises from the nonsecular parts of the dipolar coupling. They oscillate at frequencies Wl1. wzS. or w::/ ±wzs, The second sort arises from couplings of the [-spins to (HI)s and the S-spins to (Bd,- These oscillate al a frequency (wz/-wzS). Since wz/' WzS. and (WzI ±wzS) are all far from any of the energy level spacings in the rotating frame, they can be neglected. One must remember, however, that it is conceivable that should there be a quadrupolar interaction added, and should the two nuclei have similar ,'s (as do, for example, Cu63 and CuM), the frequency (Wzl - wzS) might, in fact, be close to a possible transition. If one chooses (7.95) where Hoo is a particular value of Ihe stalic field Ho around which we laler wish to make variations, then the two nuclei are exactly al resonance. Note that the two w's are negative if the 1"S are positive, representing the fact that nuclei of positive, rotate in the "negative" sense about Ho. Making use of (7.95), defining ho by Ho - Hoo '" ho, and neglecting the rime-dependent lenns of 1t', we have

1£' '" -1'rh[hoI: + (Ht),I:z] -1's1i[hoS: + (HdsS:z] +'H?t11

+'H~'S+1-l?tSS

and similarly for 'H~SS' To these may be added the pseudo-dipolar and pseudoexchange couplings when their size is large enough to be imporlanl. II is the tenn 1£3,5 which gives rise to the effects observed by Bloembergen and S.orokin [7.27] in their studies of CsBr. They found that the rapid bromine spin-lattice relaxation could communicate itself to the Cs nuclei through this lenn when the Cs nuclei were quantized along their own HI. We can view the various tenns of (7.96) as energy reservoirs of Zeeman or dipolar energy. Sioce the various tenns do not commute, they can exchange energy. Such processes may be tenned cross-relaxation in lhe double-rolating reference frame. The rate of cross-relaxation will depend on how the energy levels of the different tenns match, on the hcat capacities, and on the strength of coupling as measured by the failure of tenns to commute with one another. Thus, we note that the Zeeman tenn of the I-spins 1£Z! commutes with the Zeeman energy of the S-spins 'H.zs. However, as long as (Hlh /: 0, 1iZ! does not commU{e with either ~11 or ~,s' and we can transfer energy between 1izl and or Moreover, 1t~,S provides a coupling mechanism to transfer energy between 1tz! and 1tzs [provided (H tls /: OJ. All these remarks lead one, following RedM/d, to assume that if one waits long enough, the various pans of (7.96) will come 10 an equilibrium in which the system can be described by a common temperature 8. For some purposes, it may also be possible and convenient to assume that various parts may comc to common temperatures faster than the whole system achieves a single temperature. This is the viewpoint Lurie (7.29] adopts to calculate some cross-relaxation times. We therefore, make the assumption that when the system has achieved a common temperature it is described by a density matrix I! given as

1i311

exp(-l£'lk8) 'Ii'lk9)}

,= T, {e,p(

E = T, {,'Ii'} = _

(7.97)

(7.99)

where 1£' is given in (7.96). In lenns of I! we can calculate the average energy E and the average magnetization vector (M r) in Ihe high temperature approximation

(7.96)

It is convenient to define the Zeeman energies l£zl and 'H.zs by the equations

1i3,s'

C,[(H j )7 +

h5 + HCl + Cs[{Hj)~ + hij] 9

(M r ) = T, {e1r H } = Cr(~"'>r

(7.100a)

(7.100b)

where C, and Cs are the Curie constants given in tenns of the number of J. or S-spins per unit volume N J or Ns, and Boltzmann's constant k by (7.98)

C _ 1'yh2I(l + I)N , J 3k and where

280

etc.

(7.101)

HE is defined by the equation 28'

C,H1. • 0 • 9 = Tr {e( n dJ1 + 1tdlS + ftJ,SS) .0

.0

7.14 The Mixing Cycle and Its Equations }

(7.102)

.

Evaluating the trace gives 2

I

2

2

1'Y}NS S(S+l)

2

HL='j(Ll H)lI+(Ll H)/S+'j ,;N/I(l+I) (Ll H)ss

(7.103)

where (/1 2 H)OI!J is the contribution (in Gauss) ?f the. ,a-spins to the ,second moment of the a-spin resonance line. H L has the dimenSions of a magnetic field. Although we call it the "'ocal field", it should not be confused with the Lorentz local field. Actually, HL is introduced simply to enable us to factor Ct out of various equations. The fact thai the dipolar energy is (-Ct H[l8) makes it appear superficially thai we have taken into account only the I-spins in calculating the dipolar energy. However, thai such is nol lhe case is seen by examining (7.102) and (7.103) which exhibit explicitly the dipolar contribution of the S species 10 the expression for HL. C/He measures the tolal dipolar contribution 10 the spin specific heat. Note that although the term "local field" sounds vague, H L can in fact be calculated exactly and is to be considered throughout as a precisely predicted quantity. The only exception to the statement is found when pseudodipolar coupling becomes prominent as in higher atomic number elements. In that case one would need to know the magnitude of the pseudo-dipolar coupling tenns to make quantitative predictions. We can observe (MJ) from our oscilloscope photographs of the initial height of the free induction decay following tum-off of (HI)I' One funher expression is needed. It is the expression for the magnetization MI found after the demagnetization of I's. That is, suppose (HI)s = (H III = 0, and that M I = kMJO where (7.104) is the thennal equilibrium magnetization of the I -spins at the lanice temperature 9,_ With ho:> HL' we switch on (Hdl' and slowly reduce ho to zero. We then end up, according to (7.100) with (7.105) Note that if we were to change (H')I slowly, (Mf) would follow (HI)/, in accord with (7.105).

202

There are two general ways of doing double resonance. The first was proposed by Ha"rtmann and Halm [7.28]. The second, which is a variant of the first, was demonstrated by Lurie (7.29}. As will become apparent, the two different techniques have simple analogies in thermodynamics. Consider two bodies connected by a rod to provide thennal contact. One body, of small heat capacity, represents the low abundance S-spins; the other, of large heat capacity, represents the I-spins. Hartmann and Hahn's experiments are analogous to heating the large object by holding the small one at constant elevated temperature. The rate of heating depends on the thennal conductivity of the rod and the heat capacity of the large object (the I-spin system), but is independent of the heat capacity of the small object (the S-spin system) since we never let its temperature change. A theoretical prediction of the rate of heating of the large object would require knowledge of its heat capacity and of the thermal conductivity of the rod. In resonance language, that means we must calculate a cross-relaxation time. This cannot be done exactly. Lurie's experiment is analogous to breaking the thermal contact between the rod and the small object, heating the small object to a known temperature, disconnecting the heater, and reconnecting the rod. After a sufficiently long time, the entire system of large object, small object, and rod, comes to a common temperature. Since the S system has a relatively small heat capacity, the final temperature is not much different from the initiallemperature of the large object. However, we can repeat the cycle. In fact, if we do the thennal mixing N times, the heating of the I system is as great as it would be for a single mixing with an S system whose heat capacity is N times larger than it actually is. Since N may be made very large, a significant effect can be achieved even when the S-spins have a very small relative heat capacity. Calculation of the temperature rise requires knowledge only of the heat capacities of the pans. It is nOI even necessary for the heat capacity of the rod to be small since we can easily include its effect. Calculation of the heat capacities of the spin systems is simple and can be done exactly. We therefore have a simple. exact theory to compare with experiment. As Hahn and Hartmann's analysis shows, the effective thennal conductivity of the rod depends on the size of lhe two rotating fields. The Hahn condition provides the fastest mixing or largest thennal conductivity. The heat capacity of the spin system is detemlined in large measure by the strength of the HI '5. We can Iherefore vary the heat capacities experimentally although we must remember that when the HI ratio does not satisfy the Hahn condition, it may take a longer time for a unifonn temperature to be reached. The dipolar coupling between the two different species provides the thennal contact or "rod". As we have remarked, we can easily calculate its heat capacity. Likewise, there is a contribution to the heat capacity from the dipolar coupling of the I-spins among themselves and 283

•

!

the S-spins among themselves. All these effects can be rigorously and simply included. In the process, Lurie demonstrated that it is nOl necessary for the HI's to be large compared 10 the local fields and further demonstrated the coupling in cases where (H»/ has been turned to zero. We now tum to an analysis of the Lurie experiment. We shall assume throughout that spin-lattice relaxation can be neglected during the times of the experiment. Spin-lattice processes can be included readily, but one must ~ careful in so doing to include the sort of transverse Overhauser effects descnbed by Bloembergen and Sorokjn [7.27]. We begin by the demagnetization process. This brings (M/) down along the x/-axis, ils magnitude being given by (1.105). Let us call this magnetization (M/}j. During this process, since (Ht)S is zero, 11.zs commutes exactly w.ith the rest of the Hamiltonian. So, likewise, does 'YshSz. Therefore, (Ms) remains unaffected, and points along the static laboratory field Ho. The rest of 11./ is at a common temperature 8j which we can compute from (7.l00b) and (7.105): 0; = C/(H,)//(M/);

(1.106)

This temperature is, of course, very much lower than the lattice temperature 8/. We now tum on (HI)s suddenly. In such a rapid process, the state of the system does not change. The dipolar energy and the Zeeman energy of the I-spins are therefore unchanged. The S-spin Zeeman energy E s (1.101)

since (MS) and (Heff)S are perpendicular. The total energy of the system Ei is therefore

Eo = _ CJ!(H,)j + Hl1

8j

I

(1.108)

After a sufficiently long time, the S-spins Zeeman energy comes into thenna! equilibrium with the rest of the system at a common final temperature 8r· The final energy E r is then

_ CJ!(H,)j+H1J+Cs(H,)~ Er- 8r

(1.109)

But since the total system is isolated and its Hamiltonian independent of lime, its energy cannot change. Therefore, Ej=Er

(1.110)

This gives us that

8j 8r

_

--

where 284

C,(HI)l + Hl] 2 =-C,(H1)1+HlJ+Cs(Ht)s 1+£

(1.111)

(1.112)

£=

In the process the magnitude of (M,) drops from its initial value (M/)i to a final value (M/lr which, in view of Curie's law, is (M/)f/(M/); = 1/(1

+ e)

(1.113)

We now suddenly turn off (Ht)s. Once again the system immediately after the change has the same wave funclion as it did jusl before. The expectation value of 'Hz, and of the dipolar energies is thus unchanged, but that of 'Hzs is zero since (Heff)S = O. The tOlal energy E} is therefore

CJ!(H,)j + HlJ

(1.114)

Of

Immediately afler the tum-off of (H»s, (Ms) is nonzero. Therefore, we see that we do not have thennal equilibrium. If we wait for a sufficiently long time, the entire system will come to a common temperature 8n" In the process, (Ms) decays 10 zero. This is an irreversible decay. We shall in fact calculate the entropy increase below. When the system has reached the final temperature 8rr, the energy is Eff. Using (7.100a). we have

Ejr=-CJ!(H,)j+HlvOff

.

(1.115)

But Eff = E} since the spin system is isolated from the outside world and has a Hamiltonian independent of time. Therefore, using (7.114) and (7.115) (1.116)

8r =8ff .

Using Curie's law we see thai following the lurn-off of (Hils, (M/) does nOl change. For one complete on-off cycle, therefore, we can argue that (M,) is reduced by the factor 1/(1 + E). We can repeat the argument for another on-off cycle. The magnetization M/(N) after N cycles is thus given in tenns of its value M/(O) prior to the first cycle by

M/(N)/M/(O) = [1/(1 +e)IN When

£« I, as

(1.111)

in our experimenls, we can write this as

N!J(NVM(O) = e- N < where £ is given in (7.112). Double resonance is possible even when (H 1)/ «HL. Experimentally we accomplish this by perfonning adiabatic reduction of (HI)' after M, has been brought along (H I)' in the rotating frame. After (HI), ::::::0, /11, ::::::0 from Curie's law, however, we have retained the order in the I-spin system, the order now 285

being with respect to the local field [7.30]. (Hj)S is now cyc~ed o~ ~nd off N times. After the Nth cycle, (Hlh is adiabatically returned to Its onglOal val.ue. The resulling M[ is then observed by rapidly turning off (Ht)/ and observmg the free induction decay. TIle analysis for the case when (Ht h «. H L is essentially the same as given above; however, now, the tenn C[(H1)iI8 no longer appears in (7.108) and (7.109), and E: reduces to E:

=C5(HI)~/C/Hl

(7.118) 7

Lurje studied Li metal for which the 93 % abundant isotope Li gives the strong or I-spin resonance; the 7 % abundant isotope Li 6 is the weak or S-spin system. To observe the effect of the double resonance, he measured the amplitude of the Li 7 free induction decay afler N mixing cycles. Figure 7.15 displays Lurie's data demonstrating the effect of N on the destruction of the Li 7 signal. The solid line has no adjustable parameters apan from the N = 0 intercept.

60

.. '.... = '",," I f1/StC

50

o 1. . - t",= 1001/Stc

\ '1.~.

• • • • •

•

F1g.7.J6. In(Ah) versus (lldl II.l 1.5K (or N ..... 22, ,011. low 1,2, and IOms, (lId7 2.30G. For this value or (11.)7, (111)' =: S.tG salisfies the IIlI.hn condilion. The solid line is cakulated (rom (7.112) and

= =

=

o '.,.. - '"," 2mst>c .~

•

(7.117)

a

\. a

.'

\" 10023456

7 II U·1I 1 (gUlI$J)

')

10

Figure 7.16 shows how the spin temperature idea works even when the Hahn condition is nOl perfectly satisfied. Note that the solid line has no adjustable parameters. The deviations for larger H I )6 and shoner ton and tolr arise because there is not enough time for a common spin temperature to be established. In this example a very large destruction is produced. Alternatively, one can conclude that a much smaller fraction of Li 6 nuclei could produce an observable destruction. In this case one begins to worry about the long distance that energy must diffuse from the hot Li 6 spins to couple to the distant Li 7 's. This problem has been investigated experimentally by several authors [7.23.31.32].

7.15 Energy and Entropy

10 0

40

SO

120

160

200

Numbtf Qll'lIlst$ (N)

t·lg. 7.15. Lurit's experimentsl dll.ta showing In(M7) versus N at 1.5K. (l1d7 =:= 2..14~, (1I 1 )e = 504 G, nearly satisfying lhe lIahn tondillon. 4. =: tow =: 4 ms. The _lid IlIIe IS calculll.l«l usirl& (7.117) II.nd (7.118)

286

It is interesting to follow the changes in energy and entropy of the spins that take place during the double resonance. The essence of the experiment is the heating of the I-spins brought about by the contact with the hot S-spins. There is a net flow of energy into the system as a result of work done on the S.spins. The destruction of the I magnetization corresponds to an irreversible loss of order, that is, an entropy increase. The energy of the system is, of course, the expectation value of the Hamiltonian of (7.96). Basic theorems of quantum (and classical) mechanics tell us that the total energy remains constant as long as 1t does not explicitly depend on time. Rea.mngements of energy within the total system, even when 1£ is independent of time, we identify as a heat flow within

287

the system. Changes of the total energy due to varialion of an external parameter we call work on or by the spin system. We can follow the cycle by considering work and heatlfansfer in the rotating frame. Consider one complete cycle of (HI)S on and off. We stan by turning on (HI)S suddenly. Bearing in mind that the 5 Zeeman energy is

(HZS) = -(Ms)· (H,)s

(7.119)

and that during the sudden tum-on the dipolar energy does not have time to change, we see that it takes no work to tum on (HI)s since (Ms) is initially zero. The establishment of the 5 magnetization causes (1tzs) to go from zero to a negative value. That is, there is a heat flow from the 5·spin Zeeman reservoir to the rest of the spin system. That this is the direction of heat flow is reasonable since the initial zero (MS) in the presence of a nonzero (Ht)s can be viewed as saying that 1tzs has an infinite temperature. Associated with this heal flow between systems at different temperatures there musl be an entropy increase. Following establishment of (MS)' we tum off (Ht)S' Using (7.119) we can see that we must do positive work on 'H.zs in the process. [Note that during the tum-on or tum-off, which takes place suddenly, there is no time for heat flow, so that we can compute the work done solely from the changes in (1tZS) given by (7.119).) Following tum-off, (Ms) decays irreversibly to zero. Again there must be an entropy increase associated with the irreversibility. We are now ready to repeat the cycle. Note that we have done a net amount of work on the spin system, and thai there has been an irreversible loss in order. Had we turned on (Ht)S in the next cycle before (Ms) had been able to decay, the S·spins would have done positive work back on us. In fact, had (Ms) not decayed at all, we would have gotten back as much work as we put in when we turned off (H ,)s. We would not then have done any net work in a cycle. Moreover, apart from the effects at the original tum-on, there would have been no irreversible loss in magnetization of either spin system. We must allow a sufficiently long time for the irreversible features to occur. The entropy 0" of the system can be calculated starting from the basic equation u =

E+k81nZ 8

(7.120)

where Z is the partition function. Making use of the high-tempemture approximation, we can evaluate (7.120) to get 0" ':l!'

k[ln(Trl)-

2k~82 Tr~~2}]

(7.121)

Evaluating the Tr {'}i2}, as in (7.100a) gives (with Ho = 0)

C/[(Ht)~ + HlI + Cs(H I 20'

)1

(7.123)

(Note that (7.105), describing the adiabatic demagnelization, follows from use of (7.IOOb), Curie's law, together with the requirement that the entropy given by (7.123) remain constant.] Since we have worked out the temperature at each part of the cycle, we can use (7.123), together with the approximation that E: is small, to find that in one complete cycle starting at temperature 8 there is a total increase in entropy (1r - O"j of ur-O"i=

CS(Ht)~ 82

(7.124)

Half of the increase occurs following the tum-on, the other half following the tum-off at (HI)s. We note that the larger (H tls, the larger the change of enlTOpy per cycle. Since the existence of M, is a sign of order, we see that a large (H t)s leads 10 a large destroction of M" as was, in facl, already expressed by (l.118).

7.16 The Effects of Spin-Lattice Relaxation In Sect. 7.11, we saw that Bloembergen and Sorokin observed IWO striking ef· feclS in their pioneering double resonance experiment. As we have mentioned, they gave a full explanation of their resullS. We tum now to an ahemative description of their experiment to show how one can think of their experiments in teons of two concepts: (I) spin temperature in the doubly rotating frame and (2) Redfield theory of the equilibrium spin temperature reached in the rotating frame when spin-Iauice relaxation is included. Thus, we show that their experiment can be viewed as a combination of Redfield's experiments on saturation with the Hartmann-Hahn and Lurie·Slichter experiments. As we explained in Sect. 7.11, Bloembergen and Sorokin observed a species, [, (Cs 133 in their case) which had a long T\. The (H I )/ they used was such that one should consider the I·spins to be described by a spin temperature in their rotating frame. They observed that for the [-spins, TIl? was deteonined by the T1 of the 5-spins, leading to a circumstance that (T'e)/ «(TI )/· They observed further that if they were on resonance for the I-spins (7.125)

"IJHo=w/ but off resonance for the S-spins such that

where Tr 1 means the total number of states, and equals Tr 1= (21 + I)N'(25 + I) N s

(1 = k[N/ In (2J + I) + Ns In (25 + 1)]

hSHo -

wsl

= ''YJ(H\)/ with

(7.126')

(7.122)

"Yj(H,)~«(-ysHo -wS)2

(7. 126b)

they produced an I magnetization, M" given by 288

289

•

I

MI = ±M/(O"Ho)1s , (7.127) 1/ where M/(ot. Ho) is the I magnetization when the I-spins are in thermal equilibrium in a static field Ho with a lauice al temperature (J/. To understand how (Tt)s can determine T 1g for a case in which (Hils = 0, we follow the Bloembergen-Sorokin analysis. We consider a case in which (HI)j::> where HL is some sort of local field. Then the I-spins. if exactly al resonance, are quantized along the I-spin x-axis, the direction of (H I )/. and the S-spins are quantized along the z-axis, the direclion of Bo_ The term 'Hd/S • given by (3.55), involves products such as I;k5zp. which have matrix elements which are diagonal in Szp. bUI are off-diagonal in J;J;p- However, owing to the spin-Ianice relaxation time of the S-spins, the diagonal matrix elemenlS of Szp are functions of time with a correlation lime of (Tl)S' Defining

we can set

Ta ! = (T1 )! for the S-spins we will have. in analogy to (7.132),

aMzS Mos - M"s --/)t

HE

Ckp=

"'fnSh2

3

rkp

2

(I-3cos 6kp )

(7.133)

-

with

TIIS

(7.134)

and

(7.135)

oMzs Mzs ~=-TbS

(7.136)

where TbS ...... (TI)S. We will have three dipolar relaxation equations similar to (6.87):

~(7i )_ fJt dfJ -

(7.128)

( 7idll) T,11

(7.137a)

BlQembergefi and Sorokjfl show that

_'_=.2..l: C 2 (TI,,)1

1i 2 k,p

5(5+1)

kp

3

(T,)s

I +wrl(TI)~

(7.137b) (7.129)

The reader should compare this expression for the relaxalion of (Iz ) along (HI)z with the expression for TI of (5.372) which applies to the relaxation of (I,,) along Ho. Physically, the fact that the z-component of the S-spins is fluctuating produces a time-dependent magnetic field on the I-spins along the z-direction. A field of this polarization is ttansverse to the I-spin quantization direction, and can therefore relax the I-spin magnetization along its "static" field in the rotating reference frame. The factor TI/(I +wr,Tls ) gives the spectral density at the frequency WII which can produce flips of the I-spins quanlized along (HI)!. If (HI)s is turned on to a level sufficiently strong that the $-spins obey the Redfield spin-Iemperature condition in their rotating frame (6.84), then the magnitude of the S-spin magnetization, Nls, and of the I-spin magnetization, M" will be given by an argument such as is given in 5ecI.6.9. The calculation we described above for (Tle)l can be viewed as a calculation of Tb' defined by a modified (6.86):

z / = - M:z:d oM-

at

'I

w'lh ,

( 7idlS)

where we have neglected the tenns like (~) of (6.87). If we assume that (Tt)s <: (Tt ), the lattice will flip individual S-spins much more rapidly than individual I-spins, so we can neglect any rates dependent on I-spin flips. Therefore (7.138) and both will be much shorter than Tell - Likewise

TIl& <:Tb/ -

0 ) E = - (ho)M: s - (H,)/M:r./ + (lldfJ

+ (7i?'/s) + (~SS)

The z-component of M, will not be relaxed directly by the time variation of 1-$ dipolar coupling since that coupling commutes with I:. Thus in the equation

290

(Til)!

.

(7.140)

Then

aE

at =

at

(7.139)

In analogy with (6.91) we will then find for the Bloembergen-Sorokin case

(7.130) (7.131)

aM:J - = MOl - M:J

(7. 137e)

TeiS

(7.132)

I oM:s - (IO)S---al -

aMzl

(Hd,---at

a"~ a"~ a 0 + at ("dll) + a,!"d/s) + at (7idSS )

(7.141)

Keeping only the fast tenns which relax at rates comparable to (Tt)s, we get 291

•

I

BE = -(ho)S (Mos - M.s) _

at

(1i~15)

T as

_

(1i~SS)

TclS

We now assume the system has a common spin temperature

M

M

(7.142)

TeSS

B so

that

- Cs(ho)s 8

(7.143a)

- ClCHill

(7. 143b)

zS -

:d -

()

o

(1i dIS ) = -

CsHJs e

(7.144.)

(7.144b)

where Hys and H~s are defined by (7.144). To find the equilibrium spin temperature, we set dE/dt = 0 and substitute (7.143) and (7.144) into (7.142), gelting

()

Mos(ho)s/(ho)~ + Hl ls + Hl ss ) Tas

TaB

Tcls

(7.145)

TeSS

If one is off resonance (lto)s by a sufficiently large amount compared to the local fields, one can keep only the [enn involving (ho)S in (7.145), getting

Cs = Mos (J

iJ

1(1'0)51 MOl (ho)S

(7.146)

(lto)s

BUI, since

Ms = Cs(ho)s

8 MS=MoS

(7.147.) (7.147b)

Recognizing that

CsHo Mos=--

8,

(7.148)

(7.150)

The ratio l(ho)sl/(ho)S is either +1 or -1 depending on whether ws lies below or above resonance respectively. The two possible signs reflect the fact that 8 is positive in the fonner case [Ms parallel to (Hcff)s] and negative in the laHer [Ms antiparallal to (Heff)s]. The size of MJ is larger than its thermal equilibrium value in the ratio ishJ' This result is just the result of the Overhauser effect, hence the name "transverse Overhauser effect". In considering this result, we see that it follows very directly from the realization that if the S-spins are far from resonance

Ms =Mos

(1t~ss) = - Cs:~s

Cs =

Ml = "

(7.151)

and that the I-spins and S-spins are at a common spin temperature in the doubly rotating frame. We could achieve the same result by an appropriate adiabatic demagnetization experiment. We tum on (HI)J and we tum on an (HI)S which satisfies the Hanmann-Hahn condition at a frequency Ws far from resonance. We then sweep ws onto resonance. The two spin systems will then mix. However, since they have comparable heat capacity, the common spin temperature they reach will be holter than the temperature the S-spins would reach if the spins did not mix. So we tum off (H.)s, go off resonance. magnetize the S-spins again, and repeat cooling the I-spins several times more. In this way, we can bring the I-spins asymptotically 10 the value given by (7.150). However. we can only hold this result for a time (TI/?)! given by (7.129). The Bloembergen-Sorokin method gives one a steady-state MJ along (HI)J' Note that if one sits with the S-spins exactly at resonance [(ho)s = 0], (7.145) and (7.146a) show that the equilibrium values of Ms and MJ would vanish even if one satisfied the Hartmann-Hahn condition. Thus, to use the fast relaxation of the S-spins to generate a strong I-spin magnetization, the S-spins must be far from resonance (above or below). Equation (7.145) can be viewed as giving the spin temperature of the Sspins. If the Hartmann-Hahn condition is satisfied, the I-spins will rapidly come to this spin temperature. However. if one is far from the Hartmann-Hahn condition, the I-spins will not reach this temperature, as is shown from the experiment of Lurie et al. (Fig.7.16).

we get, combining (7.146) and (7.147), M = C/(Hl)/ = C/(Hlh CsHo = (Hdl MOl r 8 Cs(ho)s 8, (ho)s

(7.149)

7.17 The Pines-Gibby-Waugh Method of Cross Polarization

But from (7.145)

'"a (HilI = 'Ysl(1to)sl so thai we gel the Bloembergen-Sorokin result that 292

The methods of Hartmann and Hahn or Lurie et al. involve detection of a rare species by observing its effect on an abundant species. Pines, Gibby, and Waugh [7.33] pointed out that under some circumstances it is preferable to perform 293

the experiment in a manner to observe the rare species. Specifically, suppose C 13 is the rare species (its natural abundance is 1.1 %), and HI is the abundant species. as when one is working with solid hydrocarbons. Each nonequivaicni C 13 site will have its own chemical shift. One would like to record each C l3 chemical shift. We have seen (Sect. 5.8) thaI the Fourier transfonn method is then a very efficient way of collecting data. since a single C l3 7r/2 pulse will excite all the C 13 lines simultaneously. If onc observes the HI resonance, one can only measure the individual C13 resonance lines using a point-by-point search. However, if one brings the HI magnetization along (Hdll. the hydrogen HI, one can rapidly polarize the C 13 nuclei by turning on their H I> (HI )e. to the Hartmann~Hahn value. Within a cross-relaxation time, Ihe C I3 will be polarized. Since there are many more hydrogen atoms than C 13 nuclei, in this process the HI spin temperature will hardly change. Therefore

Me

-- GdHlk

o

(7.152)

But

8 = Gu(H1h,

and

(7.153.)

Mil

1'dHde = 111(Hd"

Me = M u Ge Gn

so thai

(I'll) 1'e

(7.153b) (7.154)

Suppose. now, Ihal initially MI-I has ils {hennal equilibrium value in Ho

GIIHo 0, The corresponding MC<8/) is given by MH = Mil (0/) = - -

(7.155)

Ho MdO,) ~ Ce9;

(7.156)

Ulilizing these equations in (7.154), we find

Me ~ MdO,) 711 7e

(7.157)

This is the Bloembergen-Sorokin enhancement, characteristic of an Overhauser effecl. Having polarized Ihe C13·s, one turns off Iheir H" and records the free induction decay. Meanwhile, one leaves on the proton HI for two reasons: (1) to maintain the prolon magnetization [spin-locked by (Hl)lI] and (2) to provide so-called H 1_C13 spin decoupling. This last topic we take up later in the chapter, where we see that the presence of the strong H I acting on the protons effectively wipes out the HI splillings of the C I3 resonance. By this means the C I3 resonance consists of a family of lines each of which goes with a given C13 chemical shift, but without splittings arising from Ihe C I3_H I spin-spin coupling. 294

We now lum on (HI)c again, repolarizing the C13· S along their HI. We again tum off (Hde and record the C l3 free induction decay. Each repetition of this cycle will heal the HI nuclei in accord wilh (7.112) and (7.117), so that the C13 signal gets progressively smaller until it becomes necessary to tum off (HI)}! to allow Ihe proton spin-lattice relaxation time to replenish the HI magnetization. We turn now to a few remarks about the nature of the rare spin spectra. If we recall that in general the spin-temperature approach requires that we be working with solids. we should distinguish single crystals from powders. In a single crystal a given type of C I3 chemical site might have several different bond orientations with respect to Ho. For example, in a benzene molecule (C6H6), Ho might lie along one CH bond, in which case il would also lie along one other in the molecule, but would make a 60" angle with the other four bonds. Thus we would have two C13 lines, with one twice the intensity of the other. If the sample is a powder, the C 13 spectrum of this bond would be a smear. By magic-angle spinning (discussed in Sect. 8.9) one could narrow this pauem to a single narrow line at the average chemical shifl position of C I3 'S in Ihal molecular position.

7.18 Spin-Coherence Double Resonance -Introduction The Overhauser-Pound and the cross-relaxation methods of double resonance are both dependent on the existence of relaxation processes of sufficient vigor. In the case of Overhauser-Pound schemes it is necessary that appropriate spin-lattice mechanisms are Strong. In the case of cross-relaxation double resonance. crossrelaxation times must be sufficiently rapid (relative to spin-lattice relaxation) for the method to work. We now tum to the third family of double-resonance schemes. For these schemes to work, Ihere must be a coupling between the two spin systems which manifests ilself in splillings of the spectral lines which would exist in the absence of the coupling. We observe the double resonance by producing some fonn of modulation of this coupling. Crudely speaking, we observe the effect on splittings of the I-spin resonance produced by exciting the S-spin resonance. If 8w is a typical such spin-spin splitting. the essential condition for viability of Ihis third family is that the effects of Ihe splitting not be obscured by relaxation processes. If one thinks of 8w as some sort of beat frequency, Ihe period of the bealS is 1/8w. The requirement is then physically that the coherence of this beat nOI be interrupted by relaxation. If T stands for the relaxation time (spin-spin, or spin-Ianice) which could interrupt the coherence, observation of the coherence requires that T:p 1/6w. The requirement of long relaxation times shows that spin-coherence double resonance tends to work under circumsr.1nces where the other methods do not.

295

7.19 A Model System - An Elementary Experiment: The S·Flip-Only Echo To begin the discussion of spin-coherence double resonance we are going to imagine ourselves to be working in resonance before the invention of all the varied double resonance methods which exist today. We suppose that we have observed a HI resonance which is split into a doublet by coupling to another nucleus (for example a C 13 ). Since the C 13 affects the HI spectrum, we ask ourselves is there not some way we can use the HI resonance to detect when the C l3 is brought to resonance? We first describe the model system, a pair of coupled spin ~ nuclei I and S with a Hamiltonian similar to that of (7.29) 1{ ""

-"nhHo1, -1shHoS, + 1-£12(t)

Initially we shall ignore relaxation effects, considering 1-£12 to be independent of time. For simplicity we take it to be (7.158)

This is the fonn of the indirect spin-spin coupling in liquids. If I and S represent different nuclear species, the tenns I~5~ and I1I 5 11 are nonsecular. Then we approximate (7.159)

This is also the fonn which the dipolar coupling between two nonequivalent spins takes if we keep only the secular pan, see (3.55). For indirect spin coupling in liquids, it is conventional to substitute J for A, and to speak of the J coupling. Since we wish to include dipolar coupling between unlike nuclei, we utilize A as the coupling constant. If I and S are the same species but have different chemical shifts, we can represent the chemical shift difference by defining (7.160)

where 1 is characteristic of the species. If then 1(71 - 7s)Hol:> A

(7.161)

we can approximate 1{12 by (7.159), but if the chemical shift differences are comparable to A or less than A, we must keep the complete expression AI· 5 for the coupling in a liquid. For a dipolar coupling in a solid, we would need to include both the A and B terms of (3.7). At this point we make a diversion to discuss notation. In the literature it is customary to refer to the case where (7.161) is satisfied as an "AX" case, whereas the case where the full expression (7.159) is needed as an "AB" case (do not confuse the A of AX with the A of AI· 51). Each letter refers to a given chemical shift or resonance frequency. 296

1t "" -1IliHo[, -1sliHoS, + AI,S,

(7.162)

The energy levels are

.

'Ys "" 1(1 - O's)

Letters next to each other in the alphabet have chemical shifts comparable to their spin-spin couplings. Letters not adjacent have big differences in resonance frequency compared to their spin-spin coupling, either because the chemical shifts·are quite different, or because the nuclei are different species (HI versus C I3 ). An A2 system has two nuclei with the same chemical shift. An AB2 or an AX2 system has three nuclei. In the fonner case the chemical shift differences between the identical pair are not large compared to the A-B spin-spin coupling, in the laller case (AX2) they are large. A notation AMX means three nuclei with big chemical shift differences. Note that if two nuclei are different species, they would be called an AX system. We shall treat an AX system, hence take

E""

-hwO/ml - hwosms + AmimS

(7.163)

where

WOI

-= 11HO

(7.164.) (7. 164b)

If we apply alternating fields at frequencies WI or ws (near to the resonance frequencies of the I-spins or the S-spins, respectively), the resonance condition i, (7.165.)

WI =wO/-amS ws "" wos - ami a

-=

where

(7. 165b) (7.166)

Alll

!'

Since ms "" ± the I resonance consists of two lines, spaced apan in angular frequency by a, as does the S resonance. Thus, through the couplings the presence of the S manifests itself in the I resonance. We now ask the question: Is there some way we call use this manifestatioll to ulllble us to employ the I resonallce as a means of detecting when we are producing a resonallce with the S-spins? In fact, in 1954. Virginia Roydell [7.34] was interested in just this question in order to measure precisely the ratio of the 1"S of HI to CU. We describe her experiment and that of Bloom and Shoolery [7.35] based on the theory of Bloch [7.36] in Soct. 7.20.

One simple concept immediately comes to mind. Suppose the S-spins were polarized so that they were all in a given state ms, say ms "" +~. Then the I spectrum would occur at WI "" wOf - aI2. We now stan searching for the S resonance by applying Jr pulses to them of angular frequency ws. We slowly sweep ws. After each S-spin pulse, we inspect the I resonance by inspecting 297

I-spin ~bslH"ption

(,)

Fig. 7.17. I-spin absorption spedrum for S-spins 100% l>olarizcd with (a) "'5 = and (b) "'s vcrsua I-spin scar<:h frcqucncy 1<1/

= -t

+t

before

~fter

(,)

Ibl

n

1.111 .1/2

w'I·~/2

",

I (J

n

~bsorplion

(bl 1.1 01

,,

,

n ",

wtI· a/2

Idl

I_spin

ms: 1/2

1.1 11 - a/2

-1/2

1.1'1' ~/2

n

",

wOI-~12

1.1 01. a/2

",

, l

I

(e)

the I absorption speclrum. It will be unchang~ un/~ss ws was tuned to th~ S resonance. In that case, we change ms from +~ to -'2" and the I resonan~e shifts from WI = WOI - oil to WI = WOI + 0./2. The situati?n is illustrated in Fig. 7. 17 . Simple and beautiful as this scheme may seem, It ~Ias cannot be ~sed exactly as described without first gelting the S-spins polanzed, a very difficult .~k indeed. In fact, at nonnal applied field Ho and temperatures 8, Ihe condillon "Ys hHo/ k8 <. I holds, so that to a good approximation one has an equal number of spins ms = as ms = We can reanalyze the experiment very Simply by saYing tha: half of the Ihe ot~er half have mS = -'f.' Let u,~ call I-spins have neighbors with ms = these two groups (1) and (2) respectively. Figure 7.18 shows the before and "after" spectra for the two groups produced by the 11" pulse on the S-spins. Group (I) will have a resonance at WI = WOI - 0/2 before the S-spins are flipped, and a resonance at WI = WOI + all after. Group (2) will have a "before" re~nance S·spinS are at WI -- w01 + a/2 ' which will be shifted to WI = WOI - all. after the .• flipped. We observe of course the spectrum of all the [-SPinS, which IS the sum of the spectra of the two groups (1) and (2). As we see from Fig.7.18e,f, the total spectrum is the same before and after. It looks as though our scheme has failed. . Fortunately, though the simplest version of our concept does not work, It can be modified very simply in any of several methods to be made to work. The essential point is that anyone [-spin belongs to either group {l~ or group (2). Since eirher group (I) or group (2) is affected by flipping the S-spms, both. group (I) and group (2) can tell the difference between whether or not the S-~PInS are flipped. We must do an experiment which utilizes the memory that SpinS have

+l

-l·

.

+l,

298

.

If!

n

n

1.1 01 - ~12

1.1 01.

al2

n

",

wOI-~12

n ",

1.1 01 •

~/2

Fig. 7.18. The I,spin absorption before (lcrt column) and after (right column) applying a 'II" pulse to the S-spina, for the group (I) l-spil1$ (a, b), the group (2) I-spina (c,d), and the total of alii-spina (~,f). Note that though the s.-::trum of ~ilhu group (I) or group (2) alone i$ changed by Ripping the S-spins, the loIal intensity pdtem (~.f) is unslTec:l.ed

of their prior history. An experiment which displays a spin's memory is easy to concoct. At t = 0 we apply a pulse to the I·spins with a sufficiently strong (HI)! to flip the spins in both the I absorption lines. We view the I-spins in the reference frame rotating at wO/. In this frame, we will have two magnetization vectors of equal length, one precessing at angular frequency af2, the other at -0./2. If we assume that immediately after the 7((2 pulse the I magnetization lies along the y·axis [Ihe situtation if (Hj}1 were along the x-axis], the magnetization will obey

trn.

(Mi('»

= C;i (cos (at/2) + i sin (a'l2)]

+

~oi [cos (ain) _

i sin (ai(2)]

= COi(eiat/2 +e- iat / 2)

2

Co = (MI(O,»

where

(7. 167a) (7. 167b)

(7.168)

gives the thennal equilibrium magnetization of the I-spins for the lattice tern· perature 8,. 299

Fig.7.20a-<:. p31 NMR signal versus time for diethyl phosphite I(C2IhOhPHO). (a) The precession of the p31 nuclei due to lheir coupling to the IP nuclei; (b) and (c) the precession of the p31 nuclei with the HI nuclei inverted al the indicated times. The dashed lines in (b) and (c) show whal the precession would have been had lhere been no II! inversion. (Data taken by Charles Pennington wilh assislance from Je3n-Philipl>C Anscrmel, Dale Durand, and David Zax.)

Then, at a time t1f we apply a 7r pulse to the S-spins. The group (1) spins will instantly change their precession frequency from WOf - a/2 to WOf + a/2 and the group (2) spins from WOf + a!2 to WOf - a/2. Thus, both groups will reverse their precession directions in the rotating frame. (Readers who are not content with the present informal treatment can look ahead at Sect. 7.24 for a fonnal justification of the relationships we use here.) We can utilize the complex notation to express this rotation reversal. If a vector is represented by a complex number A with initial components At: and Ayobeying Az =

Ao cos 4>

A y = Ao sin

4>

(7.169)

where Ao is the magnitude of the vector, then A = A z + iA y = Aoexp(i4». If this vector is rotated at angular velocity w, it becomes at a time t later A(t) = Ae iwt = Aoei4>e iwt

o

(7.170)

Therefore, we can apply this relationship to the two components of (Mt(t» given in (7.167b) to get the magnetization at a time t - t.,.. after the S-spin 7r pulse as

(Mt(t» = ~oi (e iat "/2 e -ia(t-t,, )/2 + e-ial"/Zei«(!-t,, )/2)

(7.171)

Thus (Mly(l» =

Co cos (at.f2 - a(t - t.)f2)

(7.172)

At time t - t 1r = t 1r (t = 2tll")' (Mfy(t» will once again be Co. In other words, we have produced an echo of the I-spins by flipping the S-spins. The Sspins so-to-speak cause the I-spin "field inhomogeneity" (a two-valued function). By flipping the S-spins we reverse this field inhomogeneity. The theoretical time development of (Mfy(t» is shown in Fig.7.19. An experimental demonstration of this phenomenon by Pennington et al. from the author's laboratory for the p31 resonance of liquid (CzHsOhPHO is shown in Fig. 7.20.

A useful way of viewing this experiment is to plot the phases 4>1 (t) and 4>z(t) of the two groups of I-spins where we define their phases at t = 0 as being given by (7.169). From (7.171), they are 4>,(1)=

(Tr/2)

4>,(t) =

o

It. S_spins inverted

300

C'f2J+ atf2

+ atll"!2 -

a(t -

t n )!2

C<(2) - «tf2 (7r!2) - at 1r / 2 + aCt - t n )/2

fOI

t
fOI

t> t1f

fOI

t

< t 1f

fOI

t

> t>r

(7.173)

These are shown in Fig.7.21. If there were more than one paicof spins I-S, each pair with its characteristic chemical shift, the situation would be more complicated, but the same principles would apply. Let us label each I-S pair by k. (k = I to N if there are N distinct pairs.) We shall assume the I-S coupling is only within a given pair. Then we have N Hamiltonians of the fonn Fig. 7.19. The y·componenl of the I-spin lllflgnetizalion versus time, t. Al t t", the S-spins are in\'erled by " 1f pulsc

=

1ik = -liWOfkIzk - flWOSkSzk

+ AkIzkSzk

(7.174)

so that, defining ak = Akl!l, 301

PhilSe iogln 4'lltl iod 4' l ll)

(7.176)

,

{Mj(O}""" = C;i l)exp [- i(!7 lk - (lk!2)t]

+ exp [- i(!7 lk + ak!2)t] Fig. 7.11. The phllSe
(7.177)

This general expression as well as the effect of a 'Ir pulse applied to the S-spins at time t" can be represented by plotting ,pu:(t) and q,2k(t) given by

for t
0,,(1) = (./2) - ({Ilk - a,/2)'

(7.178) for

t > t"

A similar expression holds for 92(/.;) except that the sign of Qk is reversed. In Fig. 7.22 we plot the ,p's for a case of two I-S groups (N = 2). It is convenient to give a name to the experiment we have been analyzing. We shall call it the

PlliU ¥lgle

S-jlip-on/yecho.

