Quantization in Space and the Relative Intensities of the Components of Infra-Red Absorption Bands

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QUANTIZATION IN SPACE AND THE RELATIVE INTENSITIES OF THE COMPONENTS OF INFRA-RED ABSORPTION BANDS By EDWIN C. KEMBI3 JIFF8RSON PHYSICAL LABORATORY, HARVARD UNIVERSITY Communicated, April 22, 1924

The purpose of this article is to show that the application of the Bohr correspondence principle to the theoretical evaluation of the relative i'ntensities of the lines in the infra-red absorption bands of diatomic gases leads, in the case of HC1 band at 3.5,u to a definite conflict with the experimental facts if the molecule is treated in the usual way as a degenerate system with one quantum condition for two rotational degrees of freedom. On the other hand, if we assume a slight precession of the orbital plane about the lines of the earth's magnetic field, thus introducing space quantization of the orbital planes, the difficulty is removed. The discussion is based on the interpretation of the HCI band in terms of half quantum numbers first suggested by Kratzerl and recently confirmed for HCI by Colby's study of the lines which appear in the neighborhood of 3.5,u when the gas is heated.2 Let n and m denote the usual vibrational and rotational quantum numbers. Treating the system as degenerate, let us consider the absorption line due to jumps from the.state n", mi" to the state n', m'. (Primes will denote the state of greater energy.) Let a, be the absorption coefficient for the frequency v and let

ettm= f8a.dv,

(1)

where the integral is carried over the absorption line in question. The integral absorption coefficient thus defined is related. to the number of molecules in the initial state N*,m and to the Einstein probability coefficient for this jump, B, m"* by the formula (2) n*,", = NnwmB X Constant. Introducing the usual statistical formula for Nn,mu in terms of the a priori probability P,t,m and the energy W(n",m") of the initial state, we obtain (3) nm= Cvpnm,,ew(-u Bi)/kTBnm Einstein3 has shown that B, is related to the probability coefficient for emission by spontaneous transition from n',m' to n",m" by the formula

p ,,mAni,M# 8irh'3Pnm'nm B",t =A8wh3

(4)

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PHYSICS: E. C. KEMBLE

Hence (3) becomes ",m" = p, ,e-W(n,mif)/kTAP,rMl',-2 -Const5 an,m n'mC

mfS. may be evaluated by Bohr's

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5

correspondence principle in terms AWYM of the coefficients in the Fourier expansion of the electric moment M of the molecule. Let X(n,r, An, Am) denote the amplitude of the harmonic component of Mx correlated with the jump in question and evaluated for an orbit with quantum numbers n and m.4 Let Y and Z denote the corresponding amplitudes in the Fourier expansions of M, and M., respectively. Then according to Bohr and Kramers5 (6) An; = 1v3(X2 + Y2 + Z2) X Const. where the sum of the squares of the amplitudes is to be averaged over the hypothetical classical orbits between the initial and final states. The method of averaging is uncertain and in fact it is uncertain whether any definite scheme of averaging will always give rigorous results. We assume, however, that as a first approximation we may write X2= [X(n,n, An,AM) ]2

(7)

with similar equations for y2 and Z2. n and m denote the arithmetic means of the values of n and m for the initial and final orbits. Equation (5) now becomes = n Const. X vpn, m,e-W(n,mif)/kT[X2(_,I) + y2(nm,) + Z2(n,m) ] (8) '

an'm

The electric polarization M is assumed to be proportional to the vector distance between the atomic nuclei. Neglecting for the present any possible precession about lines of magnetic force and assuming a nongyroscopic molecular model, the motion of the terminal point of M must be uniplanar and compounded of a vibration of frequency w, and a rotation of average angular velocity 27rw,. It is convenient to introduce two sets of axes x,y,z and x',y',z' with a common origin. The first set is fixed in space, the xy plane being the plane of the orbit. The z' axis is coincident with z but the x' and y' axes rotate with constant angular velocity 2irwc, about the z axis. The orbit in the x'y' plane is evidently periodic. If the molecular rotation took place at a uniform rate it would degenerate into a straight radial line, but owing to the variation in angular velocity witli the separation of the atoms it is actually of an elongated oval form with major axis radial. The Fourier development is conveniently written in the form 7.-

MX1 + iMyR =

Re2

vt + 6) +

R7 e-2ni(TwVt-6)

(9)

