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2.45 is not yet known, but it appears as if the texture continues to exist although the angle B is no longer single-valued (Fetter L989). One may wonder whether the precessing domain-wall solutions discussed above are also possible in a flow channel, where Î is locked perpendicular to the wall. In fact, as shown by Ho (1978b), there do exist configurations of near the surface that allow the system to absorb the continuous winding of i in the centre portion of the óhannet (see also Hu
i
1979c).
The effect of thermal fluctuations on the stability of the helical texture near the phase boundary to the dynamically unstable region and the associated decay of superflow have been discussed by Kleinert (1979a,b).
7.11
PAIR.BREAKING CRITICAL CURRENTS
The first discussion of critical currents in a superfluid was given by Landau (L94L) (see Khalatnikov 1965). Imagine a body of relatively large mass M moving at velocity z¡ through a volume of the superfluid at rest in the laboratory frame. Suppose an excitation with momentum kk and energy ^E¿ is created by the interaction of the body with the liquid. For this process to take place, energy and momentum must be conserved, i.e.
L¡ut"?:|Mul*En, Q.26aa) Mui: Mu¡* hk, (7.264b) where r¡ is the velocity of the body after creating the excitation. Since M >> m, where m is the mass of the superfluid particles, the change in the kinetic energy of the body can be neglected, and hence (7.26s) E*: hk - ui< h lkl u¡, i.e. u¡> Eolh l/<1. This leads to the so-called "Landau critical velocity"
u.":-t"
(#),
(7.266)
which ø, must exceed in order to create any excitations at all in the system. Superfluid flow (i.e. a persistent current) in an isotropic superfluid is only
/
3',¿'¿
tuperfiow and I exlures
For free particles, where E*:(hk)'lZm, u". is strictly oHe the excitation à"to. On the other hand, in the case of superfluid spectrum consists of a phonon and a roton part, so that there ate two critical possible
if u,.)0.
velocities usc,roton
usc,phonon:
:
,l,ii
A, po,
*4r:
(7.267a)
rr,
(7.267b)
where ^Á. is the roton EaP, Po is the roton momentum and c1 is the sound velocity, with c1)) U",roio,r. In a Fermi superfluid such aS a superconductor or superfluid 3He the spectrum of fermionic excitations E¡ (i.e. of Bogoliubov quasiparticles) is given by (3.32). For a (pseudo)isotropic superfluid, E¡ ís shown in Fig. 3.2. Clearly, for u," given by (7.266), one finds
U.":
A
Pe
,A up (( up, : ä; Dp similarly
(7.268a) (7.268b)
to the roton critical velocity, where Ps: hke is the Fermi
momentum. Note that the critical velocity for bosonic excitations (".g. phonons) is much higher and is therefore irrelevant since ct-u^r. In the A phase the excitation spectrum has nodes at /c ll frkF such that u*:0, i.e. an arbitrarily small velocity is sufficient to excite quasiparticles near the nodes of the energy gap. This does not imply, however, that a superflow in the A phase is completely unstable: what happens is that, at T :0 in the presence of a superfluid velocity field u,, quasiparticles are only excited near the nodes of the gap within a Fermi-surface area of radius proportional to u., leading to a finite density of excitations proportional to u]. Once the available states are filled, a new equilibrium state is reached and further decay is not possible (Volovik 1984b). The only effect of the excitation is to reduce the superfluid density component p, by a fraction proportional to u?. nt frnite temperatures the quasiparticles excited by the superflow may scatter into states aboue the Fermi energy, thus giving way for further excitations induced by the superflow. This process is clearly nonlinear in u", and thus superflow is practically stable for velocities u.11Aslpp. However, it should be noted that this does still not guarantee superflow in the A phase, since, as discussed in Sections 7.10 and 7.12, there are still other dissipation mechanisms involving textural effects (i.e. "textural critical velocities") at these low velocities. In contrast with the situation in the A phase, the Landau criterion applies to the B phase without reservation. A quantitative calculation of the pair-breaking critical velocity in this phase requires proper treatment of the effect of a frnite superfluid velocity on the energy gap itself. Indeed, a superfluid velocity defines a ne\il symmetry axis and thus leads to an axial
7.11 Pair-Breaking Critical Currents
323
distortion of the previously isotropic gap. This feeds back into the gap equation, causing an overall reduction of the magnitude of the gap.
7.11.1 The B phase The general structure of the order parameter Ap¡ in the presence of superflow along the z axis is given by
f /¡, o o\l At"j: #"'ot'rl nl 0 ar o r"-A ll .
L \o
o
,
(7.26e)
ar/Ju¡
where ß is an arbitrary rotation matrix. The order-parameter components ^Á, and 4,, are obtained by minimizing the total free energy 4., composed of the free-energy functional for p-wave pairing states, (5.4), and the gradient free energy (7.L7),
4o,:
I o', (f * f",^a),
(7.270)
î
:(T)',KAl,u!,
(7.271)
where J gtað.
-
with K=I(Kr*Kz+Kr). In the Ginzburg-Landau regime one finds (Fetter 1975a)
l&r\t :1* . . 32þrr- þ.o, K l2*\'
\;/
,,
t-1;: t"t\i) ":'
(7'272a)
4\'- 1 - 34þrr+ 3þro, K (2m\2 ^,, (7'272b) \a/ þro, løl \ h ) ":' with A= At(u":O) : ^Á,,(u.:0) and ø defined in (5.S). In the weakcoupling limit, where the þ, are given by (5.S), Ar is found to be (
unchanged, whereas Á,, is strongly reduced and vanishes at a critical velocity u"c,¡c lf")t''. At this point a transition into the planar state (5.20) takes place. For still higher values of u,, even this planar state will become unstable and the system turns normal. The nature of this transition may be expected to depend sensitively on strong-coupling and Fermi-liquid interaction effects. The supercurrent (7.32) itself is obtained as
G-f
,-ffir:),
Q.n3)
K /2m\2 8þrr* LLþro, _t_l
(7.274)
g":p"u"( where
b2:3
5þrot
NF\ft/'
7
324
Superflow and Textures
For small superfluid velocities, one finds the usual linear dependence of g. on us. For somewhat higher values, the nonlinearities in u' which are due to the reduction of Á,,, cause g" to go through a maximum before the transition to the planar state takes place. We deñne the "pair-breaking critical current" 8"," as the maximum of the supercurrent, given by
dg./du.:0, i.e.
2L /
8",":3"rbP'\l *
r\
1/2
(7.275a)
T.)
('- i)'"
(7.27sb)
This maximum is attained at the "pair-breaking critical velocity"
u',":
1, /
71rrz
tl1, \1 - r.) '
(7
'276)
For general temperatures, a full solution of the gap equation in the presence of superflow is required. For this, we note that in the presence of superflow us the quasiparticle spectrum Ek is shifted to E** k. u" (see (7.40)). Restricting ourselves to weak coupling, the gap equation is then obtained by making the above replacement for E¡ and substituting the form of the order parameter a2'fr - (( '4)'l + a'u(ft. 4)' (7.277) lAo/ruJl'
:
into the gap equation (3.46b). Eliminating the pair-coupling constant by means of the gap equation at u.:0, one finds the two equations (Vollhardt et ø1. L980, Kleinert 1980b)
[ "[ where
dpneù{+[t"nr''#* tanh';#]-
â]
p:fr 'Q and E":
utPp,
(7.27eb) (7.27ec)
ltz,
:
E2
+
E'o:
Ç2
+ Az(ur:0,
Ez = E\@J
(7.278) (7.27ea)
Í,,Qù :21't',
f'Qù: r -
: o, i : r, a,
Arr(u",rxl _
A numerical solution of (7.278)
t)
+ A?,(,", T)t t,
T:0).
(7.27ed) (7.27ee)
reveals that the qualitative behaviour found
in the Ginzburg-Landau regime continues to hold at least down to f lf"-0.5, while at even lower temperatures nonmonotonic behaviour is found. The superfluid current g. is in general given by
8r:
P%
-
8'ûr,
(7.280)
7.11 Pair-Breaking Critical Currenß
325
where the second term on the right-hand side of (7.280) is the nonlinear normal current (see (3.85))
g": ) k :
kk - û,f (Ek(u.)
- /r . u")
(7.281a)
@
hkpNç
oçf dt,l(tunr,
W-
tanh
t#).
(7.281b)
In particular, at T :0 pair breaking is only possible for superfluid velocities
A(u":0, u")_ u",o-_-=;-, T:0)
(7.282)
in accordance with the Landau criterion. Indeed, at T :0, g,, has the form (Vollhardt et al. 1930) B"
l.:o
:*trr"-
U",o)
(E?
- a?,)'''
"'"' E"(E? * a',
-
ai)'
(7.283)
where 0(x) is the unit step function. This implies that at T:0 and for u" ( u",s the superfluid current is given by g, : u"p and thus increases linearly with u" until it reaches the value us,op. As u" is increased above u",o, pair breaking sets in and begins to reduce the superfluid current. The maximum, i.e. critical, current is therefore reached onty slight\y aboue u",¡ and is itself just a little larger than uc.op. The full temperature dependence of the critical current g,," has to be obtained numerically from (7.280) and is shown in Fig. 7.3g.Ii is uppurent that the G-rlT")3/2 dependence of the Ginzburg-Lãndau regime extends to rather low temperatures. This temperature dependence of the critical
Too
d04 0.2
0
0.2 0.¿ 0.6 0.8
r
T/Tc
Figure 7.38
Pair-breaking critical current g",. for the B phase as a function of reduced ,temperature in the weak-coupling limit. The current is measured in units _of pu",o, with p the normal-state density and uc,o: A(u,:0, T :0)lpr.(After Vollhardt et al. (1990).)
7
326
Supeflow and Textures
current has been observed experimentally in a U-tube experiment by Eisenstein et al. (L979,1980b). Experiments of Manninen and Pekola (L982,1983), who measured the flow through narrow (0.8 pm diameter) channels, are in agreement with this temperature dependence only after the depression of the gap due to the walls has been taken into account. Namely, these authors find a temperature dependence of the critical current given by (1 - T lT"rr)3/2, where f"yr is
interpreted as a pressure-dependent superfluid transition temperature inside the narrow channels, with T"yt1T.,o,,,u.The suppression of 7: in cylindrical pores has been calculated by Barton and Moore (1975b) and Kjaldman et al. (1978) within a model assuming diffuse boundary scattering (see Section 10.4). Jacobsen and Smith (1987) have calculated the critical current g,." for a cylindrical channel in the vicinity of the transition and found that the temperature-independent prefactor is reduced to about one-half of the bulk value. Later investigations of the same problem in a single cylindrical channel by Pekola et ø1. (1987) have confrrmed the above findings. On the other hand, superflow in uery narrow channels of diameter 0.4 ¡rm has been found to become unstable at critical velocities two order of magnitude lower than the depairing critical velocity (Kotsubo et al. 1987). The simultaneous effect of superflow and a magnetic ñeld on the order parameter, critical current and susceptibility in the B phase has been considered by Fetter (1975a,b) in the Ginzburg-Landau regime and by Janke and Kleinert (1980) for general temperatures.
7.11.2 The A phase Similar behaviour for superflow
in the A
phase in narrow channels as discussed above has been reported by Manninen et al. (1982). It is thought that mechanisms of flow dissipation other than pair breaking are ineffective in channels with a diameter of the order of the coherence length. The pair-breaking current in the A phase for the particular case i ll u, has been calculated by Vollhardt and Maki (1978) for all temperatures. In the Ginzburg-Landau regime the same (r - T lr.)3" dependence as derived in (7.275b) for the B phase is obtained. The superfluid densities and critical currents of the A and B phases in confined geometries have been studied by Fetter and Ullah (1988) in the Ginzburg-Landau regime. It is found that both quantities are significantly suppressed by the boundaries.
7.12 DISSIPATION
OF SUPERFLOW
Superfluids carry this characteristic name because of their unique ability to sustain long-lived ("persistent") macroscopic currents. When discussing superfluids and their properties, it is therefore essential to find the possible
7.L2 Dissipation of Supefiow
327
decay mechanisms for flow and to investigate how robust the superfluidity of a particular superfluid is against decay. In the preceding parts oi this chaiter we have seen that the property of superfluidity is a consequence of the broken gauge symmetry. This suggests that we should toãt for those low-energy decay mechanisms that are connected with a change in the phase
of the order parameter. concerning the A
phase, we have alreadv mentioned the possibility of an unwinding of the phase via rearrangements of the I texture. Upper limits on the critical currents have been identified in Section 7.Ll; they are due to pair breaking. They are large enough to reduce the overall magnitude of the order parameter and can thus OJstroy the condensate itself. An extensive discussion of critical currents and flow dissipation in superfluid 3He-A and 3He-B has been given by Hall and Hook (1986); see also Cross (1983).
7.12.1 Superfluid 4He
In superfluid 4He, and similarly in superconductors, where the superfluid velocity u, is simply given by the gradient of the phase of the order
parametet, (7.5), the experimentally observed critical velocities are orders of magnitude smaller than the Landau critical velocity (7.268). This is attributed to the existence of "phase-slip processes", which reduce the phase gradient by a definite amount and hence dissipate the superflow. A typical example of such a process is the motion of a singly quantted vortex line across the flow channel, whereby the phase of the- ordìr parameter is changed by 2n. Following the discussion by Feynman (I972),lei us consider superflow with velocity u, inside a tube of rectangular cross-section bd (d <<å). We suppose that vortices are formed at the walls of the channel, with the vortex axis parallel to the wide side of the channel cross-section (length å) but perpendicular to the flow direction. (The precise mechanism of the vortex nucleation process is not yet well understood, but may be expected to depend on the properties of the wall and the geometry oi the channel.) The average separation AL of vortices nucleatìng at the wall along the flow direction is determined as follows. For the system to approach the normal-state situation, the superfluid velocity field, häving the value u, in the interior of the channel, has to vanish at the walls. This is made possible by the action of vortices at the wall with average spacing A,L: xsf u", where Ko:2nklma is the flux quantum. since the veiocity fielã of the vortices is cylindrically symmetric and decreases as Llr for a singly quantized vortex, the velocity contributions indeed cancel at the wall but add up to the required velocity u, inside the channel. We combine this result with the energy, (7.I43b), of a vortex line, whose diameter r is limited bv the geometry (r -'d): Euo,t"*
:
*,
p,(T) xtn
uft
,
(7.284)
7
328
SuPerflow and Textures
with Bf(r) the vortex core radius, where þ :'1.. The energy loss per unit time and channel length is then estimated
as
LE
usr
Lt
LLuvottex'
(7.28s)
On the other hand, the kinetic energy per unit time and channel length available for vortex creation is given by AE*r,,
L¡
- (u"bd)(Lp"u?).
(7.286)
Spontaneous vortex creation and hence superflow dissipation will take place for sufficiently large u, u"", where the critical velocity u." is obtained by equating L,E lLt and ÂE¡¡,/Âr as (Feynman 1955)
)
u"":
Ko1,
d
2n d'n
p7a)
(7.287)
(Note that the critical velocities to be discussed in this subsection will all be oFIe, denoted by ,,", irrespective of their different origins.) For superfluid the form of (7.287) seems to be obeyed for a wide range of experiments, at least for not too small d and not too close to the transition temperature (Wilks 1967). On the other hand, the prefactor seems to be somewhat larger than that given by (7.287), i.e.
u,": ,O#:
100
cm/s
(d in pm).
(7.288)
(In the above and the following discussion u, must be replaced by u" - u,, if there is an additional normal-fluid flow, unless indicated otherwise.) At velocities u" ) u," a dense system of vortex lines (a "vortex tangle") is formed. The interaction of the normal-fluid component of the system with the vortex tangle, whose origin lies in the superfluid component, can be characterized by a mutual friction force per unit volume (Vinen I957a,b,c)
L(a,- u,). 4,,: äB*oe"P! p
(7.28e)
Here B is a dimensionless coefficient of order unity and L is the average total length of a tangled vortex line per unit volume. Assuming L to be much larger than the typical linear dimension of the unit volume, the average distance Âr between vortex lines is determined by L(Ll)z: L, i.e. Lr-L-n. Since the average superfluid velocity is roughly determined by the velocity of a vortex at distanc. 2-L/2 from the core, L is related to
7.12 Dßsipation of Superflow
329
us-un by
L:o(?.(
(r"
- a,)') ,
(7.2e0)
where a(T) is a dimensionless temperature-dependent coefficient of order unity. The complex physics of vortex tangles has been investigated extensively by Schwarz (1978), who confirmed the validity of a relation of the type (7.290) by numerical calculations (see also Tough 1982). Close to the transition temperature, the energy of nucleation of vortices becomes small and vortex rings may be created by thermal excitations (Langer and Fisher L967; for a review, see Langer and Reppy 1970). Consider a vortex ring of radius R. Its energy is given by f$(n) : uxf;p"R, where ø is a coefficient of order unity. In the presence of an external superflow with velocity u. the energy of the vortex ring can be lowered if the ring moves "upstream"; it is then given by
E"(R):
Eg(R)
- P"(R)u,,
(7.zer)
where P"-E?lu! is the momentum of the vortex ring, with u!-r6/R its characteristic velocity. The superfluid velocity us for which the vortex energy (7.29I) is zero is the critical velocity u"", (7.266), discussed in Section 7.1,L. At finite temperatures T and u, ( u,", the excitation of vortex rings is possible if the thermal energy is large enough to lift the system over the energy barrier described by (7.29I). The maximum height of the barrier is
E"(R.)
:
$uzxf;p"R",
(7.2e2)
where R.: urcsf2u. is the critical radius. Beyond R", vortex rings may grow since E"(R) decreases for R ) R". A crude estimate of the lowering of the critical velocity by thermal activation processes is obtained by equating E"(R") and the thermal energy ktT, which yields
u*(r)
: o'#
(7.2e3)
This critical velocity is seen to be proportional to the superfluid density fraction p. and hence vanishes for T+ 7i with the corresponding temperature dependence.
7.12.2 The B phase The processes dominating superflow dissipation in the B phase are (i) vortex nucleation by interactions with the walls of the flow channel or by thermal excitations in channels of width larger than fs, and (ii) pair breaking and excitation of quasiparticles in narrow channels. There exist almost no explicit theoretical investigations of dissipation
7
330
Supefiow and Textures
in 3He-8, except for thermally activated dissipation of superflow for temperatures near T" and for superfluid velocities (or current densities) near the critical values, as adapted from superfluid aHe lPekola et at. LSBB; see also Soda and Arai 1981). In the absence of a more detailed theory, it is processes
therefore interesting to at least discuss the experimental situation. In persistent-current experiments using a toroidal flow channel of 3 mrn radius, packed with a powder of d:20 pm plastic grains, a temperatureindependent critical velocity of the form (7.287):
u"": o
r(6, d rn4 d
(7.2e4)
was observed for T<0.97;, with u:0.6 and B:I9, where oow Ks: 2nhl2mz (Pekola et al. 1,984a,b, Pekola and Simola 1985). The pressure dependence of u," seems to be well accounted for by the pressure dependence of 5o in the above expression. For T>0.9\, usc was found to decrease considerably. This is perhaps related to thermal activation processes as discussed above.
The influence of the size of the flow channel on u"" has been studied in torsional oscillator experiments (Parpia and Reppy 1979, Crooker et al. L981, Crooker 1983) where the flow channel contained a small aperture with a diameter varying from 2 to 100 ¡rm. Again the critical velocity was found to be temperature-independent at low T, being given approximately by u," - 4xslnd. A number of other experiments are consistent with these findings (Gammel et al. 1984, Gammel and Reppy L9B4; see also Dahm er al. 1980, Hutchins et al. 1981a,b, Brewer 1983, Ling et al. 1984a). A series of experiments investigating U-tube flow through circular channels of radii R : 100-200 ¡rm and lengths of about 5-10 mm showed an R-independent critical velocity much higher than given by (7.287), which might indicate a transition to a vortex state with higher dissipation levels (Eisenstein et al. 1980b, Eisenstein and Packard IgBz). There was one experiment (Manninen and Pekola 1982, 1983) in which the channel diameter was only 0.8 pm, so that the critical velocity for vortex creation (7.287) became comparable to the depairing critical velocity (7.268), (7.276). Indeed, the only critical velocity observed had the characteristics of the depairing critical velocity as'discussed in the last subsection. Results of an experimental investigation of flow through submicrometrediameter channels have been reported by Pekola et al. (19SS). These clearly demonstrate the presence of thermally induced dissipation and are in good agreement with theory, as discussed in detail by these authors. There has been little or no attempt to explain the dissipation occuring aboue the critical current discussed above. One might expect that well above u," a vortex tangle is formed. The resulting pressure gradient in a flow experiment would then be given by the mutual frictional force of vortices with the normal component, i.e. by (7.259). The calculation of the precise. effect of this force depends on the experiment under consideration. It is
7.L2 Dissipation of Superflow
331
interesting to note that (7.289), with a value of L givenby (7.290), gives a pressure difference of the correct order of magnitude à explain the dissipation observed in those experiments in which a larger value o'f u". than given by (7.287) was observed (Eisenstein and Packard lggz, Brewer ilas¡. Evidence for the presence of remanent vorticity in uniformly rotating 3He-B at superfluid velocities considerably below the critical veiocity has been obtained by Hall et al. (1984), who found a component of the dissipation increasing linearly with angular velocity g2, as is ìxpected from the vortex friction force (7.259). There also exists the possibility of critical superfluid spin currents in the B phase (Fomin I987a,1988). On the basis of his earlier work on the spin dynamics of superfluid 3He (Fomin 19g0, 19s1b) (see Secrion g.3.3), Fomin derived expressions for the superfluid spin current and equations of motion for the corresponding phase variable or spin superfluid velocity. The current is found to become unstable beyond a critical value of the r.rp"rfl,rid velocity, given approximately by (7.zes) where c,o is the spin-wave velocity, øris the Larmor frequency and crro is the spin precossion frequency. The theoretical predictio.tr ui" in semiquañtitative agreement with experiment (Borovik-Romano\t et at. L9B7a,b,i9g9¡.
7.12.3 The A phase As discussed in Section 7.L0, bulk superflow in the A phase is topologically unstable in the absence of a magnetic fierd. It is only u ,"rult of a "r that above numerical coincidence of the parameters in the free energy a critical temperature I sr1¡lerflow is stabilized against textural deformation by coupling of the vector / to the spin part of the order parameter. On the other hand, on the surface superflow is quantized and hènce is stable. The walls therefore have a stabilizing effect, so that superflow is expected to exist at least within a surface layer close to the wall. Even this is onìy true as long as there are no surface singularities. Since we know that all vessels, except for those with the topology of a torus, must necessarily contain,such singularities (see section 7.9), the stability of superflow again becomes questionable. As pointed out by Mermin (r977b, L97ga,b), surface singularities such as the "boojum", discussed above (7.zor) in section 7.g.5, can reduce the supercurrent in a channel by a periodic deformation of the texture (see also Ho 1978a,b, FIu I979c). A boojum is a point singularity in the I texture at the surface (Figs. 7.32c,7.39a), which carries two quanta of circulation and is nonsingular as one goes away from the surface (Fig. 7.39b). If the boojum moves once around the flow channel (i.e. on the surface, perpendicular to the direction of superflow; see Fig. 7.39c) then the
7
332
Superflow and Textures
su rf ace
(q)
l\s. (cl Figure
7.39 (a) Surface singularity in the i texture ("boojum"). (b) Cross-section through a boojum in a tube or torus. (c) Degradation of a supercurrent g" in the tube or torus by motion of the boojum around the flow channel, perpendicular to the flow. By traversing the channel once, the superflow is reduced by two quanta of circulation. (After Mermin (1e87b).)
supercurrent in the channel is reduced by exactly two quanta of circulation. Unless the boojum is pinned at the \ryall, one may expect it to move periodically around the channel, thereby causing a periodic variation of the I texture, and thus leading to a continuous reduction of the supercurrent. The existence of persistent currents in 3He-A therefore depends sensitively on whether surface singularities exist and whether they are free to move. Persistent currents inside a torus packed with powder (particle size approximately 25 ¡^r,m) have indeed been detected experimentally. For this, a rotating cryostat was used, where the dissipation of small-amplitude oscillatory flow was measured as a function of the angular velocity l) of the steady-state rotation (Gammel et al. 1985). While in the absence of a
superfluid current the dissipation was found
to be minimal at d2:0,
7.12 Dissipation of Supefiow
333
increasing as d22, the presence of a supercurrent caused the minimum to shift to a value f2-¡,, corresponding to the rotation rate for which the
superflow and the walls of the toroidal channel move in phase. Such irreversible persistent currents were generated by rotating the cryostat with an angular velocity exceeding a critical value f)". The critical velocity was then determined as usc : RQ., with R the radius for the torus. The critical velocities found thereby \ryere as large as 2.5 mm/s at lower T, and dropped towards r.. Since the observed value of u,. is one order of magnitrro" smaller than the estimate of u," for singular vortex creation, the relevant dissipalion processes is assumed to be associated with the dissipative motion of the f texture.
Orbitat dissipation simple model of orbital dissipation in 3He-A has been proposed by Hall and Hook (1986) (see also volovik L980, Gay et aI. 1983), which wilt be discussed below. It involves a simplified treatment of the dynamics of i textures to be discussed in detail in Chapter 9. Here it is necessary to anticipate some of the results derived there. The starting point is the picture of superflow collapse by creation of coreless vortices. Consider superflow through a toroidal channel. The constant superfluid velocity at the surface, where superflow is topologically stable, is thought to be screened by a layer of coreless vortices from the region of collapsed superflow in the interior (Ho 1.978a,b). This situation is reminiscent of the Meissner effect in ordinary superconductivity. The thickness  of the vortex layer ("penetration depth") is given approximately by the relation (7 .287), which specifies the typical size of vortices for a given velocity difference u, - u,, between the interior and the surface layer:
A
^kl,
tv z tr, --
2m lu"
- u,,l '
(7.2e6)
where u is a constant of order unity. The spatial variation of the relative superflow g" : g - pun is then obtained as the solution to a type of London equation, i.e.
L'Y'g": g..
(7.2e7)
As a further simplification, one may assume g, to be expressed in terms of locally averaged superfluid velocity (q) ut
8": P.r((u") -
u,,),
a
(7.2e8)
where the perpendicular component of the superfluid mass tensor p, is substituted in view of the predominantly perpendicular orientation of i to the flow direction. The rate at which I's can adjust itself to the steady state described by
7
334
Superflow and Textures
(7.297) is controlled by an orbital relaxation time Ío,u. A phenomenological model equation combining (7.297) and these relaxational processes has the form )'2V27(u"l
- u,,) :
(u")
-
u,,
* Ío.6 d(r")
(7.zee)
dl
In order to estimate ro,6, we have to make use of the equation of motion for i from Chapter 9, (9.87). The Í bending forces^in this equation may be approximated by assuming the scale of variation of I to be given by h, which makes these forces of the same order as the flow terms. Hence one has
tdit L
.,
l*=l |,0"'((") -',')',
(7.300)
þt is the orbital
viscosity coefficient calculated in Chapter 10, (10.170). The orbital relaxation rate is obtained from (7.300) as
where
*:ii((,,)
-un)',
(7.301)
where B is a coefficient of order unity. The rate is thus proportional to the square of the superfluid counterflow velocity. This is in agreement with the finding that the dissipation observed in the rotating-cryostat experiment (Gammel et al. 1985) is proportional to d22 as discussed above. Equation (7.299) may be viewed as an averaged version of the superfluid acceleration equation, as given in Chapter 9, (9.84). It is then natural to add force terms corresponding to the gradient of the chemical potential p, í.e. the pressure gradient VP: pYlt, to (7.299), which extends the range of applicability to the case of driven flow as wel[. Collecting all the terms, one finally obtains a nonlinear differential equation describing the space and time development of the locally averaged superfluid velocity field due to orbital motion (Hall and Hook 1986):
(!\'o'u r,) - u,,): ' \\-ù/ tt/ + ,p+e9-1v") -tt, ((r") \\-ù, - u.,)' \2m/
p", \ ar
p'^ l.
This equation must be supplemented by boundary conditions.
(7.302)
In
the simplest case the superfluid velocity at the surface may be assumed to be constant. In the presence of surface singularities such as the "boojum" discussed above, which may deplete the superflow by precessing around the channel, an appropriate boundary condition on d(u,)/dt must be specified. There are now several limiting cases of interest. If the thickness of the vortex layer at the surface is small compared with the channel diameter, xsfZnu^^*<
the centre of the channel), then the spatial variation of (r") may be neglected, i.e. the left-hand side of (7.302) can be put equal to zero. The
7.12 Dissipation of Supefl,ow
335
resulting equation for (u,) has also been derived by Volovik (1978a). The same approximate equation is obtained if the surface relaxation of superflow by precessing singularities is as effective as the bulk dissipation, so that superflow is dissipated uniformly over the cross-section of the channel. A different situation arises for narrow channels and small velocities, such that ]u>> d. In this case the orbital relaxation rate is governed by the bending force induced by the variation of i over the distanóe d, (7.303)
which is larger than (7.301) by a factor )rld. An estimate of the torque term in the acceleration equation for the superfluid, (9.ga) (chapter Þ), then yields
ftt
(etu,"
o,l :
e)' ffir(q) - u"r,
leading to the following equation of motion
Y :Ir, where
u'
(*';
(7.304)
for (u,):
#u,',-
u'),
(7.30s)
þ' is a constant of order unity. A simple dimensional consideration
shows that the gradient term on the left-hand side of (7.302) gives rise to a term with a similar structure as that of the last term in (7.305) (i.e. with V2 replaced by Lldz), so that (7.305) is qualitatively correct even in situations where l(r,) -u,,1 varies appreciably over the cross-section of the channel. The effect of orbital relaxation on the decay of vorticity in 3He-A has also been considered by Kopnin (1978) and Hakonen and Mineev (19g7). Experimentally, the effects of orbital relaxation on superflow have been observed in many different cases. The damping of torsional oscillations of the superfluid in a toroidal flow channel has been shown to be partly due to orbital relaxation as well as normal-fluid friction (Gay et al. L98L, 1983).
Earlier experiments, which employed wider flow channels, had only shown the effect of stationary bending of the Î texture, since the condition @ro,b< 1 necessary for orbital relaxation was not satisfied (Main et al. L976, L977, Bagley et al. L978). One of the intriguing observations of the later experiments was the strong history dependence. This has been related to the existence of varying amounts of surface singularities. With this assumption and solving (7.302), one obtains a semiquantitative description of the data
(Hall and Hook 1986). Within the model, it turns out that the surface singularities are formed for values of the superfluid velocity exceeding u,.-2mmfs, with the density of defects growing tinearly with u,-u"". It has been suggested that the critical velocity is associated with the characteristic dipole-unlocking velocity u",o, Q.z3rb). Experiments on a flow
336
7
Superflow ønd Textures
channel with an L8 ¡rm orifice yielded similar results (Parpia and Reppy r97e). A second group of experiments investigated the properties of driuen superflow. There is evidence for orbital dissipation in all of them, but several features are not yet understood. Furthermore, the effect of dissipation due to singular vortices may be estimated from (7.289) to be of comparable magnitude. The experiments of Dahm et ø1. (1980) measured the response of a volume of fluid driven by a diaphragm following a sudden change in the applied voltage, which leads to a pressure difference AP. The data seem to be explained welt by the model discussed above (i.e. (7.302) with the left-hand side put equal to zero), when values of the parameters þtlp"r:(1 - TIT)t''cm'fs and þ:1.8 are used and with the assumption of a suffrcient densily of surface singularities. A critical velocity u,":0.5(1 - T lT;)t' mm/s was observed. In later experiments by Ling et al. (1,984a,b,c), in which the force on the diaphragm was ramped in time, much higher critical velocities, as high as 0.5-1cm/s were observed. In similar experiments by Manninen and Pekola (1982,1983) two.critical currents were found at temperatures T Þ Tot. It is possible that the dissipation at lower temperatures (occurring in supercooled A phase) is different from that for T Þ Tou owing to the presence of B phase outside the channel, if a dissipation mechanism involving the precession of singular vortices at the ends of the channel is assumed (Thuneberg and Kurkijärvi 1981). A critical velocity u¡" of the order of 0.5 mm/s was also observed by Paalanen and Osheroff (L980) in a flow experiment with a strong magnetic field apptied parallel to the flow. A filter paper with 8 pm pores, used to increase the flow impedance of the channel, was positioned across one end of the channel. The pressure difference ÂP measured over the length of the channel varied linearly with us at u, ) u". and dropped near f" as G-f lf")'''. lf AP is calculated within the theory appropriate for wide òhannets, (7.302), discussed above, one finds a result proportional to ul and (1 - TlT.)-t'', i, contradiction to the experimental results. This clearly indicates that other dissipative mechanisms have to be taken into account to explain the experiment. While it seems that the simple relaxational motion of the i texture, when driven by superflow, can be accôunted for-at least qualitatively-there are obviously other types of motion (e.g. periodic or quasiperiodic) that are not yet understood. (See- for example the early experimental observation of þeriodic motions of i by Paulson et al. (1976d), Krusius et al. (1978) and Wheatley (1973).) A step forward in understanding these phenomena is the theoretióal discovery of a periodic motion of domain walls ("kinks"¡ in the Î texture, discussed in Section 7.I0.5. Although a certain qualitative understanding of flow dissipation in the A phase has been achieved, more experimental and theoretical work remains to be done to clarify the detailed structure of the complex processes that may occur in this context.
Further Reading
337
FURTHER READING
Textures and defects Anderson P W and Brinkman W F 1.97_8 in _The Physicl of Lþuid and Solid Helium, Part II, ed. K H Bennemann and J B Ketterson (wiley, ñew york), p. 177 Brinkman W F and Cross M C 1978 in Progress in Low iemperature fhysfcs, Vol. VIIA, ed. D F Brewer (North-Holland, Amsterdam), p. tOS de Gennes P G l9ll The Physics lf _Liquid Crystals (Clarendon Press, Oxford) Maki K 1986 in Solitons, ed. S E Trullinger, V E Zakharov and V L pokrôvskii (North-Holland, Amsterdam), p.435 Mermin N D 1979 Reu. Mod. Phys.51 591 Mineev V P 1980 in Souiet Scientffic Reuiews, Section A: Physics Reuiews, Vol. 2 (Harwood Academic Publishers, Chur), p. 173 Toulouse G and Kléman M 19761. Physique Lett. 37 Ll49 volovik c E 1984 usp. Fiz. Nau\143 73 fsou. phys. usp. 27 363 (1984)] Volovik G E and Mineev V P 1977 Zh. Eksp. Teor. Fiz. 72 2256lSou. 'phys. JET7 4s 1186 (1e77)l
Properties of superflow Cross M C 1983 in Quantum Fluids and Solids-lgï3 (AIP Conf. Proc. No. 103), ed. E D Adams and G G Ihas (AIP, New york), p. 325 Hall H E and Hook J R 1986 in Progress in Low Temperature Physics, Vol. IX, ed. D F Brewer (North-Holland, Amsterdam), p. L43 Mermin N D 1978a in Quantum Liquids, ed. J Ruvalds and T Regge (NorthHolland, Amsterdam), p. 195 Mermin N D 1978b J. Physique 39 Colloq. C-6, Vol. III, p. 1283 (Proceedings of the L5th International conference on Low Temperature Physics, LT-15)
Rotating superfluids
->\ Fetter A L -)
)
1986 in Progress in Low Temperature Physics, Vol. X, ed. D F Brewer (North-Holland, Amsterdam), p. 1 Feynman RP 1972 Statßtical Mechanics (Benjamin, Reading, Massachusetts) Mineev V P, Salomaa MM and Lounasmaa O V 1986 Nature lA 333 Salomaa M M and Volovik G E 1987 Reu. Mod. Phys.sg 533 Sonin E B 1987 Reu. Mod. Phvs.59 87
q\
Spin Dynamics
Experiments probing the spin dynamics of the superfluid phases of 3He provided the first specific information on the structure of the ordered state. At first the observed shift of the transverse NMR frequency seemed inexplicably large. After all, the usual spin exchange interaction is isotropic in spin space and therefore cannot give rise to any resonance shift. Spin-orbit interactions, on the other hand, which are by their nature spin-nonconserving and which thus can lead to such a shift, are very small. Of these, the nuclear dipole-dipole interaction is-relatively speaking-the largest. It was well known that in the normal phase the averagJ fot"" exerted on the total spin by dipole interaction is zero to first order. This is so because the contributions of all possible relative orientations of total spin and effective orbital angular momentum of pairs of dipotes average to zero in lowest order. In the pair-correlated state, however, the relative orientation of the spin and orbital degrees of freedom of the Cooper pairs is fixed, as pointed out by Leggett (1972, r973a), who coined the term "spontaneously broken spin-orbit symmetry" or SBSOS (see Chapter 6): owing to the broken relative symmetry, the spin-orbit forces acting between the partners of the Cooper pairs add up coherently (see also Anderson 1973, Anderson and Varma 1973). Thus the nuclear-dipole force and the pair correlations conspire to produce sizeable nuclear-spin resonance shifts. They can be measured conveniently and actually may be considered as a fingerprint of the particular pair-correlated state. As the spin resonance frequencies are generally sensitive to the preferred directions in both spin space and orbital space, NMR experiments can serve as a probe of the textures of the order-parameter field. This technique has been successfully used for determining macroscopic textures and, to some extent, textural objects such as domain walls and other nonsingular or singular defects. The preferred directions of the order parameter in spin space are of
8
340
Spin Dynamics
to the Cooper-pair spins. Large-amplitude motions of the total spin of the system will therefore induce changes in the preferred course related
direction, which in turn act back on the spin itself. This coupling gives rise to a rich and unusual nonlinear spin dynamics, offering an ideal test ground for concepts in the rapidly growing field of nonlinear dynamics in general. In the simplest case, the uniform A phase, the longitudinal spin dynamics is identical with that of a Josephson oscillator, with the dipole interaction providing the weak link between the two superfluids of up- and down-spin Cooper pairs. Tunnelling of Cooper pairs between the two spin subsystems also provides the dominating mechanism for spin relaxation, as pointed out by Leggett and Takagi (L976). As a consequence, relaxation effects are orders of magnitude stronger in the superfluid phases than in the normal phase.
8.1 DERIVATION
OF THE EOUATIONS OF MOTION
It follows from the preceding discussion that the spin motion
can be affected by the dipole interaction. For this to happen, the corresponding change of the spin part of the order parameter has to change the expectation value of the dipole Hamiltonian. The orienting effect of the dipole forces has been discussed in Section 6.3. There it was shown that in the A phase the
is proportional to orientation-dependent dipole energy density angle, but not the rota-(d. Í)', *hereas in the B phase only the rotation^/D tion axis of a spin rotation, affects /p. Owing to the smallness of the dipole energy in comparison with the characteristic energies of the pair-correlated Fermi liquid (Fermi energy €F and condensation energy Á), the timescale for dipole-interaction-induced effects is much longer than typical microscopic times such as the quasiparticle relaxation time r. On the former timescale almost all degrees of freedom are in equilibrium, the only exceptions being the conserved densities. In the present case these are the spin density and the quasiconserved variables associated with the spontaneously broken symmetry, i.e. the preferred directions. The spin dynamics for frequencies well below the gap frequency Alh and for frequencies well below the quasiparticle relaxation rate r-L is therefore described by a set of quasihydrodynamic equations for the total spin S and the order-parameter vector d(k).(Note that in the following d(k) will be taken to depend only on the direction of fr, krfr.) i.e. There remains the question of the exte.r¡t_to which the dynamics of the orbital preferred directions has to be in.eluded into the theory. In the A phase, for example, one could imagine incorporating a variable like an orbital angular-momentum density of Cooper pairs. As a matter of fact, these variables are not conserved, but rather their motion is damped on the timescale of the quasiparticle relaxation rate, r, as will be discussed in Section 10.3. For the purpose of the spin dynamics, one may assume the
k:
8.1
Deriuation of the Equations of Motion
341
orbital variables to be in the equilibrium state defined by the time averaging over the respective motion. We are now in a position to derive the equations of motion for the total spin S and the order-parameter vector d(k) (Leggett t974a,b; see also Varma and Werthamer 1974). The dynamics of S and d(k) are governed by the effective Hamiltonian
Hs:*y'Sx-'S
-
.
yS
H+
HD.
(8.1)
Here X is the susceptibility tensor. In all cases to be discussed below, X may be replaced by the one eigenvalue I observed in a static experiment. The dipole Hamiltonian He is a bilinear functional of d and d*, given explicitly by (6.e6) and (6.e7). We now consider S and d, d* as Heisenberg operators. The components of S satisfy the well-known commutation relations [S,,, S"]
: ih}
e
(8.2)
uut St .
^
in spin space and S is the generator of rotations in spin space, an infinitesimal rotation in spin space by ô0' leads to a rotation Since d is a vector
d+ d', where d' is given either by
d'(k): d(k) -
ô0" x d(k)
(8.3)
or
d'(k):
exp (å
ur' .s) a1t¡
""p
(- i ur' .s)
(8.4)
Expanding the latter expression in powers of ô0' and comparing the linear term with (8.3) yields the commutation relations ÍSp,
d"(k)l
: ik l, 1
u,^d^(k).
(8.s)
The relations (8.5) also hold with d*. If we confine ourselves to lowest order in the dipole coupling, we do not need the commutation relations of the d and d* among themselves in the following. With the aid of the commutation relations, the equations of motion for S and d(k) may be expressed as
s:-|rs,ø,t--sr#-R', åt1*¡
:-
|ta<*¡,
ã,1
: -d(k)"
H
(8.6a)
+ o(Hù,
(8.6b)
where Rp is the so-called dipole torque, which is due to the coherent anisotropic spin-orbit interaction "pulling" on S once ,S moves. It is given by the change in the dipole energy density, ô/o : ô0' . RD, when the spin
8
342
system is rotated by
Spin Dynamics
ò0'. Hence Rn
ôÉ1,-, \ : - r / _._. õ',l^ n\dt*¡ "
* c.c.
(8.7)
Note that a corresponding term in (8.6b) only leads to an additional dependence on d itself of order 1/p, and can therefore be neglected except at ultralow frequencies (Leggett 1984a). The angular brackets denote an average over the directions ii. Substituting the explicit forms for ôI{s/ôS and ôHD/ôd(fr), we finally obtain the celebrated Leggett equations (Leggett L974a,b)
S:TSxH+Ro, ./tr\
d(k)
:
d(k) x y.'II
- ;t' ),
(8.8a)
(8.8b)
with the explicit expression for the dipole torque in a general triplet pairing state obtained from (8.7) and (6.103a) as
:
-2n(yh)21'(0, T")( (t4 x d.(k)lÍq' d"(k')l* c.c.) ù n,, (s.9a) and /(0, z:) is defined in (6.94a). For an order where A:(r-n)tfr-îr'¡ parameter in a definite / state this reduces to Rp
Rp
:
6ÂDNF Re ( [ie
which for unitary states (d x simple form; see (8.40).
8.2
d*:
x d@)]lÎc' d.(k)l)¿,
(8.eb)
0; see below (3.37)) takes a particularly
NUCLEAR MAGNETIC RESONANGE UNDER LINEAR SPATIALLY HOMOGENEOUS CONDITIONS
The Leggett equations (8.8a,b) are nonlinear owing to the appearance of bilinear terms both in d and d* in the dipole torque and in d and S in the d-vector torque. For small deviations of S and d(k) from their equilibrium values 6:yHlT and d6(ft) in a static magnetic field,óÍ6, one can linearize (S.9) in the deviations ôS - S -,$ and ôd(k) : d(k) - do(k). Such deviations may be caused for example by a small time-dependent magnetic field ô.EI or a small change in the static field ^[fo. Taking the time derivative of (8.8a), ôd(fr) may be eliminated by virtue of the equation of motion for õd(k), (S.8b). This leads to a second order differential equation in time: ôS
- y ôi x Ho* Q2ôS: y,So x ôIL
In particular, in zero magnetic field (8.10a) takes the form òS + Q2ôs :0,
(8.10a)
(8.10b)
which is equivalent to the equation of a three-dimensional anisotropic harmonic oscillator with eigenfrequencies given by the eigenvalues f2r,, of
8.2
NMR Under Linear Spatially Homogeneow
the tensor sl. For general triplet pairing, dZzuu:8ÂDNF
t*.
ll
Conditions
343
is given by (Leggett r974a)
((õ*"[q ' d.(Ðilo - d.(k')l- d*(k)Q"a . d*(k,)
- [q x d(k)l*[A x d"(k,)1") al n,.
(s.11)
The interpretation of d22*u as oscillator frequencies suggests a relation to the curvature at the minimum of the di?ole energy potential. Indeed, compar_ ing (8.10a) with rhe equation for S obtained from (g.6a) by taking the second (quantum-mechanical) time derivative, one finds that túe thirdlerm on the left-hand side of may be expressed by the time derivative of -(8.10a) Rp, and hence that d22ru is determined by the double commutator (see I-eggett 1975a) ,
62 -Ys4pv:
I:XUS*,
Hol,S"l"o.
(8.12)
Here the subscript "eq" indicates that, after performing the commutators, d'* and d have to be replaced by their equilibrium values. The double commutator in (8.I2), however, is nothing else but the coefficient of the quadratic term in the expansion of the dipole energy about the minimum. This can be seen by considering the change in the dipole energy under infinitesimal rotations in spin space:
HL:*o (; ô0".s) H,exp ç;ô0".s)
: Ho -
#)
[[s' H,,l,s"l ô0"¡, ôaî,
(8.13)
where it is taken into account that the linear term is absent at the minimum. From (8.13), one finds, by comparison with (g.rz), a useful relation between the (squared) eigenfrequencies and the dipole energy density fo = (Ho) lV , errn
ð'f,
-T, aeì"ðo"u' )(
(8.14)
implying that the oscillator frequencies fJ are indeed determined by the curvature of the dipole energy at the minimum as a function of the rotation angles 0|. Let us now discuss the eigenfrequencies of (8.10a). We take the z axis along the magnetic field IIo and assume one of the principal axes of d|zuu to be along^ã16 (this is the case when the orientationof dis dominated by the magnetic field and the dipote force). Then the equations for ôs, and ôs,, ô'$ are seen to decouple. Solving the one for ôS,, one obtains the remarkable result that there is a longitudinat spin resonance (Leggett L974a), i.e. an oscillation of S, about the equilibrium value So: xHo, oi fr"qu"rr"y @ll: Qt.'
(8.ls)
One should keep in mind that in the normal liquid (or in any other ordinary
8
344
Spin Dynamics
system, for that matter) a longitudinal resonance does not occur. There, only an exponential relaxation of the excess magnetization takes place. The longitudinal mode corresponds to harmonic oscillations of the vector d near the bottom of the potential well provided by the dipole energy. The longitudinal resonant frequency is seen to be independent of magnetic field. This resonance can be observed in a CW experiment in which an RF magnetic field ô^[fo is applied parallel to Ho. The existence of this longitudinal resonance is entirely due to the broken relative spin-orbit symmetry (SBSOS) discussed above and in Chapter 6. The transverse resonance usually observed in NMR experiments is of course also affected by the dipole torque. The transverse components of the spin satisfy the coupled equations
- itttto.-ôS, : 0, ì r2lr) ô.! * iottttyô,S, : -íaøyy¡ õH,, )
(8.16)
ôs':
(#,,--#*)#*rõH.,
(8.17a)
cñ òS": -
-i--öS', @@r
(r' (rt -
r2?,) ôS,
where e)L: yHsis the Larmor frequency and we have taken the RF field to be along the ¡ axis. The solution of (8.16) is
.a'-dftr"
(8.17b)
Thus there are two elliptically polarized transverse oscillations
at
two
resonant frequencies determined by
u1:*{rtr+ q,+ firtI(rtr+ q,+
#rr),
- 4ç\,t*rrlr").
(g.1g)
The prefactors of the pole terms in (8.17a,b) determine the intensity ("spectral weight") of the respective resonance lines (see also Maki and Ebisawa L973, L974b, 1975b, Takagi I974b, Tewordt et aI. 1975b).
8.2.1 Limit of zero magnetic field Let us now apply these results to the A and B phases. In the A phase the preferred direction d is independent of /c and must be oriented perpendicular to the static magnetic field ^tls in equilibrium. The dipole energy is minimal for the preferred orbital direction i pointing along ã, say in the x direction. From (6.L04), the dipole energy density may be expressed in terms of the rotation angles 0"" as
Â/S: 3 Sr(r)[(
e"r)'+ (0",)'f
,
O; << 1.
(8.1e)
Substituting (8.19) into the general expression (8.L4), one finds for the phase
dft,:O, #,,: Q?':lfrr"rr)=
A
(8.20a)
aT.
(8.20b)
8.2 NMR Under Lineør Spatially Homogeneous
Conditions
345
One of the transverse oscillator frequencies is seen to be zero, which is generally true for any state with fr-independent preferred direction d. In this case the transverse eigenfrequency ar- is zero and carries zero spectral weight. Hence there is a unique transverse resonant frequency øf given by
(rî)t: ,?+ t*^.
(8.21)
The longitudinal resonant frequency is seen to be simpiy related to the transverse resonance shift bv aî: d}o. (8.22) The same relation would hold for any A-phase-like state with axiallv
o#'r;!"#fìi::t;î
"ä:i"i spin-orbit rotation
recar that a magneric nerd orienrs ,n"
axis ñ along the field axis. Accordingly, the only relevant angular dependence of the dipole energy is that on rotations about the z axis. Differentiating the expression (6.107) for Â,fp,
Â/B:,ES"(rX"os g) + +)', as required
in (8.14), and putting cos 0]
: -L
(8.23)
one finds
Qt*: Quu:0, ; .v-
$.zaa)
d2l,:3- S"Q)= Al. )(s
(8.24b)
There is no transverse frequency shift in the BW state (at least in the so-called "Leggett configuration", where ñllHù, but there is a longitudinal resonance at frequency
@?t:
dzr.
(g.25)
The behaviour described by (8.21), (8.22) and (8.25) is very well observed in linear cw NMR experiments in the A and B phases of 3He (see Figs. 4.4 and 4.6). In particular, the relation of the longitudinal resonant frequencies in both phases,
#: (*)'''f.(#)'",
(8.26)
has been verified near the A-B transition, where both phases may exist, if only in metastable states (see Fig. a.6) (Ahonen et al. I975c, Osheroff et al. 1975, Webb et al. r975a). The ratio of the gap parameters appearing in (8.26) may be extracted from measurements of the specific-heat discontinuity (see (5.14)).
8.2.2 Effect of the magnetic field on NMR Strictly speaking, the results for the resonant frequencies of the A and B phases given by (8.20) and (8.24) are only correct in the limit of zero magnetic field. since the order-Darameter strr¡cfrrre itself ehanöes nnder fhe
8
Spin Dynamics
influence of a magnetic field (see Section 5.4). As a consequence, the dipole energy will change too, which gives rise to a modification of the results derived above. In Section 6.3.1the dipole energy in the At and A2 phases was shown tobe given by (6.105). In the limit of zero magnetic field one has Ár: Ár and the last term in (6.105) vanishes, which brings us back to (6.L04), and hence (8.19). To determine the minimum of the dipole energy (6.105), we consider first a fixed direction of â and ã . î: cos g. The last term in (6.105) is minimal for êin the plane of âand Î(note that ê . d:0), i.e. ê.î:sin 0. In the A2 phase the prefactor of @.Ð'is larger than that of (ê -î¡'. lt follows that the minimum of Â/r is given by d ll I, with ê arbitrary. Since â x ê is along the Cooper-pair spin polarization, â and ê are oriented perpendicular to the applied magnetic field in equilibrium. . In^ the A1 phase, on the other hand, where the prefactors of (tt î)2 and (ê.1)'in (6.L05) are equal, the dependence on the angle 0 drops out and the minimum of /p is assumed for any planar configuration of ã, ê and î, i.e. for / anywhere in the plane spanned by d and â. In general, the dipole energy depends on the three angles 0'r, which may be chosen to parametrue the unit vectors d and ê as
/ - (cos g| cos 9), cos 0| sin g), sin 9|), ¿: (sin 0| sin 0] cos 0) - cos 0| sin 0), sin 0| sin 0] sin 0) * cos 0| cos 0",, - sin 9| cos 0)).
(8.27)
(8.28)
The angles 0!. describe rotations about the ¡r axis. Taking i along the x axis, one finds the general expression for A/r:
LfÈ',
- - 3 h''N''Ía?* cos2 a; cos2 o; + Á1(sin g| sin 0| cos gl - cos g| sin e)1. (8.29) The minimum of A/r is at 0|-0'r:0):0. For small deviations from the equilibrium, one may expand Â,fp in 0|
Lf #',
as
- 3 ÂnNr,l\Ø n + a ù2(0"r)' + a n a u(e'),|.
(8.30)
It
follows from (8.14) that the longitudinal resonant frequency in the A2 phase is given by (Osheroff and Anderson L974, Takagi L975b) rtt?t
:
Ê
)'oNry' X-' A n A rr.
(8.31)
There is no longitudinal resonance in thg A1 phase in this approximation for lack of a restoring force on motions of d and â in the plane perpendicular to the magnetic field (infinitely degenerate minimum). However, the dipole coupling leads to a minute admixture of the opposite-spin pairing component in the Ar-phase order parameter, which causes a finite, but very small, longitudinal resonant frequency after all (Monien and Tewordt 1985).
8-2 NMR under Linear spatially Homogeneow
conditions
34
The transverse resonant shift is given by e)2t
:
a)?+å ÂoN. 1 fon * a u)r. )(
(8.32
These predictions have been veriûed experimentally (Osheroff and Ander son 1974). The specific dependenge of a2,, and ú)2, o,n Á,, and Á¡¡ offers tht possibility of measuring the combinationC of strong-co"piing purä¡n"t" rs þ2. and B5 that enter Áx and Á¡1 (see Section 5.4). In the B2 phase the order parameter, as given by (5.66b), is characterize< by a preferred direction in spin space, which under equilibrium conditionr coincides with the direction of the magnetic field É. The dipole energj
density of the 82 state has been derived in section 6.3.1 (see (6.10gi: (6.110)). For small deviations from equilibrium (defined by ñ':ri,'e:0o,, one may expand (6.109) as
Lf*: *¿oN.{ B(Ar, - +A?,)@, - oò, -
¡, (8.33)
r, and 0s -ar^e given in (6.110) and (6.112) and rhe relarion : õñ1,, L(t- cos 0ò-t O?,rhas been used. yhpre ft,
The longitudinal resonance frequency follows from (g.14) as ,?, It'rNry' N,- t (H)(I6 Ar, l?,),
:
-
(8.34)
where x@) is the largest eigenvalue of the anisotropic spin susceptibility in the 82 phase. In equilibrium the axis i is o-riented along the magnetic field by the susceptibility anisotropy, i.e. (û . t):1. The minimum of th" dipole is located at the shifted Leggert angle ïr(H) defined by (6. tti¡. Thus, "n"rgy as theB2 transition is approached from well inside the B, phase, íh. Leggett angle decreases from 104o to 90". since Ll"": depends on the orientation of ñ, there is a transverse resonance shift (Leggett L974a). The transverse components of the tensor of eigenfrequencies d22*n are given by
dft*: #r,:
?t rNry,x-tØ1
-
A,,),
(8.3s)
implying the existence of two elliptically polarized modes with frequency r't?
In the limit of
: *{r7+ 2e,* *I@?+ 4e,)ø?lr,t}.
la,rge fields also Hasegawa 19S3)
(8.36)
(r, >> l),,) this reduces to (Greaves 1976; see
't?:{ît.2&
(8.37)
[rì
(8.38)
From the general result (S.17) on the spectral weight of these resonances,
8
348
Spin Dynamics
we conclude that the resonance at o¡? - 0 carries no weight. The resonant shift Aar, : (Ðt ttty of the Larmor frequency in (8.37) has been observed experimentatly (Osheroff and Brinkman 1977). It may in principle be used to determine yet another combination of free-energy parameters, B3a5.
8.3
NONLINEAR NMR PHENOMENA IN UNIFORM TEXTURES
Even in the simple case of spatially. uniform spin motion, the nonlinear Leggett equations (8.8a,b) are complex enough to defy general solution. In the following we shall therefore address the A phase and the B phase separately, and we shall discuss a few special cases that allow an analytical solution while still revealing the underlying physics.
8.3.1 The Leggett equations for the A and B phases Using (8.9b) and (3.68b) to determine Rp, the Leggett equations in the A phase take the form
i:ys xH*s""#lgxî¡g-î¡,
(8.3ea)
fi: yãx("- ?),
(8.3eb)
the preferred directions in spin and orbital spaces and f)o ^t" is the characteristic frequency of the A phase, defined in (8.20). It follows
where
ã
and
î
from (8.39) that S . d is a constant of the motion. The nonlinear spin dynamics of the B phase is more complicated than that of the A phase owing to the spin-orbit correlation of the order parameter d*(k): L,Rr¡ k¡. For most purposes, it is convenient to parametrne the rotation matrix by the unit vector ñ specifying the rotation axis and the rotation angle 0. Alternatively, one may employ a representation in terms of Euler angles. The rotation matrix in the axis/angle variables is given by (6.106). It is straightforward to write the Leggett equations (8.8a,b) in terms of ú and 0, using the general expression for the dipole torque valid for unitary states,
(Ro)*
> €¡"ut(AurAþu+ A"uAþt+ A"BA\B), - - ggo(r) ß],v
(8.40)
(6.8) and the relations below (6.106). The result is
i : yS x H * + A'"4sin 0(1, + 4 cos 0)fi,
(8.a1a)
8.3 Nonlinear NMR Phenomena in Uniþrm
Textures
349
.v2 e-'i.,s" x
.
1^/
ñ- - ;:tñ ¿)(-
(8.41b)
x S' + cotl? ñ x (ñx S,)1,
(8.a1c)
where .S' :,S - y-')(H is the deviation of S from instantaneous equilibrium. It should be noted that the dipole torque is along ñ, (B.4ta), and that the spin-orbit rotation angle is driven by the projection of ,s' on ñ, (8.41b).
8.3.2 Spin dynamics in zero magnetic field In the absence of a magnetic field there are no external forces acting on the spin system. If we neglect dissipation for the moment, this impties that the total energy is conserved, i.e. is a constant of motion. The spin motion is then in many \ryays similar to that of a spinning top in classical mechanics. We shall see that the present problem presents a nice example of the usefulness of differential geometry. It provides a simple mathematical framework within which it is easy to show that the spin dynamics in zero field of both phases can be solved exactly (Vollhardt 1984b). We shall focus attention on the motion of S, i.e. its space curye, when some parameter f is changed. Here t can be, for example, the time. There is, however, another parameter-the so-called "arclength" of a segment of the space curve-which is even more suitable for such a discussion and which leads to the so-called "natural parametrization". The arclength s(r) of a segment is given by (see e.g. Struik 1961)
s(r):
I,or'l#1,
(8.42)
-
and hence s(r) lSl. We can then take S to be a function of s(r), rather than of t itself, because the connection between s and r is given uniquety by (8.42). In this wây, the characterization of the space curve is considerably simplified. For example, the derivative of S with respect to s, denoted by ,S' dS/ds, i.e. the tangent vector to the space curve, is given by
:
,S'
:
s : s
s :isiso ,S' =
of
i
(8.a3a)
t'
(8.43b)
is automatically a unit vector. Furthermore, the second derivative
,S,
defines the "normal"
S":î'=no ns to the curve. It is directed
(8.44)
perpendicular to the
8 curve, i.e. to i (this is a consequence called the "curvature" of the curve:
ns:
of
lî12
-
Spin Dynamics
1). Its magnitude Insl is
Kû¡,
(8.4s)
r-r,
the radius of curvature, is the radius of the circle fitting the curve at that point. The plane passing through the tangent Î and the normal zs is called the osculating plane, because it fits perfectly to the curve at every point. A third vector can then be defined by where
6:îxño,
(8.46)
which is perpendicular to this plane and is called the "binormal". The three unit vectors form an orthonormal triad that is a new svstem of reference moving along the curve. Any change of 6 during this motion (which must be parallel to ño) describes a tilting of the osculating plane and defines the "torsion" z of the curve.
6'
:
(8.47)
-rño.
It is a measure of how strongly the curve winds out of the osculating plane. Using (8.43)-(8.47), one easily finds that r is given by T:
S' . (S" x,S'')
(8.48)
K,
The A phase
In the
case
of the A phase one obtains from (8.39a)
tño:
sin
dxl sin
, [i#
where 0 is the angle between
(8.4e)
g'
î x Qx s) + ¿],
(8.s0)
ã and i. fne binormal 6 is then found to be
6:îxfis:i,
(8.s1)
i.e. it is given by the preferred direction î. Let us
assume i to be time-independent, i.e. to be a constant vector in space. Then the torsion (8.48) of the curve described by S, being the measure of the change of 6, is identially zero.In other words, ,S moves in such yuy that the projection of Son 6:î is a constant of motion (because d(S." Ðtdt -S. 6':0),
S'Î- Cn: const. This is also evident from (8.39a), where S . l:0
(8.s2)
for H
:
0. Thus there exists
a constant vector
Ca: (S.i)i
(8.s3)
8.3 Nonlinear NMR Phenomena in Uniþrm
Textures
351
of magnitude Ca and direction i, which enables us to find an exact solution of the spin dynamics of the A phase for H: 0. For this, we make use of the vector identity
to
exDress S in cos one d which yields .
l:
I,
Sr: (S .î), + (S x î), (8.s4) terms of the angle 0 only. Taking the derivative of finds e: -ytx-t(S. î); furthennore, Sxî:Coâlsin0,
s,: l4)'r, \y" /
*l=q)' \sin 9/
(8.ss)
The total energy, as determined by (8.1) with H:0, is a constant of motion and may then also be written in terms of a single variable only: E e.
where e e:
:
#
(* . #*
d)'orin, o),
(8.s6)
Q'lx)cn. we are then lead to the equation of motion for g: ë
- e',-# +
äd:z',*sin
29
:0.
(8.s7)
The problem of solving the five coupled differential equations in (8.39) has thus been reduced to the solution of a single equation. Once th^is equation has been solved, every information about the motion of S and d can be obtained by elementary techniques. In qarticular,^S(r) is specified bV (i) its magnirude (8.55), (iÐ its projection on ô : î, . 6 : Ca, ^S and (iii) its projection into the plane perpendicular to 6, where it precesses with the angular velocity
.
9s: ld ldr ls
,Sxó
-
"a
dt2e.4 c,o
v-
(8.s8a)
cos' e
s-e"
(8.s8b)
The corresponding precession frequency of d around i is given by Ea: e olsinz 0. It is interesting to note that (8.56) and (8.57) are very similar to the corresponding expressions for a spinning top, where C^ represents the vertical component of the angular momentum, which is a constant of motion. Equation (S.57) describes a periodic motion of â (and therefore S) and can easily be solved analytically by an elliptic integral. The solution depends on the magnitude of Ca. In a turn-off experiment, when an external field Íl is switched off, Ca can be determined from (8.55), putting 0 :0. For this, the initial conflguration of ã and S relative to Î and thè external field ^tI (see Fig. 8.1) has to be determined first. (In fields H < d|oly the spin S is not aligned
8
352
Figure
8.1
Spin Dynamics
Configuration of ã and S relative to Î and a fixed external magnetic field
H.
along .ãr because the dipole coupling favours S r Í such that á is not perpendicular to.ãL) Note that once the system has reached equilibrium in the presence of the magnetic field, â and,s are stationary, lying in the plane of I and .tf. This initial condition results solely from the exclusion from (8.27) of relaxation effects that damp the motion of ã and,S, which were originally precessing under the influence of ^EL Minimizing the energy with respect to lSl and @0, the angle between S and i yields
:LIl
S
cos
where 7 is the angle between .õl and
tanZQo: When the field
^EI
(þo-
(8.se)
X),
î, and Øo (:
Ln
-
0ù is determined by
sinZy cos2y
(8.60)
- ({2olyH)''
is suddenly switched off, ó¡, is given by e
where
e:
d2oz cos þo,
(8.61)
,:*cos(@s-,) g¿nand
(8.62)
o4: yH.
The resulting periodic motion of ã and s is determined by (g.57) and depends on whether z S 1. Starting from gs, 0 initially increãsès in both cases. However, for z
Tr: g2^
4
r,i" fu
K ((
zz
-
þò'''\ *ró" / sin @o cos2
(z < 1),
(8.63a)
(z > L),
(8.63b)
8.3 Nonlinear NMR Phenomena in Uniþrm Textures
Figure
8'2 Precession of S about i
3s3
and simultaneous nutation, after turn-off of a
magnetic field FI >> eo/y.
where K(x) is the complete eiliptic integral. At z : the energy due to field ] turn-off is just sufficient to push â perpendicular to L It takes infinite time to attain this (metastable) position, i.e. ihe period is infinite.One expects that fot z - | relaxation effects become uery i-portant because of the long timescales involved. The resulting motion of the spin vector for yH >> fJA is shown schematically in Fig. 8.2: ,S precesses about i with an instantaneous angular velocity øs, (8.58b), while nutating at the same time with twice the frefiu"n"y of the 0 motion. This nutation is due to the finite spin-orbit coupting, which wants to have and. i perpendicular: the spin .s i, ,,o longer ^s a conserved quantity' As the projection of S on i and the total energy ãre constants of motion, the component of perpendicular to i will thðiefore change. Its ^S oscillation is a consequence of the periodic tunnelling of Cooper pairs between the two subsystems with spin projection ü : *-t (for a discussion see Section 8.3-3). Note that the precession frequency of S about t tr-Ñ; different from that of the 0 motión. It is mainty oetá*ined by the size of yHld¿" and the initial direction of H relative to î. rne two frequencies are generally unrelated. The B phase
In spite of the somewhat more complicated nature of the spin dynamics in phase, (8.41), the field-free case can also be solved^"*u"íly, B lhe as was first shown by Maki and Ebis awa (r976a) using an Euler-angle approach (see also Golo 1981b). within the present framãwork, we again study the
8
Spin Dynamics
space curve of S and determine the moving triad (î, ño, ¿î) UV taking the derivative with respect to the arclength s(r). V/e obtain from (8.a1a)
î-S':fi,
(8.6aa)
: ñg: $" -1. , .t
(8.64b)
or,-sñ-'i'ñ
(8.6ac)
¡Jô,
s-
where s : * ç*"Xy-' sin g (1 + 4 cos g). Interestingly, the (unit) tangent vector to S is given by the direction ñ. The binormal ó is then given by
b- ñxñ tñt
(8.6s)
Calculating the torsion z of the curve via (8.47) or (8.48), one finds that z is identically zero at all times. This implies that 6 :0, i.e. the binormal (8.65) is a constant vector in space, and hence the osculating plane is constant too! As mentioned before, zero torsion of S means that S rotates about 6 in such a manner that its projection on 6 is constant:
s.6:liÊxsl
sin]9
: Cs: const'
(8.66a)
(8.66b)
where we have used (8.41b) to obtain (8.66a). At the same time, û also rotates about 6 but with zero projection on n @ - 6 :0): its motion is planar, taking place in the plane perpendicular to 6. We have thus identified three constants of motion. They can be combined into a constant vector
Cs: (s.¡i)¿i. (8.67) This is essentially the constant vector ,4 (: -(y'lÐCs) introduced by Brinkman and Cross (1978), whose square Az is identical with the corresponding constant of motion found by Maki and Ebisawa (1976a). Making use of the vector identity (8.54) and of (8.66a), we can again express 52 in terms of one variable only-the rotation angle g:
*:(#)'(,'*(#)'
(8.68)
We note that the expressions .(8.55) and (8.68) for S are almost identical! The total energy may now be expressed in terms of 0 and 0 only, as E where e
s:
:
+lu' * ffi+
Q'l ÐC".
E aLG* cos Ð']'
(8.6e)
8.3 Nonlinear
NMR Phenomena in Uniþrm Textures
355
The equation of motion for 0 follows immediately as 0
aisin g(1 * 4cos e) :
- +ei#$ - +
o.
(8.70)
Again, the problem of having to solve a set of (here, six) differential equations has been reduced to the solution of a single one. Once this has
been done, .S can easily be determined as described below (8.57) in the case of the A phase. Here the frequency of precession of S about 6 is given by
es:tt*rTrr#, Y-uvB
(8.71a)
and that of É by
E^:!+. t0' 2sinz
(8.71b)
8.3.3 Nonlinear spin dynamics in the A phase In the presence of an external magnetic field the equations describing the spin dynamics of superfluid 'Ffe are considerably more complicated than those for the field-free case. So far, only certain limiting situations have been solved analytically (see Maki I975c). one such case is the purely longitudinal spin motion, s: (0, 0, s"). The vector á is then confined to lie in the (*,y) plane, /:(cosr/, sin qt,O).Taking i along the x axis, the minimum of the dipole energy is located at qr:0. Eliminating ,S,, one finds for rþ the equation of motion of a driven mathematical pendulum:
,i+*At, sin2p:-TÈ..
(8.72)
As suggested by Leggett (r975a), (8.72) may be considered as a Josephson oscillator equation. For this, we imagine the A phase to consist of two subsystems, the .spin-up and spin-down particles, described by order parameters .Ár * ei? and Á, o e-ie. The dipole interaction introduces a weak coupling between the subsystems, allowing pairs to tunnel from one to the other. The associated change in magnetic energy provides a restoring force for the pair current, giving rise to oscillations in the relative particle number tt.¡ - ny and hence the spin S, (see Maki and Tsuneto 1975). The "pendulum" may be excited by means of an impulse-a sudden change of magnitude of the magnetic field by LH. The subsequent "ringing" of magnetization is a special case (þo:ln) of the spin dynàmics in the field-free situation. The oscillations of ,S and, at half the frequencies, of. â, occurs at a frequency a4:ZnlT. where f, is given by (8.63á,b) with z: o4f d2^, a)L: y LH (Maki and Tsuneto L974b):
,:
[
4AÃL
K(arl
dZ
) (o\ < d2o),
l+o¡txloolo) (en< ør).
(8.73)
8
3s6
Spin Dynamics
3 LJr
QA
rectongulor covíty
l-TlTc=0.010
o
porollel plotes I
-TlTc=0.013
o
2
t:r a
t'tc . l
o I
012
¡ AH /Q4 Figure 8.3 Nonlinear parallel ringing in 3He-A at P:22.0bar in two different experimental geometries. (After Webb et al. (1975b).) Solid curve: theoretical results by Maki and Tsuneto (1974b).
In the limit ø--+0 the ringing frequency ø, tends to d2o, the longitudinal resonant frequency discussed in Section 8.2. For increasing or, i.e. larger amplitude of the pendulum oscillations, @, decreases until it approaches zero at o)L: Oa. This is the point where the impulse is just right to push the pendulum into its (unstable) maximum position. Beyond this point ú), increases with art, finally following the law o),--2e)yror ør>>l¿A. In the
pendulum picture the latter case corresponds to the rotating pendulum. This behaviour has been observed experimentally (Webb et aI. Ig75b, Ig77), except for the region close' to the "frequency dip" at oy: e,', where the system is very sensitive to relaxation effects as well as to perturbations such as walls and heat currents (see Fig. 8.3). A second case allowing for an analytic solution is the precession of the magnetuation about a strong static magnetic field ^Ho G.e. yHs>> OJ after a large RF pulse perpendicular to ãs has tipped the magnetization by an angle 0. The dipole interaction may then'be considered as a perturbation. In the frame of reference rotating with frequency THo= ø¡ about the direction of the magnetic field, the spin is a constant vector ,s' : s(sin g, 0, cos g) as long as the dipole interaction is neglected. The vector â in the rotatins
8.3
Nonlinear NMR Phenomena in Uniþrm Textures
Figure
8.4
357
Precession of the spin S and corresponding motion of the vector a spin-tipping pulse with 0:30".
d after
frame precesses about this fixed vector,S' as described by
:
d'
: -:-y2^d' x S'. )(
(8.74)
Solving (3.63), one finds â in the laboratory frame as
ã1t¡:
/ 1t -
[ \
.or
cos o) sin
ø¡r
cos @Lt\
* cosz ayt I, sin 0 sin ø¡r /
0 sin2 a4t
L
(8.7s)
representing a motion where d traverses a "figure-of-eight" shaped orbit on the surface of the unit sphere (see Fig. 8.4). The projection of the two branches onto the (x,y)-plane is a circle of radius sinzl?, given by the equation (8.76) xt + (y - cost Le)' : sina |9, which touches the unit sphere at the point i. He.e we have taken the vector as pointing along the y ãxis, such ttrãt it is perpendicular to both the static field IIo un¿ ttr" RF held that tipped the ma[netization. Substituting ã1t¡ from (8.75) into the expression for the dipole torque, and neglecting higher harmonics of the Larmor frequency ø¡-, which are small by factors dl¡la\-, one finds
Ro(r)
:#(1
+ 3 cos g)Sr(r).û.
(8.77)
8
358
Spin Dynamics
r T/Tc = 0.754 ¡ o T/ Tc = 0.854 ì zrzs oe o T/Tç = 0.930 ì v T/T" = 0.952, 466.2 Oe
I
3
o 3
00
300 600 900 1200 1500 1900
2100
Tipping ongle 0
Figure8.5 Frequency shift A,a@):o)t--ú)t- in 'He-A as a function of the pulse-tipping angìe 0. (After Osheroff and Corrucini (1975a).) Solid
curve: theroetical result by Brinkman and Smith (1975;) as given Uy (8.78).
The dipole torque Ro(r) gives rise to a frequency shift of the free-induction signal (Brinkman and Smith I975a) that depends on the tipping angle and is given by @
t:
ú)r-
*
#-rr* 3 cos
g).
