Intensities of the K-LL Auger lines

Nuclear Physics 44 (1963) 399-414; Not

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INTENSITIES

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OF THE K-LL AUGER LINES W. N. ASAAD

Faculty of Engineering,

Cairo University

Received 31 January

1963

Abstract: The variation with 2 of the intensities of the different lines in the K-LL Auger spectrum is calculated non-relativistically on intermediate coupling theory, using the transition amplitudes computed by Callan for all elements from 2 = 12 to Z = 41 and for ten additional atomic numbers in the range from Z = 44 to Z = 80. The calculations have. been made assuming the coupling to be between the two final vacancies in the atom as well as between the initial vacancy in the atom and the hole in the continuum representing the ‘missing’ Auger electron. Good agreement with existing experimental values is obtained, except for eiements with large atomic number, for which relativistic effects cannot be ignored. Deviations from pure L-S coupling are found to begin practically from Z = 25, while extreme j-j coupling is not fully approached even at Z = 80. Comparison with previous theoretical work is discussed. Also the differences introduced by treating the system as one of two electrons, rather than in the correct way as one of two holes, are demonstrated.

1. Introduction

When the intermediate coupling was applied by Asaad and Burhop ‘) to the study of the K Auger spectrum, the usual six-line K-LL spectrum expected on the basis of j_i coupling was found to modify into a nine-line spectrum in regions of intermediate atomic number, although some of the lines were found to be very weak. Experimental evidence supporting this new treatment came from the Chalk River laboratory 2-6), wh ere Graham and his coworkers observed the K-L,L, (“P2) satellite iine fir some values of Z’ (94, 62, 64, 66 and 55). From this laboratory, preliminary evidence for the existence of the other two satellite lines for Z = 55 was also reported 6). Sokolowski and. Nordling ‘) in Stockholm also found a seventh line in the K-LL spectrum of copper. Very recently, an eight-line spectrum has been observed at the Chalk River laboratory by Graham, BergstrGm and Brown *) for Z = 52 and for Z ti 53. The results of ref. ‘) concerning the relative line intensities of the K-LL spectrum were obtained using mainly the transition amplitudes calculated by Burhop 9). These had been calculated for silver using screened hydrogenic functions, the screening constants being derived from Slater’s rules ’ “). Justification of the use of these transition amplitudes, calculated for Z = 47, in estimating the relative line intensities for elements of intermediate and heavy atomic numbers was discussed by Asaad ‘l). Recently, transition amplitudes for the K-LL Auger spectrum have been computed by Callan 12) for all atomic numbers from Z = 12 to Z = 41, and for ten more 399

400

w.N. ASAAD

atomic numbers in the range from Z = 44 to Z = 80. He used screened hydrogenic wave functions, with screening constants derived from the results of Froese ~3) for the limiting screening numbers for the Hartree-Fock self-consistent field functions. The Auger electron energies needed in the hypergeometric continuum wave functions were derived from tabulated X-ray energy level values. Callan calculated also the K-LL Auger transition probabilities for the above mentioned atomic numbers, expressing his results in terms of Russell-Saunders coupling. No intermediate coupling calculations were performed, however. In the present paper, Callan's calculated transition amplitudes are used for calculating, on the basis of intermediate coupling, the K-LL Auger line intensities for the same variety of atomic numbers. Calculations reported here have been carried out assuming the coupling to be between the two final vacancies in the atom as well as between the initial vacancy in the atom and the hole in the continuum representing the 'missing' Auger electron. Satisfactory agreement with observation is obtained, except for elements with large atomic number, for which relativistic effects cannot be ignored. To demonstrate the differences introduced when the system is treated as one of two electrons rather than as one of two holes, the corresponding results are given and the agreement with experiment is found to be poor. It is of interest to investigate the ranges of atomic number over which the assumptions of pure Russell-Saunders coupling and of extreme j-j coupling are applicable. As described in this paper, it is found that deviations from pure L-S coupling begin practically from Z = 25, while the extreme j-j coupling is not fully approached even at Z = 80.