T0

To avoid undue complication the discussion is at this point limited to

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the so-called "fundamental" vibration-rotation band which. is correlated with the fundamental vibration frequency of the Fourier analysis.. Unity is then the only value of An and T to be considered and equation (9) can be replaced by

+ tM31 Mx+

+e+ R'e-2

=

(10)

Shifting to fixed axes is equivalent to multiplying by e2Tiwrt. Hence

Mx + iMy =

Re2T$[(wv + Wr)t +

I

+ RIe21i[(wvw7)t8]

(11)

If we treat the orbit as a radial line in order to get a first approximation, R and R' are equal, but an analysis of the ratio of the major and minor axes required by the law of the conservation of angular momentum shows that as a second approximation we may write R =A(

f

(A)V + COr

;

R'

=

A (X

(A)V- Wr

)2

(12)

A and xv vary slightly with the rotational quantum number of the orbit for which they are evaluated, but this variation will be neglected in comparison with that of Wr. R and R' are the amplitude coefficients for M. and M, corresponding to the values Am +1 and Am = -1 respectively. As the frequency absorbed is in the first case WV + cor and in the second cov -cor, we have X = Y = Const./v2; Z = 0. (13) Equation (8) now becomes

ao M= Const. X p ,e-W(O m')/kT Y3

(14)

X

The above equation is in sharp disagreement with the experimental facts. The two central components of the HCI band at 3.5,u are of approximately equal intensity,6 though according to (14) the high frequency component (m' = 3/2) should be much more intense than the low frequency component (m' = 1/2). The exponential factor and the factor I/O are not very different in the two cases so that the intensity ratio is nearly equal to the ratio of the a priori probabilities of the states having the rotational quantum numbers 3/2 and 1/2, which is two or three to one according to one's taste in a priori probability formulas. The trouble with the theory lies in the factor pin'' which depends on the value of m for the final state of the absorption process In order to obtain even approximate agreement with the experimental facts we must replace this factor by one depending in a more or less symmetrical way on the initial and final values of m. The desired result is effected if we assume

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the existence of a slight precessional motion about the lines of an external magnetic field and so pass to the discussion of a non-degenerate motion in which the number of quantum conditions is equal to the number of degrees of freedom and in which the a priori probabilities of all stationary states are the same.7 In analogy with the theory of the Zeeman effect the angular momentum parallel to the field is assumed to be rh/21r, where r is a quantum 'number associated with the precessional motion. It is further assumed (Hypothesis 1) that for any given m, r takes on the values 1/2, -,-3/2,.. .. -=m.8 An alternative possibility" (Hypothesis 2) to 2 (m - 1/2). The be considered later gives r the values 0, 1, 2,... molecule being presumably diamagnetic, the energy values will be sensibly independent of r and the emission frequencies for jumps with the same m' and m" but different values of r' and r", will be the same. To get the new Fourier development we may assume that the x,y,z system of axes previously considered is not really stationary, but rotates with a small constant angular velocity 27ra about the lines of magnetic force. We choose a new set of fixed axes {, , r such that t coincides with the field. Let 0 denote the constant angle between z and t. Without loss of generality we may assume that the x-axis lies in the t- n plane and makes an angle 27rat with the t axis. Shifting from the precessing to the fixed axes, we use the transformation equations

ME + iM

=

+C

(1 2{

)(MX + iM ) + (1 c2o)(MX-iM,) }e2

;

(15)

Mr = My sin 0. Hence Me + iMt,

+Cor + e)t+ t)1 + R e-2w[(wvr3T)td]} R +cos 2{ri((Wv

=

(1

+

5) + e2ri[()(1 -cos 0){Re-2wi{(wv + wr-O)t +R8

r

}

+ Ot

Mr = sin 0{R sin{27r(wv + Wr)t + 5} - R' sint27r(wv- w7)t - a .

(16)

The coefficients 1 of w, and a in the development give the values of Am and Ar for the jumps with which these terms are associated. Let HZ&'(mr, (m,r) be the {, 7, r components of the amplitudes r7(m(,r), evaluated for the mean orbit m, r. Then since cos 0 = r/m

-r/m); 1t(m,r) = 4R(1 7R'(1 + rm);

ti0(n,r) == 4IR(1 +- r/m); R'(1

Z+l (m,r)

:

rnm); I:i(r,r) a

7)

(m,r)2;(1 z^(, ZTo1(m,r) = R'V1(/_)2. Z+0m)= R sinO = Rx/1 (,/m)2; )

=

1

All other coefficients vanish.

