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The Intensities of Certain Nebular Lines and the Mean Lives of Atoms Emitting them Author(s): A. F. Stevenson Source: Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 137, No. 832 (Aug. 2, 1932), pp. 298-325 Published by: The Royal Society Stable URL: http://www.jstor.org/stable/95911 . Accessed: 08/05/2014 03:56 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. http://www.jstor.org This content downloaded from 169.229.32.137 on Thu, 8 May 2014 03:56:23 AM All use subject to JSTOR Terms and Conditions

The Intensities of Certain Nebular Lines and the Mean Lives of Atoms Emitting them

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The Intensities of Certain Nebular Lines and the Mean Lives of Atoms Emitting themAuthor(s): A. F. StevensonSource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 137, No. 832 (Aug. 2, 1932), pp. 298-325Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/95911 .

Accessed: 08/05/2014 03:56

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of theRoyal Society of London. Series A, Containing Papers of a Mathematical and Physical Character.

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298

The Intensities of Certatn Nebular Lines and the Mean Lives of Atoms emitting them.

By A. F. STEVENSON, Department of Applied Mathematics, Uninversity of Toronto.

(Communicated by R. H. Fowler, F.R.S.-Received February 12, 1932.)

? 1. Introduction.

There are certain prominent lines in the spectra of the galactic nebuli- in particular the " nebuliun " green lines, N1 (? = 5007) and N2 (X = 4959), which dominate most nebular spectra-whose identification remained till recently a mystery.t In 1927, however, Bowen proposed his now well-known

hypothesis that such lines are due to transitions from metastable states in certain ions.t The spectra involved are those of 0 III (which includes the lines N1, N2), N II, 0 II, S II, and 0 I.?

To explain the fact that such lines occur, although they are "aforbidden" transitions, Bowen postulated that there is a non-zero, though relatively small, probability of transition between the states in question. The strength of such lines is then accounted for if we suppose there is a large concentration of ions in the initial states in question due to transitions from higher levels, and that, due to the very low density in the nebula, such atoms -remain undisturbed by collisions long enough for the radiation to become effective. The non- occurrence of such lines in the laboratory (except the 0 I lines observed by Hopfield, loC. cit.) is thus explained by our inability to reproduce such conditions.

Although the agreement obtained between the calculated and observed wave-lengths is so close as to leave little room for doubt as to the substantial correctness of Bowen's hypothesis, it is evidently desirable to have additional information concerning the theoretical iitensities of such lines, and the mean lives of the atoms when in the initial states in question, and to see whether any

t For a full account of nebular spectra in general, see Becker and Grotrian, ' Ergebn. exakt. naturwiss,' vol. 7, p. 8 (1928).

t Bowen, 'Nature,' voL 120, p. 473 (1927) 'Astrophys. J.,' vol. 67, p. 1 (1928); Proc.

Nat. Acad. Sci. Wash.,' vol. 14, p. 30 (1928); 'Nature,' vol. 123, p. 450 (1929); A. Fowler, 'Nature,' vol. 120, pp. 582, 617 (1927).

? Paschen, 'Naturwiss,' vol. 18, p. 752 (1930); Hopfield, 'Phys. Rev.,' vol. 37, p. 160 (1931).

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Intensities of Certatn Nebular Lines. 299

additional confirmation is thereby obtained. Preliminary calculations to this end have been made by Bartlett,t who, however, confined himself to calculating relative intensities only. His calculations, moreover, are incomplete, due to the approximations adopted, so that his results, though in the main con- firmatory of Bowen's hypothesis, remain somewhat inconclusive.

In this paper we make calculations of the intensities and mean lives for 0 III and N II, which are the most important. These calculations are complete if the contributions of double transitions to the intensity can be neglected, a neglect which cannot yet be tested. It will be found that the relative intensities calculated here are in agreement with those observed in the nebula as far as comparison is possible, and the result for the mean lives-which, it is hoped, are fairly accurate-furnishes interesting information concerning the physical conditions in the nebulae.

The ordinary dipole term in the intensity vanishes of course for the lines in question, since they are " forbidden " transitions. A possible explanation is that the transitions are due to the presence of ionic electric fields in the nebulae; but this explanation must be rejected, since such fields must be very weak on account of the low density, and, moreover, if such were the case, a broadening of the nebular Balmer lines should result, which is not observed.4 We shall follow Bartlett in ascribing the lines to quadripole radiation; taking into account the quadripole terms sufficient to ensure the non-vanishing of the intensities. It may now be regarded as fairly definitely established that the occurrence of " forbidden " lines-unless due to disturbing external fields of some kind-can usually be ascribed to quadripole radiation (or possibly radia- tion of higher poles, though this does- not yet seem to have arisen in practice). The hypothesis of quadripole radiation in relation to forbidden lines has been shown to be in satisfactory agreement with experiment in the case of the absorption spectra of the alkalis,? the Zeeman effect in potassium,Ij and the Zeeman effect in the auroral green line of 0 I.T

In the case of 0 III and N II we have, effectively, a two-electron problem,

Bartlett, 'Phys. Rev.,' vol. 34, p. 1247 (1929). $ Becker and Grotrian, loc. cit., p. 69. Cf. also Huff and Houston, 'Phys. Rev.,' vol.

36, p. 842 (1930). ? Stevenson, 'Proc. Roy. Soc.,' A, vol. 128, p. 591 (1930); Whitelaw and Stevenson,

Nature,' vol. 127, p. 817 (1931). Segr6, 'Z. Physik,' vol. 66, p. 827 (1930).

? Frerichs and Campbell, 'Phys. Rev.,' vol. 36, p. 1460 (1931). For a summary of quadripole radiation in general, see Segre, 'Nuovo Cimento,' Anno VIII, N. 2 (1931).

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300 A. F. Stevenson.

and our calculations are based on the work of Gauntt on the triplets of helium, in which he also considered to some extent the present problem, anld to which this paper is much indebted. Gaunt extended the Dirac wave-equation to the case of two electrons, but although his calculations gave a satisfactory account of the fine structure of He, it should be mentioned that subsequent work by Breitt has shown fairly conclusively that the form of the spin interaction energy adopted by Gaunt is incorrect. Breit gives a closer approximation for it, and also works with more accurate wave-functions, and succeeds in obtaining a better agreement with experiment for He than does Gaunt. The correct form of the Dirac wave-equation for more than one electron is not yet known- and, indeed, it is not yet certain whether the equation for a single electron is entirely correct. In the present instance, however, we cannot hope for great accuracy in the calculations in any case, and it appears that Gaunt's methods, which entail less calculation than those of Breit, should be sufficient for our purpose.

After giving the requisite formula for the intensity in ? 2, the method of calculating the wave-functions by a perturbation method is explained in ? 3; the perturbation theory used is similar to that of Gaunt, but somewhat more straightforward. In ?? 4, 5 we give the calculations for the wave- functions and intensities, and in ? 6 the numerical values are given and dis- cussed with special reference to the nebulae. We summarise briefly the con- clusions in ? 7.