(8.78)
For small tipping angles, this result reduces to the transverse resonanr frequency in the linear regime (8.2L), expanded in the high-field limit. As the tipping angle increases, the frequency shift decreases and eventually becomes negative beyond a critical angle gc: cos-t (-å) - 109o. This
behaviour is due to the fact that for increasing g, the vector d oscillates with larger and larger amplitude, thereby going through configurations where the dipole torque is weaker or even reverses sign. The frequency shift given by (8.78) has been observed experimentally by Osheroff and Corrucini (1975a,b) (see Fig. 8.5). As shown by Poluéktov (1984), the Leggett equations may also be solved analytically in the case when the vector I is aligned with a constant external magnetic field. Such a configuration may be realized in a parallel-ptate geometry (see Section 7.9). An alternative and in a way more systematic approach to the nonlinear spin dynamics of superfluid 3He has been developed by Fomin (Lg76, L978a,b,c, L979a,b, L980, I984a,b,1985a,b, L986, L9B7a,b) (see also Fomin et al. (L982) and the review by Fomin (1981b)). Many of the results for the
8.3 Nonlinear NMR Phenomena in Uniþrm
Textures
359
A phase discussed so far,
as well as those for the B phase to be discussed below, may be derived using this approach. It is based on the observation that the motion of the spin variables may be decomposed into a fast precession and a slow motion of the axis (or the angle) of precession. One may then average over the fast motion to obtain a much simpler set of equations for the slow variables. The theory has also been generalized to include spatial variations and relaxation effects. In particular, it has been shown (Fomin 1984b, 1985b) that even a small inhomogeneity of the magnetic field can have a strong influence on the precession of magnetization in the B phase. It leads to a splitting into two homogeneous domains in the case of a linear magnetic field gradient. While in one domain the magnetuation is at rest, it precesses uniformly at a relative angle 0r:104" in the other. This effect has been observed by Bun'kov et al. (I9B4a) and Borovik-Romanov et al. (1984b, 1935). The energetic degeneracy of the two
possible orientations of ñ is lifted by the motion of the domain wall (Podd'yakova and Fomin 1983). Instability of homogeneous precession of the magnetization is also expected in the A phase (Fomin 1979b, I9B4a, Fomin and Shopova 19g9). The'predicted time dependence in the linear stage of the development of this instability has been observed by Borovik-Romanov et at. (I9g4a). The nonlinear spin dynamics in the A1 phase has been discussed in detail by de vegvar (1984), employing the adiabatic merhod of Fomin. Nonlinear driven longitudinal spin resonance has been investigated both numerically (Katayama et al. 1980a,b,c, 1983b, ooiwa et al. 19g1, yamaguchi et al. L982, Yamaguchi 19S3) and within an analytical stability analysis (Fishman and Folk 1984a). It has been found that behaviour familiar from other nonlinear systems, such as stable and unstable periodic motion as well as chaotic motion, may occur (ugulava 1984, Montgomery 1985).
8.3.4 Nonlinear spin dynamics in the B phase Let us now discuss the solution of (8.41), describing the spin dynamics in the B phase in a finite magnetic field. Again we concentrate on a few simple cases. As in the A phase, there is parallel ringing after a sudden change in the static magnetic ñeld ^[Io, i.e. a longitudinal oscillation of ,S driven by the dipole potential. If the magnetic field determines the direction of ñ (ñ ll H), one can find the equation of motion for the angle 0 by taking the time derivative of (8.41b) and eliminating S, with rhe aid of (8.41c): (i +Ofr sin 0 (1, +4cos 0) (8.7e)
-
- -yH,.
For small oscillations about the minimum of the dipole energy at 0:0t: cos-t (-å), (8.79) reduces to the linear longitudinal resonãnce equation (8.10a). In contrast with the A phase, the dipole energy in the BW state has two inequivalent maxima at 0 :0 and 0 : n (see Fig. 6.6). Consequently,
8
360
Spin Dynamics
there will be two values of the change in magnetic field Â.F/ for which the i'inging frequency approaches zero. These values are obtained by requiring that the kinetic energy E*,,,: âx(LH)' provided by the field change bã equal to the difference in dipole energy between the minimum and
maximum_ positions: LEr:& {*"Qly')G* cos g)2. with this, one finds y LH - (?),'' Os and y LH (ã)r,rd2, (Maki and Hu I975a). For values of
:
LH
large compared with dàrly, "rotating-pendulum" behaviour with frequency @,:2T ÂÉl occurs as in the A phase. These predictions are in qualitative agreement with experiment (webb et al. L97sb, 1977). As a second example, we consider the "wall-pinned configuration", where ñ is oriented perpendicular to H under the influence of, say, closely spaced walls (Brinkman L974, Maki and Hu I975b; for other configurations see Maki and Ebisawa L975a). The equation of motion (g.70) for e, obtained for the field-free case, allows discussion of this more complicated situation. In a field-turn-off experiment, with ñ. H:0 and 0 : 0r initially, the energy term involving Õr in (8.69) must be equal to the kineiic gained through the field turn-off, i.e. el:ã0 LH),. For small "n"rgy frelds, y LH (( 9", the dipole potential dominates in (g.70), and we may put 0(t) - 9¡ for all times. In this case ,S is stationary and ñ precesses in the p^la.n: perpendicular to the axis Cs with frequency Ea(îr): |erlsin, ä0r: (?)''ty Â1L Note that this frequency is independent of tft" Oipãt" "ouplingS owing to the special initial condition. The precession of fi couples backto if initially 0 is not precisely equal to 0t, giving rise to a traniverse ringing signal. Wall-pinned ringing of magnetization has been observed experimentally (Webb et al. L975c, 1977). obviously, the ringing mode discussed above is completely distinct from the oscillations of the vector â about the minimum of the dipole well. It may rather be compared to the free precession of S and fi, subject to the constraint ñ . S :0. It appears as if the dipole interaction is irrelevant in the latter case because the solution does not explicitly depend on it. This is not so, however: the dipole force is actually responsible for the constraint fi . S:0. Finally, we consider the precession of the magnetination tipped by a large angle in a strong magnetic field, @L>> fJs. Under these conditions, the total spin S is approximately a constant vector in the frame rotating with the Larmor frequency, which we take to lie in the (*, ,) plane. It is convenient to work in a coordinate system in which the ¡ and z axes are along S H and S + É respectively, *ú"t" S and Ê are unit vectors in the direction of S anrJ H. The equations of motion for the spin-orbit rotation angle g and the vector ñ then take the forms (Brinkman and Smith 1975b): â
:Zøt-sin ()Q)ñ*,
ñ,: ,rf- cos (+Ðñ, * cot (jg) sin GÐG - ñï1, ñ': r,tyfcos (|Q)ñ, - cot (j0) sin (Lþ)ñ,ûrl, ñ,: -ørcot(|0) sin (tO)ñ'û'.
(8.80a) (S.80b) (8.80c) (8.80d)
8.3
Nonlinear NMR phenomena in Uniþrm Textures
361
on a rong timescale, we may exp^ect the physical
system to be in a state of 0L, an¿ t á :b. 1s.aou¡, this implies ñ,:0, and .onr"qul n l! "n"" : :0 ñ, from (g.ggc,d). The .,ñ, requiremenr rhar the right-hand síde of'(gróou¡ íanish ¿eterÀinl s ñ, and ñ, as ñr:cotlettan|ç and, û,:(1.- n?,)*. Sínce the dipole energy is a miminum for this solution, no dipolé' torq* i, exerteã unJ- h"n"" no frequency shift occurs. There is a problem, however: the expression for û, becomes greater than unity for tipping ángtes > e 0r. Thus the above solution is acceptable
minimum dipore energy, i.e. 0
:
l""-ñiro
only for þ < e;. For larger tipping angles, one hãs to explicitly take into account the existence of relaxation processes, which tend io driíe the spin-oìúi, angle 0 towards its equilibrium value g¡_. This may be ]1.9rporated by adding a term -@- 9v)/t to the right-hand side"of (g.gOaj. Her" relaxation time whose significance and magniìude "'is a spin will be discussed in Section 8'6' with this modification, ttre quas'istationary ,olution of (g.70) discussed above remains meaningful even in trre regime þ > 0r: the vector i may now escape into the x direction, while staying p"rp"naiã"iuì ,o the z
axis.
A solution is then found with û.,-:0, ñ,_(0 _ ïr)/(2ø.rsinrr@) and ñr:cot)çtanle. rne stationary varue }s'of tìre spin-orbit angle has to satisfu 0o< þ) is obtãined trom the normalization (which condition O?,- 1. Tire angle 06 increases with ar.z from 0o- 0r for .¡,rt(( 1 to ** 0o- for avr >> 1'
It should be ¡oted that in general this þ consistent with the initial assumptions of a time-independentsolution is not tipping angle. However, in the limit of weak ielaxation, when o,y1ÐL, one ðan negrect the relaxation of z component of the magnetization, because fi and -the therefore the dþole toryue aie para[el to trre yäis and hence perpendicu_ lar to the magnetic field. In this case the dipole gives rise to a transverse resonant shift of magnitude (Brinkman andlorque Smirh ISZSA¡ (D
L - Q)L:
15
ffo+ 4 cos @).
(8.81)
Such a frequency shift has indeed been observed (osheroff and Corruccini, 1975b), although there are deviations from the behaviour predicted by (9.stl near @ -- ø {gee Fig. s.6). Further nonlinear effecrs have been discussed by Liu and Brinkman (1973). It has been suggested that in rongítudinal driven NMR in the B phase, nonlinear phenomena such as perioJdoubling, intermittent chaos and crisis
should occur (Katayama ,t ot. r.9g3a,b, iã-ãg""t i Lgg3, Monrgomery 1985). Note, however, that driven longitudinar ,þin ,"ronance has in fact been shown to be unstable with respect to transverse r'svesserv'ù fluctuations (Fishman \ and Folk 1984b,19S5).
8
362
13
Ï/T" = Q.7¿'3 a T/ T" = Q.$93 o
3.0
NM
to
G
t
tf)
:9
Spin Dynamics
o
o a
2.0
Oa
=J 3
o
L
J
a o
1.0
t000
t400
lg00
2200
Tipping c¡ngte e Figure
8.6
Ao : at_ - ú)L as a function of the pulse-tipping angle 3He-8. (After Osheroff and Corrucini (1975b).) Solid curve: in
Frequency shift
0
theroetical result by Brinkman and Smith (1975b) as given by (8.81).
8.4
TEXTURE.INDUCED MAGNETIC RESONANCE PHENOMENA
So far, we have concentrated on the spin dynamics of uniform textures, in particular of those textures where the orientation of the preferred directions was determined by an external magnetic field or the dipole interaction. As discussed in Chapter 7, one of the salient features of superfluid 3He is the existence of textures, both smoothly varying and strongly localized, of the order-parameter field. Nuclear magnetic resonance experiments provide a powerful tool for studying the textures related to the spin part of the order parameters. Some of the simpler results will be discussed in this section. In calculating the NMR shifts induced by a given texture, one may distinguish between two limiting situations. If the spatial variation of the texture occurs on a scale Ç¡ long compared with the dipole coherence length 6o << f¡, one effectively has a model of uncoupled "local oscillators", where each resonator corresponds to a subsystem of extension larger than Çp but smaller than 8". In this case the contributions to the NMR po\¡/er absorption simply add up. The NMR lineshape is then determined by the distribution of subsystems.-,with a given resonance shift. In the opposite limit, 6. < 6o, the different subsystems are coupled owing to the stiffness of the texture. In this case the local-oscillator picture is clearly insufficient and a more general description is required. For weak excitations the corresponding theory turns out to be equivalent to solving a Schrödinger-type
8.4
Texture-Induced Magnetic Resonance phenomena
363
equation where the kinetic energy describes the stiffness of the texture and the potential energy is due to the locally varying dipole forces in the equilibrium texture. The case of strong excitations, where a nonlinear
I
treatment is required, is naturally even more complex and so far has received not much attention. We begin by considering the linear spin dynamics in spatially uniform textures, whose orientation is not only determined by a magnetic field but also by additional effects like the container walls or a counterflow u, u,,, i.e. a heat current.
ì
8.4.1 Tilted uniform textures
I
Suppose that the i-vector field in the A phase is tilted by one of the abovementioned effects so that the angle r/ enclosed by Î and the magnetic freld H is not åæ as required by the dipole interaction in equilibrium. In this case the the tensor of eigenfrequencies d**" is no longer diagonal in the "laboratory frame", where.ãI is along the z axis, but rather is rotated bv an angle àn - ú, so that
I
i|'*u: (Rç¿2R-)r"
/
cosz
tp 0
sin r/ cos rp\
0
I
-o'ol o
1 o
\sin r/ cos @
sin2
(8.82)
p /
where Ê is a rotation matrix describing a rotation by Ln - ú about j. Here it is assumed that Îhes in the (*, t) plun". The resonant frequencies follow from (8.82) as zeros of the determin ant lotõuu - T E¡, €¡,nt Ht - Õr"l and are given by (Takagí L975a, Fetter L976a)
a)2*:
lr?+ d|'oxärr@7+
4dz2^cos2 rp)rlz
(8.83)
The associated modes are elliptically polarized mixed motions of S. In the limit of high magnetic fields, ay))d)6, andlor tþ:Lfi, the two solutions reduce to transverse (+) and longitudinal (-) excitations,
- @7+ aTG + cos2 rp),
(8.8aa)
p.
(8.84b)
a?
=
Q)2e
= o)2-:
cù2*
d)z,.sin2
The third eigenfrequency is zero and does not carry spectral weight. The transverse resonance shift (8.83) caused by the wall orientation of i has been observed experimentally (Ahonen et al. L975a,b L976b). The possibility of the excitation to spin waves with wave vector lql+0 in tilted uniform textures and other dynamic effects have been investigated by Gongadze et at. (1e80). In the B phase a rotation of the É-vector field away from the equilibrium orientation along H ll2 by the angle r/ (about the y axis, say) yields the
8
364
Spin Dynamics
\ eigenfrequency matrix Õ'*"
:"r(
r¿ 00
0
sin2
'* *o*")
sin r/ cos r/.r 0
cos2
rp /
and two magnetic resonance frequencies (Osheroff et
o)2¡: L@?+
rz'z')
(8.8s)
al.
1975)
*Ll(øL+ t8")' - arL#"cosz tplLt2.
(8.s6)
For high frelds and/or small 1þ, Ø* and a¡- can again be identified with the transverse and longitudinal spin motions respectively. Experimental studies of B-phase textures have been made in a number of geornetries, including (i) cylinders (Osheroff and Brinkman 1974, Osheroff 1974, Brinkman et al. I974b, Ahonon et al. 1974c), (ii) powders (Ahonen et ø1. I974a) and (iii) slabs (Osheroff et al. 1975, Ahonen et aI. I975a,c Engelsberg et al. L977, Freeman et al. 1988).
8.4.2 NMR shifts induced by superflow The above results for NMR in uniform tilted textures are now used to investigate the effect of superflow on the NMR frequencies. For this, the idealized assumption is made that the resulting textures may be considered as locally uniform, so that the results of the above discussion can be applied. The effect of the coupling of different regions of the sample by the gradient terms in the free energy will be discussed later. In the A phase the effect of a weak superflow is to orient the vector Î parallel to itself (see (6.120) and the discussion in Section 7.10). For orientations of the flow direction u" other than perpendicular to an applied external magnetic field ^tI (e.g. 4 ' iÌ - cos Í), there is then a competition between both orienting effects, resulting in an intermediate orientation of I with respect to H (i.e. î-ü:cosrp). When the dipole interaction is the dominating influence, i.e. for a weak magnetic^field and weak flow, the texture may be assumed to be dipote-locked, i.e. I ll d, and one finds (Fetter L975b,I976a) sin2 rp
:)lt * (1
1.
-
-
2r¡
tt
cos224
cos27¿
+ ,l')''' J'
(8.87)
where
rl:
- P"t)u? H2Lx
(P.,,
.
H and u^,, which may not be entirely correct in view of the torque exerted on I by u" (see the discussion of the helical texture in Section 7.10). For perpendicular flow (X-àn), one finds ,lt:Ln. By contrast, in the case of parallel flow (Z:0) Here it has been assumed that i lies in the plane
of.
8.4
Texture-Induced Magnetíc Resonance
Phenomena
365
än for small flow nai-b& and rl
l,:
substitutine (8.87) into (8.83). The limit of large magnetic fields has been discussed by Takagi (1975a). The theoretical predictions are in qualitative agreement with experiment
(Flint et al. L978). In the B phase the vector ñ is oriented by the combined effect of magnetic field and superflow, as given by (6.125). In the limit of weak magnetic field and superflow the orientation of ñ is found to be given by an expression identical with (8.87) except that sin2 r/ is replaced by cos2 þ and q by 2rl (Fetter L976a). The NMR frequency shifrs follow from (g.g3). The discontinuous behaviour at 2r¡: 1 that is again found for the maximallv competing case of perpendicular orientations of ^ãI and u" has been observeá experimentally in the NMR spectrum (Hutchins et ø1. 1985), along with other qualitative features predicted by theory.
8.4.3 General theory of texture-induced NMR shifts in the A phase In order to describe the spin response of a nonuniform ã texture, one has to take into account the bending energy associated with the texture. The latter acts as an additional restoring force on A, when â ¡" displaced from equitjbrium as in the case of an NMR experiment. The equation of motion for d may be derived by different, but equivalent, methods: either by a Lagrangian formulation or by a Hamiltonian approach. Both formulations have'been used extensively in the literature and will therefore be described separately
Lagrangian approach
The Lagrangian method was first formulated by Maki (1975b) for the B phase and was later adapted to the A phase by Maki and Kumar (I977b), Maki (1980) and Vollhardt et al. (I98L). Below, we shall present a general formulation of the spin dynamics in the A phase, valid for arbitrarily strong magnetic fields. We start from the Lagrangian
where
L: I
d3r
s,
9:T-V
(8.88) (8.8e)
is determined by the kinetic energy density 7 associated with the â motion and
v{d}:"f0",,¿ +
(8.e0)
^/o
8
Spin Dynamics
is the additionat potential felt by á owing to the bending of the texture, (7.20) and the dipolar coupling to l, (6.104) (if necessary, other orientational energy densities may be added to V). The kinetic energy of the â motion is written as
r:+@-,,y)x@t-,,r). ¿yHere
l
(8.e1)
is the full anisotropic spin-susceptibility tensor (3.102):
Xpu: í*(ôu" - aâuâ'¡,
with
a
(8. e2a)
- L)(lx¡,t
(8.e2b)
as the prefactor of the anisotropy term, and the external magnetic field ^ÉI is assumed to be along the z axis such that a¡-: Furthermore, the local
arí.
spin S, i.e. the induced local magnetization ¡4 of the local rotation axis ø as
:
yS,is represented in terms
': .ST ^/X''
(8.93)
ø is expressed in terms of the Euler angles (q, þ, y) as (cf. the angular
velocity in classical mechanics)
: ncosøsin p-p,sinø,
(Ðv
:¿sinøsinþ+þ,cosø
I
Q)z
:yrcosa*ur,
|
)
(Ðx
where
y,-ðylõf
(8.e4)
etc. The kinetic-energy density then takes the form
r ::F t", - afT,
*
+ þ? + yl (u,
-
- zu,ø,*
iøt) cos
2(u,
- ay)y,cos B
Bl2).
(8.95)
we must now
express á in terms of a, B and y. To this end, we generate the local vector âby applying the local rotation matrix R(o, þ, y),17.00¡, to the unit vector in the direction of the magnetic fietd Il, i.e. to â. This yields
ã: B2 : (cos ø sin B, sin ø sin þ, cos B).
(8.e6a)
(8.e6b)
Comparing this with the usual parametrization of ã given by (7.1.49), we identifu
&: 1þ, þ: 0. We may therefore replace- u and B in (8.95) by tp and 0. The
(8.e7)
angle y, describing rotations- about 4 does not enter in (8.96b), and is clearly seen to be irrelevant since ãis apolar vector. In particular, y does not appear in the potential V of the Lagrangian density 5l, (8.89), where V : V {ü;0} is only a functional of the angles tþ and 0.
8.4
Texture-Induced Magnetic Resonance phenomena
3(
of motion for â is now given by the generat Euler-Lagrang
"ol;;Tation
aag___o õg v'
ôr with
ôf ô6
(8.98r
õg
ag _ag E v ô6, a5, ô(vå,) -:
(8.e8b
and Çt:1þ,0, y. since -g depends on y only through in T, y is a cycli, variabre. The Euler-Lagrange equation (s.-gs) for y % yierds a constant o motion P = y, + (rþ, - ar) cos g, (8.ee which may be determined from the equilibrium configuration ú: úo 0 : 0o with tþ, - 0, : y, : 0 as P
Eliminatinr Tt from
r,
- -ø,
cos
go.
(g.100
(g.95), the appropriate new kinetic energy is giver Legendre by the transform (see the ¿ii",rssio. leading to (z.zgo)ï AT T*:T_y,:_ ëYt
: L x*., z?[k1,,
ar)z
sinz 0
+ e7 + z(t
-
a)p(q,,
-
o,t)cos 0]
* consr. (8.101)
The correct LagranÊian, now being a functional of q) and g only, is then given_ by (8.89) with z repraced uy r., i.e. the spin motion is described completely in terms of r¿ and 0. Introducing fluctuations around the equilibrium position of d, i.e. longitudinal fluctuations
f:tþ-1þo
(8.102a)
and transverse fluctuations
8:0-0o, (8.102b) both Z* and V are expressed in terms of and g. The linearized equations / of motion for and g f
are then found as (úoilh aïdt
fo
sin2 0o
- L g,@r(r* a) sin 20o:
,t
or. 1e81)
y' õv {.f ; 8}
xN ô/ '
g*+g@?Í1-(1 +a)coszgol qrwù ,ro! T + *f,rr(t*a)sin z0o:y2 2ltutr\L 1-alsrn zus:U"
where
Y: #. * nx H2 sin2oo.
õv*lf ;sj ôg
.
.
(8'103a) (9.103b)
(8.103c)
8
Spin Dynamics
The second term on the right-hand side of (8.103c) is a force term, which is
of the amplitude of the fluctuation. It corresponds to magnetic [-aÀ/r(Oo + g)/ôg]":0, where LJr(Ð: + LX Hz cosz 0 is the
independent
õt ttt" vector ã, 1ø.ttS). This constant term is cancelled ãrieniation "n"tgy by contributions originating from ôV/ôg since 0o, the equilibrium value of the angle I of the vector â, is determined by the Euler-Lagrange equation
*,u
+
: ^/,)
#
l":, -
?
l":o:
(8.103d)
o,
where again 0: 0o* 8. In general the two equations (8.103a,b)
for the longitudinal and transn"rr" fln.tuations are coupled. This implies that in a longitudinal NMR
experiment transverse fluctuations are excited, and vice versa. However, in the case of a strong external magnetic field ,fI the vectot â is forced into the plane perpendicular to II, so that go+ ln; thereby the constant of motion Þ, 1S.1OO), is made to vanish. Then the two equations reduce to
"
r :-rtt
8t - @t-8:
y'ôv
(8.10aa)
xN ôf '
ôy*
(8.104b)
ôg'
i.e. they decouple. Stabitity of the equilibrium â texture implies that the potentials V and V* are quadratic in / and g (see the discussion below (S.tO¡U)). We may therefore express õVlõf and òV*/òg as
V= -6,^tç-1Úf,
(8.105a)
-(K,atE,-Iz,,s,
(8.10sb)
#=
where Â/ and Â8 are dimensionless differential operators acting on / and g respectively. Using KjAS: (X,NI YIALE'" (see (7.1'8a), (7 .27), (8.20b)), (S.104) then takes the form
fo: go
In a typical NMR
-
azt
(8.105c)
-d|zoNf'
8: -
(8.10sd)
d22¡/,88'
experiment the fluctuations
/
and
I
are excited
as
periodic oscillations
f : "-t-*Í(r),
(8.106a)
g: e-i-,tg(r).
..
(8.106b)
i"
Thetlongitudinal and transverse resonant frequencias
@¿
and ø, are then
8.4
Texture-Induced Magnetic Resonance Phenomena
determined by the eigenvalue equations obtained from (8.1.05c,d),
Nfo:
)rr,ofo,
(8.107a)
Nío:
hr,*8*,
(8.107b)
with the eigenvalues
^ ^ la,¿.*\2 h€,k: Lt,k: \ o" / '
@?¡- a'y . S*o
(8.108a)
Here the subscript k denotes the kth spin-wave mode. In most NMR investigations of nonuniform textures only the lowest mode (k :0) is detected. In the following we shall drop the subscript fr in the eigenvalues )t¿,p and Â,,¿ and resonant frequencias a)¿,¡, and ú)r,¡,, unless stated otherwise,
and write
^ I @c\2 t": nt: \c ) '
In the bulk liquid hc-
- azt ' d*o
ct¡?
(8.108b)
]\:
L. However, in the presence of a nonuniform texture the eigenvalues decrease and thus determine the shift of the NMR signal from the bulk values. In general, the structure of the differential operators Â/ and z\8 is too complicated to allow an exact analytic solution of (8.107). Itt this case a variational approach, employing variational eigenfunctions/and g, allows a convenient approximation of the lowest eigenmode. The eigenvalues are then obtained by
Ia'r¡.N¡
nrt . J
-,
o",rr
o'rs*z\"g I Àr< I o',,r,'
(8.10e)
Hamiltonian approach The spin dynamics of the
A
phase with nonuniform textures may equally be derived within a Hamiltonian approach, in which the Leggett equations (8.8a,b) are generalized to account for nonuniform textures (Brinkman and Cross L978, Buchholtz 1978, Bromley 1980, Fetter L983, Vulovic et al. 1984, Ullah 1988). This is achieved by adding the spin part of the bending freeenergy density (7.20) to the effective Hamiltonian (8.1). Using the definition of the dipole healing length Ço given in (7.27) and the expression (8.20) for the longitudinal resonant frequency Qo, this effective Hamiltonian takes the form
H"tr: ây'Sx-'S
-
yS
.
H
- Ldz'o#{r, .h, -
ELK",,,) tft .v)d,l'
¡t
- ELR",,)i* tft xv¡,â*l'}, (8.110)
8
Spin Dynamics
where K,o,,,, are dimensionless constants of order unity, given in terms of the components of the superfluid spin density tensor by R,or,r: (hl2m)2p"ou.,,(y'ld2'oXE'ù. By employing (8.6), we then find that the equation of motion for the spin density S(r), given by (8.8a) for the uniform situation, acquires the following gradient terms:
i : ys x H * ïs¿'^#{(á x îl<¿. î¡ + =
63K.orr[ ¿
*
ÇfiK"o, @
x
Q
l{t2ttl].
x
(8. 111)
The equation of motion for ã, (8.Sb), remains unchanged, up to terms of the order of the dipole coupling Âo. The linear spin dynamics is^obtained by expanding the vector á about the equilibrium texture do, d- do- d', and linearizing in the fluctuating part d' . The latter may be expressed as
:
g(âox 2) + ft, (8.112) i.e. d' is composed of a transverse part and a longitudinal part relative to H, described by the fluctuations g and / respectively, in direct analogy to the discussion above (see (8.L02)). The fluctuating paft d' will be excited by a weak RF field ¡¡"-iat : (H| * H'"2)e-'-' where H|: H' x Ê is the component perpendicular to the static field ^ÉI ll â. fne equations of motion for S and ã, (8.111) and (8.8b), are now combined by eliminating S. This leads to two coupled linear inhomogeneous equations lor f and g. static external magnetic freld with @L>> d)6 the In the case of a ^strong is confined to the (x, y) plane (0o: Ln) vector âo equilibrium d,
(8.113) d6: coS üo î * sin tþo i, in which case the two equations decouple (cf. the discussion below (8.103d)). In particular, the equation for B, relevant in the case of transverse NMR, becomes (Fetter 1983, 1986)
(D + V)g
: * Wl - (rtr+ Ë¿Á-
dl'o)lg
-
iø'.yH',
- at-TH',j, (8.114)
where D is a differential operator defined by
D and
- - EtofK"orv2 + (K,o,, - R,or)(v . ixi
.
v)l
vt: -t(î " âù, + t?+ EDlR"o,(i xv4)' + K,p,(i .vrtùtlj
(8.1ls) (8.116)
Equation (8.114) resembles the Schrödinger equation for a particle moving in an attractive potential %. The general solution may be expressed in terms of the eigenstates 1þr* and the ("energy") eigenvalues ^8,,¿ of the homogeneous problem: (D + V)!tr.o:E,.*l)r.t (8.1,L7)
Note that the "local-oscillator" model is obtained from (8.117) in the limit
8.4
Texture-Induced Magnetic Resonance Phenomena
371
where 6o is much smaller than the spatial variation of the texture, such that D+0. The eigenvalues.E,,¿ describe the frequency shift of. the kth
transverse spin-wave mode,
@?,*: o?+ (t
+ nr)d*o,
(8.118)
relative to the resonance of the uniform bulk liquid, for which Et:0. Note that Er,¡, is related to the eigenvalue Lr.o, introduced in (8.107b) and (8.108b), by
Et,*:
)rr.o
- l.
(8.11e)
Since the potential V, is attractive, the lowest eigenvalues ^E,,¿ are negative, corresponding to bound states. The corresponding frequency shift is negative too. A turther simplification of (8.112) is achieved by assuming 4, (a.113), to be uniform and along say (Fetter LgB3, 1986). In the case of a strong magnetic field ^EI ll â considered here, this approximation can be expected tõ be obeyed rather well throughout most of the superfluid, except for regions of extension r < max (8", EÐ where may deviate from its asyrnptotic r¡niform configuration (for example in the soft core of the high-field vortex (Volovik and Hakonen 1-981) in the rotating superfluid). Then the potential Vr, (8.LI6), reduces to
i,
I
U: -(4+2r). A
convenient approximation by the variational expression
e'rl atr[("o, lvwl'* Âr{
(8.120)
for the lowest eigenvalue (K,o,,
- R.o,) li.v
\!1, +
(k:0)
E;'v,lvlrl
is provided
,
(8.12r)
Ia3rP¡' where the test functions W are normalizable functions. The higher eigenstates are usually difficult to detect. The longitudinal shifts may be derived similarly (Vulovic et al. 1984). The homogeneous effective Schrödinger equation has the form
(D + Vr)rl,r,o: Ee,últe,*, where now (with the approximation
ão: i)
vr:-(Ztj+tl) and,
Er,o is related to
)u¿,¡,,
(8.122)
(8.r23)
introduced in (8.107a) and (g.10ga), by
Ec.r: ]tr.o-L.
(8.124)
In the uniform bulk liquid the resonance is unshifted, so that E¿:0. The theory described above will now be applied to determine the NMR response of several characteristic non-uniform textures in the A phase.
8
372 p
ate
g eo m
etry
Pa ra
I I el -
The
i texture in a slab extending from t : -àL to z: ]L,
I
Spin Dynamics
subjected to a magnetic field in the z direction normal to the walls, has been discussed in Section 7.9 In particular, for a strong magnetic field, â li"t in the plane of the slab, say along i. The resulting nontrivial Í texture for L > 5o (ìee Fig. 7.30) is planar, i.e. parallel to the (.r, z) plane. The detailed form of the texture has been determined approximately by a variational method (Bruinsma and Maki 1979a), i.e. assuming the form
sinB:ksn(q, lk),
(8.12s)
where þ(z) is the angle made by I with the surface normal, sn is a Jacobian elliptic function (Abramowitz and Stegun 1965) with ft : sin B(0), and 4 is a variational parameter; B(0) and 4 are related by the condition K(k) : g, where K(k) is a complete elliptic integral. , To determine the NMR shifts caused by the texture, we have to obtain the operators A/ and z\8 in (8.107). This is done by expanding (7.183) in terms of fluctuations / and g of the vector
/ : (sin (90 + g) cos (rt,o + Ð, sin (gs + g) sin (rþo + f), cos (06 + S)) g.I26a) (8.126b) - ão- G(f'+ s'), f, -s) around the equilibrium texture âo: i, i.e. gs : âfr, rþo:0. The operators
zV and z\8 are thus found as
: -Eïifrt * As : -ELt[f t * N
sin2
B)#]
sin2 B)
.
#] -
sinz B,
(8.127a)
cos2þ.
(8.127b)
a
detailed treatment, the qualitative behaviour of the frequency eigenvalues îu¿ and Â., (8.108b), may easily be understood: for large plate separation (L >> 6o), the NMR frequencies approach their bulk values, i.e. )r¿- )tr: L. As L is decreased, the eigenvalues decrease, too. At L- E, the i texture resembles the so-called "bending i soliton" (Maki and Kumar 1977d), for which ]rc:0.LL6 and Lt:0.687. For even smaller L (L<< 6o), when the uniform configuration î t ¿ is assumed, the eigenvalues approach the limiting values Le :0 and Â,- -1. The above problem has also been considered by Fetter (1977), who used a different approximation for calculating the I texture and who determined the NMR response in the local-oscillator approximation, i.e. where the local spin precession is assumed to be given by decoupled local oscillators (see the beginning of
Even without
Section 8.4).
A detailed discussion of the NMR response of superfluid 3He confined in
very thin slabs (width of the order of the coherence length 5(f)) has been presented by Ullah (1988), who finds that NMR can be a very sensitive probe of the order parameter in such a geometry.
8.4
Texture-Induced Magnetic Resonance phenomena
373
Cylindrieal geometry The energetically most favourable i textures in a long narrow cylinder have been discussed in section 7.9. There it was found that, foi radii R> L0 E(T), the Mermin-Ho (MH) texture, paramerrized by (j.lgÐ, is srable. only for very narrow cylinders (R < 10f) is a disgyration-type táxture ;it; (7.197) or (7.199), energetically favoured. Their NMR signals have been calculated by Bruinsma and Maki (1978), as will be discussed below (see also Fishman 1978). In the following we restrict the discussion to the MH texture in the dipole-locked regime, Tle corresponding bending free energy, in terms of the angle 2¿madeby î:áwittr the cylindér axis, ias deriv"oîn (7.r92).The exact solution X@) is given by (7.193), where p_ erlR. Hère r is the distance from the cylinder axis and u:[2](3 - St,1lr^in weak coupling. The azimuthal angle is _denoted by g. The operators zV and ¡\s in 1's.ro1¡ to, this case are given by
ôl z a2ì * ,, ¡r_ : -o\ ^.r/Eo\'l 1 al,. * cos', 1¡)\., sin' x n ) tffirta, L'tr ðr)* ifu¡
zr'
(8.128a)
Ap: -*\,/€o\'l1al + cos". ôl'2fa2 rì * zxl] 2¡)rJ + t: (8.128b) plfæ*cos R ) t; ; lr(1 For symmetry reasons, the longitudinal fluctuation / must be radially symmetric, whereas the transverse fluctuation dependr on q according tó g(p, q) : E(p)e*tt Tlr" boundary conditions on / ãnd g at the surface of the cylinder, where x:T, may be obtained by integraiirrg (g.107) over an Âã
infinitesimal interval about the surface, with the result that
gll,:^:gll,:^ :0. ðp
(8.12e)
êp
The lowest eigenvalue of the equation for
hc:
/ is obtained for / :
const, i.e. (8.130a)
L.
The lowest eigenvalue of the operator Â8 may be determined variationally and is found as
h,:
!. r(*)',
(8.130b)
with 4 : 4.14. The gorresponding values for the radial disgyratio n (7 .I97) and the "double-half' disgyration (7.r99) are also given uv-(s.130a,b) with
:5.56
and r¡ :2.44 respectively. While the preceding derivation holds for arbitrary values of folR because of the scale invariance of the MH texture, the assumption of dþole-locking has to be abandoned for EolR < 1 or for a layer of thickn"r, near the T
io
wall of the cylinder. One may estimate the latter contribution for a wide cylinder (R > 6") by noting that the maximum angle between i and ã at the
8
374
Spin Dynamics
surface is of order EolR. Therefore the dipole energy density in this layer is of order (E"ln¡' and the contribution to the eigenvalue obtained by integrating over the cross-section is of order (6"/R)'. In the opposite limit (R << 6o) the d texture is uniform. One may then argue that the longitudinal resonance frequency for a radially symmetric I texture is zero (Gould 1979).Indeed, for.a spatially independent fluctuation (S.105a), is determined solely by the potential provided the operator ^1 (6.104), so that by the dipole interaction
f,
Af
:
)te
: l?- Py
(8.131)
where /, and l, are tle components of Î in the plane perpendicular to the magnetic field, with d ll.f. In the local-oscillator model the auerag¿ of this frequency shift over the cross-section of the cylinder therefore vanishes. On the other hand, the transverse shift for cylinders with R : 1¡rm is found as \:0.20 (Bruinsma and Maki I979d). Experimental NMR studies on narrow cylinders have been performed by Saunders et al. (1978) and Gould and Lee (1978), with different results: while the first group observed temperature-independent frequency ratios and a longitudinal resonance, the second did not. It has been suggested that the discrepancy between the two measurements might be due to the presence of persistent currents in the cylinders of the first group (Truscott reTe).
NMR response of the helical texture
As discussed in Section 7.10, the uniform Íand átexture is unstable against the formation of a helical texture within a certain range of superfluid currents and magnetic frelds. Apart from the sound-attenuation experiments described in this context, NMR experiments should provide a unique probe to this texture, since the chiral symmetry of the direction of superflow is broken. Therefore the transverse resonant frequency splits into two branches depending on the polarization of the applied RF field. The NMR frequencies of the helical texture in an open system and their dependence on the external superflow and magnetic field ,ÍI ll u. have been calculated by Vollhardt et al. (1979,1981). In an open geometry the wave vector k of the helix may adjust so as to minimize the free energy. The case where ft is not free to change (".g. in a toroidal geometry) has been addressed by Bromley (1980), who derived the NMR frequencies in terms of the orientations of I and d. In the helical texture the equilibrium ã vector is characteÅzed by the angles go: const and rþo: kz (see (7.235)); in the following the subscript zero on gs and óo will be dropped. As discussed by Vollhardt et al. (1979, 1981), the equations of motion for the fluctuations / and g around þ and 0 are given by (8.103a,b). The right-hand side of (8.1.03a,b), with V : I given
8.4
Texture-Induced Magnetic Resonance phenomena
by (7.212), is obtained
y'y: 'r* ô/
as
dzro[f""sin2 g (1
+sin2
sur p cos (u _B) Ð-fsin gsinBcos(g - o'
* g,,€ sin 29 (1 + sin2 B)], yt õv" : r¿1,{s*(1 * sin2 þ) - sÍE'coszl x¡t ôg
with
,r.rrru (1 + sin2 B)
* cos Z(0 - p)l _ f,E sin20 (1 + sin2 B)),
Æ
(8.132b.
: kEo and s : zlEo.A spin-wqve solution of the form
f' I o "i(qz-at) is determined by the dispersion relation (r'- Lud}z,.){att - ,,¡7ll - (1 + a) cosz 0l- Lnez^} - [onr(L + a) _ Lrrd|tolt where (4 : qE")
(8.133'
cos2 g
:0,
(8.134)
\ r _sinBcos(B-0).,a + sin2B¡qz, Ln:-ff+(1 " I
, Lzz:
sin2B.,) h" cos20
I
-) + (1 + sin2 p)Er,l
(8.1 :8.13s)
"i"re*
:r:ri:;Ï;Íl",.ed
An equarion ,i,',î'j,l by 980) for flow wina i torus, where k is quantized by the boundary conditionq ",o*,"r1, so that I can annot not be minimized with respect to k. In the vicinity of the phase boundary between the uniforrm texture ( region I in Fig. 7 -35) and the helical texture (region II), the anngles rre and 0 are
small and (8.134) can be solved exactly: v Íi ØLQ * o), + ezo(r, + Et + zúq)lr,t x. )ø,y(L * (g. 136) where k: | + E2 - h2 is the ratio of B and 0. Making use of the fact that the longitudinal RF field couples to the q:0 mode, while the transverse RF field couples to q: lk, one obtains from (S.136) two longitudinal resonant frequencies
,:
o),
e)(,+: [rr{*o + lofr(l + o)rlr,, x }o4(l + a)
(8.137)
and four transverse resonant frequencies
@t,.*:
+ lia,?(L + o)rlr,, x. $ør(L + a), ì (8.138) @t .t : [a1, + ia¡?(t _ o)rlr,, *. la4(L * o), I i.e. two for each helicity; here r,:rc*382. To calculate the six resonant frequencies in the full stability region of the helix, (8.134) has to be solved frrd|ro
(Vollhardt et al. 1981). The resulting frequencies give a dístinct signature of the helical texture as a function of magnétic field ior given superflãw.
8
376
Spin Dynamics
The NMR response of the ellipticalty distorted helix, which may be stabilized if the magnetic field is perpendicular to the flow (H L u,, see the discussion in Section 7.10), has been determined by Lin-Liu et al. (1979) along the phase boundary to the uniform texture.
8.4.4 Dynamics of solitons ¡n the A phase The term "soliton" was introduced into the field of superfluid 'He by Maki and Tsuneto (1975) following their investigation of the nonlinear [ongitudinal NMR in the A phase. In this problem the relative phase Ô:24t between the up- and down-spin population oscillates^("internal Josephson effect", see (8.72)); here p is the angle between d and / in the plane perpendicular to,õL The equation of motion for @ may be derived from the given in terms of (8.101) and (7.22d), (6.10a) with l-agìangian L, (8.88), î fi-¡. For a purely one-dimensional variation of A (along i, say) and neglecting dissipation, the Euler-Lagrange equation for @(x, r) is
#-,?,#- -dr'osin
@,
(8.13e)
c.p
d2o|o is the spin-wave velocity; see (LL.20). This is the socalled "sine-Gôrdon" equation, which occurs in many areas in physics (see e.g. Whitham 1974, Maki 1982). Equation (S.139) has several periodic solutionS, One of which represents a "sOliton" (also called "kink", "hump" etc. in other areas of physics), i.". a spatially confined solution where
tanl¿
:
(8.140)
e-u
with
I x-ut u: {¿oæW'
1)1c"o'
(8.141)
where u is an arbitrarY velocitY. This describes a single peak moving in the x direction. There are infinitely many solutions to (8.139), since þ- Q *ZnN, N eZ, also solves (8.139). The properties of solitons, in particular their topologicical stability, have been discussed in detail in Section 7-8. 3He-A is therefore seen to allow spatially confined The spin dynamics of magnetiõ excitations (Maki and Tsuneto 1975, Maki and Ebisawa L976c, Vtat
8.4
Texture-Induced Magnetic Resonance Phenomena
377
often referred to as "solitons" too, since they are also spatially confined topologically stable solutions of a nonlinear differential equation. Creation and propagation of solitons
One expects that pure â so[tons are created in a typical field turn-off or pulse experiment (Maki and Kumar I976b, Kurkijärvi 1978): in such an experiment the vector d is made to rotate in the plane perpendicular to the static magnetic freld inside the magnetic coil, while it remains static outside. The d texture inside the coil is thereby given a frnite winding number relative to the outside, as a consequence of which closely packed soliton-like excitations are created. The excitations will propagate with a velocity smaller than, but of the order of, the spin-wave velocity (see (8.1a1)!-a result that has been verified experimentally by a "time-of-flight" experiment for solitons (Gould et al. 1980). At temperatures that are not too low the solitons are slowed down by interaction with thermal excitations. This effect may be taken into account by adding a spin-diffusion term to the equation of motion (Maki and Ebisawa I976a). Once a soliton has come to rest after an estimated time of about 10-a s, it starts to locally distort the Î texture (which was previously assumed to be uniform) via the dipolar coupling. This is energetically favourable since the d texture is "stiffer" than the î texture (see. (7.21a)) and hence the ã texture wants to reduce its bending energy at the expense of the Î texture. It has been proposed that the array of high-density d solitons is eventually transformed into a regular one-dimensional lattice of high-density composite solitons, (7.163), which finally expands into a dilute "soliton gas" owing to mutual repulsion of the domain walls (Maki and Kumar I977a,b, 1978, Maki I977b). In the composite soliton, which has a considerably lower energy than the â sohton, ç.163), most of the reorientation from âtlî to áll -iis done by the vector L This reorientation is acted against by surface orientation effects and the presence of other solitons. These effects are difficult to estimate and therefore the explicit path for the decay of the â soHton into the composite soliton remains an open question.
8.4.5 NMR signature of solitons
¡n the A phase
As in Section 7.8, we assume a static magnetic field ^[I ll â to confine the equilibrium configuration of âand ito the (*,y) plane (see below (7.1a9)). \Ve introduce small fluctuations
/ and g around the equilibrium
angles of the describe in-plane and
soliton, f :tþ-Üo andg-e- 00, where/andg out-of-plane oscillations, respectively. The weak-coupling expressions for the operators zV and z\8, (8.105), valid for all temperatures and including Fermi-liquid corrections, have been derived by Maki and Kumar (1978) and Maki (1986a) on the basis of the free energy (7.1,53) with (7.20) and (7.23)
8
Spin Dynamics
AS
A/
: -zE"* [,t
Â8: -zEL{* [,t
o')"9]
.
L -2sin2 (ø
- o')*] . $ -
a'),t'3]
- rt)ì;,
$.taZa)
+.o,' (a -
tt),
(8.142b)
:
Î, . ,, with k the surface normal to the domain wall and a2 : (L - D(Ê, cos ø + É, sin ø)2 with L: p"pul p"pr. For the pure d soliton, where ø:0 and tp is given by (7.157) with E3 : ZIt - ÍL - L)kilED so that rp" : (6;)-1 sin2 rp, (B.t4Za,V) further reduce to
where ,
N :Âs:
-Eg#. t -Zsech2 (å)
(8.143)
Hence the operators zV and As are found to be identical in this case. The corresponding eigenvalue equations (8.107) have only one bound state, with the eigenvalues )r¿ and Â, and the eigenfunctions / and g given by
)rc: hr:0,
f:
so
sech2
(8.1aaa)
(*t)
(8,144b)
Hence the d soliton is seen to produce no longitudinal magnetic resonance, and the transverse response occurs at the Larmor frequency. This is a conseqrrence of the assumed uniformity of the i texture, over which the â soliton may then slide freely. For the twist composite soliton (7.t62b), where the surface normal fr is along the external magnetic field H (ã: 0) and which has the lowest energy of all domain walls, the operators zV and Â8 are found from (8.142a,b) as
N
: -zEtr#. t -2sech2 r¡2,
Às: -2EL#.1 -
[1
+ iotlù-zlsechz
(8.1a5a)
r¡z-
(8.14sb)
Here we have made use of the general analytic solution for the twist composite soliton, valid at all temperatures (Maki and Kumar t978), 1
u:u+-qt:const, K u:
:2tan-r (a*"),
s
-
q
/ I t x\1t2 _l:_j__:l 5;"
1þ
(8.1a6a) (8.146b)
with
\¿K/
(8.147)
8.4
Texture-Induced Magnetic Resonance phenomena
*:+;ü('-#*ll-.*
Fì 1+å^Fì(l -p,,rlp)
In the Ginzburg-Landau regime one has * recovered.
Again the operators
371
fl
(s148)
- ï and the solution (7 .I6Zb) i:
are found to have only a single boun< state, characterized by the exact eigenfunctions (Maki 19s6a) (8. L42a,b)
with eigenvalues ),,
where
:
f (z) * Au sech"- r¡2,
(8.149a'
SQ) o Au
(8.14eb,
1
z*{[(1
sech4 r¡z
+ r)(1 +
9*)lt,r- 1 - 3r],
L,: 1+K'
(8.1sOa)
(8.1s0b)
(!+e"'"'-lJ'I u:!l2L\1 **)
(8.lsla)
T(
F:1**'
(8.1s1b)
A :l 't f''' '^d Ln(|, u¡ J '
(8.1s2)
with B(x, y) the beta function. In particular, in the Ginzburg-Landau regime (f < 7i) the eigenvalues are given by
- he: +Í(65)''' -71)tr: t. with
0.53L,
(8.153a)
(8.1s3b)
decreasing temperature, the parameter r increases owing to the large Landau parameter Fì; at T:0 it would be r:È(L+år:)¡(r+åFî) - l-2. The eigenvalues ]rr,, therefore decrease with decreasing temperature. The temperature dependence of the longitudinal and transverse NMR modes of this soliton in the immediate vicinity of the transition into the A1 phase has been investigated by Kharadze and smith (197g). The intensities of the satellite resonances are proportional to the squares of the integrated bound spin-wave functions. Hence the intensity corresponds roughly to that of the total volume of the domain wall, being given by the cross-section ø(Ê) times the thickness 4-t. For other composite solitons, the eigenvalues for the satellite frequencies have to be obtained numerically. In particular, the NMR satellites of the splay composite soliton, with surface normal perpendicular to H, are found to be characterized by )rc:0.40 and Â,:0.67 in the Ginzburg-Landau regime.
8
380
Spin Dynamics
As discussed by Maki and Kumar (1978), the NMR satellites observed by Avenel et al. (L9i5) and Goutd and Lee (L976) (see also Kokko et al. 1978a) may be interpreted as the signature of composite solitons, namely of the twist composite soliton in the case of the longitudinal resonance experiment and of thè splay composite soliton in the transverse case. The existeuce of two differ"ni kittdr of solitons in these two experimental situations is indeed made plausible by the fact that Gould and Lee (1976) rotated the stationary ,rrugn"ìi" field, iather than the AC field, by 90' to perform the two exp"eriments, thereby stabilizing the twist and splay composite solitons respectively as discussed below (7-163).
The soliton lattice
In a field-turn-off or tipping experiment a large number of domain walls in the â-vector field are expected to arise. A possible scenario for the decay of these â solitons into an array of composite solitons has been discussed above. Assuming this lattice of domain walls to be one-dimensional, the spatial configuraiion .un be calculated (see (7.L65) for the twist composite ,ðhton latticã). Since the soliton lattice is expected to expand as a result of the mutual repulsion of the solitons, the dependence of the frequencies of the NMR satellites and their intensities on the spacing of the domain walls gives important information,(Bruinsma and Maki I979c). The oþerators zV and As for this case are still determined by (8.1'42a,b), the equiiibrium confrguration being given by (7.165) for the twist composite soüto; lattice' In theiase of the period LolEo* æ' the eigenvalue equation for the longitudinal resonance reduces to (8.145a), with L€ given by (8.150a). poi trlEo(( 1, wheD u- u - rþ has the form (7.168), it becomes a Mathieu equation, which picks out the spin-wave mode with zero wave vector; in t-his limit )rr: 0. For intermediate values o1 LolEp, â vâfiâtional A calculation of the ansatz for f(z) 'themay be employed to calculate A.¿.packed domain walls signal shows that for densely intensity of (Lolïo
8.4
Texture-Induced Magnetic Resonance Phenomena
381
of the satellite change with soliton separation Lu. A similar effect occurs when the spatial extension of a single soliton changes, for example during the process of soliton formation (Vollhardt and Kumar 1978). As discussed by Bruinsma and Maki (1979c) and Nakahara and Maki (1983), the above results for the soliton lattice seem to explain the findings of the field-tipping experiment of Gould et al. (1980). Immediately after the magnetization has been tipped by a large angle, a transverse resonance with negative shift, consistent with a rotation of á in the (x, y) ptane (see (8.78)), was observed in this experiment. While this resonance disappeared on a timescale of order 10-2 s, a new resonance at frequency û) @r*0.54d22^12røL appeared with increasing intensity, which soon afterwards exhausted the total absorption. The frequency of this satellite was then found to increase with time, arriving at an asymptotic frequency consistent with a single splay composite soliton on a timescale of order seconds or minutes. This time evolution of the satellite frequency and the intensity is indeed what one would expect if the nonequilibrium precession of d transformed into a dense splay composite soliton lattice, which then expanded in space until the separation was large enough for the concept of separate solitons to apply. In a similar experiment by Giannetta et al. (1981) the comparison with theory suggests that an isolated composite soliton has been observed (Kharadze and Maki 1982, Nakahara and Maki 1983). This may be a result of the time lapse between the large-angle tipping and the subsequent NMR measurements in this experiment, during which the soliton lattice had perhaps already expanded. To investigate the creation, propagation and metamorphosis of solitons,
a detailed "time-of-flight" experiment between two collinear coils was performed by Bartolac et aI. (1981) (see also Bartolac 1980). In order to control the texture, these authors confined the liquid to a stack of sheets each separated by 17 pm from the next. The vector I was thereby expected to be fixed perpendicular to the surfaces, so that the â soHtons created by the tipping could not transfer their winding to the i texture, i.e. could not decay. Since d solitons do not produce any NMR shifts ()tc: Â,:0), one would expect the evolution of the satellite frequencies to be drastically different from that in the bulk case. However, the satellite frequencies measured in this experiment were found to be essentially the same as in the unrestricted geometry. This clearly indicates that they involve a strongly nonuniform I texture, although this seems to be in conflict with the orientational effect due to the walls in the slabs. Perhaps the spacing of 17 ¡tm between the plates was still too wide to clamp i rigidty, since it is about twice as large as 6p (see Section 7.9.3). Alternatively, the Î texture in the slabs could have been a priori nonuniform, i.e. there could have existed regions with Î pointing up or down, separated by a two-dimensional domain wall, which were formed when the sample was cooled into the A phase. The boundaries would then have provided a potential for the vector â to give rise to NMR satellites. In this case an average value of Â,:0.5 might result
8
Spin Dynamics
(Sonin 1983b). Nevertheless, the dynamics of the soliton propagation and the time dependence of the NMR signal still await an explanation. Solitons in the presence of superflow
Here we restrict our discussion to the case .ãl ll u. ll â with surface normal fr.112, which corresponds to the pure twist composite soliton in the limit of zero external superflow (see Section 7.10). Since the flow induces a finite component of f along ,EI<xpressed by the equilibrium configuration (7.252) with B * Tn-the operators A/ and As take slightly more complicated forms than in (8.142a,b). In the Ginzburg-Landau regime an expansion of the generalized free energy (7.232) in terms of the fluctuations f and g yields
N
d f : -EL*ftt
Âs:
dl
+sin2B)*l .sin2B ÍL-zsin2(ø
-*{#[tt
*
+ sin2 B
(a
cosz
sin2 B)
-
4ù
- qù|,
(8.úaa)
p)!,4 #] . (1 + sin2
-
cos2
(8.1s4b)
þ.
The corresponding eigenvalue equations cannot be solved in closed form, but may be treated variationally (Vollhardt and Maki I979b). For increasing superflow p, )t¿ is found to stay almost constant at the value given by (8.153a), while Â, decreases slightly from its value 0.8 at p:0. At the critical current ptt:0.45, where the twist solution becomes unstable with respect to a purely planar, bendlike i texture (n : ,lr: 0), ¿ jumps upwards to a value of about 0.7, while Â, drops abruptly to a very small value (Â,-0.1). Further increase of the superflow with p lprr:0.625 brings little change in h¿ and l.r. Flow- i nduced sol iton I attice
For superflow p ) p,2, the uniform texture \ryas found to become unstable against the spontaneous formation of a flow-induced domain-wall lattice (see Section 7.10). The operators zV and As are then obtained from (8.154a,b), with e: q):0, as
A/
: -Eïf,ftt *
sin2 B)
#] . sin2 B,
As: -E'"i[tt * sin2 B)#] -
L
+2sinz
(8.155a)
B.
(8.lssb)
The lowest eigenvalues for )r¿ and )rrmay be obtained by explicit integration of (8.155a,b) (Vollhardt and Maki 1979a). Assuming that one deals with a uniþrm texture f.or p 1P tt, characterized by )tc : îrr: 1, the eigenvalues
8.4
Texture-Induced Magnetic Resonance Phenomena
383
are found to drop sharply (i.e. continuously but with infinite slope) &t p,2, with Â. becoming negatiue, thus indicating the appearance of the soliton lattice. Then both )r¿ and Â, decrease monotonically. At p:2''', where the
uniform texture becomes unstable, the eigenvalues have reached )uc:0.43 and Â,: -0.15.
8.4.6 General theory of texture-induced NMR shifts in the B phase The general spin dynamics for the B phase in the presence of nonuniform textures may be formulated by the same methods discussed in detail in the case of the A phase. Such a general approach was first derived by Maki (1975b) within a Lagrangian formulation (see below (8.S8)). In the following we shall discuss the equivalent Hamiltonian approach along the lines presented by Fetter (1986). To obtain the equation of motion for the spin S in the presence of a nonuniform texture, the bending energy has to be included. The linearized equation, valid for small deviations from equilibrium, may conveniently be obtained by paramefiizing the order parameter in terms of 0|, the three angles of rotation about the Cartesian axes in spin space
Ar"j:
)
R""1e"¡a!r.
(8.1s6)
Here Al, describes the equilibrium texture and the rotation matrix ß
is
expanded as
R""(0') I
:
õ*"
-)
eu"^?'^+ +f0't"0î
ô""(0")'l + . . .
-
.
(8.157)
(In the following the superrrrip, "s" on 9i will be dropped.) The angular variables 0u are taken as operators representing collective quantum variables. They are canonically conjugate to the spin density operators Sr(r), the generators of local rotations, i.e.
1.0*(r),,$(r')l :ihõr"ô(,
-
r'),
(8.1s8)
while they commute with one another. Written in terms of the variables 0u, the dipole energy density (6.107) takes the form (Brinkman and Smith 1974)
Lro=:#uru. o)',
(8.1se)
where the defrnition of fJs in (8.24) has been employed. Substituting (8.156) into the general expression (7.17) for the gradient energy in the GinzburgLandau regime (where, again, dpj:3ttzAlT¡A*,¡, two different contributions are seen to arise: those involving gradients of 0p and of the equilibrium texture Aor, respectively. Assuming the equilibrium fi texture to
8
Spin Dynamics
vary on the scale of En (which is much longer than the scale for variations in 0*, given by 5") and that the texture is dipole-locked, i.e. that the spin-orbit rotation angle is fixed at 0: 0y, y.ta may neglect gradients of Aor,. The resulting bending energy expression is given by (Smith et al. 1977, Brinkman and Cross L978, Buchholtz 1978) "fu",,¿
where
:
)4 "'"el {u ;
(y *0")'
- K' [ì
^t"
(v,0")]
],
(s. 160)
R': Kt* K3 and K:ZKt+ K' and the dimensionless parameters
terms of the coefficients 4. in (7.L7) as R¡: K¡Az(ytlXA'"83). In the weak-coupling Ginzburg-Landau limit K:€ and R;- !. The tensor quantity R!, is the rotation matrix characterizing the B phase at the dipole-locked rotation angle 0: 0t It contains the information about the equilibrium texture (see (6.106)):
Rr,
R, and K3 are defined in
nL"(")
:
- sftt ñ"+ (1s)1/2 4 ,*^o^].
å[u-"
(8.161)
The equation of motion for the spin density S(r) is derived by calculating the commutator of S with the total effective Hamiltonian iftS
:
lS, äy'S . X-r . S - yH. S +,fu""¿
* Â/ol
(8.162)
using (8.158). The result is
i : ys x H + 4 o,"gl{-K vze +ri'(FLv)[(ßLÐ . 0]] .þs*"(ñ.0)ñ.
(8.163)
The equation of motion for the angular variables 0u is not affected by the bending energy (to order Âp) and is still given by (8.8b), which translates into Ò
- -r(,
-i')
(8.164)
We mention in passing that a particular limit of (8.163) and (8.164), coíresponding to the so-called "double sine-Gordon equation", has been studied as an example of a nonintegrable partial differential equation yielding soliton solutions (Kitchenside et al. L978, I979a,b). In the situation considered here, a weak RF fietd ¡[r"-iat in the presence of a static uniform field.tIe along â excites oscillations of the texture about its equilibrium configuration. The dynamical equation for the ¡rth component of. 0,0u, is obtained by taking the time derivative of (8.164) and eliminating,S by means of (8.L62), yielding
#">
(Du" + ñpfiu
-
ôr"u)0"
- (r' -
a'ùeu +
iia,lø,y(0
x 2)u: it¡yHL. (8.16s)
8.4
Texture-Induced Magnetic Resonance Phenomena
r
Here 0* and ñu are functions of with components D*u
:
and
D"" is a tensor differential operator
E'rl -o-"xvz + K' L
) (nl^v^Xnh,v^,)1. (8.166)
j
LL'
The last term on the left-hand side of (8.165) ma¿,pe diagonalized byr projecting onto the spherical unit vectors û+:r-uz(i f ii), ño:2 u, 0*: 0 . û*, D^*,: û^Dûi",. Here m: *, -,0 correspond to the polanzation of the vector field. The remaining off-diagonal terms in D^^, are small for suffciently largg magnetic fields, such that (A"lr)r 12î (Theodorakis and Fetter 1983). In this case (8.165) separates intJ'three: inhomogeneouSdifferentialequationsoftheform
(D- + V^)0^: with D-
r/to)(ÐyäÍ@t -
Al)0^ + iayHi.l. (8.167)
-
and v^ -1 + lfr^l'. The boundary conditions on 0,, at the surface of the sample are given by the requirement that there be no spin supercurrent through the surface (Brinkman and Cross 1978). This gives rise to the Neumann boundary condition on 0^
=D^^
3
'Y0^:
g,
(8.168)
where 3(r) is the surface normal. Equation (8.167) is seen to be the analogue to (8.114) valid in the A phase, and indeed has the same structure: the homogeneous part of the differential equation (8.167) for 0- has the form of a Schrödinger equation, with D- and V^ playing the roles of the kinetic and potential energy respectively. As in the A phase (see (8.117), (8.L22)) rhe solurion of rhe inhomogeneous equation may be expressed in terms of the eigenfunctions tþ^,¡, and the ("energy" ) eigenvalues E*,¡, of the homogeneous problem
(D- + V^)rlr*,¡: E^,r1þ*,r,
(8.16e)
subject to the same boundary conditions (S.168). Assuming the eigenfunctions !t*,* to form a complete set of orthonormal functions, the solution 0^(r) may be expanded in terms of tp^.p as
0*(r):2
c^,*uþ^,t.
(8.170)
k
Substituting (8.170) into (8.167), the coefficients cm.k are obtained by multiplication of the equation by 1þ^.*,and integration over r:
Cm,k:
I "Y
d'r !,î,,o(r)H^(r) .
(8.171)
As usual, an infinitesimal imaginary part has been added to the frequency (ot- ø * i0), expressing the fact that the perturbing RF field has been
8
Spin Dynamics
on
adiabatically slowly, so that causality is guaranteed. As expected, the magnitude of c^,¡, (and thereby the contribution of rþ*,* to 0-) is determined by the overlap of the kth mode with the RF field, the resonance being strong if the external frequency ø is near the eigenfrequency Øm,k.The latter is itself determined by the eigenvalue E*.pvia switched
aT,r- t/ta)*,¡ra)t- Al:
The excitations correspond to spin waves. The quantity observed in an NMR experiment is the rate absorption. With M: TS and using (8.164), the latter is given by
#: +, Id3r Im H'*
.
(8.172)
E^,*Qzs.
of power
M(r)
: -:L armli,o: I
(8.173a)
o', e^e)1,
(8.173b)
where the RF field has been assumed to be uniform. By expressing 0*(r) in terms of (8.L71) and making use of (x - i0)-1 : Ø(Ilx) + inô(x), where Ø is the principal-value integral, (8.173) reduces to
Y-Lnx,r'lZ'*lr}
ô(a - lll(Dr * e)*,* )t
2a-m a)¡,
I
I o'r rA^*¡)l ,
(8.174)
where the RF field has been assumed to be weak. As expected, the main resonance line (k :0) is shifted and additional satellite peaks appear at frequencies corresponding to the excited-state frequencies @m,k. The lowest eigenvalue E^,s may be conveniently determined by employing a variational method, which makes use of the Hermiticity and the existence of a lower bound of the spectrum of the operator D^ * V*. A tnal function tll, satisfying the boundary condition (8.168), therefore yietds
o', w*(D^ + v^)w I E^,o4
r^d'r
I
J
For longitudinal NMR
(8.17s)
lVl"
(*:0) in the uniform bulk liquid, where Do:
Vo=0, the eigenvalue Es is zero. This is in direct analogy with the A phase, where the corresponding eigenvalua Ec, (8.124), is also zero for the uniform bulk liquid. Pa ra
I I el -
p
I
ate
g eo m
etry
As a first example, let us discuss the NMR signal from the B phase in a parallel-plate geometry as considered by Smith et aI. (1977) (see also
8.4
Texture-Induced Magnetic Resonance Phenomena
Lin-Liu and Maki L978), with a strong static magnetic field II ll.û parallel to the plates. The equilibrium ñ texture in this geometry has been discussed in Section 7.9.lnthis case the magnetization and the variables g, depend only on lzl, the coordinate along the normal to the plates (lzl < å¿). In the centre of the slab, i.e. for small values of. lzl, ñ tends to lie along the direction of the magnetic field. The orientation changes into that favoured by the magnetic surface energy, i.e. ft:5-trz1L,3r/2,1) in a layer of thickness En at the surfaces. For transverse NMR (*: +), the eigenvalue equation (8.169) reduces to an ordinary differential equation, given by (8.176)
where RL*,: û*Rt2 and here û*:2-L/2(î + i¿). The absolute square of
R!,
is calculated
where
ñ":
as
lRt*,lt --
1.
+
5Í5ñ21ñ?
-
2û.7
+ 3ñ1 -
2(15)r/2ñ,ñyñ,f,
(5.177)
ñi + ñ2".
For large plate separation (L> Er), the ñ texture is uniformly oriented along H over most of the sample. In this situation the solution of (8.176) is characterized by periodic oscillations of the magnetization. The detection of excited states in an NMR experiment is then hardly possible since the integrated magnetization averages to zero. By contrast, for smaller plate sepaiation (L< Eò, ñ!: (ñ . H)'varies quadratically with z (see (7.176b)). If one now approximates lRt*,lt in (8.176)) bV a constant then (8.176) corresponds to the equation for a harmonic oscillator. The eigenvalues are therefore evenly spaced. The intensity of the excited modes given by (8.174) decreases as ft-l. A detailed numerical calculation of both the texture and the eigenstates is found to give excellent agreement with experiment (Osherotr L977, Osheroff et al. 1977; see also the earlier work of Osheroff et al. 1975). This also shows that the qualitative treatment discussed above already includes the essential physics. The agreement may 3He-8. be taken as a proof that spin waves exist in Cylindrical geometry As a second example, let us discuss the NMR signal from the B phase in a long circular cylinder of radius R with a magnetic field along the cylinder axis. For R ) å,, the equilibrium texture in this geometry (see (7.182)) may be written as (8.178a) ñ+: 2-''t(ñ* i. iñy)
- 2-tt2 Sin B a*i(u+v), ño: ñr: cos É.
(8.178b) (8.178c)
follows that the effective attractive potential V^: -L+ ln^lt is axisymmetric and given by V+: -L+ lsinz B and Vo: -sinz þ respectively.
It
8
388
Spin Dynamics
Eigenstates of (8.167) in an axisymmetric potential can be classified by their azimuthal angular dependence e'tq.In the usual case of an axisymmetric modes are excited. RF field only the For the transverse case (m: +), the eigenvalue equation (8.169) reduces to the following ordinary linear second-order differential equation:
/:0
(-R
+
R' tR'*,t')ELt:#
('#)] t,*,*-
(1
- å sin2 þ),t,*,o:
E*,t 1þ*,r, (8.17e)
where RL*.: û *Rtî (Hakonen and Volovik 1982, Maki and Nakahara L983, Fetter and Theodorakis L983, Jacobsen and Smith 1983). The potential -1 + ] sin2 B has a quadratic dependence on r near the centre, with very small curvature, and rises to its boundary value of -t only within a surface layer of thickness f¡7 << R. As a first crude approximation, one may therefore replace the potential by a harmonic one, with å sin2 B (rlLr)', and IRL*,12 by its value at r:0, lRt*,|':]. This leaves us with the "Schrödinger equation" for a two-dimensional harmonic oscillator, a standard problem in quantum mechanics. The only characteristic length of the problem is given by r{ÇeLr)''', i... the eigenfunctions vary as rf 11. The eigenvalues are evenly spaced and given by
E+,* :
E
-L + 2(k
- äK',¡''ti ' L1 tt + 1). '
(8.180)
The corresponding NMR frequencies are then determined by (8.172).
Multiplets with odd ft do not contain s-wave states (/:0) and therefore do not couple. Within the approximation discussed here, the intensities of the satellite resonances are found to be independent of k. The ratio of the intensity of the kth normal mode, relative to the uniform-texture signal from the area frRz, is given by
:\;//96\tt2 EoL, l.rrriform ñ
(8.181)
NMR experiments in stationary cylinders do indeed show a set of nearly equally spaced absorption peaks above the Larmor frequency, but the intensity falls off as a function of increasing ft (Hakonen et al. I983a). These deviations have been explained by a numerical solution of (8.179), using the exact equilibrium ¡ô texture for various stationary cylinders (Jacobsen and Smith 1983). For small magnetic fields (H <Ë1s) and radii R < 106s, the "in-plane" texture (7.I77), (7.I78) is stable. This situation has been studied theoretically as well as experimentally by Brinkman et al. (L97ab).
8.5
NMR in Rotating Superfluid 3He
8.5
NMR lN ROTATING SUPERFLUTD 3He
389
The NMR response of the rotating liquid can be calculated by the.same method described above for the stationary tiquid. As discusr"d in Section 7 -6 the rotation generates a lattice of vortices parallel to the rotation axis. The effect of the vortices, as far as NMR is concerned, is to change the equilibrium texture.-This in turn modifies the operator D and the potential v in the "schrödinger equations" (8.117), (g.Lzz) and (g.169) and thereby changes the eigenvalues and eigenstates (see Fetter 1986, Salomaa and Volovik L987). As a result, the pattern of resonance lines is expected to vary as a function of angular velocity fJ, both in position and inteisity (see
Figs. 4.10 and
4.ll).
8.5.1 The A phase The NMR response in the
A phase is governed by the differential equations (8.117) and (8.L22) for the fluctuations of the vector â. For a iir,,"r, i texture, the "kinetic-energy" operator D and the potential V are fixed. The spin-wave frequencies follow from the eigenvalues of the operator D+V, whereas calculation of the intensity requires knowledge of thè eigenstates as
well.
Let us now
discuss the case of transverse NMR. In a relatively strong external magnetic fietd ,ãI ll â ttre â texture is confined to lie in the planã Perpcndicular to H and is essentially uniform throughout the sample (say, d.ll to minimize the bending energy. Likewise, the texture is alio essentially uniform, being oriented along â. Deviations of Î from uniformity only occur on a lengthscale 6o around each vortex core, which is much less than the separation between vortices (see Section 7.6).Therefore the potential -(lr+21,) in (8.LL7) consists of a regular array of attractive cylindrical potentials with radii of the order of 6o for both the nonsingular doubly quantized vortices and the singular singly quanti2ed vortices (Seppälä and Volovik 1983, Vulovic et al. 1984). The l,ow-jying stares of rhis
i)
i
V:
potential are well localized within a single potential well. It is therefore sufficient to look for the bound spin-wave states of a single vortex core. A glance at D * I/, shows that the potential has a range Çp and unit depth, which allows only a small number of bound states. Note that the potentiàl V, is not cylindrically symmetric, precluding a decomposition ir angularmomentum functions. A numerical evaluation of the Î texture and the eigenvalue equations (8.1L7) for the ground state at dz: 1 radls yields the eigenvalues E,,e: -0.5 and .8,.o : -0.02 and the intensities 0.047 and 0.56 relative to the signal of the bulk A phase for the NMR satellites of the nonsingular and the singular vortex lattices respectively (Seppälä and volovik 1983; see also Maki 1983a,b,c). Both Er., and the relative intensities are found to decrease with decreasing temperature (Vul ovic et al. Le84).
390
8
Spin Dynamics
NMR experiments in the rotating A phase, at29.3 bar pressure and for a range of ternperatures, show a satellite peak (see Fig. 4.10), which may be interpreted ai a spin-wave resonance located at the vortex cores (Seppälä et al. Ii84, Hakonen et al. 1985b). one finds that the eigenvalua Er,o derived from the observed NMR satellite shift is independent of the relative orientation of the magnetic field with respect to the rotation axis, and decreases nearly linearly with decreasing temperature, from Er,o: -0'4 at peak T" to Er,s: -0.1 at T lT":0.7. The strength of the satellite absorption váries iinearty with the rotation speed and hence with the number of vortices in the system, which proves that the satellites are associated with vortices. The relative intensity at d):'!. rad/s is 0.06 and increases with the tipping angle, in agreement with theory (Maki and Zotos 1985b)' These oüi"*utioñs point to the presence of doubly quantized nonsingular vortices, even though theoretical estimates indicate that singly quantized singular vortices shbutd be lower in energy in the high-fietd regime (seppälä and Volovik 1.983, Vulovic et al. 1934). It is interesting to note that ion-mobility experiments do indeed suggest the presence of singular vortices at magnetic nei¿s H>4G (Simo\a et al. 1937); see also Section 7.5. The effect of broadening of the NMR line due to spin-wave scattering on vortices with integer quanta of circulation has been considered by Fomin and Kamenskii (te82). Experimental results on longitudinal NMR shifts in the rotating superfluid, theoretically described by (8.122), are not available so far.
8.5.2 The B Phase In contrast with the A phase, in the B phase only the large-scale textures are of relevance for NMR properties. This is because the characteristic lengths are much longer than in the A phase. Vortices affect the ñ texture mai-nly through the change in the magnetic orientation energy. As discussed
in Settio, i.5, there are two contributions to the magnetic energy of
vortices: the first, quantified by the parameter Â, is due to the suceptibility anisotropy present in the cores of most vortices (see (7.t20)); the second, described by the parameter Hr, is due to the gyromagnetic effect (see (7.I25)). Let us recall that both  and H¡aare positive and proportional to the angular velocity fJ of the rotating container. The combined effect of both terms is to weaken the bulk magnetic orientation energy by a factor L-)v+HMIH (assuming the angle þ<
Superfluid iHe
8.5 NMR in Rotøting
¿'10
-2"
-3J
(\¡
c.¡
d)
3'10 "
c
-o
391
T/Tc o 0.66 o 0.73
v 0.80 -? 2'10 -
3
!
-? I'10"
G 3
o a
v 0
0.s
1.0
1.5
Q(rod/s) Figure
8.7
Frequency shift of the lowest spin-wave resonance in the B phase as a function- of rotation speed for various values of temperature. The full line is the theoretical result of Hakonen and Votovit< (1982). (After Hakonen et al. (1983a).)
Hakonen and Volovik (1982) and Maki and Nakahara (1983). This is indeed observed experimentally, as shown in Figs. 4.11 and 8.7, where the splitting is found to increase linearly with f) (Hakonen et al. 1983a). Furthermore, the NMR frequency shift is seen to depend on the relative orientation of the applied magnetic field with respect to the rotation axis (Hakonen et al. 1983c).