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PHYSICS: E. C. KEMBLE

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PROC. N. A. S.

Equation (8) retains its form except that PM',m' drops out and the squares of the amplitudes are to be summed up over all values of r and Ar consistent with the values of m and Am under consideration. As the discussion is restricted to a definite pair of values of n' and n" and as 1 are the only values of Am to be considered, the notation for the- absorption intensities can be simplified. Let a 1 (m) denote the integral absorption coefficient for the component of the high frequency branch (Am + 1) having the mean rotational quantum number m. Let a 1(m) denote the coefficient for the corresponding line of the negative branch. Then =

(M)

a

= Const.

-S(m)e

kT

(18)

,

where, neglecting a constant factor, v4S(m) represents the sum of the squares of the amplitude coefficients. From equations (17) and (12) we introduce the definition

S(m)

=

E{I(1 +/;)2+j(j

-

r/M)2+ ((1

-[-])}

(19)

-_ and are to be summed from r The terms 1 + to r = +m, while the term 1 - )2 is to be summed from -(m -) to +(m-D. Equation (19) is readily simplified and yields

s(m) =

i

+

+ (-)2+

=2mr+1-1/(4m).

_

(20)

S(m) differs only by the small term 1/(4 m) from the mean of the a priori probabilities of the initial and final groups of states involved in the absorption line under consideration. It is possible that some other method of averaging the squares of the amplitudes would eliminate this term. Equations (18) and (20) reduce the theoretical difference in the intensities of the central components of the HCl band under discussion from 320% to about 7%, and are in qualitative agreement with the observations on the outer components. The alternative hypothesis 2 regarding the possible values of r leads to the formula S(m) = 2m - 17(4 m) (21) which seems not to fit the data as well as (20). A 'more detailed discussion of this point will be given in a later paper. 1 A. Kratzer, Ber. d. Bay. Akad., p. 107, 1922. 2 W. Colby, Astroph. J., Chicago, 58, p. 303, 1923.

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279

A. Einstein, Physik. Zs., 18, 121, 1917. Each of the quantum numbers is defined as l/h times the corresponding phase integral. The quantum numbers of the hypothetical non-quantized orbits between the initial and final states are fractional. ' H. A. Kramers, Det Kgl. Danske Vidensk. Selsk. Skrifter, 8 Raekke, 3, 1919, p. 330. 6 Cf. E. S. Imes, Astroph. J., 50, p. 260, 1919; Brinsmade and Kemble, Proc. Nat. Sci., 3, p. 420, 1917. 7 Cf. Bohr, Quantum Theory of Line-Spectra, I, p. 26, Det Kgl. Danske Vidnsk. Sesk. Skrifter, 8. Raekke, IV. 8 A. Land6, Zs. Physic., 11, 357 (1922); R. C. Tolman, Physic. Rev., 2nd Ser., 22, p. 470, 1923. 9 A. Land6, Physik. Zs., 24, p. 441, 1923. 3

4

THE BALLISTIC (AIR RESISTANCE) FUNCTION By L. THOMPSON NAVAL, PROVING GROUND, DAHLGRZN, VA.

Communicated, April 10, 1924

A projectile in flight is subject to retarding forces of which the resultant may attain a value greatly exceeding the weight, particularly in the case of small calibers. It is customary to express the retardation as a function of the velocity, and the determination of this function constitutes one of principal experimental problems of exterior ballistics. It appears in the equations of motion

d (v sin e) --f(v)/C .sin 0 dt-g dt d(v cos e) = -f(v)/C cos e dt, involving the quantity C = m5o/7rr2i5, a measure of the ballistic or "carrying" capacity of the projectile. In addition to the relative air density, mass and cross section normal to the direction of flight, C includes a factor i frequently assumed to be completely defined by the shape. The variation of the air resistance R = 7rr25/6o.if(v) with changes in velocity can be represented advantageously by means of the coefficient

R/v2r25 = 7riK(v)/5o, of which v2K(v) = f(v). If a particular function, K(v), is made the basis for the comparison of projectiles of different shapes and magnitudes, the factor i, pertaining to individuals, is not a constant. This is otherwise signified by the statements Ki(v) = ii(v)K(v), K2(v) = i2(v)K(v),.....where the subscripts refer to corresponding projectiles. For velocities in excess of the velocity of sound, a rapid dissipation of energy occurs through the production of air waves. These are well shown in spark photographs of the motion. Among the best are those recently taken at the Case School of Applied Science by Professor D. C. Miller