Approximate band absorption calculations for methane

J. Quant. Spectrosc, Radiat. Transfer. Vol. 5, pp, 611-620. P e r g a m o n Press Ltd., 1965. Printed in G r e a t Britain

APPROXIMATE BAND ABSORPTION CALCULATIONS FOR METHANE L. D. GRAY and S. S. PENNER* Space Sciences Division, Jet Propulsion Laboratory, Pasadena, California

(Received19 January1965) Abstraet--A just-overlapping line model is used to calculate the band absorption of the 3.3 # bands of CH4 at various pressures and temperatures. Integrated intensities are computed in the harmonicoscillator, rigid-rotator approximation. I. I N T R O D U C T I O N THE VIBRATION--ROTATIONBANDS of methane have been intensively studied in the past. (1-19) This molecule is a spherical top with three equal principal moments of inertia. The v3-fundamental absorption band of CH4 at 3.3 t~ corresponds to a vibrational transition from the ground state to a triply degenerate state. Coriolis interactions split this degeneracy and vibration-rotation interactions affect the spacing and intensity of the rotational lines. Rigorous theoretical calculations of intensities for the methane spectrum are quite difficult to perform, particularly at elevated temperatures. II. T H E O R E T I C A L C O N S I D E R A T I O N S For band absorption calculations on CO2 and H20, PENNER and VARANASl(2°,21) have demonstrated the utility of the just-overlapping line model, in conjunction with "pseudo-harmonic oscillator", rigid-rotator integrated intensity estimates. They obtained results in good agreement with experimental data at sufficiently high pressures and optical depths where the rotational fine structure is effectively smeared out. We shall apply the same procedure to methane. An analytic representation for the band absorption of methane may be obtained by noting~" that the sum of the rotational line strengths originating from the level with rotational quantum number J is given by the relation

~'.r SJs" ~- ~gs exp(-Ej/kT)

(1)

where ~ represents the integrated intensity for the vibration-rotation band, Q~ is the rotational partition function, gs denotes the statistical weight of the ground state, Ej is the rotational energy of the Jth level, k is Boltzmann's constant, and T is the absolute temperature. For a spherical top, Qr = 7r½y- ~ 7 - 1 where 7' =

(hcB/kT),

(2)

* Consultant; permanent address: Department of the Aerospace and Mechanical Engineering Sciences, University of California (San Diego), La Jolla, California. t Compare equation (7-127) of ref. 22. 611

612

L.D. GRAYand S. S. PENNER

is lhe s y m m e t r y n u m b e r of the molecule,* B is the r o t a t i o n a l c o n s t a n t , a n d the other s y m b o l s have their usual m e a n i n g . Also,

gj

( 2 J + 1)2(~

=

(3)

and?

E j / k T ~- J(J + l b '

(4)

whence it follows that the r o t a t i o n a l line spacing is, a p p r o x i m a t e l y , ~ equal to (3 ~ 2B(1--so).

(5)

A c c o r d i n g to the usual procedure, (22) we represent the local spectral a b s o r p t i o n coefficient as the ratio of the line strength to the line spacing for large values of J, viz. J'

,]2)j13/2)~

Pj _

_ (3

and, since j 2 = becomes

exp ( __y j 2 )

ioj_~o0ie/4B2(l_~:)2, the b a n d a b s o r p t i o n A for a n optical depth

A =

f

(1-exp

16)

7r ~'B(1 -~:)

[-P~X)]d~o = 4B(1-~b,-~I2(K)

X

(7)

band

where

IdK) = f {l-exp[

- K1 , ~ e x p ( - ) ,2)]s d r ,

(8)

0

K-

B(1-~1

i_;,) '