2. Expressions for Auger Transition Probabilities The theory of the intermediate coupling and its application to the calculation of the Auger spectrum was given in ref. 1). Using the same notation, we obtain the following expressions for the Auger transition probabilities for intermediate coupling: (2s)°(2p) 6 L,

LI(1So) K{(Zsy 'So}

(2s) (2p) 5 LI Lz(Spo) Lt

L2(1p1)

L1 L3(3p1) Lt L3(3p2) (2s)2(2p) 4

LE L2( 1So) L 2 L3(1D2) L3 K3(3po) L3 L3(3p2)

K{(2s)(2p) 3po} K{(2s)(2p) 1P 1} "[a2/(1 + a2)] x [K{(2s)(2p) XPl}-K{(2s)(2p) 3p1}] K{(2s)(Zp) 3P 1} + [~2/(1 + a2)] x [K{(2s)(2p) ~P~ } - K{(2s)(2p) 3p1 }] K{(2s)(2p) 3p2} [1/(1 + ~)]K{(2p) 2' So} [1/(1 + ~2)]K{(2p)2 aD2} [a~/(1 + ~q)]K{(2p)~ %} [a2/(1 + a~)]K{(2p) z 'D:}.

INTENSITIES OF THE K-LL AUGER LINES

401

In these expressions the quantities ~ are the mixing parameters, denoted in ref. 1) by Ci("Ls). The mixing parameters are found to be 1~"

= (~211,,/2)1[G~-~21

1 +~/([G1-~2,]

2_[_

2 ½¢2~)-I,

15 y' ~1 = (x/2)~21/[~-F2 -½~2t + 4 ( [ ~ F 2 -½~21] 2 q-2~221)],

0~2 = (4"2 l/x/2)/[3F2

+ ¼~21 "[- x/([3F2 + ¼ ( 2 1 ]

2 "~ ½~21)]"

The quantities F and G used here are related to the matrix elements of the electrostatic interaction between the two holes F~(nl, nT) and G*(nl, nT), defined in ref. ~), by the relations Fz = ~-sF2(21, 21), G1 = +GI(20, 21). The screening constants used by Callan 12) for the 2s and 2p electrons of the same atom were different. Thus, in calculating the mixing parameter ~ for the P1 terms for the (2s) (2p) 5 configuration, we took screened hydrogenic wave functions in which the effective nuclear charge was the geometric mean Z7 of the effective nuclear charges ZLI and ZL2.3 given by Callan. Expressions in terms of Z for G~ and ~2~ for this configuration were obtained. These are G1 = ~

Z,

ff21 = Z4/( 48 x 1372),

atomic units being used. For the (2s) g (2p) # configuration, the effective charge ZL~. was used directly in calculating the mixing parameters gl and ~2. The values used for F2 and ~21 w e r e calculated from the corresponding expressions

F2 :

259 60ZL2,3,

(21 :

4 3/(48 x 1372). ZL2,

3. Results and Discussion

The results of the calculations of the Auger transition probabilities in intermediate coupling, using the transition amplitudes by Callan, are given in table 1, from which the line strengths relative to that of LaL 1(~So) can be readily calculated. The Auger transition probabilities as well as the relative intensities of the various lines vary markedly, but smoothly, with the atomic number Z. Moreover, the intermediate coupling theory leads to a fine structure of the K-LL Auger spectrum, exhibiting satellite lines. The absolute line strengths of this spectrum begin to differ from those given by Callan 12) on the assumption of extreme L-S coupling from values of Z as low as 25 or even less, as is readily seen from figs. 1 and 2. In these figures the transition probabilities of some lines, calculated on the basis of intermediate coupling, are shown as full curves, while the corresponding line strengths resulting from the extreme Russell-Saunders coupling are shown dotted. Fig. 3(a) shows the variation with Z of the relative line intensity, calculated in intermediate coupling. We now look into our results in more detail. First of all, we see that the L1LI(1So) line has the same streng~, whether calculated on the basis of extreme L-S, extreme

402

j-j,

w . N . ASAAD o r i n t e r m e d i a t e c o u p l i n g . T h i s line r e m a i n s u n s p l i t . M o r e o v e r , t h e v a r i a t i o n o f

its s t r e n g t h w i t h t h e a t o m i c n u m b e r Z is s l o w . I t e x h i b i t s a fiat m a x i m u m a t Z = 23, a n d t h e l e a s t s t r e n g t h v a l u e s o c c u r r i n g a t t h e e n d s ( Z = 12 a n d Z = 80 i n t h e a b o v e TABLE 1 Auger transition probabilities for (2s)°(2p) 6, (2s)(2p) 6 and (2s)2(2p) ~ LILx