7.20 Spin Decoupling

Fig. 7.22. Thc phll$C IIngles ~1 k(t) and ~H(I) versus time, t, for a ca.sc with lWO 1-5 pllit'll (k 1 and 2). At 1 I., the S·spins lire nipped by 11'. Al t 'It:r, ~u(t) ,pH(I) for bosh spin groups, indicating thllt bolh components of I magnelization for II given pair sre collinear. Nole, however, lhal since lhe two pairs have differenl chcmical shifls, lhe I magneli."aliOllS of lhe different pain are out of phase wilh each other

=

=

=

=

= Coi L(exp [ _ i(WOlk '- ak!2)t]

2

,

+ exp { - i(WOlk + ak/2)t]

(7.175)

in the laboratory. Relative to some frame TOtating at WI we have magnetization (Mj(t)"",s' Defining 302

The ideas we have just developed lead in a natural way to the concept of spin decoupling, one of the earliest goals of double resonance experiments. We will follow a pedagogical rather than a historical order. The goal of decoupling is to simplify spectra. A typical NMR spectrum consists of many lines arising from the combined effect of chemical shifts and spin-spin couplings. Decoupling is a process which effectively eliminates the spin-spin couplings. It is very important both for high resolution spectra of liquids and spectra of solids, for example to eliminate the effect of proton spins on C I3 spectra. We shall discuss removing the spin-spin coupling between different nuclear species such as HI and CU. The basic idea on which all decoupling schemes work can be understood physically as follows. The existence of two lines in the I-spin spectrum corresponds to the fact that the S-spins have two orientations, up and down. If we can cause the S-spins to flip back and forth between the up and down orientations sufficiently rapidly, we should achieve an effect much like the motional narrowing of resonance lines. That is, an I-spin will precess at a time-averaged frequency rather than at one or the other of two discrete frequencies. To study decoupling, we are going to pick a scheme which is easy to analyze. We will look at the effect on the I-spins of applying a sequence of 7r pulses to the S·spins. We shall see that as the time between S-spin 7r pulses gets shoner and shorter, the I-spin spectrum goes from two distinct lines separated by Q in angular frequency, to a single line at the average frequency. There are two general approaches we could lake to showing this result. One would be to try to calculate what the I-spin absorption spectrum Xl/(w) looks 303

like as we flip the S-spins al a progressively more rapid rate. Or, we could look at Ihe Fourier transform of the absorption line, which we do by looking at the time development of the I-spin magnetization following a rr{l pulse applied to the I-spins. In the absence of S-spin pulses, the I-spin magnetization will consist of two components, one oscillating at WOI + 0/2, the other at WOI - 0/2. So, if we choose Wf = WOf. in the Wf reference frame the components will precess at angular frequencies +a{l and -0/2. If the system were perfectly decoupled, a would become effectively zero, so both components are at rest in the WOf reference frame. and the transverse I-spin magnetization will be a constant in time. (We are not including relaxation effects in our Hamiltonian. If we did, the I-spin transverse magnetization would decay rother than remaining a conSlllnt in time.) We shall follow the Fourier transfonn approach for whi<.:h, in facl, we have already set the srage in Secl 7.19. There we analyzed what happens if we apply a single 'lr pulse to the S·spins al a time t = t .... Let us then look back at Fig. 7.19 and at (7.171), which show that at time t = 2t,.. the magnetization (Mt(t)} has returned to its value at time t = 0+, immediately after the 1r{l pulse. lllOugh Ihe magnelization al the time of the 11" pulse is smaller by the factor cos (at ...{l), the 1( pulse has produced an echo al t = 2t.... If we now think of ourselves as having a fresh start at t = 2t ... , we realize we can repeat the echo by applying a second 'K pulse to the S-spins. If we apply il at a lime t~, hence t~ - 2t... after the first echo, we witt again fonn an echo. It will occur at a time 2(t~ - 2t ... ) after the first echo, or at a time til given by til = t~

+ (t~ - 2t... )

(aJ

F1g.. 7.23 (a) The magoetiution (M,,(t» versus I for the CAlle of a ,.. pulse applied to lhe S-spins at I producing a 21 .. , followed byanrefocusing al t other,," pulse at t I~, with a sc<:ond refocusing at time I". (b) The effect on (M 'J'(t)} of applying a pair of ... pulses to the S-spins at times I.. and I = 31 .. , with nsulting echoes at t 2t.. and

=

>t.


= /'" =

=

(b)

k

'.

II

21.

lal

(7.179)

see Fig.7.23a. If we chose t~ to be given by t~ = 3t...

then

(7.180) (7.181)

so Ihat the time delay of the second echo after the first is identical to the time delay of the first echo after the initial signal. We can repeat this process again and again. The situation is shown in Fig. 7.23. The phase angles ;,(t) and ¢2(t) of Fig.7.21 will then appear as shown in Fig.7.24a. If we choose a shoner time for t,.., as shown in Fig.7.24b, the maximum excursions of ¢l (t) and ¢2(t) away from tr{l will be less, so the magnetization versus time will appear like the solid curves in Fig. 7.25 mther than the dashed curves. Clearly, (Mfy(t)} will be a periodic function as long as there is no relaxation, with period, T, equal 10 2t,... We can think of (Mfy(Y)} as consisting of a constant with a superimposed periodic ripple. Figure 7.25 shows that the shoner t ... , the smaller the ripple, and the more (Mfy(t» approaches a constant in the reference frame rotating at wOf, hence behaves as though a were zero. Thus, we have "decoupled" the S-spins from the I-spins. 304

Ibl

• ./2

i'E~-::"':;:'--:7<E:~-::"':;:'--:7i;<E:--,

".

Jo'ig.7.24. (a) The phl\.Se angles .pI (t) and .p,(I) of the two components of I-spin magnetization. The,," pulsoes: applied to the S-spins at I t .. and 31 .. cause.pl and .p, to become ~ua.l at I = 21 .. and 'It,.., correspo~ding to the maxima in the curves of (M,.(I)} shown m Flg.7.23b. (b) The effect on
=

=

305

Fig. 7.25. The effect on (Mfr(l» of halving the Lime I ... The solid-dashed curve corresponds to I .. of Fig. 7.24a, the solid-only curve corresponds Lo Fig. 7.24b. Note that ('""ffr(t» (!Ill be Vlewed as a steady value with a superposed ripple, that the ripple is periodic with period 21 .. , and that the amplitude of the ripple diminishes as I.. is shortened

/

We can eltpress the eltistence of the ripple mathematically if we wish. by recognizing from Fig. 7.25 that since (M/y(t)) is a periodic function with period 2t r it can be eltpressed as a Fourier series

=

L

(MJy(t» =

All cos (21fftt(J')

(7.182)

with

We can summarize for the case thai we flip the S-spins sufficiently frequently: following a 7r/l pulse applied to the I·spins. they precess as though the coupling to the S-spins were turned off. This method of decoopling was in fact proposed by Freeman et al. in 1979 [7.37]. They call it "spin-flip" decoupling. In their paper, they discuss its advantages over OIher methods, as well as discussing interesting variants. Another method of producing such a zero time-average z-component is to apply a constant HI at the S·spin resonance which will cause the S-spin to precess about the HI' As a result. the z-component of 5 oscillates sinusoidally at angular frequency 'Ys(HI)s, This method is historically the first method employed by Royden (7.34] and by Bloom and Shoo/ery (7.35]. To analyze it. we talee the Hamiltonian to be 1(. "" -

- ')'sMH ds(Sz cos wst - Sy sin WSt)

n=O

(7.183)

T = 2t.". The coefficient of the constant tenn, Ao. then turns out to be A = (M (9

o

J

I

» sin (at 7l"/2)

(7.184.)

at"./2

which shows that as atrn shrinks. Ao approaches (M/(UI»' The higher coefficients are An

:=

2{A1j(U,) sin (at r /2) (_ t)n+I atrn

(at r {l.)2

An cos (21rnt/2t,..) = An cos (at/2)

(7.186)

We have included the Zeeman interactions of the two spins, their spin-spin coupling, and the interaction of a rotating magnetic field (Ht)s with the S-spins. We have nof included (he interaction of (H ds with the I-spins since we assume that Ws is close to the S-spin resonance, but far from the I-spin resonance. We now view the problem in a double reference frame which is the lab frame for the I-spins and one rotating at ws for the S-spins. That is. if the original Hamiltonian ?i and wave function 1/1 obey (7.187)

(7.I84b)

(:lI"o)2 - (at r n)2

which become much smaller than the constant tenn as at r - f O. Note that if atr{l. is very large., so that there are many oscillations in the I-spin rotating frame between S-spin flips. the coefficients An are very small for small n. However, for :lI"O = at r /2. the coefficients peak. (For larger values of n they decrease again.) Substituting into (7.182) this gives a time dependence for such values of 0 of

where ?i is given by (7.186). we define a transfonned wave function transfonned Hamiltonian ?if by Vl=e~iwstS.¢

•

l/J'

and

(7.188)

which. when substituted into (7.187), gives

_~ al/J' i

at

= ?i'.I,f

with

(7.189)

¥'

(7.185)

This value corresponds to the I-spins precessing at +a/2 and -a/2 in the rotating frame. the situation without dccoupling. The transition from the split lines to the strongly dccoupled lines will therefore look much like Fig. F.3 in Appendilt F. Strongly decoupled spins require that there be no peak in the An's at frequencies .:::: ± an. hence that in (7.185) there be no tenn for n ~ I which has a denominator close 10 zero. This condition requires (atrfl) < 11". If we look a Fourier transfonn of (7.182), we would get a constant tenn in the reference frame rotating at WOJ, and side bands at frequencies ±(n/Ur ), where n is an inleger. The amplitude of the side bands diminishes relative to the constant tenn as one shortens t r . 300

"tJItHoI: - 'YshHoS: + AI;S;

H' =

- ~1"Ho[. - ~s"{[Ho - (wshs)]S. +(H\)sS.)

+ AI;S:

(7.190)

Defining (ho)s == Ho - (wshs)

and

(7.191.) (7.191b)

we get

?i = -')'JhHOI; - ')'!l(Heff}S' S + AI:S;

(7.192) '<)7

This Hamiltonian can in fact be solved exactly, but we will leave the exact solution as a homework problem. Instead, to solve this equation, we assume thai we have applied a strong decoupling field so that (7.193)

(m/mslIIz:lm~ms') = (m/IIz:lm'/)om m' s' S' so that the energy changes aE are

A.E(ms') = "UltHo - Ams' cos

(J

(ho)s

and treat the tenn AlzS z as a perturbation. It is then convenient to define axes (x', y', Zl) such that Zl lies along (Heff)S' and the y- and y-axes are coincident (Fig. 7.26).

,

•

= "'1lhHo - Ams',=~~~~

";(ho)~ + (Hl)~

(7.199)

±!.

Corresponding to the two values of ms' = there are thus two I·spin resonance tines whose spacing is a function of how far ws is tuned away from the resonance frequency wos. At exact S-spin resonance, (ho)s = O. the I-spin resonance consists of a single line. Figure 7.'2:7 shows how aE(ms') - 'nhHo goes with (ho)s/(HI)S, demonstrating the collapse of the splitting as one tunes the S-spins to resonance.

, FIg.1.26. The effective field acting on the S-spil"llll in their rotating frame as a result of (lids nd of being some..·h...t off-resonartee for the S-.pins ((ho)s ct OJ

(7.198)

A

FIg. 7.21. Spin decoupli"8' the ener&>' splitting, i1E - '"f,U/ o , seal in the I-spin res0nance as a fundion of (ho)s the amount, the S-llpin ill off resonance, showing that the two transitions collllp6e to a single transition as the S-spins approach their resonance condi· tion (ho)s = 0

An

This gives Sz = Sz' cos 8 - Sz:! sin 8 with

(7.194.)

r~(;;;ho:;)g,S=""

(7. 194b)

cos 8 =

J(HI)~ + (ho)~

The unperturbed Hamiltonian

Ko =

-A/2

-A

?to' is then (7.195)

--YJhHOlz - "YS(Heff)SSz'

with eigenstates lm/ms,) which are eigenfunctions of I z and Sz" The perturbation AlzS z then becomes AlzSz = Alz(Sz' cos (J - SZI sin 8)

.

(7.196)

But Ihe tenn in SZI has zero diagonal element in the Im/mS') representation, giving for the energy eigenvalues

E'mJm S' = -""f11IHoml - "Ysh(Heff)sms' + Am/mS' cos 8 .

(7.197)

A weak rf field tuned close to the I-spin resonance can then induce transitions with a selection rule given by 308

The exact solution of (7.193) introduces another feature, the fact that the direction of the effective magnetic field acting on the S-spins depends on mI. the orientation of the I-spins. This same problem arises in electron spin resonance and is treated in Chap. 11, (11.83-88). There it is shown that as a result the frequency of transition for .1ml = ± I is modified, and in addition transitions nonnally forbidden in which ms' changes become possible. Decoupling by applying a steady altemaling field was. as remarked previously, demonstrated by Royden [7.34J and by B/oom and Shoo/cry [7.35] based on ideas of Bloch {7.36J presented in an invited talk at an American Physical Society meeting. In Fig. 7.28 we show data from Bloom and Shoolcry, demonstrating experimentally the collapse of the splitting as the S-spin oscillator approaches resonance. 309

.'lg. 7.28. Dcmonstntion by Bloom and Slwolery of the effect on the F19 resonance of varying the p31 resonance frequency for an (11 dp of 800 Hz. L}v is the amount (in Hz) that the p31 is off-resonance. As L}v approaches zero, the splitting goes to zero. "lh~ ill the top line denotes (fft}p

Noise Modulalion

Jll1111ul1J Jl zoo Ht

"" "'

400 Ht

"" "' "" ",

~~~l J~ Jt .At JJIIJJ II11 700 Ht 800 Ht 900 Ht

No'M;;;;; lar

No double

,esona~

Fig. 7.Z9. Comparison of the dceoupling with random noise and with a coherent rr frequency. The FIB spedrum of CHFCl z is observe
As is evident from Figs.7.27 and 28, the collapse of the splittings is only perfect when the S-spins are exactly on resonance, and the size of the splitting grows linearly with the S-spin offset for small offset. As a result, it takes a large (H,)s to deal with a situation in which there are several S-spin chemical shifts present in the spectrum. Anderson and Nelson [7.38] and Freeman and Anderson (7.39] employed frequency mcxlulation of (H\)s to improve the decoupling over a broader range of (1I o)s. Ernst proposed a method of broad-band decoupling (7.40] in which he used a noise signal to modulate an rf carrier, for example, by switching the rf phase between two values (0 and 11") at random time intervals. Figure 7.29, reproduced from his paper, shows a comparison of noise modulation with an unmcxlulated (Hds as a function of the S-spin offset for the case of CHFCI 2 in which he observed the F 19 spectrum while decoupling the protons. Clearly noise modulation produces a dramatic improvement in the ability to decouple when not perfecl1y at resonance. This result follows from the fact that with a sufficiently broad noise spectrum, the residual splitting varies quadratically with offset instead of linearly as in Fig.7.27. Some other decoupling schemes for broad-band decoupling are a coherent phase alternation method described by Grutzner and Santini [7.41], in which a 50% duty cycle square wave is used to phase modulate (H,)s. Basus et al. [7.42] discuss the advantages of adding a swept frequency for ws to the coherent phase shifting. The approach of Freeman, Kempsell, and levitt can be further improved by the use of socalled composite pulses. Invented by Leviu and Freeman [7.43], they consist of a closely spaced group of pulses to achieve a desired rotation (such as a 1f pulse) 310

which is insensitive to some parameter such as the frequency offset of (HI)s. For example, instead of an X(7l") pulse, they suggest and analyze an X(1r!2) followed immediately by a Y(1I") followed immediately by another X(1I"!2). The theory of composite pulses has been reviewed recently by Levitt (7.44] and by Shako and Keeler [7.45]. Freeman has given an excellent review of the status of broad-band decoupling (7.46].

7.21 Spin Echo Double Resonance We turn now to a slight variant of the basic double resonance experiment described in Sect. 7.20. The technique was first demonstrated by Kaplan and Hahn [7.47] and by EmshwiUer et aJ. [7.48], and is called spin echo double resonance (SEDOR). This technique is important because the observation of a SEDOR proves that the two nuclei involved are physically near each other. Consider an ordinary spin echo with the echo fonned at 2T. During the first time interval T, the spins dephase, but during the second iOlerval T they rephase. Hahn pointed out that if a nucleus of spin I has a neighboring spin belonging 10 a different species, an S-spin, the neighbor produces a local field which may aid or oppose the applied field, broadening the resonance much like a magnet inhomogeneity or the existence of several chemical shifts. Of course, magnet inhomogeneities do not affect the echo amplitude since the dephasing effect of the inhomogeneity during the first interval T is exactly undone by a rephasing during the second interval T. He noted, however, that if one added a second oscillator to flip the S-spin with a 11" pulse at the time the I-spin is given its 11" pulse, the sign of the field the S-spin prcxluced at the I-spin would be opposite in the two time intervals T. Thus, if a neighbor dephased the I-spin during the first interval, it 311

_ 50~~-~~--'~~~-~~

Fret/lIt'IIC)' of Clf 1800 pl,lse mell$lfred ,e/atil'e 10 Clfi) mllill line (k1I~)

• r-'.~12~.,----_----,~~.,---__~4~.,---~ 20

~ 1.'3

'."§ ~

j

• • • • •

......•

•

40

60

•

•

'ooL

.,:-__---,4~.~ ,, ,

..

.... ....

:

'•

.;

.'

•

o Spin Echo

.~ 35

~ 30

,

~

"";;25

~20

§ PoSitiol' of 0,·)

z

,

.~

•

..

•

o

•

0 0

0

~ ••

5

• " ' ,•

.L--'-£-,--L-c--J~~',---'-.-,+c__--'-~J. 1.07

1.08

U)9

mallllllle

---'---

• SEDOR

, :

: \

~ 10

1.10

1.11

1.12

1.13

1.14

1.15

1-10'*'0 (kG / MH%)

---.J

Fig. 7.30. Boyce's observation of SEDOR in • Cu powder containing O.~ at. % Co near the Cuu resonance fT«luence of pure Cu at 9901 Gauss and 1.5 K, showing the satellite due to the 1 .... _ 1 transition of lhe first neighbor. The parameters UBed are T = 250"" 1I is :::::5Ga:" for ~"If pulse of the Cu transition.. IOO echoes 'llere averaged for each point and the dots are larger th.n the S(:a-ller and dnft. Note that t.he ?"uch more abundant Cut3 nuclei a long distance from the Co, the SOotalled Cu malll hne, do nol show up in the Co SEDOR

t .... - t

would continue to dephase during the second interval, producing a smaller echo at 2T. An example of a resonance delected in this manner is shown in Fig. 7.30. Boyce [7.49-51) studied a dilute alloy of Co in Cu. He wished to observe the nuclear resonance of Cu nuclei which are near neighbors of the Co nuclei. For such a dilute alloy, the neighbor resonances are weak in amplitude. frequently hidden under the tail of the resonance of Cu nuclei which are distant from impurities, the so-called "Cu main lines". Boyce was able to reveal the "hidden" resonance of the first neighbor by doing a spin echo double resonance in which he observed the effect on the Co spin-echo amplitude of a 1r pulse applied over a sequence of frequencies close to the Cu main line. Since distant nuclei do not produce much field at the Co, the "main line" has negligible effect on the Co echo height. But the first neighbor has a big effect. In this instance the use of double resonance can be seen as a way of selecting pairs of nuclei which are close in space. Makowka et al. [7.52] utilized pt195_C13 SEDOR to detect the surface layer of Pt nuclei for small metal panicles of Pt on whose surface they had adsorbed a monolayer of CO enriched to 90 % in C 13 . The metal particles. which were tens of angstroms in diameter, were supported on A1203, a typical supported catalyst. Makowka et al. observed Pt'95 spin echoes. Figure 7.31 shows their data. The straighl Pt 195 spin echo gives lhe line shape of the Pt nuclei in the small metal particles. This line is over 3kGauss wide! To observe the Pt195 resonance of 312

, 4.

>-

g 15

•

• •

80

• 45 '2

Fig. 7.31. Measurement ofMaki:Jwla et at (7.521 of the spin·echo (0) and SEDOR (.) line shapes of Pt lU for a sample of small particles of Pt metal supported 011 J\12~ whose surface is coal.ed with C 13 0 molecules. The metal particles have diameters of a few tens of A. The Pt lU spin echo gives the totlll line shape of all the Pt ltS in the particle. The SEDOR data, involving pt 1U _C 13 0 double resonance, and lin add-subtract method, give the NMR line shape of the PllU nuclei in clo5e proximity to the CO molecules, i.e. the surfll.Ce layer of Pt atoms

the surface Pt atoms only, they employed an add-subtract technique in which on alternate spin echo cycles they applied a C 13 'If pulse coincident wilh the Ptt95 'If pulse. For those PI nuclei far from the C t3 , the echo was unaffected by the Ct3 'If pulse, whereas for the surface atoms, the Ptt95 echo was diminished by the C I3 pulse. Thus, subtracting the Ptl95 echoes when the C I3 pulse was applied from those when il was nOl, the signal from the Pt l95 nuclei not bonded to C t3 nuclei vanishes. To analyze the SEDOR signal, we nOte that the precession angle. 8, of the I-spins off resonance by (1I o)J during an interval T is

8 = (")'(ho)1 - ams)T

(7.200)

where (ho)1 represents lhe extent to which the particular I-spin is off resonance due either to magnetic field inhomogeneities or chemical shifts. Suppose, then, that at t = 0 we apply a 1r(2 pulse with (H')I along the I-spin x-axis in their rotating frame. This puts the corresponding magnetization M«ho)J) along the +y-axis (Fig. 7.32a). At time T- just before the I and S 1r pulses, 8 has the value 9, (F;g.7.32b):

8, = 80 =f a-r(2

where

(7.201)

80"" ,/(ho)iT

The I-spin 1r pulse reflects M«ho)J) about the x-axis, producing the situation shown in Fig.7.32c. During the next time interval, T, 8 advances an angle 82 (Fig.7.32d) given by 313

(a) t: O'

(b) t :T-

,

7i(I,5) = -'I"[(l>o),I, + (HI),I.) ~ l's rl[(h o )sS: + (HI)SS~J

,

M

M

e,

(7.205b)

" 7i(I) + 7i(S)

where by 1i(f) [1i(S)] we mean a Hamiltonian which is a function only of the spin components of the I-spins [S-spins]. If this Hamiltonian acts from a time t1 to a time t2, we have

, (e) t
(7.205,)

(d) to 2T

,

,

'f(t,) = ex p ( -*7i(I, 5)(t, - 'I»)'f(tl)

(7.206)

Since 1i(f) and 1i(S) commute, we can write this as t/J(t2) = ex p (

,, , c ,,I a

l

,

,

:~,

" : e" ,

M

=ex p (

FIg. 7.31a-d. The I-spin magnetization for spin echo double resonance (SEDOR) seen in the I-spin rotating frame, M is the magnetization component off-resonance by (h o), for a particular valu!) of At ! 0, (lIJ), is applied along the ~-axis, producing a 11:/2 rotation of M onto the y-axis (a). During the time r, M precesses through a net angle 91 given by (7.201) (b), At t r, 11" pulses are applied along ~I and ~s to both the I-silins and the S-spins, so that at ! = T+, M is at the position shown in (e). During the next interval, T, M precesses through the angle 92 given by (7.202), producing the situation of (d)

=

=

(7.202) where the upper signs of (7.201) and (7.202) go logether, as do the lower signs. Thus M«ho)/) makes an angle .c18 (Fig. 7.32d) given by .c18 = 81 - 82 = ~aT

.

XJ(.fl) ... T

(7.203)

til)ex p ( -*1i(l)(t2 - til)tJ>(t1)

. (7.207)

...

[XJ(')'XS(')]'" (' -

T)

(lIo)s, to allow for chemical shifts, and use the notations of (7.176)

fh "'" l'J(h o ll -

WI

ils = Is(ho), - WS

(7.208)

Then,

(I+('» =Tc{I+,(t)} (7.204)

It is useful for future reference to derive this result using the density matrix formalism. But first we demonstrate a useful theorem. While the (HI )'s are on, we customarily neglect the spin-spin coupling, giving in the doubly rotating frame a Hamiltonian 314

-~1i(S)(t2 -

1i(S)(t2 - til) t/J(tl)

We shall assume the I-spins are off resonance by (hO)1 and the S-spins by

Since Ll8 is independent of 80 or (Ito) I, it is independent of magnetic field inhomogeneity or of the existence of multiple chemical shifts. Since ms is equally likely to be either +~ or -~, two resultant magnetization vectors, equal in length to MOI/2 form, making angles d8 of aT or -aT with the negative y-axis. Thus

(My (2T» = -MOl cos aT

til)exp (

The significance of (7.207) is that in analyzing the effect of the rf pulses, at a given time, we can treat the effects of (HI)I independently of those of (HI)S' Thus, we can treat first the I-spins, then the S-spins, or vice versa. We do not have to treat both spins at once. Of course, the conclusion of this theorem is only valid during the time we can neglect the spin-spin coupling. We now employ this theorem to analyze SEDOR using the density matrix. We consider a sequence in which the I-spin and S-spin pulses are defined in the doubly rotating frame. At t = 0- the spins are in thennal equilibrium. At t "" 0 we apply a 7r/2 pulse with (H I )/ along the I-spin x-axis. We denote it as an X/(7r/2) pulse. We wait a time T, then apply simultaneously X I (7r) and XS(7r) pulses. We then follow the density matrix e(t) and the magnetization at later time, t, such that t > T, looking for an echo at t - T = T or t = 2T. We denote this sequence with the notation

'

M

"'s.

-*

-~1i(l)(t2 -

.I

(7.209)

To find U(t), we must first know its value at U(O-), corresponding to thennal equilibrium. We thus take

- =..!..(l

0 <><)Z+

1iWOIIz+1iwosSz)

kT

(7.210)

315

I h II

with Z the partition function. In calculating (1+(0) we need keep only the teon

,,0 ) _

'!WOt

We thus have

(I

+

(t» =

Tr{I+e- iat ,S,2T(_1 )eiat ,S,2T} ( 1+(2T) = hwOI ZkT y

(7.211)

= ZkT1,

+ . ZkT Tr{1 exp(l(flt1~ + flsSz - a1~Sz)(t - T»Xt(7I")Xs(7I")

fIWO!

x exp (i(fl!1~ + flsS~ - a1~S~)T)X t(1r(2)1z X J 1(ir/2)

The ql,lantity in the curly bracket is just what we would have if (ho)t and (ho)s vanished, and the system developed solely under the influence of the spin-spin coupling, a1zS z . Thus. chemical shifts and field inhomogeneities are refocused, but not spin-spin couplings. l'sing the mjmS representation to evaluate the trace, and recognizing that

x exp(-i(flt1~ + flsS~ - al~S~)T)Xsl(1r)XJI(1r) exp(-i(fltl~

x

+ flsS z - a1zSz)(t - T»)}

(7.218)

(mjmsII+lm~m~) (7.212)

Utilizing that

=

(~II+I-1)6msm~6ml,I/26m~,_1/2

(7.219)

we get 1

Xt(1r!2)I z X/ (1r!2) = 1y

Tr {I+e -illt.S.2T Iyeiat.S.2T}

(7.213)

and insening

= 1

X s I (11' )X J (If)X t(lf )Xs(7I")

(7.214)

before Xt(7I"/2) and its inverse after X J 1(1r/2) we then transfoon

L: (~II+I ~ !)eiamST (- !msllyl!ms)eiamST

m,

I I

But

I y = ~(r - 1+)

X t (1r)Xs(1r)exp (i(fl[ 1~ + flsS z - alz S z )T)Xs (lf)X/ J (71") = exp (-i(fl t 1z + flsSz)T)exp (-ialzSzT)

(7.220)

and

(7.221)

l

(7.215)

(~Ir-I-~) = ( - W-I~) = I so that

(7.222)

and

Xt(1r)XS(lf)1yXSI(1r)XJJ(1r) = Xt(lf)1y X

r (lf) = -Iy l

.

Substituting, we then get

+

hWOj

In the high temperature approximation, Z '" 4 so that

+.

(I (t) = ZkT Tr{l exp(l(flt1z + DsS~ - alzSz)(t - T))

(1+(27) = - hWkOt i cos aT . 4·T Using for thennal equilibrium

x exp(-i(flt1z + DsSz)T)exp(-ia1~SzT)(-Iy) x exp(ia1z S z T)exp(i(fl t 1z + flSSz)T) x exp(-i(flt1z +flsS~ -

a1zS~)(t

- T»}

.

(7.217a)

This expression can be simplified immediately since the teons exp (WsS~(t -T)) and exp( ~WSSzT) on the left of (-1y ) commute with everything between them and the teons exp(iflsSzT) and exp(-WsSzU - T» to the right. Hence, they can be combined, giving a factor of unilY. We then get (I+(t» =

~~~ Tr {1+ exp (i(Dt I z X

(7.223)

(7.216)

MOt = xoHo

')'2h 2 I(I

XO =

'Yfl(Iz}

+ I)

we get liwOj

exp (-Wt1zT)exp (-ialzSzT)( -1y)exp (ialzSzT) .

(7.225)

3kT

(I,) = 4kT " 10

a1zSz)(t - T))

x exp(+Wt1zT)exp(-i(flj1z - a1zSz)(t - T))}

=

so that

(1+(2T)} = -i1o cos a7

(7.2I7b)

At t - 7 = 7 or t = 27, the factors involving flt combine to unity, leaving

(7.224)

.

(7.226) (7.227)

The -i factor indicates that the net spin lies along the negative y-axis. We can now utilize the fonnalism, in particular going back to (7.212), (7.213), and (7.215), to compare three pulse sequences, the S-flip-only echo of Sect. 7.19, the conventional spin echo and spin echo double resonance:

316 317

I

S-flip-only echo (Sect. 7.19): Sequence: X I (7r(l) ... 1' ... XS(7r) ... (t - 1'). Conventional I -spin echo:

is applied at a time T after the I-spin 7r(l pulse (T S 1'). This sequence gives. in the notation of (7.201), Dl = 00 '=F aT(l ± a(1" - T)(l

82 = 00 ± a1"(l

Sequence: X I (1f(l) .. _1' ... X/(1f) ... (t - 1').

and

so that

iJ.0 = 8 1 - 82 = T aT

(7.228a) (7.228b)

and

(7.229)

([+(2T)) = -iI, cos aT .

(7.230)

SEDOR,

For these we need to insert for (7.214) respectively Basic S-flip only:

X

S'(lI')Xs(l'I")

Conventional I echo:

X/'(l'I")X I (1'I")

SEDOR,

X

s

'(ll')X/'(ll')XI(ll')XS(7r)

so that at time 21' we will have had for the transformation of the teon exp [i(ilII:+ nss: - aI:S:)1'} Basic S-flip only:

exp [i(ill I: - ilsS: + aI:S:)1']

Conventional I echo:

exp (i(-ilrI: + ilsS: + aI:S:)1')

SEDOR,

exp [i(-nII: - ilsS: - aI:S:)1') .

As in the transfoonation from (7.217a) to (7.2l7b). we can combine the teons involving exponentials of ilsS:(t - r) or flSS:1' on the left of (-I,,) with their inverses on the right of (-I,,). What counts are the tenns involving I: or 1:5:. Thus we see that: The basic S-flip-only 1'1" pulse changes the sign of the a teon but leaves the sign of the il/ alone. Therefore, it refocuses the spin-spin coupling at t = 2r but leaves the I-spin chemical shift alone. The conventional echo 11" pulse changes the sign of both spin-spin and chemical shift terms. Therefore it refocuses both spin-spin and chemical shifts at t = 21", eliminating both. The SEDOR 1l' pulses change the sign of the chemical shift tenn only. Therefore they refocus the chemical shifts, at t = 21', effectively eliminating them. but leaving the spin-spin coupling untouched. A useful variant on the SEDOR pulse sequence was introduced by Wang et a1. [7.53.54). The time of the I-spin 7f pulse is held fixed. The S-spin 7f pulse

This sequence has two advantages. The first is that one can work at very short times. T. while keeping the I-spin echo delayed a convenient time. This feature eliminates I-spin amplifier recovery problems. The second advantage is that the decay of the I-spin echo with pulse spacing (by T2 processes) is held constant as T is varied, and thus eliminated from the result.

7.22 Two-Dimensional IT Speelra - The Basic Coneepl We now arrive at two-dimensional Fourier transfonn magnetic resonance. one of the most exciting and powerful developments since the original discovery of magnetic resonance. In this section we introduce the concepts using a very simple example, the basic S-flip-only experiment discussed in Sect. 7.19. We shall find that it already contains all the essential ingredients of two-dimensional magnetic resonance. The idea of two-dimensional Fourier transfoonation is due to JeenU. He first presented it in lectures at the Ampere International Summer School. Basko Potze. Yugoslavia, in 1971 {7.55). An excellent review by FreemlJn and Morris [7.56] describes the explosive development of the field, and states and documents that "Ernst was the first to appreciate the great potential of the method". Indeed, both Freeman and Ernst with their collaborators provided many of the fundamental applicalions. Aue et al. gave a treatment of basic significance and broad generality [7.57] which also gives a detailed treatment of concrete applications. It is a fundamental reference for workers in this field. Recent reviews include those by 8ax [7.58} and Turner [7.59}. We shall describe in later sections some of the panicular fonns of double resonance, but OUf goal in this section is to introduce the concepts using S-f1iponly double resonance (Sect. 7.19). We saw in Sect. 7.19, (7.177), that for a spin I off resonance in the frame rotating at WI by ill =wOI-wl

the magnetization was 318

319

(Mt(t» '" i~O(e-i(01+0/2)t+e-i(01-0/2)t)

(7.231)

of strength

(M/(t» '"

(where the -fl , means the sense of rotation is negative) for t < t... and

(Mt(t» '"

+ i;Obe-iOatle-iOut2(eibI2/2 +e-ibt2/2) (7.232)

i~Oe-iO/(21.)

and we would get

lal

(2t~, 2t:. elc.) 2t'lr

<,

<

<,

(t - 2t~, etc.)

<,

,11 ""'-'_,.J-

_

".

2t.

t.

<

v-

lei /

/

"

v-

<, <,

,,

/

(7.233)

< Idl

I I I I I I

/ /

In terms of t, and t2, we have that for times after t"

(Mt(t» '"

Ibl

/

<,

r...

tl '" 2t'lf

<

/'

This, as we remarked at the end of the previous section, means that at t '" 2t.... the magnetization is just what it would be if the spin-spin coupling were zero. so that only the chemical shift fl , '" WO/-WI remains. [We here assume a unifonn static field, so that (ho), arises because of chemical shifts.] The situation is illustrated in Fig. 7.33 where we show (Fig. 7.33a) that for limes greater than 2tll"' the phases 4>1 and 4>2 act as though from 0 < t < 2t'lr there were only a chemical shift, but for t > 2t1l" both chemical shifts and spin-spin splittings exist. In Fig. 7.33b we have redrawn the situation to represent the situation of a '" 0 for 0 < t < 2t'lf' H we reset t'lf to a new value. longer than t'lf' the new situation is shown in Figs.7.33c and d. We can plot phases for a sequence of values t,.., t~. t~. etc., as in Fig. 7.33e. These plots can be summarized by defining two times. tt and t2 • by

t-

(7.237)

=2t .... we then have

(Mt(t» '"

t2 '"

i~Oa e-iOJ&t'e-in/&t2(eiot2/2 + e-iaI2/2)

i~Oe-iOlt

x (e+iat./2e-io(t-t. )/2 + e-iat./2e+ia(t-t. )/2) for t > t'lr' Exactly at t

b. we could label the chemical shifts as fl/a, n,b. and then

,.•

I I

2t~

21~

i~o e-inltle-inJt2(eiat2/2 + e- iot J/2)

" !(tj,")

(7.234)

This expression is derived for t) ~ 0, t2 ~ O. If at this point we treat this theoretical expression fonnally as existing for -00 < tt < +00 and -00 < t2 < +00. we can take its Fourier transform with respect to the two time variables tl and t2, to obtain I +~ g(W\,W2) == (211-)2 e-iwltle-w;1!2 !(tl.t2)dt, dt 2 (7.235)

lei

<

J

-~

Utilizing Ihe relation 6(x) :: (1/211") L~:: exp (ixt)dt. we get

ieo g(wj,"")' 26(w, + il,)[6(..., + ill - a!2) + 6(..., + ill + a!2)J. (7.236) These dala can be represented as points in a two-dimensional plot (WtoW2) (Fig.7.34a). If there were a second chemical shift and a second spin-spin spliuing.

320

2t~

2t~

2t~·

.'~g. 7.33a~ P~1lSC:S of the two I-spin magnelization components Yersus time for the SnIp-only ex~rllnent. The two phases ~I(t) and 412(1), showing the time t. aL which the ... pulse lS applied lo the S-spins, and the formation of the echo at t '" 2t which refocuses the spin-.pin splilling. H the spin-spin coupling were zero, the ~'s wo:k1 both follow t~e dashed curve of slope 0 1 rq>resenling a chemica.! shift only. (b) The ~hll.vior of. eqUlVlllent, for! > 21 .. , to that of (a). (e), (d) Replot.1ll olea) and (b) f('$pectively for .. value t~ >~". (e) A family of phue trajectorir$ colTesponding to a family of experiments with different times I. (t~, I~, de.)

321

IbJ

la)

0. ,, I I I

-IO~-bI21

--~-------""1-

-t QI,·bI21

-----------t-II

I

iI

---+------t--

iI

I

-!n I -aI21 ---~--

_ 0,

-ttl1u12 1

I

,, ,

I I

-Q l ~ -Q h -Q~~ Fig. 7.34. (a) The lwo-dimensional plol of frequencies WI and W2, 'lhowing poinL5 in "'I corresponding to the chemical shin WI = -(af ±a/2). (b) A plot similar lo (a) corresponding to two I-spins, each coupled lo a single S-spin with corresponding chemiCl\1 shift!! aid and nf6 l\nd spin-spin couplings a and b

iCO a 9(Wl.W2) = -2-5(wl + n!a){5(W2 + !1-la - 0/2)+ 5(W2 + n la + 0/2)] iCOb

+ T5(w, + n ,b)(5(W2 + n,b - bIl) +6(w, + [h, + WZ)J

17.238)

which would give {he plot of Fig.7.34b. This plot allows us {o go along the WI-axis {O detennine the values of the I -spin chemical shifts, then at each chemical shift position (o observe in the W2 direction splitting of those panicular I -spins by the S-spins to which the)' are coupled. That is, there is a correlation of the WI frequencies with the W2 frequencies. The physical origin of this correlation can be traced back to the phase diagram Fig. 7.35, which shows that the single phase line over the time interval 0 10 2t7: splits into two lines for limes after 2t7:' For t > 2t7:' those

lines extrapolate back to an intersection at t = 2t ... The two lines are diverging from their intersection since the chemical shift frequency fl/ is split into two frequencies. n/ ± a/2.. for times after 2t... The example illustrates the general feature of two-dimensional schemes that there is some initial preparation of the system [e.g. the X t (7f!2) pulse], then the system evolves for a lime interval tl under one effective Hamiltonian (in our case, one in which the spin-spin couplings are zero), then data are collected during a second time interval, t2. for which the Hamihonian lakes on some new effective Conn (in our case the full Hamillonian). 1lle experiment must then be repeated for other values of tl_ These three intervals are often referred 10 as the preparation, the evolution, and the acquisition intervals, respectively. The experiment we have just described is in (act essentially the famous experiment invented by Maller et al. [7.60] to resolve the C l3 chemical shifts and C I3 _H I spin-spin couplings. They demonstrated the technique using n-hexane (Figs. 7.35 and 36). They utilized broad-band (noise) decoupling of the protons instead of applying a "" pulse as in our example, and they decoupled during the second time inlerval. t2. instead of during the first time inlerval, tl' as we did in our hypothetical example_ They point out one could choose to decouple during either interval, and remark that since each point tl requires a separate experiment, one needs less data if the tl spectra is chosen to have the fewer spectral lines, Le. is the time during which the protons are decoupled. In a later paper, they describe the sequence of our example [7.61]' This general method is referred to as J-resolved 20 NMR in the literature. Qoli"CH;Qol;0i;CHrQol3 abccba

'"

.,

H'

". 0'.0

I,

I,

Fig. 7.35. Schematic representation or one ronn of C 13 2D-re;olved Spedroaoopy aner Miller ct al. 17.601. The CI3 nuclei are the I-spins, Iii thfJ S-spins. For the inlerval II the fullUamiltonian acts, but during t, the spin-spin interaction i, lUnled ofT by broad-band decoupling. The situation is similar to S-f1ip-only resonance with the role of the times t l lind t2 inlerdlllllged

322

Fig. 7.36. DlIla of Muller, KUI1t(Jr, and Ernsf using lhe pulse !lCquence of Fig. 7.35 for the CI3 spectrum of n-hexane (Clb-CII,-CII,-CII,-CII,-Clb). The ""I-axis gives the combined chemical shin and spin-spin spliuing, the loo'2-axiS the chemical llhirL5. Thesoe data show how a 20 display enables one to lIl""atly simplify unraveling a complex spectrom. TwentylW() experimenL5 Vooe"" coadded for each of the 64 values of tl between 0 and 35 IllS. The authors state lhat the resolution is seve~ly limited by the 64)(64 dab. matrill: used to represent thfJ 20 Fourier lransform. The absolute values or the Fourier coefficients a"" ploued. The undecoupled I D spe<;trum is indiCllted along the ""I-llll:is and lhe protondecoul)led spectrum is shown along the W2-all:is

323

I

'I

We shall discuss funher examples in later sections. First, however, we need to take a side trip to deal briefly with a topic the reader will quickly encounter in studying the literature, the problem of 20 NMR line shapes.

we can select either the real or the imaginary part of 9(Wl) and thus either the absorption signal or the dispersion signaL If, however, we take only the real part of g(WI,W2), it involves (for A real) (7.244)

7.23 Two-Dimensional FT Spectra - Line Shapes In the previous section we developed the Fourier transfonn of the theoretical expression (7.234) by fonnally extending the meaning of t1 and t2 to negative times. This is easy to do given a theoretical expression. If, however, we have experimental data, we cannot in general guess the theoretical expression. One then encounters a problem which Ihe reader will immediately come across if he reads the literature of 20 NMR. We therefore take it up briefly. A thorough treatment is given in the basic paper of Aue et al. [7.57]. Let us therefore reexamine how to treat experimental data considering positive times only. Since the problem is one of mathematics, we assume we have the simplest fonn of experimental data given by an exponentially decaying sine (cosine) wave. We put in the decay since such decays are always present, and to avoid difficulties at infinite times. Thus, suppose we have (7.239)

Then

1 9(Wt> W2)"" (211-)2

=

J =J dtl

o

i(OI

WI)J[a2

i(02

W2)]

O't+i(OI-WI) O'2+i(!t2-w2) = (211')2 + (0 1 - Wl)2 O'~ + (02 W2)2 A

(7.240.)

0

A

"" (211')2 [al

OI

(7.240b)

(7.24Oc)

In the corresponding one-dimensional case l(tl) = Ae(in\-aJ)l\

J

(7.241)

1= g(Wt) = 21l'

e-i"'lt\ l(tl)dtl

(7.242)

o we get

A 01 +i(JtI -WI) 211' + (01 - wd 2

( ) =gWI

O'I

(7.243)

We recognize that the real pan is the Lorentzian line shape (the absorption), and the imaginary part is the dispersion. If we use quadrature detection (Sect. 5.8) 324

A Re{g(wI,W2)}=-

al

2

0'20'1

(n

+ HI-WI

)2

(7.245)

which is the WI absorption line. However, the practical problem is that since dispersion extends out farther from the line-center Ihan does absorption, the mixing of dispersion tends to decrease the resolution between adjacent lines. There are various methods of processing data to deal wilh these maners. In addition to the treatment of Aue [7.57], the review articles by Freeman and Morris [7.56], Bax [7.58], and Turner [7.59] discuss these issues.

7.24 Formal Theoretical Apparatus I The Time Development of the Density Matrix

..

dt2/(ll,t2)e-llU\t\e-I"'2t2

1

which is a mixture of absorption and dispersion. Of course, if one goes along the frequency contour W2 - O2 , one has

In discussing the S-flip-only experiment (Sect. 7.19), where we were applying 'II' pulses to the S-spins, we assumed that the effect of the S-flips was simply to shift the precession frequency of the I-spins suddenly from one ms multiplet to the other. An S-spin originally pointing up now points down, and vice versa. If, however, one applies a '11'/2 pulse to the S-spins, classically they are now perpendicular to Ho. How then do we treat what happens in a quantum mechanical system? Classically, if Sz is zero, the S-spin will not shift the I-spin precession. That would lead to a precession midway between the two I-spin lines at WOI+a12 and WOl - a/2. But there is no transition al WOl among the eigenstates of the Hamiltonian, so this answer cannot be right. The next closest thing is for the spin to act as though it now has both frequencies. Thus, a spin which precessed at WOI + a/2 before the S-pulse would have components precessing at WOl + a/2 and WOl ~ aI2 after the S-pulse. As we shall see, this latter answer turns out to be correct. We clearly need a systematic and rigorous derivation of the result. Let us therefore treat carefully a concrete problem, a pulse sequence X 1(1I'!2)· .. r ... Xs«(J)···t-r

,

where we will explore and contrast what happens when (J "" 11' with the situation for (J := 11'/2. We wish to calculate Ihe expectation values of 17; and I y in the I-spin rotating frame. We shall use the doubly rotating frame Hamiltonian. 325

We assume that the X,(1C(2.) pulse is applied at t == O. At t == 0-, just before. we assume the spin system is in thennal equilibrium. (We also treat me case of 100 % polarizalion of the S·spins.) We employ the general fonnula (I+(t)}=Tr{I+en(t))

(7.246)

where 1+ refers to axes in the rotating frame. as does llR· Since we will be working almost exclusively in the rOlating frame, we drop the subscript R to simplify the nouttion. . Then, assuming 1{ is given by (7.162) except dunng the (Hlh and (HI)s pulses, we characterize states by m, and ms. getting

(1+(t»

=

L

(m/msII+e(t)lm/ms)

m"ms ==

L: (m,mslr+lm',ms)(m~mslelm[ms) "","'S "',''''s

(7.247)

Making use of the propcnies of the raising operator and of the eigenstates Im,ms). we have (m,mslr+lm/ms) ==

and -(fl[ + all). Therefore, no matter wfu:lt rotation angle we give the S-spins, these are the only frequencies we observe. At most we can change the complex coefficients

DmsmsDml,I/26m~,-1/2(,II+I-~)

(-mSIel + ms)exp (i(fl, - oms)ttl of the time-dependent tenns exp [- i(fl, - ams)t] Since the coefficients are complex, we have at our disposal the amplitude and phase of the two frequency components by manipulaling (-mslll(t\ >l +ms). To follow what happens in the actual pulse sequence, we use the method outlined in (5.167) in which we divide time into intervals during each of which the Hamiltonian can be taken as time independent. In SO doing, we will use the resulls of (5.253) and (5.254) for the time dependence of II under the action of rf pulses. We must first, however, generalize these equations which apply to one spin for the case of two spins. In so doing, we consider the effect on II of interactions in the rotating frame. Utilizing the concept that these interactions are large compared to the spin-spin coupling, we employ (7.206) and (7.2m) to treat the interactions one at a time:

:: Dms.m'sDm,,1/26m,,-1/2 Abbreviating +~ as +,

-! as -

(7.2530)

for m[, we thus get

1{s:::: -7S',(HI)SS~

(r(t)} = L(-msle(t)1 + ms) (7.248) ms Thus, there are only two matrix elements of e needed, one for each value of ms·

. During those times when me rf pulses are not turned on, we have, usmg (5.133), that

e
(7.250)

using the notation of (7.176). As a consequence

~(m,mslelm"ms)::*(m,mslll'HS : : r; L

'Hselm/ms)

(m,msle
mi',llI s

(7.254)

(m'lmsl1islmims) == 6m7m,(msl'Hslms) we get

(-msle
(7.251)

Thus, if we know II at time tt. we can express it for later times as amS)(t -

't)J .

