The Intensity of Quadripole Radiation in the Alkalis and the Occurrence of Forbidden Lines

The Intensity of Quadripole Radiation in the Alkalis and the Occurrence of Forbidden LinesAuthor(s): A. F. StevensonSource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 128, No. 808 (Aug. 5, 1930), pp. 591-599Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/95486 .

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Quadripole Radiation in Alkalis. 591

crystal is brought into an atmosphere favourable to disintegration. Two other types of disintegration are observed in crystals of the same external form, one type characterised by a pit at each end of the C axis, the other by a cavity at the middle of the C axis. These two types are explained as due to the two

possible modes of twinning of a polar crystal on the basal plane. In view of the conclusions of Barnes, the polarity of the ice crystal is ascribed to the

asymmetric location of the hydrogen ions in the lattice with respect to the basal plane.

The Intensity of Quadripole Radiation in the Alkalis and the Occurrence of Forbidden Lines.

By A. F. STEVENSON, University of Toronto.

(Communicated by R. H. Fowler, F.R.S.-Received May 30, 1930.)

1. The occurrence of the series 1S - mD in the spectra of the alkali metals, in particular the line 1S - 3D, has been observed, both in emission and absorption, by several authors,* although such transitions are forbidden by the selection rule for azimuthal quantum numbers. Such lines are very weak, and might be caused by the presence of ionic or other external electric fields, in which case the selection rule may be violated; but they have been observed in

emission under conditions which were thought to preclude the existence of such fields, and certainly this would be the case in absorption. Another

explanation must therefore be sought. It is now recognised that the ordinary formula for the intensity of a spectral

line as given by quantum mechanics, and which yields the selection rules, is only a first approximation, which amounts to taking the " dipole moment" of the atom. A more exact treatment yields higher order terms as corrections, and these so-called " multipole " terms, though much weaker than the ordinary

dipole ones, may allow transitions which are " forbidden " by the ordinary selection rules. In this paper, the absorption intensities of the line 1S - 3D for the alkalis, lithium, sodium, potassium and rubidium, are calculated on the

supposition that this line is due to " quadripole " radiation, using Hartree's method of self-consistent fields for the radial wave-functions, and it will be

* Sowerby and Barratt, 'Roy. Soc. Proc.,' A, vol. 110, p. 190 (1926); Schrum, Carter and Fowler, 'Phil. Mag.,' vol. 3, p. 27 (1927); see also Foote, Meggers and Mohler, 'Astrophys. J.,' vol. 55, p. 145 (1922), and Datta, 'Roy. Soc. Proc.,' A, vol. 101, p. 544 (1922).


crystal is brought into an atmosphere favourable to disintegration. Two other types of disintegration are observed in crystals of the same external form, one type characterised by a pit at each end of the C axis, the other by a cavity at the middle of the C axis. These two types are explained as due to the two

possible modes of twinning of a polar crystal on the basal plane. In view of the conclusions of Barnes, the polarity of the ice crystal is ascribed to the

asymmetric location of the hydrogen ions in the lattice with respect to the basal plane.

The Intensity of Quadripole Radiation in the Alkalis and the Occurrence of Forbidden Lines.

By A. F. STEVENSON, University of Toronto.

(Communicated by R. H. Fowler, F.R.S.-Received May 30, 1930.)

1. The occurrence of the series 1S - mD in the spectra of the alkali metals, in particular the line 1S - 3D, has been observed, both in emission and absorption, by several authors,* although such transitions are forbidden by the selection rule for azimuthal quantum numbers. Such lines are very weak, and might be caused by the presence of ionic or other external electric fields, in which case the selection rule may be violated; but they have been observed in

emission under conditions which were thought to preclude the existence of such fields, and certainly this would be the case in absorption. Another

explanation must therefore be sought. It is now recognised that the ordinary formula for the intensity of a spectral

line as given by quantum mechanics, and which yields the selection rules, is only a first approximation, which amounts to taking the " dipole moment" of the atom. A more exact treatment yields higher order terms as corrections, and these so-called " multipole " terms, though much weaker than the ordinary

dipole ones, may allow transitions which are " forbidden " by the ordinary selection rules. In this paper, the absorption intensities of the line 1S - 3D for the alkalis, lithium, sodium, potassium and rubidium, are calculated on the

supposition that this line is due to " quadripole " radiation, using Hartree's method of self-consistent fields for the radial wave-functions, and it will be

* Sowerby and Barratt, 'Roy. Soc. Proc.,' A, vol. 110, p. 190 (1926); Schrum, Carter and Fowler, 'Phil. Mag.,' vol. 3, p. 27 (1927); see also Foote, Meggers and Mohler, 'Astrophys. J.,' vol. 55, p. 145 (1922), and Datta, 'Roy. Soc. Proc.,' A, vol. 101, p. 544 (1922).