? 2. Intensity of Quadripole Radiation.

Following the method of Rubinowicz,? it was shown in a previous paper (loc. cit., equation (3) ) that the intensity of quadripole radiation, per unit solid angle, in the z-direction, is

X Ir a v z (Sx + isv) dr + z (S,,-tiS) dr:]

In this, the current density is taken as the real part of Se2,r,vt, where v is the frequency of the radiation, and the integrals are taken throughout space.

Quantum-mechanically, in place of S we have, for a transition from state s to state t, a three-dimensional current density

Ss e2Lvst t ? S*e2rvtt - 2R(SSt e2r,vStt),

f 'Phil. Trans.,' A, vol. 228, p. 151 (1929), referred to hereafter as " G." Also 'Proc. Roy. Soc.,' A, vol. 122, p. 513 (1929).

$ Breit, 'Phys. Rev.,' vol. 34, p. 553 (1929); vol. 36, p. 385 (1930). ? 'Phys. Z.,' vol. 29, p. 817 (1928); 'Z. Physik,' vol. 53, p. 267 (1929).

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Intensities of Certain Nebular Lines. 301

where v is the frequency of the emitted radiation, and S,, is the (s, t) element in the matrix for S in a Heisenberg " co-ordinate system " (representation) so that the intensity, for a transition s -* t, becomest

J 4t3vt4 [ |z (Sx + jSv) dT + z (s iS)d|] (2.1)

Now in a two-electron problem, the current density j is a six-dimensional vector, and

iw=[X j dl + 0 j$, dr2] [X21 Y2, Z2) +Xf, Yi, zdj

- z, y zJ x, y z

etc., where 1 and 2 refer to the configuration spaces of the two electrons, so that

zsx d- = X (z, + Z2 jx2,) dt1 dt2,

etc. Therefore (2.1) becomes, for a two-electron problem

set= [Z1 (jxi + ijY,) + Z2 (Jx2 + ijY2)] d-L dr2

+ Z [z1 (jx, - ijY,) + Z2 (U2 - ijy2)] drt dt2 } (2.2)

On the Dirac theory, the components of jst (in the one-electron case) are given byt

(ist) = - i i (i . 1, 2, 3), (2.3)

where the o's are certain matrices. The generalisation of this to the two- electron case is obtained without difficulty; but with the wave-functions used by Gaunt, this would give zero for all components. Gaunt's approximations, however, amount essentially to using Schrodinger's wave-functions with spin factors attached, and it is evident that the same approximations will give the Schrodinger expressions for the current in place of (2.3). We therefore take?

ist = eh (4s V4*t - *t V4S))

t In the previous paper, the factor 4 was omitted; as, however, we were there con- cerned only with relative intensities, the results are unaffected.

I Dirac, " Principles of Quantum Mechanics," p. 246. ? h (not 2 h !) denotes Planck's constant.

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302- A. F. Stevenson.

where a summation over the spin co-ordinates is now included. Since

t Z+8 + tdr, dr2 I- dr,tai dtl r2) ax1 x etc., we have finally, from (2.2), for the intensity

we . (2.4) m02C

where - (Z I4jss*t + Z28+f21*t) d-1r d72

2

+ J i(Z1is*1*t + z2dA* d*t) dr2 (2.5)

I J_(t) 12 + 12(st) 2, say, and

'& k + yk k a ia (k = 1, 2). (2.6)

The integrals in I(st) include summation over the spin-factors in the wave- functions. The problem now is to calculate the required approximations to the wave-functions.

In all this theory, however, the possible contributions of double transitions to the quadripole radiation have been neglected. We shall have to be content here to evaluate the one type of contribution only; even this is a lengthy matter. More accurate discussions, including the effects of double transitions, have never yet been given so far as we know. Even Bartlett's work, in spite of its form, really proceeds from the fundamental equations of this section and discusses only the contributions of simple type.

? 3. Perturbation Theory. In Gaunt's theory we have to deal with systems whose energy levels are

nearly equal in pairs-" nearly degenerate systems " as Gaunt terms them- which necessitates a slight modification of the usual perturbation theory. We give a treatment which gives, of course, the same results as Gaunt's (G., ? 3), but which seems rather more simple and straightforward.

Let the equation L (+) = Ei (3.1)

have a number of eigenvalues Em near some value E', say,

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Intensities of Certain Nebular Lines. 303

with corresponding eigenfunctions qm We can allow for any of these states themselves being degenerate by supposing that some of the aEm's are equal, while the corresponding 4im's are linearly independent.

Now consider a perturbed equation

(L + V) =E, (3.2)

where V is small and of the same order of magnitude as the En. We may now assume an expansion for the eigenfunctions of (3.2) of the form

p E OC ? +x, (3.3)

m=lI

where X is a series in terms of other eigen- 's of (3.1) than those having eigen- values near E'; on account of the aEm's being small, we must consider the am's to be of the same order of magnitude. Further, let the eigenvalues of (3.2) near E' be

E = E' +.

Then the ordinary pertuxbation method gives the following set of equations for the coefficients am in (3.3)

p I a. [Vmn - 8mnCm ( - MEm)] = 0, (n 1, ... p) (3.4)

m=1 where

vmn J v m )*n dr Cm-j I)m 12ddr

and the integrals include a summation over the spin-factors where such are present. The zero-approximations to the wave-functions + are thus deter- mined, and the corresponding values of e are given as the roots of the deter- minantal equation

I Vmn - mnCfr (C ( Em) I. (3.5)

Suppose now that instead of using the .m's for the unperturbed wave- functions, we use a set of p independent orthogonal linear combinations of them, say,

p i s- I bijji, (i =,..,p). (3.6)

j=l

Then the form of (3.4) and (3.5) will be altered owing to the presence of the REm'S. If we neglect the aEm's, it is known that (3.4) transform to others of the same form, say,

p E a'i [V'ij-aij C> - (1p), (3.7)

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304 A. F. Stevenson.

where V'ij_ =E bimbj.VnYm (3.8)

m, n

In our case, however, we have, from (3.4), Vmn, + Amrm nC AEm ila place of Vmn, Hence, from (3.8), we must add to the V'ii in. (3.7) the quantities

E bimbjn amnCm aEm-E bimbjmGm aEm. m, n m

Thus (3.4) transform to

p p E a' [V'ij + bimb.mCm aEm -ijCj?] =0, (j = 1, ...p), (3.9)

i=1 m=-1

and (3.5) transform to p

VI V? + Em bimbjmCm 8Em - 8ijcje 0. (3.10)

This is the modification of the usual perturbation theory required; it does not, unlike Gaunt's analysis, require the calculation of matrix elements of V other than the V'ij.