!!.(m,msle(t)lml,ms):: dt

*L I

[(m/msle(t>lml,ms)(msIHslms)

m "s

-(msl1islmsXm[msle
This expression, even without specifying (-msle
(7.253b)

Since

11.:: -hfl,l~ -1iflsS~ + hollSl

(r(t» = L(-msll?<'I)! + mS)exp [- iUh m,

-1iwISS~

Thus, taking just Hs

(7.249)

where

==

(7.255)

Equation (7.255) shows that if we are driving the S-spins only (Le. 11, == 0, Hs ,;. 0), the I-spin quantum numbers do not change. There is therefore no mixing of matrix elements (m/msl/?(Olmims) with matrix elements such as (m,msle(t)lm'lms)' The problem is thus transformed to a one-spin problem. 327

As a result, we can immediately utilize the results of (5.253) and (5.254). Note, however, that though matrix elements of 7-{s are necessarily diagonal in ml, matrix elements of e are not. This is a very important point to which we will return later in discussing coherence transfer and multiple quantum coherence. Thus, under the action of 7-{s, acting for a time t, we can write (using ms for -ms) for the matrix element diagonal in ms:

(-mslg(tll +ms) = ![(-mslg(O)1 + ms) +(-mslg(O)1 +iiisl] +W-mslg(O)1 +ms) - (-mslg(O)1 + ms)] cos wIst i

+2:[(-mslg(O)1 + msl - (-msle(Oll + ms)J sin wlS'

(7.256)

We now let e
T

until time

T-

just before the S-spin pulse.

Utilizing (7.250) we get

(-msle(T-)[ + ms) = exp [- i(n , - amS)T](-msle(O+)1 + ms) =+iloexp[_i(n,_amS)T).

(7.261)

2 We now consider twO cases for the Xs(U) pulse, 8 ::: 7f and 8 ::: 7f12. Since (HI)S will not change the ml quantum numbers, and since (I+(t» arises only from teons ofT-diagonal in m/, we need consider only the effect of (HI)S on the matrix elements (-msle

(r+(t)). = ~

+

(7.263.)

(7.263) below. (7.258)

But

32.

(-mslg(T+ll + msl = W-msle(T-ll + msl + (-msle(T-ll + iiisl]

L: [(+mslg(O-)1 + msl -

T,

(-iiislg(O-ll- iiisl]

ms x exp [ _ i(n, - ams)r] exp [ - i(n , - ams)(t - T») .

(7.264l Utilizing (7.260), this becomes

(I+(t»ll"::: i~o "Ee-iI1,teiams(t-27')

(7.265)

ms

This result is exactly what we deduced in SecL 7.18 if one replaces T by the notation tll"' For t ::: 2T it displays the refocusing of the spin-spin coupling. ~e teon involving gives the chemical shift. lfwe set il, equal to zero and utlhze (7.260), (7.264) is identical to (7.171).

n,

iIo

2

(7.260)

329

For the case

e=1ffl we get

([+('»./2 =

~L

!«(+mslu
ms

x exp [ ~ i(lll - amS)T] + [(+insle1 + inS) - (-mslu
FIg. 7.33. The phllSe$ ~ of the I-spins wh)ch rault from application of a 11:/2 pulse to a 100% polarized S-spin sample (ms initially) at t = T. Bd'ore T, only one ~ is needed, corresponding to the single value or ms. The 11:/2 pullJe creates ill situation in which both ms = and _.1 arc present, hence leads to two ph_. Note that the same two precession frequcncies arc present as in Fig.7.37, but the amplitudes or the components differ corresponding to the different initial mS polarizations of the two e _

=1

•

+t

(7.266)

Again, utilizing (7.26), we get, for t > T,

(I + (t»

•

/2 =

iIo IonIt (e -e2

iflT

/2 +e- i '""/2) 2

x (e ia (L-T)/2 + e- iu (t-r)/2)

(7.267)

~is expression has a simple graphical interpretation (Fig. 7.37). The -(nIt) term gives ~e phase shifl arising from chemical shifts. The two terms exp(iatl2) and

exp (-Iai(l) show thai up umil time T there are two phases of magnetization, analogous to the ;1 and (/12 of Fig. 7.21. The product with the last bracket shows thai al t '" T. each of these phase Jines splits into two lines.

-(nl - a/2), the other precessing at -(nl + a/2) (Fig. 7.38). Thus the average phase angle advances at frequency -ill, as though there were zero coupling, but the effective lenglh of the average magnetization goes as cos(a(t - T)I2) and hence is not constanL The fact that the average phase is independent of a for the w:(1. pulse has as its classical analogy that the S-spins are pointing perpendicular to Ho, hence produce no shift in the I-spin precession. Indeed, this is the physical consequence of (7.263a) and (7.263b), which together show that even if only ms = is present after the rr(1. pulse, bolh ms :::: and ms = -~ are present after the XS(Tr!2) is applied. The sorts of considerations we have been exploring are treated in a different manner in two illuminating papers by Pegg et al. [7.62], and applied to various double resonance methods.

!

• Fig.7.37. The phllSe$ .p of the I-spins which re!lull from Ilppliulion of II. 11:/2 pul$C to unpolarized 5spir'lll at time T. Up until t = T, there are two phllSC$, one for each value of ms. At f = T the phllSe$ each split in two, but the t/opu are rela~ed so thu there are still only two pMCelSion frequencies for the 1. spins, although there are HOW four phllSe$

We can get added insight by taking the S-spins to be 100% polarized, so that ll[ t 0- only mS is to be found. Then, for t > T

=+i

=

(I+(t}}'/2 =

~[(++ Iu
(- + lu(O-ll - +)]

x exp [ - i(!h - a/2}T]

x (",PI-;(J)/-a!2)(t- T l];ex P[-i(J)/+a!2)('

T)]} (7.268l

This shows that up until t = T, we get precession at a single angular frequency, -(n l - an), corresponding to ms = but that after the 7:(1. pulse at time T, the precession splits into two components of equal amplitude. one precessing at

+!'

!

7.25 Coherence Transfer Now that we have employed fonnal methods to follow in detail the role of w:(1. or pulses, we return to one of the important applications of spin-coherence double resonance, the transfer of the magnetization of one set of spins to the other. The method of doing this transfer was discovered by Feher in 1956 (7.63,64]. In 1962, Baker discovered a related fonn of nuclear-nuclear double resonance which he called INDOR. Baker pointcd out that it was much like Feher's ENDOR experiment [7.65]. In 1973, Pachter and Wessels [7.66] invented a pulsed version of the INDOR experiment, which they named selective population inversion. leener's proposal {7.55] for two-dimensional NMR, as we shall see in Sect. 7.27 includes among other things a homonuclear pulsed version of the Feher experiment. Then a heteronuclear pulsed version, with all the benefits of Fourier transfonn, was proposed by MaJuisley and Ernst [7.67], with modifications by Bodenhausen and Fneman [7.68], Maudsley et al. [7.69J, and Morris and FnelTliln (7.70).

'If

3JO 331

Feher was studying the electron spin resonance of donors in Si doped with phosphorous. His work was part of the research that led also to his disc:overy of ENDOR (Sect. 7.7). He showed that he could transfer the electron polarization to the p31 nuclei. Since his experiment is conceptually very simple and straightforward, we begin with it. In keeping with the model system of Sect. 7.19, we assume the energy levels are those of Fig. 7.39a labeled by jm/ms), and also labeled 1,2,3,4 as shown. In thermal equilibrium, the populations of each state are given by the Boltzmann factors, which, in the high temperature approximation, may be wriuen as

p(m/,ms) =

~

[I

+

(nWOlm/ :;iwosms)]

(7.269)

where P(mb ms) is the probability of finding a given 1-$ pair in the state Imf, m~). In this ex.pression we have neglected the small spin-spin coupling in compUt1ng ~he .ener~es of the levels. The leading term, does not pr<Xiuc:e any net magnebzabo~ Sl~nals, so it is only the terms involving m/ and ms which express net polanzallons.

t,

la'

l_._

1

fbI 4

-i-

1.-0.

---i°.1

0

oA - 6

6

Fig. 7.39. The energy levels (a) and therm.al equilibrium population I)robabilities (b) of the model system. States are labeled accordlllg to the convention lm/ms)

L:

(Iz) = =

Lml [LP(ml.m s)]

hwOI

8kT ;: Ll

(7.270.)

!twos

8kT ;:,

L ms [LP(ml. ms)]

ms

t + (.1 + 6) p(-+):::: t - (.1 - 6) p(++)::::

Slate 3 P(+-}=1+(Ll-6) 332

(7.273)

III/

Now, though we have neglected the spin-spin coupling in computing the thermal equilibrium populations, we must tIOt neglect them in calculating the spectrum since they make transition 1-2 have a different energy from that of transilion 3-4. Since the" in (7 .27Ia) is the same in all energy levels, it has no effect on the intensity of any transition, and can thus be conveniently dropped in recording population probabilities, giving one the population excess above (A negative excess is a deficiency.) These are conveniently expressed in a matrix

t"

i.

mS=l

ms=-!

m/

"'-!

- .1+S

- Ll-,

m/

"'+~

Ll +'

Ll- ,

(7.274)

Thus, the resonances for both I-spins and S-spins are doublets, as shown in Fig. 7.40. For the sake of explanation, we assume the I-spins have a much larger population difference (Fig. 7.403) than the S-spins (.1:> S). So that we see a larger intensity in the I-spin absorption lines (Fig.7.40b). Feher's method consists then of two steps. In the first, he makes an adiabatic passage of frequency w/ across one of the I-spin transitions (but not the other) such as the transition 1-2 for ms '" Since the adiabatic passage invens the magnetization, it interchanges the populations of the states involved, I and 2. This step leads to the population excesses shown in Fig. 7.4la, and listed in the following matrix: ms'" ms"'!

-!

- Ll-' Ll- ,

(7.270b)

we have for p(m/,ms)

Slate 2

(7.272)

ms

ml

(S,) =

(m/msIIzlm/ms)/l(m/ms)

+!.

Defining

State I

(7.271d)

The spin polarizations are

t -A·6 1-"'06

fmjmsJ

P(--)=t-(.1+6)

State 4

(7.271.) (7.271b) (7.2710)

(7.275)

The S-spin population difference corresponding 10 a tranSition in which ms changes, such as fTOm 1- +) to 1- -) (states 2 (04), would be given by

P(-+)-P(--)=(1 -Ll+6)-<1 -Ll-6)=20 bl!fo~

(7.276.)

the adiabatic passage, but 333

Energy

Energy

F1g..':40: (a) The energy level and lhcrmal eqUlhbrlUm populations (sc.henlalic) for a CIlS(! in ~hich "'II" "'Is !lID that 11" 6. (b) The I-spon and 5-spin resonance intensities sho,,:,in~ the ~pin-spin splitting. The grealer I-spm lIItell.'llty an!leS bec:aUS1e "'If:> "'Is

la) 1- .J

""/---

r---""

+i

P(++) - P(+-) = 26

(7.r17.)

P(++) - P(+-) = -2il- 26

Population

I-Spin Absorption

S-Spin Absorption

Energy

lal

(7.277b)

which, for Ll"~, is an invened population. The population differences for both (7.276b) and (7.277b) are much bigger after this first step than before. Since (7.277b) corresponds to a population inversion of the S-spins [the upper energy state ImJms = 1+ -) has a larger population than the lower energy state ImJffis) = 1+ -)], there is no net polarization of the S-spins. To achieve a net polarization, one may therefore inven the population difference of eidlCr (7.276b) or (7.277b). Feher achieved this by his second step, an adiabatic passage of (H 1)5 across either (but not both) of the S-spin doublets. Thus, if the passage were across the 1-3 transitions, the 1-3 populations would be interchanged leading to

"s; 112 .s: -112

Energy

-!,

after the adiabatic passage. Thus, for ml = the S-spins have an enhanced polariz,ation. On the other hand, for the S-spin transition involving ffil = (the transition between states I and 3), we have a population difference before of

but after of

Population

Ib)

(7.276b)

p(-+) - P(--) =(t + il +6) - (t - il- 6) =2il+26

1--1

"'5= , "'5=-'

""1---

}---,.-,

il + 6

- il- 6

il- 6

- Ll +0

(7.278)

Using these populations, one gets readily

(5,) = !(il+6 - (-il-6)+(il-6) - (-il + 6)] = 2il

(7.279)

compared to its value before the two Steps Population

Population

(5,) =26 t-SjNl Absorption

S-Spin Absorption

Ibl

ml "1I2 ml :-1I2 li. 7,.41. (a) The ~ergy Jevel population. afler an adiabatic '--a•• 'hroo.h ,h. 1 . ,~me or rns = +.L .how· lh ,.. ..,..-spon level hassmal/u l ' I .1IIS e popu.atlon lIIverslon III which the lower energy 1+ +) popo atJ:On lh~n the hIgher eners,y +) level. (b) The J- in a d 5- . res
ms:1I2 "s·-1I2

1-

.

(7.280)

Such a scheme of magnetization transfer can be used for transferring electron (I-spin) polarization to nuclei (S-spins), or for transferring large "'( nuclear magnetization (protons) to lower..., nuclei (e.g. C 13 ). In order to be able 10 carry out an adiabatic inversion of one multiplet, one must be able to pass through the multiplet with an HI sufficiently small to inven one but not the other lTansition. Thus "(HI must be less than a. AnOlher requirement is that spin-lattice or spin-spin relaxation times not prevent the adiabatic condition in this time interval. Thus, since it takes many precession periods I/"'(H1 to have the passage adiabatic, we have ,H) :> lIT" 11T2. The Feher approach to coherence transfer using adiabatic passage can be replaced by a pulse technique, the method used by JUller [7.55J and by Maudsley and Ernst [7.67]. Although superficially (7.279) and (7.280) suggest that one is

334

335

performing an Overhauser effect, in fact, since the method does not use spinlattice relaxation, it is clearly based on different principles. We now show how to use pulses to invert one hyperfine line. Consider an WI tuned exactly to the unsplit I-spin resonance frequency Wo/. (7.281)

WI = WOI

AI t = 0 we apply an XI('If(l.) pulse (Fig.7.42a). The magnetization vectors of the mS and mS lines are shown in Fig.7.42b at t 0+. We label them MI(~) and MI(-!) corresponding to ms = and -~ respectively. They now precess in the left-handed sense at angular frequencies (WOI - amS) in the lab frame, or at -oms in the rotating frame. (This is equivalent to precession in the rotating frame frequency oms in the right-handed sense, see Fig.7.42c.) At a time, to, such that

=!

=-4

=

+!

oto = 1f

(7.282)

they are pointed opposite to one another as shown (Fig.7.42d) along the x-axis. At this time (Fig.7.42e), we apply a 'fro. pulse with (HI)! along the y-axis Ia Y/(1fn) pulse]. This puts the M I (+!) along the negative z-axis, and MI(-!) along. the positive z-axis (Fig. 7.42f). Thus, we have achieved a population inline, while leaving the ms -~ line with a normal version of the ms population difference. This is equivalent to an adiabatic passage across just t~e ms = line. There are two important points to n~te. The ~rst .is that t~e sphtting, 0, is essential since il is responsible for prodUCing the suuatlOn of Fig. 7.42d in which the two components of AI I are pointing opposite to one another. The second point is that the phase of the (H 1)1 pulses (Le. their orientation in the x-y plane) is important. If, for example, the second pulse had been another XI(1fo.), it would have been parallel to the MI vectors, and would have had no effect on their orientation. Likewise an X('lfo.), where X means (H')I pointing along the negative x-axis, could be used to obtain a state in which the MI(-!) would be inverted instead of MI(!). So far we have only inverted the ms = transition, therefore we have achieved the effect of the first of Feher's twO steps. To complete the coherence transfer, we must now do a similar thing for the S transitions_ Thus, with ws tuned to Wos, we can apply the sequence XS(1fo.) followed at a rime delay to = 'If/a by a YS(1fn) pulse to invert the Af, = transition for the S-spins. This would have completed Feher's coherence transfer and produced the populations in the matrix of (J .278). If instead of applying four pulses we apply the sequence

=+!

=

!

!

lal

L [H,.II

lei 11,11/21

Ibl~

Hl ll121

MI I-1I21

+i

"

-I

"

Idl

, ,,

"

t1 1 11/21

~

" (fl

"

(a)

Mil-liZ!

[Hll l

"

"

Fig.7.428-1. The effect of (lid! pulses tuned to resonance (WI = ,",od on the I-spin magnetization M,(ms) for the two components ms = and ms _1. (a) At I 0, (Ildl is applied IIlong the z-lIxis in the rotRting frame giving an x/('II'/h pulse; (b) I'll I = 0+, both At I (t) and M l( lie along the y-axis; (c) the two components M /(mf) precellS in o[.posite diredions at angular frequencies ±a/2; (d) at a time t 11:/0, 111'(2) and MI( - ~) point along the -z- and +z-axes respectively; (e) an (HI)I lying along the y-axis gives a Yl(lr!2) putse; (f) the I-magnetization vcdors after the Yt(,../2), with MI(-i) pointing parallel to the ~-dire<:tion, but MICt> pointing antiparatlel

+t

336

L (H,ls

1'1 1 11/2)

-!)

•

(7.283)

the I pulses will have produced a situation in which the M I (!) population is inverted, bUI not Ihe MI(-!). so the situation will look like Fig. 7.4Ia. Therefore the XS(1f(l.) pulse (Fig.7.43a) will produce a silUation in the rotating frame such

1'1[1- 112 1

" lei

X,(.12) ... (./a) ... (y/(./2). Xs(.12)) .. . observe S

/11.1-1121

=

=

Ibl

" Mi l-1I21

" Ms !1I21

(e I

=

"

"

~·Ig. 7.43. (a) (/fils is applied along the :I:-dire<:tion in the S-llpin rotating frame, giving an Xs(rr/2) pulse; (b) the SlIpin magnetization Ms(m,) jUlIt after the Xs('7f!2) putlle. Ms(_l) and Ms(+l) are antiparaUel; (c::) /lS a result of their dMerent precell6?on frequencies, M s( ~) and M s( -~) are parallel after II. time 1t!IJ

337

as shown in Fig. 7.43b in which the two components of Ms point opposite to one another initially. At later times, their relative phase will have changed, making the magnetization components, in fact, parallel to each other at time 1r/a after the XS(7f/2) pulse (Fig.7.43c). Indeed, following the XS(1f/2) pulse, there will be an S·spin free induction signal. In the lab frame, it can be thought of as two signals, one at wos - 012, the other at wos + 012, or it can be viewed as a signal at wos which is amplitude modulated as sin al/2. This signal is of course similar to what would result from using Feher's two adiabatic passages to transfer the I polarization to the S-spins, followed by a 1r/2 pulse to the S-spins to generate magnetization perpendicular to Ho. If one performed that total sequence using pulses instead of adiabatic passage, it would take a total of five 1f/l pulses (two of the I-spins, three for the S-spins). However, the last two "/fa. pulses can be omitted since the first of them rotates transverse S-magnetization to lie along Ihe z-direclion, and Ihe second rolates the magnetization from the z-direction back to a transverse orientation. In their technique of selective population transfer, Pach/er and Wessels [7.66] apply a 1f pulse to one of the I-spin transilions, invening their population difference. They use an (Hth much smaller than the spin-spin splitting. Then they apply a strong 7f/l pulse to the S-spins 10 observe the free induClion decay. The Fourier transform of the S-spin signal shows one hyperfine line invened with respect to Ihe other. and a large signal increase if "'(, »"'(5' Thus their experiment is equivalent to (7.283). For the sequence (7_283) we have required that = wo" exaCI resonance for lhe I-spins. For Ihe S-spins, XS(1fn) does nOI have 10 be exactly at resonance. It will still produce the situation of Fig. 7.43b. Then, those magnetization components will precess freely at (wos - am,), giving a signal at WO/ which is amplitude modulated at angular frequency o. The sequence (7.283) is almost the Maudsley-Emst sequence. What they do is (i) remove the requirement that WI = WOI and thereby (ii) remove the special phase requirement of the second I pulse. Then, taking a sequence

w,

(7.284)

in which t2 is the observation time. and for which (7rn), means an I-spin 1f/l pulse without a special phase condition. they make a 20 Fourier transform. The fact that WI "WOI produces a situation in which. following the initial (7f/l)/> the I-spin components precess both with respect to one another and also with respect to the rotating frame. Thus, if one is off resonance by an amount large compared to the spin-spin splitting, the pattern of Fig. 7.42f will rotate rapidly at (WOI -wI) in the frame while the relative orientation of M/(!) and M,( will change slowly. The situation is shown in Fig.7.44. The initial X,(7fn) (Fig.7.44a) puts M,C!) and M/(-!) along the y-axis in the frame rotating at (Fig.7.44b). Under the combined influence of chemical shift (il/) nnd spinspin spliltings the M, vectors rotate and separate (Fig.7.44c). A pallem of the }.th's at successive times is shown in Fig.7.44d-h. If another 1r/2 pulse with

w,

w,

338

-!)

101

L IHII I

Ml l1121 M11-1I2)

"

" r,

Idl

y, IOI-al2)1

1'1

.

Ibl

M111I2)

to l • anJt

" lei

t-

............ ~t-V21

y,

If J

MI II121

,1 M 1-1/lJ ,1

MI III2I

,

" I

I

/

Ihl

y,

191

"

,,M I-1I2)

"

I

y,

M11-l/l1 MI I-ll2)

----

-

Ml l1l2l

'.

" M1l1/l1

T

=

Fi&o 7.44a-b. The effect of the. '_spins bei~g off resonance ("" 1+'0,). (a) At t 0, (lid, is applied along the lI;.axis m the rotating frame; (b) the .)1.,(_/2). pulse puts th\t'"':O componenlll of MI(ms) along the y-axis at I = 0+. M/(t) IS the solid arrow, M/( - 2) IS the dashed arrow, (c) the componenls precess at 01 -: am~, where.OI = ""01.- "'I, un~er the combined effect of the chemical shif~ (flt) and IIpm·spm coupling (ams), (dl--:: compared to their joint rotation rate 01' An XI(_/2) pul!le would have no effect (or sltu~tlOn 00, wo~ld make M I( 1) inverted rdative to 110 for situation (e), and make M I( IIlverted relative to 110 (or ~tuation (h)

hI

-!)

(H t) J along the x-axis is applied (Fig. 7..44a). ~ere will be comple~e in.version of M,(!) for the situation (e), complete IflVerslon of M,(~!).r0r sl~unuon (h), no inversion of either component for situation (g), and partial II1verslon of ~h and M (_') for situations (d) and (t). We see therefore that the chemical M I ( "',) 1 /"'1 . ' shift frequency flj will determine the urnes at whIch the t':'"o M, com.ponents are invened, and thus they will determine the times til which a~ S-SPIfl pulse will find transferred magnetization. Of course, a will also determine the general 339

range of times for which Mj(!) and Mj(~4) point opposite to one anOlher. Indeed, Bodenhausen and FreelrUJn [7.68J analyze just this model to show that if the I-spins are protons and the S-spins are C l3 , the resulting 20 spectrum has the proton spectrum in the WI direction and the e 13 spectrum in the W2 direction. Thus by observing the C I3 spectrum one gets the indirect detection of the proton resonance frequencies, and the correlation of the proton and carbon chemical shifts. It is useful to think of this experiment in tenns of the density malTix analysis of Sect. 7.24. We have a pulse sequence Xj(1f!2) ... r ... (X j (1f!2), XS(1f!2» . .. observe S

(7.285)

in which we shall treat the X S(1f!2) as having occurred just after the second Xj(7f!2) pulse. Using our theorem of (7.207) we define r- as just before and r+ as just after the second X j (1f!2) pulse, r++ as just after the XS(1f!2) pulse. Since eventually we will observe the S-spins, we seek (mj-jg(t)lmj+) during the observation time t>r++. Now (mj_lg(r++)lml+) is prcx1uced by XS(7C!2), hence from elements of e{r+) which are diagonal in mi. At t = 0-, before the first Xj(wll) pulse, we assume f! is in thennal equilibrium, hence diagonal in both ml and ms. Hence, under the action of the two X j (1f!2) pulses, f! remains diagonal in ms· Thus, (ml-Ie(r++)lmj+) arises solely from elements of e{r+) just before the pulse XS(7C!2) which are diagonal in both ml and ms. These mamx elements can be traced back to elements (mlmsle(r-)Imlms) as well as to elements (=F msle{r-)l± ms) using (5.253):

(mjmsle
4[(mlmsle<0-)lmlms) + (mpnsle
.. + 4[(mjmsle(0-)lmlms) - (mlmsle
for the diagonal elements just before the XS(7C!2) pulse. There are several interesting limiting cases. First, if n j = 0, the result depends on cos (amsr), hence is the same for ms = and ms = This result confinns our poinl earlier thai in the absence of a chemical shift an Xj(7r!2) ... T ••• X j (1f!2) will not inven only one multiplet Instead, one needs an Xl(w!2) ... r ... Yj (1f/2). In fact, if f1[r = (7r/2), one has the case of Fig.7.44g, and indeed then and cos (ill - amS)r = sin (amsr) so the result differs between ms = ms = ~!. Thus one can inven one multiplet only. Note also that as T goes to zero, (mlmsle(r+)!mlms) goes to (mlmsle<0-)Imlms), representing the fact that two closely spaced 'I({l. pulses are equivalent to a single 7f pulse. We can now use (5.254) to get the S-spin magnetization

4

(S+(t,» = T, (s+ e
m,

=

(7.286)

and

(Tmsl,(O+)I± ms) = ± ~1(+msle
and

(7.288)

2

m,

(7.289)

(mf - le
(7.291c)

where the malTix elements of e(r+) are given by (7.290). Equation (7.29Ic) is the mathematical expression of the fact that the S-spin signal is detennined and ms = for by the population differences between the states ms = each value of ml' existing just prior to the XS(7r{l.) pulse. These population differences oscillate as a function of T, the time difference between the Xj(1r/2) pulses, at the nltes [} I - a/2 and ill + a/2. We can utilize (7.290) together with (7.275) to follow what the two X I (Tr!2) do to the population excesses. Thus we have for t = 0- , the population excesses

!

ms

We can thus relate the diagonal elements of f!(r+) to the elements of e
~ L: [emf + le
(±msl,(O+ll±msl = t!<+msl,(O-)1 + ms) + (-msle
(7.29Ib)

m,

;

(7.287)

(7.29la)

1e<',}lmf+)

= L(ml _le{T++)lml+)e-icns-amt)t2

± 1:[(-msle
-4.

+!

(± msle
- (+msle
(7·290l

=1

ms

=-,

ml=-!

-Ll-o

-Ll-o

=!

L1+6

Ll-S

ml

-4,

(7.292)

which become, for t = T+, just after the second I-spin pulse, using (7.290), 341

ms=,

-i m/ = ! m, '"

ms=-!

6 + Ll cos (ll[ - aI2)r

- 5 + L1 cos (01 + a!2)r

6 - Ll cos (II{ - 012),)

- 5 - L1 cos (n l + a!2)r

Therefore, from the point of view of the S-spins. the m/ ::

difference (+ +

31

M/(ms) =

!

and since ..

(7.293) line has a population

T[(+msle<'+lI + ms) - (-msle<'+)!- ms)J .

(7.297)

(+msle<'+)1 +ms) - (-msle
"" 26 - A.[ cos (0/ - aI2)T - cos Uh + oI2)r]

(7.294.)

-i has correspondingly

= [(+msle
Moreover,

(+msle
(- + 1e<'+)I- +) -

(- -1e<'+)I- -) '" 26 + L1[ cos (ill - a/2)r - cos
(7.294b)

Bodenhamer! and FreemtJn derive the resuil of (7.290) by use of the classical spin picture (Fig. 7.45). The X t (7Cn,> pulse (Fig. 7.45a) puts the [-magnetization along the y-direction in the 1 Jl)(ating frame (Fig. 7.45b). This precesses at (0/ams) (Fig.7.45c). The second X,(7fn) pulse r()(ates this vector down into the z-z plane, with a .:-component M,/(ms) given by {Fig. 7.45<1)

(7.295) This equation can be converted to an equation for elements of the density matrix

since

r, I: O'

(bl

(7.299)

Combining (7.298) and (7.299) we get (7.290). In summary, the pulses work by having the X pulses take (m/msle(O-)lm/ms) into (±msle
l

L

(e)

(7.296)

Therefore, using (7.295)

1e<'+)1 ++) - (+ -1e<'+)1 +-)

('I

ms)]

"( h

M./(ms) =

t '" r+ of

and the m/ ::

1~r, [(+msle
Yl

t •

y-

Idl

.,

"

(7.300)

r,

would refocus both the field inhomogeneity and the spin-spin splitting. However, thinking back to spin-echo double resonance, we see that if we also flip the 5spin by a 11" pulse simuhaneously with the [-spins, the spin-spin coupling will not refocus (see discussion at the end of Sect. 7.21). Thus, the sequence HI (lIl s 1 (OS (1'1 1 -

im s) 1

=

Flg.7.4Sa-d. The efroct of two X,(1f/2) pulses, one at t = 0, the other at I T on the I-spin magnetization M I (illS). (a) The diroctioJl of (Jh), for the X I (1f/2) pulses' (b) the magnetiulion M/(llIs) lies along the y-axis all 0+; (el Mdms) processes thr~ugh an angle (nl - (lmS)T at time T-; (d) an X 1(>onent along the I-axil! of -Al t cos (n, - oms)T

=

342

=

x/(,m ... , . .. (X/(,), Xs(,»···,··· y/(,m

(7.301)

2ra =

(7.302)

with 11"

will produce a shuationjust after the last pulse in which the M/(!) magnetization is inverted, but not the M/(-!) magnetization. 343

Then, a 7r/2 pulse applied to the S~spins will generate transverse magneli~ z.ation. It, too, will dcphase if there is magnetic field inhomogeneity, but can be refocused by an X S (7r) pulse accompanied by an X/(;rr) to prevent the spin-spin spliuing from being refocused. The same condition 2Ta = 7r of (7.287) will guar~ antee that at 2T following the first Xs(-;rr/2) pulse the magnetiz.alion will look like Fig.7.43c. We can write this sequence in a slightly different format as x/(7fn) ... T

•••

X/(1r) ... T

•••

Y/('Il"I2)

X/(7f)

i

XS(1fI2).··T",Xs ('II")··.T ... S-echo

Xs('II")

(7.3OJ)

with

+

(I (t» (7.304)

2aT = 7f

in which we utilized two lines. one for the I-spins, another for the S-spins to help keep track of the two spin systems. If the "inhomogeneity" comes instead from the existence of a number of chemical shifts of the I-spins, then one can still transfer I-spin magnetization to enhance S·spin signals: X,(7fI2) ... T

•••

X,(7f) ... T

XS(7f)

•••

Y,(7fn)

X S (1rI2)···acquire

(7.305)

in which we assume the values of a are rather similar for all chemical shifls. This sequence was discovered by Morris and Freeman [7.70] who gave it a name "INEPT' to stand for "insensitive nuclei enhanced by polarization transfer".

7.26 Formal Theoretical Apparatus lIThe Product Operator Method In our example of coherence transfer, we saw that it was important to have a formal method to carry out a calculation in order to be sure one was dealing correctly with such things as the effect on one spin (e.g. the I-spin) of applying a 7r/2 pulse to the other (5) spin to which it is coupled. The differential equations for the density matrix gave us such a method. However. the density matrix for two spins (m/msll?lm',lns) has sixteen elements, since m" ms, In J and Ins can each take on two values. 'l1lerefore, the differential equations are really quite complicated. We picked an example in which only a few elements were involved and in which we could see clearly just which elements we needed. Naturally one would like a formal method for calculating the time development of the density matrix which is straightforward to apply, and which is readily extendable to cases of more than two spins.

344

We tum now to such a method. which has been named the "product operator" fonnalism. This method was developed independently by at least three different groups: Sorensell et al. [7.71], Van den Ven and Hi/bers [7.72] and Wang and Stichter [7.53.73]. We shall present the method in connection with the two-spin system. The generalization to more spins is straightforward [7.71]. We confine ourselves to spin nuclei. The typical situation one encounters is illustrated by (7.212) which describes spin echo double resonance:

=

+

11WO,

.

ZkT Tr{I [exp(l(il/I:: + ilsS:: - uI:5::)(t - T»X/(l'l")Xs(7f)

x exp (i(il/ I, + ilsS:: - aI:I:)T)X,(7rn)I::xil(7rn) x exp(-i(il,I:: + ilsS, - aI:S,)T) (7.212) l x X '(7f)Xi ('Il")exp (-i(il,I: + ilsS: - aI:S:)(t - '7f»)}

s

The system has an initial density matrix corresponding to thennal equilibrium given by (7.210): JO-) = ~

'Z!'(t +

hwo/I: +IIWOS

IT

S:)

a21~

The effect of the rf pulses is to produce rotations. We may symbolize these in general by unitary operators R. An example of such an R is X/(9): X/(9) = ei 'z8

(7.306a)

Xil(9) = e- i 'z8

(7.306b)

Likewise, between pulses the system evolves under the influence of the time development operator T(t) given by (7.307.) a·J07b)

(7.308.) a.J08b)

(7.30&) Since T/. Ts. and T/s commute with one another, they can be applied in any order in (7.307b). We have already worked a good deal with the R's and T's. We use the symbol U to stand for R or T and U-I for their inverse. Then if Ii (I, S) and !2(I, S) are two functions of the spin components of I and S, 345

uh (I, S)!z(I, SW- I = UII (I, SjU- 1U f2(I, SW- I

(7.309)

Hence, expanding the exponential in a power series, applying (7.309) term by lenn, and converting back to an exponential, we gel Uei!J(J,S)U- =eiu/J(I,s)U-l 1

Since the same is true for in (7.315), obtaining

cos (a5 z t l)

= I _

si or S;, we can replace S; by i

(at~~)2 + (at~~)4

[51 by (-t)2, etc.] (7.319)

.

(7.310)

Now, the R's plus TJ and Ts are all simple rotation operators of the general form exp(iI:I;8), exp(i/yO), exp(i!..,O). Their effect on a density matrix e(O-) such as (7.210) can be easily followed using equations such as

cos (aSztl) = cos (atd2) For (7.316), we get d3 sin (as,'I) = S ( , _ (at + (at5!1)5 S4z + ... ) z a 1 3! s2z

(7.311) based on (2.55). We can constnlci fannulas such as (7.311) by simply noting that when the exponential on the left has a positive exponent it generates a left-handed rotarian () of the vector, I;:, sandwiched between the two exponentials. The problem arises from TIS in expressions such as T/s(t\)IyT/- I (tt) = e-illJ.S.IllyCiaf.S./l S

(7.313) then (7.312) looks much like (7.311):

(at

l _

z

2

(7.315)

3!

...

)

(7.320)

sin (aSztl) = 2S z sin (atd2)

(7.321)

Therefore, utilizing (7.319) and (7.321) in (7.314), we get e-ialzS,tlIyeialzS,tl = I y cos (atd2) - I;r.S z 2 sin (atd2)

(7.322)

It is useful in carrying out calculations to have these various results in a table (Table 7.1) in which we list U It;rU- 1 for
1.

(7.314)

This equation has an operator, Sz, as the argument of the cosine and sine, a situation which is quite acceptable if we recall thal for operatOrs the trigonometric and exponential functions are defined in terms of their power series expansions. It turns out we can simplify these expressions by starting with their power series:

(a~~)3 (~y + (a~;)5 (~y + ... ]

= 2S (atl) _ (atd2)3 +

(7.312)

which have bilinear forms (lzSz) rather than linear forms (Iz) in the exponent. If we now consider

e-ialzS,tIIyeiCllzSzLI = I y cos (aSztl) - I;r. sin (aSztl)

= Sz

Table 7.1. The effect of UloU-l for various U's and for a = "',y,z. The bottom line applies for spin only

t

I,

I.

I.

l~cosO-l.sinO

lrzcosO+lzsinO lrz cos 0 - l~ sin 0 I" cos(Oj2)

I,

I z cos 0+ I~ sin 0 I, cos 0 - I" sin 0 I. I.

I.

I.

e i9frz ei9f~

U

e i9fz e- l9fzS •

+1~(2S, sin(Oj2))

I~ I~

cos 0 + Irz sin 0 cos(Oj2)

-lrz (2S z sin(Oj2»

(7.316) We now make use of the spin! property. If we choose as a basis set functions lms), (mS = or the most general function is of the form

+!

1{;(t)

-!),

= C+(t)lms =!) + C_(t)lms = -4)

(7.317)

Explicit evaluation gives S;1{;(t) = C+(t)S;I~) + C_(t)S~1 -

!)

= iC+(t)I~) + iC-(t)1 -~) =

346

i,p(t)

We now see how to evaluate TISIt;rT1S1 for a = x, y, z. An interesling point is that it takes single spin operators, It;r, into a sum of a single spin operatOr and a product of spins It;rSjJ. We see therefore that if applied several times, it might develop terms like It;rJjJS.rS6' But such terms can be simplified as we now explain. Suppose we think of either the I-spins or the S-spins in terms of the 2 X 2 Pauli matrix representation. We recognize that spin products such as I",Iy are products of one 2 X 2 matrix with another. The result is also a 2 X 2 matrix.

(7.318) 347

Now any 2 X 2 unitary matrix can be represented as a linear combination of the three Pauli matrices plus the 2 X 2 identify matrix. Thus. it must be possible to reduce products such as Izly to linear combinations of the identity operator with la's. In fact, we have shown one such example of a product Ialp for the case that cr ::: p:

.

1~:::1

cr:::x.y.z .

We shall show that if operator. The result is iI,

2

Izly :::

Q

(7.323)

:I P. we can also easily reduce Ialp to a single spin

111 1z :::

B1 :::at,

iI,

-2

(7.324)

(7.325)

This relationship, however, also involves I y l z • so we need one more relationship involving l z I1/ and I1/Iz . Anolher such relationship arises from products such as (1z +1,)2. Let us then consider the coordinate transfonnalion from (1z,I,) to (1z" III) wilh I

I., = j2U, + I,)

(7.326)

a 7r/4 rOlation. Then (7.327)

i ::: !(1; + I;) + ~(1z1y + 1,lz) ::: !
l z 1y + lylz ::: 0

t, ... XS(7r{2) ... t2

Then

r~1-1 e
X S I (,...fl)T/f}B2 .

(7.331)

The effect of the XI(,...fl) pulse is to rotate I, into l y : XI(1rfl)1,X/I(1rfl) = 111

(7.332)

.

Next we apply TIS' Utilizing Table 7.1 we gel TlS(B 1)Iy T ls)(9,) '" I1/ cos 9)12 - 21z 5, sin BI12

(7.333)

Next we apply Xs(,...fl), a left-handed ,...(1 rotation of spin S, giving XS(,...f2XIlI cos 9 1(1- 2Jz 5, sin 9 1(1)X I (,...f2)

S

= I y cos 9i1l - 2Jz 5 11 sin 9 1f2

(7.334)

We now act on this with TIS(92):

s

T/s(B2)ly T/ 1(B2) cos 9 1(1- 2T/S(92)lzT/-s)(92)TIS(92)5yT'SI(~)sin 8,(1 = (1y cos 9212 - 2lz 5, sin e./.fl) cos 8,(2

- 2(1z cos 82(1 + 2l11 5, sin 9212)(511 cos fhfl- 21,5z sin 82fl) sin 9 1(2 "" 111 cos 92fl cos Blfl- 21z 5, sin 82(1 cos 8 1(1 - 2Iz S11 cos 2 8212 sin 9 1fl + 41z 1,5z cos 82n sin 82n sin 81n - 41y 5,5y sin 82n cos 82n. sin 8,n. + 8Iy 5,I, 5 z sin 2 82fl sin B,n.. (7.335)

We now have a number of tenns involving spin products such as 1a lp or Sa5{J. Utilizing (7.324), we reduce the various teons as follows:

.