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 44 45 46 47 50 55 57 70 72 80

LxL,

LxLs

LIL=

L,Ls

LsI-~

(1So)

(1p1)

(sp~)

(;p~)

(ap~

('So)

(~D,)

('Po)

(@,)

2442 2491 2525 2560 2583 2600 2609 2614 2619 2619 2628 2631 2629 2628 2629 2624 2618 2613 2606 2599 2601 2597 2597 2594 2590 2588 2586 2584 2591 2587 2573 2570 2568 2566 2564 2557 2551 2519 2513 2493

2774 3165 3492 3786 4030 4264 4471 4633 4784 4901 5012 5101 5182 5244 5292 5324 5336 5340 5337 5282 5257 5197 5126 5043 4944 4835 4719 4598 4477 4368 4015 3911 3815 3727 3510 3229 3137 2793 2761 2660

99 108 114 120 124 129 132 134 136 136 138 139 140 140 140 140 139 139 140 137 139 139 139 140 140 140 140 140 140 142 140 139 139 139 140 140 140 135 134 131

296 324 341 361 374 390 403 410 422 429 442 456 476 497 526 561 603 658 727 796 895 999 1118 1249 1386 1532 1679 1828 1976 2130 2525 2647 2761 2868 3154 3513 3625 4038 4077 4186

493 539 568 600 619 644 662 669 681 682 690 694 698 698 700 700 697 697 700 687 696 695 697 699 699 699 700 700 700 708 698 697 697 697 700 702 698 675 672 654

124 167 209 250 289 328 367 402 434 460 483 502 519 532 540 542 541 538 535 522 512 501 489 478 467 456 446 436 428 420 393 396 391 387 378 369 367 356 355 352

1668 2213 2758 3272 3720 4249 4733 5165 5551 5887 6174 6420 6662 6874 7032 7166 7286 7400 7535 7571 7648 7706 7760 7811 7865 7891 7910 7923 7941 7953 7992 8000 8015 8030 8073 8154 8191 8231 8222 8176

0.02 0.05 0.1 0.3 0.6 1 2 3 5 9 13 19 28 39 53 70 90 113 140 166 194 222 249 277 303 327 349 369 388 406 460 465 477 489 520 561 575 624 633 648

0.3 1 2 5 10 18 37 51 78 115 163 222 295 382 479 586 704 831 969 1099 1239 1378 1515 1651 1785 1909 2026 2136 2243 2343 2614 2694 2771 2844 3039 3305 3396 3732 3759 3833

All values are in 10-~ o f atomic unit.

INTENSITIES OF TKE K-LL AUGER LINES

403

calculations) are lower than this maximum by no more than 7.3 %. It is therefore rather fortunate that nearly all experimental results of the strengths of the different lines of the K-LL Auger spectrum have been given relative to the strength of this line. However, owing to the fact that some lines in the spectrum are more intense, some workers have tried to compare their experimental results with the theoretical calculations after relating the strengths of the different lines to that of the strongest. While this may be useful in suppressing those discrepancies between experiment and theory that result mainly from difficulties in estimating the relatively weak line strengths i'n experiments, as well as from the absence of more accurate calculations based on better approximations in the theoretical treatments, we think that it would be advantageous to uphold the current practice of relating the strengths of the other lines in the K-LL Auger spectrum to that of the L1L~(1So) line.

7°21 lo

2'o

~

,~

5'o

~o

7'o z 8~

Fig. 1. Variation with Z o f t h e intensities o f the lines (i) LaLt(tPx) a n d (ii) L1L3 (sPl) for the (2s)(2p) s configuration. (The dashed lines are for extreme L - S coupling.)