A
particularly interesting experimental finding was the discovery of a discontinuity in the spin-wave resonant frequencies at a temperature T,-0.6", as shown in Fig. 8.8 (Ikkalaet al. lg92, Hakonen et aI. 19g3a,c). The transition temperature 7î is found to be independent of angular velociiy d2, magnetic field 11 and the relative orientation of the two, and to depend only on pressure. In the low-temperature phase the gyromagnetic effect is clearly seen, i.e. the frequency shift depends on the sign of o . rL The location of the phase boundary \(P) between the two vortex states is shown in Fig. 8.9. In persistent-current experiments an abrupt change has also been observed in the critical velocity, which occurs approximately along the same line in the P-T phase diagram (Pekola et al. 1984c). Indeed, it hai been argued that the two transitions have the same origin, namely a change of the vortex core structure (Pekola et al. I984a). A ditrerent interpretation of the transition found in the flow experiment has been proposed by Thuneberg (1987b), who argues that the transitions in the two experiments are qualitatively different. Ruling out a vortex core transition or surfacerelated effects (Thuneberg 1986b), Thuneberg concludes that the transition is due to a new type of vortex, involving spin superflow rather than mass superflow around the line defect. As discussed above, rotation of the sample influences the ñ texture and thereby changes the NMR frequencies. An even stronger effect is observed
8
392
Spin Dynamícs
0.03 UI
=c =
o
ã
I
0.02
(J
c o
f o o I
o
(J
C
ó c o an o L o a 3 I
0.01
.ç oØ 0.5
T
Figure
8.8
0.7
0.6
lTc
Resonant frequencies of the three lowest spin-wave modes at29.3 bar in
rotating 'He-B (.fJ:0.6 rad/s) as a function of temperature, for magnetic field orientation parallel (O) and antiparallel (O) to the rotation axis ll. (After Hakonen et al. (1983a).)
if the magnetic field is tilted with respect to the rotation Ð(is, because in this case ñ is not parallel to H even in the bulk liquid (Bun'kov et al. 1983a,b, 1984b, Hakonen et al. 1985a). A numerical analysis of the spin-wave eigenvalue equation for several vortex cores has been performed by Hakonen et al. (L983c) in order to determine the vortex parameters  and H¡a as functions of temperature. As can be seen from Fig. 8.10,  is found to increase with decreasing temperature both above and below the vortex core transition and makes a jump of relative size 10-30Vo, which depends on the tilt angle of the magnetic field with respect to the rotation axis. In Fig. 8.11 the parameter H¡4 is plotted versus temperature. Above the transition, H¡o is seen to be very small, indicating that-if at atl-there is only a minute gyromagnetic effect associated with the vortex cores in this case. A calculation of  and H*, based on the assumption that in the
8.5
NMR in Rotating Supefiuid
3He
393
ïae
TV
:)^ tJ ¿-\)
e (l) L
)(/l
vortex
3lo
I
L
o_
r.5
1.0
Figure
8.9
Phase
2.0
2.5
3.0
Temperoture (mK) diagram of rotating. liquid 3He in a magnetic field H
(O,
H :284 G; O, ,É1:568 G), indicating a discontinuous transition betwen two vortex states in the B phase. (After Hakonen et al. (L9B3a) and Pekola et al. (1984a).)
1.0
ãA ¡¡ot_lr
-sËà4
T]
o
,ta
Ø
*'
¡<
f.
0 0.2
\P'.q
Í'
l\
0.4 0.6 0.8 T
Figure
8.L0 Vortex parameter
1.0
lTc
 charactenzing the orientational effect of axisymmetric vortices on the bulk texture in 3He-8. The values are deduced from the analysis of NMR frequency shifts for various magnitudes of the magnetic field H and orientations relative to the rotatiôn axis f2. The arrow indicates the position of the discontinuity at the vortex transition temperature. (After Salomaa and Volovik (1937).)
8
394
Spin Dynamics
0.0s
0.04
E o
0.03
l3
T
=lr -l-
0.02
0.01
o
T 0
Figure
8.L1
0.5
0.6 T lrc
t;
A 0.7
Parameter É/- describing the gyromagnetic effect of the ferromagnetic vortex cores in 3He-8. The values are extracted from the analysis of NMR frequency shifts for various orientations of the magnetic field ,EI with respect to the rotation axis f2. The solid line is obtained from a comparison of the calculated NMR frequencies with the measured spectrum in the case ^[t ll fJ. (After Hakonen et al. (1983c) and Salomaa and Volovik (1987).)
high-temperature/high-pressure phase the vortex cores are of the doublecore type whereas at low temperature the axisymmetric v-type vortex is stable (Thuneberg 1986a, 1987a, Salomaa and Volovik 1986), is in qualitative agreement with the data.
8.6
SPIN.RELAXATION PHENOMENA
Since the total spin of a strictly isolated system of individual spins is conserved, spin relaxation can only occur by coupling to other spins outside the system or by coupling to an orbital angular momentum within the system itself. In this wây, angular momentum is transferred to a heat bath, where it is dissipated. As a consequence, the spin relaxes to its equilibrium state. In liquid 'H", spin relaxation in the bulk takes place via the spin-orbit forces given by the dipole-dipole coupling of the 3He nuclei. In the normal state the spin-relaxation rate due to this coupling is proportional to the square of the dipole coupling constant (see Abragam I96L). In the limit of low temperatures the relaxation times become very long (oT-'), i.e. of order 10s s at 1-10 mK (Vollhardt and Wölfle 1981a,b). Experimentally, one usually observes shorter relaxation times (of about L02 s) due to the presence of a surface relaxation mechanism, which dominates the bulk mechanism at low T. It therefore came as a surprise when NMR
8.6
Spín-Relaxation Phenomena
395
experiments in the superfluid phases appeared to indicate a much faster spin-relaxation time, of order L0-3 s. In fact, as pointed out by Leggett and Takagi (1975) (see also Combescot and Ebisawa L974, Combescot L975a, L976, Maki and Ebisawa rg76d, Ambegaokar 1975, Bhattacharyya et al. 1975, Ambegaokar and Levy Lg76), the increased relaxation rate may be understood on the basis of Leggetús theory of spin dynamics, if one allows both the condensate and the quasiparticle systems to have their own dynamics. Following Leggett and Takagi (L975, rg77), we define the partial spin polarizations of the Cooper-pair condensate ,SP and of the quasiparticle system (normal-fluid component) So, which add up to the totãl spin polarization,S: (8.1s2) ^S: ^SP + Sc. Consider a change in the applied external magnetic field. In the hydrodynamic regime the spins of the pair condensate will adjust to the new ñeld within a time period of order hl A, i.e. essentially instantaneously compared with the quasiparticle relaxation time. We may then define an "instantaneous" value ä¡r,r, of the effective magnetic field acting on the Cooper-pair spins by 1 ,SP: ^
XPoHrn"r.
(8.183)
Here lpo is the relevant eigenvalue of the partial spin susceptibility of the condensate, given by (10.53) in Chapter 10, and the superscriptzero on the susceptibility implies the absence of Fermi-liquid corrections. The "instantaneous" magnetic field r{,.,", is given by the sum of H"ç, the external magnetic field corrected by Fermi-liquid interaction effects,
H"n: u -
,trr^t
(8.184)
(where we neglect the Landau parameters Fi for l>z), and the magnetic field equivalent to the difference in the chemical potentials of the Cooper pairs with spins up and down respectively (see (10.141b). The dipote interaction induces a weak coupling of the Cooper-pair subsystems of different spin orientation and thus provides a mechanism for spin exchange between the subsystems. In equilibrium the respective chemical potentials must therefore be equal. The field driving the motion of the vector d is given precisely by this chemical potential difference, i.e. by H"n-.Éri,,.t rather than by H-(ylx)S as in (8.8b). consequently, the equarion of motion for the vector 4 (8.8b), has to be modified as follows:
å&):
d(k) x T(H"r- rr,",,)
lvv\ : d(k) * ,(t-t*räs - ,*t,)
(8.185a)
(8.18sb)
8
396
Spin Dynamics
In contrast with the Cooper-pair spin Sp, the quasiparticle spin Sq
does not
follow the field ñI¡,,", instantaneously. Rather Sq relaxes to S.qq:;qotr[¡,,",
(8.186)
through the action of collision processes. (This is the value it would attain .úI¡,,,, were itself time-independent.) Here Xeo:
)(o
-
Xoo
if
(8.187)
is the partial susceptibility of the quasiparticle system. The timescale for this
adjustment to occur is hence given by a quasiparticle collision time rçs (Combescot and Ebisawa 1974). Taking into account the precession of ,Sq in the screened magnetic field.É["6, we may then write the equation of motion for,Sq as jc - 5c x fI.n - +(so - s"oo). (8.188) Tce
As one approaches the normal state from below [, the quasiparticle spin Sq becomes equal to the total spin S, which is a conserved quantity. It follows that z6s has to diverge in the limit T -- T". From microscopic theory, one indeed finds rceæQ-f lf"¡-trz. (For a detailed derivation of z6s in the B phase, see (L0.L49); the same qualitative behaviour is also found in the A phase.) In the usual experimental situation the value of the instantaneous field I[¡,,r, is not an externally controlled variable. Rather, ñf,,,r, is determined by the total spin S and the actual value of Sq, as seen by inverting (8.1S3) and using (8.182): .Éf¡,,,,:
#rt
-
sq).
(8.18e)
With the aid of (8.189), the local equilibrium value of ,Sq, given by (8.186), can be expressed in terms of S and Sq. Substituting this expression for Slo into (8.188), the equation of motion for Sq may be rewritten in the form
jo:
^So
x
1 (So H"n-Ttt - s3),
(8.1e0)
which describes the relaxation of Sq to the equilibrium value
,r:#r. It simply
(8.le1)
expresses the fact that the effective magnetic fields responsible for
the total spin polaÃzation S and the quasiparticle spin polarization,Sq must be equal in equilibrium. The relaxation time r¡1 was first introduced by Leggett and Takagi (L975) and is related to rçs by (see also (10.152))
TLT: xÑ tce' ¡
(8.rez)
8.6 Spin-Relaxation
Phenomena
397
Since xPo vanishes linearly in Á(r) for T+ ?i (see (10.153) for the B-phase result), r¡-¡ stays finite in this limit. In fact, may be shown from microscopic theory that r¡-¡ becomes equal to the normal-state relaxation time z$ in this case (see (10.154)). The equations of motion for s, d(ft) and sq, (8.8a), (8.185b) and (8.190), together with the constitutive equations (8.182), provide a complete description of spin relaxation in superfluid 3He. It may be inferred from the derivations given above that they should be valid as long as the superfluid (< Alk, independcomponent is in local equilibrium, i.e. for frequencies ently of the dynamics of the quasiparticles. on the other hand, the
it
,
microscopic derivation presented in Section L0.3 shows that this is not strictly true. Indeed, the quasiparticle dynamics are different in the hydrodynamic (o¡r << 1) and collisionless regimes (roz >> 1). Therefore one should expect the relaxation rates z!$ and z;.1 to be frequency-dependent. This is borne out by microscopic theory. fn fact, the frequency dependence of the relaxation rate as calculated in (10.148b) turns out to be weak. Therefore the equation of motion for So, (8.190), with a frequencyindependent relaxation time z¡a, is expected to be a reasonable approximation even for ot ) I. The derivation of the equation of motion presented above can actually be formulated in a more precise way (Leggett and Takagi L977). In fact, the reactive part of the equations of motion for S, d and,Sq may be derived from the effective Hamiltonian
H:r,,{d(/.)} -
ys
.H.;#Fôs,.:#(so), -:#(so),
(8.1e3)
by assuming that ^Sp and .Sq commute and the components of SP and obey ^Sq spin commutation relations. Let us now turn to the solution of (8.190). In the hydrodynamic regime, @t*(( L, the equation of motion for Sq can be approximated by Sq
- 53 - rtr(TS8 x Il.* x'o - -Ív¡ïRo, )(
S3)
(8.leaa) (8.1e4b)
since Sq approaches the equilibrium value S3 in the limit @ryr+Q. Equation (8.194b) follows by expanding Sg in terms of S, using (8.191) and employing the equation of motion (8.8a) for S. According to (8.194b), the dipole torque is effectively driving the quasiparticle spin polarization out of equilibrium, although it only acts on the Cooper-pair spins! Substituting (8.194b) into (8.185b), one obtains the equation of motion for d(k) including relaxation effects to lowest order in at:
å@): d(k) x
,(,
-Tt'),
(8.1es)
8
398
Spin Dynamics
where
S':S*zpRp
(8.1e6a)
' :S+zn(j-ysxII).
(8.1e6b)
Here we have defined an effective relaxation time rn bY
:_)(xqo-
(8.1e7)
I x*"t''
'*
In the limit T+ T., rrr. diverges as (L -TlD-tl'(see Fig. 10.5), but this divergence in (8.196) is over-compensated by the temperature dependence of fRpl oc1- f lT". At low temperatures the rapid increase of z¡--¡aø (see
(10.150)) for the B phase is reduced by the decrease of .lqo so that zp is a considerably weaker function of temperature than r¡'¡. We may now use (8.195) to calculate the NMR resonance linewidths in the linear regime. As can be seen from (8.8b), the only change that has to be made in the oscillator equation (8.10) is to replace ôS by ôS' according to (8.L96). This leads to the Fourier-transformed equation
l.r'-
(1,
-
itttrp)A'zl ôS
-
iy(a + iz*Q2)
ôS x
Ho: -y^So x ô^Éf. (8.19S)
For longitudinal spin motion, one finds the resonance condition
(,)2:
(1.
-
irøq)dizu,
(8.lee)
where Afr is given by d|to and Qzt for the A and B phases respectively. The linewidth of the longitudinal resonance may be extracted from (8.199) (taking into account that Olrr* << L) as
\,:
(8.200)
d2?,'*'
4' is predicted to vanish *(1 - TIT)''' as T-T, which is equivalent to a finite relative linewidth \,lQr. In the B phase at low
Note that
ternperatures (8.200) yields the temperature dependence ljt æ(Tl\)3, in qualitative agreement with experiment (Bloyet et al. 1979). A more precise expression can be derived using (10.150). ,, The width of the transverse resonance in the A phase is found from (8.198) as (Combescot and Ebisawa 1974)
ye r¡Ar -' - r
d22¡-
ut?+
#^,
(8.201)
i.e. the resonance narrows nH-z at high ñelds (rt>>fJ¡r).Note that in deriving (8.201) only the weaker condition (a - ø¡)rp (( L had to be satisfied. In the B phase, on the other hand, the linewidth is zero in the present approximation, at least as long as there is no transverse resonance shift. As can be seen from (8.194b), in the absence of the corresponding component of the dipole torque the quasiparticle spin system is not disturbed by the motion of the Cooper-pair spins, and no dissipation can
8.6
Spin-Relaxation Phenomena
399
occur. In fact, since the dipole torque Rp (and hence the dissipative torque given by (8-196)) is directed along ñ, dissipative terms appear only in the equation of motion fot 0, (8.41b), but not in the equation òf motion for ñ, (8.a1c). However, as discussed below (8.34), there is a small transverse resonance shift Àø, even in the B phase for perfect orientation of the axis rî along the magnetic field, due to the slight distortio¡ of the gap parameter by a magnetic field. This residual dipole torque will give rir" i" a small line broadening l-? * ff(¡ø rlrur)t. The above theoretical results on the CW resonance linewidths are in partial agreement with experiment. This has been discussed in detail by'sug", Lee and Richardson (1978). (see Ave\?! (Lg7s), ,t Gulry et (1976), ar. lt et al. (L977, 1978b) and Webb (1978b) for measurements of the width of the longitudinal resonance in the A phase and Bozler et at. (r97\ and Gully er al- (L975) for measurements of the transverse linewidth, aò weú as corruccini and osheroff (1975b) for measurements of the relaxation time rr.) In particular, for T,+ 1", the longitudinal linewidth in the A phase is found to increase rather than decrease as predicted by theory. This may possibly be attributed to spin sunercuucnts flowing from outside into the intårior of the magnetic coil, thereby destroying the magnetization in the coil (Vuorio 1974, 1976, sonin L979). Another possibiùty is that the assumption of a fixed Í-vector orientatiôn, which ir implicitly contained in the above derivation, fails close to T". Leggett and Takagi (1977) have derived a very useful expression for the total energy change by intrinsic spin relaxatiôn, which allows one ro calculate the global effect of dissipation even for complicated spin motions. Starting from the expression for the total energy (8.1), the time derivative of the energy may be calculated in the hydrodyñãmic approximation (g.196) AS
#:-l'**',.
(8.202)
A9 a first application of (8.202), we calculate the recovery time of the longitudinal magnetization after a sudden change in the longitudinal magnetization, as happens for example in a field-turn-off experiment (Combescot 1976, Maki and Ebisawa til6d). The time rate of change of the total energy as g11e_n by (8.202) is determined by rhe average valuã of Rfi. For a large turnoff ûeld Âr'1 along â, such that y a,H)) g,,, ih" longitudinal NMR response has been shown in Section 8.3 io be equivalent to that of a T-alhgmatical pendulum rotating with a frequency corresponding to the initial impulse. (This analogy is exacr in the caie of in" a pËur", ,"ã 1s.zz¡; in the case of the R phase, (8.79), the existence of two ineiuivaíent minima suggest the model of two pendula, coupled in such a way ihat their angles are always in the proportion 1':2).Immediately after theiulse , â therefore rotates in the (*, y) plane with a constant unguiu, velocity, the Áotion being described by some angle 0 (e.g.the relativJangle to / in the case of the A
8
Spin Dynamics
phase). In other words, the angle 0 runs through all possible values with iougÌriy equal probability. One may {herefore average Rf; over all values of 0. in it" R phut", where, according to (8.39a), lRol :(xl2y')d*esin20, the average comes out as
(n3)"
::(##^)',
(8.203)
so that the average energy dissipation (8-202) is given by
(#) :-å#,îo'-^
(8.204)
Hence one obtains a constant time rate of change of the total energy, i.e. a linear decrease of energy with time. Relating the energy change (8.202) to the initial energy äX(LH)", the recovery time Tå.,' is found from (8.204) as
rå",:o(#)'+
(8.20s)
An analogous derivation for the B phase, where -(see (8.41a)) lRol : *Qlv1g*"sin 0 (1 + 4 cos 0), so that (n3), : &Qa'"lv')', yietds
zå"
:H(#)'+
(8.206)
As a second example, the recovery time for the magnetization after a large-ang\e tipping pulse may be calculated from (8.202), if during the
motion of d the whole unit sphere is essentially covered. This is the case in the A phase, where a fixed ,èf"r"n"e direction, given by the vector i, forces ã to perform large-amplitude oscillations. Averaging RL over the unit sphere, and considering that the total energy dissipated is given by X$ - cos Q)Hf;, where F/o is the apptied static field and @ is the tipping angle, one finds a recoverY time
r !",,(þ)
:l (#)' U-
cos
t,
+
(8.207)
The above predictions on the recovery of magnetization in the A phase are in qualitative agreement with experiment (Corruccini and Osheroff Lg75b). So the linear recovery in time as well as an increase of ["" with magnetic field are clearly observed. The deviations are probably again due to the effect of spin currents. In the B phase the situation is more complicated (Brinkman and Smith L975b). As we have seen, there exists a kind of motion where the system is almost in the minimum of the dipole well, i.e. 0 - 0t, whereas S and ñ precess about a common axis. The relaxation during the time interval in which the latter mode of motion is approached is described roughly by a recovery time of the order of (8.207). The relaxation in the long-lived
8.6
Spin-Relaxation Phenomena
401
mode, however, must be calculated by taking the time dependence of Rf; explicitly into account. The above theory is in qualitative agreement with the observed spin-relaxation behaviour in the B phase after a large tipping pulse (Ishikawa et al. 1989). As an application of (8.202) to a problem involving the long-liued motion of the collective variables in the B phase, we now consider the damping of the wall-pinned ringing mode (Leggett I975b, Maki and Ebisawa 1976d, Leggett and Takagi 1977, Takagi 1977, Soda I977a, Cross 1978). As discussed at the end of Section 8.3, this particular mode is excited by switching off a small magnetic field A.F/ << Qtlh in a configuration with ñ initially perpendicular to the magnetic field. This mode is caused by a large-amplitude precession of Ê about the axis C" with frequency ø(0): (?)'''y A^FI, which excites a small-amplitude precession of S with the same frequency. Under these circumstances, the system always stays close to the minimum of the dipole energy, i.e. 0 - 0v Any function of 0 may therefore be approximated well by expanding it to lowest order in 0 - 0r. Substituting such an expansion of the dipole torque Rp (see (8.41a)) into (8.202), one may express the energy relaxation as
#---"n'*oi,-o')''
(8.208)
Substituting the precession frequency a(t) (:Er) as given by (8.71b) into (S.70), eliminating et and dropping the small term e, 0 - 0, may be related to ø(r) by
o(t)-u,:
(l:)'''l#]'
(8.20e)
In this simple problem the total energy may hence be calculated explicitly ín terms of ø(r). From the general expression (8.69) for E(t), one obtains the simple relation
E(t)
:tuþ,'r,,
(8.210)
Here we have again neglected the terms â2 and (0 - 0r)', which are small, i.e. of order (o¡l{2")'. Combining (8.208), (8.209) and (8.210), one is led to the following differential equation for ø(r):
*,",4:
-? t*aa(t)'
(8.211)
with the solution úr(0)
ro(t): [1,
+ ]r*az(o)quz'
(8.2t2)
The precession frequency is hence found to decrease with time as energy is dissipated and taken out of the mode. The relaxation time for this is given
402
8
Spin Dynamics
by [ø'z(O)z*]-t, which under suitable experimental conditions can be much longer than the longitudinal relaxation time given by (8.206). The ringing down of the wall-pinned mode as predicted by (8.2L2) has been observed experimentally (Wheatley 1978; see also Webb 1978a,c). It provides a method for measuring the intrinsic spin relaxation time ry¡. Numerical solutions to the Leggett-Takagi equations have been obtained by Katayama et al. (1980a,b,c, 1983a,b), Katayama (1981) and Ooiwa (L982). The effect of dissipation on the nonlinear spin dynamics of 3He-B discussed in Section 8.3 has been studied by Golo (1981a), Golo and Leman (L982a,b,1983), Fomin (1983), Golo et al. (1983) and Kesaev and Ugulava (198a); for a review see Golo and Leman (1989). It is found that the interplay between the Leggett-Takagi dissipative mechanism and the nonlinearity (particularly where the nonlinear ringing frequency vanishes) leads to interesting effects. In particular, a new type of propagating magnetic excitation in the B phase, i.e. a spatial modulation of nonlinear ringing of the magnetization, has been proposed by Golo (1984). The nonlinear interaction of the latter mode with acoustic modes has been studied with the aim of finding a new way of detecting second sound (Golo and Kats 1986). The existence of surface spin waves at a boundary characterized by a diffuse scattering law and the corresponding contribution to the NMR absorption have been discussed by Fishman (1979). The homogeneous spin dynamics of the A1 phase in the nonlinear regime, including spin relaxation, has been discussed by de Vegvar (1984) within the adiabatic approximation. The intrinsic superfluid relaxation is seen to be dominated by normal fi processes, leading to magnetization recoveries as in the A (i.e. A2) phase. Finally, we comment briefly on the effect of spin diffusion, which provides another mechanism for dissipation in spatially nonuniform situations. The corresponding terms in the equations of motion for the spin density are considered in Chapter 9 (equations (9.29)-(9.31)), and the spin-diffusion coefficient is calculated within kinetic theory in Chapter 10 (equations 10.138a-d)). The combined effect of the Leggett-Tagaki relaxation mechanism and spin diffusion has been discudsed by Fomin (1984b, 1985b), who thereby explained the duration of the long-lived induction signal in the B phase measured by Borovik-Romanov et al. (1984b). Spin-relaxation processes in 3He-A have been studied in particular by Bun'kov et al. (1985) using a pulsed NMR method. They found that at low tilt angles the relaxation was indeed due mainly to the Leggett-Takagi mechanism, while for tilt angles >40" an instability occurred in the homogeneous spin precession leading to large-amplitude spin waves. Their findings are in agreement with Fomin's earlier theoretical treatment of the problem of fast spin relaxation (Fomin 1979b,198La, 1984a). Early attempts to determine the spin-diffusion coefficient experimentally,
Further Reading
403
using the spin-echo method, failed (Corruccini and Osheroff 1973). In later experiments, spin echoes were found in both the A and B phases (Eska er al. 1981). These echoes were found to have a rather peculiar structure: while a single echo could be detected in the B phase at sufficiently small tipping angles, a sequence of multþle spin echoes was observed in the A phase and in the B phase at large tipping angles. Multiple spin echoes generally appear if the resonant frequency, or the spin-diffusion coefficient, vary with the spin-tipping angle. The structure of these echoes has been explained on the basis of the theory presented in this chapter for the case without dissipation (Hasegawa L982), and including spin relaxation but neglecting spin diffusion (Fomin et al. 1934). We mention in passing that multiple spin echoes have also been seen experimentally in normal liquid 3He (Einzel et al. 1984a). Relaxation of spin superflow in the Ar phase has been investigated experimentally by Lu et al. (1989). The discontinuity in the relaxation time as a function of temperature observed by these authors may be explained by the effect of the Ar-Ar. interface present in the experiment (Grabinski 1989). FURTHER READING General reviews and books
A and Goldman M 1982 Nuclear Magnetism: Order and Disorder (Clarendon Press, Oxford) Anderson P W and Brinkman W F 1978 in The Physics of Liquid and Solid Helium, Part II, ed. K H Bennemann and J B Ketterson (Wiley, New york), p. 177 Lee D M and Richardson R C 1978 in The Physics of Liquid and Solid Helium, pat II, ed. K H Bennemann and J B Ketterson (Wiley, New york), p.2g7 Leggett A J 1975 Reu. Mod. Phys. 47 331 Wheatley J C 1975 Reu. Mod. Phys. 4il 4L5 Abragam
Specialized reviews and articles Brinkman W F and Cross M C 1978 in Progress in Low Temperature Physics, Vol. VIIA, ed. D F Brewer (North-Holland, Amsterdam), p. 105 Fetter A L 1986 in Progress in Low Temperature physics, Vol. X, ed. D F Brewer (North-Holland, Amsterdam), p. 1 Fomin I A 1981 Souiet Scienffic Reuiews, Section A: Physics Reuíews, Vol. 3 (Harwood Academic Publishers, Chur), p.275 Giannetta R W, Smith E N and Lee D M 1981 J. Low Temp. phys. 45 Zg5 Golo V L and Leman A A 1989 in Helium 3, ed. rW P Halþerin and L P Pitaevskii (North-Holland, Amsterdam) Leggett A J L974 Ann. Phys. (Ny) 85 11 I-eggett A J 1978 in Quantum Liquids, ed. J Ruvalds and T Regge (North-Holland, Amsterdam), p. L67 Leggett A J and Takagi S 1977 Ann. Phys. (Ny) 106 79 Maki K 1986 in Solitons, ed. S E Trullinger, V. E. Zakharov and V. L. Pokrovskii (North-Holland, Amsterdam), p. 435 Salomaa M M and Volovik G E 1987 Reu. Mod. phvs. Sg 533
Hydrodynamic Theory
The dynamics of any macroscopic system in the limit of low frequencies an( long wavelengths can be completely described in terms of a few so-callec "hydrodynamic" variables. These variables owe their existence to thr behaviour of the system under continuous symmetry transformations (see Chapter 6)- Consequently, the structure of the equations of motion obeyec by these variables-the hydrodynamic equations-is uniquely determinec by the symmetry properties of the sysrem. Let us consider a nonequilibrium initial state of a system. Most degrees ol freedom will relax towards equilibrium within a time period of the order ol a microscopic relaxation time r. In the case of trire, is essentially the quasiparticle collision time, which has been discussed in" Chapter 2 for the normal state. The corresponding collision time for the superliuid state will be investigated in Chapter L0. This collision time is of order L0-7 s near 7., increases at lower temperatures and eventually diverges for z+ 0. Now, the important point is that a small number -of maðroscopic degrees of freedom are not affected at all by microscopic relaxation prä""rr"u-"ither (i) because there is a conservation law rendering .oìHrion processes ineffective, or (ii) because there is a spontan"o,rrly broken continuous
In the latter case the equilibrium state is energetically degenerate with respect to changes of ih" ry-*etry variable. Hence a uniform change of this variable is unobservable. The set of variables consisting of these two types, namely the conserved quantities and those describing the broken symmetry, is zufficient to complãtely determine the state of the system in the hydrodynamic regime. The existence of conserved quantitiea is intimately related to the continuous symmetries under which the system is invariant. Thus gauge symmetry, translational symmetry in time, translational symmetry in ,p".ã, and-rotational symmetry in position space and in spin spãce (negiecting ttr" weak spin-orbit coupling) give rise to the conservãtion of particle number (or mass), energy, momentum, angular momentum and spin. The corresymmetry.
406
9 Tabl"
9.1 Hydrodyra*i" turiabler
und ar.o"iat"d
Hydrodynamic Theory
.ym-ettier.
Variable
Symmetry
mass density p(r, t) momentum density g(r, t) angular-momentum density L(r, t) energy density e(r, t) spin density S(a t)
gauge
space translation orbital rotation time translation spin rotation
phase variable @ orbital rotation angles 0r9 spin rotation angles 0)
broken gauge broken orbital rotation broken spin rotation
sponding hydrodynamic variables are listed in Table 9.1 (top) together with the variables associated with potentially broken symmetri"r in thð superfluid phases of 3He (bottom).
9.1
GENERAL PRINCIPLES
The hydrodynamic variables are related to each other by virtue of the general principles of thermodynamics. To see this, one should bear in mind that a change in the energy of the total system may be caused by changes in (i) the heat content rds, (ii) the mechanical volume energy p dv, (iii) the energy of linear motion u ' dP (where P is the total mom"niu- and u is the velocity of the system), (iv) the energy of rotational motion dl . d¿tor (where Iro, is the angular momentum and ll is the angular velocity of the system), (v) the mass dM and (vi) the magnetization y dS. Obvioúsly the system's energy does not depend on a symmetry variable itself (provided the symmetry holds strictly), but it does depend on the gradient of u ,y-*"try variable (here we assume that the gradient is uniform).
9.1.1 Thermodynamic identities This leaves us with the following thermodynamic identity for the whole
system:
dE:
dS
-PdV
+
¡^tdM*u. dp+ dl .dL6t+) rîdS"
+
fþ.dVe, (e.1)
where ¡r is the chemical potential per unit mass, the ø"" are angular velocity components conjugate to the total spin and i o is a "current" asiociated with V@- The last term in (9.1) stands for all terms induced by the spontaneously broken symmetries. In the context of hydrodynamics, *" aìe obviously interested in local rather than global thermodynamic identities. Let us
9.1
General Principles
407
therefore change to a local description by introducing the densities of energy e : ElV, entropy s: SlV, mass p: MlV, momentum g: plv, angular momentum I, : LrorlV and curreñ jþ : f a lV induced by symmetry breaking. Applying the identity (9.L) to a volume element of fixed size small enough that spatial dependences of the thermodynamic variables can be neglected (but large enough that local equilibrium is established), one derives the local thermodynamic identity
de: Zdr+ l.tdp*u. dg+ A.dL*ø..dS+¡o.dVO. e.Z) This identity is valid for sufficiently slow processes, i.e. in the hydrodynamic regime. Let us next consider an infinitesimal change in volume, dv. Then it follows from (9.1) that the pressure P may be completely expressed in terms of the conserved densities: P
- -e * Ts + l.tp * u - g + d2 . L.
(e.3)
Combining the two identities, one finds for the change in pressure
dP:sdT* pdlt*g.du +L.df2+S.dar" -¡o.dVø.
(9.4)
The thermodynamic identities (9.2) and (9.a), when combined with the equations of motion for the hydrodynamic variables and the second law of thermodynamics, suffice to determine the structure of the hydrodynamic equations. More precisely, the thermodynamic identities serve to relate changes in the hydrodynamic variables to the change in entropy density. At this point it is important to remember that the entropy density is subject to constraints following from the second law of thermodynamics, which states that the total entropy of a closed system never decreases. This severely restricts the form of the equation describing the time evolution of the entropy density, and therefore the equations of motion for the hydrodynamic variables.
9.1.2 Standard procedure for deriving hydrodynamic equations The derivation of the hydrodynamic equations proceeds according to the following general rules (Khalatnikov 1965, Liu 1982).
(i) As a first step, one has to identify the conserved densities and symmetry variables of the system in question. This turns out to be somewhat subtle in the case of superfluid 3He. white in both the A and B phases several symmetries are simultaneously broken, this does not automatically imply that an independent dynamical variable has to be added to the set of hydrodynamical variables for each broken symmetry. In fact, as first pointed out by Liu and Cross (L979), when two symmetry operations affect the order parameter in the same w?y, only one independent variable
9
408
Hydrodynamic Theory
associated with the relative symmetry transformations of the two respective symmetries need be included in the set of hydrodynamic variables. For example, in the A phase a gauge transformation can be undone by a corresponding rotation in orbital space about the preferred direction (see Section 6.L for a detailed discussion). The dynamical variable accounting for both symmetry transformations is the difference of the phase angle p and the rotation angle gi (assuming ito lie along the z direction). Similarly, in the B phase the change in the order parameter under an arbitraiy rotation in orbital space may be compensated for by a corresponding rotation in spin space. Therefore, instead of the six rotation angleJ gi anã 0i, only three relative angles g!" are allowed as independent hyãrodynamic
variables.
(ii) Next, the appropriate
conservation laws and equations of motion for
the symmetry variables are listed. These equationi have the following general structure (see e.g. (2.2r)-(2.2$): the rime derivative of thè hydrodynamic variable is equal to the negative of the divergence of the corresponding current. The currents are expressed in terms ofgradients of the hydrodynamic variables, the so-called "thermodynamic -forces', or "thermodynamically conjugate fields". The behaviour of the currents under transformations to linearly moving (Galilean) and rotating frames as well as under time reversal provides constraints on the form of tiese currents. In this way, some of the coefficients in the expressions for the currents mav be
determined.
(iii) It is now necessary to employ the thermodynamic idenrity (9.2) and
to express the change in entropy density ds in terms of differentials of the conserved quantities and the symmetry variables. By interpreting ds as an entropy change in time, this may be converted into utt t* the time "q,-ruiion derivative of the entropy density in terms of the tims derivatives of the
hydrodynamic variables.
(iv) using the conservation laws, one may now replace the time derivative of each hydrodynamic variable by the divergence of the corresponding current. This yields the time derivative of the entropy in terms of spatial derivatives of the currents. To lowest order in the gråãients, i.e.
on the level of local equilibrium, entropy is conserved. In geneial, however, there is a ûnite positive entropy production. With this in mind, the time evolution of the entropy density may be written as
(e.s) where
¡ is the entropy currênt density and R is the heat-production
function. The latter is a posifive-definite quadratic form (up ìo surface terms) in the gradients of the thermodynamically conjugate variables
9.1
General Principles
(ðs lðH¡):
R:) where
Ë/¿
"(#).,.o(ãä),
(e
r
is any one of the hydrodynamic variables.
Note that only the symmetric part of the matrix of transport coefficienr r (the o"rü;;"ràutiorrr¡. t addition, the heat production R is an even function under time reversal. o the other hand, the two terms on the left-hand side of (9.5), representin the conservation of entropy, must have identical time-rèneí;i;.rperties As the first term is the time derivative of the scarar equiliuri.rir quantity s it is obviously odd under time reversal; consequentry the entropy curren densityhas to be odd, too, as one mighi expect. Theìerm, in it" equatior of motion for any hydrodynamic variible ai mary ..reactive, be or "dissipative" respectively, according to whether classified as they have the same o the opposite time-reversal property uséH,lat Writing islðtu, in (9.5) purr sufficient constraints on thè pòssible funciional structure of the currents t< determine their precise form. Before we turn to the detailed derivation of the hydrodynamic equationr for the superfluid phases of 3He, we first consider the equations of motior for the symmetry variables in generar, neglectin! oissipation.
Ltt contributes to rhe dissiparion function
9'1.3 Equations of motion for the symmetry
variabres
The spontaneous breakdown of a continuous symmetry gives rise to a modification of the hydrodynamic theory in two ways (Liu 19g2). First, the new symmetry variable leads to a new equation of motìon, which has to be
included in the set of hydrodynamic secondly, the new hydro_ dynamic variable enters the áheady existing "qüutions. hydrody"u*i. tn particular, there appears a dissipâtionress"".rír"rr, "quåtionr. proportionar to the gradient of the symmetry variable in the conservation law related to the symmetry that is broken. In local equilibriul, the equation of motion of the symmetry variable, @ say, is given by equating the material derivative to a gene ralizedforce
#:#.a.vþ:#,
:
l i
I
(e.7)
where G is the thermodynamic variable corresponding to the generator of the symmetry transformation in question. substituting (9.7) into the entiopy-production equation (9.5), one is led to a term iþ'v@E/ac¡, which is reactive. As such, it must be part of the divergence of the entropy current density on the left-hand sidå of (9.5). However, this term is not a divergence of a current itself. Consequently,
4I0
9
Hydrodynamic Theory
there must exist an additional term of the form (ðElðG)v.¡0, which completes it to give a divergence. The prefactor AE/AG in the thermodynamic density (9.2) in front of the differential dG, "pp"ã., inãicating that this additional term comes from the conservation law for G. In fact, it is generated precisely by an additional contribution -¡o to the current density of G. The ensuing contribution to the entropy equation (9.5) combines to
give an additional entropy current density jþ(ðEl}G). As a first example, let us take the broken gauge invariance characteristic of a superfluid. The corresponding symmetry variable is the phase variable rp. The generator of gauge transformations in a pair-correlated system is å¡f, where N is the particle number operator. Hencè the equation oi motion for Q, neglecting dissipative terms, is
d0
2m
dt: - k lt'
(e.8)
p:m-'ðnlðN being the chemical potential per unit mass (the change in sign, Q--þ, is conventional). In this u nondissipuìiu" with
"ãr"gauge symmetry, current appears in the conservation law associated with i.e- in the continuity equation (2.2I).It is the well-known s.fern,rid mais
current.
Similarly, the broken rotational symmetry in spin space as encountered for instance in an antiferromagnetic normal fluid ptorrid"r three additional hydrodynamic variables; namely the three rotati,on angles 0| about the three axes in spin space. The fact that these variables are globally ill-defined, because of the noncommutativity of three-dimensional rotations, has to be kept in mind when dealing wlth second- or higher-order derivatives, as will be explained later. The generators of rotations in spin space are the three components of the spin operator ,f,. According to ihe general prescription formulated above, the equations of motion for the symmetry variables 0"* are then given by
yl-ôE_ ='i' dt-:
"
asu
(e.e)
From the magnetic-energy expression
E*t:fit'-vs'H, one finds the field driving the motion
of g|
as
"\ ,\T- "u).
a'*: --lYs*
This necessitates the addition of a nondissipative current contribution
-ôE iy: ô(vq) to the spin conservation equation.
(e.10)
(e.11)
9.2 Hydrodynømic Equations for the B Phøse
4t1
Finally,"the broken rotation symmetry in orbital space occurring, for example, in a biaxial nematic liquid cyrstal is specified by three rotation angles 0r9. Th" generator of rotations i's the angular-momentum operator.L. However, as mentioned above, angular momentum and momentum are closely related. A local angular-momentum density Z contributes to the momentum densitv
as
g:p++VxL,
(e.r2)
where p is the linear-momentum density. The term åV x r is akin to the magnetuation current in electrodynamics. Differentiating the energy of motion n : ! d3r u . g with respect to the angular-momentum density.L, one finds
d09 _ôE dt
o. õL¡ --I'
(e.13)
where
a--)Yxa is the vorticity. According to the general rule described above, we expect the appearance of a dissipation-free current in the conservation law for angular momentum. However, the angular-momentum conservation law is not an independent hydrodynamic equation. Rather, the close connection of g and ,L expressed by (9.I2) leads to a nondissipative current of the form àE,Y x (V, ôE/ò(V,0")) in the momentum conservation law.
9.2
HYDRODYNAMIC EOUATIONS FOR THE B PHASE
The order parameter of the BW state as given in (6.23) is the product of a phase factor eiÓ and a real orthogonal matrix R", describing a rotation of spin space relative to orbital space. It follows that there are four additional symmetry variables: (i) the phase variable QG, t) associated with broken gauge symmetry; and (ii) the three rotation angles 0ì"(r, r) associated with the broken relative spin-orbit rotation symmetry. In the following we shall only work with the differentials
d0;":
dei
-) p
R*, de'u
(e.14)
(where Ru¡ describes the local order-parameter structure), because the variables 0]" are globally ill-defined. Even when working with infinitesimal changes in the rotation angles, we have to be careful if we are dealing with operations involving two successive infinitesimal rotations about different axes. In general, we have to observe the commutation rule (Liu and Cross 1979) (ôV
- Vò)0'": ô0'o x V0"o,
(e.1s)
9
4t2
Hydrodynamic Theory
where ô and V denote two arbitrary differentiations. In order to demonstrate the validity of (9.15), consider two successive infinitesimal rotations described by rotation angles d0, and d0, respectively, such that t+ r * r x d01 etc., where dilt - d0¡ ñr and ñ¡ is a unit vector in the direction of the rotation axis. The difference in the resulting vectors after carrying out the successive rotations in the order d0rdez and in the order d0zd01is just r x (d01 x d02), which agrees with (9.15).
g.2.1 The thermodynamic identity and equilibrium conditions Recalling that a gradient in the phase variable @ gives rise to a supercurrent (see Chapter 7), it is useful to eliminate V@ in favour of the superfluid velocity
,":loe. ¿m The thermodynamic identity for the
(e.16)
B
phase (omitting the angularmomentum term for the moment) then takes the form (Graham and Pleiner 1976, Liu and Cross 1978)
z
ds
: de - p dp- un. dg -8.. d% ôf dlv.aP\-'.. o;") d(vi -Yí a1o,ö - ? *"ô/t""¿
? d
r;o.lo;",
(e.t7)
of the B phase (7.25b). Here we have denoted the velocity conjugate to the momentum density by u,,, because it has to be identified with the so-called normal-fluid velocity, as will become clear below. The variable g, has the dimensions of a momentum density, too; i.e. it must be given by the product of a mass density times a velocity. In the frame moving with velocity u,,, the only other velocity vector left is u, - u,,, and therefore where.fu,,,¿ is the bending free-energy density
g":
p"(u"
* u").
(9.18)
Going back to the laboratory frame, the momentum density is given by
g:
pun
*
g,
:
pnun* p.u",
(e.1e)
where pn: p - p, and u^aÍe the normal-fluid density and velocity discussed in Section 3.5. The contribution p,u. is precisely the dissipationless current mentioned after (9.8). The field conjugate to VrO]" (in 9.!7) may be calculated by functionally differentiating the bending free-energy density for'the BW state, (7.25b), with respect to Vr0i":
--!-\'[ay ¡0i" + bv k0ì'+ cv . 0*1. ?4"+. : 2( -\zm/ ô(v/9i)
(e.20)
9.2 Hydrodynamic
Equøtiorx
for the B
Phase
4t3
9.2.2 Hydrodynam¡c equat¡ons for the symmetry variables We have so far assumed the system to be invariant under rotations in spin-orbit space. However,'as elaborated in Section 6.3, there is a small spin-orbit interaction-the dipole force-which acts as a kind of anisotropy field on the preferred directions. This effect is contained in the last term of the thermodynamic identity (9.11). Noting that the dipole energy density/r, (6.100), may be expressed in terms of the spin-orbit variables ñ and 0'" as given by (8.159), the conjugate field AfDlA0'":A'"[email protected]"")ñ is nothing but the dipole torque Rp discussed in Section 8.1 (Brinkman and Smith 1974).
From the requirement that the entropy should be a maximum in equilibrium, it follows via (9.17) that the temperature T, the chemical potential ¡t and the normal velocity un as well as the spin angular velocity ar" are constant in equilibrium. The vanishing of the variation of s with the phase variable @ implies V .9.:0. The equilibrium condition on the spin-orbit angles 0rl" ir obtained by taking the functional derivative of the total free energy Fn..¿: f d",fn.nu with respect to 0f": ô/u"n¿
ôor:"
= 'þi
:) voträä - I
einor
ff5]*
*,,,
(e.2r)
where
^ôf-^' tP: ñ
ê0"":
-
d22"þ0. t'".
The second term on the right-hand-side of Q.2l) comes from the commutation relation (9.15), which has to be applied when a partial integration is performed. In equilibrium the free energy is minimal and hence rþ¡-0. The thermodynamic forces driving the system towards equilibrium are given by the gradients of those quantities that are spatially uniform in equilibrium, i.e. by Vf, V¡r (or equivalently VP), Y ¡unj, Vg,, Var'u and arr. In general, the currents in the hydrodynamic equations can be represented by linear combinations involving all thermodynamic forces. However, only some of the thermodynamic forces are allowed by symmetry for each given case.
With the aid of (9.8), the equation of motion for the superfluid velocity
u
is obtained as
#. v(¡, + r¡o) :0.
(e.22)
The dissipative part of the chemical potential is a scalar of first order in the gradients. The only quantities of this type that may be formed by the conjugate fields are the divergences of the normal velocity un and the
9
4L4
Hydrodynamic Theory
superfluid momentum density g", leading to the expression
l-tr': -6rV. un - 6rV.g", Q.23) where Ç1 and Çt are bulk viscosity coefficients. The expression for pD is identical to the one for an isotropic superfluid (Khalatnikov 1965). Similarly, the equation of motion for the spin-orbit angles g]" follows by combining (9.9) and (9.13) as (Graham and Pleiner 1976, Liu and Cross Le78)
d0Ì"+ .:ft R*,r'r- å(V x u,,)¡ + orP: g.
(e.24)
4
The dissipative part d}f again must be given by a linear combination of the thermodynamic forces. The only force that has the symmetry of a vector in orbit space is r/, as defined in (9.2L), and hence
d27: -vvþ¡'
(e.2s)
where y is a kind of spin-orbit viscosity.
9.2.3 Hydrodynamic equations for the conserued quantities The remaining hydrodynamic equations are given by the conservation laws.
(i) The continuity equation has the usual form (see (2.21)), with (ii)
the
momentum density given by (9.16). The momentum conservation equation is given explicitly by ¡lo.
fr
+ Y,P
- å(v x tù, +4o o@io+ rrp) :
s.
(e.26)
Here the term -å(v x ttù is due to the reactive current in the angular-momentum conservation law, which appears as a consequence of the broken spin-orbit rotational symmetry, as explained below (9.13). The contribution Iri¡, to the stress tensor is similar to the so-called Erikson stress tensor for nematic liquid crystals: it describes the stress of an anisotropic medium, i.e.
Ir?t,:?
t#"9-
v,gi"o
*B,,ru",;.
(e.27)
Technically speaking, this contribution may be traced back to the last term in the expression for the change in pressure dp, as given by (9.4). Note that the first term on the right-hand side of (9.27) involves 0'o and hence induces a coupling of mechanical and magnetic degrees of freedom. This effect might be observable in a torsional-oscillator experiment, where the shear flo1v would give rise to a¡ NMR signal. In lddition, the symmetry of the dissipative part of the stress tensor rrf; ailows one to add a term propõrtionuì to
9.2 Hydrodynamic
Equntions
for the B phase
4T:
v'g,
to the expression for the normal fluid so that in the B phase ir has the general form
II?o: -T(y
¡un,*
* Vou,',, -
?ô,0y. u") _ (6rV
. g_
*
6zV . u,,)ô¡¿.
(e.28)
of
(iii)
the three coefficients q, Et and the bulk viscosities (1 and Ç, will be shown to be order elTò,-Çr, small compared with the shear viscosity 4, and may therefore be neglected. Note that onsager,s symmetry principle requires that the coefficient of v . g. in 1l.zs¡ ue equal to the coefficient of V. un in (9.23). The equation of motion for the spin density s is obtained as
t-
v(s x H)*
+?
t'[ãäffi
Rp¡
* a)
e,,oRt¡Yrcr+ o,,x?,l ^ l.rj
:4
Ruoqo' (g'2g)
where the first term in the square brackets is the reactive spin supercurrent, i.e. the symmetry current discussed after (9.7), and the second term is a thermally driven reactive spin curì"ní. th" dissipative spin current is due entirely to spin diffusion:
Xf;¡: -2
D**,V¡a¡î.
(e.30)
vj
The spin-diffusion coefficient is a fourth-rank tensor, which may be characterized by three independent components Dr, D, and D3:
Dp¡ui:
D164ô*u + D2R¡"tftui
+
D3Rt jR"i.
(e.31)
Employing the hydrodynamic equations and the thermodynamic -above identities (9.r7), the entropy-production equation is found to be (e.32) where
/P
is the usual entropy current
f? with
:-
urr,,
:lin,,
(e.33)
jn the heat current.
The entropy-production function may be cast into the form R
'
:;(vr),
+
+ âqZ
(y
ij
+
!rc,e, y n,¡ * ¡u n,¿ - ?ô,¡y . a.")t * vlt,
!<e,v. ¿u
(Dr+A)>
u,,
*
8",¡Bì,+
6,v . s.), *
Dr(>BÌ,)'*
(D,
Eï(v . s"),
-D.) ). Bu.B?,. ß.34\
9
4L6
Hydrodynamic Theory
where B"¿þ
:å ) i
(n.,V,øÞ
t
RB¡Y,a).
Positivity of R requires
(u) the transport coefficients K, Ez, Es,
(b)
(.) 9.3
r¡,
v
and D, to be positive;
Ere'> E?; Dr> lDrl; Dz) -â(D, + D).
HYDRODYNAMIC EOUATIONS FOR THE
(e.3s) (e.36)
A PHASE
The hydrodynamics of the A phase has been a source of considerable controversy. It is only recently that a consistent picture has begun to emerge. Following the first derivation of the linearized hydrodynamic equations by straightforward application of the rules laid down in Section 9.1 (Graham 1974, Graham and Pleiner 1975, 1976), several amendments and refinements have been proposed (Liu 1975 , 1976, 1977, Lg7B, volovik and Mineev I976c, 1981, Liu and Cross L979, Mermin and Ho 1976). rn particular, the role of an intrinsic orbital angular-momentum density associated with the Cooper pairs has attracted much interest (see also Section 7.3). This is partly because microscopic derivations of the hydrodynamic equations revealed the existence of terms that could be interpreted as being due to an intrinsic angular momentum (Combescot and Dombre 1980, Combescot L978b,1980a, Mermin and Muzikar 1980, Nagai 1980, Ishikawa et al. 1980). While microscopic theories are of importance in providing quantitative information about the coefficients in the hydrodynamic equations, they are usually model-dependent. The microscopic approach consequently cannot be expected to provide the most general form of the hydrodynamic equations admitted by symmetry. Following Miyake et al. (1981), Liu (1985) and Hall and Hook (1986) (see also Hu and Saslow L977, Ho 1977, Lhuillier 1977, Nagai 1981a,b, Sonin 1,984b), we shall therefore attempt to derive the hydrodynamics of the A phase phenomenologically, assuming this state to possess an intrinsic orbital angular momentum of density t(r, t). Note that an intrinsic angular momentum is necessarily of quantum-mechanical origin. Alternative but similar derivations of the hydrodynamics of 3He-A have been considered by Khalatnikov and Lebedev (r977a, L977b, 1980), Lebedev and Khalatnikov (1977) and Gu (I982a, b). Early discussions of the orbital dynamics of 3He-A include those of Hall (L975, L976, 1977,1978), Hall'and Hook (1977), pleiner (r977b) and Brand and Pleiner (1981a). A hydrodynamic description of 3He-A in terms of quantum fields, where in addition to the ordinary (bosonic) variables anticommuting (fermionic) variables are also introduced ("superhydrodynamics"), has been formulated by Andreev and Kagan (1987).
9.3 Hydrodynamic
Equations
for
the
A
phase
4t
9.3.1 The concept of an intrinsic angular momentum What are the changes in the hydrodynamic equations if there is a finit, angular-momentum density r? As mentioned in -Section 9.1, in the of an intrinsic angular momentum the conservation of orbital absenc, angula momentum Z does not give rise to an additional independent conservatiol law. In fact, the angular-momentum conservation law can be derived fron the momentum conservation law by employing the definition of the orbita angular-momentum density L: r xg. Note that angular momentum is onl, conserved if the stress tensor is symmetric. In the presence of an intrinsii angular momentum, the angular-momentum density with respect to anr
given centre is given by
L:rxg+f.
(e.37"
Here r is defined to be the intrinsic part of L, i.e. r is independent of the size of the system, whereas the position-dependent part r x g scales with the size R of the system as R2. The thermodynamic equilibrium state of such a system is found by maximizing the entropy as given in (9.a). The resulting siate is characterized by a uniform temperature Z, chemical potential ¡t, angular velocity ll and uniform linear velocity t¿. However, because of the appearance of g in the expression for L, (9.37), the total velocity ttn obtained by taking the derivative of e with respect to g at fixed f is nonuniform. In fact, it is given by a superposition of linear and rotational motion:
un:u*l|xr.
(e.38)
Equation (9.38) may be used to express the angular velocity in terms of the velocity un as S2:TyXu,,. (e.3e) This relation carries over to a local-equilibrium situation, i.e. into the hydrodynamic regime. It implies that lt is not an independent variable. Furthermore, the thermodynamic force y,e¡ is a second-order spatial derivative of un¡, and may only give rise to dissipative terms of higher order in the gradients. Hence additional transport coèfficients connecteã with the internal angular momentum do not appear. There are, however, ne\ry reactive terms in the momentum conservation equation. Since r x g is no longer a conserved angular-momentum density, the stress tensor is no longer symmetric. Rather, a term matching the antisymmetric part of the stress tensor appears in the equation of motion for t, so that the conservation of L: r x g + t is guaianteed. It is no\ry convenient to define a new momentum density j (Martin et al. 1972) by
j-g+lyxt, (9.40) yillr the pr-operties (Ð Í dt, j : ! d3r g up ro surface rerms and (iÐ I dt, r xj : I dt, (, x g+ r). sin.e r xjõun u" seen to be identical with the
9
418
Hydrodynamíc Theory
angular-momentum density L, the stress tensor in the equation of motion for j must be symmetric. This property facilitates the task of collecting all possible terms in the stress tensor allowed by symmetry.
The intrinsic angular momentum
r partly originates from the orbital
angular momentum of a Cooper pair, which in the A phase is directed along the vector i. this part, f6, points along î, i.e.
ts:
tsl.
(e.41)
The angular-momentum density fe is a nonhydrodynamical variable, and may be viewed in analogy to the superfluid density p", for example. To understand the significance of fo, we consider a uniformly rotating container filled with 3He-4. In equitibrium the normal fluid will move rigidly with the walls of the container. There will be a change in the free energy proportional to the angular velocity là. For a large system, this change will be dominated by the macroscopic contribution to the angularmomentum density, , x g, which scales with the size of the system as R2 x volume. However, there will also be a change in free energy induced by the intrinsic angular momentum. It has not so far been possible to experimentally determine ús (but see Hall L985, Liu 1985). For finite vorticity g¿, there is also an induced intrinsic angular momentum
tin¿:0d2,
(e.42)
where 0 is a moment-of-inertia tensor, which has to be positive for stability reasons. The total intrinsic angular momentum is the sum of the two contributions (9.4L) and p.a\:
t:
to
*
f¡r,¿.
(e.43)
The induced angular momentum f¡,,¿ should be expected to exist even in the normal phase, but its value would be rather small
9.3.2 Equations of motion for the symmetry variables We are now in a position to embark on the derivation of the hydrodynamic equations for the A phase. As discussed in detail in Chapter 6, in this phase the rotational symmetries in orbital and spin spaces and the gauge symmetry are broken, i.e. all the symmetries are separately broken. However, it has already been pointed out that gauge transformations and rotations about the vector I are indistinguishable as far as the transformation properties of the order parameter are concerned. In other words, only the relative gaugeorbit symmetry (GOS) is a relevant dynamical symmetry. The dynamical symmetry variables are therefore the three GOS angles 0Y". Their differentials are related to the infinitesimal changes in the orbital rotation
9.3 Hydrodynamic angles
ôgi
Equations
for
the
and the phase variable ò0F" --
A
Phase
4I9
ôô" by (Liu and Cross ô0i
-
/,
ô0"
1979)
(e.44)
Note that only one of the angles 0f", namely the one describing rotations about l, is a relative symmetry variable, corresponding to the u(1) symmetry discussed in Chapter 6. The notation used for the infinitesimal changes in 0i, @" and 0f" is meant to indicate that they are not differentials in the usual sense owing to the noncommutativity of three-dimensional rotations. We shall not consider the spin degrees of freedom of the order parameter explicitly, because for frequencies much lower than the longitudinal NMR resonance frequency dùt, the vector â wiil be locked in the direction along i by virtue of the dipole force ("dipole-locked regime"). On the other hand, for frequencies of order g¡- or higher, the i-vector field cannot follow, and may be considered as being in the equilibrium state, since its characteristic frequencies (orbital viscosity, see below) are much lower. It is convenient^to represent ôgf" in terms of changes in a phase variable Q and the vector I by defining
ô0: -i. ô0t",1 ôi: ô0s" x i. J
(e.4s)
Furthermore, a superfluid velocity variable % may be introduced in the usual way (see (7.10))
u':
hõó
k s
ZmË: - 2*>
IiV 0?"-
(e.46)
In a nonuniform texture the superfluid velocity thus defined will not be irrotational. This follows from the noncommutativity of three-dimensional rotations, which gives rise to the noncommutativity of two (ot more) successive derivatives of Ofo, exactly in the same way as found for the spin-orbit angle 0'" in the B phase. Indeed, by employing the relation (9.15) as applied to 0!" and making use of d0:îxdi we find for rhe curl of u. the so-called Mermin-Ho relation (Mermin and Ho 1976) (V x q),
h : - L) .*(Vrl^V ZftI ¡*m
kee-)
:L\, a*1 e,,oî.(vri xvofi.
(e.47)
which has already been derived in (7.53). We may now write down the reactive terms in the equations of motion for the symmetry variables. They are obtained by taking the time derivative of (9.44) and substituting lor dï|ldr and dþol¿t from (9.13) and (9.8). One
9
420
Hyd.rodynamic Theory
finds
d9l"
dú In terms of
f
and
I
^ 2mu _= d2¡* Iti.
(e.48)
(9.¿S) takes the form
#.î.s¿+T:0, !: ç xî. dl Note that the chemical potential in (9.a8) is the one at fixed t, i.e. This is related to the chemical potential at fixed d2, lts¿, by
þ,: ttn-
a.(*g),
(e.4e) (e.s0)
þ:
þr
(e.s1)
as will become clear later (see (9.60)). The chemical potential at fixed differs from its value at rest, po, by the kinetic energy per particle, i.e.
l}
(e.s2) ¡to+ O(t*), in analogy to the dependence of p on the linear velocity u,, given by (9.56). To first order in f2, one therefore finds
Fa:
#.('-*) î.ç*T--0,
(e.s3)
where pr: (2mlh)to is an effective mass density associated with the intrinsic angular momentum. It follows that a finite vorticity A -- iF X u,,), due for example to a mechanical rotation of the sample, will wind up the phase and will eventually lead to a supercurrent. This is the remarkable gauge-wheel effect (Liu and Cross 1979, Ho and Mermin L980, Miyake and Usui 1980b, Combescot L982a), which offers a direct link between macroscopic mechanical motion and quantum-mechanical phase. It is seen that the prefactor of ttte î.l} term in (9.53) depends on the density variation of the intrinsic angular momentum. Microscopic derivations of the hydrodynamics (Nagai 1980, Combescot 1980a,b, 1981a,b) give a strongly temperature-dependent coefficient of the î.ç term, which even vanishes for T+0. As will be shown below, the latter property may actually be derived if one assumes the existence of an intrinsic angular momentum.
9.3.3 Thermodynamic identities and equilibrium conditions In order to proceed with the derivation of the full hydrodynamic equations, \rye now establish the local thermodynamic relations. Let us consider the entropy density s as a function of the energy density e , the mass density p,
9.3
Hydrodynamic Equatioru for the A phase
421
the momentum density g, the superfluid velocity u. and the i field, as weil ar VJ¡ Td the internal angular-momentum density t. From the Galilear invariance of s, it follows that
s(e, p, gt ns,¡, VJi,f)
:s(e t
g . u + *pur, p, g + pu, u"+ u, î,Vilj, t), (e.s4)
where -z is the velocity of the moving frame. Observing that in local equilibrium the temperature T, chemical potential and normal velocity ¡r un are related to the partial derivatives of s by
ôs 1 ôs_ p ôs un a.:7' ôp: - T' ôr: -Ë, we find that the chemical potentiar transforms according to
!t,:lt-un.u_Luz,
(e.ss)
so that þ¿ may be expressed in terms of the chemical potential ¡ro in the ur,:0 frame as
þ:
þo- lr?.
(e.s6)
Putting u:-t)s in (9.5a) and taking the derivative with respect ro us, we find furthermore
-afr -s-pu,,-8".
(e.s7)
The rotational invariance of the entropy density impties s(e, p, gtns,¡,VJi,
f):s(e, Y ,1,
p, g +ôar
xg,
us
* ôa¡ x u",î +ôar x i,
+ (ôar x y)J¡ + V,(ôø x î)¡,r + ôar x
r)
(9.5g)
for infinitesimal rotation angles ôø. To first order in ôø, (9.5g) provides with the relation
t. (Tr, + ¡
[v+'# +viî",b]
*1rr"x (q -
u,,)
us
*ToX r:0. (e.se)
Here the derivatives of s have to be taken at fixed r. The local thermodynamic identity following from (9.54) takes the form
z
ds
: de - pt dp- un. ds -s". d4 -$}a
.
dî
-?
ffi
d(v,/,)
- dz. dt, (e.60)
where is the bending free-energy density (7.20) "fu.,,¿ The pressure is defined by considering a change in volume with the mass, energy' momentum and angular momentum, as well as the symmetry
9
422
Hydrodynamic Theory
variables, held fixed:
^ç P:-TÏ,dv
fs
On combining (9.60) and (9.6L),
-
€ + pp
*"
gI
*u,,.
(e.61)
d2. t.
are led to the Duhem-Gibbs relation
dP:sdT + pdl.t *g.du,.,-g..d%
* rqu. di+ .
. dd¿.
t ? uffid(v,/,) The appearance of the last term in (9.62) is one of the observable +
(e.62)
t:
the pressure consequences of an intrinsic angular-momentum density should increase or decrease in a rotating vessel depending on the relative orientation of Î and f). Even if one assumes ltl (klm)p, which is probably an upper limit on lrl, the ensuing pressure difference is quite small, namely of order L0-7 Pa for a rotational speed 1, rad/s. The conditions for total equilibrium at temperature T can be obtained by (T, p,g,u",VJ¡,î, f) sube Ts minimizing the free-ene^rgy density ject to the constraint and the Mermin-Ho relation (9.47). The In condition on the normal velocity u,, is given by (9.38), i.e. order to guarantee stationary mass and energy density, the divergence of g, must vanish: V'9,:0. The i texture in equilibrium is determined by requiring that the change
-
f: -
I.I:L
:f
V'u':0.
in the free energy with respect to a local change in I vanish. In other
words, the local torque on the texture must be zero:
îxE:g'
(9.63a)
where
,
,p,
ô.fu.r,¿ E :3ffv
.
ôf*.-., k
-o
ffii ;[î +
"
(s" . v)î],.
(e.63b)
Here the derivatives are understood to be taken at fixed ú. The last term in (9.63b) derives from the dependence of ,fu",,¿ on the superfluid velocity u,. As u, is defined in terms of an orbital rotation about I (see (9.45)), a change in i naturally leads to a change in u,. More explicitly,
h 9gr: ft \' ð,,n, (lkYi9ef): ð1,
*4¡
2*Y¡07
:
h
*Q xYtl)¿'
(9'64)
where in the last equality use has been made of d0" : î x dî. From the equilibrium conditions as stated above, we may conclude that the thermodynamic forces driving the system towards equilibrium are given by
(i) the temperature gradientYT; (ii) the gradients of the normal velocity, of which three independent
9.i
Hydrodynamic Equations for the
A
phase
42'.
combinations may be formed:
(a) the shear flow
V,,: )(Y,u,,; * V¡u,,,,) - ]ôuV . u", (b) the bulk flow V. u,,, and
(.)
(e.6s'
the superfluid bulk flow V . g.;
(iii) the torque acting on ¿ î
^
,p.
The spatial derivatives of the vorticity, which have to be taken into account as thermodynamic forces in rotating equilibrium, like v . dl, are one order higher in the gradients in the noniotating case and will be
discarded.
9.3.4 Entropyproduction In full generality, the equations of motion for the order parameter fields i and u, have the form (Ho L977) aî
-:û)xr,
(e.66)
dt
*:
-o,(,
.*î.,).*t
(s*,. v,Ð,
(e.67)
where ot: d2 to lowest order in the gradients. The form of the ø-dependent terms is determined by the requirement that the order parameter rotate as a rigid structure. The last term in (9.67) comes from the noncommutativity of the time derivative in (9.a9) and the spatial derivative in the definition (9.46) of u.: in order to derive (9.66) fròm (g.4g), the two derivatives have t.o be interchanged, which introduces a term aîrct x Vi via the relation (9.15). Equation (9.67) is usually referred to as a ..Josephson equation,,. The conservation laws for mass, momentum, energy and have "ntropy their usual form. In particular, the mass current, b"ittg identical with the momentum density ( a conserved quantity), can be expressed in terms of the 1*o local equilibrium quantities pun and g,, which have the correct behaviour under Galilean transformations:
8:
Pa'-*
9".
(9.68)
The conservation law for angular momentum may be stated as
equation of motion for the intrinsic angular-momentum density: s _> : ;t)ú jk e¡¡¡,rl¡r, _ )j ðtt
v ¡B¡,,
an
(e.6e)
E,,is an, as yet undetermined, angular-momentum current density. In writing (9.69), we have split off the term -E,o€¡¡tI4t, representing the wh.e.re
9
424
Hydrodynamic Theory
antisymmetric part of the stress tensor. This term is necessary to guarantee conservation of the total angular momentum L: r x g + t. The entropyproduction equation may now be derived from the local thermodynamic identity (9.60) by substituting the conservation laws (2.21)-(2.23) and the equations of motion for I, (9.66), and u,, (9.67):
'#'
- -V j. + LtV . g +) .
*s. . (o, *
-?
ffi
u^rY,II,,
*+ t,v,,)- %fl. (o x î) Y,(a xD¡.
?
"(?
e¡¡¡,Ir¡t +
v,lti)
(e'70)
Note that the term involving the time derivative of I in (9.67) has been written as i. @îlat xV,l: V¡(Î ' at) - î ' 7V,ot¡, using (9.15). Our next goal is to cast the right-hand side of (9.70) into the form of a sum of a divergence and terms proportional to the thermodynamic forces. This is achieved by
(i) splitting off the pressure term Pôu from the stress tensor IIt,
and
making use of the Duhem-Gibbs relation (9.62);
(ii)
expressing the gradient of the normal velocity as Y fln,j - Y,i + åôøV . u., * e¡¡*d)ú
(e.71a)
(iii) employing the Mermin-Ho relation (9.47) to write 8, . (u,, . v)r,
- u,, . (g, . V)u" - *i
.
[(r" . v)i x (g. . v)Î]; (9.71b)
(iv) making use of the following equality derived from the identity (e.5e):
-? o,, " ffi
*s" x u3 : î x q' +
*rrr,. v)i
d2 x t; ¡t v,fiL " **l d(v,r)l -
(e.lìc)
all terms
containing spatial derivatives other than the thermodynamic forces as a full divergence minus the terms that were added to make it a full divergence.
(v) writing
The result of this reformulation is
: l*o.(,"" .tÊ) -+lit"'vr + 4 ,:,yu. å? P¡,Y 'un q,)
+] nuv,o,],
(e.72)
9.3
Hydrodynamic Equations for the
where Pij
(a generarized
A
phase
42
pressure) and
Bij
(connected
angular-momentum current) are determined bätow. Here the heat current density jø is given by
io:
io.- s,(øo
.
'
*,
,") - r
>tä
(ro x D,
to the orbita
- eç. (s.73
The index 0 indicates a quantity in the frame of reference wher€ u,,:
The transformation laws are given by (9.55) and by
î.,,-î.
2m
'o+ L u''u"'
(e.74)
îxrrø:îx@o*(u^.V)î,
j, :
jor*
Q
esu,,
*
rlu,,
-
(e.7s)
är|'5,
(e.76)
where
€0:€-U.,. gr-äpU?, ¡I,,:
n?,
+
Bs,iun¡
*
un,¡gs,j
I
(e.77)
pun,¡un,i.
(e.78)
The Galilean transformation properties of the energy density e, the energy current density ¿ and the stress tensor fI,, are the same as for an isotropìl superfluid (Khalatnikov 1965), with, however, nonsymmetric Ir¡,. The quantities P,, and 8,, are found as P¿¡: If;¡ g,,
- ,ôii - g,,¿(u. -
: fiii - rrn.¡tj -
un)¡
+
rffi
fts".,î,- [i "
.
vrl,
.rå],
(e.7e)
(e.80)
9.3.5 Derivation of the hydrodynamic currents We now write the currents as linear combinations of the thermodynamic forces. Since the equilibrium state is characterized, by the i texture, the expansion coefficients will in general depend on i. It -is now important to tealize that î isproportional to ttre angulãr momentum of the Cooper pairs. Consequently,I changes sign under time reversal. This property provides us with a certain freedom to add factors of f to the entropy production that are odd (reactive) or even (dissipative) under time reversal.
(i) In the case of the stress tensor we have to allow for two reactive terms
î, ,p and to vij, (g.6s), respectively. Adding up all terms that are allowed by symmetry, the stress tensor takes the following in P,,, proportional to
9
426
Hydrodynamic Theory
form (with (V x 1),¡= e¡*tîtVn):
.v,î + gs,ius,¡ * un,¿gi + Lyo¡pi|, + piî,) A0rr) , v-,_ + y,( v x î¡V+ 7,,[(Î x v\Jil + þt(î,rþ¡ - î¡rþ,)
rr¡¡: _Põ,, __.t _.t+**
+ e,,o1o(Blv . g. +
Blv . u., * Fiîvi¡+ Bl 2* eu,v,oîo + n?,
(9.81)
where To, T t, 1¡¡ and Bl (t : 1, . . . , 4) are reactive transport coeffrcients and r/¡1 : 1þ¿ - 1,(î' rP). The dissipative part of the stress tensor, Iffr, contains the usual terms proportional to V,¡, Y. u,, and V .9., and in addition a term involving the iextural force Î, ,1,. By taking cross-products of Î with ú o, î * ,p, one can also construct antisymmetric terms. Introducing transport coefficients in a way that is as close as possible to the usual notation, one may write llf; AS
II3
:
-Qq[î¡î¡
- ,t'rõ ü')(îvÐ - 2q' à õivHô
zrt,'ll,)
tovo,ô;
* 1; --¡¡] -
rjr
(Eiô,i' + Ç\1,î)v . u^
Gi ôi + gtll,î,¡v.8, - iu,l(î x ú),î¡ + (Î x ,Ð¡î¡ 1-;4 e¡¡el-t'2(î x vÐo + h[î x (Î x 4ùlol,
Q.82)
where ôut= 6,j-î,1j, and Ttt't, q!t, hr, Lr, hr, gl't and E!)'t are the eleven transport coefficients characteruing the stress tensor. The shear-viscosity coefficients 4tt't and r¡[ are such that they reduce to the usual shear-viscosity coefficient in the timit T+7., while 4f tends to zero in this limit. The same is true for the bulk-viscosity coefficients (t'a and Çllt. The antisymmetric terms of IIf; do not contribute to the entropy production.
(ii) In the heat current density ja a reactive term appears that is linear in i loOO time-reversal symmetry). It is directed perpendicularly to VT (no dissipation). In the dissipative part there is only the usual diffusive heat current if one drops terms involving gradients in f). The total heat current is then given by
ja:
y(î xVT)
- x,YT -
(*,,
- x )[î(î - Vr)].
(e.83)
Here r,, and ¡ír are the coefficients of the thermal conductivity for heat flow parallel and perpendicular to i respectively.
(iii) In the Josephson equation (9.67) symmetry does not allow for reactive terms besides those already discussed in the context of (9.48) and (9.52), i.e. one has Í -ao:î. ç in (9.66). There are, however, dissipative terms that have to be added to the chemical potential, which are
9.3 Hydrodynamic
Equations
for
the
A
Phase
lrlPortional to V'
u,, and V'8". In addition there is a term proportional to IW, the component of v,, parallel to î; one then obtains t'tre äq,ration of
motion for u,: -å;/;ì"+
o,[r,*u,,.
where
"***(r.*) pD
î.s¿
: -EtY. g, -
+r"]:
ftt (yr, v,l,
EoY . u^- ÇrîVî.
(e.84)
(e.ss)
The coefficients Ça and Ç, are related to the transport coefficients in the stress tensor (9.82) by onsager's symmetry relations in the following way: En: \(El + ze[), (e.86)
Es: El-
E{.
(e.87)
(in) Equation for i: in the thermodynamic flux ar6 - fl multiplying the jo.qT". lx g -in (9.72) symmerry a[ows for rerms ãriu"ã uy {-teljure I x (l x !t) and by I x W on the reactive level. The first term yields a vanishing entropy production, while the second term is cancelled by the term with coefficient yo^ in the stress tensor given by (9.g1). In the dissipative part øD only î * ,p and Vî possibl,r th"r-oOynaiic forces. The equation of motion for the vector i^are consequently takes the form aî
^
xî-
yo(vî)'+ uîx
- l,rîvxî - Lorl,', (9.s8) where the superscript l- denotes the component perpendicular to i and
ä*(r".v)i:
dZ
q,
yo, &, Ât and )va are transport coefficients. Note that j,llvolr¿ is known as the "orbital" viscosity coefficient (see (10.166). In deriving (9.gg), use has again been made of onsage^r's symmetry principle in identifying the coefficient of the term îv x Î with the coefficieni LL appearing iñ the equation for the stress tensor, (9.82).