A. F. Stevenson.

seen that the calculated intensities agree fairly well with the observed, where such are available, considering the uncertainty in the experimental result. A similar explanation has been given in support of Bowen's hypothesis regard- ing the occurrence of certain nebular lines.*

As regards the occurrence of forbidden emission lines, it should be remarked that the average life-time of an atom emitting

" quadripole radiation " is

much longer than in the case of an ordinary dipole emission, and that, at any rate in the laboratory, it is quite possible that the atom would not remain undisturbed long enough for quadripole radiation to become effective.t In this case, quadripole radiation may not be an adequate explanation. Such a phenomenon would not, of course, arise in absorption.

2. A formula for the intensity of quadripole radiation appears to have been first given by Gaunt and McCrea,j using an extension of Dirac's method of treating radiation by means of quantum mechanics. They apply their

result to the case of a harmonic oscillator and a rigid rotator.

Using rather more classical ideas, an expression for the intensity of multipole radiation was derived by Rubinowicz,? who worked out the exact formula for the intensity of the Lyman series in hydrogen. More recently, Blatonil has

applied the same method to the Balmer series. The main idea of the method is that a retarded vector potential is used, and that in the classical expression for the potential, the current density is taken to be the Schr6dinger current

density of wave mechanics associated with a given transition.

Let R be the distance from the point of observation to the scource of radiation

of frequency v, and let S be the current density. Then we define the vector

potential, A, by? A l e27rt e- dr,

the integral being taken throughout space; and the intensity of radiation in

the z-direction, per unit solid angle, is given by?

J 2- (1)

where k = 27v/c, and z is a unit vector in the z-direction.

* Bartlett, 'Phys. Rev.,' vol. 34, p. 1247 (1929). t Cf. Bartlett, loc. cit. 1: Gaunt and McCrea, ' Proc. Camb. Phil. Soc.,' vol. 23, p. 930 (1927). ? Rubinowicz, 'Phys. Z.,' vol. 29, p. 817 (1928); 'Z. Physik,' vol. 53, p. 267 (1929). 11 Blaton, 'Z. Physik,' vol. 61, p. 263 (1930). ? Rubinowicz, 'Phys. Z.,' loc. cit.

592



Quadripole Radiation in Alkalis.

Denoting by r the distance of the point of observation from the centre of the atom, taken as origin, we may write R = r - z, and, omitting the time

factor,t -icr e -ikr r 1

' A=-- S eik d: -- Sdr + ik zS d+-... . (2)

r Jc cr _

The first integral in the last expression gives the dipole contribution to the

intensity, the second the quadripole contribution, etc. From (1) and (2) we have for the intensity of quadripole radiation

J C- [z (Sx + is) dd + z (S- is,)dd

' (3)

3. Let us now assume that we are dealing with a one-electron problem in a central field of force, and consider a transition from a state denoted by /1. to one denoted by d2 (we shall always use a suffix 1 to denote the initial, and 2 the final state). Then the associated Schrodinger current density is

he -S 4im (4/ grad 2*- 2* grad 4), (4)

where 4* is the conjugate of 4. From (3) and (4), and noting that

y oax d = y Z+2* axa d

etc., we have for J,

J =7he2 (X X2), (5)

where

X, = JzI+l( a i dX)2* dr 2

the integrals being taken throughout space. Taking polar co-ordinates, let 1, m denote respectively the azimuthal and

magnetic quantum numbers for the state 41, and 1', m' the corresponding numbers for the state 42,. Then we may write, with the usual notation

1P = R,PR (cos 0) e-im,

Y2a* -= R2PI (cos 0) e+imn,

t Cf. Bartlett, loc. cit.

593



A. F. Stevenson.

where R1, R2 depend on r only. We shall require the following formulae for

spherical harmonics:

e (On'+ 1) [(d R2 dR2 + 1?R)m i1 -- 2 --- R 2 dR 1' + 1 Et-- Pt' +

21' + L\ dr r d' + d'-r r

8X '-I

= ' II * i'-n'+l) (1e'- * +) " (dtd 2_ -+ R2) pl, ( I

,,>\ n . / " (

+ (1' + m')(' + m' - 1) ?- + R2) Pll j

211 dr r

P P"m' sin 0 cos 0 dO = O (' t I 1 1)

2 (-i +w(+1)! (21 (21 + 1 3) 3) (-rn) !

(1' 2 1 + 1)

(P') sin -dO = 2 (I+ )!

At -- , i 2; ?Xt = i{ .

J2 -+ 1 1) mj1 I

The first two are equivalent to some giv een by Darwin and may be proved from known properties of splerical ha; 2 he last two are well known.