<9,

a n d ~o represents the wave n u m b e r . The functions l , t K ) are discus:ed in A p p e n d i x A a n d are plotted in Fig. A-1 for n = 0, 1, and 2. II1. A P P L I C A T I O N TO T H E 3.3~ BANDS OF M E T H A N E The b a n d a b s o r p t i o n for m e t h a n e has been calculated for the 3.3/~ b a n d s as a f u n c t i o n of t e m p e r a t u r e a n d optical depth. The restllts are given in T a b l e i. The integrated intensity of the band, x, varies a p p r o x i m a t e l y as T - I (see A p p e n d i x B for details) and a value <~6) of 7. - 320 cm '~ arm 1 at T = 300 K was used. * For CH4, a = 12. ! Equation (4) is true for non-degenerate vibrational states. For the triply degenerate vibrational states, the Coriolis interaction splits the degeneracy and the rotational energy values for the threc sublevels are given by equation (IV, 78) of ref. 23. ++See equation (IV, 85) of ref. 23 for a more precise value of the line spacing. The Coriolis coefficient is small for the va-fundamental and the approximation ~ = 2B(I --~:) ~ 2B is reasonably good (`3 - 9.87 and 2B = 10.48 cm-1). For the v4-fundamental this approximation is, however, very poor and ,3 - 5.74 cnl I

A p p r o x i m a t e b a n d a b s o r p t i o n calculations for m e t h a n e

613

TABLE I. COMPARISON OF CALCULATED AND OBS,,ERVED BAND ABSORPTION FOR THE 3'3t~ BANDS Of CH4 T (°K)

p (atm)

X (cm-atm)

297

0.790 0-921 0"921 1 '04 3 "12 3-22 0-921 0'921 3.12 3"89 0.921 0.921 2.57 3 -20 5 51

30 28.2 28'2 2'5 7"4 122 28.2 28.2 7.36 147 28.2 28.2 97 30.3 209

373 473 559 567 573 673 828

Acalculated Aobserved 328 326 326 204 272 366 347 377 287 480 395 409 518 448 575

325 (18) 315 (19! 340 (16) 87.91Is~ 20Y TM 38511s) 318 (191 349 (191 204 (18) 502 (18) 368 (19) 385 (19) 600 (18) 464 (18) 678 (18)

It may be seen from Table 1 that our calculated values of band absorption are in excellent agreement with the measurements of VANDERWERV,~10) which were performed at constant pressure and constant optical depth. For intermediate values of optical depth, 20~
A:

EVALUATION

OF

In(K)

The curves shown in Fig. (A-I) were derived by direct numerical evaluation of I~(K) (with n = 0, 1, 2). The function Io was first studied and tabulated by LADENBURG and LEVY;(24) ARAKX~25) has extended the tables. The function 11 has been evaluated by PENNER and HOOKER (22) and used in the calculation of PENNER and VARANASI. (2°,21) A series representation, which converges rapidly for small values of K, may be obtained for I,,(K) by expanding the exponential in equation (8) in an infinite series and interchanging the order of summation and integration. The result is

I,~( K)

~ f ( - K)tt - ( ~ + 1)/2P[(nt + 1)/2] t= 1

(A-l)

P(t + 1)

F o r small values of K, the sum may be approximated by the first term and

In(K) ~ - ( K / 2 ) P [ ( n + 1 ) / 2 ] for K ~< 0.1

(A-2)

614

L. D . GRAY a n d S. S. PENNER [OIL

I

7 - - F T T ~

......

Y'--l--T

. . . . . . . . . T-~-["

[ T I T

T--~'~

i i ~

I

]

t

•

t 7TTX]

T

T

"" i

L

~00=

F v

?0-1"

/ ,,0~2

,

/

I0-I

10-2

I0 0

I01

10 2

1©3

K FIG. A-I. The functions I,,(K) for I1 = 0, I and 2.

I-0

i

1

"

--

0"9 0'8 0"7

_

O'G

ego( "~'~

, ~ . ~

100O)

eg I (1000)

i '

..

0'5

g 0"4

0.3

/

o-2 o.,F/

//

/ / a V_/ 0

/

x"QQ '',~r" ~eg

i-~'--.~_

//

(I)

-.-.'-.

/ I'0

\\ \ g'O y ---IP"

3"0

4"0

Fl(;. A-2. The integrands eg,Jy) for K = I and 1000 with n = 0, 1 and 2. T h u s . Q _-_ K,'r;/2, 11 ~ A/2 a n d 12 -~ K ~ / 4 l'oi s m a l l v a l u e s o f K. F o r l a r g e v a l u e s o f K, t h e i n t e g r a n d in e q u a t i o n (8) is a p p r o x i m a t e l y e q u a l lo up to the v a l u e o f v w h e r e t h e f a c t o r e x p ( + . r 2) b e c o m e s a p p r e c i a b l y larger t h a n F i g u r c A-2 s h o w s the i n t e g r a n d [ ~ e g ~ ( K ) ] p l o t t e d as a f u n c t i o n o f v for n = 0, 1. K = I a n d K - 1000, r e s p e c t i v e l y . It is a p p a r e n t f r o m this figure that, f o r K =

unity Kr'. 2 and 1000.