The intensity of the LxLx(ISo) line represents the total transition probability to the final vacancies (2s) 2. It can be deduced from table 1 that the total transition probabilities to final (2s) (2p) and to final (2p) 2 vananeies increase with Z, the increase, especially for final (2p) 2 vacancies, being large for light elements. This dependence on Z must not be overlooked when calculating the fluorescence yield as has often been done previously. For the configuration (2s) (2p) 5, the 3p line expected on the basis of extreme L-S coupling is split, when intermediate coupling is used, into three components 3P o, 3P x and 3P 2. These, together with the 1p line, constitute the four lines of this configuration. When extreme j-j coupling is approached, the last two components of the 3p line combine to form the LxL 3 line, while the first component combines with 1p to form the LIL2 line. Alterations of the Auger transition probabilities due to the application of intermediate coupling are limited to the 3P a and 1P1, leaving their sum the same as calculated on L-S coupling. Fig. 1 shows how the strength of the line component 3p~ increases, while that of the Xp1 decreases. It is to be remarked that the strength of the 1P 1 line has a maximum at Z = 29, becomes equal to that of 3p~ near Z = 52 or 53, and afterwards becomes less intense than 3p~. For the configuration (2s)2(2p) 4, L-S coupling gives rise to only two lines IS~ and ID2, the strength o f the 3p line being always zero. However, when intermediate

404

w.N. ASAAD

coupling is assumed, the components 3P o and 3P 2 begin to build up at the expense of 1S0 and ~D2, respectively, while the third component 3P 1 remains zero, again giving four lines for this configuration. When extreme j-j coupling is approached, the ~So line becomes the L2L / line, the ~D z line becomes the L / L 3 line, and the two components 3P 0 and 3P 2 combine to form the L3L 3 line. 12 ./11 // 10 /

/

/ /

/ 8

(iJ

//

6

2

/ 10

20

30

/.0

50

60

_

i 70

Z

8'0

Fig. 2. Variation with Z of the intensities of the lines (i) L~L3 (XDz) and (ii) LaLs (3P2) for the (2s)2(2p)~configuration. (The dashed lines are for extreme L-S coupling.) Fig. 2 shows the variation with Z of the transition probabilities for 1D 2 and 3P 2. Deviations from results based on L - S coupling are seen to begin from Z = 20 and to increase with Z. The intensity of the 1D z line shows saturation from nearly Z = 40, and the saturation value is about only two thirds Of that for pure Russell-Saunders coupling. At Z = 80, the intensity of the 3P 2 line becomes slightly less than half that of the XD2 line. As regards xSo and 3P 0 lines, the intensities are small compared with that of LILI(1So). Fig. 3(a) shows the variation with Z of the intensities of the lines of the K - L L Auger spectrum relative to that of Lx LI(~So). It can be seen that the most intense line in the K-LL Auger spectrum is the LzL3(~D2) line, for all Z greater than 17. For heavy elements, the lines of the spectrum can be arranged according to their intensities. In decreasing order of intensity, they are L2L3(ID2), LxL3(3Px), L3L3(3p2), L1L2(1PI), L1LI(~So), LxL3(3p2), L3L3(3Po), L2L2(tSo) and LILE(3Po). It may be of interest to indicate the differences introduced in the results when intermediate-coupling theory is applied to the system treated as being one of two electrons rather than one of two holes. The expressions listed in sect. 2 for the Auger transition probabilities, with ~2t replaced by -~2x are for the following terms in order: (2s) 2

L 1 Lx (1So),

(2s) (2p) L x Lz(3po), L 1 L3(xPI), L1 L2(sPx), L1 L3(3p2), (2p) 2 LaL3(1So), L3L3(ID2), L2L2(aPo), L2La(3p2).

INTENSITIES OF THE K-LL AUGER LINES

t ~D

tv~D

,

2o

3o

405

¢iii~

~o

50

.'o

7'o z 8;

Fig. 3(a). Variation with Z of the relative line intensity for the configurations (2s)°(2p) 6, (2s)(2p) 5 and (2s)~(2p) 4. (i) LzLx(1So) (ii) LzI.~(zaz) (iii) L1L2(sPo) (iv) LzLs(aPz) (v) LzLa(aPa) (vi) L~L2(zSo) (vii) LzLa(1D~) (viii) LsLs(aP0) (ix) L3La(aP2)

25

2.,.0 (vlii~

(i)

0,5

QO 10

Fig. 3(b). Variation with Z of the relative line intensity for the configurations (2s) a, (2s)(2p) and(2p) z. (i) LILI(ZSo) (ii) LzLz(~Po) (iii) LzLz(sPz) (iv) LzLa(sP2) (v) LzLs(zPz) (vi) LsL2(sPo) (vii) LaLs(aP~) (viii) LaLa(1Ds) (ix) LsLa(1So)

406

w.N.