(7.336)

The initial density matrix at t::: 0 is given by (7.210). Suppose we consider the ponion of e(O-) given by

~~~I,

Evaluation of Ihe portion involving 5, is left as a homework problem. 348

(7.330)

.

(7.328)

for a spin ~ panicle. Substituting (7.328) into (7.325) yields (7.324). Let us now illustrate how Ihese fonnulas are used. Suppose we analyze the 5-fHp-only double resonance for the case of a 7r{2 5 pulse:

e(O-):::

B2:::at2

•

X

plus cyclic pennutations. To prove (7.324), we nOle thai tenns such as l z l l1 are encounlered in the commutation relation

Xr(7r{2) ...

For simplicity, we consider a case in which il r ::: ils::: 0 (exact resonance for both spins). We apply an X r (7f!2) pulse al t '" 0, and an Xs(Tr/2) pulse at time tl, let the system evolve under TIS from 0 to tl, and after t1 for a time t2. We define B, and fh by

(7.329)

l y 1,5,5:I: =

'2 5 ('2i) 17; (i)

I.S 11 "'-~

When these are substituted into (7.335), the two tenns in 111 57; cancel, and the teons in IzS y combine (cos 2 8212 + sin 2 92fl = 1) giving 349

TI S(8 2)XS(1r(2)TIS (8 1)X1(7r(2)I z Xii (7r(2)T1S1(8 1)X5 1(7r(2)Ti!] (82) = I" cos 82/2 cos 8 1/2 - I z S z 2 sin 82/2 cos 8 1/2 - I z S,,2 sin 8tf2

. (7.337)

Therefore, the contribution of (7.329) to

e is

e(tl + t2) = hWOI (lz cos 82/2 cos 9 1/2 - IzS z sin 92/2 cos 9 1/2

ZkT - I z S,,2 sin 9d2)

(7.338)

These various terms tell us what elements (mlmsl,q(t)lm~ms) are nonzero at time tl +t2' For example, the first term involves I z , hence gives nonvanishing elements (± mslu(tl

(7.339)

+ t2)I~ms)

The term IzS z gives similar elements, weighted with mS:

s

(mlmsllzS z Im~ms) = ms(±IIzl=f)Oms,m Omr,±Om/,:,:

(7.340)

Lastly IzS" givcs nonvanishing terms

(± ±!e(t, + ',li'!' '1')

(7.341,)

(7.342)

Tit"', + ',)1'1' ±)

(7.341b)

However, since two nuclei are involved, with II = IS but £2r i- £25 owing to their chemical shift difference, it is better to think of this as

(±

whose further significance we discuss in Sect. 9.1. In Table 7.2 we list the other useful relationships employed in dealing with spin operators. Table 7.2. Useful relations for unitary operators U. (U is a function of I, or 5, or I and 5)

U/J(1,S)h(I,S)U-l

U/l(1,S)U-IUh(l,S)U

I

Ucxp(i/(I,S)lU- 1 = exp[iU/(I, S)U-I I For spin

!:

cos (1.8) = cos (8/2) sin (1.8) = 2/. sin (8/2)

7.27 The Jeener Shift Correlation (COSY) Experiment Since thc early days of high resolution NMR, an important goal has been to tell which nuclear resonance lines arise from nuclei which are bonded to one another. The existence of bonding manifests itself in liquids through the indirect spin350

spin coupling [7.74,75] (Sect. 4.9). The bonding might be direct, as in a C l3 H I fragment, or remote via an electronic framework as in the H-H splittings in ethyl alcohol (CH3CH20H) between the CH 3 protons and the CH2 protons. In solids, the dipolar coupling proves proximity. In liquids, thc dipolar coupling shows up through thc nuclear Overhauscr effect. One thcn distinguishes between bonding and proximity, an important distinction in large biomolecules [7.76] in which a long molecule may fold back on itself. A variety of methods such as spin echo double resonance, spin tickling [7.71], INDOR [7.65J and selective population transfer [7.66J have been employed, to utilize the fact that if I and S are coupled, perturbing the I-spin resonance will produce an effect on the S-spin resonance. All of those methods involve sitting on one line and point-by·point exploring the other lines. leener's discovery of the two-dimensional Fourier transform method converts the approach to a Fourier transform method with all its advantages. In this section we analyze the original Jeener proposal, which now is commonly referred to as the COSY (correlated spectroscopy) method since it reveals which pairs of chemical shifts are correlated by a spin-spin coupling between them. It involves a single nuclear species (e.g. HI) and a pulse sequence

XI(7r/2), XS(7r/2)··· tl ... X / (7r/2), X S (7r/2) ... acquire I(tz) and

S(t2)

,

(7.343) where a single HI of sufficient strength flips bOlh spins. We have already discussed the basic principles of coherence transfer in Sect. 7.25. There we saw that if we are observing the S-spins in time interval two, the density matrix elements (m/-Ie(tl +t2)lmr+) are responsible for the signal. These elements, however, have passed through both (mlmslfl(t)lmrms) and (=fmsle(t>!±ms) during time interval tl. Thus, the I-spin resonance frequencies modulate the S-spin signal during 12 as a function of tt. We can express these facts colloquially by saying Ihe (=f mslu(t)1 ± ms) matrix elemcnts during tl "feed" the (In, ~ Iu(t)!ml ±) elements during t2. Such a description almost suffices 10 describe the pulse sequence (7.343). All one need add is that at t = 0+ both (=fmsle(t)I±ms) and (ml =f lu(t)lml ±) are excited, so that during 12 (ml-lu(t)lml+) is fed by both (=fmslel±ms) and (ml =f lu(t)lml ±). [Note that (mrmslu(t)lmlms) during II also "feeds" (ml -lu(t)lmr+) during 12, as is shown by (7.291) and (7.290), however, since the diagonal elements of e are independent of time they do not by themselves introduce any I I dependence to e.J We could try to carry out an analysis using U similar to our approach to spin coherence. However, Ihis method rapidly becomes quite cumbersome. Indeed, one finds that all 16 elements of U are excited by this pulse scquence! We turn, therefore, to the method of spin operators. 351

Since the two nuclei are identical, we consider a Hamiltonian in the singly rotaling frame, rotating at the angular frequency w of the alternating field. The Hamiltonian includes a chemical shift difference so that il r and ils are not equal. Rather, Ihey are given by (7.344) with ur and us being the two chemical shifts. The Hamiltonian in the rotating frame is still (7.345) Both nuclei experience the same alternating field, H" so Ihat for any pulse the two spins experience the same phase for HI and the same angle of rotation, e. Thus, for example, if there is an Xr(e) there is also an Xs(e). Since we observe both I and S spins, we seek (7.346) with an initial density matrix prior to application of the first pulse given in the high temperature approximation as

(>{O-) =

~(I+ r.~:O(l,+S,»)

(7.347)

Utilizing the time development operators, T, Tr, Ts, Trs, defined in (7.307), we readily get

, H °T, {(l+ + S+)T(t,)XI (,/2)Xs(,/2)T(tl) Z·T

Expressing T(t" as T(t" = Tr(l])Ts(t])T/S(t])

we get T(t,)[yT-1(t]) = Tr(tt)Trs(tt)ciflsS~ll lye-iflsSztl T } (tl )Tr-'(t,) rf l = Tr(t,)Trs(t,)[yTif} (t,)TI- (tl)

T,{ABCj=T'{CABj

r

(7.353)

s

we pull the T '(t2)T '(t2) from the far right hand end of (7.348), giving for the first pan of the trace

Tsl(t2)Tr~ '(t2)([+ + S+)TI (t2)Ts(i2)TIS(t2)..

(7.354)

Then, utilizing eil,S l+e-ilz(};:: l+e i8

(7.355)

and the fact that I and S commute, we get instead of (7.354) (I+e- in ,11 +S+e- insh )T/s(t2)...

(7.356)

so that finally we have

+ S+(t)

J,

s

x X l(rr!2)XS(rr!2)(I: + S:)X '(rr!2)Xi l (rr/2)T-'(tl)

x X,I('/2)Xi 1('/2JT- 1(,,)}

(7.352)

Utilizing the fact that

(I+(t)

(l+(t) + S+(t» = "k

(7.351)

(7.348)

= "'(hHo Tr {1+e- in / 12 + S+e-inS(2) ZkT x [e-ia/,S~h X r(rr!2)X s (rrl2)e ifl ,I,ll e -iaJ,S,ll ly x e+ ia /. S, t1 e -in/ J, 11 Xs(rr!2)X r(rr!2)eiaJ ,S'11]}

(7.357) The expression on the right can be written as the sum of two temls, one containing the I: part of e(O-), the other containing the 5: part. We define these as (7.349) Explicit examination of the two expressions shows that one can be converted to the other if every place one has an [a one replaces it with Sa (a = x, y, z) and every place one has Sa one replaces it with an [a' Thus if one evaluates the teoo involving [: only, one can get the contribution from Sz by interchanging in the first answer Sa's with la's, [a's with Sa's, fir with fl s , and ils with fl[. Therefore, we evaluate just the [z teOll. Then X s(rr!2)Xr (rr!2)[zXi' (rr!2)X = Xs(rr!2)lyX I(rr(2) = ly

S

352

s'

(rr!2)

(7.350)

We have collected inside the square brackets the set of operators which we need to evaluate using the spin-operator foollalism. We saw in our example (Table 7.1) that the operators TJ> acting either on an la or on a product teoo [aSp, at most double the number of teoos, for example transfonning an [y to an ly cos + Ix sin An operalOr Trs acting on a single lerm la or Sp will double the number of tenns, but acting on a product will quadruple the number of teoos. The operators XJ(1r!2) and Xs(rr!2) will change an lz into an [y, etc., bllt not increase the number of teoos. Therefore TIS acting on ly will give two terms, which T J will double to four terms (two single operators, two operator products), which T/S will then convert to at mosl 2 x 2 + 4 x 4 = 12 tenns. (It turns out we get only 9.) They will be of the fonn la, or S{3, or lal{3 after we have reduced any products such as lalp, or [a[pl-y, etc. to the appropriate la" The only temlS which will eventually contribute to (l+(t)+S+(t» f involve either '

e

e.

353

(7.358) These are diagonal in one spin and off-diagonal in the other. All possible spin operators can be expJessed as linear combinalions of the 15 spin functions In. So. IoSIJ (a'" x, y. z; (J = x, y, z) plus Ihe identilY operator. But of these 16, only the eighl operalors (7.359.)

[:nSz. ["Sz. IzS%. IzS"

(7.359b)

will give matrix elements of the fonn (7.358). In fact. using the symbol 0 10 Sland for these eight funclions, (7.357) will consist of a sum of terms Tr {I+O} or Tr {S+O} which vanish if 0 is bilinear. Thus. only the terms (7.359a) of the form [:n. 1y • Sz and

5y

contribute 10 (/+0) + S+(t».

As a result, of the 12 possible terms we will get from [] in (7.357), many will not be of interest. It is therefore useful. as shown by Van den Ven and Hi/bers [7.72] to construct a table in which onc keeps only the spin products, not all the other factors, to find the tenns one needs eventually. then go back 10 get Ihe coefficients. In so doing, we will designale whal we are doing by wriling a sequence of operalors which aCI sequentially in the manner of Sorensen et al. [7.71J. Thus, to indicale that we wish to calculate the effect of lhe operators in the square brackels of (7.357) on an inilial operator I y • we write (7.360) We put these results into a table (Table 7.3) in which over the dividing line belween columns we lisl the operator which has lransfonned one column inlO the next column. If we make our table with 12 lines we should have all the space we need. Comparing Table 7.3 wilh (7.359), we see thai we will gel contributions 10

(£+(1»

Iz

(7.361)

Sz/4

(7.362)

The contribulions 10 I+ can be seen 10 arise from teons which at all stages are either I y or I z • hence diagonal in IllS. Thus they are not fed by tenns which are ever off-diagonal in ms. Therefore, these lenns do nOI involve lransfer of coherence between lhe I-spins and the S-spins. On the other hand. (7.362) shows lhal the tenn Sz/4, which is off-diagonal in ms but diagonal in m/, evolved earlier from the initial 1% to I y , to l z S% to IySy to -1,5". Hence lhe tenn I y • which is off-diagonal in m1 but diagonal in ms. feeds the lenn Sz/4. which is diagonal in m1 but off diagonal in ms. This tenn involves transfer of I-spin 3S4

------- --- -----

I

1.

1.

3

5

-a1,S,I,

X s (,,/2)

-d,S,11

, • , , ""

XI(,,/2)

0,/,11

1.

1.

-I.

I.

,,

I,

I.

I,

I,

3

1,5, _1.5,

1.5,

1,5,

1.5,

[~~:5J x [-~:/.J~

7 8

1.5,

-1,5.

1,5,

10

1,5, +1,5,1,

=

1,5, /.1,5, -(i/2)1,5. -1,5,5, (i/2lf)5. +1,1,5,5. (i/2 11.5,

= = =

5,,/4

polarization to the S-spins. We see Ihat by applying first XS(x{2). then second. X,(7f(1.). we get from I!lS, 10 -I,5,1 via the state IyS". Thi.s operator has malrix elemenls which are off-diagonal in both ml and mS' If mstead we had firsl applied X,(7f{2), Ihen XS(7f(1.). we would have gone from +ly S, to -1,5, to -I,Sll' Thus the intennediate state -I,S, would have had malrix elements which are completely diagonal. This latter Iype of Slate is the one we encountered in our discussion of coherence transfer (7.286). The fact that we go from IlJS, to -I,SlJ no matter which order we apply Xl(7rI2) and XS(7rI2) is an example of our Iheorem of (7.207) that one can interchange the order of the Xl and Xs pulses. We now need to get the rest of the coefficients of I z and Sz on lines 3 and 10 of Table 7.3. We readily find

(r+(,) + s+(t) I, =

-r::;

+ e-insl12 sin (att/2) sin (nll t )2 sin (atV2)Tr{S+Sz/4)). (7.363)

T,{r+r.) =T'{l;J :::I

L

(mlmsl1;lm, 71l S)

=4xi=1

so that

• ,, , "" 5

8

10

Explicit evaluation gives

and 10 (S+(t»I, from line 10

Line

x (e- in/ I , cos (at)(1.) sin (nltl) cos (atV2)Tr{rt l z }

I. from

line 3

Table 7.3. EffecU! of operators from (1.357) on an initial operator I,

(7.364)

(7.365) 355

In writing expressions such as (7.365), Van den Ven and Hi/bers [7.72) introduce a useful nClation which greatly cuts down on the number of lellers one needs to write down. They point out that using this formalism one encounters trigonometric factors such as cos (att!2), sin (atzf2), sin U2[t2), sin (ilst t )

(7.366,)

They write these as

C~

51

== ==

5~ == sin (at2n.) , == sin (nst,)

cos (attn.)

51

sin (nlf2)

(7.366b)

in a notation which is self-evident. One merely has to be careful to remember in the arguments involving a which is not present in those involving l the or ils. Returning to our discussion of (7.365). we now add to this the conuibution from 5~ to e(O-). which we obtain simply by interchanging ill and ils in (7.365). The final result is

"i"

n

(P"(l) + s+(t»

: ~::; {cos (atl/2l cas (at,J2)

(7.367) The various terms in this expression have simple physical meanings. We note fi~t that we have a term involving cos (att/2) cos (atz!2). This term is present even as tt and t2 approach zero. It has the terms

e- inst2 sin (ilstil

and

e-inl12sin (ilstt)+e-insf2sin (illtl)

(7.368a) (7.368b)

which correspond to oscillation near il, during both t I and t2, or oscillation near ils during both tt and t2. Of course the actual eigenfrequencies are ill ± an. and ils±a!2, but if a<: Iill - ilsl the term "near il," or "near ils" is well defined. Near t, :: t2 "" 0, this term is independent of a. It therefore does not require a coupling to exist, and represents an effect of uncoupled spins. as is evident from the fact that the precession frequency is the same (il, or ils) during both time intervals. On the other hand, the second term is proportional to sin (ati/2) sin «(lt2/2). It reaches a maximum when atd2 "" at2/2 "" Tr/2 or

.

(7.370)

The first term corresponds to a spin precessing near ils during the first interval, and near il, during the second. The second term represents precession near ill during the first interval. and near ils during the second. IT one makes a double Fourier transform of (7.366). one gets two types of peaks. From the term involving cos (atl/2) cos (at2/2), one gets peaks near WI"" W2 "" ill or ils. which lie close to the diagonal line WI:: W2 in a (WhW2) plol. From the term involving sin (aid2) sin (at2f2). one gets peaks near the points WI "" ill. W2 "" ils or WI "" nS, W2:: ill. Such peaks immediately show that the spins with chemical shifts il, and ns are coupled. Since the frequencies are always il, ±a/2. not il" and ns±a/2, not ils. the peaks are actually clusters of four peaks. Thus, the diagonal peaks occur at

ill +a/2

W2 "" il, +a/2

wI""il , +a/2

W2""nl-a/2

ill - a/2 w}:fh-a/2

W2 "" nl - a/2 W2""n/+a/2

WI""

x [e-inI12 sin (il/tIl +e-insI2 sin (nstl)] + sin (at In.) sin (atz!2)(e- in,l2 sin (nstj) + e-insl2 sin (n/tl)])

e-inl12 sin (n,tl)

time, t2, after the second pulse that the spins reach the condition of Fig. 7.43c. That coherence transfer is involved is also shown by the fact that this term also involves

WI ""

(7.371)

Likewise, the off-diagonal peaks are actually a cluster of four peaks at (7.3720) and another cluster of four peaks at WI :

fls ± af}.

•

w,:

fI/

± af}.

(7.372b)

If we think of the factor sin (atl/2) sin (at2n.) as modulating signals at il, and ils, we note that the buildup of the cross terms exp (-iil/t2) sin (nst}) or exp(-inst2) sin (il,tl) requires a time ""TrIa. In a typical molecule. a spin I will have coupling constants to both nearby and more distant nuclei. The stronger coupling constants a will produce their cross peaks in a shoner lime than the weaker coupling constants. Thus, by limiting the intervals t} and t2, one can select large or small a's. If, indeed, two spins are not coupled at all, so that a is strictly zero, the cross peaks will never arise.

7.28 Magnetic Resonance Imaging

(7.369) This is the condition we encountered previously in (7.356) as the time which optimizes coherence transfer. This is the time, t" after the first 7(/2 pulse at which the spins have precessed to the condition of Fig. 7.42d. 11 is also the 356

The use of NMR to produce two- and three-dimensional images has, since its invention in the early 1970s. increased at an explosive rate. We refer the reader to the books by MaflSfield [7.77] and by Ernst et al. [7.78] for extended treatments 357

of the many variants. We present a shan treatment of the principles al this point in the text since it enables us to discuss the method of Kumar, Welti, and Emst, which utilizes the two-dimensional Fourier rransfonn approach. Two pioneering papers, published within a few months of one another, are the proposals of lAuterbur [7.79) and of Mall.fjield and Crannell [7.80]. which independently propose use of NMR 10 ronn images. The basic concept is straightforward. We start with a sample which has a narrow NMR line in a uniform slatic magnetic field. To the uniform field we add a second magnetic field which is nonuniform. ideally one with a constant gradient. The field gradient broadens the NMR line. For a constaol gradient (e.g. in the z-direction) the magnetic field component, HI> parallel 10 Ihe uniform field is H~=Ho+z

(OH') fu

(7.373)

so that planes of constant z correspond to planes of constant precession frequency. Thus, a given frequency interval is bounded by two frequencies which correspond to two planes (of constant z). The total NMR intensity in that frequency interval is proportional to the number of nuclei in the sample lying between those planes. Thus the NMR absorption spectrum provides a projection of the sample spin density integrated over planes perpendicular to the gradient direction. From a series of such projections for various gradient directions, one can reconstruct the object. We are considering, then, a uniform field kHo to which there is added a small additional field h(r) which is static in time:

H = kHo + h(r)

(7.3740)

with

h(r) :::: ihz(r) + ihl/(r) + kh~(r)

and

Ihl
(7.374b) (7.374c)

We wish to show first that we can neglect the components h;e and h y perpendicular to kHo. The precession frequency depends on the magnitude of H: w(r) = ~H(r)

.

(7.375)

Since Ho:> Ih(r)l, the effects of ih;e and ih" are merely to rotate H slightly without the first oroer changing its magnitude, whereas h~ changes H(r) to firs! order:

H = V(Ho + h~)2 + =

hi + h~

JH~+2Hoh~+h~+I.;+h~

ll~(r)

= h:(O) + x

==

({)~~: )0 + Y (a~~v )0 + Z(aa~: )0 + ...

h:(O) + xG;e +yGl/+zG;z

(7.378)

which defines the components Gz , G", G: of the gradient G of h~(r) at the origin: (7.379) G = iG z + iG y +kG: The reader may wonder whether we can guarantee that it is possible to generate an 11 with the spatial dependence of (7.378). After all, h must obey the laws of physics (V. It = 0, V x It = 0). We explore these aspects in a homework problem. The answer is that we can achieve (7.378), but for there to be a term such as kxG z for example, there must also be terms in the i and/or i directions. However, since as we have seen the transverse components do not affect IHI to first order, we can neglect them. Let us then define two functions, a frequency distribution function, !(w), which gives the NMR intensity at frequency wand is normalized to satisfy

J

f(w)dw = 1

(7.380)

and a spin density function e(r), which gives the number of nuclear spins in a unit volume at point r, and which is likewise normalized so that

J

(7.381)

(,(r)d'r = 1

If, then, the gradient G is oriented in the i-direction,

G=

(7.382)

Z'G .

planes i = constant are planes of constant precession frequency. Since the magnetic field changes by G dz' between the planes al i = constanl and (z' + d.z') = constant, the precession frequency change, dw, is (7.383)

dw=;Gdz' Therefore

!(w)dw = dz'

f'lJ

dx 'd y"",x,y,z I"J I I I)

(7.384)

%':::consl

(7.376)

Neglecting the terms quadratic in the components of h, we get

H = HoJI + 2h~/Ho ~ HoO + h:IHo) = Ho + h~

What spatial forms can h: take? If we simply consider it to be a gen~ral function of r, we can expand it in a power series in z, y, z about a conventent origin. We assume we have shaped the field so that only the lowest terms are needed. Then

(7.377)

The integral in (7.384) is the projection of the spin density funclion e(x', yl, z') on the z'-axis. Thus, utilizing (7.383), "(Gj(w)::::

11

dx1dy'e(:i,y',/)

(7.385)

~/=consl

358

359

Since this projection depends on the direction of G, and since the magnitude of f(w) depends on the magnitude of G, we put a subscript on f(w) :

.."Gfc(w)

=

JJ

dx'dy'e(z', y', z')

(7.386)

:r'=const Since fc(w) can be measured, we can view the integral on the right as known experimentally as a function of z'. We can think of making a variety of measurements 10 delennine the w dependence of fc(w) for a variety of different G directions, thereby getting the projecled spin density for any desired projection axis, z'. It is now easy 10 show that such measurements enable one to reconSmJct e(z, y, z). Firsl we take the experimental dara for a given z' and fonn the Fourier transform of the data:

ek =

J

e-ik:r'dz'

JJ

dx'dy'e(z',y',/)

problem then becomes one of finding a suitable numerical approach for getting the most accurate value of the integral from the data as given. Of course, in the process of selecting the approach, one gets guidance as to whal data to collect. The approach of reconstructing e(r) by Fourier transforming the spin density distributions is one approach to finding e(r) from data for fc(w). Lawerbur [7.79] demonstrated a simple graphical means, and pointed out that Ihe solution to the problem of getting e(r) from spin-density projections was well known from Olher areas of science. Another experimental approach was proposed by Mansfield, based on 0bserving free induction decays. If one applies an X(ll'"{l) pulse to the spins and observes the free induction decay in Ihe reference frame rotating at wo, the complex magnelization is

(M+(tl) = (M.(t)} +i(M,(tl)

(7.387a)

= iMo

where = iMo

k = H'

(7.387b)

Utilizing (7.373) wilh Wo

ek = exp(ikwo!"'(G>

== "'(Ho

exp(-ikw!"'(G)fc(w)dw

.

(7.388)

If we know ek for all k, we can get e(1") by the relalionship 1 l'\r = (2ll'")3 -J )

j eik·.ek. cr"k· .

(7.389)

'. k,

.'11:.7.46. The paths in k-space (k = ki') eXI>lored by applying the gradient C = G i , in a variety or directions i' in a rhUle. The various directions 91, 92, ... ,9,. correspond to different directions i;,

-,

" 360

(M+(t» = iMo

so that (M+(t» =iMo

In practice, since each direction k is explored in a separate experiment, ek. is never known experimentally for k as a continuous variable. For example, if one has a two-dimensional object, the various k's would be chosen to lie in the plane of the object. Then the k values explored could be represented as in Fig.7.46. It is seen that they represent radial lines out to some maximum k, and k max • Since performing an integral such as (7.389) is a numerical operation, the

9,

e(z', y', z')e-iw(z' ,JI,z')ldz'dy' dz' f!{z', y', i)e-i.,.G:r"d:i dy' dz'

(7.39Oa)

which can be rewritten to make ils meaning clearer as

and (7.386) we can express (7.387) as

J

J J

i:;,

J J

JJ JJ

dz/e-i.,.G:r'1

dz'e- iq ''''

dz' dy' f!.z', y'. z')

dz'dy'(!{Z',y',z')

(7.3901»

(7.391)

where

(7.392)

We can rewrite \1.391) to emphasize that on the left we are observing

(M+(t» for a particular gradient direction i', and rewrite the right hand side to bring our its meaning to get

(M.+(t»j;' = je-iq • .,. e(r)d3r

.M,

(7.393)

The expression on the right of (7.393) is identical to that on the right hand side of (7.387a) if one replaces q by k. We will henceforth replace q by k. Thus, in the presence of a static field gradient, the NMR signal following a 7r{l pulse sweeps out in time the Fourier transform (?k' where the direction of k is given by the direction of the field gradient, and each point in time corresponds to a magnitude of wave vector given by (7.392). Clearly, the free induction decay is also producing f:!k along radial lines in k-space as in Fig. 7.46. In his initial paper, Mansfield [7.80] proposed using the free induction decay, giving (7.393). Since his initial idea related to crystallography, he explored the case that the spin density is given by a periodic lauice, and demonstrated the imaging scheme by making a macroscopic layered sample to simulate atomic periodicity.

36.

In Fig.7.46 we display a set of k trajectories for a planar sample. Even for nonplanar samples, it is convenient to collect data from planar slices in me sample, a sequence of slices thus providing the full three-dimensional object The methods of selecting well-defined slices are pan of the present an of imaging. We refer the reader to (7.77] for example. A simple approach to generate a slice perpendicular to the z-direction is to start by applying a gradient in the z-direction. If one applies a pulse at a panicular frequency WI, the spins at coordinate z such that

which is the Fourier transform in the plane z of the spin density. Since f z is fixed while recording (M+(t», the data are for a fixed kz • but for a continuous k y . They may be represented as in Fig. 7.47. They thus sample (}k for the z-slice along parallel lines in (k;r:. kll) space. In a full three-dimensional version, one applies the initial rl2 pulse in the absence of gradients. Then. one applies

(7.394) are perfectly at resonance. An X , (7r!2) pulse will then rotate them into the x-y plane. Spins within a Ll.z given by

(7.396,)

G=iG;r:

fo<

I.

G=iC" G= kG r

for

Iv

for

I.

(7.400)

recording the free induction decay during t r • Then

(M+(t» = iMo

(7.395) will also be flipped. but spins farther away in z wiJl be less effectively tilted into the :r-y plane. By in fact shaping the time dependence of (HI) I one can improve on the z-selection. In any event. if one now turns off Gr. one has produced an initial transverse magnetization for all x. y within Ll.z. One can then apply gradients in the :r-y plane to record the free induction decay for any direction k = G in the x-y plane. All planar imaging methods must initially carry out some such precedure to define Ihe slice Lb. Kwnar et al. {7.81] proposed a two-dimensional (or three-dimensional) Fourier transform approach to collecting data which has been adopted by many workers in the field. We start by describing the two-dimensional version in which one first exciles a slice in the z·direction. One then applies a gradient

G=iG z

J

e-izGzlze-iIlG,I'c-irG.I. p{x. y. z)dx dy dz

(7.401a)

with (7.4Olb)

""d t z = t - (t z

+ ty)

(1.40lc)

This approach gives one the Fourier transfonn of u(r) lines parallel to the k r axes fonned by the intersection of the planes k>; = constant and kll = constant (Fig. 7.48). If one is interested in knowing the density pattern in a slice of known %. it is much faster to collect the data using a z·gradient to define the slice excitation and then work with two-dimensional transfonns. An exceedingly rapid way to sample all the points in one slice within a single sweep was proposed by Mansfield (7.82J and called by him the planar echo method. In the planar fonn of this method. one first selects a slice (say

for a time called f;r:. then a gradient

"

(7.396b)

G = iCy

for a time called t y • During t y • one records the free induction decay. Using (7.170) and (7.390) one has, neglecting relaxation. in Ihe rotating frame

(M+(t» = iMoL\z

J

e-i;r:,..Gzlze-iyG,f, e{x, y, z)dx dy.

FIg. 7.47

"

Fig. 7.48

(7.397)

where t = t y + t;r: and where M o is the tolal magnetization of the sample. Defining (7.398)

k;r: = '"'{G;r:t;r:

= L\z

J

e-ikz;r:e-ik,1I e(x, y, z)dx dy

'.

F1a. 7.47. The palhs in k-llpace explored by the twu-dimerl!lional meLhod or KW1«Ir, Welli, andErltSl. For a particular lz, k z is fixed. As the Lime f, grows (7.397), k, grows as shown

one can write

(~;:t»

"

(7.399)

Fig. 7.4&. The path explored in k-space by the and Ernst

th~imensional method

of KIU1tDT, Wtlli,

363

t

at fixed z), then applies a weak steady gradient in the z-direction, and a strong gradient in ~e y-direction which is switched ahemately between two values +GO and -Go (FIg. 7.49). Thus, following slice selection one has G =

iG z + JGO/CO where

j(t)=+l

O
-I

for

Tf,
+1

for

3T,,
(7.403)

and so on (Fig. 7.49).

,.,---==='.

In order 10 ~asp the effect of such a time-dependent gradient, it is useful to follow our earlier procedure of defining a wave vector k. The approach we then take was developed by King and Moran [7.83J, BrOWII et aI. [7.84], Twieg [7.85), and Ljunggren [7.86). Since the precession frequency w and the angle a spin has precessed ,p, are related by

d4> d'

w=-

(7.404)

when the gradient is time dependent, we have t

J [-i J

W(t')dt'] eCr)d3r

(7.405.)

o = iMO

Therefore, defining

364

J

ex p [ -i'Y""

J

G(tl)dt'

t

J

G(t')dt']u
o

(7.406)

o (7.407) Thus. as t develops, k(t) moves Ihrough k space on (he trajectory given by (7.406). (.M+(t»/iMo gives f!k(t) for the successive values of k through which (7.406) takes the system. Examining the gradient of (7.403) or in Fig. 7.49, we see that the weak COIlstant component Gz gives a component kz which increnscs slowly proportional to lime. The strong gmdient G y builds up a large phase angle z,Gzt from 0 to Tb' but the sign reversal of G z which then occurs leads to this phase accumulation unwinding during the next interval 10. Thus at t = 2r" the various planes :t = conSlant gel back in phase, producing an echo that is (7.408)

fo'ig.7.49. Thc limc dependcncc or Ule lhree ~rlldicllls G z , G" aud G. cmployed III lhc planar echo Illelhod of Mmufrdd

exp

== ,

(7.402)

for

(M+(t» = iMo

k(t)

(7.405b)

In k-space, the vectorik",(t)+jl.-J/(t) follows the Inljectory shown in Fig. 7.50, in which we have assumed both G", and Go are positive. As can be seen, this method explores positive and negative k y for JX>sitive kz . One can get the resuhs for negative kz by several methods. One is to apply a strong negative G z for a short imerval after slice selection, and before applying G y • This process would move k to a point on the negative },:z-nxis. Then one begins the sequence of (7.374n). Another approach would be to utilize the mathematical result that

(M+( -k» iMo

=

(M+(k»)' iMo

(7.409)

which can be scen to be true from (7.407). This is the statement that with quadrature detection the signal of one phase gives the cosine transform, the other gives the sine transform. The practical result is that one need only collect quadrature data in one half-plane of (k"" ky). This remark of course holds true for the schemes we described earlier. We see, therefore, that the "planar echo" method samples all the half-plane in each cycle, and thus contains the infonnation for total spin-density reconstruction in each sweep. A more "physical" description of the planar echo method provided Mansfield's original motivation. He thoughl in terms of a discrele laltice (as of atoms in a regular array). Suppose we imagine an apple orchard with trees planled in equally spaced rows and columns. SupJX>se some of the trees have died and been cut down. If one stands at the end of one row and looks down that row, one cannot tell whether any trees might be missing from it. However, since one sees the next row at an angle, one can readily spot any vacant site. This phenomenon is illustrated in Fig. 7.51 from Mansfield's book [7.77]. To utilize this principle in imaging, one must produce a discrete lattice, and view it at an angle. TIle imag· 365

8. Advanced Concepts in Pulsed Magnetic Resonance

k,

0

k.

0

0

0

0

I

0

Gy

0

Hahll's discovery of spin echoes [8.1) demonstrated thaI one could defeat the effect of magnet inhomogeneity in obscuring the true width of magnetic resonance lines bec;lUse the echo amplitude M(2T) decayed exponentially willi 2r!T2. where T is the time between pulses. However, if the magnet inhomogeneity is large, and if the diffusion rale of the nuclei sufficiently great,

0 .y(j,t~

M(2r) = M(O) "p(-2r/T,)exp [-

I Fig.7.S0. The {mth in k-spa.<:e explored in the planar echo method of M(lflSfitld

Fig. 7.51. Projections of a discrete matrix filled to form the letter P. ProjectiolU in G z or 0, alone do not contain sufficient inforllllllion to uniquely determine the original object. Projection orthog<mal to the special gradient direction G. conlains all information necessary to complelely'del.crmine the original objed distribution

ing scheme of Fig. 7.50 has parallel scanning lraces which corresponds to a spatial wavelength

2.

, __ 2r __ k",

~1'GJl70

apan in k-space,

(7.410)

A

21Gyf],

This wavelength imposes discreteness (i.e. "rows'') on the spin density function

elr). Another way of saying this is that the echo train produced by the oscillatory gradient G y has a period 2Tb. hence corresponds to observations al a spectrum of frequencies spaced apan by angular frequency Llw of

L\w

"fGy

Dr']

(8.1)

where D is the diffusion constant and DHIDz the gradient in static field. In some instances, then, the r 3 leon might obscure the T2- Or, conversely, if DHlfJz is known, the T 3 term gives one a method of measuring D. The derivation of the diffusion tenn is given in Appendix G. Carr and Purcell [8.2] invented a clever scheme for eliminating the T 3 tenn when desired. 'Their proposal initated a class of experiments in which a sequence of pulses is applied 10 the resonance. In this chapter we discuss first the Carr· Purcell concept, then an ingenious modification invented by Meiboom [8.3]. and finally tum to some of the most intricate pulse melhods yet invented which have the remarkable propeny of enabling one to eliminate the dipolar broadening of resonance lines in solids. 11lese concepts were introduced by Waugh [8.4J and Mansfield [8.5], and have been extended by Vaughan [8.6J. as well as a group of very talented resonators who have spread Qut from these initiators.

(7.411)

Bm L1w implies a ..1z of LIz = -

(1 ~:)' ~

8.2 The Carr-Purcell Sequence

1 Llw = 2r,

= -

I

-

1

(7.412)

""IC" 27"b

Each frequency corresponds 10 a row of trees. The weak z-gradient produces the "view al an angle" shown in Fig. 7.51. 366

8.1 Introduction

We begin a discussion of the Carr-Purcell sequence by llssuming we call neglect the T 3 diffusion tenn. To measure a T2 by the conventional spin echo, one must make a sequence of measurements, each al a different value of T. The envelope of the echo amplitude as a funClion Of2T gives one T2. Carr and Purcell pointed out that the entire envelope could be obtained at a shot if one applied the proper sequence of pulses.

367

Suppose at t::: 0 one applies a 1r/2 pulse in which HI lies along the +x-axis in the rotating reference frame with the rf frequency w exactly at the Larmor frequency woo Such a pulse turns the magnetization Mo to lie along Ihe +y-axis. If one applies a 1r pulse at t ::: r, also with HI along the +x-axis, an echo is fanned at 2r with the magnetization along the -y-axis. If now one applies a 1r pulse at 3r, another echo will foml al 47, this time along the +-axis. In this manner, successive 1r pulses at (2n+l)r(n::: 0, 1,2, ... ) foml echoes at (2n+2)r, the echoes fanning along the -y-axis for odd 11, and the +y-axis for even n. Since all components of the magnetiz..1lion in the x-y plane are decaying exponentially with time constant T2, the echo sequence likewise decays exponentially with T2. A sequence of such pulses and echoes is shown in Fig. 8.1, with Ihe sign, positive or negative, showing whether the echo fonns along the positive or negative y~axis. The behavior we describe enables one to obtain the entire echo envelope in a single pulse train, clearly a convenience. If one is studying a weak signal, so that the signal-to-noise ratio is imponant, the Carr-Purcell sequence is a tremendous advantage as we explain. The noise depends on the bandwidth of the apparatus. One limits the bandwidth so as to pass the echo signal without undue attenuation. Clearly the same bandwidth can be used for either a Carr-Purcell train or a convelllional echo. After each single echo, one must wait several times TI for the system to recover before measuring the next echo. However, in the time il lakes to get one echo, we can obtain the full envelope with a Carr-Purcell sequence. If we take N echo signals to define the echo envelope, we could obtain N Carr-Purcell pulse trains, so each Carr-Purcell echo is recorded N times. If the spin echoes are taken to correspond to the times of the Carr-Purcell echoes, the Carr-Purcell train will have a better signal-to-noise by Jiii. We have remarked that the Carr-Purcell pulse sequence reduces the effect of diffusion on the echo decay. The reason can be seen as follows. If there is no diffusion, a spin dephases during the interval r following each echo and rephases

,

o

.

2,

_---- . -------

------ ------

),

M(2r) "" Mo exp [ _ D

------

---------- ---------

6,

7,

--------

"

(7 ~~)

2

~r3] exp(-2rIT2)

== Moa

(8.2)

When the echo is formed, we have recreated at 2r the situation which existed at t ::: 0, except the magnetization is reduced by the factor a, and points along the -y-axis instead of the +y-axis. Therefore we can consider the magnetization at 27 as another initial condition for applying the argument of Appendix G for the next interval from 27 to 4r with a pulse applied at 3r. The result is (8.3) M(4r) ::: M(2r)a::: Moa 2

If we have n cycles 2r, we therefore get (8.4)

M(n2r) ::: Moa" Therefore M(n2r) "" Mo exp [ -

(Y ~~y D(n27)~72] exp( -n2r1T2)

(8.5)

Thus, if we change r but hold n2r constant so that we are comparing two different echoes of the Carr-Purcell train a fixed time apart, we can vary the faclor involving diffusion but we do not vary the faclor containing T2. Hence we can make the effeci of the diffusion tenn negligible. It is clear thai what is involved are two facls: 1.

5,

2.

each echo cycle of the Carr-Purcell train reduces the magnetization by a factor 0', the relative contribution of the diffusion tenn compared to the T2 term depends on 7 .

Echo clll'dope

8.3 The Phase Alternation and Meiboom-Gill Methods

----------

Echo CIII'i'lop"

Fig.S.1. The Carr-Purcell sequence and tlte resulting echoes. As des<:ribed in the text, a positive echo forms along the +y-axis in the rotating frame, a negative echo forms on the -y-axis

368

during the interval after Ihe 1r pulse. It is the fact Ihal diffusion makes rephasing imperfect which causes the next echo to be smaller. In Appendix G we show that if a 1r/2 pulse at t "" 0 tilts the magnetization Mo into the x - y plane, a 1r pulse r later produces an echo at 27, M(2r), smaller than Mo owing to diffusion by

A Carr-Purcell pulse sequence may contain many pulses. If the pulse rotations deviate slighliy from 'IT, there are cumulative effecls which may become large. We illustrate in Fig. 8.2. (This figure is drawn for a negative "'/' hence involves right hand rotations.) For simplicity we take the initial pulse to be a perfect 1r(l pulse about the +x-axis, and we consider diffusion effects 10 be negliglible. At 369

J ,

(,)

-/2 M.

-----'7---r x

(b)

---0- -----

f--_ ,

'......

.• -------

r" O'

Meiboom and Gill [8.3] were the first people to find a solution to this problem. They likewise used a phase coherent method, but introduced a 90" phase shift into the radio frequency field between the initial pulse and the subsequent 'If pulses. Thus if HI for the 1f/2 pulse were along the +y-axis, Hl for the :IT pulses would be applied along the +x-axis. As a result, the echoes are all formed along the +x-axis for negative 'Y' We leave it to the reader to show that if the pulse is 1f - 5 instead of 11", the error does not accumulate.

7rn.

"\

y

"'./

, (d)

8.4 Refocusing Dipolar Coupling ---.--'-"'-"'-,,---- y

''''(31)'

Fig.&.2a-e. The effect of ,an imperfect If pulse. The initial ../2 pul$e, assunl«! perfect, ro~aleti Mo to the - y-axIS at I = 0+. At t = T - • just bo;orore the .$eOOnd pulse, a typical spm h~ Pl'('(;~ an .~~Ilt 8 in the z - II plane a.....y from the -v-
makes an angle 26,p wIth the z -" plane. In Fig. 8.2e we see that if the III of the second pulse lies along the -x-axis, the rotation direc:tiOll is reversed, so the _(,.. + 6.,> rOlalion restores the spin to ils orientation at t ::: 0+

t ::: 7 - the spins will have fanned oul in the x - y plane. a typical spin making an angle 8 with the -y-axis (for negative ")'). At t "" T we apply a ow + 6¢ pulse about the +x-axis where 6¢ gives the deviation from being a perfect 'If pulse. For a spin having 8 "" the spin lies along the x-axis. It is not affected by the pulse at t"" T. A spin for which 8"" 0 would rotate as shown in Fig. 8.2c. It will lie an angle 6¢ above the +y-axis in the y - z plane where it will sit until 2T later when the next + b¢ pulse is applied. The result is shown in Fig.8.2d. The spin now makes an angle U¢ with the -y-axis. Each successive imperfect 1'1" pulse adds another 6¢ to the deviation. The cllmulative effect becomes exceedingly serious in praclice because the HI is never uniform throllgh tile sample. COllsequclIlfy thOllgh part o/the sample may have a ow pulse, other part.r do f1ot. These sons of troubles have major implications for all multipulse sequences ~uch as those described in the later portions of this chapter. One simple approach IS to apply the H J for the ow pulses alternating along the +x- and -x-axis. Thus the sense of rotation about the x-axis reverses in alternate pulses. The result is shown in Fig. 8.2e. Instead of getting a cumulative rotation eITor of U¢ as in Fig.8.2d, the cumulative error is zero!

7rn..

7r

370

11le dipole-dipole coupling between neighbouring nuclei provides valuable in-

formation for many purposes, but on occasion it is a problem. For example, it may obscure chemical shifts, and it may cause free induction decays to be soonlived and thus difficult to see. We tum now to several interesting approaches to eliminating or effectively reducing dipolar coupling. For a group of spins an ordinary spin echo [e.g. an X('IT/2) ... T ... X(:IT) sequence] refocuses the dephasing which arises because they are placed in an inhomogeneous magnetic field. Since the magnelic dipolar coupling among neighboring spins is in some way analogous to a magnetic field inhomogeneity, one might ex:pect that the same spin echo pulse sequence would also refocus the dephasing resulting from magnetic dipolar coupling. If the dipolar coupling is between different nuclear species (e.g. HI and C I3 ). such an echo sequence does refocus the dipolar coupling. If, however, the coupling is between like nuclei (e.g. HI with HI), the echo does not work. It is easy to see why. The 'IT pulse inverts all the neighbors, so that a nucleus which was precessing more rapidly than average in the first time interval T, finds its neighbors inverted, and thus precesses more slowly in the second interval. The usual echo works because the 11" pulse converts a precession phase lead into a precession phase lag, of equal size. If at the same time the rate of precession changes, the spins do not come back into phase. Nevertheless, the fact that a well-defined relationship exists between the precession frequency when a neighbor points up versus down makes one feel that there should be some way to undo the dephasing which the dipolar coupling produces. In the following sections, we explore these ideas.