Approximate band absorption calculations for methane

615

COMPARISON OF EXACT VALUES OF In(K) FOR LARGE .g WITIt NUMERICAL VALUES ("APPROXIMATE") DERIVED FROM EQUATIGN ( A - 3 )

TABLEA-I.

K

100 I000

Io

1,

12

Exact

Approximate

Exact

Approximate

Exact

Approximate

2.26 273

2-23 2.71

2.43 2.95

2.42 2.90

2.62 3.10

2 61 3.10

the value of the integral will be approximately equal to the value o f y for which the integrand egn (1000) is equal to (1/2), i.e. I , ( K ) ± .vo where .!'0 is the value of v for which - 1 -exp [-Kv~exp(-)'o2)]

or

n(d n+l

0-782 K

-~ 2 -

)

(A-3}

\dT.,;+5 err ]' '

]/o

~1=

and (26) y

e r f y = 27r-(1/2) f exp ( - t 2) dt. 0

In Table A-1 we have c o m p a r e d the values o f I,,(K) = )'o obtained from equation (A-3) for large values o f K with the exact values. A P P E N D I X B: T E M P E R A T U R E V A R I A T I O N OF I N T E G R A T E D INTENSITIES FOR VIBRATION-ROTATION BANDS B E L O N G I N G TO P O L Y A T O M I C MOLECULES(2O.21) The integrated intensity for simultaneous transitions from the vibrational levels r~ to the excited levels v~+8~ (i = 1, 2 . . . . . n) in a p o l y a t o m i c molecule is k n o w n [see, for example, equation (7-117) o f ref. 22] to be given by an expression o f the f o r m t!

tt

~'(U1 --~ U I - ~ I ,

, ¢!

tt

U2 --->"U2"~-82 . . . . .

l! tt

vv

--~t'"-t-Sn) =- ~[V'~ + Vi +8i(i + 1,2

81r3 co Nv7 v: ...v;~ --~ - - ' " [ I - - e x p ( - - u ) ] f l ~ . o ..... 0,, 3hcp

~

.

. .n)] (B-l)

where h is Planck's constant, c is the velocity o f light, ~o is the w a v e n u m b e r at the band center for the transition, fie is the vibrational matrix element for all o f the n transitions occurring simultaneously, Nv;.v; ..... ;,/p is the n u m b e r o f molecules in the lower state per unit volume per unit pressure, u =- (hco~/kT), k is Boltzmann's constant, T is the absolute temperature, and double primes identify the lower vibrational state. We shall now consider the development o f various intensity formulae, first in the h a r m o n i c oscillator a p p r o x i m a t i o n and then for a " p s e u d o - h a r m o n i c " oscillator.

A. The harmonic oscillator approximation In the h a r m o n i c oscillator approximation, o) ~

~ (oiSi, i=1

616

L.D. GRAYand S. S. PI-.NNFR

where % is the frequency of the ith normal mode of viblation. It is also convenient to define the quantity u~ = (hc,oi)/kT whence it follows lhat t~

u = ~

u,(3~.

(B-2)

i=l

Furthermore, for harmonic oscillators, the wave function for the molecule, 4,, is given by the p r o d u c t of the wave functions ~b,, and the matrix elements l\~r transitions involving more than one normal frequency are given by rl

= I]/3£i=]

The n u m b e r of molecules in the g r o u n d level, per unit volume, is* N,,;' ,: °

,,,

N,, 'IZ[ Q,, ~1=gV7 [ e x p - - ( t i / 6 ) ]

_

~

tt

(B-41

where N r represents the total n u m b e r of molecules per unit volume, gv'~ is the statistical weight o f the ith level with vibrational q u a n t u m number c'(, and the partition function may be written as

Qo = [2I[1 - e x p ( --Ill)] -q'

(B 5)

I=1

since g~ is the n u m b e r o f times the ith normal mode appears. C o m b i n a t i o n s of equations (B-l) to (B-5) leads to the result ~[v;.' ~

z i +8,(t

= 1,2 .....

n)] t~

8 ~ r a c o [ I - e x p ( - ~ uiSi)] N,r

[~/3~;g,,;,[ex p -(vTu;) ] i=I

(B 6)

1L

3hcp

[[(1-exp(

- , q ) ] -(~'

I=i

In the harmonic oscillator approximation, tile quantities/32, are non-zero only for a i I, in which ca~e it may readily be shown, by using the procedure described in ref. 27, that 87raNTCO[1--exp( - ~ tq)] ,x[/"~ > /~'"+1(i = l, 2 . . . . . n

n)] .

l T

,,,

] - ~ ( t i + ). X

.