We now consider briefly another point arising in the perturbation theory used in this paper. In Gaunt's work, the perturbation V consists of two parts

Vz=P+S,

where P > S. Now, in general, perturbative effects are " additive " only when the unperturbed system is non-degenerate; that is to say, if we first work out the first order change in the energy levels, and the zero-approxi- mations to the wave-functions, taking V -P, and then do the same thing for the system so obtained taking V = S, we shall not get the same results as if we took V = P + S and performed the calculations in one step, unless the original system is non-degenerate. This may be seen easily by considering the equations (3.9) and (3.10) (whether or not the aEm's are included makes, of course, no difference). If, however, P > S, the perturbative effects are additive in the above sense, correct to O(S/P). This can be proved formally from (3.9) and (3.10), but this is hardly necessary, since the statement is fairly obvious from physical considerations.

In the present case, the zero-approximations to the wave-functions have already been calculated by Gaunt for the perturbation P alone, and partly also for S alone. The method adopted here is therefore to use as the initial

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Intensities of Certaitn Nebular Lines. 305

wave-functions (i.e., the 4'/ of (3.6) ) those which are correct zero-approxi- mations after perturbation by P alone and by S alone, and then to calculate the zero-approximations to the wave-functions to terms of order S/P by the method indicated above (it is necessary to include terms of order S/P in order to get non-vanishing intensities). This makes the calculation of the wave- functions very simple when the required matrix elements of S have been cal- culated; some, but not all, of these have already been calculated by Gaunt. It may be mentioned that this method gives very simply the relative intensity of ortbo-para to para-para transitions in He (G., p. 177).

? 4. Calculation of the Wave-functions.

The states that are to be considered for the nebular lines are the lowest energy states of 0++ and N+, for which the electronic configuration is (ls)2 (2s)2 (2p)2, giving rise to the terms (in ascending order of energy levels) 3Po, 1, 2

'DD IS. No transition is possible between these states for dipole radiation by Laporte's rule,t and we therefore proceed on the assumption of quadripole radiation. We treat the problem as if the two 2p electrons were in a central field of

force directed to the nucleus, and under the influence of their mutual inter- action. We adhere to Gaunt's notation as far as possible to facilitate reference. Gaunt adopts the following form for the interaction energy V:

V P+S5,

where P, S are respectively the electrostatic and spin parts of the interaction; also P = e2/r, and (G. (2.6) and G. (2.7))

S= T+T2+T U - e2h [(ro XP2) *al+( XP1)AG2]

+ (_eh \2 (y1.-2)-3(i1.r0)(f2.r0) (4.0) cm0oc )r3

where the suffixes 1 and 2 refer to the two electrons, p is the momentum, rro = r - r2, r being the distance between the electrons, and the components of the a's are Dirac matrices.

t This rule holds rigorously for dipole radiation to any order of the perturbation by P and S. Weyl, " Theory of Groups and Quantum Mechanics," pp. 201, 203 (English edition, 1931).

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306 A. F. Stevenson.

We take the following 15 independent wave-functions for the unperturbed (2p)2 state (G., p. 184):

4,2 = [10O (1) oll (2)- 01k (2) o1l (1)] Xa (1) Xa (2)

e 0b1l (1) k1l (2) [Xa (1) Xb (2) - Xb (1) Xa (2)]

4;E1 = [?E-1 (1) rlj (2) -k1-1 (2) c1l (1)] Xa (1) Xa (2)

=7_ [01O (1) oll (2) - 010 (2) k11 (1)] [Xa (1) Xb (2) + Xb (1) Xa (2)] 4^1 = [01O (1) bll (2) + ?p1 (2) k1 (1)] [Xa (1) Xb (2) - Xb (1) Xa (2)]

4J0 =D[k10 (1) oll (2) - 010 (2) kl1 (1)] Xb (1) Xb (2)

=pO [qf10 (1) qck-1 (2) - k10 (2) 01' (1)] Xa (1) Xa (2)

470 = [01E j (1) b1l (2) - 01 (2) kl ()] [Xa (1) Xb (2) + Xb (1) Xa (2)] (4. 1)

480 [01= b (1) k1l (2) + 01c1 (2) k1 ()] [Xa (1) Xb (2) - Xb (1) Xa (2)]

4eO 01= (1) o0i (2) [Xa (1) Xb (2) Xb (1) Xa (2)]

4ac1 _ [01-1 (1) fll (2) - ckr1 (2) k11 (1)] Xb (1) Xb (2)

=7 1 = [01 (1) 0k-1 (2) - 01k (2) 01fr (1)] [Xa (1) Xb (2) + Xb (1) Xa (2)]

i8t= [01 (1) 01- (2) + 010 (2) 0ki1 (1)] [ZX (1) Xb (2) - b (1) Xa (2)]

4)-2 =[c0 (1) 01f 1(2) - 1k0 (2) cf-1 (1)] Xb (1) Xb (2)

ie_ 01- (1) 01- (2) [Za (1) Zb (2) X b (1) Za (2)]J

In these functions, the arguments 1 and 2 refer to the two electrons, and (G. (4.03) and G. (4.04); we use R in place of g,,1),

0 u =3P1uR, (lul ?1), (4.2)

where R is the radial part of the wave-function, and, following Darwin,

/d \k+u (.2 1) Pku =f eif (k - u) ! 1 2)u/2 ( d ) / k/! ( = cos 0). (4.3)

The x's are spin-factors with components

Za 0?,0,10 (4.4) 0b 0,0 0, 1

The wave-functions are not normalised; we denote the normalisation factors by such symbols as C02. They are divided into five groups, corresponding to the different values of the total angular momentum (orbit + spin), and indicated

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Intensities of Certain Nebular Lines. 307

by the superscripts, which we shall call m? of the 4's. The functions with suffixes oc, f, y arise in the perturbation by S alone (i.e., the 3P states) -; those with suffixes e, 2 in the perturbation by P alone (1S and 1D). The five groups are " non-combining " for perturbations by both P and S, and can therefore be treated separately; but the sub-groups, symmetrical and anti-symmetrical in the position co-ordinates, combine when both P and S are taken into account.

As regards the perturbation by P alone, the above functions are all correct zero-approximations as they stand, except 480 and 4,O. A perturbation calculation replaces these two by 440, 4)O, where (G. (9.22))

+ ? = J^? + <Je? (4.5)

with P perturbation energies given by (G. (9.19))

-O 250 I2 A4 -: +U T5 C 7 D D , (4.6)

AE5s=~Do+ 2Th J

where (G. (9.17)),

a _ 18 (47)2 R2r2dr], (4.7)

and (G. (9.15) and G. (9.01) )

Do = 18 (47)2e2 Fi A.R2 (1) R2 (2) r22 r2 dr1 dr2, O rfnax

rmax being the greater of rl, r2' By dividing the range of integration for r2 into the two intervals (0, rl) and (rl, so ) and changing the order of integration in the second part, the integral for Do may be replaced by

Do ~ 36 (47)2e2f dr1 dr2. r1r22 12 (1) R2 (2). (4.8)

Similarly, from G. (9.12) 00 rr 4

D2-- 36 (47)2e2 dr, J dr2. -R2 (1) R2 (2). (4.9)

The wave-function 4)50 then refers to the 'So term, with extra P energy AE5; i4) and the other functions symmetrical in the position co-ordinates refer to 1D2, with extra P energy AE4; while the functions antisymmetrical in the positions refer to the 3P term, with extra P energy (G. (9.25))

/AE, (= lE2-/AE3) _ Do ID2. (4.10)

VOL. CXXXVII.-A. Y

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308 A. F. Stevenson.