On substituting from (6) in (5), we find that X1 and X2 both vanish unless ulty or I ? 2, and m' = mn ? 1. Thus the selection rules for quadripole

radiation in the z-direction are

Al 0, ? 2; A += ? 51.

The z-direction has, of course, here been specialised. Taking account of the other two directions, the general selection rules are easily seen to be Al = 0, ? 2, Am == 0, ? 1, as compared with Al =? 1, Am = 0, ? 1, in the ordinary case. These results also follow immediately when the intensity is expressed as a matrix, as in Gaunt and McCrae's result.

4. We take first the case Al = + 2 (the case Al =- - 2 is evidently deducible from this). Putting I' = + - 2 and m' = m- 1, say, we find without

difficulty with the use of (6), and inserting the normalisation for the 0, d part of the integration,

x (l+m++l)(1--n+l)(1-m+2)(l m+3)1 i 2 (21 + 1) (21 + 3)2 (21 + 5) '

- Darwin, ' Roy. Soc. Proc.,' A, vol. 118, p. 668 (1928).

594




where

I = [r3dR + (I + 3) r2R,R dr. (8)

Similarly, taking 1' = I + 2 and m' = m - 1, we find that X1 - 0, while X2

only differs from the above expression for X1 in having the sign of m changed. In the absence of a magnetic field, to find the total intensity (per atom) for

a transition Al = + 2, we take the sum of the results for Am - ? 1, sum for m from - I to + 1, and average over all initial states, i.e., divide by 21 + 1, the weight of the initial state. We thus find, from (5) and (7),

j 7h2v (l+ 1) (l+ 2) .2. (9) 2mno2cc 5 (21+ 1) (21+ 3)

Consider now the case Al = 0. Putting 1'= 1, mw' = m - 1, say, in (5), and

noting that in this case, on account of the orthogonality of the eigen functions, we have

co >r2RR2 dr 0 (Rl ? R2),

we find, using (6), and after normalising for 0 and 4),

X - (I + m) (I - m + 1) (2m - 1)2 M ( x- (21- 1)2 (21 + 3)2 M X= (1

where

M= l -- dR dr= --- - R- dr

R dr.

Jo ~'a

RJ dr

With ' = l, mi '- + 1, we find, as before, X1 = 0, while X2 only differs from the expression for X1 in (10) in a change of sign of m. To find the total

intensity for a transition A1 0, we must sum for m and average over all initial states, as before. Since Im I 1, the summation for m, in the case m' = m - , is now from (- I + 1) to 1; on account of the factor (l + m) in (10), however, this is the same as summing from - 1 to + 1, and similarly in the case m' = m - 1. We hence find for the intensity, from (10) and (5)

j h2e2v4 21 +- 1) (1) 2m0c5 ' 15 (21- 1) (21+ 3)

An S- S transition (I = 0) is thus still " forbidden," even for quadripole radiation.*

* Cf. Rubinowicz, loc. cit.

595



A. F. Stevenson.

5. We shall now apply the results (9) and (11) to compare the intensities of the simplest types of " forbidden " transition, namely S - D and P -> P, with that of the " allowed " transition S - P. Putting 1 = 0 in (9) and (8), we have for the intensity of an S -V D line, of frequency vl, say,

J 15 rzh2e 2 (12) where

I =f (r3Ri dR2 + 3rRiR2) dr = - r3R - dr. Jo\ dr ) Jo0 dr

Or, writing R P/r, and introducing " atomic " units, i.e., h2/4W2mo0e2 as unit of

length,

T12 - ( r ) 2 . (13)

Pi12 dr. p22 d

where we have now inserted the normalisation factor for r.

Now the ordinary (dipole) intensity of radiation in the z-direction for an S -> P line, of frequency v2 say, is given by*

J = 4 3e2v

V r3RR2 dr .

Or, introducing P and atomic units as before, and normalising,

4_C32 h4 \2 2

2- -

3c \4( me2) I22 (14)

where 00 \2

PP2 dr) 122 0

. (15) pl2 dr. P22 dr o Jo

From (12) and (14), the ratio of the intensity of an S- D line to that of an

S -P line is

J_ - h c(2e2)* (i *V I12V (16)

1x, I2 are given by (13) and (15). P1, P2 refer, respectively, to an S and D state in I1, and to an S and P state in I2.

* Sommerfeld, " Wellenmechanischer Erganzungsband," p. 96. To conform to the above scheme Sommerfeld's expression must be multiplied by a factor ~, due to taking a time mean. This result may also be obtained from (1) and (2) above.