.

.

.

. . 71

.

.

.

.

.

.

.

3hcp

.

i

[exp_(t;u;)]]/3o]?, .

.

.

.

VIII-exp(

.

.

.

.

.

-ui)]

.

.

.

,

1

I =: l

* In some of the following discussion, it will be necessary to consider degenerate energy levels and statistical weights differing from unity. Equations (B-4) to (B-6) refer lo this case. For a harmonic oscillator, g,.~ = ,,; = I.

Approximate band absorption calculations for methane

617

where we have set gvi' = gt = 1, used the expression (v~" + 1).I

I/3°1 = ( v ; + 1)[/3°1

(1)!(vl)!

and introduced the symbol [/3°12 to denote the vibrational matrix element for the ith of the 0 -+ 1 transitions. If we now perform a summation over all possible lower states v)', we obtain the result 87r3NT~o[l - - e x p ( - ~ u0] ~ [ v ~ - ~ v t ' + l ( i = 1,2 . . . . .

t=1

n)] =

v I =o

3hcp n

(B-7)

X

h

[1 - e x p ( - u 0 ]

i=1

since

v~'=o

(v7 +m)! (m)!(v~)! [exp(-v~u0] = [1 - e x p ( - u 0 ] - ( m + l )

For n = 1, equation (B-7) reduces to equation (19) of ref. 27. Furthermore, ~[v~--+ v ~ + l ( i = 1,2 . . . . .

n)] v

v. -o ~[v~v~+l(i-v

1-exp[ To

1,2 . . . . .

n

~,o

0

u,(T)] i=1

(B-8)

i-I(1-exp[-udT)]) i=1

where we have assumed that exp-[ui(To)] ~ 1 and have used the ideal gas law. B. The "pesudo-harmonic oscillator" approximation

We define the "pseudo-harmonic oscillator" a p p r o x ' m a t i o n as the approximation in which an expansion of the electric m o m e n t in terms of the permanent dipole, quadrupole, etc. moments is employed, in conjunction with harmonic oscillator wave functions, for intensity estimates. Furthermore, the anharmonicity contributions associated with the use of anharmonic potential functions are always assumed to be negligibly small compared with unity. This type of approximation is properly characterized as the first step in an iteration procedure. We shall first use the pseudo-harmonic oscillator approximation on the assumption that each oscillator is non-degenerate. Next we consider arbitrary degeneracies and show that the final relation for the ratio of integrated intensities remains unchanged. For non-degenerate (independent) oscillators, it is convenient to utilize the corrected treatment of CRAWFORD and DINSMORE,(28) specialize their results by deleting all terms containing anharmonicities, and treat each transition v'~ -~ v~+8 i (i = l, 2 . . . . . n) as

618

L.D. GRAYand S. S. PENNI-R

an independent h a r m o n i c oscillator. In our notation, equation (57) o f ref. 28 becomes ,3'3

~"/=0 . . . .~(0 . . -~. a,) . .

~ (v~'+aO! -= ~'i'=° 8,.'r"~,. e x p ( - ~ f u J

= [1-exp(-uOl

-~<+i~.

(B-.9)

In forming the ratio

W

x[t,

v i' = O(i = 1 , 2 . . . ,

-6i(i = I,.,~ . . . . n)],

, . r T + S t ( i = I _, . . ., n)]. x[O

n)

we note that the factor appearing on the right-hand side of equation (B-9) occurs for each value of ~, (i = 1, 2 . . . . . n). Furthermore, it follows from equation (B-6) for c. 0 and gt,~' = gi = 1 that 8~r~ co N r ~-[0

~Si{i

=

I. 2 ....

II1]

. . . . . . . . . . . .