9.3.6 Further consequences of an intrinsic orbital angular momentum
The coefficients É,8 (which should not be confused with the strong-coupling parameters {Br} in the expression for the free energy (5.4)) in ltre stress
tensor (9.81) may actually be related directly to the intiinsic angularmomentum density in the event that the induced angular momentum f,,'¿ of (9.42) can be neglected (Hall and Hook 1986). In this case r : hî ii-no longer an independent variable, but is completely determined by i and the two scalar densities p and s, i.e. f6: fs(p, s). If one now writes the antisymmetric part of the stress tensor in terms of derivatives of f as 2uruo 14*:-(å
-4
* u,, . v)tr,l,l +)v,n,, r,{*,8,¡t¡
r[t " .ã#],],
(e.8e)
9
428
Hydrodynamic Theory
using (9.69) and (9.80), one may convert the derivatives of ts into derivatives of p and.r, that is to say, into already known quantities. With the aid of the equations of motion lor p, s and I and the identity (9.7t), E¡3ei¡*II¡* can be shown to be given by the following linear combination of forces:
)j,k e,j*njt - -(1 +tou)(î x ú)¿ *
.
?
[o+
"
toyo(vb! +1,(ur9. u,,* uzY - g")
t#1,
where
&t:
P
ôfo op
*
t*, x (u, - u,,)],,
(e.eo)
.
*
'Ì
(e.e1)
ôúo
uz:;op
2m
By comparison with the general expression for II¡¡ given in (9.81), the coefficients of the antisymmetric terms are identified as follows
Éä: -+(L * tsu), I þl : änr, þl : tur,l Él: -Éf : -toqo. )
(e.e2)
The term Y¡B¡i has been neglected here, since Bri is the flux conjugate to Y¡Q,, as can be seen from (9.72), and hence is proportional to the thermodynamic forces itself. Let us finally comment on the superfluid current density gs âs defined by (9:57). It is not identical with what is usually calculated in GinzburgLandau theory (see (7.35)). Th" difference is that g., and hence g as defined in this chapter, is given by the derivative of the free energy with respect to u3: u" - un at fixed internal angular-momentum density r, whereas in a static microscopic calculation the derivative is taken assuming u,, = 0. This difference appears only in the coefficient C, of the (V x I).,_ term in (7.39). The additional contribution to C t arises because in the microscopic derivation of hydrodynamic theory one finds a term in the momentum density in addition to the terms discussed in (7.39), which is of the form (Nagai L980, 1981a, Combescot and Dombre L980, Combescot 1980a, Mermin and Muzikar 1980) Ag
: -aQ)î
xY p
(e.e3)
may be related to the superfluid mass density p,, and the coefficient C, (in the calculation where u,, - 0), given by (7.22a) and (7.22b)
where
a(f)
respectively, as
a(n:+*r0.,,+
c,-)
(e.e4)
9.4
Hydrodynamics
for Finite
Magnetic Field
429
As mentioned at the beginning of this subsection, we have adopted the point of view put forward by Mermin and Muzikar (19g0) thar (l.o+¡ i, nothing but a contribution to I from an assumed intrinsic angular-
momentum densitv
t
:2pø(T)î.
(9.9s)
The contribution (9.93) to the current is due to additional terms in the
free-energy density
Â/u.n¿
on performing A.fu",,¿
: -+(a,- u") . (i x Vro).
(e.e6)
a partial integration, this may be cast into the form
:
-T,to(u"- u,,) . (V x î¡
-
toî .
dl + surface
terms.
(g.g7)
Here the second term on the right-hand side is interpreted as the energy of a spinning top with angular momentum hî in a system rotating with angular frequency fà. We note that the first term has the same stiucture as the term multiplied by C in (7.39). This allows one to combine the two to yield ' a single term with coefficient Cr, where
et:Cr-lto.
(9.9g)
Note that the momentum density g in the frame where u,, :0 is given by differentiating /o.',o with respect to u,, at fixed gl : Ly x un:
8--(k)"
(e.ee)
The unconstrained functional derivative of .fu..,¿, which is usually taken in microscopic theories, gives an ertra term Âg : åV x f from differentiating fàin the second term of (9.97), so that -ô/o."o/ôun: j, the current defined b!
(e.40).
9.4
HYDRODYNAMICS FOR FINITE MAGNETIC FIELD
In a magnetic field the nature of pairing changes as a result of the Zeeman energy of the Cooper-pair spins. The resulting distortion of the order parameter has been discussed in Section 5.4. In small magnetic fields and if one is not too close to the transition, the order-parameter structure may be approximated by its zero-field form. This approximation has already been used in most parts of Chapter 8 and also in the derivation of the spin hydrodynamics for the B phase in Section 9.2. In this section we wanr ro explore some of the qualitative differences in the hydrodynamics brought about by the modification in symmetry of the ordei parámeter due to a magnetic field.
9
430
Hydrodynamic Theory
9.4.1 The A phase in a magnet¡c field A particularly drastic difference exists between the order parameters of the A phase and the A1 phase given by (3.68) and (5.59). It this case a change in the phase of the 111 ) Cooper-pair \ryave function can be undone not only by a .òrr"rponding rotation about the axis Î in orbital space but equally wetl by a rotation about the axis/in spin space (see Chapter 6). Here one has a spontaneously broken relative spin-orbit-gauge symmetry (SOGS) as compared with the broken gauge-orbit symmetry in the A phase in zero magnetic field (Liu I979a).In the A2 phase SOGS is broken for both spin components, lff ) and ll"l ), of the condensate.
The proper dynamical variable characterizing the broken gauge symmetry is therefore not the phase variable @1,r itself, but rather the variable describing the difference of the phase variable and the rotation angles about i and / in orbital space and spin space respectively. For a small change in this variable, one therefore has
: dOÎ - î' do"-,Î' do",l dór : d,óÎ - î' do" + j' ao'.) dór
(e.1oo)
The sign of the last term in the expressions for d@t and d@1 is T1, according to the different axes of spin polarization, If, in the two cases. In the A2 phase it is physically meaningful to also introduce the total phase @ and the relative phase of the two spin populations, @"o: dQ
: L@Q¡ + ¿@r), I
d4"p: i@Q¡
-
doJ.J
(e.101)
Employing the general relations (9.8), (9.9) and (9.L3), the equations of motion for @t,, are obtained as
dór,r: dt
Ft,t-î.s¿Tî.r',
(e.102)
where þ1,r:m-r1Ae¡An¡,t)r,",... are the chemical potentials of the (1,J) spin species and the term involving ar" (see (9.9)) describes the restoring force in a situation where the spin ,S is out of equilibrium. For the At phase one has only one of the equations (9.102), say that for @t. The last term in (9.102) implies that a supercurrent (see (9.105)) is generated by a gradient in the applied magnetic field (magnetic fountain effect). This has been observed experimentally by Ruel and Kojima (L985), although unexpected nonlinear behaviour has been found (Ruel and Kojima 1986). The equation of motion for the total phase @ has the same form as in the
A
phase:
Y:-ryp-î's¿, dth with ¡r : LQt¡ +
f¿r).
(e.103)
9.4
Hydrodynamics
A new equation
for Finite Magnetic Field
43.
appears here, namely that for the spin phase:
m. 99":_î u.a,-¡\h-ltù. dt
(9. to¿
Apart from the last term, this is the longitudinal part of Leggett's equatio (5.27b) for the vector á. B,rt this last t.r* vaniihes in loãíl equilibriun when þ¡: þy The hydrodynamics of the A, phase may be looked upon as a three-flui dynamics (Liu 1980, Gurgenishvili and Kharadze 1980; see also Baramidz et al. 1983), the three interpenetrating fluids being the two superflui components of the up- and down-spin populations and the normal con ponent. In the following we consider only the case of unifor^ î- an ;f-vector fields. The theory for the general case has not yet been completel worked out (see, however, Pleiner and Graham L976, pleiner rg77a,b Saslow and Hu L981, Pleiner and Brand 1981, Brand and Pleiner 1981a.b 1982, Dombre and Combescot L982a).
we start by defining superfluid velocities u1,, for the f , I
ul,J:
subsystems:
ft
(e.10s
z*YQr,t.
The superfluid velocities u, and u,o driven by the gradients in the total phasr Q and the spin phase 4"p âre defined by
u": ä(ur+ ,,) u,p
:*ye:*+
: ä(u¡ -ur) :
ñivñi
*ror"r: *âuvêu.
It follows from the last equation that V x u, and y x %n satisfy Mermin-H<
relations (9.47) involving the î and j fields respective¡i. Noté rhat u,o doer not transform like a velocity under Galilean trahsformations; in fact it doer not change at all! The mass (spin) currents in the normal-fluid rest frame (u":0) ar( obtained by adding (subtracting) the contributions from the 1, l
subsystems:
gs: p{ut-
h.
u")
. i"p:;Lo{+ -
* púat- un), l t píutu,,)
-
"")1. J
Here p¡,¡ are uniaxial tensors oriented along Í. (9.107) in terms of % and u.o:
It is usefur to rewrite
- u,,) I pvu"p, I i,o: fi,Ír"ou,o + pr(u, - ,Jl,J g":
(e.107)
p"(%
(e.108)
9
432
Hydrodynamic Theory
where P¡,
(9.109a)
PIt: P¡- Py
(e.10eb)
Pt: Ptp: P¡*
The equality of p. and psp only holds if the two subsystems are strictly decoupled. In the real liquid, however, the exchange part of the Fermiliquid interaction provides a considerable coupling, and this causes psp' (7.23c), to deviate substantially from ps, except near [. The difference between p¡ and pr is due entirely to the different magnitudes of the gap Á1,¡ (see Section 5.4), and may be calculated within weak-coupling theory. It turns out that pru is sizeable only very close to the A, transition and, of course, in the A1 phase, where PN:-pr. lt is interesting to note that in the region where pru is comparable to P",u" no longer has the meaning of the superfluid vetocity of mass flow (see (9.108)). Rather, the characteristic velocity changes from u, to u, as the A1 phase is approached. We shall not reproduce the equations of motion for the transverse parts of the orbital and ipin symmetry variables i and Î, ot the conservation laws (except for spin conservation), since they are similar to the equations for the A phase discussed in the previous section. The longitudinal component of the spin conservation law on the reactive level is given by
*.
v . (s,u" +i'o)
: Lf o?,Q,o,
(e.110)
where the dipole-torque term has been included on the right-hand side.
9.4.2 The B phase in a magnetic field Just as in the case of the A phase, it is also useful here to consider the spin components separately. There are three spin components in this case, which have the orbital structure given in (5.66). Each spin component is invariant under a particular combination of a gauge transformation ettn, eieu and an orbital as well as a spin rotation. These degeneracies are eliminated from the dynamics by introduction of the following phase variables for the three condensates:
: dE' + î. d0" -,Î. d0',1 d@u : dqu - î ' do" +i' do', I d@r: L@En + dEu). )
d@n
(e.111)
the last equation takes into account.that the phase relation between the 1I and 11 or l"l, components of this r,éäl-valued order parameter has to be preserved (see (5.66)). It also fo[bws that dóru : +@A¡ + d@u). Hence there exist two independent phase variables here, which may conveniently
9.4
Hydrodynamics
for Finite Magnetic Field
43.
be chosen as the total phase rf and the spin phase rp,o:
d0: d4,p:
L@þtt + døu) = d@r, +@Am
(9.Ir2a (9.rrzb
- dóu).
The corresponding Josephson equations are given by
do
(9.113a
-:dt
9f:î.dt-j.a"-Çtu,-þ),
(9.113b
where þ¡,t:m-'Aelðn1,, is the chemical potential of the f, I populations The term I - A in the equation for @"o implies that there is a gauge-whee effect for the spin-phase variable rather than for the total phase variable ar is the case in the A phase. In equilibrium, a f,nite excess magnetizatior (ø'* 0) causes the sample to rotate ("Einstein-de Haas effect"). Equiva lently, a solid-body rotation of the sample (A + 0) causes a magnetizatior ø'("Barnett effect"; see Combescot 1980a, Volovik and Mineev 1984). The last term of (9.113b) would again be zero in local equilibrium were i not for the fact that we have to allow for an intrinsic orbital angula. momentum density related to S. According jo microscopic theorr (Combescot 1980a), a finite spin polarization 5: d/gives rise to an intrinsir angular-momentum density ¡ - B"Sî, (e.114)
:
where B, is a coefficient that varies between L and 0 as one goes from T
to
:
C
7".
We now recall that the chemical potentials ¡rr,r in (9.113) are defined at fixed t. However, in a rotating vessel the quantity characterizing the equilibrium state is the vorticity d2, not f , and only the chemical potentials ltt,tþ) at fixed dI are equal. If we then express ¡r1,1(f) in terms of ¡rr,r(.fJ) by
p¡,ít): þ¡,íQ)-
a'#r,
using (ðEr,rlðt)o: Q, and take the derivative of
ôS +:-ß' dpr,r ""^:Tz*þ'l'
(e.11s)
f, assuming r: ro(S)i
k
as
(e.116)
we are led to the Josephson-Leggett equation
\ : -î.(,' dt '\ts-YH)+(1
gfu
The Barnett effect is seen to disappear at T
-PJî'a'
:0,
when ôro/ôS
(e.LL7)
:
1.
9
434
Hydrodynamic Theory
FURTHER READING Review articles and books Brinkman W F and Cross M C 1978 in Progress in Low Temperature Physics, Vol. VIIIA, ed. D F Brewer (North-Holland, Amsterdam), p. L05 Forster D 1975 Hydrodynamic Fluctuatioru, Broken Symmetry ønd Correlation Functions (Benjamin, Reading, Massachusetts) Hall H E and Hook J R 1986 in Progress in Low Temperature Physics, Vol. IX, ed. D F Brewer (North-Holland, Amsterdam), p. 143 Khalatnikov I M 1965 An Introduction to the Theory of Superfluidiry (Benjamin, New York) Putterman S J 1974 Superfluid Hydrodynamics (North-Holland, Amsterdam)
Articles
\
Ho T L 1977 in Quantum Fluids and Solids, ed. S B Trickey, E D Adams and J W Dufty (Plenum, New York), p.97 Hu C R and Saslow W 1977 Phvs. Reu. Lett.38 605 Liu M lg82 Physica 10!)/110 b + C 1615 (Proceedings of the L6th International Conference on Low Temperature Physics, LT-1,6) Mermin N D and Muzikar P 1980 Phys. Reu. B.2l98O Miyake K, Takagi H and Usui T 1981, Prog. Theor. Phys.65 11L5 Volovik G and Mineev V P 1981 Zh. El<sp. Teor. Fiz.81 989 lSou. Phys. JETP 54 s24 (1e81)l
10 Transport Properties The last two chapters dealt with the phenomenological theory of dynamica properties (spin dynamics and hydrodynamics) óf ,op"rnii¿ ,rt"rrri, theory, however, is still incomplete since it contains a number of as yet un. known coefficients. Furthermore, the hydrodynamic theory is restricted to thr regime of small frequencies (øø << 1) and wavenumbers '(qr.r<< 1). In thir and the next chapter we therefore want to address these questions withir the framework of a semimicroscopic theory. At present such a theory har only been explicitly worked out for the caie of linear response and within
the weak-coupling model. Quite generally, the dynamics of a system possessing long-range order is determined (i) by the structure of the elementary excitaiions and their interactions and (ii) by the laws of motion governing the order parameter (which, in a sense, describe the ground-state correlations). In the case of a
Fermi liquid the most important elementary excitations are the singleparticle excitations, the so-called Bogoliubóv quasiparticles, which are fermions. At very low temperatures (r<0.1?") gapless elementary excita_ tions of the Bose type, like phonons, (i.". those that do not -ugrrorr, require a minimum energy) will eventually b".o-" important. In view of the temperature range that is experimentaliy accessible uì pr"r"nt and in the foreseeable future, we shall confine our disðussion to Fermi excitations. One may distinguish four dynamical regimes in a pair-correlated Fermi
liquid:
O fhe hydrodynamic regime.In the case of very slow disturbances, the fluid of thermal excitations and the Cooper-pair condensate are both in a state of local thermodynamic equilibrium, and hence are completely described by a few collective- variables only. This is the hydrodynamic regime considered in chapter 9. For the thermal excitations to be in rocal equilibrium, the frequency @ and the wavenumbet q of the disturbance are subject to the conditions ø, upq1lllt, where ris a typical quasiparticie collision time and u. is the Fermi velocity. The corresponding conditions for the' fair condensate are @, upe << alk, where Á iJ the averaged temperaturedependent energy gap. These latter conditions are much less stringent, except for a very narrow temperature range below the
436
10 Transport Properties
(1i) The macroscopic regime. For somewhat larger frequencies and wave vectors such that r-1<< Ø, upe 11alh, the pair condensate is stilt in local equilibrium, but the quasiparticle fluid is not. In this "collisionless" regime the description of the quasiparticte fluid requires the specification of infinitely many variables-namely the components of a distribution function in momentum space. This is so because in this case quasiparticles from different portions of the Fermi surface may respond quite differently to an applied ñeld. (i11) The microscopic regime. In the case of rapidly varying disturbances such that o> alh, upe=- alh, the fluid of thermal excitations and the Cooper pairs are no longer well separated, because the external field may induce transitions between the subsystems. Then the pair condensate is no longer characterized by a few symmetry variables, but rather by a full distribution function in momentum space.
(iv) Finally, for ø, upq)) Alh, that part of the available phase space where pair-correlation effects matter constitutes only a small portion of the total phase space accessible. Under these conditions the system reacts as a normal Fermi system.
In the present chapter we shall concentrate on the hydrodynamic region (case (i)). However, rather than deriving the complete hydrodynamic equations from microscopic theory (a task that would, to some extent, duplicate the derivations given in Chapter 9), we shall confine ourselves to the calculation of only some of the more important transport coefficients. Indeed, the phenomenological equations established in Chapter 9 would be of little value without at least some information on the phenomenological coetficients involved. These coefficients fall into one of two categories: (i) the generalized susceptibilities and (ii) the transport coefficients. We have already calculated most of the susceptibilities in Chapters 3 and 7, for example the normal and superfluid'densities p,, and p,, âs well as similar coefficients for the A phase: the supercurrent, the specific heat and the spin susceptibility. The susceptibilities are equilibrium quantities and as such are given as derivatives of the free energy. On the other hand, as exemplified in Chapter 2 for the normal Fermi liquid, the transport coefficients are much harder to calculate, because they characteñze properties of a nonequilibrium state of the system. In setting up a theory of the dynamical properties of superfluid 3FIe, we may again take advantage of the fact that the transition temperature to the superfluid state is much lower than the Fermi temperature. Consequently, the energy spectrum of the liquid is modified only in a small shell of order I about the Fermi surface. This of course does not imply that the change in the physical quantities is small, as we have seen. 'we may, however, hope that a kinetic theory of interacting quasiparticles might provide a unified and intuitively appealing framework for describing such diverse dynamical
L0.1 Kinetic
Equations
437
phenomena as stationary transport properties and high-frequency collective modes. This expectation is nourished by the splendid success of Landau's kinetic theory of normal Fermi liquids. Compared with a completely microscopic theory, such a theory has the advantage of being much more tractable, since all the simplifying features of the problem have been incorporated at an early stage. The starting point of a kinetic theory is the kinetic equation, a generalized Boltzmann equation for the quasiparticle distribution functions. In the pair-correlated state this has to be supplemented by the equation of motion for the order parameter. As shown in Section 10.L, kinetic equations in the microscopic regime (case (iii)) may be derived from rather elementary considerations, although their mathematical structure is somewhat more complex. We shall consider these full equations in Chapter 11, where we study collective modes in the microscopic regime. In the macroscopic regime the kinetic equations can be considerably simplified by applying a Bogoliubov transformation and a set of gauge transformations. One arrives at a Landau-Boltzmann equation for Bogoliubov quasiparticles, with the hydrodynamic condensate variables appearing as external driving forces. The collision integral requires special consideration and will be discussed in a separate section.
10.1
KINETIC EOUATIONS
In the framework of BCS mean-field theory the equilibrium state of a pair-correlated Fermi system is specified by two momentum-dependent distribution functions n¡ and F¿, which are 2x 2 matrices in spin space, defined by their elements
ftknþ: (alpqo*|, F*op: (a-*pa*,|.
(10.1a)
(10.lb)
Note that n¡nB is diagonal in u, B except for situations involving transverse spin motion. Under the action of external disturbances, these distribution functions will deviate from their equilibrium values, i.e. n¡,nB(r, t) and Fonp(r, r) will in general become functions of space and time. As long as the scales of the disturbance in space and time are larger than the Fermi wavelength ÂF:2nlkp and the inverse Fermi frequency kles respectively, the wave nature of the quasiparticles is not essential and a local quasiclassical description is meaningful. It is now convenient to introduce yet another 2x2 matrix representation, this time in particle-hole space, even though this introduces a certain redundancy. Fotlowing Nambu (1960), we may then combine the two distribution functions ft¡,nB ãrrd F¡oB into a 4 x 4 matrix distribution function
*:(.;
,*-.)
(I0.2a)
10 Transport Properties
438
(1 is the
2x2 unit matrix), which may also be expressed as tr.
(aiøa*') - (\(a[Ba!o,)
\ þ-oBo!o,) )' (o-opaeo)
(10.2b)
Consequently r¿ and Fe are referred to as diagonal and off-diagonal distribution functions. Employing the results of Chapter 3 (see (3.43)), the equilibrium distribution function ulr! takes the form J{oo
(f
: +lt -
r'r
;t""h
#:Ì]
(10.3a)
is the 4 x 4 unit matrix), where we have defined the 4 x 4 equilibrium
energy matrix
,?:
(tr!"J -îr::u,,)
(10.3b)
For simplicity, we consider only the case of zero magnetic field (f¿.: Eo) for the moment. For unitary states, the square of the energy matrix is just given by eo¡eoo: Bzo1. Using this relation, the equilibrium distribution function may be interpreted as a Fermi function î(*): (exlkøt + 1)-t of the matrix energy
J{2: f (eoù,
(10.3c)
defined by its power series. In the following we present the derivation of a matrix kinetic equation along the lines formulated for usual superconductors by Betbeder-Matibet and Nozières (1969) and for superfluid 3He by Wötfle (I976a). An alternative but equivalent formulation has been given on the basis of the so-called "quasiclassical Green's-function technique" (Eckern L98L, Serene and Rainer 1983).
10.1.1 Matrix kinetic equation In the presence of a disturbance with wave vector q the distribution function .Ár acquires nondiagonal matrix elements (/c + Lql J{ lk - iq) in momentum space, since, for example, the expectation value (al*ora*-q/z> +0. Also, the energy matrix will be nondiagonal in momentum space. The energy matrix may actually be considered as an effective Hamiltonian in the restricted space of states spanned by single-particle excitations only. The time evolution of the matrix distribution function may then be written in shorthand notation as
in],r:fn,el,
(10.4)
where .l1r is a kind of density matrix of the subspace considered. .ff and e are matrices in particle-hole and spin spaces as well as in momentum space.
1-0.L Kinetic Equations
439
In the case of weak disturbances we may linearize the kinetic equation in terms of the deviations of ¿f and e from their equilibrium values. Defining
(k + äql.if lk - äql : ôopl{o*+ õJfo(q, t), (k + iql elk - äql : ôo,oe? * òe¿(q, /), one obtains the linearized matrix kinetic equation
ik;õ.|{k(q, t) -
(r?_ 0¡ro
(k ¡àq
-
(10.sa) (10.sb)
fr*)
- ôJrÉ €?_) + (Jf|_ôeo - ôe¿ ,f [_) : õg*.
(10.6)
This equation is reminiscent of the Landau-Boltzmann equation for normal Fermi liquids. The time rate of change of the distribution function is seen to be given bV (i) the free motion of the system (the second and third terms on the left-hand side of (10.6), corresponding to the streaming of particles in real space in a normal system), and (ii) the change induced by forces applied either externally or generated by the mean fields in the system (the last two terms on the left-hand side of (10.6)). A third source of changes of the distribution function is due to the collisions among the quasiparticles. A corresponding term has been included on the right-hand side of (10.6). A derivation of the collision term will be presented in Section 10.2. The change in the energy matrix caused by the application of external fields is given by the sum of a direct term and a mean-field term induced by the change in the distribution function, as in Fermi-liquid theory. The latter contribution can be expressed quite generally in terms of a linear and local functional relationship þetween ôeo and ô"^fr, involving a generalized Fermi-liquid intera ction /r*t, ; ôe*(",
r):
ôei' +2/oo,õtt¡,,(r, t).
(10.7)
k
However, the relation (10.7) can be simplified if one observes that cross-terms connecting the diagonal energy ôåo with the off-diagonal distribution ôFe (and vice versa) are necessarily proportional to the gap Á. On the other hand, the reference energy for the Fermi-liquid interactions is given by the Fermi energy. We may therefore assume that the off-diagonal parts of the generalized Fermi-liquid interactions/oo,are order_f"lTi small. The same holds for the changes in the diagonal parts of / induced by the pair correlations (for a justiñcation from microscopic theory, see Leggett I965a,I966). Hence, to lowest order in T.lTe, the changes in the quasiparticle energy matrix components are given by (10.8) and
õA*np:Z k'
voo,õF*,np
(10.e)
10 Transport Properties for the diagonal and off-diagonal components respectively. Here fooB,o,n,B,is the normal-state Fermi-liquid interaction (2.6) and V¡¡,,is the pair interaction. on making the same assumptions about the /c dependence of f**, as discussed before (2.6), the first terms in a Legendre expansion of the diagonal energy change õEo*p may be written as
õE**p: ôoBNr.tFi ôn + (+kNò-'lot
)
(o*)*pS*
l.t
* ô."Bp-Lå kk
.
s+
st,#
oo4 @)*ej'*.
(10.10)
Here ôn, su, g and jfl are the changes in particle density, spin density, momentum density and spin current density respectively, which wilt be defined in Section 10.1,.2; So:lhn is the maximum spin polarization. The matrix kinetic equation (10.6), supplemented by the relations (10.8)-(10.10), will be solved in Chapter L1. In the presenr chaprer the matrix kinetic equations will be cast into a form suitable for the calculation of transport properties. First, however, we need to know how the physically observable properties are actually obtained from the matrix distribution function.
10.1.2 Gonserved quantities and conservation laws The expectation values of conserved quantities can be expressed in terms of the diagonal distribution function n¿, which is defined as the average of the general single-particle operator product a[*o,r,Ba*-qrz,o in a given nonequilibrium state. Consequently, the deviations from equilibrium of the conserved densities of particle number, spin and momentum are given by ôn
: L2 t o(to ôlfo) : ) kk
s*: lk)
kk
g
-
tto @*
(ôoo),
õJ{¡): LnZ to (o*ôn¿),
LzZ n* tro (ô"Á[*) kk
where "tÍo" and
tro
:)
n* rro
(ôno),
(10.11a)
¡t: L,2,3, (10.11b) (10.11c)
"tto"
denote the traces on 4 x 4 particle-hole matrix quantities and on 2 x 2 spin matrix quantities respectively. In addition, the energy density contains an off-diagonal contribution: òe
: LZ t o[(r? + ¡tl) õJt¡1. k
(10.11d)
10.1 Kinetic Equntions
441
The 4 x 4 matrices rp, lt:0, r,2, 3, introduced in (10.11a,b) are defined ar direct products of the unit matrix t (¡¿ :0) or the pauli matrices or (p:r,2,3) in spin space with the pauli matrix o3 in particle-hole space:
'
: (å _l),
(î ": -i,),
(r0.L2"
where superscript T denotes the transposed matrix. The conservation law¡ for particle number, spin, momentum and energy may now be derived br multiplying the kinetic equations (10.6) by ro, äkt*, hk and eoo+ ¡"rL, theí taking the 4-trace and finally summing over Ic. In this way, the particle. number conservation equation, for example, takes the form
a õn -
å
>k
tra [(ree!.
-
e?_to) ôÍrÉ
- (q"y?_ - J,l"o_"o)ôe¿] : 0,
(10. 13)
where use has been made of the cyclic invariance of the trace and a Fourier transform with respect to time has been taken. The collision integral drops out because of the particle-number-conserving nature of collisions. The trace expression cancels for a spatially homogeneous system (q :0) by virtue of the equilibrium gap equation. On the õther hand, the terms linear in q arc interpreted as the divergence of the current density j:
j: > u¡tr.- (ò"í.). k Here
u*:hklm*
(10.14)
is the normal-state quasiparticle velocity, and
ôn[:
ôn¿
* QLõeo,
(10.15a)
where
þL:*(h,r) t¡:
(10.1sb)
(E*l2k"T). Note that, owing to Galilean invariance, the momentum density is equal
and
tanh
to the bare mass times the current density:
g:
mj.
(10.16)
The equality of (10.16) and (10.11c) is guaranteed by the effecrive-mass relation (2.16), which continues to hold even in the superfluid state. In analogy, the spin current density jfl, the momentum current density II,, and the energy current density j" are found to be given by
ii,:
+k2 rofto (o* õnL),
(10.17a)
k
II,¡:Z hk,u¡,,,tr, (ôri),
(10.17b)
.Ps'r Je:-$t
(10.17c)
k
) Ç¡uptro (ô¿j.).
10 Transport
Properties
Evaluating the /s sum in the usual wây, namely by taking integrals over the energy f¿ and the angular variables, the spin current, for example, may be written as
ii,: änG + åFî) 2k ,o õn*,u.
(10.18a)
Here we have made use of the integrat
ll_-otreL: I__*r(H,r*
ffi#)
:,
(10 18b)
and introduced the notation
õn*,p: tto (ou ôn¿).
(10.18c)
The last term of (10.10) has in fact already been expressed in terms of the spin current via (10.18a).
10.1.3 Kinetic equation for Bogoliubov quasiparticles As already mentioned, the matrix kinetic equation may be simplified in the low-frequency macroscopic regime, where the pair condensate is in a local equilibrium state. In this regime transitions between the normal liquid and the condensate rarely occur. The Bogoliubov quasiparticles (BQP) may then be considered to be stable entities. Under these circumstances, a description in terms of a distribution function for BQP rather than by a matrix distribution function appears promising. This can be achieved by performing a particular Bogoliubov transformation on the matrix distribution function. In the present matrix notation the Bogoliubov transformation is defined as the unitary transformation diagonalizing the energy matrix: Qt'oeooQtf: E*ao,
(10.1e)
oll*: ( uo*' 'o'u),
(10.20)
where the components of otl¡,
\-u*nþ u*np/
have been calculated in (3.25). There we found
*knþ:
u"rl:çt*fi))''',
rrknþ:
,"*,1:(-*)l'"
(t0.ZLa) (10.21b)
The Bogoliubov transformation of the equilibrium distribution function is
10.1 Kinetic Equntions immediately obtained
443 as
oìrrltooolri
: f (E¡r^\o'-\ : ( f {nùt
o_
. ), o f(-Eùr)' _
(r0.22)
f @) is the Fermi function. We are now in a position to diagonalize the matrices multiplying ò.ff and ôe in the matrix kinetic equation. For this, we multiply (10.6) iro¡¡ the left by oVo* and from the right by oùIi_, which yields the result ot õ.f¡,- E**toô,in + õ.i¡,rsÛt _+ f (toÛo) õ8o- aÉo¡@o\o_) - aìr**õ.çkoui_. (10.23) Here ô.ri¿ and õ8e are the Bogoliubov-transformed matrix distribution where
function and energy change respectively. They are defined by õirk
:
õ8 k
:
k
* õt'k oùtl- = (y;;:, _u:Ë:_u),
$0.Zaa)
¡
* õe¡, etl - -
(::;:
(r0.24b)
ott
oìt
An expansion of (10.23) in small q leads new (diagonal!) distribution ôf¡:
_
3ft:'u)
to a Boltzmann equation of the
- (q . Ve^EÉXôfÉ -/i ôEo) - (q õg 0lr*)**, (10.25a) where a suffix ++, +- etc. denotes components in particle-hole space; i.e. + + is the first diagonal element while + - is the upper off-diagonal ar ôf¿
element, etc. Furthermore,
rr:ß:9!o
: -!
êEo
ðto
(10.2sb)
zaÛk'
Y*E*:+ry. L*ffl'
(t0.25c)
In (10.25c) a contribution proportional to VoÂr has been neglected, since it is order T"lTi small. Note thatY¡E,.r, the group velocity of Bep, vanishes at thç Fermi energy E*:0. In (10.25a) the change in the Bep energy is
found
as
ôEo: tt/ ' t, 6o\ oÇ*' "'. - +(t+) -\ Ek/ :-^e *) lakl, ',
.
+,(ôÂo
^f
+
ò^f).
ôEre
^; (10.26)
^e As we shall see later, ôI9¿ is related to ôf¿ in a more complicated way than in normal Fermi-liquid theory, because ôÇ¿ and ôA¿ are defined in terms of ôn¿ and ôF¿ rather than ôf¿ and ôh¿ appearing in (I0.2a{. Indeed, the kinetic equation for the new off-diagonal distribution function ôh¿ is found to take quite a different form: (ot - 2E*) ôho - ,e ôf& : (A õg aU)*_. (10.27)
M4
10 Transport
Properties
For frequencies a¡ much less than the average gap frequency Alh (and also for collisi,irn rates r-r < Alh), (L0.27) may be solved for ôh¿ to give
ôho:
-hôro+
o(qr,r-r).
(10.28)
Equation (10.28) must be interpreted as the local equilibrium value of öho. The new off-diagonal energy change is calculated as
ôrÉ:
-(ôËo Â*
*
^*
ôEIft)
.E#ae(ô^;
h-#4(ô^¿
Âr
-r-
^e
- af
ôae)
a; ô^fr) + o(q').
(r0.2e)
In order to close the system of equations, one also needs to express the old distribution functions ôn¿ and òF¿, entering the expressions foi ôÇ¡ and ôA¿, in terms of the new ones. This is easily obtainèd from the reverse Bogoliubov transformation. One finds ôno
- å(r **)
u,,.
- +(t - Ðh^frôrre^';
- *-r(laol'ôEo
. ôFo
&E
+
orô^;
ðErfr
^É
+ ô^ft
:*zL*fuço * * L*ôflo) . +Li .
_
^f)
^;)
+ o(q),
(10.30a)
_Âo * Âr ôEro) - z,k 8,, uo-.
&,(ôEo
hft
(1o.3ob)
In principle, one would now have to eliminate ôh¿ via (10.28) in favour of ôf¿ and ôA¿ in all expressions. This is a rather cumbersome procedure, because it turns out that all the expressions have to be evaluated at least to order q. After all, we have to remember that the hydrodynamic variables associated with the order parameter are given by gradients of the angles specifying the rotation or the gauge of the gap parameter and not uy ttre
angles themselves.
Fortunately, there is a way to circumvent this laborious task. The clue is that the change in the order parameter in the hydrodynamic limit is describable in terms of gauge transformations of the ordei parameter. By performing the gauge transformations directly on the matrix distribution function, one is able to generate the terms coupling the condensate to the quasiparticle system in a natural way. The price that one has to pay for this is that one loses the equation of motion for the order parameter. This, however, is easily remedied: as we have seen befot", hydrodynamic "u"iy ôn the other variable of the condensate corresponds to a broken symmetry. hand, every one of these symmetries gives rise to a conservation law in which the particular hydrodynamic variable enters. Thus, by enforcing the conservation law, one is able to fix the value of the hydrodynamic variable.
10.1 Kinetic
Equations
Let us demonstrate this procedure for one particular example-the phase of the order parameter.
10.1.4 Gauge transformation of the kinetic equation A
change in the phase of the gap parameter
;
A*:
eiõaA1o
(10.31)
is accomplished by a unitary transformation of the energy matrix
t¡,:
aìlçeekoìli
:
ef,
+ ôef;,
(10.32a)
where
%o: ôef,:
exp (jit6 ò@),
(10.32b)
ô0 (roeå -
(10.32c)
åi
efuo).
Here the superscript r@ indicates the fixed gauge. Because a phase change ô@ in .Á¿ carries over to the nondiagonal distribution function F¿ as well, the matrix distribution function also transforms as
J{*: %otrl%i.
(10.33)
we may now choose the
phase change ô@(¿ r) at any point in space and time such that ee has constant phase. In other words, we may "gauge away" the phase. For spatially varying phase õþ(r, t), or equivalently for its Fourier transform õQ@, ø), this is accomplished by substituting õJ{o(q,
ø)
: õ$lþ"tîfllï) : ô,llry + åi ôø (.oJ{oo_- Jfoo*r"o)
(10.34)
into the matrix kinetic equation (10.6). After rearranging terms, the kinetic equation for the gauge-invariant distribution function ô"Áffl is found to have essentially the same form as before. The only difference is due to the appearance of a number of additional terms in the change in energy: ôe¿+ ôef;':
q:
ôeo
- iiha ôó to + +i õþ (r?*"o - toe?_). (10.35)
0, the last term is exactly equal to the change in õe under a change of phase by -ô4 as described by (I0.32c). (Note that (10.34) is equivalent to a phase transformation of ôJf* by -ô4.) We therefore define the gauge-invariant energy change by For
ôe[:
ôeo
- ôeão'
(10.36)
Furthermore, we introduce the notation
ôfr: )ika
õE,
'":ft4 õQ'
(10.37a) (10.37b)
446
n
Transport properties
which is borrowed from the phenomenological theory that relates the time derivative of the phase to the chemical potential ¡r and the gradient of the phase to the superfluid velocity u" (see (g.g) and (9.16)). Nõte that in this chapter ¡r is the chemical potentiat per particle. This identification can be verified later. Expansion of the last term in (10.35) to first order in q gives an expression for the change in the energy matrix in the gauge-invariant formulation:
ôef,"': ôrå - ôp ro +
+
h(k . u,)I-.
(10.38)
The degrees of freedom of the order parameter that relate to rotations in spin space or orbital space may be treated in an analogous way (see Combescot I974a). For example, the rotation of the order paiameter in spin space can be compensated by a non-Abelian gauge transformation ur: exp $i Eyo*ôji):.where ô0Ì, are the rotation ãngles in spin space and ou are the three Pauli matrices. This allows an interpretation of tire time and the space derivatives of ôg) as an effective -ugtt"ii" freld HE,
ttoHru: $iha
õ0"u,
(10.3ea)
and a spin superfluid velocity, u"p,r,
:
fi
õ0"u, - iq ou;r zmry
(10.3eb)
respectively. In this way, one generates additional gauge-dependent terms in the energy change, ôelni"
:)
p
[-p oH"*t* + + h(k . u"p,¡")trrsl,
(10.40)
which have to be added to the right-hand side of (10.3g). It remains to express the observable quantities, i.e. the densities and currents, in terms of the gauge-invariant distribution function ôJVï. For this, we first rewrite the diagonal elemenrs of ô"Áf* in terms of ô.fi 1seË 1to.s+¡;:'
õft*np:
õnek,p+
*lô*ur"+ ì
(or)
*ur,o,r].
a*þL.
(10.41)
Equation (10.41) is then inserted into the expressions for the conserved densities. one finds that in the equations (10. L'l.a,b,d) for ôr, and ôe the distribution function ôn¿ is simply replaced by ôn[. Í{o*"u"r,{, ?to.1Lc) for the momentum density acquires an additional term such that ks
(t0.42)
The momentum density g is henee seen to be given by the sum of the momentum density in the rest frame of the superfluid, àescribed by ònfl, and the contribution from a Galilean transformation to the system with
u"*
0.
L0.1 Kinetic
Equations
Since the energy change induced by the Fermi-liquid interaction involves the ungauged distribution function ôn¿, this part contains gauge-dependent terms too. Inserting (10.41) into the definition of ôe¿, (10.7), and combining all the gauge terms, one finds for the diagonal part of ôeli',
ôEil':
) [ttt, tr, (ônfl,) +2 fî,*,o*tro (o" o"t.)] -
ô¡r
l-
po}
n"*ou +
h(k. u,)l + yD [email protected],o,")ou.
r-"
ffi"
(10.43)
tt
The effect of replacing the ungauged distribution function by the gauged one in (10.43) is to remove the factor mlm* in front of /r . u, (in compliance with Galilean invariance) and to renormalize the prefactor of the spin supercurrent term (last term effective mass by
in (10.43)). Here we have defined a "spin"
'ffi,
_1 + åF'ì
m
(10.44)
I++Fî'
The Bogoliubov-transformed kinetic equation for the gauge-transformed distribution function ôffl still has the form (10.25a). However, the change in the BQP energy ôE[ is now given by (10.26) with ôÇ¿ replaced by ôÇf;'.
10.1.5 Gauge-invariant densities and currents
In order to
calculate the transport properties from the kinetic equation (I0.25a), we need to know how to express the currents in the hydrodynamic equations in terms of the gauge-invariant BQP distribution function ôffl. For this, we have to gauge-transform (10.30), which amounts to replacing òn¿ by ônfl and ôf¿ by ôffl as well as ôe¿ by ôeli'. Then we insert this gaugetransformed equation as well as (10.41) into the expressions (10.11)(10.18). In this wây, we find for the particle-number density ôn --
4r,"l|"r-;*tal'oet]
+(#)"rr, (10.45a)
where
(*9o)":à#.taot''
(10.4sb)
As is apparent from (10.45a), the quantity (ðnlðu)op relates a change in the chemical potential to a change in density at fixed BQP distribution (i.e. ôf¿ :0) and in the absence of Fermi-liquid interaction effects. Thus it only describes a part of the total density response and is therefore proportional to the so-called "bare partial compressibility" r$. An equivalent (and very
10 Transport Properties
448
useful) form of (10.a5) is given by ôr¿
:
(ôrfl
# [tr' }Etr,
-/; ôEå) * òr],
(10.46)
where ðnlðp, is the normal-state compressibility (see (2.19)). The momentum density g is obtained from its definition (10.11c) by using (10.18b), (10.30) and (10.41):
g:>
hk fto (ôfÐ + pu".
(10.47)
Similarly, the momentum current density is found to be given by
¡¡,,:\
kk,uo,""l*,(ôrå
-/; un;)1 * 6,¡n õ¡t,
(10.48)
where (10.18b) has again been used. The energy density and the energy current density are obtained ôe
:) k
E¡,tro (òfå)
* ¡t õn,
as
(10.4e)
: ) uo$¡ tro (òfå + QL õEò + I "1, k-m
1
:_
jo *
(10.50a)
ttg
1
*lrg.
Note that in a system with strict particle-hole symmetry the term involving ôf¿ is zero. The first term on the right-hand side of (10.50a) may be interpreted as the heat current. It consists of a sum of a convective and a diffusive term (10.sOb) Zsu,, * 2 roEotr, [òffl + f L(hk. u")1],
jo:
k
where u,, is the local drift velocity of BQP, i.e. the normal-fluid velocity. The corresponding expressions for the spin density S and the spin current jo are slightly more complicated because of the spin structure of the order parameter. By inserting (10.30) into the definition (10.1lb), one finds
^ S*
: h=-(
e.
*" õf o, " -
tk lAolt-
-
I
- ttoH"") ],
(10. sl ) ^p" where a notation analogous to (10.18c) has been employed. In the presence of an external magnetic freld Hs" is replacedby He"+ Hi'. Here s¿,"., is the BQP spin weight factor, which specifies the contribution to the spin component S, of a BQP with momentum /s and spin along the v direction. It is given by
ià
It
o,
r;t4
[ô",
(fr
)](ô E o,,
ç,9,
sk,t"u:
where
*.D¡,u*,* (r\ - *)n*"(o), Lp/
(I0.sza)
A*"(k) is the projecto¡
A*"(k):di(E')d"!k)
d*.d
'
(10.s2b)
10.1 Kinetíc Equations
449
In deriving (L0.52a), use has been made of the relation
:6ffi
o lo *L,e(o")'Âfl
:
6*u
-
zLpu.
(10.52c)
In analogy to the partial compressibility r$, we may define a Cooper-pair partial spin-susceptibility tensor by
)i"?:
r\\hffi[ô""
- A*"(k)|.
(10.s3)
Equation (10.51) allows one to calculate higher-order Fermi-liquid corrections to the thermodynamic spin susceptibility, i.e. those beyond the usual molecular-field correction considered in (10.10). To this end, õfo,* is replaced by its local-equilibrium form
õf*,u:
f
(10.s4)
LõEo,u,
where, using (I0.26), the BQP energy change is derived
õE*,p:
)
ro,,,,,(ô5o,"
-
as
(10.ss)
ltoH,).
Here Hu: H",:- + Hs", with.ðr'*t the apptied external magnetic field. In the B phase only the I :2 Fermi-liquid correction enters besides the / : 0 one, owing to the fact that only the weight factor s¿,"" and the projector A*"(k) have an angular dependence. The resulting spin susceptibility is given by (Czerwonko 1967)
)("
-*: rr
Í,o 1,
+ Fôio'
where
i,o:? + âyo+ &Fi!
¡t
:
"1,,2,3,
- yò'
(10.56a)
(1o.s6b)
and Y6 is the Yosida function defined in (3.101c). The spin current j'u, defrned in (10.17a), may be written in terms of the gauge-invariant distribution function as
ii::r-**lo
-,* õr L" -
htaot'
n*(ulo,"
+
.
+
n* ' ,,,.,)] (10.s7)
"rt'sou"p'¡''
In (10.57) we have defined pk,¡"u, the contribution of a BQP of spin component v and momentum /c to the spin current with spin orientation ¡,1 given by P*,p": Ur" -
(t -*)n.".o''
(10.s8)
450
1.0 Transport Properties
In local equilibrium the BQP distribution function is again given by (10.5a); however, this time only the part of õE*,p that is odd in k contributes, i.e.
õE*,t":) po,*u(ut*," + + n* . ,"o,,).
(10.59)
This may be substituted into (10.57). With the aid of (10.18b), it yields the local-equilibrium expression for the spin current:
i'í"' : #ì [t - år'r ,fror]"0åî,u"p,,, = )
pfl"u"o,".
(10.60)
Here the spin superfluid density in the absence of Fermi-liquid corrections is given by
o.
-
ys(Ê; r)l[ôu, A,"(Ê)l] (10.61) þ'Ð:1"":3so(Ê,Ê¡lr Note that j"*J* is equal to the spin supercurrentjlp defined in (7.16).
10.2 COLLISION
¡NTEGRAL
10.2.1 Matrix operator for binary collision processes The time rate of change of the matrix distribution function caused by binary quasiparticle collision processes is given by the golden-rule expression involving the scattering amplitude Z:
Ili*': - +: Ji L k2k3k4
"ri......1ri
,r,rt,y4:tr
x (ft3s1, kqs1lT lk,,s'r, kzsLl(2n)oô'(?'0,) U(?'
y,Eo,)
(
x{ô"",[Nli,i",NLi,t",(g'o:-#'o1,)"rt@i:,-Jüi)"*,^
- @ii- ttol),¡,@i,:-
" + (t,."í \s'ësJ)Ì)
JtlT)"*,NI1,,","rN1i,"4s¿l
$0.62)
To simplify the notation, the particle-hole index p and spin index ø have been combined into a single 4-component index s : (p, ø). Since the components of the distribution function belonging to the energy eigenvalues *Eeand -E¡have to be treâted separately, an additional index y: tl has been introduced in (10.62). Only in this way can we take energy conservation during collision into account, as expressed by the delta function with argument Ei y,Eo,: TtE*r* yzE*r- TsE*r- y¿E*o.This generalization leads to a further technical complication, since \rye now have to introduce a
10.2 Collision Integral quantity
451
9f,
which is the probability matrix for the state l/c) (with energy TE*) to be occupied or empty. This is not equal to ,rnity, since the quasiparticle may be in state lfr) energy +Ee or oi ,o,rrr", the gf, adds up to unity -Ek. sum of the spectral weight factors TF Erby: 7. The conservation of momentum is incorporated by it r O"ti" function with argument E'¡k¿:kr+kz-k=-ko.of course, the collisions also conserve
the total particle number and spin. This implies that the scattering amplitude (frrrr, kzszlT lk3s3, kqs+) is nonzero only for p1 * pz: ptlpq
p1u1* pzuz: pz&z* p¿uq, where p, is the partiôle-hoirirrá"" and ø¡ is the spin index of state |fr,s,). The scattering amplitude represents the sum of interaction processes involving momentum transfer over the entire Fermi surface. The pair correlations, on the other hand, are effective only in a small region of momentum space near the Fermi surface. Hence to order ntr; the scattering amplitude I is given by the scattering amplitude r in the normal state. More precisely, the components of z may be iãentified with and
: T----: t(I,2;3, 4), I T*-*-: T-*-+ : t(1, T****
-4;3, -Z), I T*--*: T-++-: t(L, _3; -2, 4), )
where the subscripts *, - refer to the particle-hole indices p,, and we have used the abbreviations L =(kr, ut), etc.'The factor 1/3! -2-(-kr, -ez), in (L0.62) accounrs for the fact that all 3! peimutaiions of (2,3,4) in rhe sums in (10.62) are equivalent and should be-counted only on"". Finally, the matrix structure of Jft introduces an asymmetry, for exámple into the first term in the curly brackets in (10.62), which is removed bv taking half of the symmetric combination (for a derivation from microscópic theory, see Wölfle L976a)- The collision integral has been investigated, in particular in the limits T-7. and T+0, by perhick et al. (Ig75,lgle, Ig77i. Next, we derive explicit expressions for the lineanzed colliiion integral
!-ø in (10.6) using (10.62). The equilibrium distribution function *16, describing the distribution of quasiparticles with energy rE*, must be proportional to f(yEù. It is then easy to decompose the complete equilibrium distribution function tîoe, (10.3a), into componenrs
J{2:
Q0.6aa)
Y:*T
with
rüo:)(t.#)r*uu. The linearization of
(10.64b)
Jf[ is done about the equilibrium value J{f,o:
JtI:
Jt"oo
+
õJtI.
on the other hand, the distribution of
(10.65)
quasiholes with energy yE¡, is
452
10 Transport Properties
proportional to
L-ffuEo) s:o
and hence
- rt': +(r .#),
- r@Eùt.
(10.66a)
From (10.64b) and (10.66a), the equilibrium spectral weight function gf,o is found as
eto
: +(t *#),
r: lrt.
(10.66b)
a projector, i.e. (glo)r:g|o, and Øfo and g¡o are ^it orthogonal, 9[o9¿o:0. Upo{ substituting the equilibrium distribution functions ,Á[f,o, (10.64b), and 9f , (10.66b), the collision integral (10.62) is found to be zero. This follows from the fact that, owing to energy conservation, the combinations of Fermi factors in the out- and in-scattering parts, frfr(r-.ÊXl - fo) and, (1 -,txl - fr)rtfo, have exactly the same value. cãñs"q"enuy, the two parts of the collision integral are found to cancel each othet. ihir is true not only in global equilibrium, but also for any local equilibrium distribution functions, i.e. distribution functions of the form 1rb.o+u¡ and (10.66a), where eo and E¿ depend on space and time. The linearized collision integral takes the form Note that 9Io
ô1¿1"",
: - +: 3!2
kTkok^
x (krsä,
""i;r, ,,,r,,y4:rt
kqs,al
T lk6,1, krsl)(zn)ao3(), t) U,(?, y,E n,)
xf î,if ",¿(r-f î))e-f i,:l{rii?¡,p\1f"*,p"o1!,"f x [ôóIl"i"
,+ õe?i"t",- õþ8"*t-
l^^?^,t
)],
-F I ".*"í õþIi","il '4ù4t \s, *sr/J'
(10.67)
where ô@ is a combination of linearized distribution functions,
ôøL,
: lflT - fÐPL,,l-'[ôNL
f[:f(yE*)
,
- fi"ôpL.
-
pk,,f Ly ôE*1, (10.68)
is the Fermi funcrion. The last term in (10.6g), which is proportional to the change in ener9y õE*, comes from the linearization of the energy variables in the energy delta function in (L0.62). Because the collision integral is zero in local equilibriuffi, â change in energy òE¿ in the delta function is equivalent to a . change -ôE¿ in the õrresponding equilibrium distribution function. Tþiq leads to the additional changè in thã distribution function given by rhe lä$fi,terms of (10.6g). The linearued collision integral (10.67) is seen to act on four distribution functions, õJü and õ9'[,, T : x.L, instead of the one function ô"Áf¿ we started from originally. However, these functions are not independent. ihis may be and
L0.2 Collision Integral
453
seen by looking at the- kinetic equations for these four functions, which are obtained by decomposing the kinetic equatiol(ro.-o) for ô,Áf¡ irrto it, parts. To do this, one simply repraces $ooby rnepdua,ior* for
ï: i1 ô.fl , are then given ttr". kinetic equations for ô,Áf¿ *itn ¡rl ,"pr"."¿ or-*f !v. and the full collision integral trfff
replaced by the cõmpo.r"rri- Oy?. m order to derive the equation for õgl, one starts from the mãtrix kinetiË equation for the quasihole distribution function ô gi- õJti. This -equatiol is gü"r, by the normal kinetic equation with replaced by yf,o-¡1o. TÃe collision 'f[ the one in integral has the opposite sign from ttre quasiiarticle equation because adding a quasiparticle to a quasiparticle siate'i. eqrriualent to removing a quasihole from the same state. The sum of the two kinetic equations for quasiparticles and quasiholes with energy yE¡ yierds the kinetic equation for õ9!,:
a õ9f,-
rfr*
õØ[+ õØf,e¡_:
ô€*
gil- g[lõeo.
(10.69)
Note that here the collision integral has dropped out. On inserting Øf,o into (10.69), the right-hand side of (10.69) is seêñ ro be proportional
to
hence
õgI:
T
õ9*.
y,
and,
(10.70)
From the kinetic equations for õ.lt|, ô"Ál¿ and õgl, it is straightforward to derive õJf[in terms of ô"Áf¿ and õg¡ alone:
õ"ifI:
+(g|_ôJfo + õ"r.fk gl,_) + ty(õg¡ J{** * .if k_ õgk). (10.71) It follows that only two of the four distribution functions,.q.rutìparticles ô"^a and õg¡, arc independent. While ô"Ál* describes the distribution of and cooper pairs in momentum space, corresponding to both branches xE¡, of the excitation spectrum, the additional function ôØo determines the relative spectral weight of the two excitation branches. When substituting Q0.7L) into the collision integral (10.67) the small momentum changes *lq due to the external field b" neglecíed; as they only introduce correction terms of order elke into "un the collisi,on integral.
10.2.2 Gonservation properties of the collision integral The conservation of mass, momentum, energy and spin is reflected in corresponding integral properties of ô9. one finds that for any distribution
function ôJA,
k,v--+l k,v--+.1
(10.72a)
(r0.72b) (10.72c)
k,v:+L k'v:tL
(10.72d)
10 Transport Properties
454
10.2.3 Gollision integral for Bogoliubov quas¡particles In the macroscopic regime, i.e. when the condensate is in local equilibrium, the matrix collision integral (10.67) assumes a simple form. Let us first perform the Bogoliubov transformation (10.20) on the distribution function På",ô0L, appearing in the linearized collision integral (10.67). The result is
pä", ôoä"
,-vi!
- f'o)l-':2 ((t:.]]u*\ ôU -(1
un1
- ), - y)õt-¡l'
(10.73)
where òfr,
:
ôfk
-/;
-
:
ôho
+;:
ðho
ôEÉ, tr-
LL*
òfe.
$0.7aa) (10.74b)
However, according to the discussion below (L0.27), ôh¿ may be put equal to zero in the macroscopic regime, so that the transformed matrix distribution function PL",ô@l becomes diagonal. Also, only one of the diagonal elements is nonzero for given y. In deriving (10.73), small terms of order q and ø have been neglected and the solution of (10.69) for ôØf (more precisely its Bogoliubov transform), i.e.
õØk:å('lt has been employed.
u;-)
.o(ry,i)
(10.7s)
equation (10.67) for the distribution function of Bogoliubov quasiparticles may now be written as the sum of an out-scattering and an in-scattering contribution (Einzel and Wölfle 1978, F;inzel 1984):
The collision integral
of the scalar kinetic
(10.76)
Here the Bogoliubov quasiparticle relaxation rate is obtained from the Bogoliubov transformation of the term proportional to ôQ¿ in (10.67). Employing the fact that at low T all momenta lie on the Fermi surface, the energy and angular dependences may be separated (see the discussion for the normal state after (2,35)). Owing to the anisotropy of the condensate, the relaxation rate is also anisotropic in general, i.e. it depends on the vector kt. Furthermore, the orientation of the momentum vectors is specified by three angles instead of two, as in the normal phase: (i) the angle 9 enclosed by ft1 and k2, (ii) the angle @ between the planes spanned by (ftr, k) and (kr, ko), and (iii) the angle @2 describing rotations about lc1.
10.2 Collísion Integral
455
The result of these reformulations ir (6, = E*,, E¡= E*,, f,=fo,)
Inf* ;,: t*"(l x
+ e-E*/k"\Jo dErdEroío
*,à,*. :*1
(ô(E' * pzEz- þEEs-
fr! - f)G - fòA(kr, kz, kz, kùl ".
þ¿Eq)
(10.77a)
Here we have defined a generalized transition probability by Q(kr, kr, kr, ko) : W - Wo"[Di . Dt* Di . Dol - W@i . DÐ(Dr. Dq) + zw@i . Dr)(Di . Dò + 2w4(Di . Dò@i . Dr), (10.77b)
where the quantities ry are combinations of normal-state quasiparticle scattering amplitudes given by (2.39) and (2.49) and Wou(o, e)
:
(Ao
+ A')(Bo + B t) + 4At Br .
(10.78)
The quantities At(e, @) and Bt(e, Q) are the singler (S:0) and triplet (S:1) components of the quasiparticle scattering amplitude introduced in chapter 2 (see oot!l'*r"(krar,
kquo),
kzeú kz&', Npa(ktut, -k4- ugk3d3,
B:
-kz- uz),
(10.79a)
(10.79b)
where e¡: LI are spin indices. The vector quantity D¡ is the normalized equilibrium gap parameter,
Dj:
þ¡d(k¡)lE*,, and the angular brackets in (I0.77a) denote the average,
(r2)"-/'o{"o,;øf
*"
Xf
Dj in
(10.80)
(10.81)
(10.77b) come from processes in which a quasiparticle transforms into a hole by condensing into a Cooper pair (or vice versa). The quasiparticle or quasihole character of a momentum label k¡ may therefore be different for either of the two scattering amplitudes, which leads to the asymmetric combinations in (10.78). The in-scattering part ôÍi" has a very similar structure. We only give the result for the spin-symmetric part:
The terms involving
ôi:1,'"
: iå*xf xrrfr(L
f* Jo
-
otrd|rdEo fr)(L
r,2,*,(õ(Er* :*1
pzEz- þzEz- pzEs-
trqEq)
- rò[a(kt, kz, h, k4)(õQ¡ * ô0o - ô4r)]),, (10.82a)
where fJ is defined in (I0.77b). The distribution function appears in the combination
ôó, - r! :
L
(L
z
+
p¡)
õf*,ro,(L
(1'
- p) õf -*
- r*,)
'' þ¡: tL'
(10.82b)
10
Trønsport Properties
Equation (10.82a) is valid in the case that ò/¿ is an even function of f¡. In the opposite case the quantity in square brackets has to be replaced by Eo
u'1,ÍW(õV¡
with
lrtz,
* ô?o -
õtþr)
-
Wou(Di . Dò
ôr/rl
(10.83a)
Wo" and D, deñned in (2.39) and (10.78) and
s
¡rrl.
¡
1.
Ç*, -,
(l\ + 'F) ò/o, + (t -
't zgkj --
p¡) õl-.o,
fo,(L-fo,)
(10.83b)
The complete collision integral for the spin-antisymmetric case has been derived by Einzel (1984). The collision integral (10.76) does not conserve Bogoliubov quasiparticle number and spin. This is due to the fact that Bep may be gènerated or annihilated during a collision process. Ffowever, momentum and energy are still conserved. This gives rise to the following integral properties of ôi:
)k ft tt, (ôio¡ :
g,
(10.8aa)
2k notr, (ôio¡ :
g.
(10.84b)
At the same time, the collision integral functions of the form
õf'fåB: f L(anr - hk.
un
vanishes
for any distribution
- urf)u,u,
(10.8s)
where un and ôI must be interpreted as the local velocity and a change in local temperature of the BQP. This implies that collisions do not alter a distribution of the form (10.35). Consequently, the distribution function (10.85) must correspond to a local-equilibrium state of the Bep system.
10.2.4 Relaxation-time approximation The collision integral for Bogoliubov quasiparticles derived in Section L0.2.3 is complicated enough to defy any general solution. It is therefore necessary
to introduce an approximation that simplifies the collision operator sufficiently to allow an exact solution for a large class of transport problems. At the same time, this approximation must be able to describe the essential physical properties of the system under consideration. This task is achieved by the following relaxation-time approximation (RTA): (10.86a)
10.2 Collision Integral Here the sum on cr: s, â extends 9ver, spin-symmetric and -antisymmetric components, and the sum on r--f(-1)': f 1 runs over even (f :+1) and odd (r - -1) functions in 6. For the spin-symmetric case the projectors p are defined by
pi^,(k,
k,):
OiG)
|f
x Qi(E')
,rr,*
' flr- ,r,-, *.r1,1y,^qîe,)1,]
1^
-
T¡,,
Y^(¿')pi,
(10.86b)
where
lf F / F\r oi@ : ;Lt * n,(-1)'(1 fi) ]no-o",
(10.86c)
and Y¡^(k) are,spherical harmonics. The operators Pro project onto the space of functions ôf¿ with spin symmetry o (:s, a) and particle-hole symmetry
t:
r(-L)t.
It is easy to verify that the above relaxation-time approximation guarantees the two conservation properties (10.84) for momentum and
In the limit a+0, i.e. T-7",
"rãrgy. (10.86) reduces to a relaxation-time approximation for the normal-state collision integral, which yields results for the transport properties within one per cent of the exact results (IVölfle l978a,b). In the opposite limit, i.e. for ,-->0, only quasiparticles with energies close to the gap edge are excited, such that Ço<< (^), where (Á) is the average energy gap. In this limit the in-scattering part of the spin-symmetric collision integral vanishes more rapidly it un the outscattering term owing to factors of Et - krlÁ. This property is also shared by the RTA collision integral (10.86). On these gronnds, we can expect the RTA to be a reasonable representation to the collision integral.
10.2.5 Bogoliubov-quasiparticle relaxation rate The main features of the collision integral (10.76), such as its anisotropy and its energy and temperature dependences, are contained in the first term, the "out-scattering term". This term describes relaxation of the distribution function ôfo with a rate llr*, which we shall refer to as the "BQP relaxation rate". Actually, the BQP quasiparticle lifetime is twice the relaxation time tp, because Llce is the sum of the decay rates for a particle and a hole, which are equal. Let us discuss the relaxation rate Ll:.afor the A and B phases in detail, in order to estimate the relevance of collision processes. The B phase
In the B
phase the dependence on energy and angular variables can be completely separated because of the isotropic excitation spectrum. One
10
458
Transport Properties
finds
;:å
[4o)
-
voQP +
Ir\+
ôo/f;)],
(10.87a)
where z$ is the quasiparticle relaxation time at the Fermi level in the normal phase given by (2.38). The quantities lf;) are dimensionless energyand temperature-dependent functions defined by
It,
: If* dírd1rdïo J6
p,!Li,{14
x ô(E, * ptzEz- ltzEzf*
poÛo)K@)
E),
= Jg| dErJ"(Er,
(10.S7b)
with
K(o): 1, ltz nz
¡¡(r)-L
E, Er- '
y(z')
(10.87c)
*, -** Lz Lq
y(z)-++#*o^. E, E, Er Eo
The dimensionless parameters y, and ôs depend only on the quasiparticle scattering amplitude in the normal phase and are given by
Tt:
(Wo"crP,(.r))" (w> "
ôo:
(-wrcnsz
e
, l:0,I,...,
(10.88a)
+Wc?+woc|)"
(w>"
(10.88b)
'
where
c3,4: Êr,' frr,o:
cos2
äe ¡sin2 l0 cos @ : frr-
Êr,o,
(10.s8c)
and P¡(z) are Legendre polynomials. For later purposes we have defined y¿ for as well. The cross-sections W, W(0, rp) and Wouhave been given ín (2.39), (2.49) and (10.78) respectively, and the angular brackets denote an angular average as defined in (10.81). It is worth mentioning that the collision integrat (10.87) describes three different types of collision processes:
l*0
(i) the usual process'CIf"'scattering of two quasiparticles into two final momentum states (e.g. for þz: ltz: l-tq:1-)) (ii) the decay of the initial quasiparticle into three final quasiparticles (e.g. for -þz: pz: pq: 1);
L0.2 Collision Integral
459
(iii) the coalescence of three quasiparticles into (e.g. for þz: -þz: ltq:L).
one final quasiparticle
The latter two processes are possible because the quasiparticle number is not conserved. Two quasiparticles may combine into a Cooper pair, or conversely a Cooper pair may split up into two quasiparticles during a
collision process. In the limit T+ [ the functions /") for factors of. A2 and Aa respectively, whereas
fl:
4:):1+(Jïon)'as
r,2, 3 vanish owing to the
r-rl.
(lo.Be)
The result (2.37) for the normal-state relaxation rate is recovered in this limit. In the opposite limit of low temperatures, all functions K(") tend to zero as e-E*lk,t The quasiparticle relaxation rate in this limit is found as
IL
llm-:T+O T¡
T
:*þ
#^(#)'''"-o'o,'(t*n"T)
as
z-+Q,
(10.e0a)
where
Wo:I-?yo*ô0,
(10.eOb)
e":1- ôo -w"åyo
(10.eOc)
The first finite-temperature correction in (10.90a), proportional
to
kBT I a(0), as well as that in (10.112a), (L0. L25a) and (1.0.13la) (see below), has been calculated exactly by Einzel (1984). The rate r¿ 1 is seen to be proportional to the density of excitations (taking into account the T2 dependence of (z$)=1):
fl"*:2
:
(10.e1) ]qzn¡1'2n(Ak'T)r/2 e-a/k'r as Z+ 0. *Ep This behaviour, reminiscent of that of a classical gas, is obtained because the occupation numbers fo arc'very small and the excitation spectrum is quadratic at low temperatures. The Fermi blocking effect is of little importance in this case. Given the relaxation time, which may be interpreted as the mean time between successive collisions, one can derive the mean free path I of BQP, which is defined as the product of the velocity u..t, and the mean collision time 2.,,. As the group velocity of BQP depends strongly on energy (and in the A phase also on direction in fr space), we shall use the r.m.s. average over the thermally excited states for the definition of u",,: fo
uuu:{[> ro.u. yrL]ll> rLf]'"
(t0.eza)
460
L0
The mean collision time relaxation rate Í
^"
2...,
:{
Transport Properties
will be defined by the inverse of the
average
ll }r, r] ll? rt] }-',
(10.e2b)
so that the mean free path is given by
l:
(10.e2c)
uuur^u.
This yietds a low-temperature limiting behaviour
-,_(zn)''' (J
12
3 \^ö)
1
(10.e3)
*[u'zR'(r,,)1""'''
The normalized mean free path is seen to increase exponentially for T+0 (see Fig. 10.1). It should be noted that this normalized mean free path l(T)l\D depends only weakly on pressure. It is of the order of centimetres for T :0.2T.In this temperature regime, and even more so at still lower T, the collisions of quasiparticles with the wall of the container dominate the collisions between the quasiparticles. This is the so-called "Knudsen r(r)
m 101
0.2 Figure 10.1. Mean free path temperature.
0.t, /(f)
0.6
0.9 T/Tc
10
of Bogoliubov quasiparticles in the B phase versus
10.3
Transport
Cofficienß
46I
regime", or "ballistic quasiparticle regime", some aspects of which will be considered in Section 1-0.4. The A phase
of the BQP
in the A
phase is more not separate in this case. At low tempÞratures, where the A phase may exist in the presence of a magnetic field, only quasiparticles with momenta in the neighbourhood of the two nodes of the gap, i.e. along tÎ, where Ak<
The evaluation
relaxation rate
cumbersome as energy and angular integrations do
,r.#)
(10.e4)
is satisfied(00 is the angle between /c and l) divided by the total area, i.e. [kBTlAo(T)12. In the cass where both initial momenta satisfy the condition (10.94), momentum conservation requires the final momenta to satisfy this condition, too. Since all quasiparticles taking part in a collision process of this type have energy gaps much less than the thermal energy, their contribution to the relaxation rate is essentially the same as in the normal state. The relaxation rate of BQP with momenta /<,, parallel to Î is hence found approximately as
1
1 f krTl' :"R(?:) I /T\o ,^:"3¡r¡ L^fo)l U:/
(10.e5a)
Since the group velocity of these BQP is equal to that of normal quasiparticles, i.e. it is given approximately'by ,., their mean free path in the direction / is roughfy
l:
upt¡ru.
(10.esb)
10.3 TRANSPORT GOEFFICIENTS As a first application of the kinetic theory formulated in the preceding sections, we shall now calculate the transport coefficients that have been introduced phenomenologically in the hydrodynamic equations of Chapter 9. Quite generally, the transport coefficients may be expressed as limiting values of the various curr.çnt correlation functions for frequency a¡ and wavenumber q tending þ zero (for a formal discussion see Brand et al. 1979, Dörfle et al. 1980). The correlation functions, or equivalently the expectation values of the currents, may be conveniently calculated from the
10
462
Transport Properties
quasiparticle distribution function, as will be discussed below. This amounts t-o soiving the kinetic equation for slowly varying disturbances, i.e. those with frequency o and wavenumber q small in comparison with the relaxation rate t-r and the inverse mean free path /-1 respectively. Of course, the conditions for the condensate to be in local equilibrium (ø, upq 11 Alk) have to be satisfied as well. The scalar kinetic equation (10.25) may then be solved by expanding it in powers of ø and q (the Chapman-Enskog procedure well known from the transport theory of classical gases). V/e shall follow the discussion given by ninzêt and Wölfle (1978), Wölfle and Einzel (1978) and Einzel (1984). In the first step we write the distribution function ôf¿ (in the following we drop the index ';g" referring to the gauge-transformed quantity) as a. sum of the local equitibrium distribution function ôff" and the deviation ôfi:
ôfrr: ôft* + ôfl.
(10.e6)
The correction term ôfi is order-(ø, q) small, since in a time-independent (r:0) and spatialty uniform (q :0) state the system is in local equilibrium, i.e. ôfi:g. Actually, ôfi describes the approach to local equilibrium. We shall show that the transport coefficients can be expressed in terms of ôf[. We now insert ôf¿, as given by (10.96), into the kinetic equation (10.25) and expand to first order in ø and q. Since the "streaming terms" on the left-hand side are already proportional to ø and 4, wa may drop ôfi there' On the right-hand side, i.e. in the collision integral, the local-equilibrium part ôff" drops out. (Recall that by definition a local-equilibrium state is not changed by collisions.) We are then led to the following inhomogeneous integral equation for ôfi: ar
ôfli"
- (Y oEo. qxôff" -/i
õEo)
: ôÍ{ôfl}.
(10.e7)
This equation describes time-independent transport phenomena involving the stationary flow of mass, momentum, energy and spin as well as time-dependent phenomena in the macroscopic regime. None the less, even for stati,onary transport problems, the first term on the left-hand side is not necessarily zero, because it contains terms proportional to the time derivative of the density or spin density, which by the continuity equation are equivalent to the divergence of the respective currents. Using (10.10), (10.26), (10.55), (10.59) and (10.85), one may write @
õf
P:p: af LaEi9'åB
: ,rLl*-ffå ôn LE*
õp)ô*B+
) (o,) ,urr,-(?r/ôs' - t "H1) pv
+2 @ò,upo,r"(s;'#hk 'iî + +ru ' ,"".")]- (10-es) In a state of local equilibrium, changes in density and chemical potential are
10.3 Transport Cofficients
463
related by a thermodynamic derivative, i.e. ôf¿
: * r": on
Ne'(1 + Fi) ôn.
(10.eea)
Likewise, the effective gauge field ^EIs is related to the spin polarization S by the magnetic-susceptibility tensor 1, which may be replaced by its largest eigenvalue; when the magnetic field is the dominant orienting force: "l
H"u:;Ot- - XHi:') _!
(10.eeb)
a)"u.
x,
Here ø'u is the thermodynamic field introduced in (9.9). The last term in (10.98), which is proportional to tÐu"o,*, may be expressed in terms of qø"u using (10.39b).
Exploiting the conservation law for mass and spin in order to replace a õn by *-tq. g and øS, by q . j'*, the transport equation (10.97) may flnally be written explicitly as
1 Lf äl-| ùo,r,o'8. * t(k ' oo -
,,lnEo
f
¡
k'-up)q ' u^
ôrl (PoY\ E*u*' qT -ikvue*)* ì^o*to.,"\î )"^ø 'i'^ - + >^ o*po,*"k(+Fî#so'pir
+
+ a"^t)ø(ff1^,,t.Ì
: ôÍ{ôri},
(10.100a)
where the tensor Vis defined by (see (9.65)) V¡
:
äi(q¿uni
*
Qiun¡
-
36,¡9' u").
(10.100b)
Equation (10.100a) is an inhomogeneous integral equation for ôfi, which cannot be solved in general terms because of the complexity of the collision operator in a superfluid Fermi liquid. Exact results can be found for some of the transport coefficients close to T. and in the limit of low temperatures (Pethick et al. 1975, L976, 1977, Bhattacharrya et al. 1977b, Einzel L984). This is due to the fact that for some of the transport quantities the leading correction term to the normal-state result just below 4 is of kinematic origin and may be calculated from the normal-state kinetic equation with the BQP group velocity inserted in place of the normal QP velocity. At low T the back-scattering part of the collision integral, i.e. the one involving integrations, vanishes in some cases more rapidly than the out-scattering part, so that the simple relaxation time approximation becomes exact. In general, the integral equation (10.100) can only be solved
L0
Transport Properties
approximately, for example by assuming the kernel to be separable (Wölfle and Einzel L978, E,irløel1984) or by employing a variational method (Ono e/ al. 1977, 1982, Hara et al. L980, Ono and Hara 198L, Hara 1981). For a somewhat different formulation based on the Green's-function technique, see Houghton and Maki (1977) and Dörfle et al. (1980).
10.3.1 Transport coefficients of the B phase
In the B
phase the task of solving (10.100a) is greatly simplified by the isotropy of the energy spectrum. This allows one to separate the dependences on energy and angles, as in the normal state. The mathematical problem may then be reduced to the solution of integral equations in the energy variable. The angular dependence is easily obtained from the left-hand side in the kinetic equation (10.100a) for each particular case.
Shear viscosity
In response to shear flow, characterized by q¿un¡*O, i+j, the spinsymmetric distribution function, hereinafter denoted by ô/i, assumes an angular dependence (Y eE)rkr. Separating out this behaviour as well as the main dependence on energy, ô/i may be written in terms of a dimensionless function ,lrr(E): õf
L: i )ij "R/i(Y ¡E¡)¡k¡q,u,¡úr(E*).