8.5 Solid Echoes Powles and MallSfield [8.7] discovered that the free induction decay of a coupled pulses pair of identical spin ~ nuclei can be refocused by applying a pair of shifted in phase by ow/2 with respect to each other. Such a sequence might be denoted by

7rn.

371

X('rr{2) .. . r . .. Y('II"/2) . .. tt

(8.6)

with the echo occurring when tl := , . This sequence has come to be known as the solid echo. Powles and Mansfield demonstrated it experimentally for CaSO'1 ·2H20, and showed that it followed theoretically. The pulse sequence perfectly refocuses spin nuclei which interact only in pairs, but does not refocus larger groupings of spins perfectly. In this section, we explain the echo for coupled pairs. This same pulse sequence refocuses the first order quadrupole coupling of a spin I nucleus for reasons explained in Appendix H. It is therefore imponam for deuterium and N t4 NMR. The existence of echoes related to quadrupole splittings was first demollStrated by Solomon [8.8] for a case of I := ~. It appears to have been Davis et al. [8.9] who recognized that for the 1:= I case sequence (8.6) gave perfect refocusing. The fact that one needs exactly sequence (8.6) may seem surprising at first glance. Why does one need to phase shift the second pulse by 11:/2? Why does one use a '11"/2 pulse instead of a 'II" pulse? To help give a feel for the answers to these questions, we shall derive the result Ihree ways. The first method is a semigraphical one, utilizing a special value of r. The second derivation generalizes the first for arbitrary r. The third derivation utilizes the spin operator method. We start with the simple physical picture. However, we must present it rather carefully. Consider a sample consisting of pairs of nuclei. That is. each nucleus has one and only one neighbor which is sufficiently close thai only the dipolar coupling to it matters. In the presence of a static field, the nuclei are weakly polarized, there being a slight excess with magnetic moments parallel to the static field, H o. If there are N nuclei and if we denote by p+ and p_ the probabilities for a given nuclear moment to point either parallel (p+) or antiparallel (p_) to H o, we have then that there are N+ and N_ nuclei pointing parallel or antiparallel, respectively, given by

4

(8.7) The population difference, n, then obeys n:=N(p+-p_)

(8.8)

p+ and p_ are related by the Boltzmann factor p+/p_

:=

e"tldfo/kT

(8.9)

so that, if ",(hHo «kT, P+:= ~(1 + "'(fiHo/2kT)

p_

:=

~(1 - "'(hHo/2kT)

(8.10)

Note that the expressions for p+ or p_ do fU)( depend on the orientation of the other nucleus in the pair, because we have assumed that the only field acting on a nucleus is Ho. This assumption should be valid if Ho d> "'(1lIr 3 , where r is the distance between neighbors. Now, it is only the excess of nuclei N+ over those N_ which can give rise to a NMR signal. To follow what happens, we need 372

only follow what happens to the excess, since Ihe remaining nuclei carry no nel polarization. Let us therefore in a thought experiment reach in at random to examine the pairs. We take n/2 pairs in which both nuclei point up, and n{2 pairs in which one nucleus points up, and the other points down. This will give us a net n nuclei pointing up. The olher N ~ 2n nuclei in the sample must then be equally pointing up and down, hence be unpolarized. We neglect them. Now, among the pairs with one nucleus pointing up, Ihe other pointing down, we label the up nucleus blue, the down nucleus red. Among the pairs with both nuclei up, we also label one blue, the other red (it does not matter which of the pair we choose for which color). We thus have n{2 red nuclei with spin up, and n/2 red nuclei with spin down. Then the blue nuclei give the magnetization, since the red nuclei have no net magnetization. We represent the spin-spin coupling by a term 1t'spin-spin of the form (8.11)

!

For pairs of identical particles of spin this is exactly equivalent to dipolar coupling, as explained in Appendix H. We assume that apart from 1t'spin-spin we are exactly at resonance for any pulses we apply. Then, following an X(1r{2) pulse, the blue spins lie along the +y-axis in the rotating frame. Now, although we have flipped both red and blue spins by 1r{2, the red spins at this point have no net polarization and thus their density matrix remains diagonal with the two diagonal elements equal to each other. Thus, we can describe the situation at t := 0+, just after the pulse, as in Fig. 8.3b, in which we represenl the blue spins by vectors I and 3, lyir]g along the y·axis, and the red spins by vectors 2 and 4, one pointing up, the other down, along the z-axis. Spins I and 3 will now precess at rates -a!2 and +a/2 (in the left-handed sense) respectively about the +z-axis. The explanation of how the echo fonns becomes very simple for a particular value of " the value which makes arl2 := 1r12 (Le. a, := 1r). For that time, the spins will be aligned as in Fig.8.3c. At this time, the y('II"!2) pulse will produce the situation of Fig. 8.3d. Now, during the next time interval" spins 1 and 3 will not precess (they lie along the z-axis), but spins 2 and 4 will precess at rates +a/2 and -a/2 respectively so that at t:= 2" 2 and 4 will lie along the +y-axis. An echo therefore occurs at t "" 2,. It is interesting to note that in this process we have transferred the net polarization which was initially in spins 1 and 3 10 spins 2 and 4. However, we have not made an unpolarized system become polarized. Thus, we can remain confident that the other N - 2n spins, which we have said are initially unpolarized, do nOI become polarized by the echo sequence. If one chooses a , other than Ihe one for which arl2 := '11"12, the explanation becomes more complicated. Then one needs to utilize the ideas of Sect. 7.24 in which we discuss the time development of one spin when it is coupled to another spin which is in a mixture of the spin-up and spin-down eigenstates. By choosing a,/2 := 'II"!2 we have made this mixture be in fact pure eigenstates. 373

However, a true echo means thai the refocusing condition is independent of the strength of the source of the dephasing. Thus. if spins are dephased by an inhomogeneous magnet. all the spins get back in phase at the echo. independent of how far the inhomogeneity has shifled their resonance from the average field_ In our case. that means the existence of an echo should be independent of a, hence of the PrOOuC[ aT, hence of T. Let us therefore reexamine the situation for a more general T, using Fig. 8.4. We start (Fig. 8.4a) with the magnetization along the z-axis, and apply an X('K(l) pulse to prOOuce the situation in Fig. 8.4b. A time T later the vectors I and 3 have rotated in the % - Y plane through angles (aT(l) and (-aT(l) to produce Fig.8Ac. Then the Y(1I'/2) pulse produces the situation of Fig.8Ad in which magnetization vectors 1 and 3 are in the y - z plane, making angles -aT(l and arl2 respectively with respect to the y-axis. In Fig. 8.4e we show the projections of the four vectors onto the x - y plane. Spins 2 and 4 lie entirely in the plane. We define their lengths as Mo. Spins I and 3 have projections in the x - y plane of Mocos(aT(2.). Now, using the concepts of Sect. 7.22, we realize that each of the four magnetization vectors is coupled to a spin which is in a mixture of up and down states. Since the quantum states of spins 2 and 4 are equal mixtures of up and down. spins 1 and 3 will each break into two counter-rotating components of equal amplitude (Fig. 8At). Spins I and 3 have z-components of -Mo sin(ar(2.) and +Mo sin(aT(2.) respectively. Thus they have an excess of m = (for spin 1) and m = +i (for spin 3). We need to calculate what fraction of their stales correspond to spin up and spin down. Denoting Ihe occupation probabilily of slate m for spin i as (mIUilm) we have that

h: O'

, y t: 0'

l 4

,

,IT: "

,

Idl 2

-!

y I: '['

,

,

(8.12)

from nonnalization. The z-component of magnetization of spin I,

y

lei

2

,

t :

M" = Mo (~I",I~)

2'[

expressing the fact Ihat if the spin is entirely in respectively. But

I'Ig.S.Ja·e. The fonnation of a solid echo for two pairs of spin!! (I and 2, 3 and 4) for a particular value of the time T" at which the refocusing pulse is applied. (8) The thermal equilibrium magnetization at 1 0- showing the vector sum of the four magnetization veclors. Spins 1 and 3 are the blue sl)ina, spins 2 and 4 are the red spins. (b) Just after the X(Tr/2) pulse (t 0+), the blue spina (1 and 3) lie along the y-axis while the red spins (2 and 4) can be taken to point along t.he positive Il.Ild negative ",-axC!! respeclively. (c) The magnetization vectOrs at times L = T- just before the Y(lr/2) pulse, for the particular T such that (JT/2 lr/2 (or or = lr). (d) The magneti:u.tion vectors at time 1 1+ immediately llfter the Y(lr/2) pulse. (e) The magnetization vectors at time t 2r, showing that there is a net magnetization IIlong the y-axi!! from spin!! 2 llnd 4. Note that initially spill!! I and 3 carried the mllgnetization. At t = 2,., the magnetiZll.tion arises from spins 2 and 4

=

=

=

- Mo (- ~Ied -~)

=

=

M~l

= -Mosin(ar(l)

M~t>

is (8.13)

i or -i. M

z1

is Mo or -Mo (8.14a)

so we get

(~Ied~)

- (-~Ied-~)

=-s;n(aT/2)

(8.I'b)

Solving (8.12) and (8.13) we get Well~) = ~[I - s;n(aT!2)]

(- ~Ied -~)

=

1(1 + s;n(aT!2J]

(8.15)

In a similar way, we get 374

375

,

Fig.8.4 Caption see opposite page

(- ~I",I-~)

lal

, ,

Ibl

,

,

,

2

2 all2

--

,

,

~o [1

,

4

1

11

/

/

,

lei

,

(8.18)

N~O [1

- sin(ar/2)] sin(ar/2)

~O [sin2(or/2) + sin2(ur/2)]

(8.19)

Therefore

L

t~,'

No

4

M y (t+2) = Mo[sin 2(or/2) + cos 2(or/2)] = Mo

,

Igi

(8.20b)

,

M

(os(aTl2)

•

Fig.8.4a·g. Formation of a solid ocho for two pairs of spins (I and 2, 3 and 'i) for a general value of the spin-spin coupling conslant u. (a) The thermal equilibrium magnetization at t = 0- showing the vector sum of the four magnetization vectors. Spins 1 and 3 are the blue SpillS, spills 2 and 'I are the red spins. (b) The magnetization vectors at 1=0+ immediately following the X(1r/2) pul$(!. (e) The magnetization vedors at time t = r-, just before the Y(1r/2) pulse. (d) The magnetization vectors at time t r+, immediately after the Y(tr/2) pulse. (e) The projections of the magnetization vectors on the :l' - Y plane at time t = r+, immediately after the Y(tr/2) pulse. (f) The projection of the magnetization components of a spin 1 on the 2: - Y plane at later times, showing how its Illagnetization breaks into two counter-rotating components whose amplitude is determined by the orientation of spin 2. (g) The projection of the magnetization components of a spin 2 on the z - !I plane at later times, showing how its magnetization vector breah into two counter-rotating components whose amplitude is determined by the z-components of spin 1

=

spin 2

spin

("'T---'

(8.20')

By a similar argument

Ho

-f

~o [1 + sin(or/2)] sin(or/2)

=

=

_-'-;:,-...L__ ,

If I

M y2

_

~--Mo(os(aTl2l--- 3

1

~o cos2(aT/2) + ~o cos2(aT/2)

,

,lTn /

M y1 =

,

,

Idl

- sin(ar!2)]

as shown in Fig. 8.4f. We now add up the y-components, My; (i = 1,2,3,4) of the spins at time -r after the Y(7l"/2) pulse:

"-k'::..o" )3 aT 12

,

(8.17)

while the counterclockwise component of 2 will have amplitude

,

4

,

I,}

~o (1 + si,(arl2)]

3

,

(8.16)

= HI - si,(arl2)]

Thus', as spin 2 precesses, it will break into two counter-rotating components whose amplilUdes are detennined by the relative amounts of spin 1 in the up and down states. That is, the component of 2 which rotates clockwise (looking down on the x - y plane) which comes from the down component of 1 will have amplitude

,

376

W",I~) = ~(I + si,(arl2»)

M. T

l1-sin(aT1211

377

This result is independent of a, therefore corresponds 10 a real echo. Indeed, (8.20) shows that in a powder sample (in which a would take on different values for each crystal orientation if it were representing magnelic dipolar coupling), spins of all the crystallites would refocus at time 2r. This exercise has been long, laborious and exhausling, but it does serve to show how coupled precessing veclors behave in quantum mechanics. Now we lum to a much more elegant approach to analyze the same problem: use of the product operator method. It has the advantage of giving us a simple way of figuring OUI what the second pulse should be in order to produce the echo. (See Seci. 7.26. See also [8.10, lIn. We start with a system in thermal equilibrium and work in the rotating reference frame. The density matrix JUSt before the first pulse (t = 0-) is (8.21)

where A is a constant. Following the pulse X(1r(2), the density matrix becomes

~~=~+~

~~

The density matrix at time t = r- JUSt before the Y(ll"/2) pulse is then e(r-) = e- ia1• S ... e(O+)e+iol.S.T

Now, utilizing (8.23) we have e+iolzS... e(O+)e-iaI,S, .. . = (Iy + Sy)cos(ar/2) + ([ZSl + I z S;l:)2sin(ar/2)

(8.29)

which, using (8.24), is equal to

Re(r-)R- 1

(8.30)

and using (8.23) gives

R[(ly + Sy}cos(ar/2) - (l;l: + S;l:)2 sin(ar/2)]R- 1

(8.31)

Comparing (8.29) with (8.31) we see that R must satisfy the relations

I y + Sy = R(Iy + Sy)R- 1

and

(l;l:SZ + IzS;l:) = -R(lzSz + IzSz)R- 1 We automatically satisfy (8.32a) if we make R be a rotation about the satisfy (8.32b), the rotation must be 1r/2:

(8.32a) (8.32b) y~axis.

To

(8.33)

(8.23a) a result which follows from the relationships (Table 7.1) such as

which, utilizing Table 7.1, gives (8.23b) Now, we wish to apply a pulse which produces some rotalion R which will cause an echo. The effect of R is to change e from its value at t = r- 10 a new value at t = r+ given by (8.24) We want e(r+) to be such that during the next interval r, e returns to its value at t = 0+, just after the first pulse. Thus, the condition for an echo is e(2r) = g(O+) but e(2r) = e- ial• S ... e(r+)e+io1• S , ..

(8.25) (8.26)

If the effect of R on e(r-) were equivalent to changing the sign of a during the interval between the X pulse and R, so that Rg(r-)R-1 = e+ io1,S, .. e(0+)e- ia1 • S... (8.27)

then we would satisfy (8.25) and produce an echo, since putting (8.27) in (8.26) we would have e(2r) = e- ioJ• S e(r+)e+ ia1 ,S,..

Y(1r!2)I y y- I (7r!2) = I y Y(1r/2)I;l:y- 1(71/2) = +Iz Y(1r/2)lzY(7rI2) "" -Iy

and

(8.34) (8.35)

Before leaving the solid echo, we should remark on one more matter. In Fig. 8.3c we show the spins 1, 2, 3, and 4 at a time ar!2 = 1r/2. We want to refocus the spins to their condition in Fig. 8.3b where 1 and 3 lie along the +y-direction. Looking at Fig. 8.3c we note that if we simply reverse vectors 2 and 4, vectors 1 and 3 will reverse their precession directions, and thus refocus as desired a rime r later along the +y-direction. All that is needed is a 7r pulse about the x-axis. So why do we use a Y(7r/2) pulse instead of an X(7r) pulse? Clearly the idea will work if Gr!2 = 7r/2. But to give a true echo, the scheme must work for other values of a, hence of aT. The easiest test is to look at (8.32). An X(1r) pulse will leave I z and S;l: alone, but will reverse I y • S", I l , Sz. It will therefore satisfy (8.32b), but it willllot satisfy (8.32a). Note, however, in (8.29) that when (ar/2) = 7r/2, the coefficient of the lenn (Iy + Sy) vanishes, so we will refocus this particular product ar. We will not, however, refocus for either a general value of a or of r. Thus an X(7r) pulse will not produce a true echo.

= e- iol• S

Re(O+)R-1e+io1,S,T = e-iol.S, .. e+ial. 5... e(O+) e -iol. 5... e+ io1• S ... = ,,(0+)

378

(8.28) 379

The resulting magnetiUltion is shown in Fig. 8.5c. So far, everything we have done is similar to our first steps in discussing the echo using Fig. 8.3. Examination of the four magnetization vectors in Fig. 8.5c shows that their resultant is zero. We appear to have lost the magnetic order. However, that cannot be true since we know that a Y(71/2) pulse will enable us to recover the magnetizalion. The secret of the Jeeller-Broekaen pulse sequence is the recognition that the spins in Fig. 8.5c have spin-spin order. This can be understood by analyzing what happens if one applies a Y(1I"/4) pulse, producing the result of Fig. 8.5d. If now 380

Fig. 8.Sa-e. The Jeener-Broekaert method of creating dipolar order from pairs of sl)ins (I and 2, 3 and 1). (a) The thermal equilibrium magneti~ation at ! = 0- showing the vector sum of the four magneti~ation vectors. Spins I and 3 are the blue spins, spins 2 and 4 the red Sl)ins. (b) The magnetization vectors at t 0+ immediately following the X(1r/2) pulse. (c) The magnetization vectors at time t T- ,just befoN! the Y(1r /4) pulse, for the case (JT/2 = 1r/2(aT = 1r). (d) The magnetizatiOll vedors at t = T+, immediately after the Y(1r/4) pulse. (e) The magnetization vc<:tors a long time after! = T, The transverse components of mllglletization have decayed to zero from spin-spin couplings to Qlher pairs, leavmg only the resultant componenls along the ~·direction which are 1/V2 shorter than the vectors of (d)

= =

381

we have in our ensemble of spins a weak coupling of one pair with another, there will be a slight spread to the precession frequencies ± aI2. Thus, at a much later time. the transverse components of the magnetization will decay, leading to the situation of Fig. 8.Se. The remarkable poim about this state is thai every spin has its neighoor pointing in the opposite direction. This is to be contrasted 10 the condition al t = 0-. before the first pulse, when half the spins had a neighbor pointing parallel, half had their neighbor pointing anti parallel. Thus, the dipolar

engergy

haI~S;,

Indeed, because I~S~ is. apart from a conSlanl, the dipolar energy, we can see that the part of /? proportional to I~S~ is a measure of the dipolar order. If the dipolar system now comains some order, one should be able to "read" the order by convening it from dipolar order to ''Zeeman'' order, i.e. to net magnetization. Juner and Broekaert do this by applying yet one more Y(.,../4) pulse. which produces an echo a time tt later given by il ;; T. It is not surprising that an echo results since the Jeener-BroekaeIt sequence then of

which was initially zero, now takes on a nonzero value as in

an adiabalic demagnetization experiment. We employ the operator approach using a derivation due to Wang et aI. [7.73]. The density matrix at time T after the X(1f!l) pulse but before any other pulse is given by (8.23b)

(;'(,) ;; A[(Iy + Sy)cos(a,fl) -

(1zS~

+ I~Sz)2 sin(aTfl)]

(8.37)

The pulse Y(1I"/4) then rOtates the spin veClOrs 10 give

e{r+) = AY(>/4)(ly + 5 y)Y-' (>/4)

can be seen, in the limit T _ 0, to be X(rm ... r

J2

y

J2

J2

2(1~S~

- IzS z ) sin(aTfl)]

We now invoke the argument that couplings to more distant spins. not included in the Hamiltonian (8.11), cause the ofT-diagonal components of ~, which arise from the lenns (Iy + Sy) and IzS z in (8.38), 10 decay. Thus at a later time T we can drop the off-diagonal elements of ~, so that f!{T + T) is then

(J{T + T) ;;

-2AI~S~ sin(a;f2)

.

(8.39)

This Hamiltonian has zero net magnetization, as can be seen easily by explicit evaluation of Ot;;z,y.~ ;;Tr{Iau(,+T)}

(8.40)

However, the average dipolar energy

;; -Tr{haI~S~/?(r+T)} ;; -2AhaTr{I; sin(ar/2)} ;; -Alia 2 sin(aT/2) .

<7-l(,+T»

(8.41)

This is maximized in magnitude for (a,/2) ;; 11"/2 at an energy of (-Aha/2). This result is to be contrasted to its value at t;; 0- before the X(1I"/2) pulse:

<1{(0-»

= Tr{rlaI~S~e{O-)}

=

0

.

.

(8.45)

u(r + T+) ;; l"llr/4)( -2AI~S~)y-l11"/4) sin(aT/2)

;; -

•

;;

2A (I,

-

I.) (5, - 5.) . ( (2)

j2

-A[I~S~

-

.j2

IzS~

-

Sill

I~Sz

ar

+ IzSz ] sin(a,/2)

(8.46)

If this is acted on by 1{ for a time t I. then we get e{T

+ T + tt>

;; e-iol.S.tl e{, + r)eial.S.1 1 ;;

-A[I~S~ + IzSz - (IzS~ + SzI~)cos(atl!2) - (IyS; + Syi;)2 sin(ailfl)] sin(ar/2)

Utilizing the fact that

(8.47)

fi and S; are !' we get

e{, + T + tl) ;; -A [l~S~ + lzSz +2 Sy) Sin(at l

(IzS~ + Szl~)cos(att!2)

/2)]

sin(a,/2)

.

(8.48)

= i: sin(ar/2) sin(at l/2)Tr{(1y + Sy)2}

(8.49)

Of these terms, only the last one will contribute a signal:

;;Tr{(I++S+)~(r+T+tl)}

This signal is clearly maximized if we choose r and tl equal to each other, and both of such duration thai

;; Tr{haI~S~A(I~ + S~)} 382

-2AI~S~ sin(ar/2)

_(Iv

S;

(8.44)

it

So

J2

(8.38)

',

which is just the Powles-Mansfield dipolar echo. To show that this happens in the general case, we take /? just before the second pulse to be given by (8.39)

X sin(aT!2)

;; A[(Iy + Sy) -

T ••• Y(1I"/2)

e{r + T-);;

5 )_ 2A ((I. + 1.)(5, - 5.) + (I, - 1.)(5. + 5,»)

y

y(r/4)Y(>/4)

;; X(1I"fl)

- AY(7r/4)2(IzS~ + I~Sz)sin(ar/2)y-t(7r/4) = A(I +

(8.43)

x(>m ... r ... Y(r/4) ... T ... Y(r/4) ... ',

(8.42) 383

arfl ::: attfl = 7C(2

•

(8.50)

Then

'H.::: --yhhoI~ - -yhH) I~ + L Bij(3I~iI~j -1; .lj)

Comparing this signal with that of the free induction decay following an X(7C(2) pulse,

= T,{(r+ + S+)AUv + Sv» = iA n{(Iv + Sv)2}

(8.54b)

i>j

We now define an effective field in the rotating frame

(8.51)

Herr:::iH, +kh o

we see the maximum signal is one half that of a pure X(1rfl) pulse. If there is a distribution of values of a, there is still a signal at tl ::: r since (8.52) sin 2(ar!2) > 0

(8.55)

We shall concern ourselves with cases in which Herr is much larger than the local fields. In Ihal c~se, it is appropriate to quantize the spins along the effective field. Defining this direction to be the Z-direction, we make a coordinate transfonnation (Fig. 8.6) for all spins, getting

Thus, we can create dipolar order by applying X(7C!2) ... T ... Y('ll14) with T chosen such that a T ~ 7Cfl. and wecan inspect the dipolar order by later applying a Y(7C/4) pulse to produce an echo a time T later. The inspection pulse can be used to study the decay of the dipolar order resulting from either conventional spinlauice relaxation mechanisms (e.g. coupli.ng to conduction electrons in a metal), or from motional effects which modulale the strength of the dipolar coupling. We have treated the problem for coupled pairs. JCClicr and Brockacrt treat the general dipolar case in their classic paper [8.12].

Iz ::: I",cos8 + I",sin8 Ix :::I~cos8-Iysin8

Iy = Iv

•

(8.56.)

giving the inverse transfonnations for an individual spin j

Izj::: I Zj cos 8 - I Xj sin 9 IZj::: Ixjcos8+Izjsin8 Iyj::: IYj

(8.56b)

Subsriruting into 'H., we gel

8.7 The Magic Angle in the Rotating FrameThe Lee-Goldburg Experiment

+2

'H.::: -"fhHcrrIZ +

L

A/l1(8)'H/l1

where

(8.57)

/1,1:::-2

Another imponant concept concerning dipolar order is connected with the famous Lee·Goldburg experiment (8.13] 10 which we now turn. Suppose one has a number of spins all with the same precession frequency, coupled together by dipolar coupling. We keep just the secular tenos of the dipolar coupling, giving a Hamiltonian in the rotating frame of

'H.::: -

- "fllH II~ 2 "f2h + :L -,-(1 - 3 cos28ij)(3I~i l~j - Ii' Ij)

'H.o:::

,

"fhhoI~

i>j

(8.53.)

where as usual

r

h,

etc.

ho::: Ho -wl-y

(8.53b)

It is convenient to collect all the radial and angular lenns of (8.53a) in a simple symbol by defining I "f2h2 B;j=zT(I-3cos2 8ij)

Ii' Ij)

(8.58)

z

----

T ij

I~::: r:,I~j.

L Bij(3I~iIzj i>j

,

I I I I HrH ! I I H,

, x

Fig. 8.6. .The erre<:t;,·e Held in the rotAtin« rl1lme, showing the AXes X,Z WIth respect to ~,z

(8.54a)

OJ

384

38S

1-£+1 =

L

3BijUt IZj + IZirt}

i>j

1-£+2 =

L

3BijUt It}

,

i>j

1LI

= (1-£+1)", 1-£-2 = (1-£+2r

and

(8.59)

-'0(9) = !(3cos 28-1) (8.60)

-' ± 1(8} = -! sin 9cos 8 -'±2(8) = sin 28

-i

The 1-£/I.1's satisfy a commutation relation

lIz, 1iM]- M1iM

(8.61)

as is easy to show by explicit calculations of matrix elements. From (8.61), or by examination of the explicit form of the 'HM's, we see that only 1-£0 commutes with the Zeeman interactions of the effective field. In the limit of large effective field, we can then as a first approximation drop all terms but the term involving 1-£0. We get then a truncated Hamiltonian 1-£ = -yhHefflz + !(3 cos 28 - I) Bij(3IziIZj - Ii' Ij}. (8.62)

L

i>j We can compare this with the secular part of the Hamiltonian of the spins in the lab frame,1tIl\b' in the absence of an alternating field, H,: 'H'l"b = -"'('iHolz +

L

Bij(3IziI:j - Ii' Ij) (8.63)

Equations (8.62) and (8.63) are identical in form, except in the rotating frame the dipolar term has been multiplied by the factor 3,os 29 _ I 2

(8.64)

['This is juSt the tenn -'0(8)]. 8 is delennined by the relative size of ho, the amount one is off resonance, with HI, the strength of the rotating field. Lee and Goldburg noted that if they chose HI and ho properly, they could make cos 28 (8.65)

=!

in which case the dipolar term in (8.62) vanished. In this manner they could effectively eliminate the dipolar broadening. An effective field for which (8.65) is salisfied is said to be at the magic allgle. We can rephrase their result by saying that if the effective field is at the magic angle, a spin will precess in the rotating frame solely under the influence of Neff, without suffering a dephasing and consequent decay of the components 386

(8.66) The failure of the dashed theoretical curve to intercept the origin arises because of slight inhomogeneity of H1. BarntUJl and Low [8.14] made an extensive study of the systems in which HI was exactly at resonance. In this case 3cos29 - I (8.67) 2 They did experiments in which they turned on HI for a time T, then observed the free induction decay a time t after H 1 is turned off. They solved the problem of an interacting pair exactly. For an HI along the x-axis they found My(t) = M(O)

i> j

:= 1-£Zeemlln + 'H'dipolllr

of magnetization perpendicular to Herr as usually occurs when there is dipolar coupling. To make a precise test of this concept, they did pulse experiments in which they observed the decay of the magnetization with time following a sudden tumon of H t at a frequency w somewhat off resonance. They varied the angle 8 and the strength of the effective field. To measure the effective dipolar strength, they determined the second moment from the transfonn of the decay curves. They studied the F ID resonance of CaF2. Figure 8.7a shows their measurements of the square root of the second moment versus (3cos 28- l)n.. AI the magic angle, the second moment should vanish. It does not quite do so owing to the nonsecular terms involving 'H± I and H±2. The effect of these tenns should vanish in the limit of infinite Herr. In Fig.8.7b we show their measurements of the faJl-off of second moment, nonnalized to its value in the lab frame, versus llw; where

~ sin(flT)cos [ (3:,: 2 ) (t + T!2)]

(8.68)

where

n' -_ (3BI2)' - - +w,,

"

4~

(8.69)

In the limit of large HI, this expression agrees with the result of the truncated Hamiltonian: My(t) = M(O) sin(wl T) cos [ (3:,:2 ) (t + T!2)]

(8.70)

A striking feature of (8.68) is Ihal the oscillation after the tum-off of HI is identical to what it would be following an X(7r/2) pulse except for (1) a slighl amplitude correction and (2) a change of T in the apparent zero of time t. In a beautiful set of experiments, Barnaaf and Lowe showed that for CaS04 ·2H20, CaF2, and ice the first zero crossing of the free induction decay moves 10 later times as T is increased. The delay in zero crossing is T/2. for values of T which are up to about half of the nonnal free induction decay zero crossing.

387

I

(oF,IU doptlll H.1I111lJ

.. ~ 0.6

>J A

."

0.'

v

0.'

~

"-

s

A

."

., ~

pulse a time T later as suddenly lransfonning the Hamiltonian to be the negative of the actual Hamiltonian. Then, over the next time interval T, the time development of the magnetization unwinds, returning the magnetization to its value just after the initial pulse. In discussing spin temperature (Chap. 6), we talked about irreversible processes connected with the complexity of a system of many coupled dipoles. We now tum 10 a remarkable discovery which shows how one can run backwards the dephasing produced by dipolar coupling, thus showing that it is possible, even after the free induction decay is over, to recover the initial magnetization. This experiment shows that the spin temperature approach does not in all cases accurately describe the evolution of a spin system. In the process an echo is fanned. This sort of echo has become known as a magic echo. The first intimation that one could refocus dephasing arising from dipolar coupling was noticed by Rhim and Kessemeier [8.15,16], who discovered the effect experimentally, and showed theoretically by an approximate method Ihat an echo was fanned in which the loss of signal owing to dipolar dephasing was recovered. Following this work, done at the University of North Carolina, Rhim went to MlT where, in collaboration wilh Pines and Waugh, he extended and perfected the experimental techniques and the theoretical analysis [8.17]. In order to refocus spins which are defocused by dipolar coupling, one needs 10 be able [0 reverse the sign of the dipolar Hamiltonian. Then, as we have seen, we can effectively run the system backwards in time to undo Ihe dephasing. The trick is closely related to Ihe Lee-Goldburg experiment. The critical equation is (8.62), the truncated Hamiltonian in the rotating frame:

H, : 5.1!O.2 Gauss

/"

0.'

/:

'"

0.2 0.1

/'

~OO~5;:;-0::;.';:;-0:::J~-0:::.2~-0:;.1---'~0'--;0:;.1--:0::.2:-:0::J:-:0~.'---'0:':5,---J0.' ]cos'e~l

lal

2

• ';

-v., , -.,

.10. 1

//

,

/

•

/ /'//

A

~

/

/

2

A

/

•/0

//

~

v

/

/

/

(oF1lU doped I

Ho lll110l 2 /3(05 2

&.,/

~

0.01

11. = -",lill wIz + ,ell

0:'0--;:-;-----;;7'-----:':c--.J 0.1

Ibl

0.2

0.3

0.4

Il&.ll'lwli....:

F1g.8.7a,b. The ~ Ilnd ?oldburs experimental results for F" relIOnance in CaF, sins1e ct)'st.I~. (a~ The normllh~ second moment. of the Fit resonance for flo par.. llel to the ~lll J dued,.on, as a function of (3 COlI" - 1)/2, where B specifiCil the orientation of H

c:r

III the rotatmg rr.~ (see Fig. 8.6). (b) The normalized second moment iLS II function the .second moment III the lab frame < (4....') > LAn divided by",,' where .... = 11 e' e " elf110 1$ parallel to the crystal (1101 direetion

8.8 Magic Echoes In Secrs. 2.10.and 5.~ we saw thai one way of understanding how a spin echo comes ~bo~t IS to view me first pulse as initialing a time development of the

(3cos 26-1)

,,,

'"' B·1)·(3[z1z· I J) -1·I - I·) )

2~.

(8.62)

In this equation, () is the angle between the static field Ho and the effective field, Heff, and the Z-direction is the direction of the effeclive field. If Heff is nearly parallel to Ho,cos8 = 1 and the angular factor, (3cos 2() - l)Il, in front of the dipolar tenn is + I. However, if () = as when H t is exactly at resonance, the angular factor becomes 3cos 2(} _ 1 I (8.71) 2 =-1:

trn,

Thus, it has a negative sign. The magic echo makes use of Ihis negative sign to unwind the dipolar dephasing. Let us then rewrite (8.62), the truncated Hamiltonian, exactly at resonance, using the coordinates x, y, z in which HI lies along the x-axis: 1 1i = -"'thHII~ - 2" BjJ.:(3J~jI~k - I j .IJ,;) (8.72) ;>k

L

In addition to Ihese tenns, there is the nonsecular tenn

magneUZ3tJon under the influence of the Hamiltonian, and to view the second 388

389

1{nonsec =

~

L

3D jk

(lt rt + I j- Ii:)

,

(8.73)

j> k

which we are for the moment neglecting. How can we utilize the negative sign to unwind dipolar dephasing? For the moment, consider an experiment in which we produce some transverse magnetization [for example, by a Y(nl2) pUlse]. Let it dephase under the action of the dipolar system, for a time T, then turn on H I to produce a negative dipolar coupling. Will this refocus the dipolar dephasing? If the density matrix (in the rotating frame for our whole discussion) is initially (t = 0-) given by (8.74) where A is some constant which we set equal to I for convenience, the pulse sequence would produce at a time t a density matrix e(t)=ex p [

L

~1{zzT) Y(-Ir/2)Iz {inv}

(8.76) Bjk(3IzjIzk - Ij' h)

j>k

e{r+2T

)

=ex p [

x exp

-* (r!lHII~-~1{XZ)TI] [-

X exp ( =exp

X exp ( 390

*(-~!lHI[~ *

-

~1{~z)

*

2T

T ]

(8.77)

(8.80)

where tl is the observation period. Picking T ' = r will cause the signal at tl = 0 to correspond to the full magnetization. It is easy to extend the discussion to show Ihat if one were to hold T I fixed and reduce T, the dipolar refocusing would occur at a time tl given by

tt +r=T'

(8.81)

so that if r goes to zero, there would be a dipolar echo at t I = r ' . Note that if T goes to zero, the Y(n!2) and y-I(lT!2) pulses just undo one another, so both could be omitted. This is exactly what Rhim and Kessemeier did in their first experiments. From (8.78), it is clear that

= ex p ( I

')]

1iZZ T) Y(-Ir!2)Iz {inv}

(8.79)

Clearly when T I = T, the dipolar dephasing has vanished. Our pulse sequence is thus (reading left to right)

y- 1 (n/2) ... r,/!]

1{zzr) Y(-Ir/2)[z {inv}

[-1ii( - 2'it..

1{;z2T')] y- I (1T/2)

Y( n /2) ... T... Y -I( n /2) ... TIIl I ... TI-III ... Y(lTl2) ... tl

This expression has two dipolar temlS with opposite signs in front, but one is 1{~~, the other 1{zz, so they are not the negatives of each other. Moreover, there is also a term involving Ih which induces precession around the x-axis, whereas the initial dipolar term involves only 1{zz, the dipolar dephasing. It is easy to get rid of the precession, as was shown by Solomoll [8.18] in his paper describing the rotary echo. He showed that he could get rid of dephasing from precessing in an inhomogenous HI by suddenly reversing the phase of HI so that the spins precess in the opposite sense. We employ the same idea here. We keep HI on for a time TI then reverse it for an equal time r'. Then we get I

*(_

(8.75)

Bjk(3I~jI~k - Ij' h)

L

I e{T+2T ) = Y(1T!2)exp [_

=exp(*1i zz T')ex p ( -*1izzT)Y(lT/2)Iz{inV}

j>k

1{zz:=

Therefore, we add two pulses, a Y(1T/2) and a y-I (1T!2), where the latter is simply a Y( -1T/2) before and after the time interval 2r' giving

xex p ( -*1i zz r)Y(1T!2)Iz {inV}

where {inv} is the inverse of the operators to the left of I z and where we define 1{~~ ==

(8.78)

-i( -'''HII<-~'it..)(t-T)]

exp ( -

X

\

This expression is still not quite what we want since it still has two different dipolar Hamiltonians. However, examination of (8.76) shows that 1{u; and 1i zz differ simply by a cOQrdinate rotation of z into x or x into z. Such a transfoooation is produced by a rotation about the y-axis. In fact

... T I -

-*(-1t

I ZZ T

»)

Ifl

...

Y(1T!2) (8.82)

Rhim, Pines, and Waugh label such a sequence a "burst". In fact, they argue that effects of the nonsecular teoo (8.73) can be reduced if one makes a single burst by putting together many pairs of (HI,T') and (-HI,T I ) using very short TiS instead of using a single HI, - HI pair of longer duration. Their argument, the details of which they omit, appears 10 be related to their thinking about so-called average Hamiltonian theory. Of course, the most obviolls way of reducing errors from neglect of the nonsecular teoo is 10 make HI very large. Rhim, Pines and Waugh used an HI of 100 Gauss for CaF 2 with T'S of the order of 1.25 p,s. 391

In their work, they also implied that one should pick ,HIT = mr where n is an integer, although in fact they demonstrate data which violates this condition. Takegoshi and McDowell [8.19] have shown experimentally that this is evidently not necessary. It is worth noting that there is a strong experimental reason for reversing HI many times at short intervals rather than once at a long interval. The requirement of applying a large HI for a long time often places a great strain on the power supply of the rf power amplifier, causing the amplilUde of HI to droop with time. For a "burst" involving only one phase reversal of the rf, mere may be significandy lower HI during the second half (the phase reversed period) of the burst Ihan was present in the first half. If, however, there are many short cycles (H .. -HI), the fractional difference between HI and -Ht will be greatly reduced. It is interesting to note that if one has a group of nuclei with several chemical shifts the shifts can be revealed by a magic echo. During the time the HI is on, the chemical shift fields, which lie along the z-axis, can be neglected since they are perpendicular to the much larger HI. During the time T when HI is off, the full chemical shift acts. Since that is only one-third. of the time, the precession frequency is displaced from w by one-third of the chemical shift frequency. Rhim, Pines, and Waugh demonstrated that one can apply a train of magic echoes which, similar to a Carr-Purcell sequence, will refocus the magnetization again and again.

In Sect. 3.4 we described experiments by Andrew and Eades on the use of line width studies to reveal the presence of rapid molecular motions. This effect had actually been discovered by GUlowsky and Poke [8.20). For the case of rapid motion, Gutowsky and Pake had shown tht the angular factor (1 - 3COS 20jk) should be replaced by its time average as discussed in Chap. 3. That average is given by (3.61)

, , (3COS 2"Yjk - 1) 2

(1 - 3cos 8jd&Y. = (I - 3cos 8)

(3.61)

where (J' is the angle the molecular rotation axis makes with the static magnetic field, and "fjk is the angle made by the internuclear vector rij with the rotation axis. This expression suggested independently to Lowe [8.21] and to Andrew et al. [8.22] that one could produce the rotation artificially by turning the entire sample. In that case, the angle fJ' would be the same for all pairs of nuclei throughout the sample. Then if one chose fI to satisfy the condition

I - 3 co"9' : 0 392

z=rcos8

,

x=rsin8cos9

,

y=rsin8sin9

.

(8.84)

We recall that me Yim's are related 10 solutions of Laplace's equation

v'(h,m)= 0

(8.85)

[The fact that the unnormalized column of r1Yim's satisfies (8.85) is readily verified by expressing '\12 in rectangular coordinates). This is the reason that, for example, 1=2 functions are made up of linear combinations of x 2 , y2, %2, xy, %%, yz, and do not include terms such as x or xy2. Table 8.1. Listing of the normali:ted spheriul hllrmonks (or f = 0, I, 2 and the uoorlllalized forms

8.9 Magic Angle Spinning

,

the time averaged dipolar coupling would vanish [8.21, 22b]. This value of 8' soon became known as the magic angle, and such a method of line narrowing is called magic angle spinning (abbreviated as MAS) or magic angle sample spinning (MASS). It is historically the first of the methods for narrowing dipolar broadened lines. The effect of spinning on dipolar coupling is only one of several important uses of spinning to eliminate unwanted couplings. It can also be used to eliminate chemical shift anisotropies and first order quadrupole splillings. These are all interactions which involve angular functions made up of the 1 = 2 spherical harmonics, Y'm' To analyze these various methods, it is convenient to begin with a millOr digression about spherical harmonics. Table 8.1 lists the normalized Yim's for I = 0, I, and 2 as well as unnormalized forms involving the coordinates x, y, % where

Normalized spherical harmonies Yo

-

0.0 -

, "7f;

l"l,dB"P) =

constan~

-fl; sin Bel.

# cos (B, 4» = #

Y1,0(B,4» =

Yl,-l

Unnonnatizcd r'Y'm in rectangular coordinates

VIm

Y2,2(B,4» = Y2,1 (B, 4» =

Y2,o(8, 4» =

z+iy

,

B

z- iy

sin Be-i.

a --/* a(3

sin' Be,l. sin 8

COlI

Bel.

+ iy)

=z.: + iy.:

cos 2 8 - 1)

Y2._1(B,4» = V*sin B cos Be-I. Y2._ 2(8,q,) =

.:(ot

j"ji; sin 2 Be- 21 •

.:(z - iy)

=:u -

(ot _ iy? =

:>:2 _

iy.: y2 - 2izy

(8.83) 393

'.

Suppose now we consider two coordinate systems x, y, z and x', y, Zl, with corresponding angles 0, tP and 8', ¢I defined by (8.84). The Yim(O, tP)'s form a complete set for a given I, which means that we can always express }'i fi (8', tP') in tenns of the Yim(O, ¢)'s: }'jfi(e', tP') =

Lafim l'im(O, tP)

and

(8.86a)

m

Y,m(B,~) = L:em"Y,,(B',?,)

,

lal

(8.86b)

It is then slTaightforward, utilizing the onhogonality propenies of the l'im's when integrated over 4lf solid angle, to show that

x'

z'

".

I

Ibl f f / l / /

//

f y/

'. 1(1 f

I

I

Idl f

'.

L: Y,;.(e;., ~o)Y,m(B', ~')

Yo /'

/'\

I

(8.89)

01

m

(This theorem can be derived by making an expansion of a 5-function lying along the z-axis. One utilizes (8.86), the fact that the 5-function is axially symmetric about the z-axis, and equates the expansion in the lI",(D.?)'s to that in the

Yim(8', tP')'s] . We now wish to consider a spinning sample. To do this. we wish to define some axes (Fig. 8.8). We define first the laboratory z-axis. ZL, along the static field, and an axis zit about which we will eventually rolate the sample. The angle between ZL and Zit is 00 (Fig.8.8a). We can then, without loss in generality, define the laboratory axis XL to be perpendicular to both zL and Zit (Fig. 8.8b). We define XIt to be coincident with x I, (Fig.8.8c). The axes xn.. !JR, and Zit are fixed in the laboratory frame (Fig. 8.8d). We then define axes xs, YS, and zs which are fixed in the sample with zs coincident with Zit at all times, and Xs and YS coincidem with Xlt and Yit respectively at time t = 0, but making an angle nt at" later times (Fig. 8.8e). n is the angular velocity of sample spinning. Utilizing these definitions, we can specify the orientation of Ho in the coordinate system Zit' YIt, Zit (at angle 00R, 4toRl and in the system zs' !IS' zs (at angle Dos, ¢os)·

/

"

(8.88)

They can also be found using the formalism of the Wigner rotation matrices (see for example the text [8.23]). Let us now specify the orientation of the z-axis as being at an angle (9'0' ¢o) in the primed system. Then there is a famous theorem, the addition theorem for spherical harmonics, which tells us that

/

/

/

"

'. "

= b31 x + b32Y + b33Z

Y,o(O,O)Y,o(B,~) =

'.

/' /

)//

=b II X+bI2!J+b 13 Z

y' = b:ll x + b22!J + b23Z

'.

"

'.

'. "

(8.87)

Noting the forms of the Yi", 's in rectangular coordinates, we realize that the coefficients all'" or Cnll' could be found by explicit substitution of the relationship between the primed and the unprimed coordinates:

"

I

'. --

I

J../

/

Yo y,

'. '."

Flg.8.8a-e. Axes imporh.nt ror sample spinning. (a) The laboratory z-axis, ZL, is chO$en along the static field /lo. The axis ZR makes an angle 80. Together. ZL and ZR define a. plane. (b) The lab axis %L is defined as lying perpendICular to the (ZL, ZR) plane. (e) The axis %R ill coincident with %L. (d) The axes lIL And YR lie in the (:R,zd plane. Af{ the axes %L.1IL,rt. and %R,YR,zR are fixed in t~e laboratory rrame. (e) The axes %s,VS. and Zs fixed In the sampl~ and rotate with il, Zs coincidins chosen to be coincident with ZL With ZR. At t = 0,Z5 and ZR, so that it makes an angle Ot with them at later times

" u:e " "

I'

From the figure we see °OR=OO,

Oos = nOR = 00 '

(8.90,) ¢os =

tPOIt -

nt = 7rn - nt

(8.9Ob)

We now consider the interaction between a pair of spins j, k which we have previou.sl~ w~tten as proportional to 3cOS 2 0jk - I (3.7). We now, however. must dlstlOgUish between coordinate systems. hence we replace (Jjk by OLjk> writing the coupling 2 3COS 0Ljk - 1 = JI67r/5Y20«(JLjk, tPLjk) (8.91) Then, we can utilize (8.89) to get Y20(O, O)Y20(OLjk, ¢I,jk) = LY2"m(00s, ¢OS)Y2m(OSjk> tPsjk)