The next step is to obtaini zero-approximations to the wave-functions correct to order S/P, taking into account the total perturbation P + S, by the method of ? 3. This is facilitated if we first find the correct wave-functions anti- symmetrical in the positions after perturbation by S alone.

We consider in detail first the group m 0 O. For this group, the calculations have already been made by Gaunt (G. (9.63) ). He finds for the wave-functions (since there is an arbitrary multiplying factor in each)

20 _L0 + O

40 ( 4O11

i3? 0- - 2ia? + 2Q?0 -e J1

and (G. (9.61)) for the S perturbation energies (giving the fine structure of the triplet):

a 31+ 2 4C 15 ME2- I + 7)2 + 4 C l (4.12)

0E3 2 =1+ 4I2 +2 {

where 2 are the spin energies for a single 2p electron in a central field of force, and (G. (9.53) )

T - y2a2e2 3 (4t)2 . dr1 j dr2. r2 R2 (1) R2 (2), (4.13) where

27Ce2 h2

hc 4TC2M oe2

The terms 1, 2, 3 are ap2, 3P1, 3P2 respectively. We now take i10, ..., 4)O for the independent orthogonal linear combinationls

4/i of (3.6). The 4)m of ? 3 are the 4), ..., 4) of G. (9.31); the bij of ? 3 can be read off from G. (9.62) (allowing for the factor by which 4)0 has been multiplied in (4.11)); and the 8Em's are given in G. (9.37). The C(m of (3.10) are given by G. (9.36), and the C', which we write C! ?, ..., C5? in the present notation, are given from (4.5), (4.11), and G. (9.34) as follows:-

C1 -:a +C130 + Cy 12C, and similarly

C20 = 4C), C30 =C C40=24C, C50 =-12C.

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Intensities of Certain Nebular Lines. 309

There remains the calculation of the matrix elements V.0 -P ?0 + S3 0, which are the V'j of (3.10). The Pi? are, of course, the AE's introduced above, and PijQ _ 0 (i 4 j). The Si O can also all be calculated from the results of Gaunt. We have, in fact, from (4.0)

Scac-c2 (Ti)aa0 + Uaa0 etc.

and (TJ)a2 ..., Ue,0 are given from G. (9.52) and G. (9.54). S3. (i, j 1 ..* I , 5) are then given by (4.5) and (4.11). We thus findt

S31- SaaO + SO0 + Syy? - 2S,0- 2SayO + 2S,yO -33T, and similarly

S220 = 15T, S330 = 60T, S140 30T, S15 = 6T, S340 - 12T, S330 = 36T, S12 = S130 - S230 S24 = 0 S440-S450 S550 =0.

The determinant of (3.10) now becomes

33T+24C(5E+1,) 0 0 30T+8C(%1-,2) 6T

f o 1 15T+4C(Qql+?'q o 2 !

o o 60T+16C(,ql+2-q 2) - 12T 36T? 16C(771-,72) (4.14) |~~~~~~~~~+4(JI 0 _EI

30T+8C(ql1-q2) 0 -12T o6C(2,q+72) , +24C(AzE4-E)

j 6T ! 0 36T+16CQq1-q2) 0 ?C(2(qu+q 2) +12C(lE.5-,E)

The third order sub-determinant at the beginning of the principal diagonal is that arising from the triplet, the fourth diagonal element that from the ID state, and the fifth diagonal element that from the ISO state. From the manner in which it has been obtained, it is evident that to a sufficient approxi- mation the roots, s, of (4.14) equated to zero are obtained by equating to zero the elements of the principal diagonal, which gives, of course, just Gaunt's values for the energy levels (disregarding an unimportant spin correction to the singlet terms).

According to the method of ? 3, the zero-approximations to the wave- functions are then given correct to order S/P by, say,

5

TJl -.E aj,O (i _ 1, .., 5), ==j

t Not all these matrix elements need be calculated for our immediate purpose, but the complete calculation affords a check on the work.

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310 A. F. Stevenson.

where I aijI is the determinant (4.14). Retaining only terms of order S/P, this gives for the root

= AE1 + XEl (3P2):

a,1 [30T + 8C (%_ - 2)1 + a14. 240 (AE4 - AE1) 0

a,,. 6T + a,5,. 120 (AE5 - AE1) = 0 a12 = a,3 0.

Hence, using (4.6) and (4.10)

a14 _125 T 25 0 %

a,, 24 D2 18D2

a"15 5 T a1l 6 D2

We may therefore take

T, 0 =10 L 125DT 25-8D (1 2)4

5 T 0 L24 D 18D2 6DT2Y

Similarly

= AE1 + E2 (3P1):

= AE, + 8E3 (3Po) 25 T 0 T 200 1

e = XE4 (lD2):

4 i4 +L 12 D2 +9 D2 (N ) l 12 D2 i ? l AE5 ('so) :

T50' =4~0 + 5 [S5T +100c( %

5 +6D2 +02D2 9 D2

We now carry out an exactly similar process for the groups m = ? 1, + 2 of (4.1). For these groups, the S matrix components have not been calculated by Gaunt, except SPP2 - - -1--' T (G. (9.77) ). Many of the wave-functions in these groups, however, differ only from those with m 0 O in having different spin factors. This enables some of the matrix elements to be found in terms of those with mn = 0, by utilising the formulee for the effect of T1 and U on

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Intensities of Certatn Nebular Lines. 311

the spin factors given in G.., ? 5, and the results of G. (9.50), G. (9.51), G. (9.52). For instance, we thus find

2 (T1)pp2 - 2 (Ti)aaO =-ST

UP32 U? =a-a T.

We obtain in this way the following matrix elementst

SPP2 =Sa-2 -11 T, 5yy1 S W1 S == SJ-1- 2T, 1 (415)

S713--Sa,y-1 13T, See2 See-2-S881 = S88-1 O J

Moreover, the integrals for T1, U are in all cases real, so that they may be equated to their conjugates. By considering the relations between the wave- functions for m 1, and for m A 2, it may thus easily be shown that

S3 - Sa 1 S l - - S_/8-1 536 - (iS32. (4.16)

Thus the only integrals that need be calculated explicitly are (T1)P81, (TJ)Y81, (T1)0,2 (the corresponding U's vanish).

The calculation of these integrals is laborious; the method has been suffi- ciently indicated by Gaunt, and we only quote the results 4:

Se2 = __ - S SJ 7T. (4.17)

Since the P matrix elements have already been found in connection with the m - 0 group, we now have all the required matrix elements.