(10.101)
substituting this into the collision integral (10.76), (10.82a), (10.87a), Vn is found to satisfy the integral equation
ú,(8,): r(ç,)[1 .
I: dE,&(8,, Ð(Z)',Þ,G,)],
(10.102a)
i(E): t*ldu
(10.102b)
; is the BQP relaxation time (10.87a) divided by the normal-state relaxation i';'tate z$, (2.1S). The integral operator t in (10.102a) is a special case of a more general operator S, (i.e.the one for l:2):
ü(å" E):
)'ì*Jo(Er,
E)
- üz(&, E),
(10.103)
which is determined by the functions.Is and J2 defined in (10.87b) and by the parameters Â|* and y¡ introduced in (2.51)-(2.53) and (10.88a) respectivety. This follows from specializing (10.83a) to the B phase. To proceed, we equate the off-diagonal components of the stress tensor (10.48) and their hydrodynamic expression (9.28): ¡7,i
:2 k
F onò,k¡
õf
L: -iîle¡uni,
i + j.
(10.104)
10.i Transport Substituting
Cofficien*
469
for ô/í. from
(10.101), the following expression for th
shear-viscosity coefficient is found: T
:
rfl,
2 F onn)?k'kiçf L)rp,Gù: ,r(#rþ,)
,.
4o: !nm*u!"R,
(10.105a
(10.10sb
and we have deûned the energy average
(# r,),= l-
d'o Gr Ð
f;r,re-t'
(10.105c
Note that (10.105a) does not depend on i or j for symmetry reasons. In tht immediate vicinity of 7: the temperature variation of 4 is dominated by thr change in the quasiparticle excitation spectrum, which introduces a factor EIE into the expression for th_e group velocity (10.25c) (Shumeiko 1972) The ensuing factor of (Ela¡', for example in the energy average ir (10.105a), gives rise to a square-root dependence of the change of in thr 4 superfluid state: qQ) - qQ) * - (?: - T)'''. This may be seen from the limiting behaviour of the f integral over the expression
IxÉ):Hà
t#*)
:r -naõ(Ç)
(10.106)
if we recall that Á *(7.- T¡rrz for r+ Zi. The integral equation (10.102a) for rþr(5) -ay be solved exactly to first order in A, using the eigenfunctions of the transport problem in the normal state, as discussed in Section 2.3 (Pethick et al. 1976). In fact, the only correction linear in A (and hence
nonanalytic in the "natural variable" Az) comes from replacing the singular term ETln3in (10.102a) by (10.106). The relaxarion time t(Ç2)ãs welt as rhe kernel sr(Er, Sr) possess analytic expansions in a', as may be seen by
inspection, and
to order A may therefore be approximated by their
normal-state expressions. Introducing a function
þ(x) where
x:
(n'+
:
)
V,(Ç)
sechz
(10.107)
)x,
ElksT, the integral equation (10.102a) takes the form x')Q@)
- )tr[ dx' K(x, x')e@,) :
sech2
]x
- nt 21*r*,0)O(0) (10.108)
(here and in the following we write )u2as shorthand for Â)*). To first order in a, the quantity @(0) is approximated well by its Á :0 value. using the eigenfunction expansion (2.6r) of the integral kernel K(x, x'), (10.10g) is easily solved. The result is conveniently expressed as
ô(x'l:
ón@\ + ôó(x).
110.109a)
L0
466
Transport Properties
where Qou)
¿odd
J
is the solution in the normal state and
: l_ ut1', dx
f v-
xrz
(10.10ec)
eo(o)loo(o).
The shear-viscosity coefñcient is obtained by inserting (10.109) into (10.105a), using the relation (10.106):
rt: r¡.t-!
nrÍoo(o)l'
#.
o(a'),
(10.110a)
where TN is the normal-state result (2.66). The quantity @s(0) may be obtained as a rapidly converging series whose first few terms are
oo(o):L*(r.i&.äh;
)
(10.110b)
Turning now to the limit of low temperatures, we observe that the integral term in (10.102a) vanishes somewhat faster than the term on the left-hand side owing to the extra factor of EïlEïæTlA in the limit T+0. For T lT.(( 1, we may therefore neglect the integral term. The function r/ is then given by
,l'r$):
t(5)[1 + oQ lr)1.
(10.111)
Substituting this into (10.105a) and using the low-temperature result for t given by (10.90), one finds a remarkable result: the viscosity coefficient tends to a finite value as I+ 0 (Pethick et al. L975, Einzel 1980, 1984):
fot rtrl
: cnl(rl)(t * o,T),
(I0.LLza)
where
2n I ks\12 1 c': T L^(o)l wof,(hy
(10.112b)
un:4u"
(I0.ILzc)
W
and Ws, fr(Lr) and u" have been defined by (10.90b), (2.68) and (10.90c) respectively. Estimates of the coefficient cn using realistic values of Á(0), I4zs and h2 yield values of 0.1-0.2, depending on pressure. The behaviour of q at intermediate temperatures may be calculated either by solving the integral equätion (10.102a) approximately using a variational method, or by approximating the integral kernel in (10. L02a) by a separable
10.3
Transport Coeficients
467
kernel as given by (10.86), which then allows an exact solution of the integral equation. In the present problem the symmetry is specifred by I 2 and r: *L (even function in 5), and the problem is symmetric in spin space. The integral kernel in (10.I02a) may therefore be approximated by
slto(6r, Er.): lr,L t(Er) (1,1-r(Ë))e -t(E)'
(10.113)
where Z(f) is the normalized relaxation time given by (10.102b) and the energy average ( )e ir defrned by (10.105c). The integral equation (10. L}Za) with the separable kernel (10.113) is readily solved. The viscosity coefficient calculated from (10.105) using this solution is then obtained as
,:
rt?z
(E'lE')'e
\
r,L\þ-r(Ð)
(10.114) -r(Ð) E+ ^z (U e - ),r((Ez l E1l i> El' Note that the result (10.114) becomes exact in the low-temperature limit and yields the correct behaviour near 7" as given by (10.110), with @s(0) approximated by the first two terms in the parentheses on the right-hand side of (10.110b). A numerical evaluation of (10.114) with parameter values appropriate for pressures of 0, I0,20 and 30 bar but neglecting surface effects is shown in Fig. 10.2. The initial drop in 4 below 7" is well described theoretically, whereas the low-temperature plateau is not borne out by experiment. Qualitatively, the reason for this can easily be understood: it is due to the deviations from the hydrodynamic behaviour associated with the increasing mean free path at low 7- The latter may eventually become larger than the characteristic dimension of the sample used in the experiment. Consequently, scattering processes of the quasiparticles at the wall become more important than those between the quasiparticles. This effect is not included in the theory described above and will be further discussed at the end of this chapter.
Bulk viscosity The hydrodynamic equations for the B phase derived in Chapter 9 contain three coefficients of the bulk (or "second") viscosity: Er, Er. and (3. These are dissipative coefficients multiplying V . u,, and V . g" in the expressions for the stress tensor and the dissipative part of the chemical potential (9.28) and (9.23). On the other hand, the trace of the dissipative part of the stress tensor is obtained from (10.48) by inserting the deviation from local equilibrium of the distribution function: uf r* tDi n3: å >ka k. uo* E¡,
:
-e ty. g, -
nm õ¡P
Ery . u,.
(10.11s)
L0
468
Transport Properties
0¿ rl (T) rl(Tc)
03
o9o fo$
oor
0.1 0.2 Figure
0.4
0
6
0.8 l trc 1.0
10.2 Shear-viscosity coefficient q(T) for tHe-B versus temperature at pressures P:0, 10, 20,30 bar as calculated from (10.114) (after
Einzel 1984); neglecting surface effects. Experimental values are from vibrating-wire measurements of Carless et al. (L983a,b): V, 0.L bar; A, 9.89 bar; O, 19.89 bar; E, 29.34bar.
The dissipative part of the chemical potential, i.e. ô¡,lo: ôpt"" - ôpto', where ô,,rt"" is the local equilibrium value, can be determined with the aid of (10.46), which expresses the density õn in terms of ô¡r and ô/*. By definition, the local-equilibrium value of the density change ôF¿ro'is equal to the exact value ôn, which implies that the dissipative part on the right-hand side of (10.46) has to vanish, and hence öpo
: - 1¡r.'^ >+ õf L: - 6rV . un - 6rV .g.. m ãE*
(10.116)
Here the factor Ilm accounts for the fact that ô¡rD is the chemical potential per unit mass. It is clear that the two terms in (L0.115) may be combined into a single expression:
+> nÌ i
: ! nZ (k . uo ku
keue)
*L¡, uf'r
(10.117)
10.3 Transport Cofficients Now the quantity in parentheses vanishes at the Fermi energy. Therefore only particle-hole-asymmetric terms contribute to the viscosity coefficients 6r and Ç2. Since each particle-hole-asymmetric term is order-ft/ft,small and since one needs two of these to get a nonvanishing result for the energy integration in (10.1-Ll), e, and (2 are at least order-([ l|*)'smaller than, for example, the shear viscosity,. The coefficient Ç2 multiplying V'u' acquires yet another factor of (?:/f.)'from the driving term in (10.100a), which is of the same structure as the term multiplying ô/i in (10.117) (Shumeiko 1972). The remaining coefficient Ez describes the friction arising from a normal-superfluid counterflow with nonvanishing divergence, which exerts a force on the superflow. It is again convenient to split off the dependences contained in the driving term by defining
ò/l
:
*ff
tr*,q'
(10.118)
s"t'e(E*).
The dimensionless function ?E is found to satisfy the integral equation
,þeG): ?(å,)[1
.I
d5, so(6,
,t'rfrrÞeG,)f,
(10.11e)
which is rather similar to the corresponding equation (10.I02a) in the case of the shear viscosity except that the kernel is now given by ,S0 (see (10.103)). From the experience with the shear-viscosity calculation, we expect to be able to calculate (3 exactly near T" and for T+0. The general result for Ët is obtained by substituting (10.119) into (10.116):
E,: es(#úr(6))u
(10.120a)
Here
tS:#urz$:l#r,
(10.120b)
sets the scale and the angular brackets denote the energy average defined by
(10.105c).
In the limit Á+0 there is again a nonanalytic contribution from
the factor of ETIEZ in (10.119). Replacing 7 and So by their normal-state limits and introducing a new function
Q@): where
(n'
x:
Elk"T,
jrr<e)
sechz
(r0.Lzt)
)x,
one arrives at the following integral equation:
+ x')e@) - a.' K(x, x)lr [
n
#
urr'rlOf''l
:
sechz
lx.
(10-122)
10 Transport Properties
4t0
However, this time one is not allowed to treat the term proportional to Á in perturbation theory, because the coefficient Âð* appearing in the definition (10.103) of 56 is unity and is thus equal to the eigenvalue kt- L of the first of the eigenfunctions tþ"(x) introduced in (2.56). This simply expresses the fact that particle number is conserved in the normal state (Âå* : L !) but ceases to be conserved in the superfluid state, where the number of quasiparticles is no longer equal to the number of atoms in the system. As one approaches 7: from below, the coefficient of the eigenfunction It r(x)
:#
sech2
lx
(10.123)
in the expansion of @(x) in terms of eigenfunctions ú"(x) diverges as .Á-1. The leading term in the bulk viscosity (3 is easily found by employing the eigenfunction representation (2.6I) of the kernel K(x, x'); it is given by Wölfle I977b)
E,: ESI:*T."o,,,t]
as
r+
r"-
(r0.124)
In other words, (3 diverges as (1 -f lf)-t/z as T+ [. The corresponding term in the Josephson equation (9.22), -6rV'g", still vanishes as it should,
:
-
because g, pr(u. u") approaches zero as A2, and therefore overcompensates the divergence of (3. In the opposite limit of low temperatures the integral term in (10.119)
again may be neglected compared with the out-scattering terrh. For the same reason as discussed in the context of shear viscosity, Ç, is found to tend to a constant:
E,(D:
ceeS(r")(r
*
"rT
as
r+ o,
(10.125a)
where
,e
:
2n 1. /k"T"\'
T
\^(0)/ yo &E:4U, - 1I4r,
,
w,
(10.12sb)
(10.125c)
At intermediate temperatures we may approximate the integral kernel in (10.119) by the relaxation-time expression (10.113) with L3*:\. The resulting expression for (3 is found to be
s,:ce[(
#rrt):ffi]
(10.126)
Equation (10.126) tends to the correct limiting values for T+ 7" and T + 0. In Fig. L0.3 the result of a numerical evaluation of Ç3 according to (10.126) is shown versus temperature.
10.3 Transport Cofficients
47t
.=t(¡l
î qt
800
1¡l
E
l¿
(v)
lJr
600
/.00
o oo o
200
o
"
0
Figure
0.8
o.?
o
o o
0.9 1 t Îc
l.o
10.3 Second viscosity coefficient CtQ) for 'He-B versus temperature
at as calculated from (10.126) (after Einzel 1984). Experimental values (O) are from the vibrating-wire measurements at "1..28 bar of Carless et aI. (1983a,b).
P:0
The coefficient Ç3 is not easily accessible experimentally. In principle, it determines the attenuation of fourth sound, as will be shown in Section tL.L, but in practice this contribution is very small compared with the one from mean-free-path effects. It has recently become possible to extract f3 from vibrating-wire experiments (Carless et al. 1983a,b). A comparison of the data with theory is shown in Fig. 10.3. Thermal conductivity We begin by splitting off the ft dependence of the driving term in (10.100a) from the distribution function: õf
L:izfl./Í.
Z+
@o-
q)ôr,/.(5).
(10.127)
It can be seen that ô/i is odd in f as in the case of 4 and (3, but is also odd in k, in contrast with the previous cases. The function qr"(Ð is found to satisfy the integral equation
ú-(E): r(F,)[1 .#,Il o;,sf-)(6,,
tU#,,Þ.G)f,
(10.128)
where one factor of E-L has now been shifted outside the integral, owing to
1.0 Transport Properties the extra factor of E in (10.127). T.aking into account the oddness of ò/¿ with respect to k+ -k, the kernel sÍ-) is obtained from (10.s2) and (10.g3) AS
sÍ-)(5,,
E):
Âì-/5-)(6 r, Er)
- yrrs)Gr, Er).
(10.129)
Here /f-) is defined by (10.87b) with, however, an additional factor of I,tr. in the Ír2 sum. The parameters Âi- and yr are given by (2.52) and (10.8ga) respectively. Inserting (L0.127) into the expression (10.50b) for the heat current and comparing with (9.33), one obtains the thermal conductivity as
r:
rs(r)
(10.130a)
@h(E'rt,*)e,
with
ro(z)
: åCN(r)ufr$(r).
(10.130b)
The thermal conductivity varies smoothly near [, because the factor E7ln3 (giving rise to the nonanalytic behaviour of 4 and fr) has been replaced by the nonsingular term E3lnr. Nevertheless, the slope of r changes discontinuously at T". In the limit of low temperature-s the integral term in (10.128) is again negligible owing to the factor 87. The various factors of T from the
relaxation time 7 and the energy averages (accidentally) happen to combine to a T-r law as in the normal state. The complete result is
¡r(r) :c*xs(rl(r *
""T)
as z->Q,
(10.131a)
where
21 n
ü*:4ut
(10.131b)
W6'
1"__Zy,
-l -'Hu,
(10.131c)
with l4ro given by (10.90b) The relaxation-time approximation for the kernel SÍ-) should take into account the fact that {-l1Er, E) vanishes for S1 : 0 and Ez:0 in the normal state. This property is readily incorporated by introducing factors E, and E, such that
sÍ-)(Er, 6r) l*ro: åÂì-
h
E2
(Ez liG))
e
¡G)'
(10.132)
With this approximation, excellent results are obtained in the normal state.
10.3
Transport Coefficients
473
1.2
l-lro l-l¡-'
t'r-=
o.s
Ylx 0.8
0.1
o'ob
Figure
'l l'lc l0
3He-B versus temperature at P :20 bar according to (10.133) using Âi- values from 0.5 to 2.O (after Einzel 1984). Experimental data points (x) are from heat-pulse measurements at 20bar by Wellard (1982) (see Hall and Hook 1936).
10.4 The scaled thermal conductivity r(T) of
For the B phase, a simple calculation yields
K(r) : *,
@f.ul<e'rfe>)e
+
åÂì-
l (10.133)
At low temperatures
(10.L33) reduces to the exact behaviour (10.131). A numerical evaluation of (10.133) shows that Trc(T) is roughly constant (see Fig. 10.4). This is essentially borne out by experiment. However, at low temperatures the experimental curve (Wellard 1982, Wellard et al. 1982) falls below the theoretical results. This may again be caused by finite-meanfree-path effects. Additional scattering processes involving phonons and, at high pressure, fluctuations of the solid phase have also been considered but have been found to be small (Geilikman and Chechetkin 1975). Spin diffusion
The spin-diffusion coeffrcient describes the dissipative spin current, which is driven by a gradient in @p, the deviation of the spin density from its equilibrium value (see (9.29) and (9.30)). Therefore the last term in (10.100a) is the relevant driving term. For simplicity, we assume the Landau parameter Fi to be equal to zero. When we discussed the spin density and spin current density in (10.51) and (10.57), we already noticed that the coherence factors here are rather complicated. Indeed, the spinantisymmetric component of the collision integral is so complex that we
10
474
Transport Properties
it here. A complete treatment has been given by Einzel (1980, 1981). Here we only consider the simplest relaxation-time approximation, where the in-scattering part is completely neglected. The shall not attempt to discuss
distribution function is then given by (see combescot 1975c) s
. Í¿oI r¡l L,p- i'--:! L qÍL@o. q)(o*)*Bp*,¡"u{D"n.
,.,
õf
(10.134)
Inserting this into the definition of the dissipative spin current,
(i"-t)"
: ä nZ u*rp*,¡"utr, (o" ôfi) kv
(10.13s) vl
one finds the spin-diffusion coefficient )
D,,;u; : fr 2 \. t - f L), ¡,u -trtvt
x-x,
¡¡(pz¡,)
*ú *.
(10.136)
u¡ing the definition (10.58) of p*,¡"u and the projector property of A2: A, one has (p,o)
*"
:
õ *u
-
4 #R*,Ê,R,¡Ê¡,
A¡,u,
(10.137)
whgre &, is the rotation matrix characterizing the B phase. As anticipated in (9.31), it follows that the diffusion tensor consists of three independent components: D,"ru¡: D1õ,"u6i¡
+
DzRpiR"¡
*
D3Ru¡Ru¡,
(10.138a)
with (10.138b)
Dz: Dz:
-+oo(ifetfi)r,
Do:åu!"|n Z& x
.
(10.138c)
(10.138d)
we already know that the factors E'ln'produce nonanalytic dependences proportional to A near [. Hence D.r3 vanishes as Á for T - 7., whereas D, tends to the normal-state coefficient Ð ". Just below T, D, falls, which is similar to the behaviour of the viscosity r¡, but the effect is smaller here, owing to the factor of ! in front of E'/8,. In the limit of low temperatures the spin-diffusion coefficients of (10.136)
10.3 Transport Cofficienß
475
are found as
Dt:
-4Dr,:#D"#;ffi
(10.13e)
since Do(T) is proportional to r$, 1z.l_s), whose low-temperature dependence is given by r$(f): rR(?:XT.lT¡', the spin-diffusion coefficients are seen to diverge as T-1 for T+ 0. The qualitative behaviour near I and as T+0 shown by (10.136) is in agreement with a more refined calculation (Einzel 1981). The spin-diffusion coefficient is difficult to measure quantitatively, because the dipole torque and spin supercurrents tend to mask the effects of spin diffusion (see the discussion of spin-echo experiments at the end of Section 8.6).
10.3.2 lntrinsic spin relaxation In this subsection we discuss situations in which, by some means, normal and superfluid components are driven out of equilibrium with respect to each other (see Section 8.6). For example, we might envisage increasing the density of the normal component (instantaneously) by injecting hot quasi-
particles into the system. This type of experiment has been carried out on superconductors, where a suitable device is available-the tunnel junction. One expects the system to relax towards equilibrium within some intrinsic relaxation time. In superfluid 3He the equivalent of a tunnel junction is not available, but there is none the less a way to prepare a nonequilibrium state of this kind for the spin densities, as discussed phenomenologically in Section 8.6. There we argued that the dipole torque acts only on the Cooper-pair spins, which adjust to this field within a time of order klA. The quasiparticle spins, on the other hand, relax towards the mutual equilibrium state via quasiparticle-number-nonconserving collisions in which quasiparticles are converted into Cooper pairs and vice versa. The corresponding intrinsic spin-relaxation rate is typically of the same order as the quasiparticle collision time and hence much longer than hlA, except very close to 7i. In the expression (10.51) for the spin density S" one can identify the quasiparticle contribution as
Sf:Lh2to,*"õfo,u,
(10.140)
kv
where s¿,u., is the BQP spin weight factor defined in (10.52a) and a notation as introduced in (10.18c) is used. The remaining terms in (10.51) describe the polarization of the Cooper pairs by the gauge field Hs", the external field and the Fermi-liquid molecular field. In the approximation where the Landau parameters Fi with
Hi'
10 Transport Properties
476
l>-2 are neglected, the Cooper-pair spin density is given SË: y-t
Z
by
(10.1ala)
Xf"lt""u,
where lPo is the cooper-pair partial susceptibility defined in (10.53) and
HïÏ: Hï, + HE"-,ä ott
(10.141b)
is the total effective magnetic field acting on the Cooper-pair spins. As
usual, this field is screened by the total spin polarization S, via the Landau parameter Fô. The effective field Hin may be expressed in terms of the spin densities by inverting (10.141a):
Hi"u:
yZ 0oo);,1(s" - sT),
(10.1a1c)
where we have defined the quasiparticle spin density S! - Sr, SË. The quasiparticle spin density in a local-equilibrium state, described by the distribution function
ô/fi, : f L(õe*,*- ttoHï'_ _f LttoH"*u, is found by substituting
ttoV"*)
(10.142)
(I0.I42) into (10.140a): S¡'t".
:
y-rZ
(10.1a3a)
x?"o"Uîu.
Here we have defined the quasiparticle partial spin susceptibility by xf"ou:
p3> Gf k
ÐG?)*",
(10.143b)
where ,i',:
(s?)u,:
(*)'u*"
*ff
ru",
(10.1a3c)
wrth Apu given by (10.52b). Of course, the two partial susceptibilities add up to the usual spin susceptibility 10, discussed in (3.102)-(r.t-o+): xo*u:ï¿Xo"+
xf,.
In order to calculate the relaxation time
(10.144)
zçB introduced in (8.188) from the kinetic equation, we need an approximation for the relevant part of the collision integral. To neglect the in-scattering part completely, aJ we did in the case of the spin-diffusion co,efficient, is not allowed here, because spin is a conserved quantity in the normal state. As one approaches ?: from bélow, the Cooper-pair density S! vanishes and Sl- S". However, in the absence of spin-conserving interactions, the spin-relaxation time is infinite in the normal state, and one therefore expects tçB to diverge for T-+ ?i. This
L0.3
Transport Cofficients
behaviour is brought about by a cancellation of the in-scattering and out-scattering parts of the collision integral. A simple relaxation-time approximation along the lines of (10.68a), which keeps much of the structure of the exact collision integral, is given by
ô/r,r,
---;['"-?t'"-
E s*,,u¡,õfL,.^lro
k'L
E
.
f L"lro
(10.14s)
The sum over k' in the last term of (10.1a5) may be related to Sl. Multiplying the kinetic equation (10.25a) in the limit Q:0 (we consider spatially homogeneous relaxation processes here) by E*s*,*ntr,(op. -.) and summing over k, one finds o,*"õf 1,") : iøsr ]!!----"E,{r-to)fL,lto, f / r¡, = ã\
(10.146)
s.
In deriving (10.1a6), *e have replaced (s?)r" by
-:#) f s,*)*u = ôpusz*: õ*u' "l('-#,1 (t
(
(B phase), (10.147)
(A
phase).
This is an exact result for the B phase. In the case of the A phase we assume that Sfl is approximately parallel to S[, i.e. perpendicular to d. Then the term propottønal to A*u in (10.143) does not contribute to s2¡,. The equation of motion for Sä is obtained by substitution of the local-equiiibrium solution (L0'r42) into ôfi: ôft - ôff" in the first term of the collision integral, then solving for ôf¿ and finally integrating as before. The resulting expression is so simple that we need not specialize to the hydrodynamic limit øz << L, but may keep the a¡ dependence, valid for ltot < Á, as a)by-product:
ø,sf,:
-ãb(s;-sË''"').
The complex-valued relaxation time 2"6(ø) is found
r"n(a):
E tïf kk
E (1 k
-
L
s'òf
E s?f
(10.1a8a) as
fuol(i+ an¡)
J--
Llq
E sïf k
Lt(ir
tttr¡)
.
(10.148b)
ren may be identified with the relaxation time introduced phenomenologically by Combescot and Ebisawa
In the hydrodynamic limit @r¡111,,
10 (1974).
By taking the limit rrl->0 of
Transport Properties
(10.14Sb), we flnd
the explicit
expression for the B phase: nB úcE
3(1-!d2ln2), , -
- 2((A2l E2) I 4n¡l
(
(1
u
- 3ñ I n')r(E)) e
(10.148c)
The first term in (10.148c) clearly diverges for A-+0. Using (10.106), the singular term is easily evaluated as
zB,(r):*"*
ffi
as r-+[.
(10.14e)
According to (10.L49), the intrinsic spin-relaxation time at fixed effective field Ë/"tr diverges as (L -Tlf)-t'z for T-7". The result (10.149) agrees with an exact calculation (Bhattacharyya et al. 1975, L977b; see also Combescot L975a). In the limit of low temperatures, z6s is found from (10.148c) as
råe: Ccp.T as Z+
(10.1s0)
0,
i.e. zçE is essentially given by the BQP relaxation time (10.90a). From (10.148c), the coefficient C6s follows as l, whereas an exact treatment of the collision integral yields a somewhat different combination of angular averages of the scattering amplitude than that appearing in z (Bagnuls et aI. 1979, Dombre et aI. 1980, Einzel 1984). Similar expressions may in principle also be derived for the A phase. In most experiments the spin relaxation does not proceed at fixed effective Ë1"tr, but rather at fixed total spin density S". It is a simple matter to rewrite the relaxation equation (10.1a8a) under these conditions by expressing Hi" in terms of S, and Sþ with the aid of (10.141c):
ø^sp:
-*tq-(s3),1
(10.151a)
The new equilibrium quasiparticle spin density (Sg), is then given by
su:Ës,
(10.1s1b)
where Xqo and yo are the relevant eigenvalues of the tensors 2(qo and 10. The new relaxation time z¡-¡ was introduced by Leggett and Takagi (1975). It is related to zçs by
Tt:t: )(Ñ rce. (10.1s2) ¡ The ratio of susceptibilities appearing in (10.152) vanishes as Á for T-7. and therefore cancels the divergence of r6s. More precisely, from (10.53), one finds for the B phase
#
: 3l**otrh#:or#"
as r
--+
r..
(10.1s3)
10.3 Transport Cofficients
ø
479
t'L
(o
o I
r.0 cc
þ
0.8
0.6 Figure
0.7
0.8
0.9 tk T
1.0
L0.5 Spin-relaxation time z*(f): Lr(XoolXÑ)XlXo defined in
(S.197) versus temperature at P:20 bar (after Einzel 1981). Experimental data are from Webb et al. (I975a) (O) and Eska er al. (1982) (L).
Consequently, ty, is found to become equal to the quasiparticle relaxation time at the Fermi energy in the normal state
ãr-r: zS as T +
\.
(10.1s4)
Equation (10.154) is also valid for the A phase (Bhattacharyya et al. 1975, L977b). This important result implies that the normal-state quantity z$, which is not experimentally accessible in the normal state, may be measured in a spin-relaxation experiment in the superfluid state close to 7" (see Section 8.6). In the limit of low temperatures one has XPolXo'-l and consequently rrr+ zce. A comparison of Í¡a âs calculated from (10.148) and experiment is given in Fig. 10.5.
10.3.3 Transport coeff¡c¡ents of the A phase The calcutation of the transport properties of 3He-A is considerably more complicated than for the B phase. Technically speaking, the anisotropy of the gap A¡, i.e. the explicit /c dependence of A¡,, forbids a separation of the energy dependence and the angular dependence. Consequently, analytic results may only be obtained in particular cases, such as the first-order
480
rc
Transport properties
corrections in Á to some of the transport-coefficients near ?i. Indeed, the results for these corrections obtained in the preceding section for the B phase may immediately be taken over by simply replacing the isotropic B-phase energy gap a by the angular average (aù a in ttre À phase. Besides the additional mathematical complications, the anisõtropy of the A phase in orbital space also leads to many more transport coefficients than in the B phase. In the limit of low temperatures these behave in quite different ways, depending on the relative orientation of the orbital axis I to the direction of transport or to the spatial orientation of the quantity transported (e.g. momentum). In the following we shall discuss several important transport coefficients, employing the simplest possible relaxationtime approximation.
Viscosity
Let us first consider the shear-viscosity coefficients ?,', Tt,4f and the bulk-viscosity coefficients E,l't, €l),t appearing in the for the "*pr"rrion (9.82). rr,,, From general the j"nsor definition of Iru as obtained in :ll"r.r
(10.48) and the expression for the dissipative part of the ch'emical-potential change (10.116), it follows that the dissipative component of II,, is given by
:> #Lk h(ue,k¡ - !ô,,urk ) ò/í. ku E
rI3
(10.1ss)
The change in the spin-symmetric BQp distribution ô/i due to velocity fields u,, and u" is immediately obtained from (10.100a) in the approximation
where the collision integral is assumed
as
i o^' ôÍo:-¿oro' with õf
L:
Íþo\lr
. V . uo+ åi(fr
. u¡,
-
kçue)q
.un- i
kpup
3p
o
'r"]. (10.1s6)
It
follows from (10.155) that the trace of rrf; is of order (TlTï)t, i.e. it is small owing to the factor of u¡. k - upkp. This gives rise to the foltowing relations among the viscosity coefficients in (9.S2) (for the first relation seè Combescot L975):
qtt-qt-q'
:
(10.157a)
"[(ä)'], 2E!
+er
: o[(
är] (i: L,2).
For all practical purposes, one may put the right-hand equal to zero.
sides
(10.1s7b)
in
(10.157)
10.3
Transport Cofficien*
481
substituting (10.156) into (10.155) and defining a funcrion
A*: (-¡Llrr(*\', \Í-k/
(10.1s8,
one ûnds the following result for the transport coefficients in the simple relaxation-time approximation (see also shahzamanian 1975):
rlt
:)
,toul,,,t ku
(10.1sea)
3,
: ) Aeuz¡,,k1,, ku qtL : @ rkr)rà Ao(Ê', - +)', ?,,
tr:TZ u(Ê?- å),
(10.lseb) (10.1s9c) (1o.1sed)
where the z axis has been chosen to be along /. The coefficients 6i,r are order g ITF)z small. The transport coefficients er, Eo and Ç, in the Josephson equation (9.g4), defined in (9.85), are calculated similarly by evaluating t'he expression (10.116) for the dissipative contribution to ttr" ðn"tnical polential. As Ça and Ç5are related to 6ll'1 by (9.86) and (9.g7), we need onty consider Ç3, which is obtained by inserting (L0.1,56) inro (10.116):
t,:(ry)'ànr In the limit T (10.105b),
one finds q,
(10.160)
: q' :
ql_: 4s(?:), with 4o defined in and 6l'': E!,':0. The coefficient Çr'diveiges like Z;r, u, given by (10.124) with A reptaced by ( An) t. At low temperatures Bogoliubov quasiparticles tend to be concentrated in two small areas on the Fermi surface of relative extension ksTlAo around the nodes of the gap (chosen to be along tâ). Hence the momenrum sum in (10.158)-(10.160) gives rise to a factoi( T lT")r, whereas each facto r of kf; or.fi under the k sum conrributes an additionai factor of (TlT")t. togethe, yitfr lhe leading temperature dependence of the Bep relaxatiol rime given in (10.95a), one finds in the limit f -+ 0, t T¡2 TL
q'(r)-(;l \ rc'l
qoT.),
qtt(T)-const xtto(T),
qLQ)-
(Ë/
ro(T),
-*: ni,, o lhkrur\z 6,-(Ë)rL. -
(10.161b)
t T:,2
ç, hkpup
Çi
(10.161a)
I
(10.161c) (10.161d) (10.161e)
10
482
Transport Properties
The coefficients n!-, çil and E, are seen to diverge as T-z at low temperatures. Measurements of the various viscosity coefficients are made diffrcult by the existence of nonuniform i textures in any given sample. Nevertheless, it
has been possible to determine the coefficients qll and 4' quantitatively (and the remaining coefftcients at least qualitatively) by means of a torsional-oscillator experiment, in which the frequency shift and the damping are measured as functions of temperature. In order to relate these data to the transport coefficients, a theoretical analysis is required that involves the calculation of the possible l-textures in a slab geometry (Hook L989, Hook et al. 1989). Thermal conductivitY According to (9.83), the diffusive thermal transport is charactefized by the thermal-conductivity tensor K¿i, which in the A phase is completely determined by the two eigenvalues rll and rt for transport parallel and perpendicular to Î. fne diffusive heat current in the kinetic theory is given by (10.50b) as
jB:2
uoEo
ku
af
where particle-hole symmetry has been employed. thermal-conductivity coefficients are given by K',,-
-
#¿Gf
(t0.L62)
L,
L)roí'o(r,l;'),,
It
follows that the (10.163)
where ul,'t is a component of the velocity vector hklm* parallel or perpendicular to L At low temperatures, using (10.95a), Kll'' are found to obey the power laws (note that (Eù - f),
.rr(r)
¡('(r)
-# "-(e),1
(10.164)
-fi *tcl.J
Orbital relaxation
In the equation of motion for î, (9.87), four transport coefficients have been introduced. Among these, the coeffrcient Âa relating the bending force rlr to ôrÎ is of particular importance (Brinkman and Cross 1978). It describes the effect of viscous forces, caused by the dependence of the quasiparticle energy E¡, on Î. ttt" remaining coefficients appear to be small (Nagai 1980)An infrnitesimal rotation of I changes the energy of the quasiparticle system
10.3
Transport Cofficien*
483
by the amount
ôr"-àut.*uo,
(10.16s)
where ô/¿ is the deviation of the quasiparticle distribution from equilibrium (corresponding to a.quasþarticle energy spectrum with gap axis ãlong the instantaneous direction of I). Since the quasiparticle system needs a finite time to adjust to a new gap orientation i the quasiparticle distribution always lags behind the motion of. Î, i.e. ô/¿ nA,î. This .,normal-locking,, effect (Leggett and Takagi t976, I97S) produces a viscous force (Cross and Anderson 1975; see also Combescot L975d, 1978a)
ò^E" La ôt--?rni"":-¡t, where
h:llhc
(10.166)
is called the "orbital viscosity".
According to (10.165), this force may be expressed in terms of the quasiparticle distribution function ô/¿. Employing the simple relaxation-time approximation for the Boltzmann equation
ô,ô/o:-louro,
(r0.167)
and replacing ô/¿ on the left-hand side by the local-equilibrium expression
ô,ô/f.:fL?ff.a,î, one finds as an approximate distribution function
(10.168)
in the hydrodynamic
regime (10.16e)
On gombining (10.165), (10.166) and (10.169) and using AE¡IAî: -k(k ' l)A"olEk, one finally obtains the transport coefficient ¡r¿ in the form
pt:*attZf'r<*.îY(+\' \Ek/ k
(10.170)
The limiting temperature dependence is easily found to be given by (T
-
TL),
(10.171)
(r+ 0). As shown by Pethick and Smith (L977), the coefficient ac may be calculated exactly, yielding ø": l28ln2 .
L0 Transport
484
Properties
The motion of i is dominated by the viscous interaction with the normal fluid. This is confirmed by experiments (Paulson et al. L976c) in which the relaxation of Î, fottowing a sudden change in the orientational force exerted on Î by a magnetic field via the dipole interaction, has been studied. The observed relaxation rate is in good agreement with the theoretical estimate (10.170). The limiting value of pt as T+ ?i provides yet another measurement of the quasiparticle relaxation rate L/zR¡ at T..Long-lived oscillations of the I fiet¿ with typical periods of the order of seconds, rather than pure relaxation, have also been observed (Wheatley 1978). These are probably to be explained as nonlinear periodic solutions of the hydrodynamic equations.
10.4 FLOW IN RESTRICTED
GEOMETRIES
We have seen in Section I0.2 that the mean free path / of Bogoliubov quasiparticles may become very long at low temperatures. V/hen / exceeds the dimensions of the sample cell or some other characteristic dimension like the size of an object immersed in the liquid (".9. u vibrating wire or a torsion pendulum), the state of the quasiparticle system is determined by the interaction of BQP with the conñning boundaries rather than by the collision processes within the liquid. It goes without saying that the flow behaviour expected on the basis of bulk hydrodynamic theory will be altered considerably in such a situation. A theoretical interpretation of flow experiments thus requires the consideration of quasiparticle-boundary scattering in addition to interquasiparticle scattering in the bulk. At present very little is known about the interaction of quasiparticles with a solid wall. A convenient model of a rough surface is provided by the assumption of diffwe scattering, whereby quasiparticles leaving the wall are assumed to be in local equilibrium with the wall. At a free surface of helium the assumption
of.
specular scattering is probably more adequate.
10.4.1 Slip correstion to hydrodynamics Bulk hydrodynamic theory is valid in the limit llR-+0, where R is a characteristic dimension of the boundary. The boundary condition on the fluid velocity freld in this limit is given by the "sticking" condition: the tangential velocity of the normal-fluid component at the wall must be equal to the velocity of the wall. In contrast, the superfluid velocity and the component of the normal velocity perpendicular to the boundary need not satisfy the sticking boundary condition. In fact, a normal-superfluid counterflow 8" perpendicular to the boundary (at zeÍo mass flow) is possible-provided that heat may be transferred through the boundary into the adjoining solid. A first-order correction in I I R to bulk hydrodynamics may be obtained by allowing for a modifred boundary condition, i.e. fluid slip at the boundaries.
10.4 Flow in Restricted Geometries
48s
In order to calculate the amount of fluid slip, it is sufficient to consider the simplest situation (Couette flow), i.e. shear flow parallel to an infinitely extended plane wall. In the usual bulk hydrodynamics-valid in a samplè sufficiently far from the wall that collision processes in the liquid dominàte over surface scattering-the velocity profile will be linear. This is a consequence of the hydrodynamic equations. The linear behaviour, however, need not extrapolate to zero velocity at the wall (assuming the wall to be at rest). Rather, there will be a finite intercept of the linear extrapolation to the velocity profile at the wall, i.e. to some extent the fluid slips along the wall, depending on the nature of the quasiparticle scattering at the wall. The slip effect may conveniently be characterized by the so-called slip length Ç, which is defined as the distance behind the wall at which the velocity extrapolates to zero (see Fig. 10.6). The stip length is usually of the order of the mean free path. Although the actual velocity field does not remain linear close to the wall (as assumed by this extrapolation), but falls somewhat below linearity, the effect of fluid slip on the bulk properties is described correctly by the slip length. Namely, the (transverse) velocity profile u(z) in the bulk shifts by the constant amount
",: €(y)_
(10.r72)
This effect may easily be incorporated into the hydrodynamic description, by replacing the usual boundary condition ø(0): ür*.n by the slip boundary condition
ø(0):
t(*).:, * Ø*"n.
(10.173)
Examples of the application of this boundary condition will be discussed below.
Figure
10.6 The velocity profile u(z) near a wall (schematic). The extrapolated velocity uo and the slip length
f
are indicated.
10
Transport Properties
The slip length may be calculated within the kinetic theory discussed in this chapter by adding a boundary condition on the distribution function ò/¿ to the kinetic equation (10.25). In general, the boundary condition connects the distribution functions for particles approaching and leaving the wall,
ô/[-) and ô/[*)
respectivelY: (10.174)
Here V*:Y*E¿ is the BQP velocity, ñ is the surface normal, and the ft' sum is restricted to states with ñ ' V*,10. The transition probability (fr| R lr{') satisfies the detailed-balance relation
lñ.Vo,l (ft| R lk'):lñ'Vol(-k' lR as well as the sum rule valid for elastic processes
l-ft)'
(10.17s)
(10.176) k
(ñ.vk>o)
which ensures quasiparticle conservation. For the normal Fermi tiquid as well as for an isotropic superfluid (B phase), the slip length E may be calculated exactly by assuming diffuse boundary scattering and employing a relaxation-time approximation of the collision integral that is consistent with momentum conservation (F,inzel et al. 1983). Foi the normal Fermi liquid, one finds E :0.579/, where I : upt. For more general types of scattering, variational methods aÍe useful, allowing one to obtain bounds on the slip length. For example, a lower bound on the slip length, which is within one per cent of the exact solution for diffuse scattering, is given by
,r(Lz-
Kz) + (L'+ K")(Lz/\-r --'1" . zLz(Lr* Krr)
Kz)(L'-
K..-)
(10'r77)
where
Ln
: k'>' ef þkf;vo,,@vo,')-' , k
K^n: tP>' 2' ef L)k"k''vk,,"(rv*,")^-'(rvo',")- t(tl n ak'), kk' and the prime on the summation sign indicates Vo,") 0
ot V¡,',,> 0.
10.4.2 Andreev reflection
At a boundary the order parameter is distorted within a layer of thickness
EQ), the coherence length. This leads to a modification of the scattering of BQP off a boundary. In conventional superconductors this effect may occur at a normal-superconducting interface and will lead to important changes in
L0.4 Flow in Restricted Geometries
487
the transport properties, as frrst discussed by Andreev (1964). Indeed, the thermal conductivity of a superconducting metal in the intermediate state is dominated by the scattering of quasiparticles off the interfaces between normal and superconducting regions, at which the gap parameter tends to zero within a distance EQ). A similar effect occurs at a boundary in superfluid 3He, where the gap parameter varies in space. The gap distortion induces an interaction of the BQP with the boundary. This effect, called Andreev reflection, acts in addition to the microscopic scattering process right at the wall. It is similar to the reflection of a quantum-mechanical particle by a potential threshold in the case of a particle approaching the threshold from the top of the step. The Bogoliubov quasiparticle is turned back by means of an elastic transition from the particle-like excitation branch with Eo> 0 to the hole-like excitation branch with fo
10 Transport Properties
488
quasiparticles may exist (Privorotskii 1977). An experimental determination of the order-parameter healing length has been reported by Ichikawa et al. (1937); see also Chainer et al. (1984). Given the gap profile, the probability for Andreev reflection, R(E), may be expressed in terms of the usual quasiparticle amplitudes u¡, ãÍrd u¡,, defined in (3.18), which determine the reflected and incident currents. Using (3.16), it may be readily shown that the stationary "quasiclassical Schrödinger equations" for u¡ and up, known as the Andreev equations, take the form (Andreev 1964)
(n r (u r
- ir,#)*rQ) : ^(z)u rØI * i",
*)'
-@)
:
(10.178)
a(z)u ¡(z),)
where z is the distance from the surface and A(z) is a given gap profrle. In the case of diffuse scattering at the wall the transmitted quasiparticles are necessarily scattered into different /c states, which means that they are no longer able to interfere constructively. Such a process may be modelled by assuming the quasiparticles to propagate from the threshold to infinity. The differential equations (10.178) may then be solved easily. At the gap edge, i.e. for E*-0, total reflection is found to occur, with corrections linear in Eo. A useful interpolation expression is given by
Ro(E):
E-E E-uE'
(10.179a)
with a coefficient u l L depending on the details of A(t) (Kieselmann and Rainer L983, F;inzel et ø1. 19S4b). A comprehensive treatment of Andreev scattering using the quasiclassical Green's-function formalism has been presented by Kurkijärvi and Rainer (1989). The transition probability in the general boundary condition (10.174), which takes account of Andreev reflection, is thus given by (kl R lk'):NFlRA(Ee)ô(ie
- fr)o(Eo+ E*).
(10-r79b)
The expression (10.179) for the transition probability (fr| R lfr') may conveniently be substituted into the general formulae for the slip length obtained by variational methods, ,and into similar expressions for the integrated Poiseuille flow, the surfâbe impedance and other quantities. One finds that Andreev reflection leads to an increased slip length and a reduced effective viscosity. 3He-A with nonuniform i texture. Andreev reflection also occurs in bulk Greaves and Leggett (1983a,b) predicted that if a quasiparticle arrives at a point where E-
10.4 Flow in Restricted Geometries
489
Furthermore, the small momentum transfer occurring in the Andreev reflection process at the phase boundary between the bulk A and B phases (Yip 1985, Yip and Leggett L986, Leggett and Yip 1989; see also palmeri 1989) may explain the surprisingly high propagation velocity ues of the A-B interface observed experimentally (Buchanan et al. 1986). A somewhat similar theory of the dynamics of the interface was developed by Kopnin (1987), while Markelov (1987) used a hydrodynamic approach. A different aspect of the moving A-B interface has been studied by Salomaa (1988), who finds that the moving superfluid A-B interface undergoes a phase transition associated with a change in the discrete symmetry and topology of the energy gap (see also Salomaa and volovik 1988). Such a transition should lead to a discontinuity in duas/dT, i.e. in the slope of uo". This effect has not vet been seen.
10.4.3 Poiseuille flow The simplest example of boundary effects on fluid flow is provided by stationary flow through a channel of constant cross-section. For example, in a parallel-plate geometry (of width d) the total flux is given in terms of the pressure gradient ðP lðx along the channel by
G_
AP ¿I ðx l2r¡"u
(10.180)
Here T.n is an effective-viscosity coefficient; it reduces to the usual shear viscosity q in the limit lld+0. The first correction in the so-called "Knudsen number" Kn=lrld may be calculated by employing the slip boundary condition (I0.L73). Here /n is the viscous mean free path, defined by lr: llÍL - )'r(EtlEt>El (1)sl, with / and ),2: ),î* defined by (L0.92c) and (2.51) respectively. The corrected expression is given by
q
flen -t+61. A
(10.181)
For general Knudsen number, the total flow has been calculated using a variational solution of the kinetic boundary-value problem for normal and superfluid Fermi liquids assuming diffuse scattering (Højgaard Jensen et al. 1980; for the case of cylindrical geometry, see also Topsoe and Højgaañ Jensen 1984). As expected, the scaled integrated flow Gld2 is found to decrease with decreasing plate separation d, reach the so-called Knudsen minimum and rise again in the Knudsen regime (dlln << 1). At dll,t<2 the slip approximation deviates significantly from the full result for general Knudsen number. Experimentally, the flow of normal 'He through a cylindrical tube was studied by Eisenstein et al. (1980a), who found reasonable agreement with the slip theory. Extending earlier experiments by Parpia et al. (1978) and Archie et al. (1981), Parpia and Rhodes (1983) measured the damping and frequency shift of a torsional oscillator with
10 Transport Properties
490
parallel-plate geometry, which also relates back to Poiseuille flow at the low frequencies used. In the normal state they found a slip length that at zero pressure is 40Vo smaller than the value calculated for diffuse surface scattering whereas at melting pressure good agreement with experiment is obtained (see Einzel and Parpia I987a). They also found a minimum in the integrated flow, which was, however, displaced towards a higher value of dll when compared with the theoretical curve. Deviations from hydrodynamic flow have also been observed in gxperiments on spin transport along narrow capillaries (Sachrajda et al. t984). In the superfluid the effective viscosity T.u falls off rapidly with decreasing temperature as shown in Fig. 10.7 (,Einzel and Parpia 1987a,b). Figure L0.7 also shows the dependence of 4en orl the Knudsen number Kn:l,tld. This behaviour can be explained quantitatively at high pressures and semiquantitatively at lower pressures by assuming diffuse and Andreev scattering at the boundaries (Einzel and Parpia L987a,b, Zhang and Kurkijärvi 1988). The effect of the additional scattering of the quasiparticles due to friction 3He between the normal and superfluid components in rotating superfluid has been investigated by Sonin (1981). In narrow channels and at high flow
Kn
0.5
(r¡
02
0.r
(T)/ 1 (T.))5u¡¡
l0-1 (I)
= .o
--
1o-
l-
Ë
'o-
¿ Figure 10.7
3
2 ltr
Effective-viscosity coefficient I"u(T), introduced in (10.180), versus T"lT and Knudsen number Kn : lr ld at P :2O bar. The full curves are the theoretical results for diffuse scattering only (curve L) and for diffuse scattering plus Andreev scattering from a steplike orderparameter profile near the surface (curve 2). The open circles represent the data points. Also shown (dashed line) is the theoretical result when surface effects are neglected. (After Einzel and Parpia (1e87a).)
L0.4 Flow in Restricted Geometries
49r
rates critical velocities associated with pair breaking have been observed in the B phase (Eisenstein et al. !979, Maninnen and Pekola L9g2); see Sections 7.IL and 7.I2\.
10.4.4 Vibrating-wire exper¡ments The first experiments to determine the shear viscosity in superfluid 3He \ryere performed by measuring the response of a vibrating wire immersed in the liquid, i.e. the damping and resonant-frequency shift caused by the liquid (Alvesalo et al. 1974, 1975). The method has since been used extensively to study normal and superfluid 3He (Carless e/ al. I983a.b, Hall 1983) as iell as dilute mixtures of 3He in aHe (Guénault et al. 1983a). The physics of a vibrating wire is characterized by three typicat lengths: the radius R of the (cylindrical) wire, the viscous penetration depth ô: (Zqlp,a)1/2 (given by the wave vector for shear flow) and the mean free path /. In the limit / << R, hydrodynamics is valid. For this situation, the force per unit length exerted on the wire, including the slip correction of first order in //R, has been calculated by Højgaard Jensen et al. (1980). A solution of the problem including the Knudsen regime (/ << R) has not yet been given, although a variational theory may be formulated (Beyer 1986). The mosr recent measurements show that the vibrating-wire technique provides a sensitive probe of the quasiparticle dynamics even in the ballistic regime (Guénault er al. 1986, Coates et al. 1986). Extreme nonlinear damping by the quasiparticle gap was observed in these experiments (Carney et al. L989). Distortion of the order parameter by a moving surface has also been detected in these experiments (Guénault et al. I983b, Castelijns et al. 1986). This was inferred from the observation of a sharp critical velocity u",., above which high dissipation sets in. Comparison with the bulk valu" ,1" of the depairing critical velocity showed u,,": äu?,", suggestive of a 50Vo depression of the gap parameter near the surface.
10.4.5 Sound propagation and other flow problems When sound propagates in restricted geometries, its velocity and attenuation are affected by the interaction of the particles in the medium with the walls. In a normal fluid the friction of the sound wave at the walls of a channel of width d leads to a complete damping of sound for d (( ô, where ô is the viscous penetration depth. By contrast, in a superfluid system the superfluid component is still able to flow (provided that d > E, the coherence length) and may carry a density wave, so-called fourth sound (see Section 11.1). Since the viscous penetration depth increases rapidly with decreasing temperature, a transition from first sound to fourth sound may be achieved by simply cooling a given resonator. In the transition region the sound attenuation is expected to go through a maximum, while
10 Transport Properties
492
the sound velocity drops by a factor (pJ p)''' (Højgaard Jensen et al. 1983). The transition is particularly sensitive to the liquid-boundary interaction. It is plausible that in the regime dll << L the sound dispersion relation is governed by the stationary Poiseuille flow properties (see (11.10)). On the
other hand, for dll>>'L., only a thin layer at the walls of the cavity is disturbed by the boundary scattering, and the momentum transfer to the wall is described by the transverse surface impedance Zr:Il,,lul"", as described by (11.3). The surface impedance is perhaps the physical quantity that reflects the properties of the boundary surface most directly. The transverse surface impedance has been calculated from an exact solution of the model kinetic phase by applying equation for the diffuse-scattering boundary condition (Einzel et al. 1983; see also Ebisawa 1980) and using a variational approach for a general scattering law (Einzel et al. 1987). Within the slip
B
a
approximation (valid for at 11I), the transverse surface impedance is readily found from the hydrodynamic equations as
Z,=#:(1 -i)3(t-?r),
(10.182)
where 5 : (2r¡ I pn )t'' is the viscous penetration depth and Ç is the slip length. The surface impedance in the A phase has been discussed by Hook and Zimmermann (1981). Another flow situation in which boundary effects are very important occurs in a so-called U-tube oscillation experiment. There the two vertical columns of a U-shaped vessel are connected by a narrow channel, such that only superflow connects the two parts. When the level of liquid in one of the columns is raised by means of an electric field applied to the liquid, and if the field is then switched off suddenly, one might expect oscillations to occur about the equilibrium position of the liquid levels in both columns. However, experimentally, only a slow relaxation back to equilibrium is observed (Eisenstein and Packard 1982, Eisenstein 1983). This finding may be explained by taking into account superfluid-to-normal-fluid interconversion near the free surfaces and the friction of the normal flow at the walls of the vertical tubes (Brand and Cross L982). From the hydrodynamic equations supplemented by the usual boundary conditions on the normalfluid velocity field, u': usu.ra.. at the free surface and u,,:0 at the walls of the channel (plate separation d), one flnds a pressure change associated with the velocity i of the moving free surface:
ôP: p(l2rtEr)'''L d' At lower temperatures,
(10.183)
7., the mean free path is no longer small in comparison with the diame{er of the tube. In this ôase the viscosity 4 must be replaced by the effective viscosity coefficient r¡"o relevant for Poiseuille ,T,<(
flow (Einzel and Parpia L987a).
10.4 Flow in Restricted Geometries
493
The boundary condition on the normar-fluid velocity componenr pe{pendicular to a boundary surface depends in principle on the pioperties of the superfluid itself, as well as on the nature of the surface. Ñormal-nuid flow carries entropy, and therefore the physical quantity that should determine a boundary is the heat je current through the boundury. Àrìir"urr"d !" ?t in chapter 9, the heat current in a superfluid ñas two additive contributions: a convective one and a diffusive one, i.e. jo: Zsu,, _ xyT. Thus, in order to determine the normal-fluid velocity fromã giuen'heat current íhrough the boundary' one needs to know the temp"r"t,.,." gradient at the boundary. The general hydrodynamic boundary condition, ão, an isotropic superfluid relating je and the superfluid flow g, to the temperature discontinuity az and du^ldx at the boundary (surface normal paraùel to Í) ur" girn"r, uy
(f) :(": íîi"')çï,1),
(10.184)
where
n - ân - 2Efl
*
Ez+ Etpz,
(10.18s)
and r¡, Er, E, and f3 are the viscosity coefficients of an isotropic superfluid defined in (9.28) and (9-23) (Grabinski and Liu 1987). For u o"""ription of a general boundary condition, three so-called "surface onsager a, b and c aÍe required- The coefficients have been calculated "ãiffi"i"n,r,, from kinetic theory for the case of diffuse scattering at the boundary 3He-B in al' (1989)' who find that the coefficieni a dominates. This implies by sun er that, to a good approximation, u,,:0 and jn: at -tcyT the boundary and justiûes the choice of boundary condition for the above u-tube osciilation experiment. By contrast, in superfluid fie one has js : Tru,, and __ VZ 0 at the boundary. This drastic difference in behavi-Ñrr is associated with the characteristic length  governing the temperature variations in a superfluid in the presence of a srationary hear flow. From (g.zz), (g.zi) uÃa p.n¡, one finds that the spatial variation of the temperature and the normal-fluid yglogity component along the direction of heaì flow i are given by (saslow reTL)
T
- L:
T" e't^ + T{e-,/^,
sun:#,f, - L) *Et i.e. there is a
temperature jump
(10.186) (10.187)
at the boundary, followed by
an
exponential relaxation extending over a distance Â. The characteristic length )u can be expressed in terms of ihe transport coefficients as
^ n:
/
uK\Llz
\Æ)
'
(10.188)
with ødefined above. For superfluid 3He, Â is of order (TilZ")t, i.e. much longer than the mean free path l, whereas for s.rp".ti.,ij'i¡g1", ],"-1.
10 Transport Properties
494
The temperature jomp at the boundary in the case of 'He is found to be negligibly small, and hence the so-called "Kapitza resistance" is also small. (In the case of aHe the temperature jump and the exponential relaxation may be combined into a single effective temperature jump.) The heat resistance associated with a plane boundary is then given by is the difference in the temperaRr= L:I.o,lie:s-r(ulKT)t'', where ^7w tures of the superfluid far from the boundary and at the (solid) boundary. We note that the mean free path drops out of the ratio ulx. In the B phase, where s c. s-atksr, the thermal boundary resistance is seen to increase exponentially at low temperatures, in agreement with experiment (Parpia L985, Castelijns et aI. 1985). If the channel connecting the two reservoirs in a U-tube experiment is sufficiently narrow that only tunnelling of Cooper pairs is possible, one should expect the analogue of the Josephson effect to occur, as found for superconducting weak links (Josephson L965, Anderson 1967). Josephson 3He-B may be flow oscillations across an orifice connecting two reservoirs of induced by changing the pressure on one side. The optimal conditions for such an experiment, i.e. the shape and dimensions of the orifice, have been investigated theoretically by Monien and Tewordt (1986). These authors numerically solved the corresponding Ginzburg-Landau boundary-value problem for the order parameter and found the optimal length of the channel to be given by about seven Ginzburg-Landau coherence lengths. The Josephson flow through an aperture of radius R much less than the coherence length has been discussed by Kopnin (1986a). aHe An unambiguous observation of Josephson oscillations in superfluid was first reported by Avenel and Varoquaux (L985, 1.986), who were able to identify individual phase-slip events. They were then also successful in 3He-B at zero pressure (Avenel observing near-ideal Josephson effects in and Varoquaux 1987; for a discussion of a Josephson experiment see also Lgunasm aa et al. 1983). These findings gave flnal proof for the actual of this important phenomenon peculiar to superfluids. Phase "*þr"" slipfi,'äge has in fact also been measured in the spirz supercurrent (BorovikRomanov et al. I987a,b,1988). Hence one may expect that the corresponding Josephson effect is also observable (Markelov L988).
10.4.6 Superftuidity in
3He
films
Following the first discussion of the effect of boundaries on the superfluidity of a 3He film of thickness comparable to the coherence length by Ambegaokar et at. (I97$¡, the variation of the transition temperature with the channel width between the'a.p.*ates was calculated in more detail by Kjäldman et ø1. (1978) (see alsci:,:)Ìe5anovió and Valls 1986a,b). Thermodynamic stability of various phases in narrow channels had been studied earlier by Barton and Moore (1975b). As already mentiotrsd, â realistic
10.4 Flow in Restricted Geometries
49f
calculation of the effect of boundaries on the superfluid state requirer formidable numerical effort, except for the simplest case of specular scattering and the somewhat more realistic case of diffuse scatteringby the confining walls. An extensive treatment of the dynamical properties oi superfluid 'He films using the quasiclassical Green's-function ãpproach has been given by Kopnin (1986b). The superfluid transition tempèiature of a film may even be higher than that of the bulk liquid owing to enhanced pair attraction in the vicinity of the surface. This hypothesis is not implausùle, since the first few layers of 3He at a solid surface are known to be nearlv ferromagnetic (Ahonen et al. L976a, L977a, r97ïa, Bozler et al. rg7g, Godfrin et al. L978). Theoretical models that account for the observed behaviour have been developed by Béal-Monod and Doniach (1977) and Béal-Monod and Mills (1978,1980). In later experiments on superfluid 3He films of 0.3 ¡rm thickness confined between a stack of mylar sheets, it was found that the A phase is stabilized down to zero pressure and down to the lowest temperatures of about 0.5 mK (Freeman et aI. 19sg). Superflow in adsorbed films of thickness (0.5 ¡^rm has been studied by Davis et al. (19gg). These authors determined the suppression of T" with decreasing film thickness and found the critical current to scale with temperature as
(t _ T lL)',r.
So far we have considered 3He films with thicknesses of the order of the
coherence length. These are still three-dimensional systems. A truly two-dimensional 3He film, i.e. a monolayer or less of 3Hè, is expected to form on the free surface of a dilute solution of 3He in aHe (Andreev L966, Zinov'eva and Boldarev 1969). Estimates of the pair interaction indicate a p-wave superfluid transition (Edwards et al. L977). The fascinating properties of such a two-dimensional Fermi liquid have been studied by Stein and Cross (1979), Fujita et al. (1980), Brusov and popov (1981), Ashida and Takagi (1981), Miyake (19s3), Tesanovió (1984), Korshunov (19g5) and Nakahara (1986). It is found that there exist two unitary phases corresponding to the A and B phases of the three-dimensional liquid, whictr are, however, energetically degenerate in the weak-coupling limit because both phases have isotropic energy gaps. Stein and Cross (1979) point out the possibility of an Ising-like phase transition in the "4" phase in addition to the Berezinski-Kosterlitz-Thouless vortex-binding transition expected for a two-dimensional superfluid (Berezinskii 1970, Kosterlitz and Thouless L972, 1973, Kosterlitz L974). Korshunov (1985) has calculated the strong-coupling corrections following Rainer and Serene (L976) and finds that the "4" phase is favoured over the "B)' phase if the particle-hole scattering dominates over the particle-particle scattering, and vice versa. He also stresses that the symmetry group of the "4" phase is U(1)/Z2because states differing by a rotation in the plane by æ are equivalent (see the discussion on the topological classification of defects in the A phase in Section 7.4). As a consequence, the discontinuity in the superfluid density at the vortexbinding transition is four times the conventional value.
10
496
Transport Properties
As discussed by Volovik and Yakovenko (1989), two-dimensional films of superfluid 3He may exhibit novel properties that are due to the existence of a topological term ("Chern-Simons term") in the action of the corresponding quantum field theory (for a discussion of topological terms see Wilczek and Zee 1983). This term implies that in a film of 3He-4, particle-like solitons in the d-vector field will obey either Fermi-Dirac or Bose-Einstein statistics, depending on the film thickness. Furthermore, Volovik and Yakovenko (1989) predict a specific quantized Hall effect for the spin current in 3He-4. In the planar phase of the tHe film the nonsingular 4æ spin disclination is found to have a fractional fermion charge (equal to the number of particles contained in the defect) of Ln (with n an integer), while the spin disclination in the A1 phase has both fractional spín |hn and fractional fermion charge f,n. Most interestingly, these "particles" (spin disclination) in the A1 phase obey "fractional statistics" with parameter 0 : Ln, i.e. the state vector of the system changes by u factor et"' upon interchange of two "particles".
10.4.7 lon mobility The motion of ions in liquid'He is determined by their interaction with the Bogoliubov quasiparticles in the liquid. The order parameter is essentially unchanged because the ion diameter is much smaller than the coherence length. In an applied uniform electric field ^B the electron acquires an average drift velocíty u proportional to the field (for sufficiently small fields): (10.18e) ll : þønE, where ltri.,n is the ion mobility. The mobility depends strongly on the effective mass of the ion. Indeed, there are only two kinds of ion that are not firmly bound at the container walls: (i) positive helium ions, which form a sort of solid "snowball" of mass M*:30m, and (ii) those negative ions consisting of a free electron in a bubble established by the electron's zero-pojnt motion (M* -300m). In both cases the ionic diameter, being of order Im;{, is much smaller than the quasiparticle mean free path.
The mobility is determined by the force on the ion moving with velocity u due to collisions with the quasiparticles in the liquid. This may be expressed by the Golden-Rule formula (Josephson and Lekner 1969)
pãl:
:rà
h(k
-
k').uw¡¡,,fo(r - fo)s(q,
,,),
(10.1e0)
where W¡,¡,' is the scattering cross=section for quasiparticles with momentum ft to be scattered into state k' , Í* is the Fermi occupation factor and ø, is a frequency depending on the velocity u (see below). In contrast with the boundary problems discussed before, the mass of the moving object cannot
L0.4 Flow in Restricted
Geometries
497
be taken to be infinite in this case. Rather, the scattering process is inelasti,
-l I
owing to the finite recoil of the ion. This implies that tñe quasiparticle-ior interaction is retarded. The dependence of the transition piobuUitity on thr transferred energy is given by the dynamical structure facior S(q, ø) of thr ion in the system moving with velocity u, i.". q - k ft, and i* _ hú), = Ek - E*, * ku ' q. In general, many quasiparticle collisions have to be tul"n into account within the characteristic recoil time. This is the hydrodynamic regime, in which the ion performs a random diffusive motion. Tht dynamical structure factor is then given by S(q, a):Tql(ø2+y?), whert Tq: Dqz describes the diffusional broadening oì the rpeätìu function. It ir very important to realize that the diffusion constant D is connected with tht mobility Ir¡o,, by way of the Einstein relation D : (p¡o,,le)k"T.This maker (10.190) a self-consistent equation for the mobility. For the electron bubble, one may estimate To to be much less than thr thermal frequency kBTlk. This may be verified by substituting the experi. mental value for ¡r¡o', and using the fact that typical momentum transfeis hc, are of order hkr. rt implies that s(q, ø) is a sharply peaked function on the scale of the occupation numbers /¿ and may be replaced by an energy. conserving delta function, so that the scattering is essentially elastic (fetiél and Kurkijärvi L977a,b, Baym et al. L977a,b). The mobility in the normal state is then found to be temperature-independent, as observed experimentally (Ahonen et al. I976b, I977b).
For the lighter positive ion, a much higher mobility is measured experimentally (Roach et al. 1977, Ahonen et al. Lg7gb, Kokko et ø1. 1978b). In fact, the condition hyo l ktT for elastic scattering is violated
below about 100 mK. A full solution of the self-consistent equation (10.190) yields an increasing mobility at lower temperatures, in agreement with experiment (Bowley 1977, wölfle et al. 1980, wölfle 19g0; see also the diagrammatic theory of Kondo and Soda 1983, Soda 19g4). In the superfluid phases the mobility is expected to rise. Assuming the scattering to be elastic and the transition probability Woo,to be given by its value in the normal state (apart from coherence factors due to the Bogoliubov transformation), the expression (10.190) may easily be evaluated for the B phase; one finds (Bowley 1976; see also Soda Lg7s, tg77b)
pt",(T)
p["
:
llsdr*"r + 1).
(10.1e1)
The increase in p¡o,' reâr [, proportional to a, is clearly seen in the experiment, but the agreement is not quantitative (cf. the experimental references cited above). In fact, the Born approximation is not sufficient to describe the quasiparticle scattering off the ion, because of the diverging density of states N(E) n E lE. An exact calculation of the scattering amplitude for elastic scattering shows that the differential cross-section is strongly peaked in the forward direction (Baym et al. r977a,b, 1979,
498
n
Transport properties
Salomaa et al 1980a,b, Salomaa rggz). At the same time, the total cross-section remains roughly at its normal-state value, implying that the rate of momentum transfer is reduced (see also Bromtey tgsi). The interaction of ions with vortex structures is of õonsiderable interest since ions are a useful probe of both the structure of individual vortices and a vortex lattice. Calculations of the interaction potential of ions and singular vortex cores have been performed by Rainer and Vuorio (Lg77), Thuneberg et al. (1981a,b, L9B2) and Mineev and salomaa (19s4) for the o_ and v-vortices of the B phase. It appears that ions are trapped by the vortex cores as in the more familiar case of superfluid oue loonnelþ 1965). An ion-mobility experiment would therefore provide iniormation about the thermal excitations in the vortex core region and hence the local order parameter. There may even exist bound quasiparticle states, giving rise to resonant interaction processes (Thuneberg et al. 19g1b). In the A phase the mobility is a tensor quantity characterized by two eigenvalu?r, fr,, and ¡rr, for motion para[el ãnd põrpendicular to the gap axis l. Since at low temperatures thermally eicitèd quasiparticles are concentrated in regions on the Fermi surface aroúnd Ê: 1.î, ions ^the travelling along I are much more effectively scattered than those moving perpendicular to î, i.e. þt) rf,,. The ditrérence þt- rtu increases with decreasing temperature. This qualitative picture is borne out by detailed calculations (Salomaa et al. I980a,b, Salmelin and Salomaa tgSZa,U) as well as by experimental observation (Ahonen et al. lg76b, rg77b, lg7gb, Roach
al L977). Ion-mobility experiments are a powerful tool for investigating the î textures in the A phase, as the anisotropy of the mobility tensãr leads to a focusing effect similar to the focusing of light in a medium with spatially varying index of refraction. Since the mobility is lowest for ions moving along l, the path of an ion will tend to follow the "potential lines,' of / in a given texture. This effect has been exploited for the-investigations of vortex structures in rotating 3He-4. As discussed in Section 1.6, one of the candidate states, a nonsingular vortex structure, has î í" the plane .perpendicular to the axis far from the vortex core, flaring out along thô axis ãt the vortex-core centre. Ions drifting along the rotation axis are expected to be captured by the vortex cores, where they rnove more slowly-. This effect has been seen (Simola et at. 1986), and provides an important clue to the nature of vortex cores in rotating superflui¿ (see the discussion in 'He-a Section 7.6). In later experiments two types of vortices were identified (Nummila et ql. 1989). As discussed by Salmelin et al. (1989a), a moving ion will feel the orbital angular momentum of the coherently aligned cooper pairs in the A phase. This leads to a measurable "internal Magnus effectl' on the ion, maniiesting itself as an asymmetry of the associated quasiparticle scattering. Pronounced nonlinear behaiiiour of the mobility is obseried in the B phase at ion velocities above about 5-10 cm/s (Ahonen et ql. 1976b). This et
Further Reading
499
may be associated with pair-breaking processes, which set in at recoil energies (!),".: hq - , exceeding the threshold value 2A. The corresponding velocity of the ion ("Landau critical velocity") is given by ," : Af pp, as discussed in Section 7.1,L. The nonlinear dependence of þi.,n on the drift velocity has been considered by Fetter and Kurkijärvi (I977a,b).
FURTHER READING
Review articles
'
'l
Brinkman W F and Cross M C 1978 in Progress in Low Temperature Physics, Vol. VIIA, ed. D F Brewer (North-Holland, Amsterdam), p. 105 Hall H E and Hook J R 1986 in Progress in Low Temperature Physics, Vol. IX, ed. D F Brewer (North-Holland, Amsterdam), p. 143 Kurkijärvi J and Rainer D 1989 in Helium 3, ed. V/ P Halperin and L P Pitaevskii (North-Holland, Amsterdam) Leggett A J and Yip S 1989 in Helium 3, ed. W P Halperin and L P Pitaevskii (North-Holland, Amsterdam) Salomaa M M 1982 Contemp. Phys.23 L69 Serene J W and Rainer D 1983 Phys. Rep. 1:0l22l Smith H 1984 Physica l:268+C 267 (Proceedings of the lTth International Conference on Low Temperature Physics, LT-17) Smith H L987 in Progress ín Low Temperature Physics, Vol. XI, ed. D F Brewer (North-Holland, Amsterdam), p. 75 Wölfle P 1978 in Progress in Low Temperature Physics, Vol. VIIA, ed. D F Brewer (North-Holland, Amsterdam), p. 191 Wölfle P 1979 Rep. Prog. Phys. 42 269
Articles Betbeder-Matibet O and Nozières P 1969 Ann. phys. (NÐ 51-3gz Combescot R 1974 Phys. Reu. 410 1700 Hara J and Nagai K L979 J Low Temp. Phys. y 35L I-e,ggett A J and Takagi S 7977 Ann. Phys. (Ny) lM79 Nagai K 1980 I. Low Temp. Phys. 38 677 Pethick C J and Smith H 1977 Physica 908 + C, L07
11
Gollective Modes The dynamic behaviour of a many-body system on a macroscopic scare i governed by its coilective modes. These modes involve u *"ukry dampe and consequently long-lived coherent motion of a rarge fraction of particle in the system' In a n-ormal liquid, for example, there exists one well-define, collective mode: orf,inary rongitudinar souìd. By contrur,, ,up.iË"i;'T;l: can support a number of different collective modès, some type never before encountered in nature. The existence or *Å¡cn are of of these modes i intimately related to-the peculiar type of macroscopic order present in thr superfluid phases of 3He. Quite generally the collective,modes can be grouped into two categories (i) the so-called gapress modes (or "massrers .îd"s,,, as they are ca¡ed ir the general context of field theory), whose tends to zero ar the wavelength becomes infinitely íong (q : "ig"nr."quency 0; and (ii) modes wirh nonzerc eigenfrequency, which have a gup ìn- -tne 'áispersion reration for e : c ("massive modes"). According to a theorem åf Goldrton" (1961), in a system with short-range interactions a gapless collective (so-cìlt"o ..Goldstone") mode appears whenever a continuous symmetry is broken in the ordered state' This. may be easily understood if one recalls that in the case of a spontaneously broken continuous symmetry the free energy is degenerate with respect to the spontaneousry preferred varue of the associated symmetry variable (see bection 6.ri. Thus a long_waverength excitation of the symmetry variable does not iivolve the ordËr-parameter structure itself and only costs the bending energy of the orde._parameter field, which is proportionar to the square of the *ãu.nur ber q. By contrast, the massive modes do involve the internal structure of the parameter. In the case of superfluid 3He they may be envisaged asorder phase-coherent oscillations of the cooper pairs ubout the equilibrium .onfigurution. The vibration frequency may be expected to be of the order oT the binding energy of a Cooper pair, o: Alh. In the next section we consider the collective modes in the hydrodynamic regime (',<<1). Since the collision rate z-ris much less than Á except for a tiny region close to 7,, and since u,t((r, onry gapress modes are accessible in this regime. However, gaps may appear in the collective mode spectra owing to symmetry-breaking tèrms intió¿u.ed by a magnetic fierd i
11 Collectiue
502
(reducing rotational symmetry (lifting spin-orbit symmetry).
in spin space) and the dipote
Modes
interaction
In Section LL.z we present a group-theoretical classification of the order-parameter collective modes, which provides a framework for microscopic calculations of the collective-mode frequencies performed in Section 11.3. In Section ll.4 the question of observability of these modes is addressed. V/e shall present the detailed theory for the currently most powerful detection method, namely ultrasound propagation. 11.1
HYDRODYNAMIC MODES
The hydrodynamic modes are obtained as eigensolutions of the linearized hydrodynamic equations. Consequently the number of modes equals the number of hydrodynamic variables. Not all of these modes are well defined, in the sense of long-lived wavelike excitations. Some are overdamped or even purely diffusive. Let us begin with the sound modes familiar from isotropic superfluids.