3hop I-~ [1 - e x p ( - u , ) ]

~

i=1

,

}[

i,__I711<1 ' ,-exp (" - 7" -where the quantities I/9~12 represent the squares of the matrix elements for the transitions 0 ~,~, (i = 1, 2 . . . . . n) and the exponential factor is the usual induced emission term. F r o m equations (B-9) and (B-10) it is now apparent that 3O

'"+8i(i = I , 9

J,

.

v i = O ( i = 1 , 2 . . . . . n)

n)l

.

=

[l-exp( 77 ~ / 0 9

- ~ uiS,) l

n

....

Nr[]~]/3~;12 ':~

3hcl'

- ' -cxp[u,( r)]l a

~t

I Ill

IB 11)

I=1

and, therefore. /__,

7.[t i

.ti+8t(i

1,2 . . . . .

n)] 'r

t

',l-cxp[-

/

v i ' = O d = 1 , 2 . . . . . n)

T0

~ tfi(T)~i]'~ i = 1

.....

]

v'{ = O ( i = l,'2 . . . . . n)

-exp[-,

i=1

t

rl (B -12)

if we assume again that exp-u,(T0) <~ 1. For the p s e u d o - h a r m o n i c oscillator with degeneracy, it is necessary to evaluate directly the quantities /3]~ in equation (B-6). Expansion of the electric m o m e n t / , in the normal (degenerate) coordinates q~ leads to the relation n.

9i

t~ = #o+ ~ I~t ~ qy,+ i

1

y=l

n

~ i.]=l

9;

tqj

~ 7,7'=

qT,qT'j+ . . .

(B-13)

t

where only terms t h r o u g h the quadrupole terms have been given explicitly. Examination

Approximate band absorption calculations for methane

619

of the matrix elements arising from the terms in equation (B-13), in conjunction with harmonic oscillator wave functions, shows that (t, 7 +g~ + 3~ - 1)! I/3~,1 ~ -

< v';t~l v7 + 8, > ~ oc

(B-14)

(vT+g~-l)!

Furthermore, if the ith mode has a degeneracy of gi, then g~i' represents the number of ways of forming v~ from g~ indistinguishable vibrations. This number turns out to be (29) (v~ + g i - 1)! gv7

(gi - 1)l v~!

Hence 00

I-~ flozNd [exp - ( v ~ u )] vi" = o(i = 1,2 ..... n)

I=1

(v~+g~+~-l)!( i=1v~=o(~=1.2 ..... ,o I~I ( g * + 3 * - 1)! *=z

( g , - 1)!

~+g~-I

( 1) ttf + g f + ~ , - 1 ) !

V~=0(/.=1,2

.....

n)

exp( - v;ui)

(g~-l)!v~!

(v~+g~-l)!

[exp( - z ,'ui)]

v;l(g, +8~ - 1)!

I~I ( g ~ + 3 i - 1)! [1 - e x p - (ui)] -(g,+a).

(g-F)!

(B-15)

Introduction of equation (B-15) into equation (B-6) again leads to equation (B-12). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