Consider now, for example, the m c 1 group of wave-functions. We must first find the zero approximations after perturbation by S alone for the functions anti-symmetrical in the positions. We find

CA= Cl-l = 4C. (4.18)

Omitting the n factors, since we merely wish to find the wave-functions, the S perturbation applied to the functions tp1, 4yl gives the secular equation

2T-40Cs 13T =0 13T, 2T - 4Ce

t The elements (T1)ppl and (Ti)aa1 are not given directly in this way; it can be seen that they vanish on account of the integrations with respect to 9,5 02.

I It may be mentioned that the coefficients in the expansion of inverse powers of r in a series of Legendre polynomials, required for the evaluation of the integrals, are given very simply by some formulas given by Routh, 'Proc. London Math. Soc.,' vol. 26, p. 481 (1895), and do not require the special artifices used by Gaunt for finding them.

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312 A. F. Stevenson.

with roots z= 15 T an so that 41, +' are replaced by linear com-

binations

(4.19) i21 C t4,81 _ A }} 1 J

These refer to 3P 3P2 respectively. From (4.18) and (4.19),

l ' C2= 8C,

and from (4.19), (4.15), and (4.17)

S3ll = 30T, S221 22T, S81 14T, S12l= SI8 = Sa1 - 0.

We now take 41'L 4l , 42 1 for the 4"j of ? 3. - We next require the bij, ME and C. of (3.10), that is, we must express our wave-functions in terms of the fundamental Dirac solutions for a single electron. Analogously to G. (9.30) we fid

4p1-y+24' ' ds=4)8 + i9 + 3410 (4.20)

4'+Y8 - 4)9 + 3410 where

4). -41 (1) 4-2 (2) - i-' (2) i)-2' (1) )

+9 ~1_4 (1) 4-21 (2) - 41-' (2) i-21 (1) (4.21)

ilO +_-2 (1) 10 (2)- 4)20 (2) 4),0 (1).

In the right-hand side of (4.21), the notation is that of G. (4.00); the suffixes here refer to Dirac's j. From (4.21) we have

=ES 2= , 2 E9 = 8EKo =1 + '22 (4.22)

Also from (4.19) and (4.20)

t2 =3 (+9 + i10) T(.3 4),'=3(4)9+4),o) ~~~~~~(4.23)

2=2%8 + 9 - 310

Solving (4.20) for %., %9, 410, and using (4.18), we fid

C8s4 C) 98= 3C, CO 2 . (4.24)

The bi can now be read off from (4.20) and (4.23); the KM are given by (4.22), and the Cm by (4.24).

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Intensities of Certain Nebular Lines. 313

The determinant of (3.10) now becomes

{30T + 8C (% -2) 0 , 0 + 8C (E1-) J

0 {-22T + 8 C (5%1+ 2)} 14T+8C(%1-t2) + 8(C (AE -)

0 , 14T?~G(~1 -'~2) f "s" terms | ~ ~0 14T + -8 Q% -2 Ti+O0(dE-) 3 1+ 40(AE4 - )

By the method used for the case m 0, we find for the wave-functions:

3P1: T=l

2 L [24D218 9i -24 D2

In a similar way, the groups m - 1, ? 2 are treated; we shall only give the final results

n - - 1.

3p ,: 'T-1 = P1-1

3P2: J2- ) = )21 [?J2 T 25 DC 2 'ID2:'iv' ~p -?L D7 2 D2: + [24 D +8 8D2k (1- _2)] i}2 2

where @ -1 _ tl - 4 1 2 7 cJ' + QJ' (4.25)

m =-2.

3P2: 42 T t2r L+ 1 T 25( 2

1D:?e J24[ D2 + 18 D2- %)

ID Te2 42 + 175 T 2 50C 1 29 b2

m =-2.