11.1.1 Sound modes In an ordinary liquid the sound mode is a density wave. In a superfluid the density is not only coupled to the longitudinal component of the normal velocity, but also to the superluid velocity (which in the B phase, as well as in the A phase with uniform t texture, has only a longitudinal component, i.e. is curl-free) and to the temperature. This follows from the hydrodynamic equations for the B phase (2.2L), (9.22), (9.26) and (9.32). The two types of eigenmodes of these four equations are given bV (i) a simultaneous oscillation of normal and superfluid components (un : u"), so-called first sound, and (ii) by a counterphase oscillation with total mass current density g: pnun* p,u":0, so-called second sound (V/ölfle I973b; see also Saslow L973,1976a). The dispersion law of first sound is found as
(,)2: r?n'(,
- $r#r)
(11.la)
Here contributions due to second viscosity and the coupling to entropy fluctuations, which are both of.-higher order in TITï, have been neglected. In (11.1a) r,: (ðPlðp)t/2 is the ordinary sound velocity, which may be expressed in terms of Fermi-liqui¿ parameters as c7::¿,ù,oçL+
which
FåXl +
årì),
(11.1b)
is identical with the normal-state result (2.20). The damping
parameter 4 is equal to the shear viscosity 4 in the B phase.
1L.1 Hydrodynamic
Modes
50:
In the A
phase with uniform i texture n is given by the followinr combination of viscosity coefficients ?t, 4ll and defined in (9.82): 4ll n:rt!,cosa0 +(3rtr' - rlil.)sin20cos20 +l(3q' +TL)sinag. (11.2)
Here I is the angle between the preferred direction i and the sound propagation direction f. At low temperatures the sound attenuation q which is proportional to n, becomes very anisotropic. using the low temperature limiting behaviour of 4t,- r?-roand calculated in (tõ.t6t), ont 4[ finds a maximufr,&^u**(T"lT), for Q ll i and-a minimrr* ,*":const for 0 : cos-l 3-1t2 -55" (Combescot 1975b). From (11.1), it might appear as if first-sound attenuation measurementr would be an ideal probe for determining the shear-viscosity coefficients unfortunately, as one goes to lower temperatures, the quasiparticle collision time increases rapidly (see Section 10.2), so that the hyãrodynami< regime is limited to rather low frequencies and l,ong wavelengths. . Experimentally, the dissipative losses caused by the intJraction of the liquid with the wall dominate the first-sound attenuation even at the lowesl frequencies attainable for a given resonator geometry (Eska et al. 19g0, 1983; see also Kodama and Kojima 1981). In this case the sound dispersion for a parallel-plate resonaror of width d is given by (Nagai and wöm¿ 19g1)
(r)2: r'rn'lt where Z t(a)
-ræ],
(11.3)
is the surface impedance
discussed (10.182) et seq.). The dispersion law of second sound is obtained as
a)': cïq,(r where the velocity c, and the relaxation
in
Section 10.5 (see
i¡,oq),
,!"
ú:t!l&). PCv \p^/'
"rare
(11.4)
given by
cu r. q:#ltn * p'8,+ä/e)]
(11.s) (11.6)
Here s and p are the entropy and mass density, cy is the specific heat, and p" and pn are the densities of the superfluid and normal The bar "o-pon"nt. is used to denote the component of the respective tensor quantity in the direction of sound propagation A.For example, rt is an un"r"g" of the shear viscosity as defined by (IL.z) for the anisotropic A phase and? : QxQ is the relevant component of the thermal-conductivity tensor. In (11.6)-6, is ttre second-viscosity coefficient. Second sound is essentially a wave-like excitation of the entropy density. Since the entropy of the Fermi liquid 3He is smaller than that oÌ th" Bose liquid oHe by a factor of r lTr, tñe second-sound velocity turns out to be about three orders of magnitude smaller than the Fermi velocitv. The
504
1.1 Collectiue Modes
damping, on the other hand, is quite large. Although this makes second sound difficult to observe, it has been detected by Lu ãnd Kojima (19g5). In the limit T f. the term in t2 involving the thermal conductivity dìverges as Q.- f)^t.This divergence is removed in the dispersion taw (ir.+¡ Uy ttre factot cl, so that (11.4) goes over into the dispersion law for thermal diffusion for T >7":
a: -iDrez
(1r.7)
where Dr: xlCv. Fortunately, there is a more effective way to induce oscillating superflow. Suppose that the normal-fluid fraction is clamped by the sãattõring of quasiparticles with the surface of a very narrow channel, as for exampleln a porous medium. The superfluid fraction is barely affected by the cónfining walls, provided that the channel diameter is greater than the coherence length_ EQ), and thus may move freely. The oscillatory motion of the superfluid in such a confined geometry (superleak) is cailêd fourth sound. The dispersion- law of this collective mode is obtained from the continuity equation and the Josephson equation upon putting the normal-fluid velocity equal to zeto. Note that the momentum conservation law does not hold here since momentum is continuously transferred from the quasiparticle system to the walls. one finds the linear dispersion law
(Ð': clq'(l
-
i¡a'ro),
(11.8a)
where the velocity of fourth sound is given by
c7: c?þ,1p. (11.8b) Thus the ratio of the velocities of fourth and first sound is given by the
square root of the superfluid density fraction, which vanishes urlr _ r ir)r,t for Z-+ 7i and tends to unity for T-+0. Here B" is the diagonaicomponent of the superfluid density tensor in the direction of sound piopagati A. The damping of fourth sound is determined by the reláxatìoñ time ro. In the B phase and in the uniform A phase ra is given by
";
tq:
PEzlc?,
(11.e)
where (3 is the coefficient of second viscosity appearing in the Josephson equations (9.22) and (9.84). Fourth sound has been observed "rp"ri*"ntally in a number of experiments (yanof and Reppy 1974, Kojima et al. 1974, 1975, Chainer et al. 1980). Measurements of ttre fourth-ronnd velocity provide a convenient technique for determining the superfluid density component P". In principler'. it should be possible to determine the second-viscosity coefficient frlom the damping oÌ fourth sound. In practice, however, this intrinsic contribrution to the damping is often obscureã by the so-called "normal-fluid slip", i.e. the effect of a small residual motion of the normal-fluid component. This contribution may be related to the effective viscosity describing
L1.1 Hydrodynamic Modes
505
Poiseuille flow through narrow channels discussed in Section 10.4 (see (10.178) et seq.). For a parallel-plate geometry the dispersion relation of fourth sound including viscous losses at the channel walls for arbitrary ratio of channel width d to mean free path is found as (Højgaard Jensen et al. 1e83) (,)2
: rin'(, - iøta-
å
ä
#l
(11.10)
Here ô.n: (Zrt"ulp,a)t" ir the effective viscous penetration depth and 4.u has been discussed in Section L0.3, (10.180). The additional damping contribution deriving from the last term in (11.L0) comes close to, but is still somewhat less than, what is observed experimentally (Yanof and Reppy 1974, Chainer et al. 1980). Another possibility of observing a kind of second sound is as entropy waves in the At phase. Since superfluid mass current and spin current are identical in 3He-41, â supetfluid density wave associated with second sound is equivalent to a spin wave. It is therefore stabilized by the spin stiffness, which is much stronger than the entropy stiffness. This mode will be discussed at the end of Section ll.l.z.
11.1.2 Spin-wave modes superfluid 3He the Cooper pairs carry spin leads to well-defined spin waves in both the A and B phases. Crudely speaking, the spin waves owe their existence to the stiffness of the order parameter against local rotations in spin space. The spin waves are the Goldstone modes associated with the spontaneously broken rotational symmetry in spin space. While this is strictly true for the A phase, where one has a preferred direction d in spin space, in the B phase only the relative spin-orbit symmetry breaking is relevant for the dynamics (see Chapters 6 and 9). The spectrum of spin \ryaves has been discussed by Brinkman and Smith (1974), Combescot (I974a,b), Maki (L973, 1975b, I976b), Maki and Ebisawa (L974a,b,c), Maki and Tsuneto (L974b,1975), Graham and Pleiner (L975), Pleiner and Graham (1976) and Pleiner (L977a,b). The equations of motion governing the dynamics of spin waves are the spin conservation law and the Josephson equations for the spin phase variables 0'and 0'" respectively. Using (8.8), (9.29), (9.110) and (1,0.60), we may write the equation of motion for the spin-density components Su as
The fact that
in
*-
y(s x Ho)r
*? oo;.Tr"p,',* V . X3 :
Ro,*,
(1 1. 11)
where p'p is the superfluid spin-density tensor, Rp is the dipole torque and Xf is the dissipative part of the spin current, for example the spin diffusion current. The equation of motion for the spin superfluid velocity u"o," takes
11 Collectiue
506
Modes
the general form
*r"0,"
. *v(ri,+
of;¡
: s,
(r1.12)
where Y Qo, is the dissipative force. Substituting (11.12) into (11.11) and neglecting dissipative effects, one flnds the spin-wave equation of motion
# - t!vp"i"vs, : * ^,,-.
(11.13)
The A phase
In the A phase the spin superfluid density tensor is given (see (L0.60)) bV
p;1": (ô*" where âo it the equilibrium and perpendicular to I are ^SP Ptt,t
- âo*âou)p"o,
á vector and the components of
1-Yrr,, _t+år'î - ¡=Fi 11 tp"ry,,_uo, "
(11.14) p"P
parallel
(11.1s)
is the saturation spin polrrizawhere Yr,r:3
+: where do
r2itôs
- 4(os . âo)1,
(11.16)
is the equilibrium A vector (usually oriented along Ð
ôS: S-y-'xH.
and
On Fourier-transforming (11.13), one obtains the dispersion law for spin waves of longitudinal and transverse polarization relative to the magnetic field:
a?t: c¿L+ c?p(A)q',
a1:
oft +
atft.
(tt.L7a) (11.17b)
where the spin-wave velocity c,o is given by
,,o(4) =
(Hor"ol)'
(11.18)
The velocity of spin waves depends on the orientation of i with respect to the direction of propagation. Nonlinear spin waves in the A phase have been discussed by Poluéktov (1983).
1L.1 Hydrodynamic
Modes
s01
The 4., and A2 phases
As already mentioned, the hydrodynamics of the A1 phase has a peculia structure because it is a superfluid magnetic liquid. Here one encounters ¡ broken relative gauge and spin rotational symmetry (the simultaneoush broken rotational symmetry in orbital space (see Chapter 6) is ,rol important for this discussion). Therefore the two associated Goldston< modes, second sound and longitudinal spin waves, are expected to appear ar a single mode, which combines properties of both of the abovementionec
modes (Maki I975a, Liu 1979a,b, L980, Gongadze et al. !978a,b, Gu rgenishvili and Kharadze 1980, Takagi 1981a,b). At the transition to the A. phase this mode has to split again into the two ordinary Goldstone modes. We now wish to calculate the eigenmodes of the system. To do this, wr combine the hydrodynamic equations (9.110), (9.108), (9.104) and (9.106) Neglecting the weak coupling between the mass-density oscillations and the entropy and spin modes, the mass current density may be put equal to zero, This condition allows the normal-fluid velocity to be expressed in terms ol the superfluid velocities u. and %o by t(prr" + pruurp). ur,: (11.19)
-pr,
Thus one finds the equation of motion for the spin component along the preferred direction .,¡f : a)
#t¡ dt-
's)
+
þt4,î' ^,
h ," : *{(Vp-o)V(.f
. a") + p[V(prp"')]Vô,r¿]. (11.20)
Here we have introduced Þr
:
P.p
*
PruP,r
tPtl,
(tL.21)
which may be interpreted as an effective superfluid spin density. The angular frequency ar" is defined in (9.11). The first term on the right-hand side of (1I.20) describes something like the "stiffness" of the spins. In the absence of ò¡r, the spin-wave frequency is obtained as
^ a!o:
hy2
ffit|t"rg
*
d2'o.
(1r.22)
However, the pure spin-wave frequency is not an eigenfrequency of (11.20) (although it is almost so). Rather, the spin density couples to the temperature field by virtue of the last term on the right-hand side of (LL.20) via
ô,rr: -ps ôf,
(rr.23)
where s is the entropy density. An equation of motion for the temperature fluctuation ôT may be obtained with the aid of the entropy conservation law
11. Collectiue Modes
508
(9.27) and by substitutitrB u,, using (LL.2I); this yields 9"
(#tt)
- 'v . [p"'(p"q * pru"o)] :0.
(rr.24)
Now the Josephson equations (9.103) and (9.L04) are used to replace the superfluid velocities % anq u"o in (LI.24) in terms of gradients of the chemical potentials ö¡r and.f . t". Finally, by combining (L1.23) and (11.24) and after performing a Fourier transformation, one is led to the dispersion law
(r' - o?)(o' - ú)Z)z':
Y *@p 2m7¡ lðT '' ðs
ap^'q)'.
(tr.2s)
Here the usual second-sound frequency has been introduced:
,s2,
o¡3':
p ð,larq(P'P"
t)q'
(r1.26)
Far away from the A, phase transition, one has p ru (( p" and the right-hand side of (II.25) is small. Hence spin waves and second sound decouple into separate modes with frequencies crr"o and @2 respectively. On the other hand, in the A1 phase one has pN: -p"p:-p" and fJa:0, whence the right-hand side of (11,.25) becomes @[email protected] the relevant eigenfrequency in the A1 phase is
(r)': ø'rr+ ol.
(rr.2t)
The second solution, a)':0, carries zero weight and may be discarded. The first mode is a combined temperature-spin wave with linear dispersion and velocity approximately equal to the spin-waue velocity, because @?o>> @7. On moving into the A2 phase, the second eigenfrequency begins to grow until it reaches the value a2 not far from the transition. This mode, which \ilas predicted by Liu (L979a), has been observed experimentally by Corruccini and Osheroff (1980), using an oscillating superheat transducer. The efficiency of such transducers in superfluid 3He has been investigated theoretically by Liu and Stern (1982) and Johnson (L982). A different method for detecting spin waves in 3He-4, using the scattering of light from the associated magnetization fluctuations, has been proposed by Golo and Kats (1984). As pointed out by Stern and Liu (L982), the damping of spin waves may be dominated by spin-relaxation processes at the walls of the container rather than by spin diffusion (Nagai 1981b).
The B phase
As in the A phase, we are again interested in determining the eigenmodes of the equation of motion of the spin density, in this case given by (11.11). Here we neglect the dissipation, i.e. the term Y . Xo* in (11.11). In the B
\
11.1 Hydrodynamic
Mod.es
phase the situation iS somewhat more complicated owing to the coupling of spin and orbital spaces. This is, for example, expressed in the specific structure of the superfluid spin density as given by (10.60). However, in the representation where the spin coordinate system is rotated relative to the coordinate system in position space by R¡,¡ (the rotation matrix characterizing the B phase), the superfluid spin density tensor has the simpler form (p"o)'l,o
:
P
tô
¿¡ô pp
i
p2õ
¿çõ,0
+
(11.28)
p3õ ¿*ô¡u,
where ¡t and V label the Cartesian components in the rotated spin space. In the absence of Fermi-liquid corrections, the coefficients k in (11.28) are obtained from (10.60) as
p?: -4p3: -4p3:
åSo(1
-
yo),
(rr.2e)
where % is the Yosida function defined in (3.88). The respective renormalizations of the p¡ ãrc then readily found from the general expression (10.60) for p'P:
, n,4ar* a2 Pt: lPál--7---î al- at t
(11.30)
4a2
na1* -> -> t Pz: Páalat
(11.31)
(rt.32)
Pz: PZ al+ a2' where
1'
at:l*!-^(' .t\t
a,:
ffi"
^o\
*x*Ut¡,
lFiffioï
Substituting (1,1,.2L) and the expression for the dipole torque, d .*^" : -fuñ(fi
.s),
(11.33)
into the spin-wave equation (11.13), one is led to
afs
-
iøy(S
xIf') -
arqzs
- azg(e.S):
A2Bñ@.
S),
(11.34)
where the stiffness parameters ø1 and u2 are given by
&t::-
hy'
¿m)( hy2
(11.3s)
Pt,
uz: ? (pr+ ¿mx
pt).
(11.36)
The eigenfrequencies of (11.34) are conveniently calculated in a coordinate
11
510
Collectiue Modes
system (i, i, â) with É along 2 and i both perpendicular to wave-propagation direction f (Brinkman and Smith tg74). In the limit of zero magnetic field the mode spectrum is found
at?:
stez, *
@7,s: +îa""
I
2a1q2
r @L - 4u2d22"q?,)r,rl, J
fi
and the
as
(1r.37)
where Õ'u:A'"+azez (note that u2<0). Here ø, is the frequency of a spin wave that is linearly polarized in the direction orthogonal to ñ and Q, whereas the remaining two modes are elliptically polarized in the plane spanned by É and f. For large q, (11.34) reduce to the linear dispersion laws
@¡:cìPQ,
j:r,2,3,
(11.38)
with the spin-wave velocities
clo: ql'', cìPs:
(nrtlnrl)''t
(11.3e) (11.40)
A
static magnetic freld 116 introduces further gaps into the excitation spectra. The result for small q (such that c'pq 11o"), but arbitrary frequency ú0L compared with 9s, is @?'
: L{a?, 4õL - 4urqz(a,,q' * orq?)|,,,},
o?: a'"+
urez
* arq'",
(11.41) (1.1.42)
where õ¡?: @?+2arqt + uzetr. Spin waves are difficult to detect experimentally because there is no analogue to a sound transducer or absorber used in sound-wave measurements. Furthermore, an NMR pick-up coil, which might appear to be an ideal detector, only allows one to measure the integrated magnetízation in the interior of the coil, which averages to zeîo in the usual limit of wavelengths small compared with the size of the coil. None the less, in the case of inhomogeneous textures and fields, standing spin waves do lead to detectable NMR resonances provided that the wave train is sufficiently modulated in amplitude that it has a net magnetization. As discussed in Section 8.4, spin waves in a nonuniform fi texture have been investigated by Smith et al. (L977). The spectrum of standing spin waves in a stack of parallel plates has been observed experimentally (Osheroff et ø1. 1977). The results are in close agreement with theory. Spin waves in the collisionless regime are discussed at the end of Section 10.3. The collective modes of the B phase in a strong magnetic field (i.e. such that the distortion of the gap is not negligible) have been discussed by Pleiner and F.,¡pnd (1983).
1.L.2 Order-Parameter Modes in the Collisionless Regime
511
11.1.3 Other hydrodynamic modes The preceeding discussion did not include all hydrodynamic modes. In fact. according to the general classification of hydrodynamic modes pr"r"ni"J earlier in this chapter, there must be several other such modes. In the linear hydrodynamic regime (i.e. where the hydrodynamic variables enter only linearly, as is the case for weak excitations¡ there is a rigorous correspondence between the number of hydrodyn"-i. variables and the number of possible modes. For example, in chapter 9 we found that the hydrodynamic properties of the B phase are described by the following set of 12 hydrodynamic variables: {p, T,,f,, u,,, u,, u",l; ¡t: L,2,3h here the four superfluid velocities u. and ule hàve each been counted just once because only their longitudinal component enters. As discussed above, the corresponding modes are (i) two first-sound modes, at -- Lcvq and two second-sound modes úo : Xcze associated with the four variables p, T, u" and the longitudinal part of u,,, and (ii) six spin_orbit wave modes ot: rc,q connected with the six variables .f, and ule. The two remaining variables are the transverse components of u", which iive rise to two diffusive shear modes with dispersion a : iDqz An analogous classification of modes holds for the A phase. There the " /-vector field provides two more hydrodynamic variableJ. The associated "orbital-wave" modes attracted a great deal of interest in the early stages of research on superfluid 3He (de Gennes L973, Anderson Lg73, wáme1wq, volovik L975a, saslow L976, combescot and combescot L976a, cross 7977a; see also Brinkman and cross rg7g, Leggett and Takagi lgTs). However, as shown in Section 1,0.3, the motion of iat finite tempeìatures is severely inhibited by the strong coupling of Î to the quasiparticle sysrem, giving rise to the so-called "orbital viscosity" (Cross and Anderson Lg75). Thus orbital waves are overdamped except in the limit z+0, where they merge with the so-called "normal-flapping" mode that is discussed in the next section.
11.2 SYMMETRY CLASSIFICATION OF ORDER-PARAMETER MODES IN THE COLLISIONLESS REGIME
In contrast with a BCS superconductor, the Cooper pairs in the superfluid phases of 3He have an internal structure. This in turn allows collective modes of the order parameter. The nature of these modes is determined by the equilibrium order-parameter structure itself. In the A phase, for example, the unit vectors fi and û characterizing the orbital part of the
order parameter may oscillate about their equilibrium configuruiion ñ I ñ by vibrating within the plane spanned by these two unit vectors. This is the so-called "clapping mode", the origin of whose name is self-evident (see Fig. LI-Z). The frequencies of such order-parameter modes are of the order
512
11
Collectiue Modes
of the gap frequency (only the "normal flapping mode" is an exception-see below)- In addition to these modes, there -are Goldstone modes with frequency tending to zero in the limit as the wave vector q+0. Depending on the symmetry of the equilibrium order parameter, the-eigenfrequenciei of different Goldstone modes may or may not be equal. Accoidingty, tne ra components of the order parameter, which are the dynamical variables of
this problem, may be grouped into multiplets. This fact is familiar from atomic and solid-state physics. As in atomic physics, the multiplet structure can essentially be derived on the basis of symmetry considérations. The classification is then used to perform the actual micioscopic calculation of the eigenfrequencies in an economic way. The multiplet structure of the collective-mode spectrum is completely determined by the symmetry group H of the order parameter, i.e. the set of symmetry operations, both continuous and discrete, under which the order parameter is invariant (Volovik and Khazan 19S3). The group .FI is itself a subgroup of the overall symmetry group G : so(3; i so(:¡ x u(1) describing the free-energy functional of p-wave superfluidity (see Chapi"r O¡. The degeneracies of the eigenfrequencies (if any) aie' given by the -gro.rp dimensions of the different representations of the ,{. The iepresentations are labelled by the eigenvalues of the Casimii operators of the group ÉL Although in the cases of interest there exist infinitely many representations, the problem of selecting a particular set of representations is uniquely determined by the dimension of the space of states of dynamical variables. Hence in the case where only p-wãve order parameters are
considered this space is l8-dimensional.
11.2.1 The B phase discussed in section 6.3, the symmetry group H of the B phase is SO(3)¿+s, i.e. the order parameter is invariant under simultaneous rotations in spin space and orbital space. In addition, the order-parameter components are real, apart from an overall phase factor. This implies that d, d'it' the real and imaginary parts of the change in the ordå parameter, ?n9 õd*¡, ate_not coupled. Here we have introducedã shorthand noìation, i.e. dþ=|e [ôdrr(r, t)1, d'i¡=Im [ôdrr(a r)]. The complete group ,F/ is rhen given by
As
Hs:
SO(3)¿*s.
(1r.43)
We are now interested in the Casimir operator of the group ,FI because it commutes with all elements of lhis symmetry group urr¿ because its eigenvalues label the different irredù.*oþle representationi. The casimir operaror of so(3С#i* is j2: (L + s)2, with eigenvarues J(r+t), where J:0, 1,2,... is the total angular-momentum quantum number of the p-wave Cooper pair. The representations labelled by .t are
11.2
Order-Parameter Modes in the Collßionless Regime
513
(U + L)-dimensional. The 18 order-parameter variables may be grouped into 9 real and 9 imaginary components, which fall into a singlet (i:gj, a
triplet (/:1) and a quintet (J:2) (Maki 1974). The classification scheme is shown in Table 11.1. The different modes listed in column 1 are labelled bv three quantum numbers (columns 2 and 3): the angular momentum | ; quantity K taking the values (+) and (-) for the real and imaginary parts respectively, and the magnetic quantum number m¡:0, tl, . . . , tJ. The transformation properties associated with these quantum numbers lead to the linear combinations of order parameter components shown in column 4. The square of the eigenfrequency in the limit q-+0, as obtained from weak-coupling theory (see below), is shown in column 5, and the coefficient
of the (rrq)'contribution is shown in column 6. The Fermi-liquid correction factor given in column 7 renormalizes the bare frequency ,r(ù.The order to which the order parameter couples to spin or density fluctuaìions is given in column 8 (up to order qz). Finally, in column 9 the magnitude of the magnetic-field splitting is shown, where ø¡ is the Larmor frequency and the g factor is discussed below. From the table, we can see that there are four Goldstone (i.e. gapless) modes: the imaginary r: 0 mode and the three real J: 1 modes (ieìativé spin-orbit rotations), which manifest themselves as second or fourth sound and spin-orbit waves respectively. For the purpose of discussion, we have neglected the dipole interaction. The frequencies of the remaining modes are of order AQ) ("pair vibration modes"). In the presence of a magnetic field the overall symmetry of the free-energy functional of the system is reduced to the group G : u(1)", x so(3)r_x u(l)o (see chapter 6), and the subgroup ^É1" is õorrespondingly reduced to
Hs: U(1)r,*",.
(tr.44)
The latter group has only one-dimensional representations, so that the degeneracies of the m¡ sublevels are completely tifted. For small magnetic fields, when the distortion of the gap itself may be neglected, the splitting is symmetric and linear in the field. Similarly, in a spatially inhomogeneous situation characterized by a finite q vector, the symmetry group of the system is reduced to G: so(3)s x * U(1)0, where U¿ is the group of rotations about the direction of q 9r plus the reflection on the plane perpendicular to q. Since the rotations do not commute with the reflection operation, the group U¿ is non-Abelian. The subgroup Ë1is then reduced to
Hn:
fJt,s,
(11.4s)
where !z,s is isomorphic to {J¿, but contains relative spin-orbit rotations about the axes q and Rg, respectively, plus the reflection operation. The group has one- and two-dimensional representations, such that the eigenvalues of I,: tm states are still degenerate. The situation is similar to the
(¡r
5
Table
11.1 Collective
Mode
J,K
mJ
modes of the B phase. Variable
(r)'
Dispersion
az(q)
-
az(o)
Fermi-liquid correction factor
Coupling to
Magnetic-field splitting
observables
(rrq)' 0* Spin waves
L*
Real squashing mode
2-
Sound
01-
Squashing mode
2-
0
dl+
0
dlz- dLt dâr- dltri(diz- dL)
r1
diz+
dL3
0
d'tt+ di2-2d5,
r1
dh+ dhxi(diz+
T2
dir-
0
d'ir+ dir+ di,
0
d'ír-
t1
d\r-
dLr)
dLzLi(d,rz+ dL)
d5,
d'irxi(dä2- dir)
0.24 -10.3
0
z
.l
d'ir+
r1
d\l+ d'hri@äz+ dir)
dh- d\rr.i(d'iz+ d,l)
5
Density, O(o¡lEe)
(1+3FÐU+o(Fî)l
Spin density, O(qo)
ayftt¡g¡
t + $tr1
Spin density, O(q'); density, O(q2alEr)
0ytTl¡92+
(1+ FåXl+ åFì)
Density, O(qo)
5
Et'
-0.44
tlt
å(r
+ ft3ttr¡
-
å€r,)
0
1
4A2
0.11-i0.17 0.35-i0.33
dir-2di,
0
i.2
4A2
Et'
J
å(3
-
Spin density , O(uolEr)
å8,r,)
-0.42 !1r + Jfr,¡
t + ft1,n",
Density, O(q'); spin density , O(qza
lEr)
@umt8t-
aumtSz-
ñ R
È
I
L1.2
Order-Parameter Modes in the Collßionless Regime
515
coupling of an electric field to an atom in atomic physics, leading to doubly degenerate Stark levels.
11.2.2 The A phase The order parameter of the A phase, (3.68), is given by dp¡: Asï*(î¡ ,yr) if we take the preferred directions in orbital and spin spu.ä, frr, ñ and tâ, io lie along the r, y and z axes respectively. The symmetry group of the order parameter in this case is (Volovik and Khazan 1983)
He:
uSlJo
xuf),ø.
group USiJo comprises (i) the group of rotations l!e. p1, U(1)+, and (ii)
(11.46)
about the z axis, the combined symmetry operation which consists of a rotation by ru about an axis perpendicular to z carrying á into -â and a gauge transformation by ¡, which compensates for the minus sign. The effect of the gauge transformation is seen to be equivalent to a space inversion. (Note that for a discussion of the bulk free energy of a homogeneous system as in Chapter 6 the symmetry transformation & is irrelevant, so that, instead of ul]]a in (11.a6), one only needs u(l)r..¡ itre operator Pt:exp(irus")P", with P, the inversion operator in spin-space, does not commute with the operators exp (ig].s,) of u(1), (the cómponents of the spin operator, ^S' are invariant under space inveision). Hence the group g(t) must have multidimensional repiesentations. The Casimir operator of U(1) is 57. For a triplet ordér parameter, the possible eigenvalues of 52, are 0 and 1. The representation labelled bt s, :0 describes the transformation of the p : z component of the order parameter and is therefore one-dimensional. The nontrivial representation, tãUeneA Uy lS,l : 1, is two-dimensional. It mixes the p: x, I components of dr, undir the transformations of the group ,F1o. The group Ut),o of the orbital and gauge degrees of freedom contains (i) the group of gauge-orbit rotations U(l)r,*r, and (ii) the discrete symmetry operation P2: exp (inL-) K consisting of a rotation in orbital space about the x axis by n, carrying .f + ij into i - ig, and the operation óf complex conjugation K restoring the original order parameter. As in the case of the group USllø, the additional discrete symmetry transformation P2 is irrelevant for the consideration of homogeneous bulk properties, so that ug),oU(I)r,-o in Section 6.2. Since the operator p2 does not commute with the generator of U(1)¿" +t, i.ê. Pr(L, - I)Pz: -(L, - D, the group U(2) is again non-Abelian and must have multidimensional representations. It follows from the simple transformation property of L, - 1 under p2 stated above that the Casimir operator of U(2) is given by (L, - Dt. since the possible eigenvalues of L, for a p-wave order parameter are given by 0, tl, and the relevant eigenvalue of the generator of gauge transformations I : -iAlAQ is
516
1.1 Collectiue Modes
*1,
there appear three representations labelled by lL, - Il:0,I,2 in this case. The representations have dimension one for lL" 1l :0 and two for lL, - Il: I,2. The two-dimensional representations are a consequence of the operator P2 mixing real and imaginary parts of a given linear combination of order-parameter variables drr. The representations_of the grg.up H are obtained as direct products of the representations of u(1) and u(2), and may be labelled by ilr, - 4, ls,l). Fxcept for the singlet representation with lL. - /l :0, i.e. (0,0) un¿ (0, l)-which necessarily appears twice, both for the real and imaginary parts of the order parameter-the representation for lL"-4+0 õ."rrrc only once. Accordingly, there are four doublets, labelled (1,0), (2,0) and (0, 1j, and two quartets, (1, 1) and (2,1). The multiplet spectrum of collective modes is shown in Table L1,.2. As in Table L1.L for the B phase, the respective modes, their quantization numbers and the associaied orderparameter variables are shown in the first four columns. The next two columns contain the weak-coupling results for the eigenfrequency in the temperature limits T+T" and T+0. Next, the coupling to order qt is indicated. The set of eigenvectors in column 4 is of couise not uniqìely determined, since the eigenvectors within each multiplet transform among each other under the group .F/. In principle, the eigenvectors may bé determined by diagonalizing the Casimir operators, but in the present case they may simply be found by inspection. As is evident from Table LI.z, some of the weak-coupling collective frequencies are equal, although they belong to different multiplets. This implies an additional symmetry of the order parameter in thii case. To identify the additional symmetry transformations, we recall that in weakcoupling theory the two Cooper-pair systems of up- and down-spin pairs are independent. Hence the orbital structures of the two types of pairs may be oriented arbitrarily with respect to each other. This is expressed by the more general order-parameter structure
dr,¡: *AoÍ(â* + iêt)Qñri + inl) + (â,
-
iêr)Qhu +
iñr)|,
(LL.47)
d, ê, fi¡,t and..fir,l ?re unit vectors and ã. ê:0, ût¡. ñ¡:0, ñl ût:0. strong-coupling corrections in the free energy, however weak, where
will tend to align the orbital structures of spin-up
ano þin-down cooper pairs, such that ñ1:ñt:ût, ñI-fi!:ñ. In this case the vector ê drops out of (LI.4). Nevertheless, the collective-mode spectrum will exhibit tñe additional degeneracies associated with the large¡ symmetry group of the order-parameter structure (rL.47). (These degenera"i", ar", of-course, only approximate because of the aforementioned strong-coupling corrections.) Let us now find out what the additional symmetries correspond to mathematically. Using (LI.47), it is clear that any rotation about the axis d x ê in spin space by an angle u may be compensated by a corresponding rotation of fi¡ and ñ, about the axis firxñ, by an angle -u and, a
Table
11.2 Collective
modes of the
lL"-4
Mode
ls,l
A
phase.
Variable
F Weak-coupling frequency ø
TnT. Sound
0
Spin waves
1
di,
Pl
T--+0
d'i, d',,
t\)
Coupling
urQ
3-'''rrq
I
Spin density, O(qo)
0 0
d'r,
ffi
d'|., 1
Orbital waves
d'j, d'.o
0
Flapping mode
aorønir/2q2
oiq3
I
d'-o
@nn=
dL,
d!'t
-1
di,,
Density, O(qza/Er)
dL,_,
d'io
d"r,_,
d't,-t
2
dl,_,
1
1.22Ao(T)
Spin density, O(qza
I Er)
-1
d!,o
¿
Þ
Spin density, O(qr) Spin density , O(q2alEr)
d'ro
dio
G
dL,-, I
d'),_,
1
d'r,-t di,_,
1
d':, -l d';,
*
oõ
Density, O(qr)
1,.23Ao(T)
s(\
F
Density, O(qza/EF)
0
s
dLo 1
d']o
t.SOaolO¡
(\
R
1
Spin density, O(qr)
a"o:2A¡(T)
s s G
è;
1
dlo djo
Clapping modes
1
(ê)1/2/'oe) æl
d'ro
Flapping modes
d,,, d'r,
-Density, O(ø2)
d'lo
*,
s-
d'i,
-1
1.
Density, O(rt¡/Ee)
None
o *
(field along y) Density, O(qo)
æ
Variables in a magnetic field
-1
d'r,-, dj,_,
(/¡
11
518
Collectiue Modes
ñtx fir by an angle *ø. Now if (frr¡, triads ñ¡, ût, x frr) and (frrv ñt, the relative orientation of the ñrx nr) is irrelevant on account of the independence of spin-up and simultaneous rotation of
ñ¡
and fi¡ about
spin-down Cooper pairs then the order parameter (LI.a\ is invariant under rotations in spin space about any axis perpendicular to d, in addition to rotations about the axis d considered in the strong-coupling case. The symmetry group of the weak-coupling ABM order parameter'is then given by
II*¡": SO(3)s xut),o
(11.48)
(for a somewhat different discussion, see Volovik and Khazan L983). Since the Cooper-pair spin is ,S: 1, the relevant representation of SO(3) is three-dimensional. Including the orbital part, which has either onedimensional (lL, - /l :0) or two-dimensional (lL" - 1l : 1, 2) representations, one finds the following degeneracies of the collective-mode spectrum (see Table LL.Z): there are two threefold- and two sixfolddegenerate states. The four additional Goldstone modes, which may be interpreted as spin-orbit waves, result from the degeneracy of the S" :0 and lS,l : L modes in the lL" - 1l : 1 multiplet. These frndings are in accordance with the results of the explicit calculations presented in the next section.
In Table Ll.2 three different frequencies have been entered for the case lL, - /l : L. As will be shown later, the transcendental equations determining the eigenfrequencies are nonlinear in o¡2 in this case, and therefore allow more than one solution. Finally, we want to discuss the influence of a magnetic field on the symmetry. In the presence of a magnetic field .Ef along j the rotational symmetry in spin space is reduced to U(L) so that G: U(l)s, x SO(3)¿ x U(1)o (we retain the direction of d along the z axis; consequently d ' H :0 in equilibrium). The subgroup 1{ is then reduced to H:Zzxu?),o, where the group Z2 consists of the two elements P1 and L. The modes are then only labelled by the quantum numbers lL" - 1l :0, I, 2 and Pt: LL. Since the group Z2has only a one-dimensional representation, the degeneracies in the spectrum are lifted to the extent that only singlets and doublets remain. This is shown in detail in Table I1.2. We do not consider the small effect of residual interactions like the dipole interaction and particle-hole asymmetry here, but they may be readily included in the scheme (Volovik and Khazan L983). Having listed the various possibilities for the existence of collective modes allowed by symmetry, we now wánt to calculate the mode frequencies and the order-parameter response functions explicitly. This can only be done within a model theory. Under the given circumstances, where [ << Zp, the obvious model is weak-coupling theory.
11.3 Time-Dependent Mean-Field Theory in the collisionless
Regime
519
11.3 TIME.DEPENDENT MEAN.FIELD THEORY IN THE COLLISIONLESS REGIME
In weak-coupling theory, i.e. to leading order in the small parameter TLIT,., the dynamics of a pair-correlated Fermi system in the r"gi*" of external frequencies and wave vectors ha 11Er, lql( k., is compleiely described by the matrix kinetic equation (10.6) if supplemented by the definitions of the change in the diagonal energy, ôEo, and of the order parameter, ôao, defined in (10.8) and (10.9). The mean fields ôÇ¿ and ôÂo are functions of the position on the Fermi surface, Îr, and of spin, but they are independent of the magnitude of the momentum lfrl : frr. up to order f"l1'. The l/rl dependence may therefore be neglected. Nevertheless, in order to keep thl notation simple, we adhere to the notation ô^É, ôÇ¿ (rather than Â(É) etc.), where it is implied that lfrl :ftr. The ^e,neglect oi tt" ¡t¡ dependence allows one to cast the matric kinetic equation into a set of coupled integral equations in É for the mean fields. In order to achieve this goal we define a four-component vector distribution function bv
* òn-o ôno - ô.-o ôn¿
ôNe
:
1
(ôno
¡a*¡ 1
la*¡
^å (òro
+
ôF;:
(11.4e)
^fr
Li -
^"
òF;
where each element represents a 2x2 spin matrix. A vector of the quasiparticle energy change, ô€¿, rrâ! be defined in complete analogy with (1,L.49), i.e. with ôno and ôFo in (11.49) replaced by ôEo uni ôÂn respectively. In the nonmagnetic case, when ôno and öfo are proportional to the unit matrix in spin space, the matrix kinetic equation can then be written as a system of four inhomogeneous linear equations for the components of ôN¿:
t ,
ll¿ôN¿:
M¿ôe¿.
(11.50a)
The matrix of coefficients flo is given by
Ì
7?
hot
Þ- >+ hø
0
0
0
hot
2A
0
f-
5+
0
-(6*
+ 6-)
-rr':r:;
)
,
(11.50b)
1L
520
Collectiue Modes
and similarly
lo
rr I E*r*- E-orvr¿:l o
I
\-¿(e* + 0_)
E*0*- E-e-
0
0
A(o* - o_)
A(0* - e_)
0
-a(0** 0-\ E*o*l u-r- I
ol
E*0* + E-e-
0
(11.50c)
Here we have used the abbreviations
l&ol, I E+: Eo*oo, 0*:9ono,r,l tr:+t^h+k.. ) A:
(11.s1)
Also, we have restricted ourselves to the collisionless regime, i.e. we have dropped the collision integral in (10.6). An extension of the theory including collisions has been given by Wölfle (1978a,b). The system of equations (11.50a) may be readily solved for ôN¿:
ôNo: (Slo)-lMo ôeo,
(11.s2)
and, consequently, also for ôn¿ and ôF¡. Employing the definition of òÇ¿, (10.8), and the decomposition (11.59), the spin-symmetric component òÇ¿ is found after some algebraic manipulations and expansions in q as õEp
-ô5f'
:l f",rrt"{ãhT - t,o)òEo - L},o@Eo+ ôE-o) + +k(@ + tl),[&(ô^ o Li +
!¡uei+ E*(a+
- ae ô^f )
ø)¿l(ôao
Âf *
a¿
ôl;)Ì
(11.53a)
The spin-antisymmetric components õEo,u are given by an analogous equation, where /)* is replaced by f"oo and tr, {. . .} is replaced by tr, {or(. . .)}. Furthermore, using the definition of ôFo, (10.30b), one obtains
ôA, +
4, *, oôÂo
:
4
r*{"-lor,+
+,[n[41f,
ø¡ ( r
*&^)
r ^
ôao
*, :e;]
oe-
* Â* ôa; ^e
- *k'(,' -
ô(t.'#)u"l
* o*oiô^ÉÌ.
(11.s3b)
L1.3 Time-Dependent Mean-Field Theory in the Collßionless Here we have defined fi
n:___ k m* 4o*
Regime
. q,
(11.5aa)
oL:;; , ëE*
(11.s4b)
h*: -2H[(kut)2(0*+ 0-) + (g* -
:
52t
0-)(E*
- E)]
lAol' îo,
(11.55a)
(11.ssb)
where
D*:l@t)' -
(å*
- E)'llØ@)'-
(å* +
E)'l- 4(ho)2laol'. (11.55c)
The quantity Lk is a kind of Cooper-pair susceptibility (Tsuneto 1960). Its poles at ko¡ : l2Uk + O(q) describe the excitation of pairs into two Bogoliubov quasiparticles of momenta k and -k. In (11.53a,b) we have retained terms asymmetric with respect to an interchange of particles and holes (i.e. terms odd in 6r). In spite of their numerical smallness, they are nevertheless quite important in some instances, as will be seen later.
The time-dependent mean-field equations (11.53a,b) still contain the
complete dynamics of the system in the collisionless, linear-response regime, but they are much easier to solve than the original matrix kinetic equation. Equivalent equations may be derived by Green's-function methods (Serene 1974, Maki 1974, Ebisawa and Maki 1974, Nagai 1975, Tewordt L974a,b, Serene and Rainer 1983) and by functional-integral methods (Alonso and Popov 1977, Kleinert 1978, Brusov and Popov 1980a,b,c, 1984a,b).
Let us first consider the collective modes of the order parameter, neglecting Fermi-liquid corrections. They are given by the eigensolutions of (11.53b) at fixed diagonal mean field ôf¿. In the case of a pure spin-triplet order parameter the equation of motion for ôA¿ may be rewritten in terms of the vector order-parameter fluctuation õdu(Ê;e, @), defined as usual by
õA*,,p(g, in the
ot):) \a*(c; q, @) (o*io2)nB,
form / (
r;¡É¡^(
{,[ ç¡¡qnø¡'z
- u'l õdp(k)
- LIG)Z ud:"(ùf +
(11.s6)
t''
õdp(c)
zdop(ùdî,(¡e) o¿l(¡e)l
-
Íd.'"(k)l' aa;g¡
+ 4l(Ðoa"tÉl]),
: z(vi^1*¡-t qîc¡lqnat+ øl(r + z^^(Ò xla".ç*¡ô6o
-
i)
e
*"ñX¿l
#) . ze ^<*>"fif
ogr,^])r.
Gt.s7)
11 Collectiue
522
Modes
Here do*(k) is the equilibrium order parameter, and we have allowed for pairing in an arbitrary odd-/ state. The pair-interaction constant has been expressed in terms of the gap parameter by means of the equilibrium gap equations, leading to the term
E(i,):lorrler-+ffi1
(11.s8)
Furthermore, the diagonal energy has been expanded in spin matrices
as
3
õËo,,p: ôår
ônþ
* )
ôåo,,,( op)op.
(11.se)
lL:l
The angular brackets denote the average over the Fermi surface. The quantity- dî€r) is the complex conjugate of d*(î<; -(1, -ar). We have kept the particle-hole-asymmetric terms in ôS¿, because they can lead to a small but important coupling of ôÁ¿ to ôÇ¿, but have neglected corresponding terms in òÁ¿, since they cause only a negligible frequency shift. The quantities Â(É) and L1Îc¡ are defined by
^f h(k\: I dEo)'o J
= la*l' L(ò,
(11.60)
and the particle-hole-asymmetry functions h,.(Ð and ga(É) are given by
In the limit
g+
(i) For T +
],^(k):hlor,&^r,
(11.61a)
o,,ô):#lotrtort|f-,.
(11.61b)
0 the function Â(i0) has the following limiting values. T", the main contribution is given by
Ak tanh (ko l4kBT) rr L\K):nkr@, nz
indicating that
)r()
whether o¡)2lÁ¿l to zero as
takes on imaginary or real values according to or vice vers¿. For fixed a, and A*+0, h(k) tends
),()-Oo# (assuming ha
(1,1,.62a)
<<4k¡T).
i
as
r+r"
(Lr.62b)
L1.3 Time-Dependent Mean-Field Theory in the Collßionless Regime
(ii) At T :0
one frnds
^.i, : h(k) where y
523
:
(kct
L
L
.2y-1-zÍy(y-L)l''' , _ñr"ffi ily|
l2Aù'. It fotlows that at a :0
one has
(It.62c)
L(ò : I.
11.3.1 Gollective modes in the B phase To calculate the collective modes in the B phase, we begin by substituting the corresponding equilibrium Eap parameter do*(î): AÊ* into (11.57) (remember that spin space is rotated relative to orbital space by the action of the dipole force; this is, however, irrelevant for the present calculations as long as we neglect the tiny effect of the dipole interaction on the mode frequencies). Defining the real part and the imaginary part (up to a factor i) of the fluctuation variables by
l[õd*(k; q, @) + ôd](ie ; -8,
-ø)l :
[vo'u' 14o''u'
for (+), (11.63)
for
(-),
we are then led to the set of equations
(hro)zd*-
f-l
tt'lô*t4
o'""+ dh+ d;.1
: z¡kaaZ [(kr)' -
4azld'i¡ +
e
td'la*,4 or" t
:
¡"¡u(68*,"> k
d'i¡
+ 4tu^a'õ*¡(õEù a, Q'r.6aa)
* di.f
2ha õpjt(õEk) ¿+ 4ilu6&z\ e,"i"(õEo,)
a-
(11.64b)
For simplicity, we have assumed g :0. The function ¿ is isotropic for q- 0 and has dropped out except for the particle-hole-asymmetric terms, where it appears in the parameter Âa defined in (11.61a). Note that, owing to the isotropy of the B phase, Âa does not depend on ft. Similarly, the quantity E has disappeared because of the isotropy of the quasiparticle energy. Since (Ll.64a,b) are linear in a2, there exists exactly one eigenmode for each variable. The set of equations (11.64a,b) is diagonalized by constructing the linear combinations of order-parameter components listed in Table LL.1'. This immediately yietds the 18 eigenfrequencies also given in Table LL.1 (see Maki 1974, I976b, Serene L974, Nagai 1975, Tewordt et al. L975b, Tewordt and Einzel1976, Jacak 1977, Chechetkin l976b,c,d, Flasegawa and Namaizaw a 1982).
LL
524
Collectiue Modes
(i) Goldstone modes. There are altogether four Goldstone modes with ,à'q. One of them (with quantum number (/, K):0-) is associated with
Eud;Ju, the variable corresponding to the total phase of the order pìrameter. This is the well-known Anderson-Bogoliubov mode of isotropic neutral superfluids (Anderson 1958b, Bogoliubov et al. 1959). It may be excited via its coupling to density fluctuations. The other three gãpless modes (with J : I, K: *L, i.e. L* in the classification of Table 11.1) are associated with the variables L*,¡ e "*td', (v : I,2,3). They correspond to relative rotations of spin and orbital variables and may be identified as spin-orbit waves that couple to spin-density fluctuations.
fofr-uibration modes. The "I:2 modes are found to have eigenfreqoàícies hto2*:(g)t,,A (real order parametet, i.". K: *1) sftd har-: K- -1) in the limit q-0.Each (+)t,rl -these(imaginary order parameter, i.e. modes is fivefold degenerate, in compliance with the symmetry of classification. The "I :2 modes of the B phase were first calculated by Vdovin (1963), i.e. long before the experimental discovery of the superfluidity oi'H", but the results later fell into obliviorl. Of these ten modes, the 2: modes couple to inhomogeneous (i.e. finite-q) density fluctuations, such as sound *uu"r, while the 2+ modes couple to spin fluctuations with finite q. lnfact, the latter also couple to sound waves through particle-holeasymmetric terms, but the coupling is much smaller than that of the 2*od"r (Koch and wölfle 1981). The 2- modes and 2* modes have been dubbed the "(imaginary) squashing modes" and "real squashing modes" respectively, because they describe an oscillatory, quadrupolar mode in which the spherical gap parameter is squashed in its imaginary and real
þ¡
parts.
It is interesting to note that the sum of the squares of the frequencies of
the 2+ modes is èqual to the square of the pair-breaking frequency 2A, i.e(hrr-), (har*)z (2A)2.This is a consequence of a supersymmetry of the èffectíve Hamiltonian (Nambu 1985), which relates the energy gap of the basic fermionic constituents and the frequencies of the bosonic collective modes for each angular-momentum channel. The eigenfrequencies of the four remaining modes 0* and L- are exactly at the pair-bréaking edge, i.e. a --2A. They couple to homogeneous density ãr spin-density variations through particle-hole-asymmetric terms
t
-
only. Finite wave vector, lql + 0
The generalization of (11.64).,1p nonzero wave vectors is straightforward (Wölfle L977a, Brusov and Pb.bov 1980a,b, Combescot 1982b). The variàUt"r decouple in the (J,J,) representation, as noted before on the basis of symmetry cònsiderations. At targe values of q such that hupq >> A, all the
11.i
Time-Dependent Mean-Field Theory in the Collisionless Regime (¡)
525
breoking
2L
squoshing
(El',','.L
\ l-/
(8)"',A
)
Zeemon sptitting
reoI squoshing zero sound
112
collisionless spin waves
l/E
q
Figure 1L.1. Collective modes in the B phase (schematic). The transition from the hydrodynamic to the collisionless regime is indicated by a broadening of the excitation branch. The dashed lines indicate the splitting in a magnetic field. The collective-mode frequencies at q:0 indicated in the figure are the weak-coupling results.
modes possess linear dispersion. The teading qt correction for small q is listed in Table Ll.L. The numerical coeffrcient in some cases involves the quantity Â1, which follows from expanding )'(r) to first order í, q', and is defined by
)'r[r(q:
0)]
4az 1481,,_(kr),^***ll ::l!^,or,^[._ oço14EodE) lo:o' Luh rnry J
(11.6s)
@æL4ET- ØæooThe frequency in the argument of )', is the rnode frequency at q :0. A schematic plot of the collective mode frequencies as functions of q is shown
in Fig. LI.L. Effect of a magnetic field
An external magnetic field may be incorporated into the description by allowing for the Zeeman splitting in the normal quasiparticle energies: E*": Ç¡,Xs¡tsH,
s: tl,
where f¿o is the nuclear magnetic moment. The
LL Collectiue Modes equations then become quite complicated, so w9 do not reproduce them hére (see Tewordr and Sóhopohl 1979b, Schopohl and Tewordt 1981). The magnètic fietd induces a spin-singlet component of the order-parameter fluctuations, which is responsible for a linear rather than quadratic splitting of the mode. Such a splitting has already been calculated for the spin-wave spectrum, i.e. the 1* mode, in (11.37). -" :2 collective-mode For small magnetic frelds the linear splitting of the J frequencies is given bY (11'66) @*ù(T, H): o2*(T) + mrgz*(T)av(T, H),
where ffit:Q, t1, t2. The g factors may be expressed in terms of the yosida function Ys defined in (3.101c) and the function  as
8,,(r):ílr*#], and
at(T, H):
yH
r+Fi(3+å%)
(t1..67a)
(11.67b)
real is the renormalized Fermi-liquid Larmor frequency. The splitting of the aI. squashing mode (I :2+) hai been observed experimentally (Avenel et rôso) urrã ir in full agreement with the theoretical prediction by Tewordt and Schopohl (1971Ui. T.ne splitting of the squashing mode is harder to observe b""u.rr" of the stronger coupling of the sound to the collective mode (Meisel et al. 1983c). It has nevertheless been detected in measurements of the group velocity (Movshovích et aI. 1938) by employing the theoretical anaiysis of this experimental situation developed by varoquaux et al. (19g6). Ri trigtrer fields the splitting deviates substantially from linear behaviour (Shivaram et aI. 1,983, 1986) because the equilibrium gap iò itself distorted by the magnetic field (Schopohl et al' 1983' parameter 'Brurou and Popov 1984a,b,c, Fishman and Sauls L986, 1988b)' The possibility of inducing transitions between the sublevels by an RF field has L""n disóussed by Dombre and Combescot (1982b). The collective-mode similar to freq,uencies are also affected by superflow. The effect is somewhat of a magnetic field (Kleinert 1980a, Sauls and Serene 1984, Brusov
thai
1985a,b).
11.3.2 Gollective modes in the A phase In the axial state the equilibrium gap parameter is given by t dou(it) :,fr\n)r' A o âo*Yr rçt¡ .
(11.68)
Y11, We shall take âo*to lie along the z axis. Since do*çft¡ is proportional to
is convenient to
expand the order-parameter fluctuations
it
in terms of
1-1.3 Time-Dependent Mean-Field rheory in the collisionless Regime
52i
spherical harmonics Y1*:
õdp(c):
_ì*,
õdr"* Yr^(Ê).
(11.6e)
Furthermore, we introduce real and imaginary parts_of õd¡, , d,*. and d.,i,^, in analogy with (11.63). Subsrituting (11.69) inro (11.57), weïnd in îti limit q+ 0 the following set of equations:
l(X)' ^- - z^^ + +a^fa; + 2ô^,^*(o;^ - zâ".4 â"¿'"^) ^ ' \-\ r*^â?(ôåo,^)¿, :-Ziha¡ ô^rtrà (11.70a)
f
l/hat\'. L U" / ^^
.-l-..
2h^ + +a^]a';^
/
- 26*t)"^(o';^ - ,¿",4 â"¿i^)
:zry ô^t)Ltâot"(o6*)¿.
(11.70b)
ao
Here we have defined
L^:
I oo, lYr^(k)l' t (k), tr^: oo, lv,^(Ê)l'ffi I E*: oor lYr^G)lzE@).
^rÌ
(11.70c)
I
In
deriving we have exploited ^(11.70a,b), functions )'(Ê) and E(k) abour î for q->9.
the axial symmetry of
In contrast with the B phase, different angular
the
averages of t () are involved in (11.70a,b,c), so that Âo does not simply cancel as in (11.64a,b). A further difficulty arises from the frequency dependence of )r(c), as defined by (11.55). The eigenvalue equations, obtained by putting the right-hand sides of (11.70a,b) equal to zero, are now nonlinear in ar2 and may have more than one solution. As usual, a nontrivial solution of a system of linear homogeneous equations exists if the determinant of coefficients vanishes. A numerical analysis of the eigenvalue problem for (11.70) yields well-defined zeros of the real part of the determinant at real frequencies (WöIfl e I976c, !977a, Tewordt and schopohl r979a). However, the imaginary part of the determinant does not vanish at these frequencies (in contrasì with what happens in the B phase). This is a consequence of pair-breaking processes, which may occur at any frequency owing to the two nodes in the energy gap in the axial state. Consequently, one might expect a zero of the determinant to exist somewhere in the complex frequency plane. But this is not the case,
LL
Collectiue Modes
as will be discussed below. In fact, it may be shown that-strictly speaking-the determinant does not have any zeros in the complex frequency plane, except for ø :0 (Dombre and Combescot 1981). Hence in the A phase there are no well-defined modes at all-there are only "resonances". Nevertheless. we shall call these resonances "collective modes".
The limit løl
:0
Following Table 1I.2, we group the collective modes according to the label ffi:0, t1 of the order-parameter fluctuations di- and d'i"^ (at least in weak-coupling theory).
(i) m : L. For m: tl,
the angular average of .E'(É) defined in (11.70c) vanishes, i.e. ,E*t :0. It turns out that in this case there is at most one collective mode per variable. The order-parameter variables d',r, d'r, and d')r, are associated with rotations in spin space about the ¡ and y axes and the gauge-orbit symmetry respectively. Their eigenfrequencies are zero (they are Goldstone modes) and the corresponding dispersion laws are linear in q, as will be discussed below. The remaining variables d',r, d'], and dlr do not even show any resonance because the real part of the eigenvalue equation
(T)'^,-
4)",:s
(Lt.7L)
does not have a solution for real a (at least in the absence of Fermi-liquid corrections). Note the simplicity of (11.71), which is a consequence of
E^:+l:0'
(ii) m--1. The six variables dL,_r, d'i,_, (p:*,y,2) are found to oscillate at a frequency
. hrr"t:
12 Re Â,1r/2
\ ú'/
Ao:
L.23A0(T).
(1r.72)
Hence they correspond to pair vibration modes. Since the motion of the order-parameter structure, i.e. the motion of the unit vectors râ and á, resembles the clapping of hands, as explained in the beginning of Section r1.2, this mode has been called the "clapping mode" (see Fig. ll.2).lt is remarkable that ø.¡ scales with the energy gap A¡(T) at all temperatures within one per cent accuracy (see Fig. 11.3) in spite of the presence of the  factors in (I1.72). Among the six modes, only d'i,-, co,uples to sound waves (neglecting the coupling due to particle-hole-asymmetric terms), whereas d',,_¡ and d'r,_, couple to spin waves.
11.3 Time-Dependent Mean-Field Theory in the Collisionless
I
Figure
Regime
529
qx¡ol stote
11.2 A pictorial representation of the order-parameter modes in the A phase.
The case m:0 plays a special role because here there exist three resonances for each of the variables d'*,s and d'i",o. This is due to the nonzero value of go, given by
(iä)
m:0.
Eo
: -)
[ ao, vß(îù ^(fr)f I "r;#
(r1.73)
(*)'^-2)'o*
(1t.74)
(here a partial integration with respect to cos 0 has been performed to bring Eo into the form (11.73)). A small admixture to the gap parameter proportional to Yro(k), as given by these variables, causes the preferred directions ñ and ñ to oscillate out of plane. This motion may be visualized as a flapping of wings (see Fig. lL.Z)-hence the name "flapping mode" (Wölfle I975b). The m:0 eigenvalue equation is given by
4Eo:
s
More-or-less stable oscillations are found to occur at two different nonzero eigenfrequencies. In addition, there is a solution at zero frequency corresponding to a diffusive mode (see below). Let us discuss the solution of (1I.74) in nlore detail. There is one solution with a strongly temperature-dependent eigenfrequency: ha¡nn
_T
(tt.7s)
Ao r"' More precisely, in the limits hann ao
: (t)'''
T-T.
and
i.l, - #
hr,¡no: (!)t/znk, r(rn
f
T+0
Ee)
*
the analytic results are
#l
"on,
,)-'''
", r
-+ r.,
as T-+
0.
(IL.76a) (11.76b)
L1 Collectiue
530
Modes
rlrc 064 084
*r99P
normol flopping
0.8 06 0.4 0.2 lttl?vz (a)
Figure
0
1
0.8 06 0.¿ il-T/Tc)
0.2
1t2
(b)
11.3 Collective-mode frequencies of the A phase versus temperature in
units (a) of the temperature-dependent gap An(T) and (b) of the gap at T :0, ^n(0).
The complete numerical result is shown in Fig. 11.3. The physical origin of this mode may easily be visualized. It arises from the fact that the flapping of the vectors ñ and ñ is equivalent to an oscillation of the vector i about its equilibrium direction (remembe^r that î: rh x ñ). Since i specifies the axis of the energy gap, a motion of J will change the quasiparticle spectrum and hence the equilibrium quasiparticle distribution fi(Ð. The quasiparticles then have to adjust to the new equilibrium state by means of collision processes. Hence the relaxation to equilibrium will take a characteristic time r¡, which is of the order of the quasiparticle collision time z. As one may expect, r¿ is closely rela.ted to the orbital-viscosity coefficient Âo calculated in 1fô.fZO¡. Now, if the oseillation of i is too rapid for the quasiparticles to follow, the quasiparticle dïstribution will not be changed much. In this case there is a wêll-defined direction Î* given by the time averaged axis of the energy gap, about which the superfluid component of the system oscillates. The restoring force that maintains this oscillation is provided by the potential well formed by the energy of the quasiparticle system. Therefore
11.3 Time-Dependent Mean-Field rheory in the collisionless
Regime
531
the oscillation frequency will decrease with temperature and will eventuallr tend to zero as l--+0 in the same way as the density of quasiparticles goes to zero in this limit. Because of the important role played by the normal-fluid component in the mechanism of this mode, it has been callec the "normal-flapping mode". It is interesting to note that this mode ir exceptional since it owes its existence to the thermal excitations in the superfluid and hence is not a property of the superfluid condensate itself. In contrast with the "normal-flapping mode", the frequency of the other flapping mode scales approximately with the gap, similarly to the clapping mode. It may therefore be viewed as a ground-state property of the superfluid as in the case of the squashing modes in the B phase. Hence we shall refer to this mode as the "super-flapping" mode. The ratio of its eigenfrequency to as(T), i.e. ha"olao(T), is found to decrease from z at T, to approximately I.56 at T:0 (see Fig. 11.3). The leading correction for T - T. is given by ht lru
ao(T)
:rl'-:#ee)#l ", r-r"
(rr.77)
Pair-breaking effects The anisotropy of the energy gap in the A phase, l\&)l: Aosin g, implies that the minimum energy required to break a Cooper pair into two quasiparticles varies from zero to ZAs, depending on the position on the Fermi surface. Hence pair breaking via absorption of collective-mode quanta may occur at arbitrarily low frequencies. However, at low frequencies (Ð this effect is reduced because of the limited phase space available. Mathematically, the pair-breaking process is described by the vanishing of the energy denominator in the expression for ),e given by (11.55), which gives rise to an imaginary part of ).(*), as evident from (LL.62a,c).
To determine the frequency eigenvalues, we have so far omitted the imaginary part, because it is relatively small as long as ha 1ZA. and has
little influence on the real part of the eigenfrequency. However, one has to keep in mind that the collective modes discussed above are damped by pair-creation processes. Clearly, the higher the frequency, the more damping will occur. We defer a discussion of the damping until later, because in cases of practical interest (e.g. coupling of collective modes to sound waves) the broadening of the absorption peaks caused by the collective modes is also influenced by the imaginary part of the coupling function. Therefore it is more sensible to discuss directly the physically observable quantity (".g. sound absorption) rather than collective-mode frequencies themselves.
There is yet another problem associated with pair breaking, which has already been briefly mentioned. Even though the small imaginary part in
11
532
Collectiue Modes
(1I.72) and (11.74) induced by pair-breaking processes may not be of great quantitative importance, it is conceptually relevant. Namely, it implies that, strictly speaking, there are no collective poles in the complex frequency plane corresponding to the clapping and flapping modes. These modes are only peculiar resonances on the real frequency axis, and are not related to a complex pole. The argument runs as follows: In the absence of collision effeóts, the left-hand sides of (11.71) and (11.74\ are functions of ar2, expressing the fact that time-reversal invariance is strictly valid in this limit. Then, for any given zero of, sâY, (11.71), at frequency @+:@ct -if (assuming f >0), there is a corresponding zero at o-- -@ct +ill which lies in the upper half-plane. However, any collective pole in the upper plane violates causality and causes an instability of the system. Hence there can be no complex zeros of (11.71) and (11.74) other than a¡ :0. An investigation of the functions on the left-hand sides of (11.71) and (1L.74) (Dombre and Combescot 1981) has shown that this is indeed true. Finite wave vector, lql + 0 Even at lql + 0 the different spin components of the order parameter decouple. This is a consequence of the fact that the spin and orbital degrees of freedom separate. On the other hand, the respective orbital components become mixed. The eigenvalue problem thus reduces to a set of six 3 x 3 matrix equations. In thã following we only consider the special case q ll î for simplicity, where the 3 x 3 matrices assume diagonal form. The dispersion of the Goldstone (i.". gapless) modes with m:l associated with the variables d'!, and d',t, d'rt is given by the single expression
a),: ("rq)'lri\
((k . Ð'^(¡el )¡ ( r(re )
(11.78)
)¿
In the limit T--+0 this reduces to
ú):
(rt.7e)
*.3-t/zurQ.
gapless mode, with ffi:O, is obtained for the "flapping variables" d'u,uandd'i,.o as a solution of (11.74) in the limit a¡, q--0. The  functions in (11.74) are highly singular in this limit and have to be analysed with great care. One finds (11.80) o¡ : a(urq)' as T ---> 7.,
:,
A further
where
a-h(r.-rl##-u*]
,
while in the low-temperatiire :iimit
a:-.i.j-(*r)'"(#) (,.q)' as r-+0.
(11.81)
11.i
rime-Dependent Mean-Field rheory in the Collisionless Regime
533
This mode is the so-called "orbital wave", i.e. the Goldstone mode associated with the breakdown of rotational symmetry in orbital space. It can be seen that this mode is overdamped at all temperatures because of its strong coupling to the quasipar(icle system (see the discussion on orbital relaxation in Section 10.3). The dispersion of the normal-flapping mode follows from (11.74) as
oÅ"(q): a?n(q:0) + u?@ .ï'.
(11.82)
where (11.s2) now holds for arbitrary directions of f and i. nt T:0, where a^n (g - 0) : 0, the dispersion of the normal-flapping mode comes out to be linear. This behaviour is changed into a quadratic behaviour owing to particle-hole-asymmetric terms in the very-low-frequency regime ho¡ << k"T"(T.l Tr) (Combescot and Combescot 1976a, Cross t977a). The collective modes discussed above are all listed in Table 11.2. The effects of the Fermi-liquid interaction, i.e. Fermi-liquid corrections, have been calculated (Ashida and Nagai 1984), but are not included in the table for lack of space. The effects of the dipole interaction on the collective modes have been explored by Tewordt et al. (1977\. Effect of a magnetic field
In contrast with the rather complicated situation in the B phase, the effect of a magnetic field on collective modes in the A phase may easily be obtained. This is true at least in weak-coupling theory, where the two spin species are uncoupled, and follows from the fact that spin and orbital structures are separated in the A phase. Therefore the preceding results remain valid for the individual spin-up and spin-down systems. One simply has to take the expressions obtained so far and insert the appropriate gap functions A¡Q) or A¡(T). As a result, one finds a mode splitting (e.g. for the clapping mode) proportional to Á, - aL. There is only one mode of each type in the A1 phase. A second mode appears below the transition to the A2 phase (Koch 1980, Schopohl et al. 1985). Another interesting effect involves the motion of the vector i i.e. the so-called "flapping modes". If strong-coupling are taken into ^effects^ account, the relative orientation of the I vectors J, and l, of the two spin subsystems is no longer undetermined, but acquires a fixed value. This gives rise to a further collective mode consisting of relative oscillations of Î, and i, (Wartak 1978, 1980a,b).
11.3.3 Effect of residual interactions Corrections to the collective-mode spectra of both the B and A phases are not only due to Fermi-liquid effects but are also induced by higher-/
11
534
components
Collectiue Modes
in the pair interaction, i.e. Vo-o,:E,(21+L)VrPr((.ft') for
(see (3.a6a)). In the B phase a:finite value of the /:3 and /:5 pair-interaction parameters V3 and % leads to corrections to the mode frequencies of the order of a few per cent. They are comparable to those induced by the Landau parameters F)'^ (Sauls and Serene 1981c, Fishman and Sauls 1986). In the case that Vz and V5 are sufficiently close to V, there
lfnl, lEo,l<e"
will even appear a new mode in the gap (i.e. for ho <2A) with total angular momentum J :4. The effect of a small f-wave interaction component V3 on the A-phase collective modes has also been considered (Wojtanowski and Wölfle 1986a,b, L987, Wojtanowski 1986, Sauls 1986, Hirashima 1987). The effect of an attractive s-wave pairing constant on a p-wave state has been discussed by Chervonko (1976). Another type of residual interaction effect is due to collisions among the quasiparticles. These tend to destroy the coherence of the collective motion. This is particularly important in the B phase, where the squashing modes are not damped by pair-breaking processes. As a rule of thumb, the imaginary part of the mode frequency generated in this way is of the order of the quasiparticle relaxation rate z- I multiplied by a factor proportional to the density of excitations:
øårr:
# orrr.*
Here yo: (E)t'' o. (8)t'2 and the relaxation rate
(11.83a) z"ol,
is found as (V/ölfle
L975a, I976b)
*:!,>en)Aå;#
(11.83b)
In the A phase the collision-induced broadening adds to the pair-breakinginduced broadening (rWölfle and Koch 1978). A strong-coupling theory of the collective modes has not yet been worked out, although it is in principle known how to calculate the leading correction in T.lf, (Serene and Rainer L978, 1983); see, however, Wartak (1978, 1980a,b, 1981, 1982). Judging from the smallness of the strong-coupling corrections left in the thermodynamic properties once the strong-coupling gap parameter is inserted in the weak-coupling theory, one may anticipate that genuine strong-coupling corrections will be small.
11.3.4 Observability of collective modes Among the set of collective modes found in each of the phases, only the Goldstone mode associated with broken gauge symmetry (which is essentially a sound wave) may be excited and detected directly by experiment (Wheatley 1978). Besides that, it has also been possible to observe standing
11.4 Collisionless Sound
s35
spin waves using a gradient magnetic field (Masuhara et al. 1934). These Goldstone modes are in turn found to couple to some of the orderparameter pair-vibration modes. This allows for an indirect detection of the latter modes via absorption and dispersion effects on the Goldstone modes. In particular, sound waves are very well suited for this purpose since the sound velocity is high compared with the Fermi velocity, implying that the sound-excitation branch will intersect the pair-vibration branch (see Fig. I 1.1).
This is not necessarily the case with the spin-wave branch. So far,
collective modes have only been seen in sound-propagation experiments. In fact, sound waves may be used as a spectroscopic probe to investigate the pair-vibration modes in superfluid 3He, much as electromagnetic waves are employed to study the electronic-excitation spectra of atoms and molecules. In the next section we shall therefore give a relatively detailed account of the theory of sound propagation in the collisionless regime. A general discussion of the symmetry aspects of the coupling of collective modes to observable quantities has been given by Serene (1983).
11.4
COLLTSTONLESS SOUND
The detailed discussion of order-parameter modes in the preceding section has equipped us to calculate the physically observable effeôts of p-wave pair correlations on the high-frequency dynamics of 3He. Our starting point will be the matrix kinetic equation (10.6) and the conservation laws for particle number and momentum (2.2I) and (2.22), together with the definirion of the momentum current density Ir,, given in (10.17b). Combining (2.2r) and 2.22), we find at once the Fourier-transformed equation of motion for the mass-density fluctuations õp(q, ,) in terms of the momentum current density: @' õp
*
iøqlI.q
:
(11.84)
Q.
The second term in (11.84) may be rewritten by substituting the expression for II, (10.17b); this yields
qfiq
çr2
:
*4
@ . q)'(õn*
+
eLô60).
( 1 1.8s)
Using (r. Ð':?P2((. Ð + {, where P2 is the l:2 Legendre polynomial, and employing the integral property (10.lSb) of QL, this may be cast into the form
qnq::#
@,ù'Le+
Fi,) ôp + zm(t +
*Få)> P,(Ê-4> a"rf. (n.86) k
It follows from (11.84) that the dispersion law for sound waves is (,)2:c1q'[r+22Ç(a)1.
(r 1.87)
11
536
Collectiue Modes
where
.E )e
P,(k- Q) õro
E@¡): 2
Z_
Eônn k
(11.88)
'
2t + +F) 51+Fi)
(11.89a)
ln fact, if small terms of order (urlr,)a are neglected, the coefficient Z
is
given by the relative difference between the velocities of zero sound c,, and first sound c, in the normal state: ct
7:co-
(11.8eb)
C1
The response function f(ø) may be expressed in terms of ôf¿ and ôÁ¿ by taking the l:2 component of the diagonal mean-field equation (11.53a):
4
,,
x,(r1*.ø{hl - Â(¿)l ô6o - Â(ie) ô6e + ih.a.l,(kxô^o Ai +
^^(k)î.(k)(õak
- Ak õAi)
ai * a*uo;lÌ),
(11.e0)
We now take advantage of the fact that the Fermi velocity is much smaller , than the sound velocity. This allows us to calculate ôf;¿ and ôÁ¿ in an expansion in powers of ulo¡:(r. Q)urlrr, which immediately yields an expansion in powers of. uef cr-
The sound attenuation u(a) and the sound velocity c(a) may be expressed in terms of the imaginary and real parts of Ç(ar) respectively as
a(a): -qZ Im f(ar), I c(a):
c,[1
* Z Re å(r)].
(11.e1)
J
11.4.1 Phenomenological model Before we embark on the microscopic calculation of the response function Ek¡) (see Section 11 .4.2),let us have a look at a simple phenomenological model (Wölfle I978c, L979). While simple, it nevertheless contains the basic physics and is able to reproduce the main features of the results obtained from microscopic theory. Furthermore, it not only applies to the collisionless regime but even covers the hydrodynamic regime. It will enable us to express the momentum current density II in terms of the density p so that (11.84) becomes a self-contained equation of motion for the density.
11.4 Collisionless Sound
s37
The model is based on the assumption that the longitudinal momentum current density qtlq may be decomposed into (i) a local-equilibrium term given by the pressure P, (ii) a viscoelastic term If', and (iii) a contribution If from the internal vibrations of the order parameter:
QIIQ:P+I'"+ff.