ASAAD

The transition probabilities for these two-electron configurations, and the corresponding line strengths relative to that of La L~ (1 So), have been calculated. The results are shown in fig. 3(b) which corresponds to fig. 3(a) in the case of the two-hole calculations. Perhaps the most important difference is that, in the two-electron treatment, the larger component, (iii) in fig. 3(b), of the L~ L 2 line is always less than th e larger component, (v) in fig. 3(b), of the Lt L3 line, in disagreement with experiment; whereas in the two-hole treatment the curves of the corresponding components, (ii) and (iv) in fig. 3(a), intersect. This intersection means that the L t L 2 and L 1L3 lines have equal strengths for some intermediate atomic number, and moreover, shows that for larger atomic numbers the L~ L 2 line is even more intense than the L~ L 3 line, a result which is well established experimentally. (See for example refs. ~4, ~)). it is also of interest to compare the results of intermediate coupling with those obtained on the assumption of extremej-j coupling. The importance of this coupling is due to two facts. Firstly, observed Auger spectra, until very recently, have almost invariably been analysed in terms ofj-j coupling. Secondly, it is this coupling which gives the non-relativistic limit of relativistic calculations such as "those carried out by Asaad 11). ~Ihe expressions for the transition probabilities for the K-LL Anger spectrum in jzi coupling can readily be obtained from those for intermediate coupling when the mixing parameters given in sect. 2 of this paper are allowed to take their limiting values. These limiting values are attained when the ratios G~/~21 and F2/~2~ vanish. Theoretically the j-j coupling will be attained only when Z goes to infinity. Thus, for j-j coupling, we put

~,'-=~I=2,

~=½,

in the expressions given in secL 2, and obtain the following expressions: L~L~

/C{(2sy 1So},

[,1L~

K{(2s)(2p)aPl} + .tK((2s)(2p)Ip,},

L1L3 2K{ (2s)(2p)3p~} + 2-K{(2s)(2p)tP~ }, L2 L=. ~x{(Zp) ~ 'So}, LzL3 L3 L3

~K{(2p) 2 'D2}, 2 c r~ " 2 1 -xK~(zp) So} +~K{(2p) 2 'D2}.

I t is to be ~oted that the above expressions will also be obtained from those of intermediate coupling on the assumptic.n of a two-electron system when we put

Therefore, it is immaterial, both in the L-S couNing and in thej-j coupling, whether the system is looked at as one of two holes or as one of two ele~rons. !r~serting in the above expressions the values computed by Caftan 12), we obtain

407

INTENSITIES OF THE K-LL AUGER LINES TABLE 2

Auger line strengths in

j-j

coupling

Z

LxL2

LxLs

L~L2

L~La

LsI-,s

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 44 45 46 47 50 55 57 70 72 80

1221 1378 1505 1622 1716 1809 1889 1949 2008 2049 2094 2130 2165 2193 2219 2242 2258 2278 2301 2301 2329 2343 2360 2377 2390 2402 2413 2422 2431 2449 2459 2465 2471 2477 2501 2528 2533 2547 2548 2544

2441 2757 3009 3245 3431 3618 3779 3897 4016 4099 4188 4260 4330 4385 4439 4483 4516 4556 4603 4602 4658 4687 4721 4784 4779 4805 4825 4844 4862 4898 4918 4930 4942 4954 5003 5056 5066 5094 5096 5087

41 56 70 83 96 110 123 135 146 156 165 174 182 190 198 204 211 217 225 229 235 241 246 252 257 261 265 269 272 275 284 287 290 292 299 310 314 328 329 333

1112 1476 1840 2185 2486 2845 3180 3477 3753 4001 4224 4428 4638 4837 5007 5168 5326 5487 5670 5780 5925 6056 6183 6308 6433 6533 6624 6706 6789 6864 7071 7129 7190 7249 7408 7639 7725 7975 7987 8006

639 849 1059 1259 1436 1642 1837 2009 2169 2313 2443 2562 2684 2799 2899 2992 3084 3178 3284 3349 3433 3510 3584 3657 3730 3789 3842 3890 3938 3982 4104 4138 4174 4209 4303 4440 4490 4644 4652 4670