596




(16) assumes, of course, that the lines S -- D and S -P are both emissioni lines. For the case when they are absorption lines, using the well-known relation between the Einstein coefficients, namely, A =- 8rchV3/c3. Brs, we find for the ratio of the absorption intensities, in place of (16)

i 2 (2re) 2 '2 (17) J1_ ,2ie2h 2 Vx

I_._ 2

(17)

Taking now the case of a P - P transition, of frequency vl, say, (11) gives

=-Ch2e2Vl 4 T

2J 4 'Ihev1 T 2 Jl =74rh 13,

where

1r i,P idr 2

J dr -}P2rr I32 O. D

Pi 2dr . | p 22dr o Jo

For the intensity of a P - S line (instead of S -~ P) of frequency v2, the expression (14) must be multiplied by -, since the weight of the initial state is now 3 instead of 1. Hence the ratio of the (emission) intensity of a P - P line to that of a P -S line is

J1 2 (2^e ) (18)

I2 being given by (15). 6. Using (17), the numerical values of the absorption intensity ratio of

1S - 3D to that of IS - 2P have been calculated for the first four of the

alkalis, Hartree's method of self-consistent fields* being used to obtain tabulated values of the radial wave-functions P, the integrals 11 and I2 being then calculated by numerical integration; the experimental values of v were used. The radial wave-functions for Rb have been calculated by Hartree, those for Na by McDougall, and those for the S and P terms for Li by Har-

greaves,t the solution for hydrogen being used for the D term for Li as a sufficient approximation.

The necessary functions for K were calculated by the author from the self- consistent field obtained by Hartree. The calculation involves incidentally the determination of the eigen values e (e = 1/n2 where n is the effective

quantum number). The approximate calculated and observed values are

* Hartree, 'Proc. Camb. Phil. Soc.,' vol. 24, pp. 89, 111 (1928). t Hargreaves, ' Proc. Camb. Phil. Soc.,' vol. 25, p. 15 (1929).

597



A. F. Stevenson.

given in Table I, the mean of the two term-values being taken for doublets. The agreement would be improved by adding in the spin and relativity cor-

rections, though even then, as pointed out by Hartree, a smaller value than the observed is only to be expected on account of the fact that the polarisation of the atom-core by the series electron has been neglected.

Table I.-Approximate Calculated and Observed Eigen values (E) for Potassium.

Term. 1S. 2P. 3D.

Calculated ................................ 0 261 0-171 0-114

Observed .................................... 0 319 0-200 0 123

The values of I12 and I22 as found by numerical integration for the alkalis

considered, together with the intensity ratio calculated from (17) and the

experimental values for K and Rb, are given in Table II. The result for

hydrogen, as calculated from Rubinowicz' formula (loc. cit.) is also given for

comparison.

Table II.--Ratio of Absorption Intensities of (1S - 3D) to (1S - 2P).

Element. Li. Na. K. Rb. H.

I2 ........ ......... 2335 2-627 3312 3.338

122 ................................. 18-60 23-89 38-19 25-25

J1/Ja calculated ............28 2810-6 20x10-6 1 -5x 10-6 2-1x10-6 1 lx 10-

J1/J9 observed ................ X- -x 85X10 -6

It will be seen that the calculated intensity ratios all lie close together. As

regards the experimental values, the first value for K and that for Rb are taken from Sowerby and Barratt's paper (loc. cit.), while the second value for K is a result for the relative number of dispersion electrons given by Rasetti,* since the number of dispersion electrons per atom associated with a given frequency is proportional to the corresponding absorption intensity. Values for Li and Na do not, unfortunately, appear to be available, though, as stated

before, the absorption line 18 - 3D has been observed for all the alkalis.

* Rasetti, 'Atti Accad. Lincei,' vol. 6, p. 54 (1927).

598




While the agreement is far from good, it may perhaps be regarded as fairly

satisfactory, considering the difficulty of estimating the relative intensities

and the fact that the experimental values for K differ widely. Rasetti esti-

mates the probable error of his result as 50 per cent. The relative emission intensity of 3P -> 2P to that of 3P - 1S has been

calculated from (18) for the case of Rb only. The values found were:-

I12 = 1-288, 12 = 04949, J1'/J2' = 16 X 10-6.

The corresponding ratio for hydrogen as calculated from Blaton's results

(loc. cit.) is 6-7 X 108. The series P -> P does not appear to have been observed experimentally in

the alkalis; a few lines of this series are given in Fowler's "Report on Series

in Line Spectra " for lithium, but these were doubtless due to external electric

fields. In any case, since this series only occurs in emission, it seems probable, for the reason mentioned earlier, that this numerical result has not the same

experimental value as those given for absorption.

I wish to express my thanks to Mr. R. H. Fowler for suggesting this problem, and for his helpful criticism; I am also indebted to Dr. Hartree and Messrs.

McDougall and Hargreaves for supplying me with the radial wave-functions mentioned above.



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The Intensity of Quadripole Radiation in the Alkalis and the Occurrence of Forbidden Lines