(1r.e2)
While at low frequencies.the term 11' accounts for normal-fluid friction, at high frequencies (ot > r- r¡ it describes the elastic behaviour of the medium due to the Fermi-liquid interaction (see the discussion of collective modes in the normal state in Section 2.3). At low frequencies the normal-fluid friction term If is simply given by the hydrodynamic expression
ilhva-
-i
_ _i
iir.r Tf,, u,,
(11.e3)
where in the second equality the continuity equation (2.2L) has been used. At higher frequencies, ¿o Þ r-', the collisions are no longer effective enough to establish complete local equilibrium. Rather, the normal component II. relaxes towards the local-equilibrium value at a rate given by the viscousrelaxation time zn:
toffn:-lf¡I'-fTnru). Tq
(11.e4)
We omit the usual divergence of a current term in (1L.94) because it gives rise to correction terms of order uçlct only. At still higher frequencies, ka 7 A, the excitation of pair vibration modes takes place, which in turn generate a contribution to the longitudinal stress proportional to the time derivative of the collective variables ô^i"", where / labels the different collective modes:
rr:
+4 A,G)ô^;",,
(11.es)
Here ô^;'" represents any one of the order-parameter components d'u,, d'i,, or combinations thereof that we have associated with a finite frequency at 8:0. The A¡(q) in (11.95) are dimensionless anisotropic coupling parameters, which depend on internally preferred directions (vector i in the A phase) or an external field (magnetic ñeld in the B phase). We have already discussed the equations of motion for the collective variables ôÁ;"tt in the absence of collisions. In the case of the B phase these were found to be given by harmonic-oscillator equations. If collisions are included in a phenomenological way, the equations may be expected to
11
538
Collectiue Modes
change into ones that describe damped harmonic oscillators
(r' * ,O,- ri) oa¡'t : iAíaflro . r
(11.e6)
Here r¡ is a relaxation time, which, by comparison with (11.83), may be identified with 22"o,,, and @¡ ãre the collective-mode frequencies, which in principle contain an imaginary part originating from pair-breaking processes. The force term on the right-hand side of (11.96) is proportional to the divergence of the mass current, i.e.q.g. It must be linear in A, since 6¿corr is expected to scale approximately with the equilibrium gap. For dimensional reasons, a further factor with the dimension of a frequency must appear on the right-hand side of (11.96). Since this cannot be the external frequency itself, the gap frequency or the relaxation rate, it must be given by the characteristic frequency of the normal state, Erlh. The dimensionless coupling constants Ai and,zI¡ will be of order unity except close to I or if the coupling is due to particle-hole-asymmetric terms only. Combining (11.84) and (1I.92)-(11.96), one is led to the following general structure of the sound dispersion relation: (,)'
:,?r'1, -
:#,å,,,,- : ef
4
n,nl
(Ð'
t.s.) o¡2+io¡lr,-a7 ] ,t
(cf. (2.76) for the normal state). This expression contains the main features of sound propagation in superfluid tHe. The first correction term, involving the shear-viscosity coefficient 4, describes the transition from ordinary hydrodynamic first sound to collisionless zero sound at onr:1. Substituting the result for 4 as obtained from simple kinetic theory, q:!p^u'24, one finds that in the limit Z--+0 the zero-sound velocity co approaches the first-sound velocity roughly as co- ctq pn, with pn the normal-fluid density.
The last term accounts for the excitation of collective modes. which are damped both by collision processes and by pair breaking (contained in the imaginary part of ø¿). Incoherent pair-breaking processes give rise to an imaginary part of the coupling functions ,21,. [t is clear that ,ut, vanishes proportionally to A as T - T.. A similar model description may be employed to analyse the acoustic impedance (Ketterson et al. 1983a).
11.4.2 Sound propagation in the B phase Longitudinal sound is characterized 'by,u single vector, i.e. the propagation direction ti. The relevant fluctuations of the order parameter in this case are described by two components dq, and'd,, defined by õdu¡(q,
@)!õdi,Gq,
-o):{"ï"::i1i,i:,å7 _?:l ill
[1ì
(11 es)
11.4 Collisionless Sound
s39
For f along 2, the components of the order parameter appearing in (11.98) are given by dí;" :4Tr d;,;" and d'i" :|(Zd:;', - dli, - O;;,,). Remember that the real parts couple to density waves only via particlehole-asymmetric terms. In the limit q--+0 the solution of (11.64) yields the order-parameter
fl
uctuation
dk -L - 'uA
d'i¡:
ga2
(h')'-
(ô6o ) U",Ì
4a2
(1l
4A
ee)
^(ôEo)6ri,
where ÂA << 1 is the particle-hole-asymmetry parameter. Substituting the g:0 solution into the q'terms on the left-hand side of (11.57), one obtains, to first order in q',
321
di
15 NF
A2
+ (Âo + go)
d'í:
l^"(o#+(T|,,
(hr)' - ll'
rr] u,r,
#
(1
l.100)
t haA tN, (haf - nr 7,õn' 16
where ôø is the particle number-density fluctuation caused by the sound wave. In deriving (11.100) we have used (10.10) and the continuity equarion to express ôf¿ as ô6r
: 1l-
N. LFi,*
Fì -:@^ I i:' A
¡}1
(11.lol)
la".
Also, we have replaced (rrq)'by ,'lllfi,(l + åFì)], using the zerorh-order sound dispersion relation. The result (11.100) may now be substituted into (11.90), using the relation
j tr, [ôÂn
^;
+ aÉ ò^;]
:
':l!
|lo,,i,l for :" l[di',+ Pr(k. q)d'íl
ax{lo,::
Í*ì' (-),
(r1.r02)
which finally yields
Ek¡): t -
(hr)'^[; (hr)' -
4A2 E
¡'
t6 12
- 7/lt
a2
" (hr)' - 2¡,)'
(1 1.103)
The coupling constant to the real squashing mode is given by
lro:
rFi()""+0ì-2A"
3
Note that in the limit lto
1++Fì <<
Á one finds
),a:"[(+)']
E: 7 -
(1 1. 104)
Â, which tends to zero
as
11
T+0. In
Collectiue Modes
other words, the sound mode approaches the Anderson-
Bogoliubov mode in this limit. The frequency variable ø in (11.103) contains a small imaginary part in order to guarantee the correct causal behaviour of the response function $. A more general theory based on the matrix kinetic equation including the collision integral (Wölfle 1978a,b) shows that the main effect of collisions is to broaden the infinitely sharp collective resonances in (11.103). In addition,
LIOUID
he - B
19.6 bqr 15.15
MHz
.E
()
a
r
- T/ïc
LIOUIO 3He - B 19.6 bqr 15.15
o
MHz
Ix d
ìs (-) I
c) OO
t
0.r0
Figure 11.4
t
))t_ 002 'nol r - T/Tc
(a) Attenuation and (b) velocity shift of 15 MHz sound in 3He-B versus temperature. Experimental data after Paulson et al. (I973b); theoretical curve after rilölfle (L976b).
541
11.4 Collisionless Sound 200
"lo ol o J=2'
,t
I
?
oo
c-^ L o I(J
: :"
?o ooo
J=2*
I
é20
o o
oo $o
I
F
'S
ãro
ooo
I
I t.83Â
2 0:12
Figure
0.8 îtlc
0.9
1.0
11.5 Attenuation of 44.2MHz sound in the B phase at 2.44 bar pressure.
(After Saunders et al. (L983).) The contributions of the pair-breaking continuum (solid curve) calculated from (11.103) and the positions of the collective-mode peaks are indicated. The additional structure at ha:1.83^Bcs is as yet unexplained.
collision-induced background. Both effects may be crudely described by replacing ø in (11.103) by (Ð:or*ift".,1¡, where z"o¡¡ is the relaxation time discussed in (11.83). The result of an evaluation of the sound attenuation (11.91a) and velocity change (11.91b) including collision effects in a more rigorous fashion (Wölfle 1976b) is shown in Fig. II.4, together with the experimental data of Paulson et al. (1973b). A comparison with data of Ihas et al. (1985), who used 5 MHz sound, appears to indicate additional absorption at lower temperatures. Data at still higher sound frequencies are shown in Fig. 11.5.
there is
a
Three main contributions to the attenuation are clearly discernable, although the centre portion of the second peak could not be measured owing to the high absorption level. The round peak just below 7i is caused by pair-breaking processes, while the other broad peak at a still lower temperature 4q, defined by ha:(+)t''A(7:o), is due to absorption into a squashing mode (Meisel et al. L983a). Finally, the real squashing mode leads to a sharp peak ât ?i,q, the temperature where the resonance condition for the real squashing mode, ho¡:(E)"'A(T,,o), is satisfied (Koch and Wölfle 1981). In spite of the small coupling of the latter mode to sound (due to particle-hole-asymmetric terms) it was detected experimentally by Giannetta
542
11 Collectiue Modes
et al. (1980) and Mast et al. (1980). The sound velocity ,o@t): @le, i.e. the phase velocity, also shows considerable structure. This structure is seen to be even more pronounced in the group velocity of sound c"(a):èatlðq, in
particular in the neighbourhood of the collective modes, where values as low as one-third of the phase velocity have been detected (Calder et al. 1980; Halperin 1982). Accurate measurements of the collective-mode frequencies at low temperature have been performed using a dynamic pressure sweeping technique (Adenwalla et al. 1989). The propagation of sound pulses at frequencies near the real squashing mode and at low temperatures, where the damping is negligibly small, leads to interesting beat patterns, even in the linear regime (Avenel et al. 1983, Varoquaux er
al.
1986).
The collective mode with quantum number J : r- and frequ€flc! a¡ : (see Table 11.1) is barely observable in zero magnetic field (Oaniels er al. 1983), but becomes clearly observable in a sufficiently high magnetic field (Saunders et al. t983 Ling et al. 1987). That is, the field has to be iufficiently high that the frequency (split by the magnetic field into three substates) is moved below the (likewise shifted) pair-breaking threshold. The observed magnetic-field dependence is in qualitative agreement with theory (Schopohl et al. 1983; Schopohl and Tewordt 1934). In general, the experimental sound data are described well by theory. This is also true for the magnetic-field splitting of the ,/:2+ modes mentioned in Section lL.4.l, although there are still some unexplained features, which may perhaps be related to textures (Ketterson et al. L983b). In particular, the central peak in the fivefold-split attenuation peak structure is observed to split again into two peaks at higher magnetic fields (Shivaram et al. 1982). This effect has been explained on the basis of a local resonance shift of the collective mode induced by the É texture (Volovik 1984c, Volovik and Khazan 1984, Mineev
2lln
1985c, Fishman and Sauls 1988a,b). In fact, the agreement between theory and experiment is so good that one can even extract values of the quasiparticle collision time z (Wölfle I97Ba),
the Landau parameters F)'' (Engel and Ihas 1985) and the ratio of the pair-interaction constants vzlv, by fitting the above theory to the data (Sauls and Serene !982, Fishman and Sauls 1986, Meisel et al. lggT). A
contribution from the squashing mode to the longitudinal surface impedance has been detected experimentally (Avenel et al. 1981a). While the above derivations made use of an expansion in the small parameter (urlcr)2, expressions for the complete density and current response functions have been derived by Czerwonko (1980a,b, L9BZ,1985). The effect of vortices on sound propagation in the rotating B phase has been investigated in a recent experiment (Salmelin et al. 1989b). The five collective-mode peaks, associated with the real squashing mode in a magnetic field, are found to depend strongly on the angular velocity of rotation and the preparation of the system, in qualitative agreement with theory (Salomaa and Volovik 1989).
11.4 Collßionless Sound
543
11.4.3 Sound propagation in the A phase The situation in the A phase is slightly more complicated owing to the orbital anisotropy of the liquid. On the other hand, spin and orbitaidegrees of freedom are decoupled here, if we neglect the tiny spin-orbit interaction. Furthermore, since all the collective modes couple to density fluctuations even without the aid of particle-hole-asymmetric terms, the iatter may be discarded. The relevant combinations of order-parameter variables are then d"r^, ln:0, *1, as defined in (11.69). Following the procedure described in Section LI.4.2, one finds to order
(upq lot)z
A, 1 d,! u:2trz3 *z'\' " ha F¡
(Íar)j
sin É cos É (cos'z
I
Â(É) ô6¿)¿
<(k@),h
,
)il _r4,, r (hot)2 sin2B (sin2 0 uz.-r ^(Òô6¿)¿ n
,
(11.10s)
cosp:A-Î,^cosg:Ê.i and i is the axis of the gap. we want to assume a uniform i configuration. As discussed in Section 1I.3, d,i.u and where
d'1,-, possess resonances as functions of ot, i.e. the flapping and clápping modes.
substituting the results (11.105) into the expression one finds (Wölfle 1973a)
E@¡): f,, cosa p + zE.sin2
Bcos, B
where
g&)coso 8,,: rr -| -+l 4 L\.,\._/_""
E^: 5c-r| -f g lleil,sin2 L-
_ (1,(*)
_ _ _J,
cos2
¡"tk
g
+ E rsina
(11.106)
B,
(L(k) co::s' o>i1
erfl)a
J,
g cos2 o)¿
e) aQ,,G) s¡nz o)
(t(Ðla
a
cos'
O)za
as F ^ w/*-- I - [i ç(k)sina g, 5-L-r -(t'(*)
s:n' o>',
Q'(k)
(Â1É¡f <1(Ò sin2 e)rt
L6L2\/t\^.irùrrr
for E@), (1l.gg),
I
(1 1.107)
11
544
Collectiue Modes
As expected, the sound-propagation properties are anisotropic, depending B between the sound-propagation direction Q and the preferred direction l. The anisotropy is characterized by three functions f ¡¡, f" and fr, where Ç,, accounts for parallel and f, for perpendicular orientation, whereas all three functions contribute for intermediate orientation. The anisotropic sound attenuation u(ø) can then be obtained from (11.91), yielding
on the
ang^le
(11.108) þ+Zu.sin2B cost B* drsinaB, where u, : -qZ Im Ç(ar), i : ll, c, l- . In general, the temperature dependence of u(a), (11.108), has to be evaluated numerically. However, in the limiting cases I--+0 and 7--+?l it may be calculated analytically by
s(a¡):
ø,, coso
employing the respective results for Â(/<), (II.62).
Thus for sound propagation parallel
to l, none of the flapping and
clapping modes are excited, which is^easily understood from the schematic representation in Fig. 11.2. For q I l, on the other hand, only the clapping mode is excited: a compression wave perpendicular to i will lead to a local widening and narrowing of the angle between the preferred directions râ and ñ, which is the clapping mode, while leaving the angle with respect to Î* unchanged. In Fig. 11.6 the result of a numerical calculation of the three components of the sound attenuatiorr, u,, &. and ør, is compared with experimental data of Paulson et al. (I976b,c). The theory includes both pair-breaking and collision effects (Wölfle and Koch 1978). The peaks associated with the clapping mode (ør) and the normal-flapping mode (u.) are clearly borne out. The contribution of the super-flapping mode is masked by pair-breaking processes (Ashida and Nagai 1983, Einzel and Wölfle 1984), but a corresponding feature in the sound absorption has nevertheless been observed (Meisel et al. 1983b, 1985). The remaining discrepancy between theory and experiment, especially in ø,,, is thought to be related to the incomplete orientation of the I texture in the experiment. [n fact, the strong dependence of úv on the orientation of / may be used to monitor the time dependence of the i texture in a nonequilibrium situation (Wheatley 1978). Unfortunately, this probe is not local but samples the texture over a region of the size of the acoustic penetration depth a-'. The attenuation of lower-frequency sound (v: 5 MHz) has been measured by Berg and Ihas (1985). Sound-attenuation measurements well below I and in the pairbreaking regime may in principle be used to determine the value of the gap parameter (Piché et al. L982). The re-entrå$ce of the normal-flapping mode at lower temperatures (see Fig. 11.3)-has been observed by Ling et al. (1989). Analysis of the latter experiment provides evidence for weak f-wave pairing effects and nontrivial strong-coupling corrections. Experimentally, there appears to be some excess attenuation even above f. (Paulson and Wheatley 1978c), which might be interpreted as pairfluctuation effects. and various contributions to the fluctuation-induced
11.4 Collisionless Sound
545 I
3H.-A t
5
2l,.1bar
o.,.
^ dc -o
t4l '
dlr
;tr
t'
1^t, E JJ
\i
c
.9
-']
¡
d l
/
C
93
ns/
I T
d E c :t o
&o.
.l,/ '/,
lotr
v)
/n ¡o-
I
/tf -rcDt,/ ,/
,/^d,..^ C't'2 -/
i-v ./ 0.20
0.15
0.t0,,. (1-T
Figure
I
0.05
0
Tcl',.
sound attenuation in 3He-4. (After Paulson et al. (1976b,c).) The curves are the theoretical results of Wölfle and Koch (1978).
11.6 The three components of the anisotropic
attenuation have been discussed (see e.g. Emery 1976, Samalam and Serene 1e78).
It
appears to be worth drawing explicit attention to the astonishing fact that sound absorption in 3He-A is not isotropic, owing to the existence of the preferred directiòn i. Rs we know, this direction may be oriented by a weak (approx. 30 G) external magnetic field (via the dipolar coupling between and d), and will thereby change the level of sound absorption. This anisotropy is in marked contrast with the intrinsic isotropy of the 3He liquid, which consists of spherical-hence structureless-atoms; it finds its origin solely in the direction of the spin of the 3He nucleus. So the hadronic nuclear spin and its direction are ultimately responsible for the anisotropy of
i
macroscopic properties like sound absorption. Here \rye encounter particularly fascinating example of macroscopic quantum coherence.
a
546
11
Collectiue Modes
11.4.4 Nonlinear efiects of sound propagation Interesting nonlinear behaviour of sound propagation has been observed in both the A and B phases. In the B phase the propagation of sound pulses near the real squashing mode at low temperatures and at correspondingly high frequencies shows the characteristics of self-induced transparency, i.e. weakly damped propagation of soliton-like excitations (Polturak et al. 1981a,b, de Vegvar et al. 1981). A simple model, based on the analogy with two-level systems in an optically resonant medium, with the Cooper-pair ground state and the pair-vibration state in question taken as the respective two levels of the system, explains some of the features observed (Sauls 1981; see also Ruoff and Varoquaux 1983, de Vegvar, 1985). However, the model does not account for the necessary strong coupling of the sound wave to the order parameter. Earlier, Paulson and Wheatley (1978b) had observed satellite peaks in the sound attenuation in the B phase at lower temperatures. As shown later by Engel et al. (1985), the effect is probably not an intrinsic one. In the A phase nonlinear effects due to saturation of pair breaking have been observed (Avenel et al. 198Lb, 1985) (see also Pushkina and Khokhlov (1976) for a discussion of ñonlinear sound propagation in the hydrodynamic regime). It is likely that both effects are a consequence of resonant, i.e. coherent, pair breaking and recombination (Kopp and Wölfle 1987). The generation of higher harmonics by nonlinear coupling has been considered by Serene (1984) and McKenzie and Sauls (1989).
11.4.5 Transverse sound and spin waves discussion' similar to the preceding one for longitudinal sound can be given for transverse zero sound and for spin waves. The salient results are
A
that transverse zero sound continues to exist in the superfluid phases provided that it existed in the normal state, i.e. provided that the condition
Fì > 6 is satisfied. In the superfluid the / : 1 molecular field is produced by the quasiparticles alone, in contrast with the case l:0, where the rigidity of the phase takes over the role of the Landau molecular field. Consequently, the coupling parameter F] is effectively replaced by the temperaturedependent quantity F|Y,(T), where Y,Q) is the Yosida function. Below a temperature I defined by FìX,(4):6, transverse zero sound ceases to exist (Combescot and Combescot 1976b, Maki and Ebisawa I976b, 1977) (see Leggett (1966) for the case of s-wave pair.ing). By contrast, spin waves become more and more well defined'as one enters the superfluid state, because here the phase rigidity helps to stabilize the mode and eventually supports it entirely (Combescot I974a,b). The possibility of excitation of pair-vibration modes by spin waves has been investigated by Schopohl et al. (re82).
Further Reading
547
Transverse zeÍo sound has been detected experimentally in the acoustic shear impedance by Roach and Ketterson (1976b), while collisionless spin waves have been studied in the form of standing waves localized by a nonuniform magnetic field (Masuhara et al. 1984).
FURTHER READING Brinkman W F and Cross M C 1978 in Progress in Low Temperature Physic.s, yol. VIIA, ed. D F Brewer (North-Holland, Amsterdam), p. 105 Brusov P N and Popov V N 1984 Collectiue Excitations in Superfluid Quantum Liquids (University of Rostov, USSR) (in Russian) Halperin W P 1982 Physica 109/U0B + C 1596 (Proceedings of the 16th Inrernational Conference on Low Temperature Physics, LT-16) Ketterson J B, Shivaram B S, Meisel M W, Sarma B K and Halperin W P 1983 in Quantum Fluids and Solids-1983 (AIP Conf. Proc. No. 103), ed. E D Adams and G G lhas (AIP, New York), p. 288 Kleinert H 1978 Fortschr. Phys. 26 565 Serene J W 1983 in Quantum Fluids and Solids-1983 (AIP Conf. Proc. No. 103), ed. E D Adams and G G Ihas (AIP, New York), p. 305 Wheatley J C 1978 in Progress in Low Temperature Physics, Vol. VIIA, ed. D F Brewer (North-Holland, Amsterdam), p. 1 Wölfle P 1977 Physica 908 96 Wölfle P L978 in Progress in Low Temperature Physics, Vol. VIIA, ed. D F Brewer (North-Holland, Amsterdam), p. 191
12
Amplification of Weak lnteraction Effects due to Macroscopic Ouantum Goherence
The concept of macroscopic quantum coherence has been touched upon several times already: for example within the context of persistent currents (Chapter 7), in the calculation of the expectation value for a two-particle operator (Section 3.5) and when this result was applied to calculate the magnitude of the dipole-dipole interaction in the condensate (Section 6.3.1). The latter example provides a particularly striking demonstration of how a weak microscopic interaction may be amplified to macroscopic size by the specific long-range order in the system. This effect is clearly very similar to macroscopic quantum coherence in superfluid oH., where only gauge symmetry is broken. Indeed, both systems are characterized by a macroscopic wave function expressed by a complex-valued order parameter. In view of the fact that Cooper pairs are composed of two fermions with a total spin S :0 or .S : 1, these pairs may in a way be looked upon as "composite bosons". Furthermore, the pairs are all in the same quantum-mechanical state and have zero total momentum in the ground state. Hence it is tempting to view the transition to the superconducting state (or to the superfluid state in tH") as one where the formation of Cooper pairs and their Bose-Einstein condensation coincide (i.". Cooper pairs aÍe automatically Bose-condensed). In this wêy the macroscopic quantum coherence of the pair-correlated state and the amplification of the microscopically small dipole-dipole interaction in superfluid 3He may easily be visualized. On the other hand, one has to bear in mind that a Cooper pair in superconductors or superfluid 3He is neither an elementary boson nor (at least) a tightly bound composite of two fermions with an overall bosonic excitation spectrum etc. Its spatial extent f1, is very large compared with the interparticle distance a. Consequently, a given Cooper pair overlaps with
5s0
12
Macroscopic Quantum Coherence
many others (Schrieffer 1964), their number in order of magnitude being given by (Enlo)t{r.lrr¡:10-3(GlT.)', which is about 10s-106 in the case of clean conventional superconductors and about 102 in the case of superfluid 'He. Furthermore, we already know from the discussion in Section 7.3.I that it makes a difference whether the Bose-Einstein condensation involves overlapping or nonoverlapping diatomic molecules (Mermin and Muzikar 1980, Mermin 1980, Volovik and Mineev 1981). In spite of all these objections, a theory of superconductivity based on a Bose-Einstein condensation of pairs of particles ("Schafroth condensation") may indeed be formulated, but is mathematically very complex (see Blatt (1964) and also Baranger (1963)). Altogether, the concept of Bose-Einstein condensation is appealing since key features of superconductivity like the Meissner effect, flux quantization and superfluid mass currents in conventional superconductors are naturally implied. Since the theory of conventional superconductivity is firmly based on BCS theory, the concept of a Bose-Einstein condensation of Cooper pairs traditionally did not receive much attention (or was even considered to be downright wrong). However, within the context of superfluid 3He, this notion was taken up again by Leggett (1980a,b), who argued that tightly bound Bose-Einstein-condensed molecules on the one hand and Cooper páirs on the other may be viewed as the extreme limits of the molecule picture. This concept, which also applies to electron-hole pairs in semiconductors, has been further investigated by Nozières and Schmitt-Rink (1985) within several models. Both Leggett (1980) and the latter authors found that the transition between the two limits is srzooth. From the point of view of critical phenomena we should therefore conclude that the formation of Cooper pairs and the Bose-Einstein condensation of tightly bound molecules is indeed not something distinctly different. This view is also implicit in the theoretical approach to high-7" superconductivity, based on the "resonating-valence-bond" (RVB) approach, which has been advanced by Anderson (1987a,b). In this case superconductivity is viewed as the Bose-Einstein condensation of "preformed" (in the insulating regime) Cooper pairs, once the system is doped. An offspring of this idea is the concept of a gas of hard-core dimers, which may undergo Bose-Einstein condensation and thus lead to superconductivity (see e.g. Rokhsar and Kivelson 1988).
12.1
PROPERTIES OF THE BCS PAIR AMPLITUDE
Apart from the overall conceptual equality of Cooper pairs in conventional 3He, based on BCS theory, there are also superconductors and in superfluid distinct qualitative differences (Leggett 1980a,b): (i) Cooper pairs in superfluid 3He form in a S : 1, I: I state, i.e. they have an internal structure; (ii) their constituents have a strong mutual repulsion at short
12.1 Properties of the BCS Pair Amplitude
551
relative distances; (iii) these constituents are not pointlike as in a superconductor, but have a physical structure themselves. In these respects Cooper pairs in superfluid 3He resemble diatomic molecules much more than do those involving electrons in a superconductor. Besides that, the coherence length fs is much shorter than in a clean conventional superconductor (see below (7.18b)). According to Leggett (1980a,b), the picture of cooper pairs as diatomic molecules be helpful in gaining insight into some properties of -may superfluid 3He. In particular, he pointed out that such a modei provides guidance for a more realistic evaluation of expectation values of two-particle operators in superfluid 3He than is possible in conventional BCS theory, where one only works with model wave functions. Let us compare the results for the expectation values of a two-particle operator W(r, or, o) in the case of a diatomic molecule and of Cooper pairs respectively. Here r is the relative coordinate, which is supposed to be the only important coordinate. For molecules, each with a pair wave function tþ(r; or, az), the result is
(wl:
f
" ,ld3rw(r)lq'Q)l',
(t2.1)
which should be taken as a schematic notation since (W) will in general be a more complicated object owing to the spin dependence. In the case of Cooper pairs the result was calculated in Section 3.5, and is given by (3.119a) and (3.120): f (w>:ld3rw(r)lF(")l', (12.2) f' where again complications due to spin have not been written explicitly. The two results are identical in form. On the other hand, (L2.2) does not contain a two-particle wave function, such as the wave function r/(r) used in (3.10a) to discuss the Cooper problem for two fermions and which may equally be used to construct the total N-particle wave function wn, (3.11). Rather, it is the pair amplitude (i.". the order parameter) F that enters. That is why F is sometimes called the "pair wave function"f. on the other
hand, although F and V are quite different quantities (after all, F is obtained from the solution of a complex many-body problem), they are clearly related, and in fact have the same transformation and symmetry properties. This implies that F and rp have the same dependence on spin and on the direction î, i.e. the orientation of the relative separation r of the atoms in the pair. However, they will differ in their radial dependence, i.e. T Of course this is partly a matter of definition. But, as pointed out by Mermin (1978a), one should in fact carefully distinguish between an order parameter and a pair wave function; for this purpose, he discussed a particular two-particle wave function given by the order parameter. that leads to a wrong result when used to compute the quantum-mechanical current density
(7.32).
12 Macroscopic
Quantum Coherence
dependence on the magnitude of r, owing to many-body effects, choice of momentum cutoff etc. entering in F. Let us separate the angular- and spin-dependent part of F from its radial part and write
in their
F(r; or, az): Fo(î; ot, or)F(r). This is a good approximation as long as the energy eigenvalues (3.9a) of the paired state for different / components are sufficiently separated, and becomes exact for T--+ [. The first part, Fe, reflects the structure of the order parameter in spin and orbital spaces and is known for the different superfluid phases. It can therefore be integrated over in (12.2) to yield a constant factor. The radial part, however, poses a problem because its actual form is not known. In BCS theory, i.e. mean-field theory, the pair amplitude has a particular model form, which can be obtained by transforming (3.43b) back into real space. For r (( 5,,, F1r¡ is given approximately by the radial wave function of two free particles in a state of relative angular momentum / at the Fermi surface (Leggett 1975a), as discussed below (3.10b). For r)6u, i.e. r larger than the maximum size of a Cooper pair, F1r¡*e-'tE', although the explicit fall-off depends on the choice of the cutoff in the momentum integral. This r dependence of F1r¡ is not only unrealistic but also insufficient for two distinct reasons. (i) The operator W(r) in (12.2) typically contributes mainly for small r. The dipole interaction, for example, has a r-t dependence; other interactions decrease even more rapidly (see below). So the dependence of F(r) is particularly important at small arguments. (ii) On the other hand, in the case of 3He, where atoms have a finite diameter R1¡, the BCS-model form for F is especially bad at small r since it does not take into account the microscopic interaction (it even contributes for r < R,,). In order to improve on this point, Leggett (1977a,1980a) suggested an ansatz for F(r) that modifies the BCS form at distances r: Ro. It is guided by the results for the molecule picture, where the radial pair wave function at short distances is given by the relative wave function of a simple diatomic-molecule. Note, however, that such an approach requires one to go beyond the regime of validity of the quasiparticle picture. Therefore it cannot yield a controlled description of the effects involved, but may only be expected to be qualitatively correct. In superfluid 3He F(r) may then be expected to be similar to E,(r), (3.10b), for r (( 6,,, i.e. to be given by the pair wave function for two free 3He atoms in a relative /: 1 state at the Fermi surface. A possible form is (Leggett 1977a,1980a)
.::
F(r):
sin,k(r
{: which cuts out the region for r
-
s
,',
R1¡)
(r < R,,),
(R.,
(r2.3)
Rs and assumes free-particle behaviour for
12.2
The Permanent Orbital Magnetic Moment of the
A
Phase
( r(
553
:
f1¡. (Note that this is actually a form describing an / 0 rather than 1 pair state; but since kRr¡:2.3, this choice is not expected to introduce a substantial error (Leggett L977a,b).) This model form for ^F is then used to Ro
/:
calculate the expectation value <w> in (12.2). The prefactor .4 can be obtained by calculating the expectation value of the dipole interaction energy (H") in (6.88) or the dipole-coupling parameter
so(r)
: +r'h'l d3r r-3l1(Òl',
(12.4)
which is known from experiment. This yields
A2:L
R3
2n Yzhzt(kni
8o(r)'
(12.s)
where
r(x):r*## For kRq¡:2.3, one obtains /:0.2. ln this way, all expectation values may be expressed in terms of the dipole interaction in the superfluid.
(w)
12.2 THE PERMANENT ORBITAL MAGNETIC MOMENT OF THE A PHASE Because of the filled ls shell, an isolated 3He atom has a perfectly spherical shape. This is why helium is chemically extremely inert. In such a case the geometric centre of the electronic charge distribution coincides with the nucleus and therefore the atom does not possess a permanent electric dipole moment. On the other hand, if two 3He atoms are brought closely together,
the electron shells will be distorted owing to the mutual polarization-an effect of purely chemical origin. Following Leggett (1977b),let us consider the Cooper pair in 3He as a sort of diatomic molecule of two 3He atoms. Since the electronic charge distributions will be slightly distorted in both atoms, their centres will no longer coincide with those of the positive nuclear charges. Since the "dimer" is rotating (it has an orbital angular momentum l: l), there exists a finite magnetic moment due to the (opposite) charge currents of the circulating nuclei and electron distributions, which no longer cancel. To estimate this magnetic moment, we consider the 3He nuclei in the pair to have separation R and mass M, and we imagine the atoms to rotate around an axis perpendicular to the plane of rotation (parallel to the relative orbital angular momentum ,). Let the centres of the electronic charge distributions be shifted by a distance P(R) away from the nucleus as a result of the polarization. The orbital magnetic moments of the electrons
12
554
Macroscopic Quantum Coherence
and the nuclei for the two 3He atoms are then given by
¡t":4¡lsL"î, m^
Pn: -4Ps -mn
(12.6)
LnI,
where ps is the Bohr magneton and L. and L,, are the magnitudes of the angular momenta, with L.: ffi" lul (jR + f) and Ln: àrfln lul R respectively. Here u is the velocity of the charges. Since the dimer is in a state of l:L, it is given by lul: klm^R to lowest order in the small parameter m"f mn. The magnitude of the total magnetic moment Ft,,t: F" * Fn along / then follows as
,,,ffi"P(R) lPtotl : *P" ^^
(t2.7)
R
So the polarization P(R) of the electronic charge distributions leads to a finite orbital magnetic moment. In a normal system this has no macroscopic consequences, because not only are these magnetic moments extremely small, they also average to zero because they are randomly oriented. Even a fantastically high magnetic field would not be able to align them. In the case of 3He-4, however, Î is the same for every Cooper pair, so that the magnetic moments ¡r,o. of the Cooper pairs will add up coherently. Consequently, this orbital magnetic moment is permanent and has a macroscopic magnitude. The difficulty lies in going beyond such a qualitative statement and actually calculating the size, as was done by Leggetf (L977b). For this, one first of all has to know the actual polarization P(R) within the Cooper pair. If one assumes that P(R), measuring the distortion of the electronic wave function, depends exponentially on the separation R of the two atoms then one can write P(R)
: a*c,"*o [- rcã&)]
(12.8)
Here ø,, is the Bohr radius (the characteristic length in a problem dealing with electrons), Rs:2.5 Ã is twice the hard-core radius, i.e. the diameter, of the 3He atom, and c,¡ and ), are constants, which may be determined by means of an approximation for the distorted electronic wave function (co: -4.4 x 10-3 and Â:I.4). Using (I2.2) to obtain the total spontaneous magnetic moment M(T), the moment for one Cooper pair, F*,,(R), has to be weighted by the probability lF(R)l' of finding the two 3He nuclei ar a distance R aparhand then has to be integrated over all R:
WQ)l:4nAz+s
k
mn
ourrf- o^ JRu
sin2 [k (R R
-
R,,)l
"*pl-g;Ð]
ezs)
12.3
The Permanent Electric Dipole Moment of the B Phase
55s
Note that P(R) only contributes for small separations R Þ Ru of the order of 2.5 and drops off exponentially for R >l Rn. Now it is crucial that a Cooper ^ pair is not a molecule: if it really were a molecule with a more or less fixed separation Çn (the maximal spatial extent of the pair, which is several hundred ångströms) then there would be no effect at all. It is only because the position probability density of the two partners of a Cooper pair has an appreciable weight at small R : Ru even for p-wave pairing (see (3.10b)) that the integral in (12.9) gives a non-negligible conrribution. Inserting the expression for 4nA2 in (12.5), Me) can be calculated. Actually, it is more instructive to convert it into an equivalent magnetic field H"q: M ly¡1,,.where lN is the normal-state spin susceptibility. Using the fact that 5 x 10-tt',,r" p"t atom corresponds to a field of 1G, one finds
H",(r)
- -H, (t - ä)¿
(12.10)
where Ht:0.03f(k) (in gauss) and f (k): [1 + (Àl2auk)z]-r. This calculation therefore predicts that there exists a spontaneous (permanent) orbital magnetic moment in the superfluid A phase, oriented opposite to the i direction in the sample (Leggett 1977b). It corresponds to an internal magnetic field of about L0-20 mG. We note that this magnetic moment will couple to an external magnetic field. The enerïy E - -M. rI is given by E
:
M(T)H .î.
(12.11)
Hence there is a linear coupling between the external field and the direction of Î that prefers Î to li" antiparallel to ^EL For the first time we encounter an orientational effect on the Î texture that distinguishes between î and -Î, or-put differently-that depends on the polarity of the external field. Consequently, the ground-state energy of the system in external magnetic fields H and -ñf is different, leading to a slightly different equilibrium i texture when the magnetic field is reversed. . Indeed, Paulson and Wheatley (1978a) showed that the orientation of the Í texture does depend on the sign of the applied external magnetic field. By means of sound-attenuation measurements, they determined the prefactor of this linear coupling term .lï. Î and found it to be in overall agreement with the theoretical prediction. The macroscopic orbital magnetic moment makes 3He-A a liquid ferromagnet.
12.3 THE PERMANENT B
ELECTRIC DIPOLE MOMENT OF THE
PHASE
Perhaps even more spectacular may be yet another effect, which is expected to be caused by the quantum-mechanical amplification of ultraweak interaction effects within elementary-particle physics. As is well known, the interaction between an electron and a proton (more generally, a hadron)
12
556
Macroscopic Quantum Coherence
contains, besides the usual parity-conserving Coulomb interaction, a very much weaker parity-violating part due to the exchange of a neutr al Zlt bosoi ("weak neutral currents"). This latter interaction is predicted to lead to a macros.copic effect in 3He-8, i.e. to a permanent eiectric dipole moment þ": c(Z x S) of the Cooper pairs along the preferred direction ¡ô (Leggett 1977a)- Its existence may be understood by considering the basic p.op"rti., of an electric dipole moment under time reversal Z und rpu." reflection p. Let us therefore take a look at the energy Eu: F". of an electric dipole ^E in an external electric field E. Since all electromagnetic interactions are pand T-invariant, the energy must also have this invariance. To illustrate the following arguments, the symmetry behaviour of various physical quantities under the P and Ioperations are listed in Table 12.1. The electric field ^E is invariant under I, but not under p. Consequently, an electric dipole moment must have transformation properties identical with those of E, since the energy E has to be invariant under both p and 7. Now the spontaneous electric dipole moment is a vector, which is something very fundamental for a single particle. In fact, in the case of a single particlé at rest (e'9. an elementary particle or an atom) there is onlv one such characteristic vector: its spin S or some orbital angular moment;m J. Thus the electric dipole moment has to point along this one fundamental vector. However, the spin s (or J) is odd under z, and therefore the energy E: -þ"'fio-^s.^E' would change sign under time reversal. This is forbidden and hence frc must be identically zeÍo. (Demanding p invariance leads to the same result, i.". lt.:0 follows from both p unã f invariance separately.) This is the reason why "simple" systems, characterized bv at Table 12.1 Symmetry of various quantities under parity transformation p and time inver_ sion T respectively. Physical quantity Space
r
Momentum p (: drldt) Angular momentum L: r x p Spin S
ZxS
Electric field E : -dV ldr Magnetic field ÍI Electromagnetic energy E
p.S
E'L
,'¡
E.(Lxs),
H.L H.(LxS'¡
12.3
The Permanent Electric Dipole Moment of the B phase
557
most one vector, do indeed possess no electric dipole moment. (In more complicated systems rike morecules (e.g. H2o¡ trrii argument is no ronger valid, since there is more than one characteristic direãtion; in fact, many molecules do have an electric dipole moment.) Let us then consider an object characterized by two independent vectors, say the spin S and angular momentum L- Then one can construct a vector Z x.S thai is invariant both under P and T- T invariance alone would therefore allow a dipole p.(¿x^s. However, p invariance of E:-p".Eq.1zxi¡.ø moment is still violated and if all interactions were parity-conserving (like the electromagnetic and strong interactions), ¡r. would have to be zero after all. Here the weak interaction becomes important. Its theoretical formulation within the unified theory of the Weinberg-Salam model implies the existence of .,weak neutral currents" mediated by the exchange of zo bosons. such currents are, for example, generated by the electron_nucleon interaction within an atom, made possible by the finite probability amplitude for electrons at the nucleus. In the case of the atå- there is a part to the interaction involving an electron and the 'He neutron, which can be written as (Bouchiat 1977) V..^(r)
- -Z-r,rGrC on) 0". S"ô(r) * Herm. conj. mc
(IZ.l2)
Here Gp and Con are the coupling constants of the usual weak interaction and of the specific lepton-neutron axial-vector interaction respectively, while m" is the electron mass, p" is the electron momentum and S, is the spin of the neutron. The delta function expresses the fact that one considers a contact interaction of pointlike objects. Obviously, the above interaction is P-violating but Z-conserving. Its existence removes even the last symmetry argument against a finite static dipole moment F": c(L x s), where c must be proportional to the coupling sìrength GFCA". Now that we know that such a dipole moment may exisi, we have to check whether the vector L x,S makes sense at all, i.e. whether ^L and ,S are really independently conserved quantities. In a normal atom this is not the case, because L and S will couple to a total angular momentum J owing to some hyperfine interaction. Even if this were not the case, the spatia-í average !L*S) would average to zero in a macroscopic system. In this respecr 3He-B is an exception since the relative orientatìon of Z and S is the same for every Cooper pair-a consequence of the broken relative spin-orbit symm^etry. (Note that in 3He-A (n rS) * î x (S) also vanirt es- because only / and d are macrosco.pically defined, while (S) :0.) In rhe B phase one finds that (Z x S) : -4rh, sin g,- É, where gr: cos-' (_å) is the Leggett angle. We now have to calculate the magnitude of this electric dipole moment p"q(Lxs). In fact, it could still be zero for a symmetry reason not yet considered-and if the electrons in the shells of the 3He átoms in
12 Macroscopic
558
Quantum Coherence
the Cooper pair were in an exact ls state, as in the isolated case, it would 3He atoms slightly indeed be eiactly zeto. But in a Cooper pair the two polarize each other's electronic shells, giving rise to an admixture of 2p
atomic orbitals
to the wave function. Owing to this finite angular-
momentum component, the electronic motion can couple to the rotation of
the Cooper pair. The induced distortion of the electronic distributionagain a chemical effect-
þ":
as
-2-3/2GpC¡ne(R)z x
S,
(12.13)
the where €(R) is called the "electrohelicity". It describes the details of 3He distortion of the wave function as a function of the separation of two atoms in the Cooper pair. Its actual calculation is quite difficult, but it appears reasonable to make an ansatz as in (12.8) with an exponential R dependence:
€(R)
:â*,o".p [-1ß^4d],
(r2.14)
where elhao is a characteristic unit of €(R) in the case of an atom. Again 3He atoms at different using results of atomic-orbital calculations for two separations, e6 and  have been estimated by Leggett (I977a) as €0: 3 x 10-4, Â:0.3-0.5. To obtain the total electric dipole moment D in a sample of 3He-8, one may now proceed exactly as in the previous case of the magnetic moment: ¡r"(R) is multiplied by lF(R)l', and is then integrated over R. One obtains
D
:2:
Grc onesin
g,- ñ,
(r2.ts)
where
O: Id3r e (R) lF(R)l'
(12.16)
To get a feeling for the size of this spontaneous electric dipole moment, we introduce a unit Dn of the dipole moment that in an electric field of L V/cm leads to an orientational energy equivalent to the thermal energy ksT at
LmK, i.e. Do:8.6x10-8ecm. Using the numerical values Gçlhaï: 1.6 x 10-tu, Co,:l for the coupling constants and (12.3) for F(r), and proceeding as in Section I2.2, Leggett (1977a) obtained an order-ofmagnitude estimate for
lDl in terms of Do:
lÐJ.i"lo-'o'
(r2.t7)
3He-B is hence seen to be The permanent parity-violating dipole moment in very small. As in the case of the spontaneous orbital magnetic moment in
12.i
The Permanent Electric Dipole Moment of the B Phase
559
3He-4, the dipole moment D leads to a linear coupling with an external field, this time with an external electric field E. This coupling is sensitive to the polarity of .8. Reversing the electric field will change the energy of the system and will therefore modify the equilibrium ñ texture. In principle, the change may be detected by a shift in the NMR frequencies of the texture. Although the effect will be very small---even in a strong external electric field-such an experiment does not seem to be hopelessly outside the accessible range. A positive result would show, for the first time, the consequences of parity violation of the weak interaction on a macroscopic level.
The weak interaction between electrons and hadrons not only violates parity, it also has a T- (i.e. CP-) violating part. The typical coupling constant for the T-odd interaction is about 10-3 times smaller than that of the P-violating part. CP violation has so far only been observed in the decay process of the neutral K meson. Clearly, it would be desirable to find evidence for this interaction by a different experimental technique. As discussed above, an electric dipole moment (EDM) of an elementary particle will violate both P and T invariance. Hence if such an EDM were indeed to be measured, it would allow one to determine the strength of the T-odd interaction. However, in spite of the long history of the search for EDMs of elementary particles (Ramsey 1982), no evidence for EDMs has emerged. There are many suggestions how to improve the sensitivity of EDM experiments, particularly using macroscopic samples (see e.g. Bialek et al. 1986). Based on Leggett's prediction for an EDM in the B phase, Khriplovich (1982) suggested that the A1 phase of superfluid 3He might be 3He atom (i.e. of electrons and neutrons) used to measure the EDM of the by again exploiting the macrosc<-rpic amplification of this subatomic effect due to the condensate. In this context several other similar effects may be discussed. For example, in the A phase a permanent electric dipole moment on every Coopêr pair (which must be along i) would lead to an energy E : - þ"' B xf ..8' when placed into an electric field E. In this case the expression is again odd under P and Z; so the magnitude of p. will be determined by the smaller of the P- and T-violating parts of the weak interaction, i.e. the part 3He-4. Its odd in Z. Consequently, an electric dipole moment can exist in magnitude is proportional to the Z-odd interaction strength and may therefore be expected to be even smaller (by u few orders of magnitude) than the one predicted by Leggett for 'He-B on the basis of parity violation. Similarly, we may consider a spontaneous magnetic moment ¡¡-o (L xS) in 3He-8. The energy of such a moment in a magnetic field is given by E^: - p^. H o (L x S) . ^ÉL Evidently, this energy is even under P but odd under Z. The magnitude of any such ¡r- is therefore again determined by the strength of the T-violating part of the weak interaction.
560
12 Macroscopic
Quantum Coherence
FURTHER READING Leggett A J 1978 J. Physique3g Colloq.C-g,vol. III, p:L264 (lroceedings of the --tSttt International Conference on Low Temperature Physics, LT-15) Leggett A J 1980 in Proceedings of the XVI Karpacz Winter School of Theoretical --Physics (Lecture Notes in Physics, Vol. 115), ed. A Pekalski and J Przystawa (Springer, Berlin), P. 13
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Author lndex
Note: page numbers in bold refer to reference section Aarts, J. 589: see also Steglich, F. 22 Abel, W. R. 28,45,561 Abragam, A. 23,394, 403, 561
Abraham, B. M. 576,586: see also Ketterson, J. B. 45. Roach, P. R. 104 Abramowitz, M. 278, 372, 561
Abrikosov, A. A. 34, 36, 59, 92,561 Abud, M. 165,561 Adams, E. D. 571: see also Flint, E. B. 365 Adenwalla, S. 542,561 Aharonov, Y.245,561 Ahonen, A. I. 94-6, 101-3, 188, 345, 363-4, 495, 497-8,561-2,572: see a/so Giannetta, R. W. 105
Ainsworth, T. L. 49,562,
5g
Akimoto, T. 575: see also Ichikawa, K. 488 Aldrich, C. H. 46,48,49,51, 55,57,562 Alexander, P. W. 592: see a/so Wellard, N. V. 473 Allen, J.F.2,562 Alonso, B. 521,562 Alpar, A.22,171, 5E5 Alvarez, L. W. I, 562 Alvesalo, T. A. 98, 106, 491, 561,562, 563,574: see ¿/.ço Ahonen, A. I. 363, 364.495: Archie, C. N. 106,430; Halperin, W. P. 98. 100 Amar, A. 56E: see also Davis, J. C. 435
Ambegaokar,Y. 14,117, 131,165, 185-6,193, 203,281 ,289, 395, 487 , 494,562 Amend. B. 570: see also Eska, G. 403 Amit, D. J.36,562 Anderson, A. C. 561: see also Abel, W. R. 28,45 Anderson, P. W. 6, 7,ll-12, 22-3 , 52, 54 , 66 , 73 , 82, 92, ll4, r23,136,150, 172,228-9,250,263, 265,266,280,289,293, 294, 320 , 337 , 339 , 346, 347 , 403, 483, 487 , 494, stt,524,550, 562-3, 565-6, 568-9, 5E4, 591: see also Blount, E. I. r95,196,203; Brinkman, W.F.127; Brueckner, K. A. 5; Engelsberg, S. 181, 183; Vollhardt, D.52,54 Andreev, A. F. 416,487, 488,495,563 Andres, K. 94,563 Andronikashvili, E. L. 109,
257,5ß'
574: see also
Hakonen, P. J. 109 Aniola-Jedrzejek, L. 5t6: a/so Roach. P. R. 98
.re¿
Anufriyev, Yu. D. 94,562, 563: see a/so Alvesalo,
T. A. 106 Arai, T. 291, 330, 563, 5E9 Archie, C. N. 106,490, 563, 574: see a/so Halperin,
w.
P. 38,95, 100 Ashburn, J. R. 593: see also
Wu. M. K.22 Ashida, M. 135, 495,533,
s44. ir63
Avenel, O. 105, 108, 380, 399, 494, 526, 542, 546, 563, 5E2,5E5,587, 590: see also Bloyet, D. 398;
Meisel, M. W.542; Piché,
L.
544; Salmelin,
R. H. 542; Varoquaux, 8.526,542 Babu, S. 49,563 Bagley, M. 335, 561,572: see also Gay, R. 107, 133, 335
Bagnuls, C. 478, 563, 569: see also Dombre, T. 478 Bailin, D. 128, 263, 280, 299, 563
Bais, F. A. 155,5E8 Balachandran, A. P. 251,563
Balatskii, A. V. 205,208, 563,591 Balian, R. 6, 7, 11, 80, 117,
128,165,167,59 Balinskii, A. A.234,5U Band. W. T. 580: see also
Main, P. C. 106,335 Baramidze, G. A. 431, 564 Baranger, M. 550,564 Bardeen, J.4,61,U,5U Bartolac, T. J. 381,564,565, 573 see also Bozler, H.
M.495; Gould, C. M. 377,381 Barton, G. ll4, 117-18,
120-1,157, r59, 165, 170,326, 494,59 Bashkin, 8. P.21.,59 Batenburg, P. 208,sEt Bates, D. M. 311,5(A Baym, G.26,36,59,497, 564.5E7: see also
596 Baym,
Author Index
G.-{ont'd.
Bowley, R. M. 479,565 Bozler, H. M. 102, 103,399,
Salomaa, M. M.497,
495,59,565, 573, 576,
498
Béal-Monod, M. T. 53,495,
5g
Bedell, K. S. 49, 51,59,123, 131,562,564, 5E6: see ¿/so Ainsworth,
T. L.
49
Bednorz, I. G.22,5g Berezinskii, V. L. I23, 495,
5& Berg, R. F.544,564, 569, 575: see a/so Engel, B. N. 546; Ihas, G. G. 541 Berglund, P. M. 565, 587: see also Bloyet, D. 398;
Salmelin, R. H.542 Berk, N. F.7,36,53,Sffi Bernier, M. E. 563, 565: see abo Avenel, O. 380, 399; Bozler, H. M. 102, 103, 399
Berthold, J. E. 106, 107,5(A, 584: see also Parpia, J. M.490 Bertsch, G. F. 46,564 Betbeder-Matibet, O. 438,
4gg,5g Betts, D. S. 24,56E,575,
579,5t7, 593: see also Dahm, A. J. 330,336; Hutchins, J. D. 330; Ling, R.-2. 330,336; Saunders, J.374 Beyer, J.491,5(A Bhattacharyya, P. 196, 298, 300-2,395,463, 479-9, 564,5E5: see also
Pethick, C. J.451,463, 465,466 Bialek, W. 559,564 Black, W. C. 561: see also
Abel, W. R. 28,45 Blaha, 5.292,5U Blatt, J. M. 550,564
Blount, E. I. 150, 195, 196, see qlso Brinkman, W. F. 186, 187, 364, 388
203,íffi,565:
Bloyet, D. 398,565 Bogoliubov, N. N. 68,524, 56s
Bohm, D.245,561 Boks, J. D. S. 495,594 Boldarev, T.^.2,576 Borisov, N. 5E3: see also Neganov, B. S. 93
Borovik-Romanov. A. S. 331,359, 402, 494, 565 Bouchiat, C. 557,565
57t: see a/so Bartolac, T. J. 381; Bates, D. M. 311; Gould, C.M.377, 381; Israelsson, U. E.
98,253; Lawson, D. T. 104
487-8,56é Buhrman, R. A. 574:
see also
Halperin, W. P. 95 Bullough, R. K. 577: see also Kitchenside, P. W. 384 Bun'kov, Yu. M. 359,392, 402, 565, 566, 574, 576:
Brand, H. 198, 416,431,46I, 492,510,565, 569, 5E6: see also
Buchholtz, L. J. 185, 289, 290,291,313, 369,394,
Dörfle, M. 461,
464
see also
Borovik-
Romanov, A. S. 331, 359,494; Hakonen, P. J. 254, 391,392, 394; Ikkala, O. T. 109,391 Burckhardt, T. W. 46,56ó Butcher, K. J. 578: see also Lea, M. J. 45
Bredl, C. D. 589: see aßo Steglich, F. 22 Brenig, W. 7,36,565 Brewer, D. F. 331,568,575, 579, 5E7: see also Dahm, A. J. 330,336; Hutchins, J. D. 330, 365; Ling, R.-2. 330,
Calder, I. D. 542,566, 5t2: see also Mast, D. B. 105,
336, Sachrajda, A. S. 490; Saunders, J. 374
Campbell, C. E. 47,59, 566,
Brinkman, W. F. 7, 11,
542 578
22-3,36,52-4,82,92, 102,114,123,126, l2g,
Candela, D. 582: see also Masuhara, M. 45,535,
736, L82,184, 186-8, 200,204,223,229,231, 263-6,289,337 ,348, 354, 358, 360-2,364, 369,384,385, 388, 400,
Capel, H. W. 135,5E3: see c/so Nijhoff, F. W. 135 Carless, D. C. 468, 471,491,
403, 473, 434, 482, 499, 505, 510, 5ll, 547, 562, 565, 56E-9,5E0, 5t4, 589: see a/so Blount, E. I. 195, 196,203; Engelsberg, S. 181, 183, 364; Osheroff ,D. D.32, 102, 103,283,345,364,
H. l8ó, 282,284,384,510 Bromley, D. J. 369, 374,375, 498,565 Brooker, G. A. 38,40,565, 387,510; Smith,
5E9
Brown, G.8.49,59,562, 563,565: see also Ainsworth, T. L.49 Bruder, C. l2l, 150, 161, 163, 167 , l7t, 5({t Brueckner, K. A. 5, 46,Sffi Bruinsma, R. 278, 287,3I3, 373,374,380-1, 56ó, s81
Brusov, P. N.495, 521,524, 526.547,56(t Buchal, C. 5E4: see also Owers-Bradley, J. R. 22 Buchanan, D. S. 489,56ó Bucher, E. 104,563
547
56ó
Carlson,
J.46,5U
Carneiro, G. M. 36,56ó,5E5 Carney, J. P.491,56ó Carton, J. P. 135,567 Castaing, B. 54,567
Castelijns, C. A. M. 491, 494,567
Caudrey, P. J - 577: see also Kitchenside, P. W. 384 Ceperley, D. M. 46,567 Chainer, T. 488, 504,567 Chapellier, M.572:- see also
Godfrin, H.495 Chechetkin, V . R. 228, 473, 523,567, 572 Cheng, 'f . P. 172, 567 Chervonko, E. 534,567 Chester, G. V. 579: see also
Lee,
M.4.46
Chocolacs, H. 5E4: see also
Owers-Bradley, J. R. 22 Choi, C. H.206,567 Chu, C. W. 593: see also'V,lu,
M. K.22 Clark. W. G. 572: see also Godfrin, H. 495 Clarke, G. R. 580: see also London, H. 93 Coates, K. F. 491, 567: see
Author Index a/so Castelijns, C. 434
597
A. M.
326,330 den Breems, A. 56é,5t3:.çee
Coffey, D.37,567 Coleman, S. 129,567 Collan, H. K. 562: see also Alvesalo, T. A. 106,491
alsoCapel, H. W. 135; Nijhoff, F. W. 135 de Bruyn Ouboter, R. 56t:
Combescot, M.511,533, 546,567 Combescot, R. 86, 195,
de Gennes, P. G. 15, 92, 182, 189, 196,227,231, 286, 289 , 298 , 319 , 337 , 511, 562, íffi: see also
205-6,208,251,395, 396, 398-9, 416,420, 428. 431,433,474, 4779,480,493,499, 503, 505,517,524,526,528, 532-3, 546, 563, 565, 567-9, Stl:. see also Avenel, O.542:' Bagnuls, C. 478; Bloyet, D. 398; Dombre, ^f . 478 Coombs, G. J. 58E: see also Scherm, R. 45 Cooper, L. N. 4, 5, 61, 564, 567,56t: see al.so Bardeen, J. 64 Cornog, R. 1,562 Corruccini, L. R. 31, 102, 105, 358, 361-2,399400,403, 509,56E,584: see also Osheroff. D. D. 102, 103,345,364,387 Cowley, R. A. 5E7-9: s¿e ¿/.ço Scherm, R. 45; Stirling, W. G.45 Crooker, B. C. 330,56t, 576: see a/so Israelsson,
. u. E.98,253
Cross, M. C. 10, 23-4,128, 146-7, 154, 157, tgz, 184, 188, 196-7,2(n, 203-6,210,223,231,
263-6,299,300-l,320, 327,337 ,354,369,394-
5, 401, 403, 407, 4lt-12, 414, 416, 420, 434, 4923, 492, 495, 4gg, 571, 533, 548,565, 56E, 580,
5E4,5E9: see abo Paalanen, M. A. 182 Czerwonko, J. 90, 449, 542,
5(ú
Dahl, D. A. 182,184, 56t Dahm, A. J. 330, 336,568, 575: see a/so Hutchins, J. D. 330 Daniels, M. 8. 542, 56t, SEE: see abo Saunders, J.542 Das, P. 93,568 Davis, J. C. 495,56E, 5t5: see also Pekola. J. P.
see also
Das. P. 93
Ambegaokar, V. 14, 185,186, 193,203,299, 487,494 de Vaard, A. 565: see also Borovik-Romanov. A.
s. 494 de Vegvar, P. G. N. 359, 402,546,56t, 587: see ¿/so Polturak, E. 546; Sagan, D. C. 98 Delrieu, J. M. 182,568 Dianoux, A. J. StE: see also Scherm, R. 45 Dmitriev, V. V. 565, 566: see ¿/so Borovik-Romanov.
A.
S. 331 ,359, 494; Bun'kov, Yu. M. 359, 402
Dobbs, E. R. 24,51ú, 57E-E0,
5ß:
see also
Daniels, M. E. 542; Lea, M. J. 45; Ling, R.-2. 542,544:' Saunders, J. 542
Dombre, T. 205-6, 208 ; 251,
416, 428, 43t, 478, 526, 528, 532, 563, 567-9: see a/so Bagnuls, C. 478 Doniach, S. 7, 36, 53,495,
564,569
Donnelly, R. J. 257,498, 569,572 Dörfle, M. 198, 461,464, 565, 569: see also Brand, H.46r Dörre, P. 590: see also Tewordt, L. 127,523 Dow, R. C. M. 321, 569 Duru, I. H. 311,569 Dy, K. S. 56,569 Dyugaev, A. M. 55,569 Dzyaloshinskii, I. E. 92,561 Eastop, A. D. 575: see also Hook, J. R. 287 Ebisawa, H. 272, 278, 344,
353-4,360,376-7 ,3956, 398, 401,477,492,
505,521, 546,563,567, 569,5E1: see also
Avenel, O. 105.526 Eckern, U. 438,569 Eckstein, Y. 586: see also Roach, P. R. 98
Edwards, D. O. 495,569, 570,575, 5E2: see also Feder, J. D. 98; Hoyt, R. F. 98; Masuhara, M. 45,535,547
Efetov, K. B. 128.569 Eilenberger, G. 72, I 18, 569 Einzel, D. 403, 454,456, 459, 462-4, 466, 469,
471,474-5,479_9,496, 489,490, 492-3,523, 544,590, 593: see also Tewordt, L. 127,523 Eisenstein, J. P. 98, 326,
330-1, 490,491,492, 569, 5E9:.ç¿e ¿/.ço Swift,
G. W. r82 Emery, V. J. 5, 6, 545,569 Engel, B. N. 32, 542,546, 569,575: see alsolhas,
G. G. s4l Engelsberg, S. 7, 36, 53,
tgt-3, 364,565, 569, 5t4, 5t9: see also Osheroff, D. D. 102, 103, 345, 364,387; Smith, H. 186,282,284, 384, 510 G. 102, 105, 403,479, 503.570-l: see also
Eska,
Einzel, D. 403; Fomin,
I. A.
403
Essmann,
U.
Fak. B. 58t:
109, 570
see also
Scherm, R. 45
Fal'ko, V. I. 488,570 Fantoni, S. 5t1: see also Manousakis, E. 47 Faraj, E. 575: see also Hook, J. R. 287, 289,492 Fay, D. '7, 127,570, 57E, 590: see also Tewordt, L. 127,523 Feder, J. D. 98, 569,570: see ¿Iço Edwards, D. O. 495 Feenberg, E. 47,53, 59, 570 Fetter, A- L. 23, 92, 135, 180, 184, 197-8,233, 239,246,248,256,259, 262-3,286-91,301-4, 308, 311-13,319-21, 326, 337 , 363-5, 369 , 371-2,393, 385,388-9, 403,497,499, 56ó, 570, 590,592-3: see also
Author Index
598 Fetter, A.
L.--{ont'd.
Freeman, M. R.364,
Vulovic, V.2.233,243, 389, 390 Feynman, R. P. 257, 327 262,
37'.1.,
-8,
337,570 Fisher, M. E. 329,57E Fishman, F. 290, 359, 361,
402,570-l Fishman, R. S. 45, 526,534, 532,571 Flachbart, K. 565: see also Borovik-Romanov. A.
s. 359 Flint, E. B. 365,571 Folk, R. 359, 361,570 Fomin, l. A. 23, 182, 186-7, 205,283,331, 358-9, 390,402-3,570-1, 573, s86
Fomin, N. V. 244, 262,589:-
Golo, V. L. 402 Ford, P. J. 574: see alsoF{all, H. E.93 Forster, D. 188, 434,57t Franz, W. 589: see also Steglich, F. 22 Frautschi, S. 165,571 Fredkin. D. R. 564: see also Béal-Monod, M. T. 53 Freeman, M. R. 3(A,495, see also
571
Friedman, L. 5E7: see also Sagan, D. C. 98 Frossati, G.572: see also
Godfrin. H. 495 Fujita, T. 238, 251, 259, 261, 495,571,5t3, 590: see ¿/so Nakahara, M.26L; Ohmi, T.237,245,254, 292;
Tsuneto,T.26l
Fulco, J. R. 575: see also Ho,
T. L. 208 Fulde, P.22,571 Furukawa, F. 576: see also Katayama, '1. 361, 4Oz
Z. 5,571 Gammel, P. L.46,107, 108, 190, 330, 332,334,566, 571.574: see also Hall, H. E.331 Gao, L. 593: see ø/so Wu, M. K.22 Garg, A. 206, 251, 572, 5t9: see also Stone, M. 205 Garibashvili, D. I. 574: see a/so Hakonen, P. J. 109 Gay, R. 107, 333, 335, 572 Geilikman, B. T. 473, 572 Germain, R. S. 571: see also
Galasiewicz ,
Giannetta, R. W. 102, 105, 381, 403, 54r,564,572: see also Berthold, J. E.
A. 8. 298, 572,
575: see ø/so Hoffberg,
M.22 Glyde, H. R. 46,572 Godfrin, H. 495,572,
SEE:
R. 45 Goldman, M.23,403,561 Goldstein, L. 55,572 Goldstein, S. 57t: see also Lawson, D. T. 103, 104 Goldstone, J. 501, 572 Golo, V. L. 150,291,353, 402, 403, 508,572,573 see also Scherm.
Gongadze,
A. D.
183,252,
253,254,277 ,288,363, 507,573 Goodkind, J. M. 104, 105,
584,sEt Gorkov, L. P. 22, 92, 150, 561,573,591 Götze, W. 593: see also Wölfle, P.497 Gould, C. M. 374,377 380-1, 564'573,576: see abo Bartolac, T. J. 381; Bates, D. M. 311; Israelsson, U. E. 98,253 Gould, S. G. 575: see also Gully, W. J. 182,399; Hook, J. R. 287,288, 482
Grabinski, M. 403, 493,573 Graham, R. 412, 414, 416, 431, 505, 565, 569, 573, 586:. see
43r Guernsey, R. W. 106,573 Gully, W. J. 96, 182,399, 565, 570, 573,57t, 584:
106, 107
Ginzburg, V. L. 127,572 Glaberson, W. I. 257, 572 Glassgold,
Castelijns, C. A. M.
49I,494; Coates, K. F.
495
¿/so Brand. H.
461; Dörfle, M. 461,464 Greaves, N. A. 183, 347,
498,573 Greytak, T. J. 96, 573,592: see also Webb, R. A. 98
Greywall, D. S. 28, 29, 31-2, 36,54,55, 59, 95-9, 573 Grilly, E. R. 589: see also Sydoriak, S.,"ç. 1 Gros, C. 5ß: seelalsp Seiler, K. 37, 55Gu, G.-Q. 416,573 Guckelsberger, K. StE: .see ¿Iso Scherm, R. 45 Guénault, A. M. 491,sffi, 567, 57t: see also Carney, J. P.491;
see
abo Bozler, H. M.
102, 103,399; Feder, J.
D. 98; Lawson, D. T. 103, 104; Osheroff, D. D. 18,99, 1(n, 101 Gurgenishvili, G. E. 431, 507,56/.,566,571: see alsoBaramidze, G. A. 431; Bun'kov, Yu. M. 392;Gongadze, A. D. 183,252,253,254,277, 288,363, 507 Gutzwiller, M. C. 52,53,573 Haavasoja, T. 561,
562:. see
A. I. 363, 495; Alvesalo, T. A. 98 Hahn, K. D. 57E: see also Kotsubo, V.326 ¿/so Ahonen,
Haikala, M. T. 561: see also Ahonen, A.f..94,96, 363,364 Hakonen, P. J.15,24, 109-11 ,129,233,242, 252,254,2&,285,335, 371, 388, 390-4,5(6, 574,576,5t5, StE, 591: see also Ahonen, A. I. 94,96; Bun'kov, Yu. M. 392; tkkala, O. T. 109,
391; Pekola, J. P. 330, 391,393; Seppälä, H. K.
241,243,390 Hall, H. E. 23, 93, ll1, 134, 190, 196, 205, 2l'1., 298, 320,327,331, 333-5, 337,416,478,427, 434, 473,49r,499, 563, 566, s7l., 572, 574-5, 577 , 5t0,592: see also Bagley, M. 335; Carless, D. C. 468, 471,491; Gammel, P. L. 107, 190, 330; Gay, R. 107, 133, 335; Hook, J.R.287, 288,482; Kiewiet, C. W. 106; Main, P. C. 106, 335: Wellard, N. V. 473 Halperin, W. P. 95,98, 100, 542, 547 , 5(ú, 574, 577 , 5t2,SEE: see also Calder, l. B. 542; Ketterson, J. B. 538, 542; Mast, D. B. 105,
Author Index 542; Meisel, M. W. 526, 541,544; Shivaram, B.
s. 526,542 Ham, T. 8.312,574,575: see also
Hu, C. R. 294
Hammel, E. F. 5t9: see also Sydoriak, S. G. 1 Hara, J. 4(4,499,574, 5t4: see abo Ono Y. A.4U
Harris, C. G.299,574 Hasegawa, Y. 247, 347, 403, 523,574 Hebral, B. 568: see also Crooker, B. C. 330 Hernadi, S.I. 46,572 Herring, C. 53,574 Higgs, P. W. 172, 574 Hilton, O. A. 589: see also Stirling, W. G.45 Hirai, A. 576, SEE: see also Ishikawa, O.401; Simola, J.T.1I0,263, 498
Hirashima, D. S. 534,574 Hirayoshi, Y. 569: see also
Einzel,D.492 Hirschfeld, P. J. 294, 569, 575: see also E,inzel, D. 492
Ho, T. L.17,196,210, 229-30,261,298-90, 321,331,333,416,4r920, 423, 434,5&, 572, 575,582: see also Bhattacharyya, P. 196, 298,300,301,302; Gammel, P. L. 107, 108, 130,322,334 Hoffberg, M.22,575 Højgaard Jensen, H. 38,40, 489,491-2,505, 569, 575, 590: see also Einzel, D. 486, 488,492 Hong, D. K. 129,575 Hook, J. R. 23, 111,134, 190 , 196, 205 , 211. , 287 8, 298,312, 327, 327 , 333-5,337, 416, 427, 434, 473, 482, 492, 499, 563,566, 569,572,5745,5E0, 592: see also Bagley, M. 335; Carless, D. C. 468, 471,491; Gay, R. 107,133,335; Main, P. C. 106,335; Wellard, N. V. 473 Hor, P- H. 593: see also Wu, M. K.22 Houghton, A. 464,575 Hoyt, R. F. 98,575, SE2: see
599 a/so Masuhara,
M. 45,
535,547
Hsu, W. 22,575 Hu, C. R.292-4,301, 304, 3r2-r3,331,360,416, 43r,434,574-5, 5t1, 5E7
Huang, K. 66, 172, 188,575:
Wu, M. K. 22 Huang, Z. J.593 Hubbard, J. 52,575 Hutchins, J. D. 330, 365, 56t, 57E: see also Dahm, A. J. 330, 336 see also
lchikawa, K. 488,575 Ihas, G. G.32,285,541, 542,544,564, 569, 575, 589: se¿ a/so Engel, B.
N. 546 Ikkala, O. T. 109, 391,574, 576: see ¿/so Hakonen. P. J. 109, 110, 129,399, 391,392,393 Ishii, C. 576,ífd/'593: see also Katayama, T. 359, 361,402; Ooiwa, K. 359; Yamaguchi, Y. 359 Ishikawa, M. 205, 416,576 Ishikawa, O. 401,576 fslander, S. T. 574, 576: see ¿/so Hakonen, P. J. 109, ll0, 129,388,391,392, 393; Ikkala, O. T. 109, 391
Israelsson, U. E. 98, 253,576 Jacak, L. 135,523,576 Jacobsen, K. W. 285,326,
388,576 Janke, W.326,576 Jarió, M. V. 150, 165,576 Jiang, Q. 580: see abo Lu, S.
T.
403
Johansson, B. 137,188, 582 Johnson, D. L. 508,576 Johnson, R. T. 106, 573,576, 585.593: see also
Greytak, T. J. 96; Paulson, D. N. 96, 103, 540,541:' Webb, R. A. 98; Wheatley, J. C. 93 Jones, D. R. T. 128,576 Jones, R. B. 117, 12L,165, 168,576 Josephson, B. D. 108, 494, 495,576 Kagan, M. Yu. 416,563
Kalos, M.
H. 46,567,579,
5E8: see also Lee,
M. A.
46
Kamenskii, V. G. 390, S71 Kamerlingh Onnes, H. 2, 4, 576
Kanamori, J. 52, 576 Kane, J. W. 562: see also
Amit, D. J.36 L.2, 4, 576 Katayama, T. 359, 36I,402, Kapitza, P.
576,584, 593: see also Ooiwa, K.359; Yamaguchi, Y. 359 Kats, E. I.402,508, 564, 572:. see
¿/so Balinskii.
A.4.234
Kaul, R. 128,576 Kawamura, K. 574,584: see also Hara, J. 464; Ono, Y. A.4& Keen, B. E. 45,576 Keesom, W. H. I,2,576 Keith, V. 573: see aßo Guénault, A. M.491 Keller, J. 571: see also Fulde. P.22 Keller, W. E. 59,576 Kennedy, C. J.573: see aßo Guénault, A. M. 491 Kesaev, V. I. 402,576 Ketterson, J. B. 45, 104-5, 538, 542, 547,561, íffi, 576-7,5t2, 5t6-8: se¿ a/so Adenwalla, S. 542; Calder,I. D. 542; Mast, D. B. 105,542; Meisel, M. W. 526,541,544; Roach, P. D. 497,498; Roach, P. R. 104; Shivaram, B. S. 526, 542
Khalatnikov, I. M. 34, 36, 59,327,407,416, 425, 434, S6t,577-E Khanna, F. C.46,572 Kharadze, G. A. 379,38L, 431,507, S&,sffi,573, 576-7: see also
Baramidze, G. A. 431;
Bun'kov, Yu- M.392; Gongadze,
A. D.
183,
252,253,254,277 ,288, 363,507; Ikkala, O. T. 109, 391
Khazan, M. V. 245,512,5I5, 518, 542,577,591 Khokhlov, R. V. 546,586 Khriplovich ,I. B. 559,577
Kieselmann , G. 488,577
Kiewiet, C. W. 706,577,
Author Index
600 Kiewiet, C. W.-{ont'd. 5t0: serr a/so Main, P. C. 106,335
Kim, J. S. 150, 165,571,577 Kim, N. SE3: see also Movshovich, R. 526 Kinsey, B. B. 583: see also Oliphant, M. L. E. 1 Kitchenside, P. W. 384,577 Kivelson, S. A. 550,5E6 Kjäldman, L. H. 326, 494, 577
Kleb, R. 585: se¿ a/so Sköld, K.45 Kleinberg, R. L. 99, 311, 575,576,577, 5t5, 5E7, 592: see a/so Johnson,
R. T. 106; Paulson, D. N. 103; Sager, R. E. I02,399; Webb, R. A. 101,102,103 Kleinert, H. 128, 301-4, 321, 324,326, 521, 526, 547 , 576-7 Kléman, M. 213, 216, 337, s90
Koch, V. E. 524, 533, 534, 54'1.,
544, 545, 577, 593
Kodama, T. 503, 561, 575, 577: see ¿/so lchikawa, K. 488 Kojima, H. 105,430, 503,
567,575,577, 579, 585, 586: s¿e ¿Iso Chainer. T. 488, 504, 505; Ichikawa, K. 488; Lu, S. T.403; Paulson, D. N. 96, 101 Kokko, J. 380, 497, 561, 562, 577:. see
ø/so Ahonen.