P2: Ta1-2 4,2 + ? 5 TX

+2 2)] 25 L 24 ID 2 -75 iY25

ID ~/-2 4~- - 175T +250 C -2

~~~~~e-2~~~

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314 A. F. Stevenson.

Collecting our results, and using (4.5), (4.11), (4.19), (4.25) to express the wave-functions in terms of the original functions (4.1), we now have the follow- ing complete list of zero-approximations for the different states, correct to terms of order S/P:-

3p 2: O - 4yO + C1I08+ C2Ve (12C)

_ -

C34h1 (80)

Pa 1 + 4j 1 - C3C 1 (8C) - '234?2 (2C)

2 ?a +2C3K2 (2C)

3P1 4a? + %O0 (4C) + 1 (8C) 1

(J J) (8C) , (4,26)

3p - 24,O + 2pO ?- OY + C44h0 + C54?0 (24C)

ND2: ?0 + 4+,O + C3+u0 - 3%0 + c4Y+O (24C) , + -92C3 (%1 - 4H) (4C)

WI + lC3 (4i71 ? 4-1) (4C)

42 + c3%2 (40) 2 - C3 qj2 (4C)

1s3: i- 244? - C7440 + C7%0 + IC64J0 (12C) J where

145 25 11.5 50 175 25

35 20 55 +40 25 25 120 9 > s 3 > + , c12_9

C4- -~ tm c6 -- ,

25 20 6 9

and

I D -

2 ( 72)' (4.27)

The normalisation factors-disregarding terms in , 'n-are given in brackets. These may be compared with the futnctions given by Bartlett (loc. cit.).

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Intenstties of Certain Nebular Lines. 315

? 5. Intensities and Mean Lives.

We now calculate the integrals I1 and I2 of (2.5) for the quadripole intensities for transitions between the various states. For the 4's of (2.6), we have the formule due to Darwin (G. (5.20))

(PR) L(dR R )2 - (1-u) (dR + 2 R) pu+ 1

0 * (p u&R) = l L \dr ( -R) P2M(d +2 r ( J From a consideration of the integrations with respect to the azimuths ki, k21 it can be seen that the integral 1J(sJ) vanishes unless rnt =' m + 1, where mn, mt denote the r's of the initial and final states; and that the integral 12(st) vanishes unless mt = Ms - 1. This simplifies the calculations considerably, since only one of the integrals I1 or 12 can contribute anything in the com- binations between the various wave-functions of the different states. Moreover, functions symmetrical in the spins can combine only with functions also symmetrical in the spins, and likewise for functions anti-symmetrical in the spins. The best plan is to calculate first the values of Il, 12 for the only possible combinations of wave-functions b from (4.1) that can occur. The integrals always reduce to the product of two simple integrals, of which one is of the form J zo1', (+1X+1)* dr, (p =-1, 0), (5.2)

or

J ZI14l* (0P-1)* d-r, (p = 1, 0) (5.3)

and the other of the form J 01Vp 12dr. (5.4)

Using (4.2), (5.1) and (Pku)* = (_ l)U Pkg-u, (5.2) becomes

(1)P+I 3 2n f dO dr [Pi P2- sin 0 cos 0 (r3RR'- r2R2) 0 0

(p + 1) (p ,- 2) P jPPoP sin 0 cos 0 (r3RR' + 2r2R2)] (5.5)

where RI - dR/dr, and PIP stands for P1P as defined by (4.3) without the b

factor. By an integration by parts 00 3 rx J r3RR' dr = - - r2R2dr 0 2 c

and the integrations with respect to 0 can be performed without difficulty.

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316 A. F. Stevenson.

We thus find for (5.5) the value

12T jr2R2dr=A, say, (p- 1, O), n

where As = IC by (4.7). Simnilarly (5.3) has the value - A, (p = 1, 0). We easily find that (5.4) has the value A for p - 0, anld 2A for p _+ 1.

We can now make a table of valuies of J(st) for the various possible pairs of the +'s of (4.26), including the summationi with respect to the spin factors. The total intensity of a transition such as ID 3P2, for instance, is then obtained by summing the values of J(s8) where s is any wave-function of the, 1D2 state and t any wave-function of the 3P2 state, after inserting the normalisa- tion factors for the wave-functions. We shall omit the details and give only the final results:

'D2 3P2: I -6 5 (209752+ 768j + 80_2) 3*123 29~ 78 -8w

ID

3Pl.9:

I=

125 (198902 + 768j + 80n2)

ID 3po :I 25 ,2 'D2-t3P0: 48 L (5.6) iS 3P2: IJ 252

ISO 3Pi,: I _ 25 (390 + 16n)2

1-2-.324 so 3po I-0

ISO -01D): I 0

The value 0 for 'So 3Po is a special case of the result that the transition j 0 -Oj = 0 is forbidden for any multipole radiation (Bartlett, loc. cit.). The value 0 forISO >- ID2, however, has only been obtained because the functions j tO, 4P in ISO anld +81, 4s' in ID2 (which are the only ones that arise) occux in

certain combinations; it is thus, so to speak, accidental, and it is clear that by taking account of higher order terms in the wave-functions we should get a non-vanishing result. It is only the perturbation by P which comes in question, and this perturbationi is large when both electrons are in 2p orbits; in fact, it will be shown in the next section that the separation ratio is not in agreement with experiment, so that the wave-functions with which we work cannot be very good approximations.

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Intensities of Certain Nebular Lines. 317

To take account completely of the second order perturbation by P would require a great deal of additional calculation. We therefore make use of a modification of the usual perturbation theory which has been given by Lennard- Jones.t Let Vki be the elements of the perturbation matrix; then the first- order correction to the wave function %k, energy Ek, is ordinarily given by

E Vk i

Lennard-Jones points out that this may be replaced by

E kk + z E

Vki, (5.7) E c jk (Ek- Es)

where Sk iS the first-order correction to the energy-level Ek. The principal part of the correction is thus expressed in terms of the unperturbed eigen- function 4k, and if Ek> Ei the second part may be neglected. In the case of 0++ and N+, the condition Ek> Ef is not fulfilled, the levels from the 2s(2p)3 configuration, in particular, lying fairly close to the (2s)2 (2p)2 levels. Never- theless, we may expect that the contribution from the second term in (5.7) will be comparatively small, and that-at least as regards order of magnitude- a fair approximation will be obtained by retaining only the first term in (5.7).

In the present case, V P e2/r, so that, from (5.7), a wave-function is is to be replaced by

( E + Er) is=(1 + ?s + r is,say (5.8) where z\E8 is the P-perturbation of the state in question, and E the unperturbed energy level-say, the mean of the levels 1S, 1D, 3P. From (5.8) and (2.5)

IlS =| Z, (I + ?ts + r) 4 )sO (l + Oct + r5 i*t dr, d'72

+ ~ ~ ~ ~ ~ ~ }.. ~~~~~~~~(5.9) + Z2 (1+ aXs + r) isa2 1+ (xt + f5)+*t dr, dr2

Retaining only first-order terms in oc, 3 and neglecting the terms independent of oc, 3 since we know that they contribute nothing to the final result, the first integral in (5.9) gives

(OXS + oct) J Zjs,q&,*t dr, d72 + T J 1 4s8l4*t d-l d72 r

+ Zi Z

V1 a lt d'r d-2. (5.10)

t'Proc. Roy. Soc.,' A, vol. 129, p. 598 (1930).

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318 A. F. Stevenson.

The last integral in (5.10), after an integration by parts, gives

- i rl 4*t4)s dxl d-"2.

The second integral in (5.9) is similarly treated, so that we may put

j(st) =(?ts + at) (Z14S''14*t + z2dt'}24*t) dtl dC2 1 + i * - *tal48) d tl d72 . (5.11)

+ t 3 (YsA2 ~ t 2ta2'S) d'1 dr2 j

2I(St) has the same value with P in place of D,. The first integral in (5.11) has already been calculated; the only new ones that need be calculated are the second and third in (5.11) and the similar expression for 12(St')

Since we may expand I/r in the form (G. (9.01) and G. (5.22))

r Ef.f (rP, r2, 01, 62) eiU (01-2S r

it can easily be seen that only those wave-functions contribute to F(st) which do so when the corrections in (5.8) are neglected, and, moreover, that if one of the integrals 11(st), I2(St) does not vanish, the other one certainly does. Since we are considering only the 'So 1D2 transition, this means, from (4.26), that we need only retain the wave-functions

i80 - 24e,O for ISO 4) -1, ?1, for 1D2

The second and third integrals in (5.11) are of a type met with before in the work. There are a number of new integrals to be found. The best way is to proceed systematically with the functions b that occur in (5.12) in the same way as before. The calculations mnay be considerably shortened by observing that some of the integrals can be deduced from others by interchanging the variables 1 and 2, or equating an integral to its conjugate (since it is real). We further have in our case, from (5.8)

\AE O AE4 _

? E ' C E' - '

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Intensities of Certain Nebular Lines. 319