D. M. DENNISON and S. B. INGRAM, A new band in the absorption spectrum of methane gas, Phys. Rev. 36, 1451-1459 (1930). M. JOHNSTON and D. M. DENNISON, The interaction between vibration and rotation for symmetrical molecules, Phys. Rev. 48, 868 883 (1935). W . H . SCHAFFER,H. H. NIELSEN and L. H. THOMAS,The rotation vibration energies of tetrahedrally symmetric pentatomic molecules, Phys. Rev. 56, 895 907, 1051 -1059 (1939). H. A. JAHN, Coriolis perturbations in the methane spectrum IV four general types of coriolis perturbation, Proc. Roy. Soc. A171, 450 468 (1939). A . M . THORNDIKE, The experimental determination of the intensities of infra-red absorption bands III. Carbon Dioxide, methane, and ethane, J. Chem. Phys. 15, 868 874 (1947). H . L . WELSH, P. E. PASHLER and A. F. DUNN, Influence of foreign gases at high pressures on the infrared absorption band of methane at 3"3 t*, J. Chem. Phys. 19, 340 344 (1951). D. R. J. BOYD, H. W. THOMPSON and R. L. WILLIAMS, Vibration rotation bands of methane, Proc. Roy. Soc. A213, 42 54 (1952). D . R . J . BOYD and H. C. LONGUET-HIGGINS, Coriolis interaction between vibration and rotation in symmetric top molecules, Proe. Roy. Soc. A213, 55 73 (1952). L. H. JONES and R. S. McDOWELL, Force constants of methane, infrared spectra and thermodynamic functions of isotopic methanes, J. Mol. Spectrosc. 3, 632 653 (1959). R. L. ARMSTRONG and H. L. WELSH, The absolute intensities of the infrared fundamentals of methane, Spectrochim. Acta. 16, 840 852 (1960). E . K . PLYLER, E. D. TIDWELL and L. R. BLAINE, Infrared absorption spectrum of methane from 2470 to 3200 cm -a, J. Res. Nat. Bur. Stand., Wash. 64A, 201 212 (1960). R . E . HILLER, Jr. and J. W. STRALEY, Vibrational intensities and bond moments in deuterated methanes, J. Mol. Spectrosc. 5, 24-34 (1960). K . T . HECHT, The vibration rotation energies of tetrahedral XY4 molecules. Part I. Theory of spherical top molecules, J. Mol. Spectrosc. 5, 355-389 (1960). K . T . HECHT, The vibration-rotation energies of tetrahedral XY.t molecules. Part II. The fundamental va of CH4, J. Mol. Spectrosc. 5, 390~404 (1960).

620 15. 16. 17. 18. 19. 20. 2l. 22. 23. 24. 25. 26. 27. 28. 29.

L . D . GRAY and S. S. PENNER K. Fox, Vibration rotation interactions in infrared active overtone levels of spherical top molecules; 2v3 and 2v4 of CH4, 2v3 of CDa, J. Mol. Spectrosc. 9, 381--420 (1962). D . E . BURCH and D. WILLIAMS, Total absorptance of carbon monoxide and methane in the infrared, Appl. Opt. l, 587-594 (1962). J. HERRANZ and B. P. STO~CHEFF, High-resolution raman spectroscopy of gases. Part XVI. The v~ R a m a n band of methane, J. Mol. Spectrosc. 10, 448-483 (1963). W . A . MENARD, Band and line structure models for correlation of gaseous radiation, M.S. Thesis, UCLA, January 1963; see also, D. K. EDWARDS and W. A. MENARD, Correlations for absorption by methane and carbon dioxide gases, Appl. Opt. 3, 847-852 (1964). D. VANDERWERE, A study of the temperature dependence of the total absorptance of CO near 4.7 tt and 2.3 ~t, and CH4 near 3.3 ~, Scientific Report No. 5, Contract No. AF19(604)-6141, Ohio State University Research Foundation, Columbus, Ohio, May (1964). S. S. PENNER and P. VARANASI,Approximate band and total emissivity calculations for C()~, J Q S R T 4 , 803-810 (1964). S.S. PENNER and P. VARANASI,Approximate band absorption and total emissivity calculations for H20, J Q S R T $, 391 (1965). S.S. PENNER, Quantitative Molecular Spectroscopy and Gas Emissivities. Addison Wesley, Reading (1959). G. HERZBERG, Molecular Spectra and Molecular Structure 11. hl/?ared and Raman Spectra of Polyatomic Molecules. Van Nostrand, Princeton (1945). R. LADENBURG, Appendix I in R. LADENBURG and S. LEVY, Untersuchungen tiber die anomale Dispersion angeregter Gase, Z. Phys. 65, 200-206 (1930). K. ARAK), Numerical table of the integral expressing the anaount ol'absorption or emission of the spectral lines, Publ. Astr. Soc. Japan, 6, 1 0 9 112 (1954). E. JAHNKE and F. EMDE, Tables of Flmetions with Formulas and Curt,es. Dover, New York (1945). K . H . ILLINC.IER and D. E. FREEMAN, Temperature dependence of infrared and Raman intensities, J. Mol. Spectrosc. 9, 191 203 (1962). B . L . CRAWEORD, JR. and H. L. DINSMDRE, Vibrational intensities. I. Theory of diatomic infra-red bands, J. Chem. Phys. 18, 983 987, 1682 1683 (1950). L . D . LANDAU and E. M. LIFSCHITZ, Quantum Mechanics (Non-Relativ&tic T/woo'), p. 368. Perga men Press, London (1958).