A. I. 101, 103,363,495, 497,498 Kondo, J.497,577 Konyshev, V. A. 208,5Á1, 591: see a/so Balatskii, A. V. 205.208
Kopnin, N. E. 261, 335,489, 495,577,591 Kop, T. 546,569,577: see alsoE,inzel, D.403 Korhonen, J. S. 5E3,SEt: s¿e ¿/so Nummila. K. K. 498; Simola, J. T. 110, 263,390,498 Korshunov, S. E. 495,577 Kosterlitz, J. M. 495,57E Kotsubo, V.325,578 Krotscheck, E. 47, 52, 57, 59,57E
Kruppa, D. 575: see also Hutchins, J. D. 365
Krusius, M. 336,561, 56ó, 574,577,5t5,SEE: see ¿/so Ahonen, A. I. 94, 95,96, 102, 188,345,
363,364; Bun'kov, Yu. M.392; Hakonen, P. J. 129,254,390,391,392, 394; Paulson, D. N. 103, 104, 336, 448,544,545; Pekola, J. P. 330, 391, 393; Seppälà,H. K.241, 243,390 Kubota, M. 584: see also Owers-Bradley, J. R. 22
Kuchnir, M. 5E6: see also Roach, P. R. 104 Kumar, P. 195, 272-3, 275, 277 -80, 365, 372, 37 6-8, 390-1, 575, 57E, stl, 591: see ¿/so Hu, C. R. 292,293,294 Kurkijärvi, J. 336, 377,488, 4W, 497, 499,570, 5778,590,593-4: see also Kjäldman, L,H.326, 494; Thuneberg, E. V. 498; WöIfle, P.497; Zhang, W.487 Kuroda, Y. 123, 127, 135, 57t Kurten, K. E. 47,57t Kurti, N. 94,578 Kyynäräinen, J. M. 98,57t, 587: see ¿/so Salmelin. R. H.542 Landau, L. D. 4, 25, 27, 28,
32,33,43, 50, 59, 78, l\3, 162,321, 577 , 578 Langer, J. S. 257, 329, 57E Lawson, D. T. 103, 104,573, 57E: see ø/so Gully, W. J. 96 Layzer, A.7,I27,57t Lea, M. J. 45,578 Lebedev, V. V. 416, 577,578 Lee, D. M. 7, 23-4, 92, 95,
lll,136,188, 374, 380, 39% 403,565, 568, 5724 ; SV9, 582-4, 586-7 : see
95,96,99, 100, 101; Polturak, E. 54ó; Sagan, D. C.98 Lee, M. A. 46,579 Lee, P. A.22,579 Leggett, A. J. 6, 10, 13-14, 18-19, 22,24,59,96, 90,92, 102, 123-4, 1289, 133, 136,146,154, 174-9, r82,188,244, 247 ,298,339-43,347 , 355,395-7 ,399, 401, 403,439, 479, 493, 4879,499,511, 546, 550-7, 560, 573, 578-9, 593 Lekner, 1.496,576 Leman, A. A. 402,403, 573: see also Golo, V. L.402 Levin, K. 37,53, 56, 59, 127, 131,135,136,579 Levy, M. 395,562 Lhuillier, D. 416,579 Li, L. F. t72,567 Liburg, M. 5t3: see also Neganov, B. S. 93 Lieke, W. 5t9: see also Steglich, F. 22 Lifshitz, E. M. 32, 33, 78, 92, 113,162,57E, 579 Ling, R-Z. 330,336, 542,
544, 579,5E0
Lin-Liu, Y. R. 188,277-8, 283,304,309,
31 1, 314, 376,387 ,577, 5E0, s8l, 591: see ¿/so Kleinert, H.302,303, 304;
Vollhardt, D. 309,310, 365,367,374 Liu, M. 10,144-7,154, I57, 159, 188,210,298,3(n2, 361, 407, 409, 4lr-r2, 414, 416, 418, 420, 431, 434,493,507, 509, 5ó8,
573, 5E0,589
London, F. 4,5t0 London, H. 93,5E0 Loponen, M. T. 562: see also Alvesalo, T. A. 106,491 Lounasmaa, O.V. 15,24, 104,337 , 494,561,562,
'aliöBozler, H. M. 102, l03l 399; Corruccini, L. R. 31; de Vegvar, P. G. N. 546; Giannetta, R. W. 102, 105,381; Gully,
574,5t0,5t3,5t5, 587: see also Ahonen, A. I.
w.
P. J. 109, 110, 129,399, 391, 392, 393; Pekola, J.
J. 96, 182,399;
Halperin, W. P.95; Lawson, D. T. 103, 104; Movshovich, R. 526; Osheroff, D. D. 7, 18,
, 96, 36? , 364 , 495 , 497,498:' Alvesalo, T.
94
A.
106,491; Hakonen,
P. 107,110, 190, 330, 391,393; Salmelin, R. H.542
Author Index
601
Love, A. 128, 217, 263, 280, 299,563,576, 5E0: s¿e
D. 128; Jones, D. R. T. 128 Lu, S. T. 105,403,504,5E0 Luey, K. 565; see also ¿Iso Bailin,
Bozler. H. M. 495 Lyden, J. K. 573: see also Guernsey, R. W. l0ó Ma, S. K. 564: see also Béal-Monod, M. T. 53 Maack Bisgaard, T. 575: see also Højgaard Jensen, H. 489.491
McClure, M. G. 205,579 McCoy, R. J. 573: see also Guernsey, R.
W. l0ó
Mclnerney, M. F. 135,5E0 McKenzie, R. H. 546,580 Magradze, O. V. 5E7: see ¿/so Salmelin, R. H. 542 Main, P. C. 106, 335,563, 577,St{l: see also Bagley, 335; Kiewiet, C.
w.
5E2
Markkula. T. K. 574: see also Hakonen, P. J. 109 Marquardt, W. 5E8: see also Schopohl, N. 533 Martin, P. C. 417,ítz Mast, D. B. 105,542,íffi, 582,SEE: see also
Calder, L D.542¡' Meisel, M. W.526; Shivaram, B. S. 542 Masuhara, M. 45, 535,547, 582
Mathews, P. W. 576: see also Keen. B. E. 45
Mattuck, R. D. 137, 188,5t2 Meisel, M. W. 526,541,542, 544,547,577, 5t2, 5E6, 588: se¿ ¿/so Ketterson. J. 8.538,542; Roach, P. R.98; Shivaram, B. s. s26,542 Mendelssohn,
106
Maki, K. 15,23,182, 188, 195, 208, 230,233,2434,252,263-4, 267 ,2723, 27 5, 277 -80, 283, 285, 287, 289, 291-3, 3 13- ls, 318,326,337 ,344,3536, 360, 365,372-4, 37683, 387, 388-90, 391, 395, 399, 401,403,464, 505, 507, 5t3, 521, 523, 546, 563, 566,569, 575, 577,5E0-1, 5E3, 591, 594: see ¿/so Hu, C. R. 292, 293, 294; Kleinert, H. 302, 303, 304; Lin-
Liu, Y. R. 304, 309, 3l I, 376; Vollhardt, D. 309, 370, 324, 325 , 365 , 367 , 374
Makrotsieva, V. 565: see also Borovik-Romanov, A.
s. 494 Mamaladze, Yu.
Salmelin, R. H.542 Manousakis, E. 47,581 Markelov, A. V. 489,494,
G.
109,
257,563 Mammiashvili. G. 5t5: see also Pekola, J. P. 330, 391,393 Manninen, A. J. 578: see also Kyynäräinen, J. M. 98 Manninen, M.'1. 326, 330, 336,491,562, 578, 5t0, 5E1,587: see also
Alvesalo, T. A. 98; Lounasmaa, O.
Y.
494;
K. 1,5t2
Mendoza, E. 580: see also
London, H. 93 Meng, R. L. 593: see also
Wu, M. K.22 Mermin, N. D. 17, 24,ll4, 117-18, 129, t3t,165, 169,
l7l-2,
192, 196,
205,2t0,213-14,220, 224,229,250,263,266, 281,289-90,292,293, 295,33r-2,337, 416, 419-20, 428-9, 434, 550-1, 562, 564, 575, 5t2: see also Bhattacharyya, P. 196,
298,300,301,302 Meschede. D. 59t: see also Steglich, F. 22
Metzner, W. 53,5t2 Meyerovich, A. E. 21,59
Michel, L.
l4l,
166-8, 188,
2t3,5t2 Mikeska, H. J.7 ,565:
se¿
akoBrenig, W.7,36 Millis, A. J.74,5E2 Mills, D. L.495,Sil Mills, R. L. 56t: see also Cooper, L. N. 5 Mineev, V. P.23,24, l4l, 150, 188,204-6,208, 213,217,220,223,2301,242,250-1,254,263, 265-8,270,278,280, 292,335,337,416, 433,
434,499,542, 550, 5ú3, 574, 5t2_3,597, 5gl_2: see also
Hakonen, p. J.
254,39r,392,394; Mermin, N. D. 220; Salmelin, R. H. 498.542
Misener, A.D.2,562 Mishra, S. G. 36,5t3 Miyake, K. 205, 416,420, 434,495,576, 5E3: see a/so Ishikawa, M. 416 Mizusaki, T. 576: see also Ishikawa, O. 401 Monastyrsky, M. I. 150,572: see also
Golo, V. L.291
Monien, H. 346,494,5E3 Montgomery, R. 359, 361, 5E3 Moody, J. 564: see also
Bialek, W. 559 Moore, M. A. ll4, ll7-18, 120-1,157, l59, 165, 170,326, 494, 563-4, 576: see ø/so Bailin, D. 128; Jones, D. R. T. 128 Morel, P. 6, 7, 12, 73, 82, 562,Sffi: see also Brueckner. K. A. 5 Morii, Y. 567: see also Chainer, T. 488, 504, 505
Movshovich, R. 526,583
Mueller, R. M. 571,SfM-: see a/so Flint, E. B. 365; Owers-Bradley, J. R. 22 Muething, K. A. 570: see ¿/so Feder, J. D. 98 Mühlschlegel, B. 75i 82,5t3 Mukharskiy Yu. M. 565,566: see also BorovikRomanov, A. S. 331, 359, 494; Bun'kov, Yu. M.359,402 Müller, K. A.22,5ø Mussett, S. G. 567, 573: see a/so Castelijns, C. A. M. 494; Coates, K. F. 491; Guénault, A. M.491 Muzikar, P. 22, 204-6, 208, 242, 416, 428-9, 434, 550, 567, 5t2-3, 5E9: see ¿/so Stone, M. 205
Nagai, K. 123, 127,135,
205-6, 416, 420, 428, 482, 499,503, 508, 521, 523,533,544, 563, 575, 583-4: see also Hara' J. 4641-Højgaard Jensen,
H. 489,491; Ono' Y. A.
4&
Author Index
602 Nagi, A. D. S. 57E Nair. V. P. 572: see also Garg, A. 251 Nakahara, M. 252, 285, 261, 291,381,388, 391, 495, 571,
stl, 5ß; see also
Fujita, T. 259,261, 495 Nakajima, S. 7,5E3 Namaizawa ,H. 523,574 Nambu,Y.437,524,5t3, 5E4
Nayak, V. S. 569: see also Edwards, D. O. 495 Neganov, B. S. 93,5E4 Nenonen, S. A. 580: see also Lounasmaa, O. V. 494
Neumaier, K. 569, a/so Eska,
570:. see
G. 102, 105,
479,503
Nijhoff, F. W. 135, 584: see alsoCapel, H. W. 135 Novikov, S. P. 572: see also Golo. V. L.291 Nozières, P. 26, 28, 33, 51, 54, 59, 438, 499, 550,
56/-.567.5t4, 5E5 Nummila, K. K. 498,5t3, 585,5E8: see also Pekola, J. D. 107, 110, 190, 330, 391, 393; Simola, J. T. I10,263,
390.498
Ohmi.'r. n7,
245, 254, 263,
291,292,571, 5t3, sEE,
Fujita, T. 251,259,261,495; Nakahara, M.261; Seppälä, H. K. 241 ,243, 59n: see c/so
Tsuneto,T.26l Oliphant, M. L. E. 1,5E3 Ono, Y. ^.464,574,5t4: Hara, J. 464 see also Onsager, L. 140,585 Ooiwa, K. 402,576,5E4: see also Katayama, T. 359, 390;
402
Osgood, E. B. 104,5t4 Osheroff, D. D. 7, 18,23,
32,95-6,99-103, 105, lll, 128,283,336,3456, 358, 361-2,364,387 , 399-400,403, 508,510, 565,56t, 569,573, 5E4: see also Brinkman, W. F. 186, 187,364,388;
Corruccini, L. R. 3l; Gully, W. J.96; Paalanen, M. A. 182 Østgard, E. 46,59, 5E4
Ostrowski, G. E. SEt:.çee a/so Sköld. K. 45
Ott, H. R.22,5U Owen, J. C. 47,5U Owers-Bradley, J. R. 22, 566,5E2, 5t4: see also Calder, I. D. 542; Mast,
D. B. 105,542
A. 182,336, 561.562, 577, 583, 5t4:
Paalanen, M.
Ahonen, A. [. 95, 101, 102, 103,345, 363,3(A, 495, 497 , 498 Packard, R. E. 98, 109, 331, 492,5(8,569, 585, 5E9, 593: see a/so Davis, J. C. 495: Eisenstein, J. P. 98, 326,49O:. Kokko, J.380, 497; Pekola, J. P. 107, ilO, 190, 326,330,333, 391,393; Swift, G. W. see also
499,562, 56/,-7,569, 571,5E7: see also
Aldrich, C. H. 46,48; Baym, G.497; BhattacharYYa, P. 196, 395, 463, 478, 479;
Brown, G. E.59; Fomin, I. A. 182; Salomaa, M. 497,498 Pfitzner, M. 22, 49, 51, 52, 56,57,58, 5t5 Piché, L.544,563, 5t5: see ¿/so Avenel , O. 542,546 Pickett, G. R. 566,567,573: see aßo Carney, J. P. 491; Castelijns, C. A. M.494; Coates, K. F. 491; Guénault, A. M. 491
182
Palmer, R. G. 280,292-4, 562
Palmeri, J. 489, 5t4 Pandharipande, V. R. 47,
581,5M: see also Manousakis.E.4T Panoff. R. M. 46,5U Parodi, P. 582: see also
Martin, P. C.417 Parpia, J. M. 106, 108, 330, 336.490, 493, 494, 569, 57E,5M: see also Kotsubo, V - 326 Passvogel, 'f . n7 , 245 , 246, 247,262,584, 5t5 Pati. J. C. 173,5E5 Patton, B. R. 58, 128,5t5 Paulson, D. N. 14,96, l0l, 103-4, 182,336,484, 540-1, 544-6,555,573, 577-t,585:. see also Greytak, T. J. 96; Kleinberg, R. L. 99; Kojima, H. 105,504; Krusius, M. 336 Pekola, J. P. 107, 110, 190, 326,330,336, 391, 393, 491, 56t, 57E, 5t0, sEl, 5t5, 587: see also Davis, J. C.495; Kyynäräinen, J. M. 98; Lounasmaa, O. Y. 494; Manninen, M.T.326: Salmelin, R. H.542 Pelizzari, C. A. 45,46, 58E: see also Sköld. K. 45 Penrose,
Pershan. P. S. 5E2: see also Martin, P. C.417 Pethick, C. J. 26, 36-7, 56, 59, 451, 463, 465-6, 483,
O.
140,5E5
Pines.
D.22,26,28,46,
48-9,51 ,55,57,59, t7l,562,564, 575, 5t5: see also Aldrich, C. H. 46.48 Pitaevskii, L. P. 5, 92,579, 5E6
H. 412, 414, 416, 431, 505, 510, 565, 573,
Pleiner,
5tó Pobell, F. 5M: see also Owers-Bradley, J. R- 22 Podd'yakova, E. V. 359,586 Polturak. E. 546, 56t, 572, 587: see also de Vegvar, P. G. N. 546; Giannetta,
R. W. 105; Sagan, D. C. 98
Poluéktov, Yu. M. 319,358, 506.5Eó
Pomeranchuk, I. 104, 5t6 Popov, V. N. 495, 521,524,
526,547,562,56 Price, P. B. 5E5: see aLso
Pekola, J.P.326 Privorotskii, I. A. 290,488, 571,5Eó Pushkina, N. I. 546,5tó Putterman, S. J. 434,5t6
Quader, K. F.49, 131,562, 564,58ó: see also
Ainsworth,
^f
. L.49
Rainer, D. 23, 74, 124, 136, 182, 185, 198,206,208,
286-7,298, 300, 438, 487-8. 495, 498, 499,
Author Index
603
52r,534,562, 566, 56E, 577-E,583,5Eó, 588, 590,594: see also Ambegaokar, V. 14, 185,186, 193,203,289, 487,494; Kjäldman, L. H.326,494; Thuneberg, E. V.498, Zhang, W. 487
Ramakrishnan, T. V. 36, 583 Ramsey, N. 559,586 Rapp, R. E. 593: see also
Wheatley, J. C. 93 Rasmussen, F.
8.
574: see
alsoHalperin, W. P.95, 98. 100 Reppy, J. D. 105, 108,257, 329-30, 336, 504, 563-4,
568,571,574,578, 5E4, 593: see ¿/so Archie, C. N. 106,490; Berthold, J. E. 106, 107; Crooker, B. C. 330; Gammel, P. L. 107, 108, 190,330,332, 334; Hall, H. E. 331; Parpia, J. M. 106,490 Rhodes, T. L. 490,584 Rice, M. J. 7,5E6 Rice, T. M. 53, 150,565,
579,stt, 59O: see also Lee, P. A.22; Seiler, K. 37,55 Richardson, R. C. 7, 23, 92,
95,102, ll1, 136, 188, 399,403,561-3, 565,
5ß,572-4,577-9,5E4, 5t6:
see ø/so
A. I.
Ahonen,
101, 103,363,495,
497,498; Archie, C. N. 106,490; Bozler, H. M.
102,103,399; Corruccini, L. R. 3l; Freeman, M. R. 364, 495; Giannetta, R. W. 105; Gully, W. J. 96, 182,3991' Halperin, W. P. 95,98, lfi); Kokko, J. 380; Lawson, D. T. 103, 104; Osheroff, D. D. 7, l8ì 95, 96,99,100, 101 Richardson, R. W. 575: see ¿/so Hoffberg, M.22 Rickayzen, G. 92,5Eó Riedel, 8.7 ,36,565, 5tó: see also Brenig, W. 7, 36 Ripka, G. 47,5U Roach, P. D. 497-8,576, 5E6: see ø/.ço Ketterson, J. B. 45, 104; Roach, P. R. 104
Roach, P. R. 45,98, 104-5, 547,576,586: see also Ketterson, J. B. 45, 104; Roach, P. D. 497,498 Robinson. F. N. 57E: see abo
Kurti, N.94 Rokhsar, D. S. 550,5t6 Rosen, G. 63,586 Roubeau, P. 574: see also Hakonen, P. J. 109 Rozhkov, S. S. 208,5t6 Ruderman, M. 575: see also
Hoffberg, M.22 Ruel, R. 430,5E6
Ruoff, M. 546, 563, 5E5, 5E7: see also Avenel, O. 542; Piché, L.544 Rutherford, E. 583: see also Oliphant, M. L. E. 1 Sachdev, S.582: see also
Millis, A. J.74 Sachrajda, A. S. 490,5E7 Sagan, D. C. 98,56t,5t7: see also de Vegvar, P.
G. N.546 Sager, R. E. 102,399,587, 592: see ølço Webb. R. A. 102,283,345,356, 360,479 Salam, A. 173,sts Salmelin, R. H. 263,498, 542,5t7 Saloheimo, K. M. 574: see ¿lço Hakonen. P. J. 109 Salomaa, M. M. 23,24,128, 150, 155, 234-7, 239-41, 243-8,250-4,256,259, 263,337 ,389,393-4, 403,499, 497-9, 542, 564, 569,573,5t2-3, 5E6-t, 592: see also Baym, G.497; Hakonen, P. J.129,254, 391, 392, 394; Salmelin, R. H. 498; Seppälä, H .K.241,243,390 Samalam, V. K. 545,5E7 Sanchez-Castro, C. 131, 564 Sandiford, D. J. 563,572, 5E0,584: see also Bagley, M. 335; Gay, R.
107,133,335; Main, P. C. 106, 335; Parpia, J.
M.490 Sarma, B. K. 543,561,56ó, Sn,5E2,5ß: see also
Adenwalla, S. 542; Calder, l.8.542: Ketterson, J. B. 538,
542; Mast, D. B. 105. 542; Meisel, M. W. 526. 541,544; Shivaram, B. s. 526, s42 Sartori, G. 166,561 Sasaki, Y. 576: see aßo
Ishikawa, O.
401
Saslow, W. M. 195, 301,304, 416, 431,434,493, 503, 511, 563,567, 575, 5E7:
Avenel, O. 542; Hu, C. R. 294 Sauls, J. A. 22, 45, 56, 123, 127 , 171,253, 526, 534, 542,546,570-1, SE0, 5E3,587: see also Fetter, A. L. 197,799,262; Muzikar,P.22 Saunders, J. 374, 541, 542, 5û,572,579, 580, StE: see also Dahm, A. J. 330,336; Daniels, M. E. 542; Giannetta, R. W. 105; Ling, R.-2.542, see also
544
Schäfer. J. 5E9: see also Steglich, F. 22 Schakel, A. M. J. 155, 208,
217.sEt Scherm, R. 45, 5t7-9: see aÍso Stirling, W. G. 45
Schertler, R. 571:
Fomin,I. A.
see also
358
Schmidt, K. E. 46,579,StE: see also Lee, M. A. 46 Schmitt-Rink, S. 550, 5E3 Schoepe, W. 561, 562,570,
571,577,5ß: see also Ahonen, A. I. 101, 103, 363, 495, 497, 498; Eska,
G. 102,105, 479, 503;
Fomin, I.4.358; Kokko, J.497; Simola, J. T. 110, 263,498 Scholz, H. N. 570,575,5t2: see also Feder, J. D. 98; Hoyt, R. F.98; Masuhara, M. 45, 535, 547
Schopohl, N. 180, 262,
ll8,
128,135,
3ll , 526-7 ,
533,542, 546,569, 5t4, 5t5, 58E, 59O,591: see a/so Passvogel,T.237, 245,246,247,262; Tewordt, L. l2l,533; Vollhardt, D.324,325 Schrieffer, J. R. 4, 7,36,53, 61,92,550, 564, 575, SEE: s¿¿ also Bardeen, J -
Author Index
604 Schrieffer, J. R.--Cont'd. 64; Ho, T. L. 208 Schwarz, K. W. 329,sEt
Seiler, K.37,55, sEt Seppälä, H. K. 233,241,243, 262,389,390, 574, 58E: see also Hakonen, P. J. 390
Serene, J. W. 23, 56, 124, 127, 136, 171, 198, 300,
438,495, 499,521, 523, 526, 534-5, 542, 546, 547,565, 571, 579, 5&3,
5t6-E: see also Brinkman, W. F. 127; Fomin, [. A. 182; Lee, P. A.22; Muzikar, P. 22; Sauls, J. A.22,253 Sergatskov, D. A. 565: se¿ ¿lso Borovik-Romanov, A. S. 331,494 Sessler, A. M. 5, 298,56E, 569,572:. see also
Cooper, L. N. Shahzamanian, M.
5
A.
481,
s88
Sham. L. J.579:. see also
Lee,P.4.22 Sharma R. G. 5E0,5t1: see ¿lro Lounasmaa, O. V. 494; Manninen, M. T. 326 Sherill, D. S. 5E2: see also
Masuhara, M. 45,535, 547
Shields, S. E. 105,58E Shigi, T. 575: see also
Ichikawa, K.488 547,577,582, 5EE: see a/so Ketterson, J. B.
538,542; Meisel, M. W. 526,541,544 Shopova, D. V. 359,571 Shumeiko, V. S. 465,469,
5tt
T.
107
, 110, 190,
263,330,390, 498, 574, 583,5E5, 5ß: see also Hakonen, P. J.129,254, 391,392,394; Nummila, K. K. 498; Pekola, J. P. 107, 110, 190,330,391, 393; Seppälä,H.
102. 381
H. 186,200,282, 284-5,326,358,360-1,
Smith,
362,379,384, 386, 388, 400,413, 483,499, 505,
510,564-5, 569, 575-7, 584-5, 5E9, 593: see also Bhattacharyya, P. 196, 395, 463, 478, 479;
Brinkman, W. F. l8ó, 787,3&,388; Einzel, D. 486, 488, 492; Højgaard, Jensen, H. 38,40, 489, 491, 492, 505; Osheroff,' D. D.32,283,387,510; Pethick, C. J. 451,463, 465,466; WöIfle, P. 497 Soda, T. 291, 330, 40I, 497, 563,56ó, 577,5t9: see ¿/so Brueckner, K. A. 5 Soinne, A. T. 562: see also Alvesalo. T. A. 98 Sokolov, A. I. 128,5E9 Solomon, A. I. 118,5t9 Soni, V. 294,589 Sonin, E. B. 244, 254, 262, 337 , 381 , 399, 416, 491 , 5t9 Spencer, G. F. 285, 56ó,5E9: see also Carless. D. C.
K.241,
243,390 Simon, F. 57E: see also Kurti,
N. 94 sköld, K. 45, 46,5EE: see ¿/so Scherm, R. 45 Skrbek, L. 588: see also
Spohr, R. N. R.
also
198,262: Sauls, J. A.22,
Volovic,Y.Z.233,
243,262,371, 389, 390 Steinback. M. 573: see abo Guernsey, R. W. 106 Stern, M. R. 508,5t0,5E9 Stewart, G. R. 22,5E9 Stirling, W. G. 45, 46,588, 5E9: see ¿/so Scherm. R.
45i.
Schopohl, N. 546 Struik, D. J. 349,589 Sun, Y. 493,5E9 Swift, G. W. 98, 182,56Í, 569,5t9: see also Buchanan, D. S.489; Eisenstein, J. P. 98, 326, 490
Swithenby, S. J. 587: see also Saunders, J.374
Sydoriak, S. G. 1,5t9 Sykes, J. 38, 40, 565,5E9
Taconis, K. W. 5(ß: see also Das, P.93 Tagirov, M. S. 5E0,5E1:see ø/so Lounasmaa, O. V.
494; Manninen, M. T. 326
Takagi, H. 434, 507, 5t3, 589: see a/so Miyake, K. 416
Takagi, S. 71, 87-8, 177,
205,286,301,340,344, 346, 363, 365 , 395-7
,
399, 401,403, 478, 483, 495, 499,511, 563, 579-
E0,5t!) Takano, Y. 561, 56i2,577: see a/so Ahonen, A. [. 101, 103, 363, 495,497,498;
Tankersley, L. L. 182,590 Te5anovió,
SES: see
Pekola, J.P.326 Sprenger, W. O. 584: see alsoPaalanen, M. A. 182 Stamp, P. C. E. 53,5E9 Stare, C. l14, 117,129,165, 169,171, 5E2-3 Steglich, F.22, SEg Stegun, l. A. 278, 372, 561 stein, D. L. 294, 495, 570, 575,5t7, 5E9,592: see ¿/.ço Fetter, A. L. 197, 253;
Stone, M. 205, 572,5E9: see also Garg. ^.251 see also Streckwall, H. 588:
Kokko, J.380,497
491
Spohr, D. A. 57t: see also
Kurti, N. 94
Shivaram, B. S. 526, 542,
Simola, J.
Simola, J. T. 110, 263, 390,498 Smith, E. N. 403, 572: see ¿Iso Berthold, J. E. 106, 107; Giannetta, R. W.
Z.
494, 495, 590
Tewordt, L. 127 , 180, 311, 344,346, 494, 521, 523, 526-7,533,542,5E3-5, SEE.590: see øLço
Passvogel,'1. 237, 245, 246, 247, 262; Schopohl,
N. 526, 533,542,546 Theodorakis, S. 246, 248, 256,385,388, 570, 590 Thompson, K.574: see ølso
Hall, H. E.93 Thomson, A. L. 565: see also Bozler, A. L.495 Thouless, D. J. 5,495,578, 590
Thoulouze. D.572:
see also
Godfrin, H.495 Thuneberg, 8. V.248-9, 251, 336,391,394, 498, 571,590, 594: see also Freeman, M. R.364, 495;'Zhang, W.487
Author Index
60s
Tilley, D. R. 24,590 Tilley, J.24,Sm Tinkham, M.,92,590 Tkachenko, V. K. 258, 259, 590
Topsoe, F. 489,590 Torizuka, K. 578,587: see also Kyynàräinen, J. M. 98; Salmelin, R. H. 542 Torng, C. J. 593: see abo Wu, M. K.22 Tough, J. T. 251, 257,329, 590
Toulouse, G. 213, 216, 228, 250,266,337, 563, 59{l Träuble. H. 109,570 Truscott, W. S. 374,56E, 575,5E7, 59O:. see also Dahm, A. J. 330,336; Hutchins, J. D. 330; Sachrajda, A. S. 490; Saunders, l-374 Tsakadze, J. S. 574: see also Hakonen, P. J. 109
Tsubota, M. 576: see also Ishikawa, O.401 Tsuneto, T. 238, 261, 272,
278,355-6,376, 505, 521.571,5t3, 590: see ø/.so Fujita,'1. 251, 259, 261,495: Nakahara, M. 261; Ohmi, T.237,245, 254,292
Tvalashvili, G. K. 5t7: see ¿/so Salmelin, R. H. 542
A.1.74 Varoquaux E. 108, 494,526, 542,546,5ú1, 565, 5t23, 585, 5E7, 590: see also Avenel, O. 105,380, 399, 526, 542, 546¡' Bloyet, D. 398; Meisel,
M. W.542; Movshovich, R. 526; Piché, L.544; Salmelin, R. H. 542 Vdovin. Y. A. 6,80, 524, 590
Veuro, M. C. 561, 562: see ¿/so Ahonen, A. I. 495; Alvesalo, T. A. 106,491 Vibet, C. 5úi, 565:. see also Avenel, O. 380,399; Bloyet, D. 398 Vinen, W. F. 257, 328,590 Vollhardt, D. 24, 52-4, 59, l2l, 150, 16l, 163,167, 171, 309-10;, 314-15, 317- 18, 324-6, 349, 365, 367,374-5,381, 382, 394.5ffi,5t0, 582, stE, 590-91: see also Lin-Liu, Y. R. 304. 309, 3l1,376:' Seiler. K. 37,55; Tewordt, L.533 Volovik, G.8.21, 23, 150, 155,204-6,208,213, 217.220,223,230-t, 233-7 . 239-48, 250-4, 256,259,261,263-8, 270,278,280,285,292, 322, 333, 335 ,
Ueda, K. 150,5tt, ø/so Seiler, K. 37,55 Ugulava, A. l. 402, 576, 590 590: s¿e
Uhlig, K. 570: see a/so Eska, G. 102, 105,479,503 Ullah, S. 326, 369,372,570, 590
Unal, N. 311,569 Usmani, Q. N. 581: see also Manousakis , E. 47 Usui, T. 205, 420, 434, 576, 5t3: .çee ø/.ço Ishikawa. M.416; Miyake, K.416 Valatin, J. G. 68,590 Valls, O. T.37,53, 56, 59, tzl. l3t,135, 136,494, 5?9,590 van Roosbroeck, W. 5t4: s¿e ø/so Osheroff, D. D. 32, 283, 387, 510; Paalanen,
M. A. 182 Varma, C. M. 117,339,341, 5lxi, 590: see alço Millis,
337
, 37
389-91, 393-4, 403,
I,
4 16,
433, 434, 489,495,496, 511,5t2,515, 518, 542, 550, 5Ái-4, 574, 576, 5E2, 583, 5t5, 587, stt,
591-3:
see
abo Balatskii,
A. V. 205,208; Balinskii, A. A.234:' Hakonen, P. J. 109, ll0, 129,254,388, 391, 392, 394,399; Ikkala, O. T. 109,391; Mermin, N. D. 220;Pekola, J. P. 330, 391, 393; Seppälä, H. K.
241,243,390 Vulovic, V . Z. 233, 243, 263, 369,371, 3t9-90, 592 Vuorio, M. 186-7, 283, 399, 498,571,5Eó, 592 Wagner, H.562:
see also
Amit, D. J.36 Walecka, J. D.92,570 Wang, Y. Q. 593: see ølso
Wu. M. K.22 Ward, P. L. 568,588: see also Daniels, M. E. 542: Saunders, J.542
Warkentin, P. A. 5E7: see ølro Sager, R. E. 102, 399
Warnke, M. 584,Stt:
see
also Passvog el , '1 . 262;
Schopohl, N. 526,542 rilartak, M. S. 533, 534,592 Webb, R. A. 98, l0l-3,283, 345, 356, 360,399,402, 479, 576-7, 592: see also
Johnson, R.
T.
10ó;
Kleinberg, R. L. 99 Wellard, N. V. 473,592 Wennerström. P. 562: .ree a/so Alvesalo. T. A. 106 Werthamer, N. R. 6, 7, I l, 80, 117, 165,167,341, 5ó4, 590
Wheatley, J. C. 4, 14,23-4,
30-1, 36, 54, 59, 92,93, 95,98-9, 104, il1, 136, 182,336, 402, 403, 484, 534, 544, 546, 547 , 555, 561, 566,573, 576-8,
5t5,587, 592-3: see also Abel, W. R. 28,45; Buchanan, D. S. 489; Greytak, T. J. 96; Johnson, R. T. 106; Kleinberg, R. L. 99; Kojima, H. 105,504; Krusius, M. 336; Paulson, D. N.96, l0l, 103, 104, 336,484,540, 541,544,545; Sager, R.
8.102,399; Webb, R. A. 98, 101, 102, 103, 283,345,356,360 Whitham, G. B. 376,593 Wiedemann, W. 570: see also Eska, G. 102, 105,403, 479,503 Wilczek, F. 496, 564, 575, 593: see a/.ço
559; Ho,
Bialek, W.
T. L. 208
Wilkins, J. W. 575, 579: see also Højgaard Jensen, H.38,40; Lee, P. A.22 Wilks, J. 24, 59, 328, 576, 593: see ¿lço Keen, B.
8.45 Willers, H. G. 570: see also Eska, G.403 Williams. G. A. 109,563, 590,593: see abo Avenel, O.542;
Author Index
606 Williams, G. A.-{ont'd. Varoquaux, E. 526,542 Williams, M. R. 259,303,
311,319,570,593 Wiringa, R. B. 47,584 Witten, E. 173,593 Wojtanowski, W .73, 534, 58{1, 593: see also Ling, R.-2.544 Wölfle, P. 20, 23, 43, 45, 49, 5l-2, 56,58,59,73, 103, 194, 203,394, 438, 454, 457 , 462, 464, 470, 497 , 499,503, 51 1, 520, 524, 521, 529,534, 536, 540-7,569, 570-1, 575, 577,5t3, 5t5,5E9, 591, 593: see also E,inzel. D. 403, 486, 488, 492; Eska,
G. 105,503; Fomin, I. A. 403; Højgaard, Jensen, H. 489,491, 492,505; Sun, Y. 493; Vollhardt, D.52,54 Wolfke,
M.2,576
Woo, C. W. 47,53, 593 Woods,
A. D. B.
SEE: see
a/.so Scherm, R. 45 Wu, M. K.22,593
Yakovenko, V. M. 495,496, s92
Yamaguchi, Y. 359,361, 576,593: see also Katayama, T. 359, 361, 402
Yamasaki, H. 575: see also Ichikawa, K. 488 Yan, S. S. 568, 5t7: see also de Vegvar, P. G. N. 546; Sagan, D. C. 98 Yang, C. N. 140,593
Yanof, A. W. 105,504,593 Yanof, J. M. 5t4: see also Parpia, J. M. 106 Yarmchuk, E. J. 109,593 Yip, S. 129, 205, 247, 489, 499,579,5E9, 593, 594: see aho Sun, Y. 493 Yosida, K. 89,593
,t¡
i' i. !;..
Ytterboe, S. N. 564: see also Bates, D. M. 311 Zakrzewski, S. 217,5E0 Zaringhalam, A. 58, 565,
5t5: see ¿/so Brown. G. E. 59 Zee, A.496, 593 Zeise, E. K. 572,5tó: se¿ abo Giannetta, R. W. 105; Polturak, E. 546 Zhang, W.487,490,594
Zhao,Z.56l:
see also
Adenwalla, S. 542
Zhu, Y.-Q. StS:
see ølso
J.P.326 Ziercher, E. L. 587: see also Sagan, D. C. 98 Pekola,
Zimmermann,W. 492,575 Zinov'eva, K. N. 495, 594 Zotos, X. 230, 233, 243, 263, 2&,390,5E1,594 Zwicknagl, G. 185, 487,566, 571: see ¿/so Fulde, P. 22
Subject lndex
A
phase 7, 95 see also ABM state Bogoliubov-quasiparticle relaxation rate 46L collective modes 526-33, 515-19,
543-6 definition 7,95,8L dipole interaction 178 dissipation of superflow 331-6 finite normal density at T :0 206-8 free energy 120 gap parameter 82,'/-.L6 gradient free energy 195-8 hydrodynamic equations for 4t6-29 invariance properties of L64 Leggett equations for 348-9 linear defects 223-33, 240-5 longitudinal NMR linewidth 399 in a cylinder 288-9 in a magnetic field 430-2 in a slab 286-8 in a sphere 292-4 magnetic susceptibility 87, 101, 133 NMR 348, 355, 363, 365-83 NMR in rotating superfluid 389-90 nonlinear spin dynamics 355-9 normal fluid density 84 orbital magnetic moment of 553-5 order parameter 7,81, L48 orientation induced by an electric or magnetic field, and by superflow 182
pair breaking critical currents 326 permanent electric diPole moment 5s9
permanent orbital magnetic moment 553-5
persistent currents L07, 203, 294-320, 331,-7
planar defects 267-70 planar soliton 272-8 point defects265-6 rotating 109-1 1, 259-63, 389-90 solitons in 376-83 sound propagation 102, 502, 542-5 spin dynamics 342-5, 348-59 spin susceptibility 87, L01, 133 spin-wave mode 506 supercurrents L6, 203-8, 294-321, 331,-7
superfluid density 87, 198 superfluid helium 3 26L-3 surface energies and boundarY conditions 186 symmetry classification of collective modes 515-18 symmetry of order parameter 148, 158 textures 223 -33, 240- 5, 286-32L texture-induced NMR shifts in362-
76,389-90 thermodynamic identities for 420-3 transport coefficients of 394, 4L6-34, 479-84 vortices in 223-33, 240-5 A, phase 96,98,99,130-2, L59, L60, 169, L1l definition 7,148 longitudinal NMR resPonse 346 hydrodynamics 430 nonlinear spin dYnamics in 359 order parameter 7, L32,'l'48 second sound in 508 specific heat t32 spin-wave modes.507-8
608
Subject Index
Ar phase 96, 98, 99, L30, 133, 158, 430 hydrodynamics 430-2 order parameter 7, L30 specific heat 133 spin-wave modes 507-8 A, state lzt,169 see also At phase At state L21 see also A2 phase A transition 95, 96 A-B interface 489 A-B phase transition 99,128,247 ABM state 6, 7-8,12,157 see also A phase anisotropic gap parameter 83 equal spin pairing state 82 specific heat discontinuity for 84 spin susceptibility of 89 Absolute temperature scale 95 Acoustic shear impedance 105 Adiabatic demagneti zation 93 - 4 Aharanov-Bohm interference effects 245 Alpha (ø) state 164, L68, L69,I72 Anderson-Bogoliubov mode 524, 540
Anderson-Brinkman-Morel
see
ABM
state
Anderson-Higgs mechanism 172 Anderson-Toulouse (AT) textu re 228, 230,233,26L lattice 262 nonaxisym metric 243 Andreev equations 488 Andreev scattering at boundaries 48690
Anisotropic superfl ui dity 7 Attenuation of sound L03-4,540 see also sound attenuation Average energy dissipation of spin motion 400 Axial state see ABM state, A phase Axiplanar state249 Axisymmetric vortice s 234, 235-8, 240-
1,245-9,252-6 B phase 7,98,99, 160, 169 see also BW state Bogoliubov-quasiparticle relaxation rate 457-6L collective modes 502-IL, 512-15, 524-7 , 534, 539, 546 definition 7 , 8I, 95, 147 dipole interaction L79-80
dissipation of superflow 329-31 electric dipole moment 555-9 free energy 120 gap parameter 82,116 gradient free energy L99-200 hydrodynamics equations for 41,1,-L6 hydrodynamics for finite magnetic freld 432-3 in a cylinder 283-5 in a slab 28L-3 Leggett equations for 348-9 linear defects 220-3, 233, 245-57 , 259-63 magnetic susceptibility 89, 102 NMR 348, 359, 363, 383-9,390-4, 40L NMR in rotating superfluid 390-4 nonlinear spin dynamics 359-61 nucleation of L29 order parameter 7, 80, 147 orientation induced by magnetic and electric field and by superflow L81, 183-5 pair breaking critical currents 323-6 permanent electric dipole moment s55-9 persistent currents I07, 208. 323, 329 planar defects 270-2 planar soliton 278-80 point defects 264-5 sound attenuation in 104, 502,541 sound propagation in 502, 538-42 specific heat 83-4 spin dynamics 353-5 spin susceptibility 89, L02 spin-wave mode 508-10 supercurre nt 203 , 208 , 323 , 329 superflow dissipation in 329-31 superfluid density 87, L94 surface energies and boundary conditions 186-8 symmetry classification of collective modes 512-15 textures 220-3, 233, 245-57, 259, 264,
279,290-6 texture-induced NMR shifts 362-75. 383-8, 390-4 thermodynamic identities for 412 transport coefficients 464-7 5 vortices in 245-5L topological propertie s 220-3 B, phase (planar phase) 98, L34, B, phase (B phase in magnetic field) 98, 99, 134, L35, 157, 170, I7g, 1gl, 182.347
1
Subject Index
order parameter 7,134 B-transition 95. 96 Backflow 48 Balian-Werthamer, see BW state, B phase
Ballistic quasiparticle regime 461, 484, 491.
Barnett effect 433 BCS pair amplitude, properties of 67, s50-3 BCS pairing theory 61 BCS superconductor 141,-2 BCS theory 4,5 BCS weak-coupling theory 64 Bend spatial change of order parameter 196
Bending 189 of the order-parameter field 15, 196 Bending free energies L95, 199,206, 21L,238 in A phase 196-8 in B phase 199-200 Berezinski-Kosterlitz-Thouless vortexbinding transition 495 Beta (B) state 159, 169 Binary collision processes 450-3 Bipolar state 164, 168 Bogoliubov quasiparticle (BQP) 76, 85, 206 collision integral lor 454-6 definition 69 distribution function 72 energy change in 443 energy spectrum 70 kinetic equation lor 442-5 mean free path 46L,484 relaxation rate 454, 457 -61, scattering of, off at boundary 487 velocity 443 Bogoliubov transformation 68, M2 Bohr-Sommerfeld quantization rule 209 Boltzmann equation 443 for Bogoliubov quasiparticles distribution functions 437 for normal phase 38 Boojum 250, 266, 293, 294, 331, 332, 334 Bose condensate of diatomic molecules 205 Bose-Einstein condensate 4, 66, L40, 549
Bose-Einstein statistics 4 Bose system 66 Bound quasiparticle states 488, 498
609
Boundary condition 211 on the distribution function 486 Broken symmetries 9-12, 137-74, L3g, 14L-9, 179, 1gg chiral symmetry 306 discrete symmetries 162-5, 243-4 gauge symmetry 65, L41,189,327 ,
4t0 in BCS theory 141 in ferromagnet 142 in high-energy physics 172-3 in superfluid 4He 141 relative gauge-orbit symmetry 295 relative symmetry 144-7, L54,173 rotational symmetry in orbital space l44,4Ll in spin space L42,144,4L0 spin-orbit symmetry 11, 19, L46,L54, 339
Bulk energy 238 Bulk viscosity 33, 4L5,467-7L Bulk-viscosity coefficients 426, 480 BW state 6, LL,80-2 see also B phase Catastrophic instability 320 Chaotic motion 359 Chemical potential 26, 406, 468 Chiral symmetry, breakdown of 302 Cholesteric liquid crystals in a magnetic field 319 Circular disgyration 289 Circulation 16, 208, 209, 214, 226 Clapping mode 245, 5ll, 544 Coherence length 64, 195 Collective modes L8, 42, 245, 501-46 in the A phase 502-L1,515-19, 527-35 in the B phase 502-LL,512-15,524-7 collision-induced broadening 534 frequencies, splitting 526 observability of 534-5 residual interactions, effect of 533-4 spectrum, multip^let structure of 5L2 Collision integral (l) 33,37 , 450-61. for Bogoliubov quasiparticles 454-6 conservation properties of 453 for normal quasiparticles 37 linearized 38-40,452 Collisionless sound 535-46 see also zero sound Complex order parameter 6L, I90, 549 Composite bosons 549 Composite soliton 273,275-7 ,377
Subject Index
610 Composite soliton-Cont'd. creation of273 global instability of 316 Compressibility 30-2,502 Condensation energy 79 Conservation laws 32,405 Conserved densities 27,440 Continuity equation 32 Continuous degeneracy of ground
Diffuse boundary scattering 1.85, 484, 487
Diffusion constant 497 Dilution refrigerator 93 Dimensionality of defects 2L6 Dipole coherence length 20L,217 Dipole coupling parameter 177 Dipole-dipole interaction 13-14, 102, state 169, L74-9, 18L, 201 65,272 in A phase 178 Continuous mappings, theory of 213 in B phase 179 Continuous symmetries 150-60, 407 Dipole length 259 Conventional superconductors 74 Dipole-locked regime 218,231 Cooper instability 61-4 Dipole torque L9,342,358, 505 Cooper pair 4, 63,I37 Direct interaction 49,51 as diatomic molecules 551 DisclinationZZL internal degrees of freedom 9 Discontinuity of specific heat C" 83, internal structure 8-9 L32-3 Cooper-pair susceptibility 521 Discrete degeneracy 267 ,306 Cooper-pair wave function 64 of the ground state 270 Coreless vortices 333 Discrete symmetries 148, 150, 158, Correlated basis function (CBF) 47 162-5,223,267 Correlation length 64 DisgyrationZ42 Couette flow 485 circular 289 Critical angular velocity 258 Dissipation of superflow 326-36 Critical current 110, 309, 370,3L2, 3t7 , Domain wall 208, 2L2,267 ,268 definition 15 327 edge222 Critical field 135 Critical supercurrent 315 effect of boundaries on277 effect of superflow on 313-19 Critical spin supercurrents 331 global instability of 318 Critical superfluid velocities in the A and B phases 108,32L-6,329-36, lattice of 270 491 local stability of, for small currents Critical temperature 76 3L6 planar 269 Current density 27,193,44I propagating3T6 Cutoff energy 74,76 regular array of320 spontaneous formation of 318 d vector in A phase 82, 148 Double-core vortex 248,249 d(k) vector 70 Double-half disgyration 289,29I,373 dsoliton 273,275,377,378 d-wave pairing L14,170 Double sine-Gordon equation 384 Defects of order-parameter field Doubly quantized nonsingular vortices dimensionality of 216 390 linear 219-57 Duhem-Gibbs relation 422 planar 267-80 Dynamical pattern selection 321 point 263-7 Dynamical structure factor 497 topological investigation of 2LL-I9 Dynamical symmetry variables 418 topological equivalen ce 21ó Degeneracy space of order parameter Edge of a domain wall222 141., 212, 2L7 , 219, 220 Effective Hamiltonian 72-3, 341, 369 see also order parameter space Effective mass 27, 54 Density of states 26 Effective-mass relation 29,31 Density response function 48, 535-8 Effective pair interaction 73 Diatomic molecule 92,55I Effective-potential approximation 55
ffi
IË
IF
Subject Index
611
i,
i Ë,
Effective relaxation time 398 Effective-viscosity coefficient 489, 490 Effective viscous penetration depth 505 Einstein-de Haas effect 433 Einstein relation 497 Electric dipole moment (EDM) of a Cooper pair 558 induced 781-2 permanent 555-9 Electrohelicity 558 Electron-nucleon interaction 557 Elementary excitations 25 Energy-conservation equation 32 Energy current 34, M[,448 Energy density 440,448 Energy gap, drawing of 81 Entropy 28-30, 7 6, 407 -8, 420 Entropy current 408, 415 Entropy-production function 415 "( phase" 157 Equal spin pairing state 81 Equation of motion for hydrodynamic variables 32,341, 409 derivation of 340-2. 406-9 for intrinsic angular-momentum 423 for 1423.427 for spin density 341, 415, 505 for spin-orbit angles 414 for spin superfluid velocity 430-3, 505 for symmetry variables 409-11 for u. 4I3,427 e state 170 Euler angles 366 Euler characteristic 280 Euler-Lagrange equations 27 4, 367 Exchange symmetry 51 Extended Fermi-liquid theory 49-52 f-wave pairing 114 Fermi-Dirac statistics 4 Fermi energy 4,26 Fermi function 28 Fermi gas 4,25 Fermi liquid 25-60,84 see øbo quasiparticles collective modes 42-6 compressibility 30 extended theory 49-52 Landau parameters 27 , 29,31 Landau theory 25-46 specific heat28 spin diffusion coefficient 36, 42 spin susceptibility 30 thermal conductivitv 36, 42
thermodynamic properties 28-32 transport properties 32-42 viscosity 36,41 Fermi-liquid corrections 85, 89, Ig7,
rgg,200 Fermi-liquid interacti on 6, 27, 86, 89, 198,323 Fermi momentum 25 Fermi statistics 25 Fermi temperature 4 Fermi velocity 26 Ferromagnetic spin fluctuations 6-7 Field-turn-off experiment 399 Finite normal density at T :0 in A phase 206-8 Finite orbital magnetic moment 554 First sound 42,502 Flapping mode 529 Flare-out texture 284 Flow in restricted geometries 484-98 Flow-induced domain-wall lattice 318. 382 Flow-induced dynamical textures 3192L
Flow-induced precessing domain walls 327
Flow-induced soliton lattice 382-3 Flow through narrow channels 326 Fountain pressure 105 Fourth-order invariants 1 14 Fourth sound 105, 491, 504, 513 Free energy 76-9, I43 invariants of l22 symmetry and stationary points of 165-9 Free energy per unit length of a vortex, 239,252 local minimum of 301 Free-induction signal 358 Freedericksz transitio n 286 Galilean invariance 28, 421 Gap Á in the excitation spectrum 70, 83, 84 average magnitude 79 Gap equationT3-4,75 Gap parameter'10, 73, 7 4, 75, 82-4, 135 of the A phase, anisotropic 83, 120 for the B phase 83,120 near 7i 79,82-3, l\7 at zero temperaturc 79, 82-3 Gapless modes 501 Gauge invariance 65
Subject Index
612 Gauge-invariant densities and currents 447-50 Gauge-invariant distribution fu nction
45
Gauge-invariant energy change 445 Gauge-orbit symmetry 10, 4L8, 430 Gauge-orbit symmetry angles 418 Gauge symmetry L0, 15L, 407 in the theory of electroweak interactions 173 Gauge transformation of the BCS wave function 66 of the kinetic equation M5-7 operator for 151 of the order parameter 14L Gauge-wheel effect 420, 433 Generalized BCS wave function 65-7 Generalized Fermi-liquid interactions 439
Generalized pairing theory 64-71 Ginzburg-Landau free energy functional 79,lL3-18,149, L65, 195 parameters of t'1.6, L26-7 B including magnetic freld L29-36 including gradient terms 194-200 Global gauge symmetry, spontaneous breaking of 4 Global symmetries L72 Goldstone mode 139, I72, 5O5, 513, 524 Gradient free energy 194-202, 213, 238 Ground state, degeneracY of 65, L39 Ground-state energy 47 of normal phase 47 of superfluid phases 79 Group-theoretical approach to collective modes 513-19 Group-theoretical classification of defects 2ll-36,267-72 Gutzwiller-Hubbard lattice gas model 52-s Gyromagnetic ratio 18, 25, 30, 17 5 Gyromagnetism of rotating superfluid tHe Lo9, 252-7, 259-63, 392
Half-quantum vortice s 244-5 , 250 , 25L Hamiltonian approach to spin dynamics 365,369-71,383 Healing lengths 200-2, 27 0, 281 induced by dipolar interaction 20L induced by magnetic freld 202 induced by surfaces 202 Heat current 425-6,448 through boundary 493 Heavy-fermion superconductors 150
Heavy-fermion systems 22 Hedgehog order parameter configuration 2ó5 Heisenberg spins 213, 214
in aplane2lT Helical texture of I field 303 for qll^tÍ306-11
foru,
IH3ll-Lz
stability regions of 309, 310 time-dependent 320 unstable 319 Helium 3 (3He) 1-4 condensation 2 dilute solutions in helium 4 21 films, superfluidity in 494-5 layers at a solid surface 495 phase diagrams 2, 3 production of 1 superfluid 2 symmetry group 143 two-dimensional model lM-7 Helium 3-A (3He-A), see A Phase Helium 3-AL, see A, Phase Helium 3-n (3He-n), see B Phase Helium 3-helium 4 dilution refrigerator 93
Helium 4 (oHe) 1,-4 condensation 2 dilute solutions of helium 3 in2I phase diagrams 2, 3 Helium physics, early history of L, 5-7 Higgs-potential minima 165 High-energy physics 165 High-7i oxide superconductors 22, 550 Homogeneous spin precession, instability in 402 Homotopy theory 213-4, 216, 2L9-20, 263-4,280 Hubbard model52 Hydrodynamic equations 32-3, 405 for the A phase 41,6-29 for the B phase 411-16 for paramagnetic fluid 32 Hydrodynamic modes 502-t2 Hydrodynamic regime 18, 435 Hydrodynamic sound propagation 42, LOs,502 Hydrodynamic theory, microscopic derivation of 435 Hydrodynamic variables 405, 406 Hydrodynamics for finite magnetic freld 429-33 slip correction to 484-6 Hyperbolic texture 230
tç
ìI Subject Index
Induced electric dipole moments L8I-2 Induced interaction model 131 Induced intrinsic angular momentum
4t8 Inert state Il7,154,156, 160, L64, 168, 169,172,234 in plane texture 283,284 Interpenetrating square lattices of vortices 261 Intrinsic angular momentum 205, 4L618,422,429 induced 418 Intrinsic spin relaxation 394-403, 475-9 energy change by 399 Invariants of free energy 122 Ion mobility 495-8 effect of interaction with vortices 498 experiments 110 lsing spin 214,2L7 line-shaped defects for 218 pointlike defects 218 Isotropic superfluid 1,40, 257 -9, 327 Jastrow factors 47 Josephson effect 20, I08,494 Josephson equation 426, 433 transport coefficients in 481 Josephson flow oscillations 494 Josephson oscillator equation 355
Kelvin waves244 Kinetic equations 33-4, 437 -50 for Bogoliubov quasiparticles 422-5 gauge transformation of 445-7 matrix 438-40 Knudsen regime 460-1, 484, 489-91
i texture in a cylinder 288-91 in a slab 286-8 in a sphere 292-4 relaxational motion of 336 ^ I disgyration 227, 230, 289, 290, 293 I monopole 293,294 I vector in A phase 82, 148 Lagrangian formulation of spin dynamics 365-9 Landau-Boltzmann equation for Bogoliubov quasipar ticles 437 Landau critical velocity 321,327 Landau damping 43 Landau molecular fre\d,27,85, 88, 103, 440,462 Landau parameters 27 , 3L,54, 85-6,
6ti lgg, 440, 462, 520, 534, 542 Landau's theory 6 see also Fermi liquid Large-angle helix 311 Larmor frequency 18 Lattice-gas model of normal liquid 52_5 Lattice of planar domain walls 3tZ Lattice of vortices 258 Leggett angle l7g, lBB,2Zl Leggett configuration 199 Leggett equations 342, 34g Linear defects 2lg-57 see also vortices Local equilibrium 37 -8, 456, 462 approach to 462 of the BQP system 456 Local oscillator model 362 Local identity 406, 4IZ, -thermodynamic 421.424 Localization transmission 54 London limit 195 Long-range order 137-49, 174 Longitudinal driven NMR 361 Longitudinal (first) sound 42, 102, 502 Longitudinal NMR frequencies 345, 375 Longitudinal NMR linewidth in the A phase 399 Longitudinal resonan ce lO2 Longitudinal spin resonance lg, I0Z, 343 Longitudin al zero sound 43, 535_46 see also collisionless sound Macroscopic orbital magnetic moment 55s Macroscopic quantum coherence 549-Sg of the pair-correlated state 549 in superfluid helium 4 549 Macroscopic quantum effects 2 Macroscopic quantum state 190 Magnetic energy density 129-35, 1g0 Magnetic healing length 201-2, Zl7, 254,259 Magnetic resonance frequencies 364 see also nuclear magnetic resonance Magnetic susceptibility 181 see also spin susceptibility in the A phase 87, I01.,I33 anisotropy 87 , 180,201 in the B phase 89, L34 in the normal phase 30 see also spin susceptibility Magnus force 266 Manifold of internal states 14L,272
Subject Index
614 Mapping of a contour 213 Mass current 32,84,202, 412, 440, 448 Mass density 27,406 Mass supercurrent, see supercurrent Massive modes 501 Matrix distribution function 437 Matrix kinetic equation 439,5L9 Matrix of transport coefficients 409 Mean free path 36,460,484 Mean-field Hamiltonian 67 -9 Mean-field solution, effect of fluctuations on 128 Meissner effect in a superconductor 257 Mermin-Ho-tv{T vortices, lattice of 263 Mermin-Ho relation 210, 250, 295, 296, 422
Mermin-Ho (MH) texture 229,230, 233, 243, 250, 261, 262, 289, 291, 3L3,373 Microscopic derivation of hydrodynamic theory 428,461-84 Microscopic theories for the normal liquid 46-8 Mixt-twist (MT) texture 230, 26I, 262 Model states 80 classification of 1 18-23, 149-72 relative stability of 122-3 Molar volume 31 Molecule picture of Cooper pairs 552 Momentum conservation equation 32, 4L4,425 Momentum current densitY 4L4,425, 44r, M8 see also stress tensor Momentum density 27, 441, 448 see also mass current Monopole order-Parameter configuration 15, 265, 266, 292 Monopole-antimonoPole Pair 294 Multiple spin echoes 403 Multiplet structure of collective mode spectrum 5L2
ñ vector in B phase 146,199 fi soliton 272,279,280 effect of superflow on 3L4 ñ texture induced by surfaces 280-6 Neutron scattering 45 Neutron stars2Z bulk matter of 171 NMR, se¿ nuclear magnetic resonance Noether's theorem 172 Nonaxisymmetric AT texture 243
Nonaxisymmetric vortices 240, 241-3,
248-50.256-7 Noninert state 155, 159, 160, 168 Nonsingular point defect 265,266 Nonsingular structure 227 ,228 Nonunitary state 89 Normal-flapping mode 5lL, 53L, 544 dispersion of 533 Normal-fluid density 80, 84-6, 194,198, 207
Normal-fluid slip 504 Normal phase 25-46 Nuclear demagnetization 94, 96 Nuclear dipole interactions lO2, L74 see aßo dipole-dipole interaction Nuclear magnetic moment 30,175 Nuclear magnetic resonance (NMR) 18-20,95, L00, 339-403 frequency shifts 342-8, 362-90 effect of magnetic field on 345-7 effect of superflow on 364 effect of tilted uniform texture 363-4 limit of zero magnetic field 344 longitudinal 103, 345-7, 363 transverse L03, 345-7, 358, 3ó1, 363,369,371 Hamiltonian approach 369 Lagrangian approach 365 linear in nonuniform texture A phase 365-83 B phase 383-9 longitudinal driven 361 nonlinear in uniform texture 348-62 response of the helical texture 374-5 resonance linewidths 398 tHe 388-93 in rotating superfluid satellite peak 109, 110 signature of solitons 377-82 under linear spatially homogeneous conditions 342-8 Nuclear magnetic susceptibility, see magnetic suscePtibilitY Nucleation of B phase 128 o vortices 238, 240, 245-9, 251 Oblate state 157, 170
Off-diagonal distribution function, kinetic equation for 443 Off-diagonal energy 68 Off-diagonal long range order 140 Off-diagonal mean field ó8,70 Onsager relations 409
IE lç
Subject Index Onsager's symmetry principle 415 Orbital dissipation 333-6 Orbital magnetic moment of the A phase 553-5 Orbital relaxation 335, 482-4 Orbital supercurre nt 204 orbital viscosity 320, 427, 483, 5L1 Orbital wave 511, 533 Order parameter 9-12,71, 137 , 139, L40,236 complex-valued 6'1., 190, 549 matrix form 70 of A phase 7,8L, L48 of A, phase7,132,1.48 of Ar phase 7,l3O ofB phase 7,80,I47 of Br phase 7 , L34 orientation of. L92, 238 induced by dipole interaction 17480
induced by electric field 181-2 induced by magnetic field 180-1 induced by superflow 182 induced by surfaces 185-8 preferred diiection 1î¡ tt+ spatial variations of 185 see also textures symmetry of 144-7 transformation prope rties L44-7 vector form 70 Order-parameter manifold 212 see also order-parameter space Order-parameter modes, symmetry classification of 512-19 Order-parameter space 1,47, 212 see also order-parameter manifold, degeneracy space Orientational energies of orderparameter field 174-88 Orientational transiti on 286
P-T-H
phase diagram 99
Parity-violating interaction 557 p-wave pairing 80, 114 Pair amplitude 67, I40,557,552 Pair breaking20,326,329, 487 , 53L-2, 541
'e &
g Ê ìgi ,Ë
ffi
saturation of 546 Pair-breaking critical currents 32L-6 in the A phase 326 in the B phase 325 Pair-breaking critical velocity 207 ,322, 324 Pair condensation, energy due to 69,79
615 Pair-correlated state 6L, t40 Pair interaction 5, 57-9,74,76, \77 , 534 Pair vibration modes 20,5L3,524 Pair wave function 8, 62,92, 55L, 552 Pairing theory at finite temperature
7t-9
Pam-Am texture 289 Parallel ringing 359 Paramagnon theory 53 Parity-violating dipole moment in 3He-B 558
Partial spin susceptibility of the condensate 395 Particle-hole asymmetry 130, 469, 523 Particle-like soliton 266, 268 Particle number current 28 Particle number density 27,447 Particle physics 2L, 251 Persistent current 106-8, 190-4, 209, 294-321,326-37 see also superflow Phase diagram 95-7, 109, I22, 287 experimental investigation 95, 311 of 'He 3 of aHe 3 of helical texture 311 of rotating liquid'He 393 Phase
rigidity 4,6L,66
Phase-slip processes 250,327 , 494 Phase vortex 214-7, 220, 258 see also pure phase vortex Planar defects 267-80 topological classifi cation 267 -7 2 Planar soliton 267 in the A phase 272-8 in the B phase 278-80 Planar state 119, 150, 157, L&,168, '/..69,249, 25L, 323 Point defects 263-6, 2L6-8, 263-6, 33L see also monopole order-parameter
configuration in the d-vector field of '}]re-A266 in the l-vector field of tHe-A 265,331 in the ñ-vector field of t}le-B 264 Poiseuille flow 489-91, 505 Polar state 118-19, L55,160, L69, 242
Polarization potential theory 48-9 Polycritical point 96, 102 Pomeranchuk effect 94, 95 Preferred direction of order parameter, see long-range-order Pseudo-isotropic state 11, 81 see abo B phase and BW state
Subject Index
616 Pure phase vortex in the
A
phase 227
,
230-r,240,241,242 Q balls 129 Quantization of circulation 208-1 1 Quantum of circulation 209 Quasiparticle 6 see also Bogoliubov quasiparticle collision processes 450 compressibility 30-2 concept 25-8 definition 25,26 distribution function 33 dynamics in ballistic regime 491 effective mass 26, 29 energy 26,62 entropy 28-30 interaction2T lifetime L7,34-7 , 479 scattering amplitude 34, 49-52, 451 phenomenological model of 55-7 velocity 34,443 Radial disgyration 230, 242, 373 Real squashing mode 524,539,541 Relative gauge-orbit symmetry t57, L60 Relative gauge-spin symmetry L59, 160 Relative orbital angular momentum 8, 63
Relative rotation of spin and orbital spaces 147
Relative spin-orbit symmetry 154 Relative symmetry 146-8, 160 Relaxation time 396, 476,478 Relaxation-time approximation 456 Remaining symmetry 149 Remanent vorticity 331 Resonating-valence-bond 550 Rigidity phase 4,6L,66 Ringing of magnetization 102,355 tHe 109-1 I, 257 Rotating superfluid 63,389-94 s-p approximation 56 Scattering amplitude 34-5, 51, 55, 451 see aßo quasiparticle scattering amplitude Scattering of light 508 Schafroth condensation 550 Second sound 105, 502, 503, 505, 508, 513
Second-order (A) transition 78, 95, 96 Self-induced transparency 546 Shear flow 485
Shear modes 5L1. Shear viscosity 33,36,
4'/-.,
4L5, 426,
464-7,480 for 3He-B 468 for 3He-A 481 Sine-Gordon equation 26T, 37 6 Sine-Gordon solitons 272 ,Single-particle excitations 61, 69-70 Singlet pairing 70-1 Singular defect 2Ll-12, 214 Singular vortices 220, 259, 329, 390, 498 Slip correction to hydrodynamics 484-6, 489
Slip length 485, 486 Small-angle helix 311 Solid 3He 7 ,94-5 Soliton 268, 272, 273, 3L3 in the A phase 272-8,376-7 in the B phase 278-80 composite 273,275-7 ,3L6, 377 creation of 377-8 energetics of273-5 NMR signature of 377-82 planar 272-80 in the presence of superflow 382 Soliton lattice 277 -8, 380-3 Soliton propagation 382 Sound attenuation 20, 42, 44, 103, 536 in the A phase 104,545 in the B phase 103, 541 Sound dispersion relation 44,538 Sound modes 42-6, 102-5, 491-4,
502-5,535-46 first sound 42, 105, 49I, 502 fourth sound 105, 504 second sound 105, 503, 508 zero sound 44, 102, 49I, 535-46 Sound propagation L02-5, 491-4 in the A phase 542-5 in the B phase 538-42 nonlinear effects 546-6 in the normal state 31 ,44,I04 in restricted geometries 491-4 Sound velocity 3I, 44, 536 Specific heat 80,83-4,98-100, 117 in the A phase 84 in the At.r phases 132-3 in the B phase 83-4 discontinuity of 83, 100, ll7,132-3 in the normal state 28-30 Specific-heat curve, experimental 98 Specific-heat discontinuity 100, II7, 132, 133
for the ABM state 84
Subject Index Specular scattering of quasiparticles from a plane boundary 185, 484, 487
Spin conservation law 32,432 Spin current 34, 198, 431,44I,449 Spin density 27,448 continuity equation for 32, 505 equation of motion for 341, 415, 505 Spin diffusion 402, 415, 473-5 Spin diffusion coefficient 33, 36,42,415, 474 Spin dynamics 18-20, 102,339-402 in uniform texture of the A phase 342,348 ofthe B phase 343,348 nonlinear regime 348-62 in zero magnetic field 349-55
in nonuniform texture 362-89 of the A phase 365-83 of the B phase 383-9 including spin relaxation 394-403 in rotating superfluid tHe 389-94 Spin-echo method 403 Spin effective mass 447 Spin fluctuations 6,53, L24 Spin fluctuations model 123,I27 Spin-orbit angles 413 Spin-orbit forces L3, L74,341,394 Spin-orbit-gauge symmetry 430 Spin-orbit interaction 78, 174, 341, 394 Spin-orbit rotation symmetry 10 Spin-orbit waves 5L3, 524 Spi n - re ax ation 393 - 402 Spin-relaxation rate 394, 479 Spin supercurrent 194, 204, 33L, 403, 449-50 Spin superfluid velocity 194,446 Spin susceptibility 30, 80, 87-90, 180, 252,256 see also magnetic susceptibility Spin transport along capillaries 490 Spin-triplet pairing 70 p-wave 8, 80, 114 Spin wave 505-10, 546 damping of 508 standing 282-3,570 velocity 506, 510 Spin zero sound 43,45,546 Splay composite soliton 277 ,3L4-t6, I
r
i
N
I ? L
dt
tåt )L'
380 Splay spatial change of order parameter 196 Spontaneously broken continuous symmetry, see broken symmetries
617 Squashing modes 524, 547 Stability of superflow 190, 294,321,326 Standing spin waves 282-3,510 Static helical configuration of i-texture 302 Stress tensor 32, 33, 414, 417, 424-7, 464 Strong-coupling corrections 77 , IZZ-T, 198.300 SU(5) grand unification theory 165 Superconductivity 4, 61, 7 5 Superconductors, high- [ oxide 22 Supercooled A phase 247 Supercurrents 16-7, 193, 198, 202-11,
4t0 Super-flapping mode 531, 544 Superflow 16-17, 106-8, 193, 198, 202-11 collapse 333 in a cylindrical geometry 313 decay mechanisms 327 dissipation 326-36 effect of 182-3 on orientation of order parameter
t82 on domain walls or solitons 313-19 global stability 296 in a slab geometry 312 stability of in the bulk liquid 295-301 and textural transitions 294-32I Superfluid current 195 see also supercurrent, superflow Superfluid density 87,107,194, t98 Superfluid mass currents 16-17, 198, 204,410 see abo superflow Superfluid spin current 331 see also spin supercurrent Superfl uid-to-normal interconversion 492
Superfluid velocity 16-17, 174, 191, 193, 4r2, 4L3, 4L9, 431, Superfluidity 2, 4, 189, 190-4 in a Bose system 66 SuperheatedB phase247 Surface defect 266, 280, 281, Surface dipole energy 187 Surface-dominated orientation 283 Surface energies 217, 282 and boundary conditions 185-8 Surface healing length 202,283 Surface impedance 492, 503 Surface-induced textu r es 280-94
Subiect Index
618 Surface singularitie s 292-3 , 331 , 332, 335 see also surface defects 88 Susceptibility susceptibility magnetic see also
SymmetrY and order-parameter structure 141-73 discrete symmetry transformations 1,64
reduction due to a magnetic field or spin-orbit couPling t69-70 of a state. connection with energy as stationarY Point 165 Symmetry breakings, sequence of fi2 see also broken sYmmetries Symmetry classification 137 -49 of equilibrium states 149-72 of order-parameter defects 2Ll-38, 263-72 of collective modes 511-19 3He 143, Symmetry group of superfluid
t69-70 Symmetry variable 407
T. sptitting 131 Temperature-dePendent coherence length 185 Temperature-dePendent mean-field Hamiltonian 72 Tensor order Parameter 70 Textural critical velocities 322 Textural instabilities due to superflow in A phase 294-319 Texture-induced NMR shifts 362-89 in the A Phase 3ó5-83 in the B phase 383-9 see also nuclear magnetic resonance Textures 15, 189-200, 208-319 see also defects of order-parameter field induced by surface in 'He-A 286-94 ''He-B 280-6 3He-A 294-321 induced by flow in Thermal boundarY resistance 494 Thermal conductivity 33, 36, 471-3, 482 of 3He-A 482 of 3He-B 473 Thermal expansion coefficient 98 Thermodynamic identitY 406 for the A Phase 420-3 for the B Phase 412 Three-dimensional rotation of order parametet L42
Time-dependent mean-field theory 52035
Time-dependent texture 319, 320 Topological charge 15, 214, 220, 226, 232,267,268,273 Topological classification of defects
2ll-19,233 Topological ProPerties óf n-þnase vortices at large distances 231-3 of A-phase vortices at small distances 223-3r of B-phase vortices 220-3 Topological quantum number, s¿e topological charge Topological space of order parameter 142,1,43
Topological stability 216 of linear defects 2L9-20 of planar defects26T of point defects 263 of a singularitY or defect2l3 Topologiðally equivalent states 216 Topology 141 ôf nod". on the Fermi surface 250-L Torsional oscillator 106, 490 Transition temPeratures 7 5, 97 dependence on magnetic field- L29-36 vaiiation of, with channel width 494 Transport coefficient 426, 436, 461-84 ofthe A Phase 479-84 of the B phase 464-75 in the JoiePhson equation 481 matrix of 409 Transverse NMR resonance 344,345, 347 , 36L, 363, 375, 398 see also nuclear magnetic resonance Transverse sPin-wave modes 390 Transverse zero sound 43,45,105, 546
Triplet pairing 6,70-I Tritium 1 Twist comPosite soliton 277,314, 316-18, 378 insHe-A276 Twist spatial change of order parameter
196
ì
Two-dimensional Fermi liquid 495 Two-dimensional suPerfluid 216 Two-fluid model 16
U-tube oscillation experimen t 326, 330' 492
Ultralow temperatures, experimental techniques for attainin g 93-5
619
Subject Index
Ultrasound attenuation, see sound attenuation Ultrasound excitations 20-L Ultraweak interaction 555 Uniform I texture global stability of 295-8 helical instability of 302-t2 hydrodynamic stabilitY of 298 local stabilitY of, in Presence of superflow 298-30I tilted NMR oT363-4 Unitary states 71,118 Unwinding of the Phase 327 uvw-vortex 238, 241, 247 v-vortex 238, 24L, 243, 246, 247,
249-5r.256 van der Waals Potential 2, 49 Vibrating-wire experiments L06, 491 viscosity 33, 4'1.,105, 415, 426, 464-80 in A phase 105, 481 in B phase I05,468,47I in normal phase 33,36, 4L second (or bulk) viscositY 33,4I5, 426,467-7 1, 480 shear viscosity 33, 36, 41, 415, 426,
t I
I
É
464-7,480 Viscous penetration dePth 491 Vortex-antivortex Pair 217 Vortex core hard233 soft232 spontaneous magnetization density in 237,254 structure 250-L Vortex-core transition 25t, 391, 392 Vortex lattice 257,259 Vortex line2l7,226 see abo linear defect in superfluid helium 4,212 Vortex nucleation 327 ,329 Vortex ringsZ1l,329 Vortex tangle 328,330 Vortices L5, 109, 2L9-57 in the A Phase 223-33,240-5
associated with the phase @ (pure phase vortex) 219, 222, 226 axisymmetric 235-8, 240-1, 245-8,
252-6 in the B phase 220-3,245-51 magnetic proPerties of 252-7 energetics of238-57 half-quantum244-5 nonaxisymmetric 241-3, 248-50
vorticity 221,
23'1.,
232, 233
remanent 331
w-vortex 238, 24'l', 243, 244, 247 Wall-pinned configuration 360 rilall-pinned ringing mode, damping of 40r Walls influence on superfluid 174 and orientation of orbital angular momentum 14 Weak neutral currents 556, 557 Weak-coupling collective frequencies 516
Weak-coupling free-energy functional 77
Weak-coupling limit 63, 196 Weak-coupfing theory'17, 1L3, 127, L94, L97, r99 of pair correlations 61 Weak-coupling values of B parameters in free energy LL6 Weinberg-Salam model L73, 557 Wide-angle helix 320 Winding number 2I4, 215, 2L6, 221, 226,303
XY spin model t40,2I3 Yosida function 86, 89
z-vortex lattice262 Zeeman energy 30 Zero-point motion 2-3 Zero sound 43,45, L02,538-46