(8.92)

m

395

where 0Sjk, ¢Sjk give Ihe orienlation of the internuclear vector Tk in Ihe coordinate system xS' Ys. zs which is fixed in the sample and where Oos and ¢os are given by (8.90). Substituling the explicit expressions for the Ylm'S from Table 8.1, we get

1

3 COS 20Ljk _

= (3COS

2~OS -

1)

We turn now to a discussion of the effect of spinning on a pulse experiment in which we observe the free induction decay of a spinning sample following a -;r/2 pulse. Then we might express the dipolllr Hamiltonian, 11.d as

(3cOS20Sjk _ I)

j
(8.93)

We see that the second and third tenns on the right vary as f}t and 2f}t respectively, whereas the first tenn on the right is independent of time. Thus if one can average the time-dependent tenns, we recover the result of (3.61) with the notation

Cs = 3 CO!;200S - I

(8.95)

2 the/orm of the absorption line should be identical in the two cases except for a scaling of the frequency. Thus, if i(w - wo) is the nonnalized intensity function without spinning, Ihe normalized function with spinning is(w - wo) becomes fs(w - wo) ,

1

C/(Cs (w - wo))

FOjk

=

Fljdt)

=

for large 'Y nuclei, since spinning speeds are typically a few kilohertz, comparable 10 typical dipolar line widths.

(8.98b)

(3COS200S - 1)(3

2

'" COSVSik- l )

6 sin Oos cos 80S sin 0Sjk cos 0Sik COS(¢Sjk - ¢os)

(8.99)

F 2jk (t) = ~ sin200s sin 0Sjk coS 2(¢Sjk - ¢OS) Since ¢os = -;r/2 - f}t from (8.90b), we see that FOjk is independent of time, but F1jk and F2j~' oscillate in lime at f} and 2f} respectively. Explicil examination of the spin functions G jk shows Ihal two different funClions which have one spin in common, for example the jth, such as Gjk and Gjl (f of- k), do not commute. This makes it difficult in general to determine the time development. Such mailers are discussed in Appendix J dealing with time-dependent Hamiltonians. Here, we are dealing with Ihe case of a Hamilitonian composed of a sum of noncom muting parts each of which has a lime-dependent coefficient. This is Case II of Appendix 1. However, the lime dependence is periodic, so we can consider what happens over a single period. Suppose that the period T(::= 21f/f}) is sufficently shan that not much happens during one cycle. Then, as we show in Appendix I, we can relate the wave function at the start of a period to its value at the end of Ihat period !>y the unitary operator U(T) given by

,p(T)' U(T),p(O), exp ( - i7tT) ,p(O)

(8.l00)

where

l'

-11. T'fo =

(8.97)

3Izj l zk )

2

(8.96)

At the magic angle, the line theoretically becomes a O-function. In practice, it is difficult to achieve the condition

-

is a function only of Ihe XL, YL, and ZL components of the spin operators Ij and Tk' and

~~

For steady-state experiments (i.e. if one talks about the "frequency" domain, perhaps with data which are the Fourier transform of data from a pulse experiment), the tenns involving f}t and 2f}t give rise to frequency modulation at frequencies f} and 2f}. Such a modulation gives rise to multiple sidebands spaced in frequency by f} aboul the ncnnal fequency wo. For frequencies f} much bigger than the dipolar line width ~Wd' these sidebands will be displaced by an amount large compared to the width associated with the time-independem tenn. If we neglect the sidebands to focus our attention solely on the main transition, it will appear narrower. Indeed, since the time-independent tenn of the dipolar coupling for the spinning case is simply reduced from its value for the nonspinning case by the faclOr Cs where

396

where 2 ·2 o" Gjk(Tj , h)::= -,-(Ii' h

r jk

(8.90b)

~k=~k

(8.98a)

j>k

Since ¢os obeys (8.90b), 10s ' ,/2 - fit

1 jk

, I: [FOj' + F1j'(') + F'j,(.)jGj,(Ij,I,)

+ 6 sin OOS cos Oos sin 0Sjk cos 0Sjk cos (¢Sjk - ¢os) + ~ sin200s sin 20Sjk cos 2(¢Sjk - ¢os)

...,2h2 3 COS 20Sjk)-.-3-(3Izjlzk - I j ·Id

E (l -

'H<J =

But

(8.lOl)

'H(r)dr

T

T

jF1jk(r)dr=0

jF2ik (r)dr=O

o

0

so that it is only the term FOjk which brings about a change in period.

(8.102)

t/J over an

integral

397

This is tantamount to saying that to the extent F1jk(t) and F 2jk(t) produce changes during the time interval T, they must also undo these changes over a complete period T. The resuh is that the free induction decay has periodic maxima which Lowe called rotational echoes. Figure 8.9 shows the data from Lowe's paper [8.21] for FI9 in Teflon. The rotational echoes are clearly apparent, in Fig. 8.9a. as is the fact that at the magic angle (54.7°), the free induction decay lasts much longer. The Fourier transfonn of these data (Fig. 8.9b) shows the sideband structure. It illustrates the fact that even when spinning at over 6 kHz, the sidebands are still not separated from the central line for the high FI9 nucleus.

,,---,----.----,---,-----,

•

F (I)

(aJ

,_

• , •

'H,

Is flle/s) 0.3.1,6..2 _ _

".1

"

".1

"

Andrew et al. [8.22] and Kesscmeier and Norberg [8.24] demonstrated clear separations of the spinning sidebands for lower 'Y nuclei Na 23 and p31 respectively. hKieed, removal of the dipolar coupling between like spins is done more readily by applying strong pulses to flip the spins, so-called spin-flip narrowing, which we discuss in the next section. We can understand physically the significance of the requirement (8.97) that n>6.wd and its relation to the fact that G ij and Gjl do not commute as follows. For spinning to average out an interaction, the interaction must be constant in time over at least one cycle of rotation. To average the coupling GjJ:' the local field that spin j produces at spin k must not change during one cycle. But the coupling Gjl pennils spins j and I to undergo mutual flips, thereby interrupting the coherent averaging of G jk' 6.wd is a measure of Tjl' how long spin i can maintain its orientation without such a mutual spin flip (Tjl ~ 1/6.wd ). For narrowing of the jk coupling to occur, the period of rotarion (1/0) must be less than Tjl, Tjl>

02

"

1/n where

Tjt"" I/.6.wd

(8.103.) (8.103b)

These can be rewriuen to give (8.97)

3 2

{»

'" (b)

..

I (I's/

""

f!

,I

3

P

"

"" ii'G~

2

f"'

__'_ r'W t F(11
G(WI

201684048121620

W( rod/s) x 10'4

1<'1g.8,9, Free induction decays (8) for spinning and nonspinning samples of Teflon, and their Fourier lransforma (b). The curves are corrected for instrument.l nonlinearities. The data are due lo t. L/)wt, 1/. Ktsst~itT. W. Ytn. G. Thriss, and R.E. Norbug 18.201

398

.6.wd

(8.97)

TIle case we have just discussed, which we call Case II in Appendix J, has been called an example of a "homogeneous" broadening mechanism by Maricq and \Vaugh (8.25]. (This tenn is closely related to an idea introduced by Portis [8.26] in electron spin resonance). Its essential feature is that Ihe time-dependent parts of the Hamillonian do not commute with one another. To produce line narrowing in this case, one needs n ;» .6.w where 6.w is the line width from this mechanism. In the absence of spinning 6.w = 6.wd for dipolar broadening. We now tum to a case which has a much less stringent condition, and thus is much easier to achieve experimentally, narrowing lines when the time-dependent parts of the Hamiltonian commute, Case I of Appendix J. We illustrate this case by line broadening from the anisotropy of chemical or Knight shifts. This case is called "inhomogeneous" since it is analogolls to line broadening by inhomogeneous magnetic fields. Andrew recognized that magic angle spinning would remove line broadening from sources, other than dipolar coupling, which depend on the orientation of Ho with respect to the crystal axes. The key point was that the broadening be described by appropriate angular factors arising from the orientation of the static magnetic field, Bo, with respect to the crystal axes. When the angular factors are proportional 10 Yim's with 1 = 2, there is a magic angle. Other cases are line broadening from chemical and Knight shift anisotropies, and from electric quadrupole splitlings. We treat the case of shift anisotropy.

399

Recalling (4.198), we get that, in lhe presence of anisotropic shifts,

Y2,0(9 0p, ,pop)

=

L Coms(P, S)Y2ms (90S• tPos)

'H. = - 'Y;,Ho[1 + (/( _ U)+(/(LO _ 0LO) (3COS:O- I) +

(I(TR;

UTn) sin 2ocos2 tP] l z

.

(8.104)

We consider lz to be the sum of the z components l z k of N noninleracting spins (k = I toN). We shall analyze the effect of rotation on this Hamiltonian. The angles 0 and tP of (8.104) specify the orientation of Ho in the principal axis system of the shift tensors. We shall need to keep track of various reference frames. Therefore, we shall define !he spherical angles Oop and tPop to designale the orientation of fIo in the principal axis system xp, yp, zp. We keep the same definitions for the coordinate systems XL, YL, zL (Ho lies along ZL), %R, YR, Zn, and %s, YS, Zs (Fig. 8.8). We have then that at t = 0, ZL is at Oop, tPop or expressed in R or S at 00R, 4>oR or OOS. 4>oS. which in tum are described by (8.9Oa). Therefore. we can wrile at t = 0

Y2 _2(9 0p , tPoP)

,

At later limes. owing to the rotation, fos will change with time. bUI since the rotation axis, 90S does not change, so 80 5(')

=80

(8.105)

As time goes on, Oop and t/Jop change, owing to the sample rotation. hence should be viewed as time dependent. Recognizing that the two angular functions are related to Y2rn'S apart from nonnalization conslanlS. we define lhe nonnalizations N 2,o, N2,2' N2,_2 as Y2,0 Y2,2 Y2,-2

== N2,0(3cos 29 - I) == N2,2 sin gei .p == N 2,-2 sin ge- i .p

,


=.f2 -

ill

zs is

(8.109)

.

Therefore the mS = ±2 lenns of (8.108) will oscillate at ±2n, the ms = ± 1 terms will oscillate al ± and the ms :: 0 tenns will be time independent. Collecting lerms we gel

n.

'H.= -'YhHo1z[I+(J(-U>

+ .

(8.108)

+ ( 1(1.0 - 0'1.0) "L...COms(P,S)Y2rns(8o.'ll"f2 - fI) t 2N2,0 ms

'H.= -'YhHO[1 +(l( -U)+ (J(LO ;ULO) (3 cos 290p -I) + (KTn;UTR) sin20oPcoS24>OP]lz

no, = L C-2Jl1s(P, S)Y2ms(Oos. tPos) no,

(8.106)

and express

"L...[C2ms(P,S)+C-2,ms(P,S)I ( [{TR4N-2,2OTR) ms X

[Y,no,(8 0,.f2-flT)I]

(8.110)

This Hamiltonian consists of several time-dependent parts all muhiplied by the same spin funclion, l z • Thus the Hamiltonian alone time, t, commutes with the Hamiltonian at another time, t':

['11('), '11(,')) = 0

(8.111)

.

This HamillOnian therefore belongs to Class I of Appendix 1. Indeed, it is even described by (l.28) (1.28)

'II(t) = a(t)'II.

for which the formal solution is given in (1.30) as ,p(') = exp ( - T.'II.

We can now express the angular functions in tenns of the S-system, using (8.86),

400

f

a(T)dT) ,p(0)

.

(1.30)

Let us first consider just the time-independent term (mS = 0). It will give a contribution to the Hamiltonian of

401

1tlimeindcp= --rhHQTZ{ 1 +(J( -01+ [ +

(/(Tn -

(J(L~N~,:I.O) CO,O(P,S)

a TO ) [C20(P,S) + C_, O(P,S)I]

4N2,2

x N2,0(3cos 280

'

,

1£ = - ,hHol, {I + (I( - 'i7) + AI(P.S)cos

-I)}

--yfiHolz[I + (I( - a)J

.

(8.113)

£

7{(T)dT = -'YhHo1,

C',o(P,S)

= a;,,(S,P)

Y2~O(O.O)

Y;,(a,fIj • aO,2 = yi,o(O,O)

'!>(') = exp ( -

Y2.,O(O,O)

(8.115)

where (a, fJ) gives the orientation in spherical coordinates of the spinning axis

[(I(LO-O'LO) (3Ca"a-l) 2 {1+(I(-'i7)+ P)] (3CO"Bo-I)} ' _O'TR)(,inaCOS2 + (} 2 2 .

1£timeindep= ~'YfIHo[,

(8.116)

(This expression can be checked in the limiting case that 80 = O. hence the spin axis lies along Ho. Then the spinning has no effect and the time-independent tenn is the entire Hamiltonian. Then 0' = 0 or 0' = rr(2, {J = 0 or 1r(2 correspond to.flo lying along the three principal directions. The result is that for the three cases we get shifts respectively 1(3 - 0'3. 1<1 - 0' 1, or J( 2 - 0'2, where 3, I, and 2 stand for the zp. x p , and Up directions.) The other very interesting point to consider is the time dependence when one is spinning at the magic angle. Examination of (8.110) shows that the time dependence of 1£ comes from the factors Y2ms (80, 1r(2 - nt). Ulilizing the prop-

402

(8.118)

*£

7{(T)dT) ,!>(O)

(8.119)

and recalling that T = 27f/n. we see that when

Y;,_,(a,p) °0,-2::

%s in the Xp. yp. %p (principal axis) coordinate system. The result is

\TR

"1JI

Utilizing the fact that

and the spherical hannonic addition theorem (8.89)

Y;o(a,p)

+(I( -

. [ + A,(P,S) 2il ( sm 2ilt - r2(p,S)] + sin r 2(p,S)} )

(8.114)

C_',o(P,S) = a;,_,(S,P)

ao,o =

([1

. [ + AI(P,S)( n smm-r 1 (P,S)]+sinrl (p,S)}

(8.87) = a;,o(S,P)

(8.117)

where AI(P.S) and A2(P.S) are coefficients, and rl(p,S) and r2(P.S) are phase angles, all of which in general will depend on the orientation of Zs in the xp, yp, zp system. Utilizing this fonn, we get

Thus, it is identical to the fonn taken in a liquid in which the anisotropic shift contributions average to zero. If 80 is nO[ at the magic angle we might wish to know the coefficients CO,o(P, S), C2,O(P, S), and C-2,O(P, S). These we get from the inverse relalions

CO,o(P,S)

[nt - rl(P.S)]

+ A,(P,S)co, [W. - r,(p,S)))

(8.112)

The coefficients C 2,0 (P, S) and C- 2 ,0 (P, S) can be obtained (if desired) as we explain below. But even before that, we nOie thai when 80. the angle between the rotation axis and Ho. is 3t the magic angle. we gel 'Htime indep =

erties of the Y2ms 's (Table 8.1) we see that 1£ is a sum of tenns which are either independent of time. or vary as exp(int), exp(-inl). exp(2int). exp(-2int). Keeping in mind that the Hamiltonian must be real. and that we are at the magic angle,.we can thus write

t=nT

•

n=O,I,2 ...

'!>(I) = exp ( - i'YHo1,(1 + [( -

(8.120.)

a)') ,!>(O)

(8.12Ob)

just the result we would have if the time-dependent terms were missing. Note that this result says that at the magic angle the time development over integral multiples of the rotation period is independent of the orientation of the crystal axes relative to Ho. Thus, if we applied a 7f{l. pulse to a spinning sample. the free induction decay at times given by (8.120a) would act as though there were no anisotropy to the shift tensor. If one had a powder sample which was not spinning, the spread in precession frequency arising from shift anisotropy. 6.wshif~' would cause the free induction to decay in a time 1/6.wshif~' If, however, the sample is spinning we can see from (8.120b) that the signals are rephased at times given by (8.120a). thereby producing a string of echoes. Notice that we have not placed any requirement as yet on how fast the spinning must be, in contrast to the previous siwalion concerning narrowing of dipolar line broadening. For the dipolar case we compared the rigid lauice line width ~d with n. What frequency should we compare with w in the present case? Dimensionally, there is only one relevant frequency. the total excursion

403

of the precession frequency over the spinning cycle. Note that all our equations so far imply there is a single crystal being rotated since there is a well-defined orientation of zs in the xp, yp, zp coordinate system. However, for a powder, all possible orientations occur. For the rest of our discussion we will focus on a powder. Then, the maximum frequency excursion one can have in a rotation cycle is less than 1'Ho[(J(zI'Zp - qzpzl') - [(I{:cI':C1' - q:cp:cp)]

== .6wshifL

n '"

404

•

(8.121)

where we are defining the zp and xp axes as having the maximum and minimum shifts respectively. So we ask, what happens when we vary the relative sizes of fl and .6wshift.? Suppose, first, that the powder sample is not spinning. Then, as we have noted, following a rr/2 pulse, the transverse magnetization will decay to zero in a time :::::: l16.wshift. The Fourier transform of this free induction decay would give us the line shape. If we then applied a 11" pulse at time tll" we could refocus the magnetization into an echo. If t1l':» 1/.0.wshifL' the echo and the free induction decay would be well separated in time. Indeed, we could apply a string of such 11" pulses (a Carr-Purcell sequence) pnxlucing a string of echoes. The Fourier transform of anyone echo will give us the powder line shape in frequency space. If, instead, we took a Fourier transform of the sIring of echoes, we would now have introduced a periodicity. The resultant J1ansform differs from that of a single echo in the same way that a Fourier series differs from a Fourier integral. The transform of the string of echoes would consist of 5-functions spaced apart in frequency by the angular frequency l/t rep where t rep is the time between successive rr pulses. The Fourier transform of a single echo would be the envelope function of the spikes. If now one shortened t rep , one would get to a point where t rep :::::: lI.6wshifL' Then the echo would not have decayed completely to zero when the 11" pulse is applied, so that over a single period one no longer would have the complete shape of a single free induction decay. The Fourier transform of the sequence of echoes would still have a center line plus sidebands spaced in angular frequency by 1/t rep , but the envelope of these sidebands would not be the powdeI line shape (the rransform of a single free induction decay). Stejskal et al. [8.27] realized that the same situation would arise from magic angle spinning. When n« .6wshifL, the NMR signal following an initial 11"/2 pulse decays rapidly compared to the period of rotation, hence is essentially identical to what it would be with O. However, after one full rotation there is an echo, as described by (8.120). Indeed, a train of "spinning echoes" will be formed, analogous to a Carr~Purcell train. The Fourier transform of any echo will give the 0 powder line shape. The Fourier transform of the train of echoes will give o-function spikes spaced apart in angular frequency by with an intensity envelope versus frequency identical to the nonspinning powder patlern. This situation is illustrated beautifully by Fig. 8.10 showing the data of Herzfeld et al. for a p13 compound [8.28,29].

n '"

Proton decoupled p3l spectra at 119.05 MHz of barium diethyJ phosphate spinning aL the magic angle. The figure illustrates how, in the limit of slow spinning, the spinning sidebands reproduce the shape or the Ilonspinning spectrum [8.28] .'ig.8.IO.

I'AQT- OkHz

n,

"OOT - 0.94 kHz b

I'OOT"

2.92 kHz

d

16

,

a

0

-

a

, -16

Freq (kHz)

n

Once becomes comparable to or larger than .6wshifL> the satellites occur outside the envelope of the nonspinning line. As can be seen fIOm (8.117), the coefficients A\(P,S) and A2(P,S) ale multiplied by lin, hence become progressively smaller than the lalger n. This is part of a general theorem discussed by Andrew et al. [8.22] that the second moment is invariant under rotation. Thus, the intensity of lines spaced by multiples of must fall off as l/n 2 in the limit of large n. When there are many spinning sidebands, how can one tell the central line? The easiest technique is to change n. One line will be undisplaced - the cenrral line.

n

405

The beauty of the slow spinning regime of Stejskal, Schaefer. and McKay is that it gives one simultaneously a high precision determination of the isotropic average shift, as well as a picture of the associated powder paHem. It also removes the difficulty of spinning at a rate faster than ~Wlihirt., a genuine difficulty in the strong fields of superconducting magnets when one has large shift anisotTOpies. A common use of the slow spinning regime is to slUdy low abundance nuclei (e.g. C 13 ). so that there is no like-nuclei dipolar coupling with which to contend, combined with standard decoupling to any high abundance nuclei like HI whose mutual spin-flips would otherwise broaden the spinning side bands. In some instances it is desirable to remove sidebands in order to simplify the spectra. This can be done by applying 1( pulses synchronously with the rotation as shown by Dixon [8.30) and by Raleigh et al. [8.31].

H( 1 r[5 ) =

406

S

M[ - -3-

r lS

r lS

(8.122)

We consider M[ to lie along the k-direction, and consider three cases in which S lies at a distance a respectively along the X-, y-, and z·axes (see Fig.8.ll). Application of the formula shows that when S is on the z·axis it experiences a field 2M, HI(O, 0, a) • - 3 (8.123) a whereas when S is on the x· or y-axes HI(a,O,O)' HI(O,a,O)' - M.,I (8.124) a If, for some reason, S were to jump rapidly among the three positions, spending equaJ limes in each on the average, it would experience a time-averaged field < H/ > given by

8.10 The Relation of Spin.F1ip Narrowing to Motional Narrowing The strong spin-spin coupling characteristic of solids was essential for achieving the high sensitivity of double resonance. Its disadvantage is that it produces line broadening, and thus may obscure details of the resonance line such as the existence of anisotropic couplings (chemical shift, Knight shift, or quadrupole interactions). Within the past few years, a varielY of clever techniques have been introduced which utilize strong rf pulses to eliminate much of the dipolar broadening. We now lum to the principles on which the ideas are based. The highly ingenious concepts owe their development to several groups. The pioneering experimenlS were performed by two groups, one headed by John Waugh [8.4], Ihe other headed by Peter MallSfitld [8.S]. Subsequently Robert Vaughan [8.6] and his colleagues have contributed imponantly, as have many of the scientists [8.32,33] who worked with Waugh, Mansfield or Vaughan. The essence of these schemes is to apply a repetitive sel of rf pulses [8.3439) which produce large spin rotations and which, by a process akin to motional narrowing, cause the dipolar coupling to average to zero. Each cycle consists of a small number of pulses (8 pulses per cycle is common). To achieve the effect, Ihere need to be many cycles within the normal dephasing time of Ihe rigid lattice line width. Since the rf pulses must produce large spin rotations (90" pulses are typically used), and since the rotations should occur within a small fraction of a cycle of pulses, one needs H ,'s which are large compared to the rigid lauice Ijne breadth. We have remarked on the similarity between the multiple pulse schemes and motional narrowing. We stan by making the analogy explicit. For convenience we shall name the multiple pulse schemes "spin-flip narrowing". Consider two spins, I and S. Let 1" [5 be the vector from spin I to S. The magnetic field H[ at spin S due to I is then

3(M/-rIS)r/5



(8.125)

, ,..---... , ,

,,

,,

,

/

, s (,j

,

, I

x

s

"F----

(b)

s

\, ,I

, ,,

IF----

(c)

Ag. 8.lla-e. The magnetic field or [ III S for lhree localions of S all the nme distance, a, from ,pin I. The dashed line indicates II. magnetic line of force

407

This result is the essence of motional narrowing. Though for our case we picked only discrete locations for spin S, the same result is found for continuously variable sites for which the strength of the interaction goes with angular position 81 s of rlS with respect to the z-axis as 3cos 2 81S - I. This averages to zero over a sphere. Suppose now we consider a variation in the above picture. Suppose we position S at (0,0, a), Le. on the z-axis, and consider the orientation of both M I and Ms (Fig. 8.12). In tenns of the orientation of the vector r I S with respect to MI' it is useful to introduce some names. Referring to Fig. 8.11, we denote the (c) configuration as the on-axis position of S, we denote the position of S in (a) and (b) as the side position. Note that for the on-axis position HIS is parallel to MI, whereas for the side position HIS is anti-parallel to MI and half of its on-axis magnitude. Referring now to Fig. 8.12, we see that in (a) and (b) S occupies a side position, whereas in (c) it occupies an on-axis position. Moreover, since we have taken M , and Ms parallel in all parts, for configurations (a) and (b) the magnetic energy E llIag = -Ms' H,s is:

a quick rotation of both spins to the side position of Fig. 8.12b. Wait another time 'T. What is the average magnetic energy < Em > over the interval 3'T? Using (8.126) and (8.127) we get

- .!...(-2MI MS'T MIMs'T ll'hMST)_o + + a3 a3 a3

< E m> - 3T

.

(8.128)

That is, over such a cycle in which, at stated times, we apply selected 7fn. rotations to both spins, we can make the magnetic energy average to zero. Thus by spin-flips we can cause the dipolar coupling to vanish just as we could for position jumps in Fig. 8.11. This is the principle of spin-flip line narrowing. We make the dipolar energy vanish by flipping the spins among selected on-axis and side positions, spending twice as long in the side positions as in the on-axis ones. All of the complicated pulse cycles are based on exactly this principle.

8.11 The Formal Description of Spin-Flip Narrowing

(8.126) whereas for (c) Emag)e =

-Ms' HIS =

-2NhMs

(8.127)

a3

Suppose, then, we began with the configuration of Fig. 8.12c, the on-axis arrangement with MI and Ms parallel. After time 'T let us quickly rotate M I and Ms by 7fn. about the y-axis to the side position of Fig. 8.12a. After a 'T give

,

(,)

S

~x

, (

(b)

S

(c)

S

"

" (

FIg.8.I2. The two spill!l Ai, and M. are oriented parallel to one another. In (a), (b), lind (c) their magnetic moments are parallelrespcctive!.y to the Z-, Ih and z-directiolls

408

From the previous section we see that it is possible to cause the dipolar interaction 10 average 10 zero if we cause the spins to flip between the "on-axis" and "side" arrangements. For the conventional motional narrowing, narrowing occurs when the correlation time Tc and the rigid lauice line breadth (in frequency) DwRL satisfy TcDwRL $ I

(8.129)

Withoul jumping, a set of spins precessing in phase initially get out of step in a lime ..... (l/OwRL)' The condition expresses the faci that for motional narrowing to occur, the jumping must occur before the dephasing can take place. In our example of motional narrowing, the correlation time would be on the order of the mean time spent in one of the three configurations of Fig.8.11. If T is the time in anyone orientation, roughly Tc

:::::: 3T

.

(8.130)

In a similar way we expect that spin-flip narrowing will worle only if the spins are flipped among the needed configurations before dephasing has occurred. Thus we get a condition on T:

(8.131) The better the inequality is satisfied, the longer the spins will precess in phase. We expect, therefore, that we shall wish to flip the spins again and again through lhe configuration of Fig. 8.12. This we can do by applying a given cycle of spin-flips repetitively. The basic cycle will bring Ihe spins back to their starting point at the end of each cycle. 409

A fonnal description of what takes place begins with a specification of the Hamiltonian. We shall use it to compute the development of the wave function in time. We write lhe Hamiltonian as 1i(t) = 1irr{O + 1iint

where

(8.133) (8.134)

1t3 = 2: Bij(rjj)(Ii .Ij -

(8.135)

'Ho includes the chemical and Knight shifts qUi."H~ is the secular part of the dipolar coupling, the coefficients Bij(f"ij) including the distance and angular factors [see (8.54a»). The objecl ofspin-jlip narrowing is to cause ~ to vanish while maintaining 1to nonzero. To illustrate the principles of spin-flip narrowing, it is useful to idealize the situation. Correclions to the idealization are important as a practical matter. We return to them in Sec!. 8.13. The idealization is to consider that 11rf is zero except for very short times during which it is so large that 'Hint can be neglected in comparison. This approximation enables us to say that 'Hrr produces spin rotations at the time a pulse is applied, but that between pulses the wave function ¢(t) develops under the action of the time independent Hamiltonian 'Hint. Thus between pulses the wave function at lime t can be related to its value at an earlier lime t I by (2.49) ¢(t) = exp{ - (ilh)'Hint(t - tt»)¢(t t)

(8.136)

which defines Uillt. The effect of pulse of amplitude HI and duration t w along the a-axis (a = x, y, z) at time tt is to produce a transfonnation by the unitary operator Pj for the ith pulse, 410

Pi = e i(1rj2)lo

(8.138)

if the pulse produces a 1r/2. rotation about the a(= x, y,or z) axis. Let us consider a three-pulse cycle for simplicity and concreteness. We always begin an experiment by having a sample which has reached thermal equilibrium in the magnet. We then tilt the magnetization into the :t - Y plane at t = O. We call that pulse the preparation pulse. After a time TO we begin applying the repetitive pulse cycles. Choice of TO and the phase of the preparation pulse is made in tcnns of producing a signal at some point in the pulse cycle which is convenient for observation (at a time which is called "the window" in the spin-flip literature). Let .p(tn)be the wave function after n cycles [Le., just before pulse PI of the (n + I)th cycle]. Then we can follow the wave function in time with Table 8.2. Table 8.2. Varialion of the wave function with time for. three-pulse sequence Time

3I::iI::j)

ij

= Uint(t - tlhb(t)

(8.131)

where, for example, "'{Htt w is chosen as 1ffl giving

(8.132)

We shall work in the rotating reference frame with the z-axis defined as the direction of the stalic field. 1irr{t) is the coupling to the applied rf pulses used to apply the spin rotations. It is time dependent because the pulses are switched on for very short intervals, two In principle the corresponding H) can be oriented along the x-, y-, or z-axes. Selection of the x- versus the y-axis is a question of the phase of the rf pulse. Though pulses along the z-axis can be applied in principle, in practice they are not used since they would require addilion of ,mother coil 10 the rig. However, a z-axis rotation can be achieved by two successive rotations about the x- and y-axes. (Show how this is done!) The tenn 1tin~ consists of two tenns, 1tint = "Ho+~

¢(tt) = ei..,lI,tw l o .p(ti)

I; (just before PI) (just after ~)

It t.

+ T,-

(just before 1'\)

+ Tt (just .ner P,) I. + Tl + Ti" (just before 1\)

,,= "('.)

.p=P,¢(I,,)

'" =exp( -i1ti•• 1"1 j")~ ¢(t.. ) =UI.,(TdPt.p(I.)

t.

¢ =

I.. +TI +T1+T3-

tP = Ui•• (T1)1'\ Ul •• (Tl) PI ,,(t.) ¢(I,,+d = UihtCr~)/"JUi ... (T1)P:!Ulnl(Tl)Pl\/.,(t,,) == UT ¢(! .. ) defining UT

Dust before (II

+ l)th cycle]

~Ui .. (TdP,

tP(t.)

We can thus write (8.139) where UT is independent of n. Since UT is a product of unitary operators, it is itself unitary. After N pulses, we have

,p(tN) = u!f,p(to)

(8.140)

so that the problem can be considered solved if the effect of UT can be deduced. Let us examine one cycle, and introduce the unitaty operators p j - I which are the inverses of the Pi'S. For a unitary operator Pi-I = P/

(8.141)

where the" stands for complex conjugate. 411

Then we write UT as UT :::: Uill~(T3)P.JUin~(T2)P2Uin~(Tt )Pt :::: P3 P2 PdP\-t p 2- t P3- 1Uinl(T3)P3P2 PIHP.-l p 2- t UinL(T2)P2Ptl

X [P1-tUinl(T)Pd

(8.142)

Now we showed in our example of Sect. 8.10 that a complele cycle of spinflips should get us back to the starting point so that we can repetitively flip the spins among "on-axis" and "side" configurations. Therefore the cycle PI, P2, PJ should get us back where we started. Hence (8.143) giving us

(8.144a)

:: Uinl(TJ)[PI- 1P2- 1Uinl(T2)P2PI][PI-IUinL(TI)Pl]

(8.144b)

The meaning of the individual tenus can be made evident by several transformations. Erst, consider a unitary operator P and a Hamiltonian 1t, and the corre· sponding U: P-IU(t - to)P:: p-l e -(i/l)1{(t- 1o) P

.

(8.146)

so that the effeci of p- 1 UP is 10 cause U to develop in time under a transfonned Hamiltonian. If we were to evaluate the expression in the exponent of Ihe transfonned Uin~(T2)'

Pit P2-I1tinL(T2)P2Pj

(8.147)

we could first do the trunsfonnation p2-t'HinlP2, then sandwich the result between p t- 1 and PI and evaluate that, using the appropriate expressions for exponential operatOrs. 111is order of application of the opemtors is the rever.fe of the order in time of application of Ihe spin rQlation operators. Why is that? Consider a SchrOdinger equation

_':.iN=1i'" (8.148) i 'f' We can transfonn this equation with a unitary operator R, which is independent of time, to the problem

at

412

(8.150)

as the 7t corresponding to the rQlation P- I • We can consider the following two descriptions of the effect of 1t~ following 3. 1fn. rotation: i) Use ~ untransfonned acting on a

t/J which

is rotated

+wn.

ii) Leave the spin function alone bUI rolate the spin coordinates in ~ by

-wn., the inverse of lhe spin rotation in (i). Equation (8.144) corresponds to (ii).

(8.145)

By expanding the exponential, insening p-l PC:::: 1) between factors, and regrouping, we find that p-I U(t - to)P = e.p[ - (Uh)(P-'1iPXt - to»)

which leaves the problem lhe .fame. Thus, if R were a spin rotation operator giving a +1r{2 rotarion of the spins about some axis, R1tR- 1 musr transfonn the coordinates of Ihe Hamiltonian corresponding to the same rotation. If 1t:: -..,hHoIt and the spin is in a Slate corresponding to the spin-up state, a transformation, R, which rotates Ihe spin function into the up spin state along the +y-direclion requires rotating 1t Ihe same way, which is done by replacing I: by I" in 1t. We can therefore interpret p- l 1tP:: (P- I )1t(p- I )-1

UT :::: (P1- 1P2- 1P;'Uinl(T3)P:! P2P j][P1- 1P2-IUinL(T2)P2Pd X (PI-IUinl(TI)Pd

(8.149)

If a rolalion is made up of several rolations in succession, the inverse consists of lhe inverse rotations perfonned in the opposite order. Thus, if the spin is flipped by P2P\, the inverse transfonnmion Q is (P2PI)-I, so the 1t transformed by the inverse is

(8.151) The prescription for finding the transfonned 1t conesponding to the ith interval Ti of an n pulse sequence is to take the spin rotations PI, P2 Pi, which preceded the interval and transform the coordinates in the Hamiltonian by applying rhe inverse rotations in the reverse sequence (e.g., first p;-t, then Pj-=.ll , ••••... , lastly PI-I). We now define the three rransfonned Hamiltonians 11.", 'HB, and 'He as 1t" :::: PI-I1tintPI 1tB :: P l- l P2-I1tinLP2PI

(8.152)

11.e :::: 1tint On expanding the exponentials we get

413

('H~)B : 'L Bij(Ii ·Ij - 3lzi l zj )

UT =ex p ( -*1iCT3)eXp( -*1iBT2)eXp(-*1i"'11) = 1-

i<j

i

(~>c

1i (1iCTJ+1iB1'2+1-£",Tt)

• L: B(I; . I j -

3I,;I,j)

(8.158)

i<j

and

- (*Y(1iCT31iB1'2 + 1tC7'J1-£", 11 +1iB1'21iA11)

T) :

7"2 : TJ.

(8.159)

2 2 +'H. 2 7"22 + 'H.",7"I) 22 + ... +-l(i)2 - (1-£C7"3 B 2 I,

(8.153)

If the 7"'S are short, (8.154) where by 1(I'HA,B,clhl)1 we mean a quantity of the magnitude of typical matrix elements of the transformed Hamiltonians. Under these circumstances, UT is well approximated by keeping only the leading two teons on the right of (8.153). The condition on 7"i is similar to the requirement on the correlation time 7" for there to be motional narrowing. Introducing the period of a cycle. t c = Tl + 7"2 +"f'J, we get

(T'J

7"2 7") ) UT = I - -i 1ic- +'H.B-+'H.Atc II tc te te

i __ = I -1i'Hinttc ';:!

ex p ( -*'H.inttc)

(8.160)

Setting w : wo for simplicity. we would then get

Ho :

liwo"" -3 ~ azzi(Izi + I y ; + lzi)

,

(8.155b) (8.161) (8.155c)

'H.in~ : -h(wo - w)Iz - liwQ 'Laui1zi + 'H~ (8.156)

The trick then is to choose the pulse cycle (PI> P2, P3, etc.) so that we eliminate the dipolar coupling but maintain Ihe chemical shift and Knight shift information: (8.157.)

-h(wQ - w)lz - hWQ 'Lazzi1zii-Q

(1io )", : -h(wo -w)III-llwo 'Lazzi1yi

(8.155.)

where (b) defines 'H.in~' the average 1iinh and (c) serves to remind us that over a cycle the system develops to a good approximation as though 1iint were replaced by its average 1iint' defined above. Now we expand. 1iin~ into its elements

" 110 + 11~

These results could be achieved if p,-I were equivalem to a 1r/2 rotation about the x-axis which would transform Izi imo [yi and if P 1- 1p-I were a 1r12 rotation about the y-axis which would transform Izi and Izi. 2 These pulses would also transform 'Ho

(8.157b)

(8.162) Equation (8.162) shows that the chemical shift and Knight shift are reduced by v'3 (i.e., multiplied by 1/Jj) from their value without averaging. We have succeeded in making the dipolar coupling vanish without eliminating the chemical shift. To the eXlent that the term in (w - wo) is included (it represems frequency offset or field inhomogeneity). it is also reduced in the same proportion. According to (8.139) and (8.155c), ¢(to) and f/J(t) are given by

,,(t) . ul( ,,(to) : exp( - (i/li)1tinl(t - to»)¢(to)

(8.163)

provided

t = to + Nt c

(8.164)

How abour ¢(t) at other times? Consider a time t in the Nih interval such rhar

Thus if

(~)A : 'L Bij(li .Ij - 3111iIlIj)

t:tO+Ntc+tl

(8.165)

i<j 414

415

The time t) could fall into anyone of the time intervals T), T2 • ••• Tj which make up the basic cycle t c . Over anyone complete cycle there is no large change in ¢ if (8.152) is true. However, each large pulse produces a sudden big spin change _ typically a 1fo, rotation. These large pulses might, for example. progressively cycle the spin along the X-. y-. and z-axis in the rotating frame. If one always observes during the ith interval. however. the effect of the big pulses in successive cycles will always have returned the nuclear magnetization to the same direction in the rotating frame. Thus we can write ,,(t) = e
'0»)"«0)

(8.166)

when t and to are both within the same subinterval Tj of tc' If they are within different intervals Tj and Tj. with (8.167)

Ntc
we should properly go back to expressions like (8.136) and (8.137). 1f we consider thm the individual T'S are so short that (8.154) is true, then 'Hint does not produce much of a change during t e , but the pulses i + 1, i +2, i + 3, ...• j - I, j do. Thus, defining P

== PjPj-I ...• Pj+2Pj+l

(8.168)

there are two expressions which are nearly equal:

,

pexp( - (i/h)1{intNte)tP(to) ';!

add either a steady-state or a pulsed spectrometer to observe the character of the resulting resonance. In practice. the amplifiers of most steady-state apparatus would be blocked by a large rf pulse at their input, with a recovery from blocking which would be much slower than the short time T between spin-flipping pulses. Consequently. to see spin-flip narrowing one goes to NMR apparatus designed to handle intense rf pulses-one uses a pulse rig. With coherent pulse apparatus. one picks out a particular component (Mz or My) of the magnetization in the rotating frame. If one is adjusted to detect M z • then one finds in general that it is largest in one panicular interval Ti of the cycle. perhaps zero in others. One could start with the system in thermal equilibrium. apply a 7(0, pulse about the x-axis to rotate the magnetization to the % - Y plane. and then start the spin-flip cycles. For simplicity of the discussion we assume we can apply 7ro, rOlations about the %-. y-. and z-axes in the spin-flip cycle. though in practice z-axis rotations are not used. A possible set of pulses to initiate and carry on spin-flip narrowing is shown in Fig. 8.13. In Fig.8.13a we see M along the direction of the slatic magnetic field (the +z-direction). AI t = 0, Po rolates M to the +y-direction. This is called the preparing pulse. At time TO we initiate the spin-flip pulses Pl. P2, P3 which successively rotate M by -rr/2 about the z-axis, +rr/2 about the y-axis. and lastly +2rr/3 about the -i + j + k axis. The rotation P3 is deduced as being the inverse of P2P) and can also be thought of simply as PI-I P2- 1 mathematically. or physically as the successive steps -7I:n. about the y-axis followed by +71:/2 about the z-axis. Note

P exp ( - (i/h)7-{int(N + I)tc) ,p(to)

(x)

,, ,,

.......- - - , Po

(8.169)

Since Nt c < t - to < (N + I)t c we can write

.p(') '" Pe
'0»)"('0)

(8.170)

relating ,p(to) in the ith interval to ¢(t) in the jth interval of a pulse. Equation (8.167) says in essence to compute the wave function evolution between t and to as though 1{inl acted the whole time, and was followed by the rotations Pj+ I, ....., Pj-I. Pj in quick succession.

x

,

!-i+J+.t

---

(d)

8.12 Observation of the Spin-Flip Narrowing How can one experimentally observe the effect of spin-flip narrowing? Ordinary motional narrowing can be seen either by steady-state or pulsed apparatus as very narrow lines in the former case. and as long Bloch decays or slowly decaying echoes in the laller. To produce spin-flip narrowing large rf pulses must be applied to the sample. Assuming one has the equipment to apply such pulses. one could then in principle

,

i

' -, P,

,, ,

axis

i i .......... ----...., I'J

i

\

416

(b,

,

i

,, ,,

(l')

,

Fig.8.13a·e. The erred of ~he various pulses 8S llCCll in the rota~ing frame. (a) Thc initial thermal equilibrium configuration wi~h lhe ml
417

that ahhough P J looks superficially like a rOlation of -1(/2 about the x-axis. it is not. as can be verified if one considers rotating a three-dimensional object in PI, P2. P3 instead of just a vector M pointing initially along the y-axis. This distinction is imponant because lld is a second rank tensor, thus it should be thought of as an ellipsoid rather than as an arrow. If the apparatus were adjusted to measure M z • and if it were tuned exactly to resonance. including any possible chemical and Knight shifts. there would only be a nuclear resonance signal between pulses PI and P2. If there were a resonance offset (as with a second class of nuclei chemically shifted from the first), M z would gradually change in the PI-to-P2 interval for successive pulses. undergoing an oscillation (see Fig.8.14) as we show below. It would appear during the PJ-to-Pt window as well as time went on. Let us tum to a mathematical formulation of the calculation of the observed NMR signal. For concreteness, we compute the time development of M z in the rotating frame < Mz(t) >. We will use the language of wave functions rather than density matrix. Given the wave function !/J(t), we have < M,(t) > =

< ,p(t), M,,p(t) >

= '''<,p('),1,,p(r»

(8.171) We assume that we have a pulse sequence similar to that of Figs. 8.13 and 8.14. At t = 0- we let the wave function be t/J(O-); then the preparation pulse gives us ,p(0+) = Po,p(O-) (8.172)

The wave function then develops under 'Hin~. If we ask for it in the interval PJ to PI. there will be an integral number of cycles PI followed by P2 followed by PJ, so that. using (8.166).

,p(t) = e -(1M"; .. ' Po,p(O-) If we ask for

t/J(O later in

(8.173)

the cycle. we use (8.170)

,p(t) = Pe-(I/.),,;o,'Po,p(O-)

(8.174)

where P = PI or PIP2 depending on whether t lies in the interval Pt-to-P2 or P2- to-PJ. We now need to express the fact that the initial wave function 1/J(0-) cor· responds to the initial thermal equilibrium of the sample. To do so we express y,(0-) in terms of the laboratory frame 4»(O-} instead of the rotating frame. In general the rotating frame wave function t/J and laboratory function ifJ are related

by 1/J(t) =e-ilo.ltl·ifJ(t)

(8.1750)

so that at t "" 0- we get (8.175b)

Taking I n) to be the eigenstates of the secular Hamiltonian 1(. in the laboratory frame. we then have (8.176)

"

Thus <Mz(t»

=
='Yh E cnc~(m I Po-le(i/.)'Hiall P- t IzP ",m

xe-i/.)'H;a,IPo!n)

.

(8.177)

We now make a statistical average over the coefficients Cnc:" utilizing the fact that the states are occupied according to a Boltzmann distribution of the Hamiltonian 1(.. which describes the system before application of the pulses:

e- E.. lkT (n le-'HlkT I m) cnc m = Onm Z = Tr{e 'HlkT}

_._ (,)

P,

P,

p,

(8.178)

p,

Substituting (8.178) into (8.177) we get

,10 Tr( - p-llzPe-(ilh)"H,nt - t Poe- 1ilkT ) < Mz(t) > = Z Po-te(i!h)"Hinl! (8.179)

In the high temperature approximation we can write (II)

/'0

1',