and (4.6) may be used to express AE4, AE5 in terms of the integrals Do) D22

We again omit the details, and give only the final result:

'So-- ID2 :I 216 CE)2 (5.13)

This now replaces the last of (5.6). The relative intensity of lines with the same initial level, ISO or 1D2, are

given, from (2.4), by the ratios of the values of I given by (5.6) and (5.13) multiplied by v4, where v is the frequency of the line.

For the mean lives of the 'So and ic2 states, we require the absolute intensities. The result (2.4) is the intensity per unit solid angle in the z-direction. For an individual transition between two wave-functions the radiation is, of course, polarised,t but the total radiation for a transition between two states is evidently unpolarised and the same in all directions. The total intensity per atom is therefore obtained by multiplying (2.4) by 47 and dividing by the weight of the initial state (1 for 'So, 5 for 1D2). The probability of transition, per unit time, is obtained by division by a further factor hv,t. The total probability of transition, per unit time, from 'So or 1D2 to a lower level, is obtained by summing the probabilities so found over all lower levels, and are thus, from (2.4), (5.6), and (5.13), given by

A (15o) = 47r2he2 (V1\ 216 ID 2 mo2c2~~~~~~ A ('So) m 2c2 c 625 CE~~> (*14

AQD)47t21e2 32 1252'Wh A (1D2) =4m2o2 . 9. 312 (8604,2 + 3072> + 320N2)

where vj, v2 are respectively the frequencies of So - 1D2 1D2 -- P. W e have neglected the contributions to A ('So) of the 1S -> 3P transitions, since they are relatively very small.

The mean life, -sy of an atom in state s is defined by

1=E A r

where As,-r is the Einstein A " coefficient, and the summation is taken over all states r of lower energy than s, so that if no influences other than spontaneous

t By considering the field given by the retarded potentials, it can be shown that the radiation is circularly polarised parallel to the (x, y) plane. Cf. Rubinowicz, 'Z. Physik,' vol. 61, p. 338 (1930).

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320 A. F. Stevenson.

emission are at work to increase or decrease the number of atoms in state s, that number will be

N No ,/t

after time t, where No is the number at time 0. The mean life of an atom in state iSp or 1D2 is therefore the reciprocal of the corresponding expression in (5.14).

? 6. Numerical Values.

So far the calculations have, of course, been general and apply to any atom or ion which can be treated as composed of two electrons in equivalent p orbits. We now wish to apply these results to the special cases of 0++ and N+, with reference to the nebular lines. The combinations of integrals of radial wave-functions which remain in the results can all be calculated from the experimental data. For, from (4.6) and (4.10), the separations between the different multiplet terms are given by

ID- 1S - AE-AE4 E l

(6.1) 3P-ID AE4 -AE1 - 6 Dj

while from (4.12) the triplet separations are

3p, 3 M ~13 T 2 - ) -1 3P2 E, - a2 = 2t + 1 t2)

(6.2) 3p 0-3p,

_ 2 Sa T I.

so that from the experimental values for the separations, the quantities D2/C, T/C, % - T2, and hence also i and -, can be calculated.

It is of interest first, however, to see how the observed separations fit in with the calculated values where such can be found. A discrepancy arises at once, however, as pointed out by Bartlett, for the ratio of the inter-multiplet separations given by (6.1) is 3 : 2 independently of the value of D2/C, whereas this is not so experimentally, the ratio being about 23: 20 for O++ and 17: 15 for N+. It is evident that this arises from the large value of the perturbation by P when both electrons are in 2p orbits, and the fact that our wave-functions are not sufficiently accurate (cf. the results on the intensity of :Sol --->ID2 of the last section). The necessary wave-functions, derived by the method of Hartree's self-consistent field, are available in the case of O++, having been calculated by Hartree and his collaborators. The method of the self-consistent field has been amply justified by comparison of results derived from this

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Intensities of Certain Nebular Lines. 321

method with experiment, and has recently been justified theoretically as being a good approximation in many-electron atoms.t

With these radial wave-functions the integrals for T, D2, C were calculated by numerical integration4; % - 2 iS the spin-doublet separation for a single electron in a p orbit in a central field of force, and is given by?

X0 Z 3 ZR2dr

% - _2 = ay2O22.Z a

22e2 . 0 r jI~~R2r2dr

where Z is the effective nuclear charge at distance r, so that, since Z is also obtained with the self-consistent field method, % - m2 can also be calculated.

The values thus found for the separations for 0 III, computed from (6.1) and (6.2), together with the experimental values (taken from Becker and Grotrian, loc. cit.), are given below in cm.?

Separation ratio. Observed. Calculated. Observed. Calculated.

'D - 'S ............ 22913 16600 )1 3P- 1D............ 20160 11100 1

3pi - 3P2...... 193 22.1 3P0-3p1 .... 116 125 } 1*66 1*8

The initer-multiplet separations do not agree well with the observed values, as might be expected in view of the above, but the triplet separations come out as well as can be expected. On the whole, then, we may expect our theory to yield reasonably accurate results for the intensities and mean lives, in spite of the inaccuracy of the wave-functions used after perturbation by P alone. For the numerical calculations, we shall now use the experimental data as explained above, taking the mean of the two values for D2/C obtained from (6.1). The use of the experimental values for the quantities involved in this way may reasonably be supposed to increase the accuracy of the calculations, though this cannot be said with certainty.

We give, in Table I, the results for the relative intensities of lines from the same initial level, and the mean lives, as calculated from the formule of ? 5,

t Pock, 'Z. Physik,' vol. 61, p. 126 (1930); Dirac, 'Proc. Camb. Phil. Soc.,' vol. 26, p. 376 (1930).

t The values of D2, C were supplied by Dr. Hartree. For the other integrals, a straight- forward method of numerical integration was actually adopted, although in similar cases, owing to the inverse powers of r in the integrand, it may sometimes be better to use a more indirect method. I am indebted to Dr. Hartree for pointing this out to me.

? Dirac, 'Proc. Roy. Soc.,' A, vol. 117, p. 624 (1928).

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322 A. F. Stevenson.

for the spectra 0 III, N II, and the similar spectrum C I (added for com- parison), and also the observed values for the intenisities of the nebular lines. The observed values of the separations in 0 III and N II, and of the average nebular intensities, are taken from Becker and Grotrian (loc. cit.) ; the data for C I are taken from a paper by Fowler and Selwyn.t

Table I.-Relative intensities of lines, and mean lives of metastable states, arising from (2p)2 configuration. The intensities are relative for lines in the first group and for lines in the second group, but not for two lines in different groups.

/ ~~~~0111. NIl1. cI1.

/ ~~~~~~~Relative Relative Relative / ~~~~A. intensity. A. intensity. A. intensity.

1D2_1SO 4363-21 1 748 1 1 8732-09 OI s

3P2-ID2 5006 -84 (Nj) 3 2 6583 -6 3 2 10348 -7 3 -

3P1-1D2 4958-91(N2) 1 1 6581 1 1 1031 3-

3P0-1D2 4931 93 7 x 10O4 0 6528-2 7 x 10O4 0 10303.6 1 x 10-4

Mean life 0 -10 seconds 0 1 1 seconds O -12 seconds 'So state

Mean life 26 seconds 3 -I minutes 58 minutes 1D2 state

The results for the relative intensities of the nebular lines corresponding to 3P2, 1, 0-ID2 are very satisfactory ; in particualar, the calculated value of 3 : 1 for the relative intensity of the nebulium lines N1, N2in 0 III agrees quite as well as could be expected with the average observed value of 2 : 1. Indeed, this ratio, which differs somewhat in the different nebulae, should be more

nearly 10 : 3 according to Campbell and Moore. Similar remarks apply to the red nebular lines, 2 . 6583 - 6 and ? . 6548 -1, of N IIL The extreme weakness of the line 3P 0-1D2 in both 0 III and N II accounts for these lines not being observed in the nebulae. Bartlett's results (loc. cit.) for the relative