1'1

I'j

P,

1'1

1',

P,

1'1

Flj;.8.14a,b. Grllph of ~he OllCilloseope signal of the resonance venus time for ~he pulse sequence when (a) apP"ra~us is se~ exactly a.~ re&Onan<:e, or (b) II chemical shifl. causes ~he M z signal to change with time appearing in ~he PJ·to-p. inter....11llI it diminishes in ~he p'-to-f\ window

.,8

e-"/kT

Z

(I -7{JkT)

"" (21+I)NG

(8.180)

where there are No spins of spin I. The leading term vanishes, so that the nonvanishing term gives a I/kT contribution. 419

If we then wrile

::: NO'Y2h2I(l + l)Ho

1£ ::: ~'YhHoIz

ttansfer the

PO-I

3'0'

(WOU"')]

from the lefl to the right in the ttace, using

1 2 :::Mo [ 3+3cOS ~

Th{ABC}: T,{BCA}

(8.182)

and if we pick Po 10 make

PolzPO- 1 = I z

(8.183)

which rotates the spin from the z·axis to the x-axis. we get

2 H {~ } <M (t»:::'...,2h °Tr e(i/1I)'Him,tp- I I Pe-(i/Il)ni·'(I z (2I + I)No kT z z· (8.184) A useful exercise for the reader is to derive (8.184) using the density matrix language. For simplicity we take P ::: I (Le., an integral number of pulse cycles PI. P2, P3 pass between t ::: 0 and the time of the measurement). Now, 1£int = -

[.!.. + ~ 3

3kT

(8.181)

m;o 2;:Uni(lzi + Izi + IIJi)

(woont)] J3 (8.189)

Mo

where is the thermal equilibrium magnetization. We see, therefore, that the chemical shifl-Knighl shift tl zz shows up as an oscillatory contribution to the < Mz(t) > ,reduced in frequency by 1/../3 from the normal frequency shift W1)Ou one would see if one could resolve il in a steady-state resonance. Jf more Ihan one kind of chemical or Knight shift is present, the resultant < Mz(t) > 's add, Ihe respective strengths being in proportion to the number of nuclei possessing that chemical shifl or Knight shift through the Mo faclOr. The form of (8.189) is that of a conStant term plus an oscillatory term. By taking a Fourier transform one can deduce a "reduced chemical shift frequency" (WOtl u/../3). II is possible to utilize Fourier transform NMR attachments therefore to decompose a transient signal when there exists man: Ihan one Knight shiftchemical shift to unscramble the discrete frequencies.

•

(8.185) where [z'i we get

=

1£inl::: -

(1/J3)(lzi + IIJi + IziJ. Suppose all spins have the same

(Tn'

hwo~zIzl

Then

(8.186)

Now in terms of the prime coordinates we have, in an obvious notation,

I z ::: Iz'cos(x, x') + 1IJlcos(x. y') + Iz'cos(x, z') Izl ::: Izcos(x, Xl) + IIJcos(y, x') + lzcos(z, x')

Our discussion up to now has been directed 10wardS explaining the principles of spin-flip narrowing. Trnnslating these notions into prnclice. we encounter certain problems. We discuss these briefly, Ihen list a few of the pulse sequences which have been developed to cope with them. An excellent discussion of these problems has been given by Haeberlen in his NATO Summer School lectures. I Vaughan and colleagues [8.38,8.39] give a thorough theoretical discussion including explicit fonnulas of use. See also the references in the Selected Bibliography.

(8.187)

8.13.1 Avoiding a z-Axis Rotation

so that exp (

')I

-iwouutIz J3

zexp

::: cos(x, x') [IzICOS

+ cos(x, y')

[-l

+ cos(x, zl)Iz'

(iwouutIzl) J3

(WOjizt) + III,sin (wojizt) ]

z 'Sin (w0./izt)

+ Iy'COS( W0.,ij-zt) ] (8.188)

Using the fact that Tr{Iz ' III'} = 0, and Tr{I;,} ::: Tr{~I} we get, after evaluating the traces, 420

8,13 Real Pulses and Sequences

Pulse rigs apply HI'S in the x - y plane. By choosing the rf phase, a single one of these can produce rolations about any arbittary axis lying in the z - y plane, but not about the z-axis. To produce a z-axis rotation with a single pulse would require adding a coil wilh an axis parallel to Ihe z-axis plus the electronics to energize it. In order to avoid so doing, one can go to a four-pulse technique. That in a sense is equivalent to composing a z-axis rotation from successive x-axis and y-axis rolat.ions. Ordinarily, however, the two pulses are spaced apan in time so that the basic timing intervals can be maintained more simply. 1 The 1974 NATO Advanced Study Inslilule held in Leuvcn, Belgium fcalur«! a number of userul lectures on NMR in solids./fa,,/)#!rl"/1 aoo Van /f«~ in p"rticulllr lreat spin-nip nflrrowing in [8.401.

421

To write sequences, it is convenient to introduce a notation based on the fact that1r(l pulses are nonnally used. For example, consider the following expression which defines one four·pulse scheme. X(-r,X, T, Y,2T, Y,"',X,T)

(8.190)

.

The first "X" represents the preparing pulse, a 1fn. pulse about the +x-axis. The parentheses enclose the pulses of a single cycle, which in the example indicated is considered to stan with a time interval T prior to application of a 1((1. pulse about the -x·axis (or a -1fn. pulse about the +x·axis). This sequence could also be written as X,T(X,T,Y,2T,Y,T,X,2T)

,

which simply chooses to call the staning point in the cycle as the time just prior to the first -:I: pulse in the chain. This notation makes explicit the presence of two intervals of 2T and twO of T. The first notation is useful in proving certain theorems which utilize the time symmetry of a single pulse cycle. 8.13.2 Nonidealily or Pulses Pulse Duration. We have utilized instantaneous rotations in the discussion up to now, which would require an infinite HI of zero duration. Actual pulses have nonzero duration. One's first guess might be that there would be major problems as a result of the nonzero duration. Theoretical investigation of this point reveals, however, that it is not serious [8.34,35,38,39]. lbe point is that slow rotations can still maintain the property of averaging out the dipolar coupling. They are in this sense equivalent in motional narrowing to allowing (3 cos 20 - I) to average to zero over a conrjmwus variation in 8 as opposed to the discrete variation of Fig. 8.1. The important experimental consequence is that much lower If power is needed that one would at first suppose. Inhomogeneous HI. As we discussed in Sect. 8.3 it is always difficult to make HI homogeneous over a sample, panicularly when one is attempting to keep the rf coil small to cut down on the power requirements (for a given Q and Hj, the power needed is proportional to coil volume). This problem is a bad one in that it causes the rotation angles to vary throughout the sample. It can be removed to first order, however, by using the trick of Mejboom and Gill or of pairing positive and negative rotations about any given axis, thereby achieving a refocusing effect. Phase Modulation. Turning on an HI which points along a given direction in the rotating frame sounds easy, but in practice is hard to achieve. Usually the phase of the HI swings during the buildup and during the decay. Such swings can be labeled as modulations of the phase angle tP of the rf. They are equivalent to being off-resonance L\w = dtPldt during the time that dtPldt=l-O [8.41). Their

effect can be eliminated to first order by having l[1>/dt during turn-on and dtPldt during turn-off equal and opposite. Making l[tPldt zero is not practical, but the balancing during turn-on and tum·off is. Among the various pulse sequences used or referred to in the literature, we shall list three. They are known as WAHUHA, HW·8, and REV-8. These mysterious symbols are made up from initials of their inventors plus numbers which tell how many pulses there are per cycle. The WAHUHA is named after J.S. Waugh, L.M. Huber, and U. Haeberlen

[8.34J. \( is X, T(T,X, T. Y,2T, Y,T,X, T)

This sequence can be sampled at the times corresponding to just before the parenthesis by measuring M lI • The name WAHUHA refers to the contents of the parentheses. The preparation pulse can be chosen to give convenient sets of timing. The panicular sequences shown have the property that the X,T, at the start are in fact like the last pulse and interval of the cycle; hence no special timing intervals are needed at the stan. That is, the preparing pulse in this case is pan of the cycle. For the preparing pulse shown, the magnetization is initially turned to the y--direction. Thus, if one detectS My at the inverval between the X and -X pulses, one will get a signal. We can write the HW-8 [8.35] as (T,X,2T,X, T, Y,2T, Y, T, Y,2T, Y,T,X, 2T,X, T)

There are a number of equivalent variations. The MREV-8 [8.38] sequence is

X, T(T, X, T, Y,2T, Y, T,X,2T,X, T, Y,2T, Y, T,X, T) This can be viewed as pair of pulses X, Y separated by T followed at 2T later by their inverse y, X. MREV demonstrate the highest resolution obtained with their MREV-8.

8.14 Analysis of and More Uses for Pulse Sequence The importance and utility of Illultipulse techniques goes far beyond spin· flip line narrowing. We turn now to several examples, as well as to illustrations of some calculations of the effects of the pulse techniques. Before beginning our discussion, it is useful to introduce a notation and 10 demonstrate some useful relationships. We shall be following the effect of pulses on individual components of spins such as I:r:, I y • or I: and on dipolar tenns. For the dipolar tenn we introduce the notation 'Hon(a = x, y. z) defined as

423

1i&& =

L

(8.191)

Bik(/j ·lk - 3I&jI&k)

i>k Explicit calculation then shows that

1iu +1iIlIl +1iu =O

~4~J

.

(8.192)

In general. the effect of pulses Pi will be rotations. To discuss the effects of nonzero pulse lenglh:i. one must treat the Pi 's as time-dependent operators such as (8.193) where t is the time variable during the pulse. For our present purposes. however. we shall Ireat the pulses as negligible duration (so-called 6-function pulses) and moreover treat them as all being 1tll pulses. We then have to evaluate expressions such as

P1- 1I: P,

(8.194a)

P,-t HuPt

(8.194b)

But once we know what (8.194a) is. it is easy to get (8.194b). For example. let

Pt = X

~.I~

using right-handed rotations. For a pair of pulses PI followed by P2. then the lefl-handed rotations

PI-tP2-t1&P2PI

(8.I99b)

would be replaced by the right-handed rotations Pt.l'2. etc. p)

P21&P2-J p)-t

(8.199<:)

Using either approach, one can work out a table for (8.198) which is lhen useful for all applications in which we list 1a across the top and in each row list first the operator PI. then the effect of Pt on 1&. The result in the table is the same whether we treat PI as a left-handed rotation and use (8.198), or a right-handed rotation and use (8.199a). Rotation operator

Operator being rotated

Iz

(8.195.)

then

Pi] = X

PI = X then P 1- J = X) and then using the left-handed rotations or by instead pretending that one is calculating

III

I:

x

then

(8.195b)

-I, Iz I,

y

Z

X1:X = - ly X1i u X =

L

or expressed as coordinates x, y.

Bij(X Ii' [/cX - 3X I:jl:/cX)

%

i>k =

L

X

Bij(lj·Ik-3XI:jXX1:kX)

y Z

i>k =

L

Bii(lj ·lk - 3IlI jIlJk)

i>k = 1iYIJ

%

z y

Y

,

y

%

,

(8.200)

2

Y

.- ,

(8.201)

(8.196)

where the bar indicates a negative. Suppose then that in the sense of (8.198) one has the operator HYII acted on by PI = X. Then the table shows us that

(8.197)

(8.202)

Thus. if

Pi] I&PI

= kl{J

Pi 1 H&&PJ

k= ±1

then

Let us now illustrate this with a calculation of the effect of a very useful = Hpp

whether k = +1 or - 1. The effect of a (Iefl-handed) rotation PJ in the expression

Pit I&PJ

(8.198)

can be worked out once one knows PI either by first computing P 1- 1 (e.g. if 424

pulse sequence utilized by Warren et al. [8.42] to refocus dipolar dephasing. They employ it as pan of their method of generating multiple quantum coherence of a particular order, a topic we discuss in the next chapter. In discussing magic echoes, we showed that one could refocus the dephasing arising from dipolar coupling by creating a dipolar coupling of negative sign. In particular we showed how we could create a dipolar Hamiltonian 1id given by 425

1{d =

-~1{n

(8.203)

We now show how Warrell et al. created such an 1{d by pulses. The basic approach is founded on (8.192) 1{z",

+ 1i yy + 1i u = 0

Thus

-11.",,,, = 1i yy + 71.:u

(8.204)

Therefore, if we create a Hamiltonian which spends equal time as 71. yy and 1{u, its average value will be }inn. Since without pulses 1id = 1i u , we simply need pulses to switch 1i u to 71.'11'11' Clearly, all we need is X's or X's for the Pi'S. The simplest thing one can do is to switch 1i u to 71.'11'11 with PI = X, then switch it back, P2 = X, so that (8.205,)

have omitted the "T' and the "1{" since only the subscript and sign is needed to specify .the entry. On the third iine we list P2, then the product P2 PI, which we simplify to the extent possible. The last line labeled "sum" contains the sum of the time intervals, and the sums of the products of the operators with the time intervals for which each operator acts. The average values are then just the weighted sums of the operators divided by the duration. Using (8.204) we see that the pulse sequence does produce an average Hamiltonian (8.206) Of course, the pulse sequence must be repeated continuously for as long as one wants to satisfy (8.206). A pulse sequence which gives the same average dipolar Hamiltonian but eliminates the chemical shift tenn and has some other advantages is given by Warren et a!. [8.42]: X,~X,~X,T,X,T,X,T,X,~X,T,X,T

for a sequence (8.205b)

(X,T,X,T)

To follow the action, we make a table. We need to follow two tenus in the Hamiltonian: the dipolar coupling and the chemical shift. Ir we simply wish to create a negative 1id for all spins without regard to their chemical shifts, we need a pulse sequence which will make all chemical shifts zero. We will find that our simple (X, T, X, T) sequence has problems with chemical shifts. We construct a table in which time progresses downwards (Table 8.3).

We evaluate it in Table 8.4 utilizing (8.201). Table 8.4. Sequence (X, T, x, T, X, T, X, T, X, T, X, T, x, T, x, T) Duration

X

, 2

Average

X

X'

X

X

(-l y +l.) Iy)

tU. -

11..

1i

yy

1i

yy

.,

"

,

"

X

yy

X

X'

X

X

.,

" yy

y

+ H .. hHJY + H.,) H

X

yy

Sum Average

In Table 8.3, the "duration" column is measured in units of

T.

8

, 0 0

4(h u

"

+ H yy )

1(H.. +HJY)

The column

PI ... Pi lists PI, PI P2, PI P2P3, etc. as in (8.199c). Remcrl"!ber, the first operator

to apply is the right hand one, but its effect is to be computed with right-handed rotations. On line I we list Pl' Then on the second line under I z and 71.u we list the operators produced by the action of Pion I z and Hz%> as well as the duration of the time between application of PI and the application of the next pulse P2. We 426

X

I,

y

yy

Sum

X

X

I,

P; P j =X

P, ... Pi

X

Table 8.l. Sequence (X,T,X,T)

Duration

P;

The same approach can be used to analyze the spin-flip line narrowing sequences discussed in Sect. 8.13. For example, in Table 8.5 we analyze the fourpulse sequence (X, T, Y, 2T, Y, r, X, 2r). As remarked in Sect.8.l3.l, this pulse sequence is equivalent to the three-pulse average we employed (rotations about the X-, y-, and z-axes) in Sect. 8.12. Utilizing Table 8.5 we see it produces a zero average dipolar Hamiltonian. 427

Table 8.5. Sequeoce (X,,., Y ,2T,Y, ",X, 2,.) Durll~ion

p.

y

J~

... P;

I,

X

,

j7

XYY = X

X

XYYX= I

•

..'"

y

yy

T.ilble 8.6. SBml)lc

"

Line

,

2

Sum Average

M..

XY

2

2(r. + J,

6

So if we knew what QIII was. we could immediately evaluate (8.212). Since fJ will be x, y, or z (or %. y, Z). we can handle any eventuality if we know the results of Qlz• QI", and QI~. A procedure which will provide us with these three results is to replace the single column I~ with three columns Iz,!",!z. A portion of the table would then look like Table 8.6.

+ Iz )

~(l.:. + J,

+ J.)

2(1l:... +1l,,+1l.. ) f(ll zz

+ 11" + 11,.)

For multiple quantum excitation, Warren et al. [8.42] show that it is useful to generate an effective operator proportional to 'H"" -'Hu . In panicular, they

I 2 3 4 S

~Ilble

to help ill

eVlllua~jng

f.

I,

f,

I~II

I"

I,

I",,,

I.,

I.

the

effcc~

or QP•• I

PJ ... p.. _ 1 PI ... P.. (= Q)

P, "'P"P"+ J

6

find

!('HYfI -'Hzz ) = i(2'H fIfI + 1l u

(8.201.)

)

is generated by the eight-pulse sequence

(x. 21'.

X.

T,

X. 21', X.

T,

X, 2,., X.

T,

X, 2,.. X,

T)

(8.208) I~

Pn+1I~ = Ipl

(8.207b).

After a bit of experience in calculating tables, one soon discovers a problem. Suppose one has just calculated the effect of the operator

acting to the right. Suppose it produces a result I a in the have to calculate the effect of the operator

To compute line 6. we first find the effect of P n + 1 on I z • I y • and I z . There will be three results. We look for them on line 2. Thus. suppose

column. We next

Then

Pj ... P,d/p = la' So we would enter I rr on line 6 under the I z column. This approach requires evaluating three columns. but it makes the step of going from one line to the next much simpler - one has only to evaluate two operator steps per column. hence six operator sfeps per line.

(8.209)

Now it would be easy if the operator we had to calculate was Pn+l Q since then (8.210)

This situation would let us take the result of Q. which we already have. and apply Pn+l to it. However. what we must calculate is instead QP..+1

which has the PII +l operate before Q operates. Therefore, we must go through all n + I operator calculations. a tedious process for long pulse squences. Now suppose Pn+lI~ =

IfJ

(8.211)

Then (8.212)

428

429

9. Multiple Quantum Coherence

9.1 Introduction In Chapter 5 we introduced the density matrix as a powerful tool for the analysis of magnetic resonance experiments, In analyzing the case of spin we saw that the diagonal elements of e were connected with the magnetization parallel to the static field, and the off-diagonal elements were related 10 the transverse components through the equations

!'

(5.257.)

11<

(M,(O) ~ 2[,++(0 - e--(O]

(5.257b)

In a system with two spins I and S we saw that

(I+(O) ~ L::(-msle(t}1 +ms}

m,

(7.248)

so that the transverse magnetization of the I-spins arises from elements of the density matrix which are off-diagonal in the I quantum numbers, but diagonal in the S quantum numbers. There are other off-diagonal elements of (2, for example

(-+le(OI+-) (- -1,,(01 ++)

0'

(9.1.) (9.1b)

It is a fundamentaltcnct of statistical mechanics that for a system in thennal equilibrium such tenns are zero, using the hypothesis of random phases described in the text just below (5.87). This hypothesis is based on two ideas. The first is that all observables of a system in thennal equilibrium must be time independent. This requirement really expresses the meaning of "equilibrium". It is a necessary (though not sufficient) condition for thermal equilibrium that all observables be time independent. The second idea is that since the time dependence of I! comes only in the off-diagonal elements, there is some experimental means for probing every offdiagonal element of I!. If there were not, the existence of off-diagonal elements would not produce time-dependent observables, to conflict with the hypothesis of equilibrium. Since we have just seen that resonance experiments do not directly 431

detect elements such as in (9.1), we naturally wonder if there is any experimental method for probing the existence of such matrix elements. A closely related question is whether there is any way to excite such matrilt elements if they are initially zero. We are already in a position to say that in cenain cases we can eltcite these elemenls because indeed we have analyzed in detail an eltperiment which generates them. Referring to our analysis in Sec!. 7.26 of the pulse sequence (9.2) we see that in (7.338) we produced tenns of form, are proponional to

{l

which, eltpressing

{l

in operator

These gave nonvanishing elements of

{l

(7.341b)

where in each matrix element the upper signs go together, and the lower signs go together. The matrix elements (7.341 a) join states of energy difference, i1E, given by (9.4)

and the matrix elements of (7.34lb) join states differing in energy by i1E::: ± 1i.(wOf - WOS)

(9.S)

If the two spins were identical, so that WOf ~ Wos, we would refer to these energy differences as two quantum and zero quantum and we would say that the pulse sequence (9.2) has generated 0- and 2-quantum matrix elements of (l in addition to the I-quantum matrix which we observe directly. A next question is, clearly, can we detect the zero- or two-quantum matrix elements given the fact that (7.248) shows that resonance directly sees only the one-quantum matrix elements? Before proceeding with that question, we discuss a bit more the concept of 0-, 1-,2-, and in general n-quantum matrix elements. In some cases, mf or ms are not individual constants of the motion. For example, consider two identical spins, with a chemical shift difference and a spin-spin coupling involving tems such as 1· S. If the spin-spin coupling is comparable in magnitude to the chemical shift difference, ml or ms are not constants of the motion. It is still true, however, that the operator, F::, for the total z-component of angular momentum (9.6)

commutes with the Zeeman energy and the secular part of the spin-spin coupling, and thus its eigenvalue, M. is a good quantum number. 432

~-E~ k:::t

~

the eigenvalue M of ['IT is a good quantum number. Thus, if we have a set of quantum numbers M and a, where a are all the other quantum numbers needed so~ , I,TIMa) = MIMa)

(9.7b)

we define matrix elements (9.8)

as n-quantum matrix elements where

n _IM-M'I (7.3410)

i1E::: ± 1i.(WOf +wos)

N

(M'a'Ie(OIMa) (9.3)

1z 5"

In fact, for an N -spin system with total z-component of spin I::T given by

(9.9)

We follow the convention of calling n the order of the matrix element, and say that the existence of a nonvanishing matrix element (9.8) describes n-Qrdtr cohertnct. It is immediately evident that for a system of N spins of spin I. M can range from N I to - N I, so that the largest n one can have is 2 N I. Thus, for protons, a spin system involving an H2 group (as in a CH2 fragment) should pennit generation of two-quantum matrix elements, a system involving three protons (as in a CH3 group) should pennil generalion of both two- and Ihree-quantum matrix elements. The generation of a given order of coherence thus enables one to determine what spin groupings exist if one can detect the existence of orders different from one. NOle, however, that a pair of CH2 groups contains four protons, so one may ask, when should one consider a CH2 fragment to be a pair of protons, and when should it be considered to be part of a larger group? We thus see that there are exciting possibilities to use multiple quantum phenomena to classify spin groups, but that there are also imponant things we need to understand, such as, how do we create and how do we detect the various orders of coherence? Multiple quantum phenomena are encountered in all branches of spectroscopy. Bodenllausen [9.1), in his excellent review of the field, includes a discussion of the manifestations of multiple quantum phenomena in cw magnetic resonance speclroscoPY. It was the advent of pulse methods in multiple quantum speClroscopy which caused the technique 10 become of central importance as an almost routine speclroscopic tool. In two fundamental papers, Hatallaka et aJ. [9.2,3] showed how to generate and to detect multiple quantum coherence. They sludied the Al 27 resonance of A1 20 3, which can produce multiple quantum transitions since the spin of Al27 is 5/2. Aue et al. [9.4]. in their famous initial paper on two-dimensional Fourier lransform NMR, show that, using two-dimensional Fourier transform 433

spectroscopy, one can observe both zero- and double-quantum speCtra. They describe several ways of producing the desired matrix elements. Since the publication of the basic papers from the laboratories of Hashi and of Emst, the field of multiquantum coherence has exploded_ 1n addition to the review by Bodenhausen. there is an extensive review by Weitekamp [9.5]. There are useful reviews by Emid [9.6J and by Mu/Wwitz and Pines [9.7]. As in aU aspects of pulsed NMR, the book by Ernst el al. [9.8) gives a thorough and detailed account of the principles and practice. Our goal is to explain the physical principles. We turn next to showing that it is the nonlinearity in the equation of motion which makes possible excitation of multiple quantum coherence. We initially deal with cases in which we are tuned to discrete transitions, showing how we can then progressively pump higher and higher order coherences. Nexi we deal with the much more common experimental case of applying pulses of sufficiently greal amplitude to excite a number of transitions, discussing as well how to detect the fact that we have generated multiple quantum coherence. We then lum 10 melhods of singling out in the detection coherence of a desired order, and lastly to the problem of how one can selectively excite a panicular order of coherence.

9.2 The Feasibility of Generating Multiple Quantum Coherence - Frequency Selective Pumping Consider for the moment a two-spin system. We have seen (Sect. 7.5) that we can produce two-quantum elements of f! by applying a driving signal at WI = wOJ, followed by one at Ws := Wos. Among the mauix elements produced is

(- - 1,,(')1 ++)

•

We now proceed to show that this result is closely related to the idea of the beating of the frequencies of two oscillators. Beats are produced by putting input signals into a nonlinear device so that the output signal contains tenns proportional to the product of the input signals. Thus if we had a device whose electrical propenies are characterized by the property that the current i through the device is proportional to the square of the voltage V applied across the device

i.AV' application of a voltage

434

j(O

:=

A(V? cos 2 Wit +

(9.12)

W2t

(9.13)

vl cos2 w;et + 2Vt V2 cos WI t cos w;et)

(9.14)

Utilizing the fact Ihal cos WI t cos W2t

!{ cos (WI + W2)t + cos (Wt - W2)t]

:=

(9.15)

we get i =

~ { V12 [1 + cos (2w 10] + vlp + cos (2wzO] + 2VI V2[ cos

(WI + wz)t + cos (WI - wz)tl }

(9.16)

w,

Thus the current i contains the frequencies 2w), 2w 2 , + W2, WI - W2, and 0 (which is WI - WI and w;e - W2). A similar thing happens for the densily matrix equation df!

d' •

i

,,(eli -1/e)

.

(9.17)

Since this equaJion involves a product of f! with 1l, if both f! and 1i are time dependem on the right hand side, they couple to tenns on the left hand side which oscillate at the sum or difference frequencies. We want to show that Ihis is just what we need to take a {} which has an n-quamum coherence and pump it either up 10 an (n + I)-quantum coherence or down to an (n - I)-quantum coherence by means of a time-dependent 11. which oscillates at the frequency difference between Ihe nand n + I (or n - I) quantum coherence. To make those ideas more concrete, let us consider a system with a set of nondegenerate energy levels, a static Hamiltonian 1iO' and a time-dependent drive tenn V(O given by V(t)/h:= Fe

(9.11)

+ V2 cos

would produce a current

(9.10)

which oscillates in lime as exp [ - i(wOI + WOS)tJ

v = Vt cos Wit

iwt

+ F·e- iwt

(9.18a)

where the· refers to the complex conjugate. For example, if we have a group of identical spins as in (9.7) acted on by a rotating magnetic field, we get Vet) = -,!tlll(J'Z:T cos wt - lyT sin wt)

(e eiwt

= -,"/till, [[Z'f := _

iWI

+2

)

+ ~IYT(eiwt _

,!i.ll, (ueiwt + l-e- iwt ) 2

T

T

so that, using the usual notation WI

e-

iwt )]

(9.1gb)

=..,1l I, (9.18<:)

435

Note that a rotating field applied in the opposite sense is obtained by treating

W

as negative. . Let the eigenstates of 'Ho be denoted by quantum numbers, J, k, 1 each of which stands for some sel of quantum numbers such as M, 0' of (9.7b). Then we

Substituting (9.18) for V/IJ in (9.24), we thus get

dl

exp(-iwj/t)....::l!..dI t

""

define

i

L:, ejk {exp [i(w - i

(9.19)

Wjk)tjFkl + exp [ - i(w + Wjk)t)FkI}

L: , {Fjk exp [i(w -

Wk/)t]

+Flkexp[ -i(w+wk/)l]}ekl

(9.25)

Then, denoting the interaction representation by a prime, we have

ee:O

= e -(i/A)11ot {!' (l)eCi/A)11o t

(9.20)

or

(9.21.) (9.21b) The density matrix then obeys the matrix equation d i dt {ljl = h ~ [ejk('Hoh:1 - ('HO)jk{!kEl

+ .!. L [ej' V,,(t) h ,

Vj,(t)e"J

(9.22)

If we had only the tenns involving 'Ho, (!jk would oscillate at exp(-iwjkt). We can eliminate them from the right by substituting (9.2Ia) into (9.22). We also utilize the fact that

The first two tenns on the right give the sum and difference of wand Wjh the third and fourth lenns on the right have the sum and difference frequencies of W with Wk/. This is the same situation we gOt for our example of the nonlinear currenT/voltage relationship (9.12) and (9.16) in which the product of V1(t) with V2(t) gave the sum and difference frequencies. Here what is happening is that we consider the elements of e' on the right hand (either e"k or ekl) to be the n..quantum coherence which we wish to pump by means of'v up to the (n+ 1)quantum coherence or down to the (n - I)-quantum coherence represented by ejf on the left hand side. The point is that since ell [the (n + 1)- or (n - I)-quantum coherence] is to be produced, its time derivative must be different from zero. Let us suppose, then, that at t "" 0 the matrix elements ejl and ek/' and thus ejl and e'kl' are zero, but that we have previously excited the mauix element ejko for a particular value of k, labeled ko. So, we have a nonzero ~jko' hence f!..krJ' Let us suppose funher that V(t) possesses a nonzero mauix element betkn states ko and 1. Then, we have

dt ""

de'· exp(-iWjft)

+exp[ -i(w+Wjko)t]Fkotl

(9.23)

.

' d (!'f

.

ht ""' L..(ejkexp(-IWjA:t)Vkl(t) I

.

Fkol "" 0

I

-

~ V,'k(t)eklexp(-iwk/t)]

"

(9.24)

This equation reminds us that the matrix elements of e' would be independent of time (Le. that dej/dt "" 0) if V were zero. Consequently, w~ expect that mauix elements of e' to change slowly compared to the frequenCies Wjl' Wjk. Wk/' Therefore, to a good approximation the left hand side of (9.24) oscillates at Wjf. This requires that the right hand side also oscillates at the same frequency if (dtl;/dt) is to be something other than zero.

,

Fiol i- 0

.

(9.27)

Then we would have exp (-iwjlt)

:!' "

d'

ielko F kol exp [ - i(w + Wj~'o)t]

(9.28)

Then the condilioll that F pump ejko up to Iljl is that F kol i- 0

and

(9.29.) (9.29b)

Wj -WI ""Wj -wko +w

436

(9.26)

Now, ordinarily, F is a raising operator and F* a lowering operator. Thus, recalling that ko and I will in general be specified in pan by the quantum number M, see (9.gb), we see that if Fkol i- 0, then Fiol "" 0, or vice versa. Let us assume

since the states, j, k, 1 are the eigenstates of 'Ho. We thus get exp(-tWjlt) d: =

iejko {exp [i(w - Wjko)t]Fkof

(9.29c) 437

pumpThis equation may be interpreted as saying that the beat between the wig) Wj ncy (freque ce ing term F (frequency w), and the n-quantum coheren [If ce. coheren tum l)-quan must add to give the frequency, Wj - WI, of the (n + ce.] tum coheren W is negative, the left hand side is an (n - I)-quan as written be also can (9.29c) n Equatio (9.29<1) .

Energy

Fig.9.J

....,

3

3

'·-1

2

Fig. 9.2

1--1

1-.)

W.&:o -WI::W

match The meaning of this is that to excite ejl from ejko (or ejko)' W must matrix a have must V and 1, to ko n transitio the of the resonance frequency which joins ko to 1. ce If we stan with a diagonal {J, we can first produce a one-quantum coheren two-a to up this pump can we that shows above with a 1((2. pulse. The discussion coherence, quantum coherence. That in turn can be pumped to a three-quantum transition one excites which rf a apply can etc. Note that we are here assuming we Thus, (9.25). of side hand right the on at a time since we kept only one term the out pick to weak ntly sufficie HI an if there are splittings, we are using l spectra between spacing the than individual lines. It must therefore be smaller strong A on. exciuu{ e selecliv cy lines. This mode of pumping is called frequen it in lhe HI covering all the splittings is called nonselective excitation. We treat next section. levels Let us follow our example in greater detail. We define three energy I, 2, 3 (Fig. 9.1) with the order of energies such that W3 >"'2 >wI

(9.30)

We assume that W3 -"'2

4-

"'2 -WI

(9.31)

not e32). and that we have previously excited 1!21 (hence e12), but not 1!23 (hence Fig.9.2. in spins of pair a for shown is system a such of ion An actual realizat Assuming lhey have a coupling of the form (9.32) rand 5, and are identical except for a chemical shift difference between spins ed in discuss we as cies, frequen t differen at occur the four allowed transitions Chaptet 7. Let us rewrite (9.24) by multiplying both sides by cxp(iw jlt). Then .I t del ./ i · (9.33) ~ :: h ~ [ejke+lI ·Ik/IVkl (t) - Vjk(t)e + "-'J1- I!I./] with the This is actualJy just what we would have gOllen if we had started (9.21)] ing [follow equation for I! in the interaction representation, with Vkl(t) :: Vkl(t)ci"-'k/t

etc. Then, 438

(9.34)

....

,

I•• )

2

...., h -' h ex""'---· E, E,,.. labeled 1, '1 , 3 wilh ener&ies E, " levels lergy <=eU3$ "-'I, l<.'2 ( . L_ . Threeel hFig. 9.1. ""3 -l<.'2 '# "'2 _ "'I' ""3 In til'" notallon of 9.19). Note that l<.'2 > l<.'2 > "-'I and we assume lhe lhree levels labeled Fig.9.2. The energy levels of a pair of spin-t nudf!i. We focus on 1, 2, 3

. it l

' d 1!31

--;It "" -he "-':)1

d

I

V32e21

and

~~I

.

I

""

-*eil<.'2 3tv23e; 1

(9.35)

F- the Usin~ (9.1&), we see that F involves the raising operator r+ + S+ and I + _) is two state and -) I is three lowenn g operator r- + 5-. Since stale ' we have

F32 ::: - ~I(- - I r +5-1 +-) =_wI (__ jrl+_ )

2

WI

(9.36.)

::--

2

+ ..... WI F32' -2(-I'· +5 1+-) ·0

(9.36b)

Likewise (9.360) Therefore

de;1 :: dt

,.w,2 ei(""31-,,-,)L

I e21

de21

) WI'( dt:: fTel l<.'23+"-' t e3t

(9.37)

The right hand side has an oscillation at W3-W 2-W

.

(9.38)

This causes lhe left hand sides to vanish unless 439

(9.39) If we satisfy (9.39) with an equality, then we gel

tP£'~lI

(WI)'2 £'31 - O ' dd'2£'21+(WI)2,=O t2 2 f'21 t

-;ur + "2

_

(9.40)

which we recognize as hannonic oscillator equations. Writing down the general solutions, then evaluating the constants of integration with the help of (9.37), we

gel

,

,

£'31(t)=e31(O)cos

,nd

(W"). , (0) Sin. (W") T +1£'21 T

e~!l(t) = e21 (0) COS (wt) + ie~l1(O) sin (w~

t)

(9.41)

lllUs, if we turn on Hi. e~lI will transfer its amplitude to l?31' which will then transfer it back 10 e~l later tInd so on. Note that if t is applied for a lime t w generating a 11" pulse (WI t w = 11") so that (9.42)

One can represent the basic equations (9.28) and (9.29), describing selective el(citation of (n ± 1)-quanlUm coherence from a state containing n-quantum. coherence graphically. as shown in Fig.9.3. We use a solid line between two energy levels to denote the matril( element of V joining those states. Dashed lines between states represent matril( elements of f' (or t/) between those states. Then, the requirement that to generate a "if" matril( element on the left of (9.28) demands a jkg and a kgl matril( element on the right simply means that the two dashed lines plus the solid line fonn a closed loop, as in Fig. 9.3a. Then one also automatically satisfies the energy conservation requirements (9.29c) or (9.29d). Note that although V is in resonance with the 2-3 transition, we have assumed it is not in resonance with the 1-4 transition since we did nOI connect those states with a solid line. In Fig.9.3b we show that if we tuned V(w) to be resonant at the 1-4 transition, it cannot pump f'21 into f'31 since the lines do nol fonn a closed figure. However, looking at Fig.9.3c we see that V"I will pump f'21 into ~24. the zero-quantum transition. Note also that these figures show (Fig. 9.3a) that V23 can pump l?JI> the twoquantum coherence. into l?21, a one-quantum coherence. Also (Fig.9.3c) VI4 can pump the zero-quantum coherence into the one-quantum coherence l?21. We began Sect. 9.1 with some general remarks about the density matril( of a system in thennal equilibrium, asking whether or not one could experimentally

(9.43)

l'31(tw) = iU21(O)

(al i4,(lw)=O

(since £'31(0)=01

(bl

.

Thus. the full coherence has gone from one-quanlum coherence to two-quantum coherence at time two These same equations show that if we start with pure two-quantum coherence [£131 (0) f- 0, e:u (0) 0). we can transfer it to one--quantum coherence [~21 (t w ) j. O. ~:H (t w ) = 0] by a 'If pulse (WI t w = 'If). It is interesting to note that, if one starts with pure one-quantum coherence and applies a 211" pulse (WI t w = 211") to transition 2-3 (flipping the I-spins)

2

~21(tw)=-e21(O)

(9.44)

Now,

~21 = (+

-1ll'1 + +)

(9.45)

gives the $-spin transverse magnetization. Thus. a 27r pulse applied to the Ispins reverses the sign of the S-spin transverse magnetization. This result is a manifestation of the spinor character of spin systems. It was employed by Stoll et al. [9.9, to] and Wolf! and Mehring [9.11) to demonstrate the spinor nature of a spin panicle. Mehring el al. [9.12J have utilized this fact in an ingenious version of spin echo double resonance for electron spin echoes when the electron is coupled to a proton giving a resolved hyperfine splitting of the proton.

i

i

440

\ 911.W)I

1

,

I

1

911 \

\

I

l

2

\

1 1

\

I"

,

1

'''\ I 1

\1

A

V41

(el l

1'2,(0)'; o. '3,(0) ~ OJ, •

it' W

\'{"'11

=

f'JI(tw)=O

Vn

2

("----!1\

" ,,,""" /< 1

\

,

VII

FIg. 9.3a-c The four energy levels of a pair of spin-t nudei. (a) The density matrix dements e~1 and 1<'31, which oscillate at l.o.I~1 and ""':II respectively, Rrc CQupled together by the lime-dependent perturbation V oscilla.ting at fre
detennine whether all eij (i • j) were zero for a system in thennal equilibrium. The fact that we can pump n up or down gives us hope. Thus if we want 10 inspect a panicular eij. we look for a sequence of transitions to pump which eventually enable us to pump the ij coherence (if it is nonzero) into a pair of levels kl whose coherence, t!kl, is directly observable. Alas. a simple example will suffice to show that the usual excitations will not do the job in all cases. Consider two identical spin nuclei (e.g. a pair of protons) with identical chemical shifts. Their energy levels are the singlet and triplet states (Appendix H). Designating Ihe total spin by F, we have

i

F= II + h

Fz = liz + 12z

(9.46)

=

Fz == I tz + I2z

(9.47)

has no matrix elements connecting the singlet 10 the triplet, the only nonvanishing Yij'S are entirely within the triplet. l11OS, if we wish to inspect a matrix element

(FMFlelF' MF') = (II1e100)

(9.48)

we could only pump it to stales such as (lOlt!loo) or (I - II£'Joo) in order to generate a closed set of lines. Figure 9.4 illustrates how we could use (Illeloo) to feed (IOleIOO). But (lOlelOO) is not observable either. nor is (I - lIeloo) which we could produce as in Fig.9.4b. Thus, we can only succeed in pumping our original unobservable matrix element (9.48) into other nonobservable matrix elements! In principle we could apply a spalially inhomogeneous ahemating field of strength HI at nucleus I and H2 at nucleus 2 so that we had a rotating frame Hamiltonian

Ibl

I---<

11, -1191001

11'-

II°'I---:<

lI

V

110191001

.

"':>:>-- 10.01

V

1111

--(1119 100 1

11,01

r--_ \

11019100)

"-

\

-.;1

10,0)

11,11---

Fig, 9.4. (a) An alternating potential joining the states J10) and lit) can couple (lotf?IOO) to (t1If?IOO). If we wanted to insp«tan element (tllf?IOO) whidl is itself not directly observable, we a~ uJ\$uccessfulsince we can only pump it into an element (tOI(lIOO), which is also unobservable. (b) If we produced (toteIOO), a furlher IIpplieation of \' tuned to the JI- t) -110) trllnsition will produce (t -lleIOO) which is still nOl observable 442

=

--yhhOFz + hahrh: _ -yh( HI; H2 )Fz

--yl{ HI; H2 )(lb - hz)

(9.49)

The tenn Itz - hz now connects singlet to triplet states, as can be seen by evaluating matrix elements between Ill) and Joo) using the spin-up and spin· down functions it and fJ fO express the IFM F) Slates:

III) = ,,(1),,(2) .nd

=

For the two spins, we have F = 0, M F 0 for the singlet, F I, M F = I, 0, -I for the triplet. We now demonstrate a type of matrix element of t! which we cannot probe. Since

101

1ieff = --yllhoFz +haIlzl2z --yh(HIIlz +H2I 2z)

(9.50.)

I

1(0) = j2[,,(I)P(2) - P(I),,(2)]

(9.50b)

We find that

1

(II Ill. - 12' 1(0) = - j2

(9.5Oc)

Thus, if we had initially a nonzero malrix clement (10IU
to produce

en,

(9.51) then pump the 23 transition using

W=W32

to produce e13. This would require shifling the frequency W from W21

(9.52) 10 W3to

443

something of a nuisance 10 do. In practice, one usually works with large HI's which can flip spins corresponding to many multiplet lines. Such pulses are called nonselective. They cover all the lines within an angular frequency width Aw="IHI

(9.53)

of the oscillator frequency w. We turn now 10 see how they work.

9.3 Nonselective Excitation

hence that there were no groups of three protons within a given molecule. They thereby showed that the species CCI-I3 was not present. 9.3.2 Generating MuUiple Quantum Coherence To generate multiple quantum coherence in the absence of detailed spectral information we utilize an approach which is referred to as nonselective excitation. There are many variants. We stan with the simplest. We will first describe without explanation the pulses we apply first to produce, then to detect the generation of multiple quantum coherence. Then we will explain how the sequence works.

9.3.1 The Need for Nonsclooive Excitation As we have seen. using frequency-seleclive pulses we can pump an n-quanlum coherence either up or down. Thus. slaning with the usual one-quantum c0herence following a 7fo. pulse. we can proceed in principle to produce a pure two-quantum coherence. then conven that 10 a pure three-quantum coherence. and so on. To do this, we would need (0 know the energy levels and the cor· responding wave functions. Having produced some desired. level of coherence. we could later inspect how it evolved in time by later reversing the process and pumping the n-quantum coherence to n - I. retuning the pump frequency to pump that to n - 2. etc.• evenlually gelling back to one-quantum (observable) coherence. If we studied how the observed coherence varied with the length of time the system was in the n-quantum state, we could learn about Ihe evolution of the system in the n-quantum stale. However, this approach implies we already know the energy levels of the system, and know therefore what spectral lines are connected with what pair of states. This situation would be satisfactory for some purposes - for example. if we wanted to verify that the two-quantum matrix elements of e of a pair of spins vanishes when Ihe system is in thennal equilibrium. However, frequenlly one is attempting to utilize multiple quantum coherence to understand some unknown system for which the spectml lines may nm yet be assigned. One then wishes some general approach to exciting mUltiple quantum coherences. One would then leI them evolve. and lastly inspect what happened during the evolution, to try to learn about the system. For example, one might wish to know whether there are groups of two spins but not groups of three, as in the experiments of Wallg et al. [9.13]. who studied acelylene (C2H2) adsorbed 011 the surface of Pc metal and wished to detennine whether or not any CCH3 spccics wcre fonned. They found they could genemte both two· and three-quantum coherence. By comparing the intensity of two-quantum coherence wilh that of three-quanlum coherence. Ihey showed that groups of three or more prowns were rare. By studying how the relative intensities of two- versus thrcequantum coherence depended on the concentration of acetylene. they showed that the three-quantum coherence arose from coupling between different molecules,

444

.

preparation

evolution

detection

Fig.9.5. A three.pulse sequence, sho"'inS the prc,r'rlltion, e\'olution, ods, lind defining the times 0- ,0+, T - ,T+. r;, ' I

111m

detection peri-

The simplest approach to generating multiple quantum coherence is illustrated in Fig. 9.5. A strong 7f12 pulse is applied to a system in thermal equilibrium. "Strong" means "IH} exceeds the spectral width• ..1w, of the spectrum. That is. HI is larger than the spectral spliuings arising from chemical shifts or spin-spin couplings. The system develops freely for a time T. This pericx1 is called the preparation period. During this period. as we shall see, spin-spin coherences develop so that when a second rro. pulse is applied at the end of T, it convens the spin-spin coherence to a variety of multiple quantum coherences. These coherences then develop freely for a time tt. called the development time. At the end of that time, one wishes to inspect what has happened. This is done by applying a rro. pulse which transforms the multiple quantum coherence back to one-quantum coherence for detection. The signal is recorded as a function of Ihe time t2 after the detection pulse. Frequently t, is varied as a parameter to enable one to identify the contributions of the various multiple quantum orders. Thus, the signal, S. is a function of both t I and l2. It will also depend on the preparation time, T. giving us in general a complex signal, S(T. tl, t2)' To explain how the above pulse scqucncc works, we considcr a model system of two identical spin ~ nuclei, spins [ and S, possessing identical chemical shifts. We suppose that the rf pulses have a frequency wand write the Hamiltonian in the coordinate system rolating at w. In that frame, in the absence of the pulses, the Hamiltonian is thcn (9.54,) 445

n == "(Ho -

M

(9.54b)

w

representing the possibility of being somewhat off resonance either from the existence of chemical shifts. or because the experimenter has deliberately chosen

coupling. See Appendix H. For more than

twO

spins. the form of the interaction

term ([lS:) is only an approximation even if the coupling is dipolar. However, though Ihis fonn of coupling gives only approximate results in thai case, it makes possible simple explicit calculalions. enabling us to get results whose physical

significance we can explore. We utilize the spin-operalor method (Sect. 7.26) to follow the density matrix in time. DenOling times 0-, 0+. r-, r+, etc. as indicated in Fig. 9.5 to indicate times just before and just after the pulses at t = 0, T, etc., we take the density matrix at t = 0- to be e(O-) = (l, + S,)

(9.55)

omitting the constant of proportionality. Then the development of e is given in Table 9.1 up to the time just after the second pulse. We utilize the shorthand notation cos (0,/2)

==

C;

sin (nT)

Sh '

==

etc.

(9.56)

It is useful to express the spin components I z , Ill' etc. in terms of raising and lowering operators such as

I

1

I. = -(P" + r) , I, = 2', (P" - n . 2 We have collected the results (or products of twO spin components in Table 9.2. Utilizing these results we get the last column of Table 9.1, which expresses e{,+) in terms of raising and lowering operators. Utilizing them, we see that line 1 of the table conesponds to elements (m,mslelm'/ms) for which T.ble 9.1. The lime dependence of (/ durin, Ihe preparalion period Column no.:

Drivin,lcrms: Column labc:I~: (/(0-)

2) X(1I'/2) (/(0')

/,+ 5,

4 aT/,S,

5

,

,

m, +mS

(I,+S,)ShC~

(9.57)

(9.58)

hence are one-quantum mauix elements. Line 3 has products of two raising operalOrs or two lowering operators, hence corresponds to M'=M±2

,

(9.59)

which are two-quantum matrix elements. Lastly, line 4 has I z or Sz multiplied by either a raising or a lowering operator, hence has

M'=M±1

(9.60)

again a one-quantum coherence. Utilizing Table 9.2 we see that at t = 0-, the density mauix consists entirely of zero-quantum coherence. It is converted to one-quantum coherence by the first X(1rI2) pulse. At t = T-, just before the second X(1r!2) pulse, we note £:I(T-) still consists entirely of one-quantum coherence. However, there are now (our terms (lines I through 4) as opposed 10 the single term at t = 0+. Half of these terms arise simply because we are off resonance. That is, they come from the effect of the off·resonance term n(Iz + Sz). The other half arise from the effect of the spin-spin interaction term alzSz . For lines I and 2, the elements of £:I are proportional to cos (o.T(2). For shon T, those terms survive even if a goes to zero. Thus, they would be present even if I and S were nOt coupled together. In the limit of uncoupled spins, line I corresponds to magnetization which is flipped by the first pulse into the z-y plane, and then is flipped back onto the z-axis by the second pulse. Note that as , goes to zero, the combined effect of the two 1r(2 pulses is to flip the magnetization onto the negative z-direction. They are thus equivalent to a single 1r pulse.

•

(/,+S,)ChC~

' m,'+ mS

M'=M±1

,

X(III2)

D.(/,+S,)

"1' -=

HI

are the same (M' = Ivl), hence are zero-quantum matrix elements. For line 2, the terms have

to be off resonance. Recall that this form of coupling would give exact results for a pair of spins with dipolar coupling, bUI would not describe twO spins with a pseudo-exchange

==

"

-(l,+S,)ChC~

Line

-(/,+S,)ChC~

..

~(r +s'+r +s-)ShC~ 2

__ 2..(1' S' - r S-)Ch2S' 4i

•

2

,

(/, 5, + I, 5, )Sh 25:

446

447

T.b~ 9.2. ProdllCl$ of 5pln oomponcnu cKprcs5Cd in lcrms of r.lslnl .nd lowcr;nll opcr.tors

I.S~ .. ~/,(5' +5-) 2

I,S,

_~/,{S' -5-)

5,1,

_~5,(r +r}

...

S,I,

_~s,(r -r)

I 5

"~{/'s'+rs-+rs-+rs')

IS

--~(I"s'-rs'-rs-+rs')

.

" IS .2.(r5'-rS--I·S-+r5') >'
Line 2 for a = 0 corresponds 10 two uncoupled spins which are flipped inlO the x-y plane by the first pulse and left Ihere by me second pulse (because mey point parallel to HI of the second pulse). On the other hand, Jines 3 and 4 are proportional to sin(aT/2). Therefore, if one swilches off Ihe spin-spin coupling (a _ 0), these tenns vanish. They do nOI therefore arise for uncoupled spins. In order to observe these terms, one mUSI choose a time T such Ihal sin (aTn.) is appreciable, hence T'S for which aT/2 ::::: 1rfl, 31fn., elc.

(9.61) 3 If a represenlS dipolar coupling as in solids, a ::::: I/r where r is the distance between spins I and S. Thus, by selecting T, one can pick out the values.of.a or of Ihe distances T which will contribute. For example, the proton pairs 10 a CH2 group have a characteristic H-H distance. The existence of such a pair will give rise to a characteristic splitting of the proton NMR line into a pair of lines whose separation is proportional to the coupling. This splitting, first observed by Pake [9.14] is often referred to as a Pake doub/~t. Our a.naly~is shows that the existence of the coupling, hence of the doublet, Will also gIVe nse to the possibilily of generating double-quantum coherence. The pulse spa~ing r to produce the maximum coherence depends on Ihe strenglh of the couphng, a, hence on the magnitude of the splitting. Note that one can distinguish coupling between protons in the same molecule from coupling of protons in diffe~nt molecules either by setting T to correspond to the desired distance, or by.studYlOg the effect on the Iwo-quantum signal strength of diluting the molecules 10 a nonproton-containing matrix, as in the experiments of Wang et al. Extensiv.e SlUdies of this sort have been carried out by BQllnl et al. [9.15], demonstrating large clusters of protons in a given molecule. 44.

We note that while line 3 gives double-quantum coherence, line 4 gives single-quantum coherence. The distinction between these two lines arises solely from the precession effect of being off resonance. The lines are identical except that the tenn cos nT of line 3 is replaced by sin nT for line 4. If one were exaclly on resonance (n = 0), line 4 would be zero. However, one could men excite this single-quantum tenn by making the second pulse be Y(1f/2) instead of X(7ffl). In the process, me double-quantum coherence would vanish. The point is mal if one is exactly at resonance, the phase of the second rf pulse will determine whemer one gets single· or double-quantum coherence. We discuss these points more in lhe next section. 9.3.3 Evolution, Mixing, nnd Detection of Multiple Quantum Coherence We have now seen how Ihe first two pulses produce zero-, one- and two-quantum coherences. We now wish 10 see how these develop in time. Moreover, since neither zero- nor double-quantum coherences are direclly observable, we want 10 see how me third pulse converts them to something we can measure. The conversion is conventionally called "mixing". Since we observe nuclei by singlequantum coherence. we therefore wish to see how the third pulse converts zeroand double-quantum coherence to single-quantum coherence. We merefore conlinue our study of how fJ develops in time under the combined effect of me Hamiltonian of (9.54a) and a mird (mixing) rf pulse at time tl after the second pulse. Using the spin-operator methods, we construct Table 9.3 to describe the evolution and Table 9.4 to describe the detection periods. From me tables we can follow the effecl of the differenr interactions (spin-spin and Zeeman) on me time development of the density matrix. We now discuss some important features the tables reveal. First, we back up a bil 10 express the lime development of fJ. During Ihe time interval between pulses, the Hamiltonian (9.54a) is static, hence the lime development of the density matrix obeys an equation similar to (5.85)

("1,(1)1"') = ("Ie(O)I,i)e,p(i(w,,, -

w,,))

(9.62)

where (9.63)

W mrms

= -n(m, + mS) + wnJmS = -{}M + am/ms

(9.64)

where M=mJ +ms

(9.65)

gives the total z-componenr of spin angular momentum. Utilizing (9.62) we see then that the time dependence of fJ between pulses is given by

449

Tablt:

'.J. Time devdopmml of g durilll tM r:¥OIulion and mWlII period

Number of quanta in column 1

o

,

Column no.; I Column la~l; g(r') Operalor; Line number

•

)

-(/,+S,)ChC;

(I.+S.)SoC~

•,

•

-(I, S,+I,S,)C:Sb 2S~

-(I, S,+ I,S,)Sb2S~

'Q

'Q

450

'Q OQ

-{I, S,+ I,S,)C~ 2S:S o C;

(/~ S, + I, S,)Sb2S;SbC;

'Q

-(I. S,+ I,S.)Sh2S:S o C;

'Q

-(I, S,+I, S~)C!QCO 2S;

'Q

+(1, S,- I.S.)S;QC O2S~

O,2Q

- (/, S,+ I,S,)C~C;So2S~

(I, S, + I, S,)ChC:So2S~

+ (/,S~+ l~s,.)ShC:So2S~

'Q 'Q

+ (/:S~+I. S:)2S:Sg 2S; {S~+I.)Chs:sgs:

- {S~+ 1.)S;SoS;

(S~+I.)chS;SbS;

-(S,+I,.)ShS;Sos;

Line 00.

,, ,• • , • "" .."" """ "" "" "

Q

..

Table 9.4 (conlinued)

during llle delecticn period

,

Column no.: Column la~l: Q(r+f,') Operator:

+(I,+S,)CoC;

2

•

l

(I,+S,)ChC;SbC;

-(S,+ 1,)ShS;sos:

'Q OQ

, Q(r+fl+'V

"'I{/,S,)

+ (/,+ S,)~CoC; -{I. S, + I, S.)2S:CoC~

+(I~+ S.)SbC;CoC~

{/.+ S~)C:ChC;SbC~ (I,S, + S,I,l2S:ChC;SgC:

7

'Q

'Q

(/,S,+I, S,)Cb2S;SbC;

-{/,S~+/~ S,)ShC;So2S~

)

'Q

+(I,.+S,.lCoC~

Num~r of quanta represented by column 1

- (I.+S,)S~C;SbC;

+(/, S,.-I. S,)2ChShCb 2S;

durilll evolution

'Q

•

- (Shlll c)., 2S;

Table 9.4. Time developmenl of

OQ

, g(r+ft)

(I~+S~)ChC;soc:

-(/~ S,.+ I,S~)I{C~ll

- (/.S,+ I,S.)C o 2S~

"

o>
(/~+S~)C~C;ShC;

(I, S, + I, S,)2S;SoC;

• • "

co~rC11o«

-(/,+S,)ChC; -(/,+S,)S~C;SbC;

7

Order qUilnlum

• X(II/2)

{I. + S.)C;ShC;

)

,

,

,,(HI,)

01,(/,+1,)

-{/,+S,)CoC;

,

Table 9.3 (ronlinued)

-{/,S,+ I, S,)C~ 2S;SDC~

- (I, S,+ I,S,)c:ch2S:S o C;

+(1: S,+/~S:)2S:C:1I2S:SoC; - (I~ S,+ I, S~)ClDCO 2S~

-{/:S,+I,.S:lSb2S;Ch25;SoC; .. -(S,+I,.lsbs;Chs:SbC;

-(/.S, + I, S~)C;ClQCo2S;

-(I, (I,S,+ I, S,)ChC;So 2S;

(I;S~ + I. S:)cb2S;Ch2S;soc: _ (S. + 1.)CbS;Chs;5 0C:

S: + I: S,l2S;C;QCo2S~

(I,S,+ I,S,lC;chc~So2S;

- (I. S:+ I:S~)5b2S;ClD25;

.. - (1,+ S.)SbS;C1QChchS~

-(I. S:+ I:S.)Cb2S;ChC;So2S~.. -(l~+S.)cbs;chc:sos;

- (I.S: + I:S.)2S:ChSg2S; (S~ + 1~)Chs:sDS:

{S~+I.)C;Chs;sos:

(S,I,+ I,S,l2S;chs:sos:

451

exp(i[n(M - M ' ) + a(m~mS - mrms)]t) = exp (W(M - M')t)exp (ia(m~mS - mrms)t) (9.66) where the first factor arises from the Zeeman offset. and the second from the spinspin coupling. Recalling thai n = 1M - Mil gives the order, n. of the coherence we see immediately that the time dependence of the frequency offset uniquely labels the order of coherence of the element_ This fact is evident in Table 9.3. The time dependence of the Zeeman offset is given by the terms involving and t 1. They are to be found in column 5. and are re-expressed in column 7. Since M - M' = 0 for zero-quantum terms. (M - M')ntl = O. hence the Zeeman offset term is merely a conslant independent of or 1I. This prediction agrees with the table where the zero-quantum terms (line I) have no terms depending on ntl' The one-quantum terms (lines 2. 3. 4. 5. 8, 9,10. II) all involve either Ch or Sh. The two.-quanlUm terms (lines 6 and 7) involve (Ch)2 - (Sh)2 = C~o or 2ChSh = S~n' hence are at frequency 2n. A second important fealUre of the table can be seen by comparing column 3 with column I. showing the effect of the spin·spin coupling (column 2) on ~. Examination of line 6, the matrix elements corresponding to two-quantum coherence, shows that the spin·spin coupling has no effect on ~. To get this result using the operator method, one gets eig1l1 terms. They can, however, be added together. Using relationships such as sin 2 +cos 2 = 1 and I~[11 = iI~/2, they can be reduced to a single final term identical to the starting term. One naturally wonders if there is some deeper significance - indeed there is. The effect of the spin-spin coupling is given by the second factor of (9.66)

n

n

exp(ia(m~ms - mrms)l)

•

(9.67)

which. for the two-quantum term, has mr

= ms = i

and

m~

=

ms = -i

(9.68)

or vice versa. But

am

m=a(-!l (-!l

(9.69)

Therefore, the spin·spin factor becomes and we get no difference on line 6 between columns I and 3. If one has a system with more than two spins. a similar theorem is true. Consider a general system with N spins of and a Hamiltonian in the rOlating fmme 1{ = - L hni1zi + L(AijIi' Ii + Dij[zi1zi) (9.70) ij

!.

corresponding to various chemical shifts and a generalized secular portion of the spin-spin coupling. The total angular momentum operatOr (9.71) with eigenvalues MF commutes with 1{. 452

Thus, we know that the state in which all the spins point up (with MF == N/2) and the state in which all spins point down (M = - N /2) are both eigenSlates. Using a notation Iml.m2.m3• ...• mN) for the m-values of spins 1, 2. 3, etc,. we know that the state IN/2) in which all the spins point up (i.e. with M = N!2) is given in terms of individual spin m values by (9.720) and the state with all spins point down is

1- N/2) =I-!. -!..... -!)

.

(9.72b)

It is easy to show by explicit evaluation using (9.70) that these two states have Zeeman energies which are equal in magnitude but opposite in sign, whereas the dipolar energies are equal. Therefore. the energy difference between the states. which gives the time dependence of the N -quantum coherence. is independent of the strength of the dipolar coupling. We turn now to the detection method period described by Tables 9.3 and 9.4. This requires converting the various orders of quantum coherence back to the observable one-quantum coherence, the step called mixing. Since a 1r(2 pulse initially convened one-quantum to n-quantum coherence, it is natural to ask if ~ would reverse the process. Indeed, in Table 9.3 we analyze the effect of an X(1f!2) pulse, which is just the inverse of the original X(7f/2) pulse. The result, in column 7 of Table 9.3. is labeled in column 8 according to the orders of the resulting quantum coherence. For example. line 3 represents zero-order quantum coherence. II would be undetectable. In Table 9.4 we keep only the coherences which are single quantum during the detection interval, but label the rows by the order of the quantum coherence during the evolution period. In column 3 there are tenns such as (l~S~ + I~Sz) which eventually give no signal because when we form Tr {g(J+ + S+)} the [z and Sz cause the trace to vanish. We do not therefore bother to follow these terms past column 3. In column 5. we include the effect of the Zeeman offset. 11le signal is observed as a function of 12, and is given in tenns of F== 1+ 5 as (9.73) Utilizing (9.73), we can combine various lines of column 5 in Table 9.4 to get the contributions of the zero-, one-, and two-quantum coherences to (F+). Denoting

by (9.74) the contribution of the nth order quantum coherence, we get for the zero-quantum coherence

(P+}o = ~[e-iCn+(J/2)/1 + e- i (il-(J/2)I1) x [eiCn+(J/2)T + eiCil-(J/2)T +e- i (O+(J/2)T +e- i(O-(J/2)T)

(9.75) 453

For the one-quantum coherence

{F+>' = 2e- iS7t2 cos [a(t} - r - t2)/2] cos (ntd sin (f}r)

the conversion of the initial z magnetization to one-quantum coherence for the preparation period, the fact that zero-, one-. and two-quantum coherences have been brought into existence for time interval tl' and the conversion back to single-quantum coherence for observation. We have seen that the pulse sequence X(-rr/2) ... r ... XC1r/2) will produce zenr, one-, and two-quantum coherence. As we remarked earlier. if we study column 10 of Table 9.1, we note that the zero- and two-quantum coherence terms (lines I and 3) are proportional 10 cos f}r, whereas the one-quantum terms (lines Therefore. if = 0 (exact resonance) we 2 and 4) are proportional to sin would not produce one-quantum coherence. The question, how, when we are tuned exactly to resonance. can we produce one-quantum coherence with a pair of spins. turns into the question. how do we produce odd-order quantum coherence for an arbitrary number of spins. when tuned exactly to resonance? We turn now to a simple approach. The disappearance of the one-quantum coherence for a pair of spins can be traced back to column 5 of Table 9.1 where we see the effect of the spinspin coupling alone [i.e. before the resonance offset term f}r(1z + Sz) acts]. It produces (ly + Sy)C; on line I and -(lz:Sz + IzSz:)C:; on line 3. These tenos, if acted on by X(1r/2), produce (!z + Sz)C; (a zero-quantum coherence) and -(!z:Sy + I y S z )2S:; (two-quantum coherence). To get the one-quantum terms we allowed f}T(lz + Sz) 10 turn I z into -Iy • I y into I z , then we applied an X(1r!2) pulse. Therefore, if we have f} = 0, so that I z and I y are not rotated by 1r/2, we should get one-quantum terms by rotating the pulse from being an X(1r!2) to being a Y(1r/2) pulse. Indeed, we see that a y(7I'"!2) pulse would leave (Iy + Sy)C; alone as a one-quantum coherence. but tum -(IzSz + I z S z )2S; into (IzSz + I z S z )2S:;, a one-quantum term. In general, of course, f} =f O. Then the points we have just discussed show that production of a particular order of quantum coherence depends on the amount one is off resonance. That is frequently a disadvantageous situation. It can be remedied by making f} effectively zero by means of a spin echo, then choosing the phase of the second pulse to get the desired order of coherence. Thus. we can use a pulse sequence

(9.76)

which, in complex fonn, gives

(F+)I = ~[e-i(n-aj2)12(ei(!Haj2)tl +e-iC l?-aj2)1 1)

x (eiCS7+aj2)T +e- i(S7-a/2)T) + e-i(J?+a/2)t2(ei(l?-aj2)t1 + e- i(l?+aj2)t 1 ) x (e i(S7+a/2)T +e-iCS7-a/2)T)]

nr.

For two-quantum coherence we get

(F+)2 = _2ie- int2 sin (at2/2) cOS(2f}tl) cos (f}r) sin (ar(2) = ~ [e-icn+a/2)t2 _ e- i(l?-a j 2)t2]

8

x (e2int1 + e-2int1 )(ei(S7+a/2)T _ eicn-aj2)T +e-iCS7-a/2)T _ e-icn+a/2)T)

(9.77)

Referring back to (9.62), we recognize thac the elements of the density matrix will have time variations

e+ i(l?±a/2)t

(9.78.)

for the states M - M' = +1, will have e- i(l?±aj2)t

(9.78b)

for M - M/ = -1, will have e 2int

(9.78<:)

forM-M/=2,and e~2int

(9.78<1)

for M - M/ = -2. We know from the Hermitian property of f! that whenever we have generated an M - M/ = p matrix element, we have also generated an M - M' = -p element. We note next that all three {F+)fl'S have the factors exp [ - i(f} + a!2)t2] and exp [ - i(f} - a(2)t2]. This corresponds to the fact that during t2 the spins are precessing in a given sense. We can identify the role of the preparation interval by looking at the rdependent terms. Note that these all involve single-quantum terms since their time dependence involves the frequencies (f} ± 012). We note that the time dependence tl tags the evolution period, and the order of the coherence is immediately obvious from the f} dependence which is exp(±Oif}tl) for zero order, exp(± Iiiltd for single order, and exp (± 2intl) for double-order quantum coherence. Thus, if one studies how (p+) varies with tl. one can isolate the zero-, one-. and two-quantum coherences. The expressions for (F+)p show clearly

n

X(,/2) . .. T/2 ... X(,) ... T/2 . .. [Y('/2) or X(,/2)] to produce the multiple-quantum coherence. One can also then eliminate the offset effects during the detection period by introducing a 1r pulse Cfor example at t2 = T12) to give an echo (at t2 = T).

9.3.4 Three or More Spins We have seen that a pulse sequence X('rr/2) ... r ... X(1r/2) will produce zero-. one-, and two-quantum coherence from a pair of spins if T is sufficiently long. What happens if there are a larger number of coupled spins? Will this sequence produce higher orders, or must one add more pulses? We shall show that the sequence still works.

454

455

1

i

We approach this problem two ways, first by considering three spin nuclei with a coupling of the form haikljrlh. which we can solve exactly, then we will look at the general case which we will treat by an approximation method. Let us stan then with a Hamiltonian for three coupted spins which generalizes that of (9.54a):

1t:::: -hillzT + h(aI2I%1Ir 2 + a23Ir2Ir3 + a3lIr3Irl) where IaT :::: Icrl + I cr2 + In 3 a:::: :to y, Z • (9.79) (Note, we shall assume below that the coupling between any pair, such as al2. can also be wOllen as a2l.) As can be seen from the case of two spins, it is the spin-spin coupling terms which lead to multiple quantum coherence. Therefore we need to focus on the second term of (9.79). Then, as with (9.55), we take e(O-), just before the first pulse, as (9.80)

,,(0-): I'T so that at t :::: 0+, just after the first X(.,./2) pulse,

(9.81)

e(O+) :::: IlIT

M::::M'±1

-*1tu t) IzT exp (*1l u t)

where

Thus all are one-quantum operators. The frequency off-set term in 1i, -hillrT' will cause a precession of the x- and y-eomponents of spin, as can be seen from column 7 of Table 9.1. It will leave the Irk (k:::: 2,3) terms alone. but will conven 1%1 or Iyl inlo linear combinations of IZI and II/I so that we should replace Izl by

(9.82)

exp (-io 12/r I I r 20 exp (-ia23 Ir2Iz3t) exp (-ia31 Ir3/%1 t)/1/1

(9.84)

In (9.84) the exponentials involving I r2Iz3 commute with Illb hence annihilate one another so that (9.34) reduces to exp (-iaI2/r 1I r20 exp (-i03 I Iz3Ir 1t)Iyl X exp (i031 Ir3Irl 0 exp (i012Iz I Iz2t)

(9.85)

We can now apply either of the (I,3) operators to get a result similar to column 5 of Table 9.1. Then we apply the (1,2) operators 10 that result. TIle final result is

(*1i t) u

: cos (a31T/2) cos (a12T!2)IlIl -

cos (a3IT/2)2 sin (aI2T/2)I%l/z2

- 2 sin (a31T/2) cos (a12T/2)I%l/r3 - 2 sin (031T/2)2 sin (012T/2)Iyllr2Ir3 456

(9.88)

at + 1%1

sin at

(9.89)

(9.83)

Since IlIT :: IlIl + 11/2 + 11/3. and since all three spin-spin terms commute with each other. we have to compute quantities such as

x exp (ia31 Ir3Id)exp (i023Ir2Ir3t) exp (iaI2I%1/r20

m

and replace I lIl by II/I cos

?t,,:IE h(0l2IzlIz2 +023Iz2Ir3 +03IIz3Izl)

ex p ( -*1iut)Izl eXP

(9.87)

1;1:) cos ilt - I yl sin

Then we wish to calculate ex p (

We have written (9.86) in a manner intended to help indicate how, using the spin operator formalism. each term arises. We note first of all, that only spin I has either x- or y-components. Secondly, we note that only coupling constants 0t2 and 031 involving spin I occur. This is because terms such as 023Ir2Ir3 commute with Iyl' Thus. if we had four spins, we would get from I yl terms involving 012, 013. and al4 but no terms involving 023, 02~, a3-l. We note that we can easily find the contributions from I y 2 and I y 3 from (9.86) by cyclic permutation of the indices. Expressing Izl and Iyl in terms of raising and lowering operators shows us that all terms of (9.86) connect states M and M' which satisfy

giving us for the contribution of I zl to e(t) cos (031 T(2) cos (0I2T12)[ cos (GOlyl + sin (ilOI%d +cos (a31 T/2)2 sin (OI2T(2)[ cos (at)I zl I r2 - sin (ilt)III1 I z 2]

e(0)l:.. 1

:

+2 sin (03 I T/2) cos (012T/2)[COS (ilt)I>:\I z2 - sin (nt)Iyl l z 3] -2 sin (031 T/2)2 sin (012T/2)[ cos (ilt)II/I Ir2Ir3 + sin (nt)Iz1Ir2Ir31 + tennsobtained by permuting the indices 1,2,3

(9.90)

This Hamiltonian is now acted on by the X(rr/2) pulse. In order to see what spin coherences are thereby produced, we need only examine what happens to the spin factors. TIle result is a table of operators before and after, and the order of quantum coherences which can arise from each term. Denoting the values

M-M'::p for nonvanishing (M O'lulM' 0'1), we get Table 9.5. The various possible orders of quantum coherence can be deduced by expressing the spin operators as linear or (k '" 1,2,3), then multiplying them to get terms combinations of I r /,:, such as for line 4

It, 1';

(9.86) 457

(9.93)

Tlblt 9.S. Effect of X{1l/2) pulst on operators Linc

Operator tltfOTe X(lll2) pulsc

Operator after X(>rI2) pulsc

,

I"

-I"

l

• ,

Ordcr!; of quantum collcrencc, p

0

,

I"

Ix, .. !(I,'+/,)

1.,1,2

,x,/~2" 2.. (I,' + I n(l; -/,-)

"

-1"/~l· -~/,,(1" -/,-)

1~1 "l

lx' I,l •

lx' 1<)

2.. .;

(I" +

/,-)(11' -11>

1,,/<)

,

-I"/~I" -~/,,(/1'-ln

I" /,//'1

-I"

8

I", 1,1/'1

6

which, using (9.92), is produced more quickly. 1J1erefore, if all spins in the 3 system are close together, we can produce the onc-, two-, and three-quantum coherences when the strong coupling Qjk'S satisfy

1~2 I~I· -~ I" (l i

-1 2- )(I{ -Ii)

Ix,IY1IY1·!(/,' +1'-)(/2' -/ 2 )(/1' -Ill

"

ajkT/2

:1:2,0

';;t

(9.94)

1r/2

If, however, two spins are close, the third far away, two of the three coupling constants are small, so that while two-quantum coherence can be produced quickly, three-quantum coherence is produced slowly. We can also analyze the two-pulse method for nuclei whose spin is not restricted to I = ~,and for arbitrary fonns of spin-spin coupling, such as dipolar coupling instead of coupling such as aIzSz , by using a power series expansion. Suppose then we have a Hamiltonian in the rotating frome

"

:1:2,0

"

:1:2,0 ±l, ± 1

1-l = -hfllzT +

8

L

1-ljk == 1-l: + 1-l88

(9.95.)

j>k 1 + It 4i II 2

p= +2

0'

.!-I+ r 4i 1 2

p=o

0'

1 4i II Ii

p= -2

where the fl represents being off resonance and 1-l88 and 1-ljk'S are the secular part of the spin-spin couplings. That is

[1-ljk' IzTJ =0

,

(9.95b)

[1-l 88 ,I:Tl =0

For example, for dipolar coupling

1-l j k = B j k(3I:jlzk - Ij' h)

(9.91)

(9.96.)

= B j k(2Iz jlzk - IZjIzk ~ IyjIyd

Once one has made such a table, one can work out the result in one's head since an Izk or an Iyk gives ± I, so a product of two such operators gives p values of 1 + I = 2, 1 - 1 = 0, -I + 1 = 0, -I - I = -2. We see that the three spins give us p values ranging from +3 to -3, as we proved earlier. Now, however, we can see how long T must be to produce Ihe coherence. Note that if a12 ~ GI3, that the three-quantum coherence will develop in a time when sin (aI2T/2) ~ 1 or UI2'/2 ~ 7r/2. If al3 «: al2, then we must wait until a13T/2 ~ 1r/2. In the process, as time develops, sin (aI2T/2) will have been through several oscillations. Thus, we must wait for the weakest coupling. Suppose, however, we had three spins in a line so that spins 1 and 3, on opposite ends of the chain, are weakly coupled. Then we expect (9.92) Then our expression (9.90) would imply GI3 would control how long it would take to generote a three-quantum coherence. Thai conclusion would, however, be wrong since, had we analyzed the time dependence of I z 2 instead of I zl ' we would have gouen a three-quantum coherence dependent on 458

(9.96b)

As in (9.7b), we then have quantum numbers Al and

IzTIMo:) = MIMo:)

and

0:

where

1-l ss IMo:) = E(> IMo:)

(9.97)

Then, the time development up to just before the second X(1r:I2) pulse is given

by {?(r-) = exp (

-*

1-lr) X(1rI2)I:TX(rr/2) exp

(*

=ex p ( -k1-lT) IyT exp (*1-lr) = exp (ifhIz) exp ( -*1-l88 T)IYT exp

1-lr) (9.98)

(*

1-l88 T) exp (-iflr I:)

The spin-spin exponential can be expanded using the theorem (see for example [9.16]) eABe- A = 1 +[A,BJ+~[A,[A,BlJ+

(9.99)

to give 459

e(r-) = e+

+

iI7r1

• { fYT - *r['Hss, IyT)

(.!.)2 ..!..-[1i,u, [1i [yT]J + ... }e tl 2! 88

•

(9.100)

-inrI,

Consider now the first commutator

[H.., IyTI =

L ,

(9.101)

[Hjk,!,,]

i>k

Equation (9.107) is, however, the condition for one-quantum coherence. In a similar way, one can show that every tenn in the series (9.101) generates onequantum coherence, a fact to which we will return. Returning to the two-spin Cllse, (9.100 and 103), we can now apply the Zeeman operators exp(inTI.:T) to get for the density matrix at time Te(T-) "" [yT cos DT + [zT sin DT -

~ fl I2[Izl [.:2 cos DT

- [yI I t 2 sin Dr+I.: I [z2 cos Dr-IdIY2 sin Dr]

If 1 is different from both j and k. the commutator vanishes. Thus, we have typical lenns such as (9.102)

(9.108)

Then, the X(1r!2) pulse will produce e(T+) "" - I.:T cos DT + IzT sin DT

3.

- 2'flI2[(IxIIy2 + I y 1Iz 2) cos DT

which are just whm one would have for .a pair of coupled spins. It is straightforward to evaluate the commutation for a specific choice of H 12. Thus, for dipolar coupling, we get

[1i12,(IyJ + [y2)] = BI2 [2Izl I z2 -

1:0:1/2:2 - [yl[y2,(1y1

+ [y2)]

= B I 2[2(Izl,!yl)Iz2 - Uzl,Iy d I :t2

+ r)J 2

(9.103)

From (9.103) we see that this commutalOf at this stage has one-quantum coherence. Indeed. we can see this in general by calculating a matrix element of

[H u ' I yT ] (Ma,!,H s8 I y'l'

"" L:

-

IyTHuIM'a')

(9.104)

0"

(9.105)

so that

I.)It2] sin Dr} ..

(9.110)

8S '

['H s8 , IyTll

(9.111)

Here we will have teons in the double commutation '

(9.112)

[Hjk> ['Hlm,(IYIII + I yl )]]

(9.113)

Now, if neither j nor k is 1 or m, the outer commutator will vanish. Thus, we get either two-spin terms such as

Thus M and M t are states which are joined by IyT' But IT)

- Ii) +ui -

DT

but since either for m must equal n, this will be of the foon

I

"" L:(M al'HsslMa")(M a"IIyTIM 0")

~i([,t -

II Ii) cos

We see clearly the one- and two-quantum coherences. The fact that the B I2 tenn is proportional to T is analogous to the small r behavior of the tcnn sin (aT!2) of lines 3 and 4 of column 10 of Table 9.1. Clearly, we can use a similar approach to the next Icon in the series of (9.100)

[Hjk, [Hln!' [ynn

which, using (9.97), which says 'H S8 is diagonal in AI, gives

Iy'f ""

+ [I.: lui

'12 (.I,: )' [H

(Mal'Hs8IM"a")CM"a"IIyT1M'a')

/If'' ,a"

- (M alIyTIM' al/)(M'a" IH8IM' a')

Expressing the [z and I y operators in tenns of raising and lowering operalOrs, (or using Table 9.2) we have

3i T{ + + ~ 2'Bt22i (II I 2 -

= -3iUz1 l z2 + 1:1 [z2)BI2

3.

(9.109)

e(T+) "" - ItT cos DT + Ix'!' sin DT

+2(Iz2.Iy2)Iz1 - (lzz,!y2)Iz d =-"2BI2[(I\+ +[1- )Iz2+ Jzl (1+ 2

+ (Itl I y2 + I y1 I z 2) sin Dr] ...

(9.106)

['H 12' ['H 12,Uy l + I y 2)ll

(9.114)

or three-spin terms such as

M""M'±l '60

(9.107)

(9.115) 461

In this mannerlwe see that each higher order or T in the series (9.IOO) adds one more possible spin to the couplings. We can now see another userul point by looking in detail at (9.103). AI· though lhe top line involves products of three spin operators, the commulation step reduces the prodUCI to two operators (e.g. ZI with [22)' In a similar way, one finds that the three·spin tenn (9.115) involves three spin operators, such as [zll1/2Iz]. In general each higher tenn in the series involves PrOOtiCts of the previous tenn Wilh a pair of spin operators, bUI lhen the commutator reduces the num~r by one. Thus, each successive lenn has one more spin operator in the product. Since it takes three spin products to get a three-quantum coherence, the T 2 lenn is the first one in the series which can give three-quanlUm coherence. The T] tenn is the first one which could give four-quantum coherence. Note that we say "could", implying that il does nO! necessarily do so. Thus, if one has two spin-j nuclei, the highest-order coherence one can produce is second, hence the T 2 , T , and higher tenns cannOI produce more lhan second-order coherence (double-quantum coherence) in this case. To find what coherences are aClually realized, one mUSI look in detail at the lenns or use the general rules about the maximum coherence 2NI of a group or N nuclei of spin I. Suppose in (9.115) one has a three-spin prodUCI involving coordinates of spins 1,2, and 3. Expressing the 11:1: and lyA: 's in tenns of raising and lowering operators, one might ask, will there be tenns in ()(T-) Uust berore the second. X(1fn) pulse] such as

I

ItriI:

?

(9.116)

The answer must be "no" since this is a three-quantum tenn whereas we have said each tenn in the infinite series prior to the second X(7fn) pulse has one-quanmm coherence. Permissible terms then would be of the general fonn

IiliIt

or

(9.117a)

It 1:2[,]

(9.l17b)

These may be thought of as grouping a raising (or lowering) operalor (e.g. with either zero-quantum operators or with paired raising and lowering operators (e.g. Ii It)· When these are operated on by the Zeeman operators. they simply multiply or I; by a phase exp(inT) or exp(-Wr) respeclively, or leave the the I:k's alone. Thus. the second X(7rn) pulse still acts either on expressions such as (9.117a or b).

Ii)

I:k

r:

Now

xIi X: X(l~t + i/y 1)X: 1",1 X/ 1X: /"'1 + il,l

so

il:1

X(It Ii !jJX = (1z1 - iI: I )(lz2 + il:2)(I",3

+ i/", I 1:2 /",] + 1"'11:21:] - iI: 11",2 1 z3 - 1:1 1z2 1:] + J:t /:21~3 - il:II:21:3

.= 1",1/",2/",3 - i/",t 1",21:3

(9.118)

When we express [zl> 1",2, and /",3 with raising and lowering operators, we see explicitly how the three-quanlUm coherence arises. TIle lenn (9.ll7b) will give

XIi h:IJ:X

: (lzt - iI,t)ly21yJ : [.1;\ I Y2I y] - il: 1l y2 / Y:1

(9.119)

TIle tenn l"'t1Y21Y3 contains three-quantum coherence. From our analysis of the power series expansion, using (9.9a), our con. c1usion is. then. that the pair or pulses X(1:(l) ... T •.• X(7(fl) will generate all orders of quamum coherence permitted. While it is a question of proper choice of T to achieve lhe desired orders, in general, all allowed orders will be presem. We thus have IwO problems yet to consider. (I) How can we detecl a particular desired order? (2) Is it possible to generate a particular order without generating other orders? We tum to these IOpics in the next two sections. 9.3.5 Selecting the Signal of a Particular Order of Coherence We turn now to a discussion of how [0 select the signal arising from a panicular order of coherence. We have already mentioned one approach at lhe end of Sect. 9.3.3, where we noted that if the signal is Fourier analyzed with respect to the evolution time t], the coherence or order p gives rise to lines at frequency pil, where is lhe amoum one is off resonance. Thus. introducing a deliberlarge compared to the spin-spin splillings ate resonance offset by an amount produces groups of lines in which the spectra of the different orders are well separaled. This approach, and a variant or it (lime proportional phase incrementation, TPPI) in which the offset is actually zero but is made 10 appear nonzero, has been widely employed. especially by the Pines group [9.5,7J. We discuss TPPI at the end of this section. AnO!her powerful method was imroduced by WotOIUi and Ernst [9.I7J. We turn to il now. Their method is based on a recognition that the phase of the multiple quantum coherence depends on the phase of the exciting pulse in a manner which depends on the order of the multiple quantum coherence. Their concept also underlies the scheme for production of a selected order of coherence which we take up in the next section. The basic Wokaun-Emst scheme may be described as follows. Let us denote by Xf>' a 7(/2 pulse applied about an x'-axis in the rolating frame where the Xl. axis lies in the x-y plane. making an angle ¢ with the x-direction according to lhe equations

n

x' "" x cos ¢ - y sin ¢ 462

- il:])

n

(9.120.)

(9. 120b)

I;,:I = I;,: cos ¢ - I y sin ¢ = eil.t/J I;,:e-il.t/J

(9.12Oc)

Note Ihal Xl is rotmed from x in a left-handed sense about I z by an amount Wokaun and Ernst apply a sequence Xt/J ... r ... X",,,.tl ... X ... t2

1J.

(9.121)

recording the complex signal 5(T, 1>,tl, t2) as a function of t2, for fixed values of the parameters T and tl, and for a succession of appropriately chosen phases 1>. Let us write the signal for ¢ = 0 as a sum of contributions G p from the various orders of coherence p (p = - N 10 +N)

1 I

Then, utilizing the fact thm X = eiJrTff/2 we have Xt/J = exp [i!(l;,:T cos ¢ - I lIT sin

L

G"

-!...

G =

h

P

J

5W,e-; pod.

.

(9.123)

o

Since in practice Ihere are 2N + I values of p, measurement of 5(1)) for 2N + 1 values of 1> should suffice, as we show below. To understand the Wokaun-EmSI theorem, we need to

1. 2.

examine Ihe general form of the contributions of a given order of coherence, examine the effect of a phase shifl on the differenl orders of coherence,

,nd 3.

show how to ulilize Ihis knowledge to pick OUI the contributions G p of the particular order of coherence. We consider a system characterized by a Hamiltonian in the rotating frame 1{ =

-

E hDkI:k + E k

1{jk

==

'Hz + 'H u

(9.124a)

X", = exp(iRI;,:TR- I 1r!2) = ReilrT1r/2 R- 1

= eil.Tt/J Xe-il.Tt/J

where for the spin-spin interactions 'Hjk we keep only the secular terms so Ihal (9. 124b)

Then, we have the quantum numbers NI, a such that (9.1240)

'HuIMa) = EnlMa) Note Ihat (9.124a) allows for a variety of chemical shifls. 464

(9.125d)

We now wish to express the ¢ dependence of the signal, 5. Abbreviating 5(T, 1>,tt> i2) as 5(t/J), we have S(t/J) = Tr {

If [e -(i/h)'H"J2 X e -(if h)'H"tl J£:I(T+, t/J)

[inverse]}

(9.126)

where (I(T+,t/J), defined as f!{,,+, t/J)

== Xt/Je-(i!h)'HT X",IzTX;le(i/h)'HT X;I

(9.127)

is the density matrix jusl after the second 1r/2 pulse, for the case that both 11' pulses have phase shifts t/J. The tP dependence of S(t/J) therefore arises in £:I(T+, ¢). Expressing Xt/J utilizing (9.125d) and utilizing the faci Ihal I zT ' and thus exp (iIzT¢), commutes with H, we get f!{T+, t/J) = eil,Tt/J Xe-(i/h)'HT X IzX-leCi/h)'HT X-le-i/,Tt/J

= eil.Tt/J £:I(T +, O)e-il.Tt/!

(9.128)

We recalllhm (1(,,+,0) is the density matrix just after the second X pulse, hence it contains all the multiple quantum coherence. In general, for a given spin system, £:1(,,+,0) consists of a number of terms representing not only the different possible orders of coherence, bUI also the various ways of generating a one can generate p = 0, given order. Thus, for a three-spin system, of spins ± 1, ± 2, ± 3. Consider p = I. It could arise from a variety of operators such as

!'

k>j

['Hjk> IzT] =0

(9.125c)

we gel

p=-N

,.

(9. 125b)

R=ei/.Tt/J

(9.122)

and then consider 5(1J) to be known as a continuous function of 1J. The essence of the Wokaun-Ernst method is then that the Gp's can be found as transfonns of 5(.)

1»]

Defining

+N 5(.=0)=

(9.125')

It

It12:

It h.hz

ItIiIi

(9.129)

plus all the operators one can get by permuting the labels 1,2, and 3. Utilizing a symbol f3 to designate all such various ways of generating a given order, we can clearly break £:I(r+ ,0) into a set of components Yp{J of given p

"T+,O) = Lgpp p,p

(9.130)

465

Now, a given order, p, is distinguished by having p more raising operators than lowering operators. (If p is negative, there are more lowering operators than raising operators.) We can then easily evaluate g(r+,4J) as i ' ,1'.;P9p13e -i/,T.;P (9.131) e< r +, f) "" eiT'TtI> g( r+, O)e -it ,T.;P =

l.:e

P.P

But,

ei/'T h±eif,.;p ""

h

±e±i.;p

eif'T Izke-i/'T "" Izk

by p, and in which the coefficients of the hannonics are the Gp's. Therefore, if one knew S(tjJ), one could deduce the Gp's by taking the Fourier transfonn of 5(<1). Thus, suppose we have deduced S(f). (We can do this by measuring at a number of discrete values fi and using a smooth interpolation procedure to estimate S for values between adjacent pairs of values fi') Then consider I, the transfonn of S, 2.

(9.132) 1=

Taking expressions such as (9.129), inserting factors exp (-ilzTf) exp (ilzTf) between each pair of spin operators, and using (9.132), one gets results such as

e i/'T4>

J5(
Substituting (9.1 36a) for S(f) we get

It h ze- iT""4> "" It h ze i4>

2.

I = L:G p

(9.133) That is, the effect of the f rotation for any operator with p "" 1 is simply to multiply that operator by exp (if). In a similar way, the rotation operators acting on a p "" 2 operator merely multiply it by exp (2if). In general, for the operators

= 21rG

I

p

+ '" G p [e 21l"i(p-p') - I] ~ itp' p) p f. pi

= 21rGpl

9p/3'

eif,T4>Yp/3e-i/.T4> "" 9p/3e iP4>

t

11<,+.<1)=

p=-N

(9.134)

fl· (9.135)

(L:9pp)e;po /3

where N is the maximum allowed value of p for the particular spin system. Substituting this expression into (9.126) we get the signal as a function of f to be 5(<1) = pEN e;poT{r:t [e-O/")"" Xe-Olh)'Ht, ] (1gpp) [iowse]} N

"" L

G p e iP 4>

where

(9. 136a)

p=-N

G

p"" Tr{ r.t

[e-(i/h)"Ht 2xe-(i/h)1"f.t l ]

(~yp.B)

[inverse]}

. (9. 136b)

The individual tenns 9p/3 depend on r, on the ilk'S and the strength of the spinspin couplings as illustrated in the tables we have worked out for g(r+) for the two- and three-spin cases. Examination of (9.136a) shows that it says that S(tjJ) can be represented by a Fourier series in tjJ in which the harmonics are labeled

466

Je;(p-p')Od
p

Note that this result is independent of We can utilize (9.134) to get

(9.137)

0'

(9.138)

2.

Gp =.2.. 2.

J5(
(9.139)

o

In some systems, such as a dilute solUlion of some molecule of interest, there is an upper limit to the quantum order [the quantity N of (9.135)]. In Olhers (for example an effectively infinite solid of dipolar coupled nuclei, such as CaF2) N has no limit. Suppose we had the fonner case. Then, there are 2N + 1 values of p, consequently 2N + I Gp's. Clearly 2N + 1 measurements of S(f) should suffice to deduce the Gp's. Suppose, then, we pick the 2N + 1 values of f, which we label fk' as


2.

= '2N + 1

(9.140.)

k=0,1.2..... 2N

(9.140b)

Note if we chose k "" 2N + I, we would have a value of f "" 21r, which produces the same shift as ¢ "" 0, a value we have already obtained from k "" O. Then 5(<1,) ,,5, = L:Gpexp(ip
+N =

L:

Gpexp [ipk2p/(2N + 1)]

(9.141)

p",,-N

This is a finite series. Consider then the transforms

467

I:S(1,)e>P (-ip' 1,) k

L: G e>p [i(p - p')1d k,p " = L: G p L: "p [i(p - p')kh/(2N + I)J

5(0)+5('11") = L:Gp(eipO+eip1r)

=

p

p

(9.142)

p

=

Now, defining

2N

=

1.=0

1.=0

1,,=0

=

=

(

L:I'

)

L:

I' I

f2N+l

-1-1

G 1(1 +ei1r/2 +ebr +e i31r / 2)

if

1. 1

(9.144)

1,,=0

Examination of (9.144) shows that as long as p 1= pi the exponent in neither zero nor a multiple of 21l', hence

f

f

is

=F 1. But

(9.145)

f2N+l = e i (p-pl)211" = 1

I

p

=

L. '1'1. e -;P'¢' ( - -I - )"'5(")

2N+l

with

,

(9.146)

k=0,1, .. .,2N

Therefore, measurement of S(¢k) for these 2N + t values of