intensity of 3P2, IL, 0-1D2 are :

39 :25~ 0. for 0IIIL 39 :25: 0 forN il,

11: 7: 811: 7:81

t 'Proc. Roy. Soc.,' A, vol. 118, p. 34 (1928).

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Intensities of Certain Nebular Ltnes. 323

the three different results being obtained by assuming three different " inter- mediate states " in the calculation of matrix elements.

It will be seen from (6.2) and (4.27) that when the 3P triplet is nlot far removed from a " normal " one (separation ratio 2: 1)-as is the case in all the spectra here considered-the ratio i/ is small (- = 0 for normal triplet). From (5.6), we see that it is the small value of 4/ri which accounts for the relative weakness of 3PO-1D2' As the triplet departs farther from normality the line 3Po-1D2

becomes stronger, though, owing to the nature of the coefficients in (5.6), the ratio of 3P2-1D2 to 3P1-1D2 is always close to 3: 1. The relative intensity of 3P2, 1, 0_1D2 for a normal triplet is exactly 3: 1: 0 according to the present theory. It will be observed that the summation rules are not obeyed in these intensities.

The weakness of the lines 3P1, 2- S0 compared to 'D2-'So would also account easily for their not being observed; but we have in addition the fact that these lines lie in the ultra-violet and so would not be observed in any case in the nebulae. The line 3P1-lS0 (which the calculations show is much the stronger of the two) has, however, been observed in the similar spectrum Pb L.t It should be pointed out, incidentally, that the value for 1D2-1SO is somewhat unreliable owing to the way in which we dealt with the first-order approxi- mations to the wave-functions for this transition in ? 5.

What is perhaps the most interesting result of the calculations from an astrophysical point of view is that for the mean lives of the metastable states ISO and 1D2, especially the latter, in 0++ and N+. The figures for the ISO state are not very reliable for the reason mentioned above, but those for 1D2

should be fairly accurate. It is usually very reasonably assumed that the physical conditions in the nebule must be such that the mean time between collisions suffered by the gaseous ions must be at least as great as the mean lives of the ions in the initial metastable states; for otherwise the ions would be knocked out of these states so quickly by collisions of the second kind that the radiation would not be effective.t It is also necessary to suppose that the radiation from the exciting star is so weak that the induced absorption does not seriously impede the emission.?

t Sur, 'Phil. Mag.,' vol. 2, p. 640 (1926); Gieseler and Grotrian, 'Z. Physik,' vol. 34, p. 374 (1925).

: This should not, however, be assumed without question, since emission lines from metastable states, as, for instance, the auroral green line of 0 1, have been observed in the laboratory where this condition would not, presumably, be satisfied. Frerichs and Campbell, ' Phys. Rev.,' vol. 36, p. 1460 (1931), suggest that the occurrence of this line in the laboratory is due to the strong excitation in laboratory sources.

? Eddington, 'Mon. Not. R. Astr. Soc.,' vol. 88, p. 134 (1927).

VOL. CXXXVII.-A z

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324 Intensities of Certain Nebular Lines.

We may therefore (probably) assume that the mean time between collisions in the nebu'le is at least about 26 seconds in regions where the nebulium 0 III lines are emitted, and about 3 minutes in regions where the red lines, X=6583 6 and 6548X1, are emitted. The greater time necessary for the latter is con- sistent with the fact that the N1, N2 lines occur strongly in almost all the nebulee, whereas the red lines occur only in a few; the nitrogen spectrum, however, occurs less frequently than the oxygen in the nebulhe, and this may not have any significance. The mean time between collisions depends on the temperature, density, and effective collision area of the colliding particles in the nebulae, and these are not at all accurately knowni. Very rough estimates give a mean time ranging from 104 to 106 seconds for the diffuse nebulaT, and from 7 to 700 seconds for the planetary nebule.-t

The nebulium lines N1, N2 occur also in the spectra of novee a certain time after their appearance. If we accept the theory that these lines are emitted by a gaseous shell which expands radially outwards from the nova, then, as pointed out by Elvey,J the density of the shell at the moment the nebular lines are emitted must have decreased to such a value that the mean time between collisions is just sufficient for the radiation to become effective-about 26 seconds according to the present theory. If the density of a nova shell at the moment of emission of the nebular lines could be found accurately,? this, then, should give fairly precise information concerning the temperature (or vice versa).

? 7. Conclusions.

We are justified in drawing the following conclusions from the present work: (1) The only point at which direct comparison of the calculations with

experiment can be made is in the relative intensity of the lines 3p2. 1 D2;

the agreement is very satisfactory. (2) Assunming the validity of our results and of the hypothesis that the mean

time between collisions suffered by the emitting atoms in the nebulwe must be at least as long as the mean lives of the atoms in the initial states, the physical conditions in the nebule must be such that the mean time between collisions is at least about 26 seconds in regions where the nebulium N1, N2

t Becker and Grotrian, loc. cit., p. 74. 1 Elvey, 'Nature,' vol. 121, p. 12 (1928); Pike, 'Nature,' vol. 121, p. 136 (1928) ;

Elvey, 'Nature,' vol. 121, p. 453 (1928); Gerasimovic, ibid. ? Elvey's estimate is 10-17 gr. cm.-3, about the same as the estimated density in the

nebulhe, but the data on which it is based can hardly be considered reliable. See Menzel, 'Nature,' vol. 121, p. 618 (1928).

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Diffraction of Elasttc Waves. 325

lines are emitted, and at least about 3 minutes in regions where the red lines, X = 6583 *6 and X = 65484 1, are emitted. (3) In the case of novae exhibiting the N1, N2 lines in their spectra, the mean

time between collisions should be about 26 seconds in the region where these lines are emitted at the moment the lines first appear strongly marked.

(4) All in all, this paper, as far as it goes, confirms Bowen's hypothesis concerning the origin of certain of the nebular lines, anid the further assumption that these lines are due to quadripole radiation. The final step in the con- firmation would be the production of these lines in the laboratory if it should become possible to reproduce in some degree the nebular conditions (as has already been done in the case of the 0 I nebular lines by Hopfield, loc. cit.).

In conclusion, I should like to express my warm thanks to Mr. Fowler for his interest in this work, and his helpful advice; also to Professor Hartree for supplying me with the numerical data for the radial wave-functions for 0++.

The Diffraction of Elastic Waves at the Boundaries of a Solid Layer.

By J. H. JONES, Ph.D., Chief Geophysicist, Anglo-Persian Oil Co., Ltd.

(Communicated by 0. W. Richardson, F.R.S.-Received February 25, 1932.)

[PLATES 16 and 17.]

1. INTRODUCTION.

When a disturbance occurs near the free surface of a solid medium or near the interface of two solid media, many remarkable effects arise from imperfect reflection and refraction due to the curvature of the elastic wave-fronts. In the case of a solid with a free surface, Lord Rayleigh predicted the existence of surface waves travelling with a velocity lower than that of the distortional waves. These waves are usually very pronounced on earthquake records.

It has been demonstrated by Nakano that a large percentage of the energy of an earthquake will appear in the Rayleigh waves if the focus is near the surface, and conversely, for a very deep focus there will be little or no energy in these waves. Another kind of surface wave was discovered by Love. This wave exists when a layer of finite thickness rests on a deep solid layer and when the velocity of the distortional wave in the upper layer is less than that in the lower layer.

z 2

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