All values are in 10 -6 of atomic unit. The LIL 1 line strength is the same as that of LIL~(tSo) in table 1.

t h e t r a n s i t i o n p r o b a b i l i t i e s in

j-j

c o u p l i n g . T h e s e a r e l i s t e d i n t a b l e 2. Fig. 4 s h o w s t h e

v a r i a t i o n w i t h Z o f t h e r e l a t i v e i n t e n s i t i e s . F r o m t h i s f i g u r e w e see t h a t , f o r Z > 50, t h e i n t e n s i t i e s o f L 1 L 1 a n d L I L2 a r e n e a r l y e q u a l . T h e s t r o n g e s t line i n t h e s p e c t r u m is L2 L3, a n d t h e w e a k e s t is L~ L 2. Fig. 5 s h o w s t h e v a r i a t i o n w i t h Z o f t h e i n t e n s i t i e s

408

w.N. ASAAD

of the lines L1L 2 and L x L a. These intensities increase smoothly with Z and show saturation after Z equals 45. In this figure, the corresponding Auger transition probabilities of the stronger component of each line, calculated on the basis &intermediate coupling, both on the correct assumption and on the assumption of a two-electron sys-

3.0

f

~

2.5

2.0

(~i~

O.51

" ~ Ow

20

~o

,~o

~o

~o

7'0 z

Fig. 4. Variation with Z of the relative line intensity in j-j coupling. (i) LjLx (ii) LIL= (iii) LILs (iv) L=L= (v) L2Lz (vi) LzLs.

z

tiD...... --

"~

(1)

,o

2'0 ~

~o

~o

6'0

¢o zs~

Fig. 5. Variation with Z of the intensities of the lines (1) LzL=and (2) LIL8inj-j coupling. (Dashed lines are for intermediate coupling (i) LxLz(zPx)(ii) L1L3(sP1)for the (2s) (2p)~ configuration, (i)a LzL=(sP0 (ii)a LzLa(zPz) for the (2s) (2p) configuration.)

tern, are shown dotted for Z => 50. This is done for sake of comparison, and from the figure it can be seen that, even at Z = 80, when the correct assumption is assumed, the 1p~ component of L1 L2 is about 5 ~ stronger than L l L2 calculated onj-j coupling, while the 3P t component of L1 L3 is weaker than the corresponding line by about 18 ~o- Fig. 6 shows the variation with Z of the intensities of the lines L2 L3 and L 3 L 3. Again, the intensities increase smoothly with Z, but saturation is not attained till Z exceeds 60. In this figure also, the corresponding Auger transition probabilities of

I N T E N S I T I E S OF T H E

K-LL AUGER

409

LINES

the stronger component of each line in-intermediate coupling is shown dotted, and once again it can be seen that the extreme j-j coupling is not fully approached. This has been confirmed experimentally (see ref. a)). Calculations on a two-hole system assumption give, at Z -- 80, a transition probability for the 1D 2 component of L2 L3 higher than that of the L2L3 line by about 2 ~ and a transition probability for the aP 2 component of L3L 3 lower than that of L3L 3 by about 18~o. ti)

8

2

~o

2o

30

ho

s'o

do - ~-

z ~'o

Fig. 6. Variation with Z of the intensities of the lines (1) L~La and (2) LaL~in j-j coupling. (Dashed lines are for intermediate coupling (i) LzL3(1D~) (ii) L3I.~(3P~)for the (2s)2(2p)4 configuration, (i)a L~La(sP~) (ii)a L3Ls(1Dz) for the (2p)2 configuration.) The above analysis shows the necessity of introducing the intermediate coupling theory in the relativistic calculations of the K - L L Auger spectrum. Until this is properly done, it is possible to give an estimate of the expected values for Z = 80 by modifying the author's results 11) obtained for mercury in the ratios found in the above comparison for the non-relativistic treatments. The modified values are given in table 3. Only the main lines or their larger components are given. (These new values are still in good agreement with experiment.) TABLE 3 Modified relativistic values of the K-LL Auger line intensities for Z = 80 Original values L1Lx LxL2 L1L3 L2L2 L~Ls L~La

1.00 1.44 0.82 0.09 1.46 0.66

Modifiedvalues eS0) eP1) (3Pt) (1S0) (1D~) (ap~)

1.00 1.51 0.67 0.095 1.49 0.54

The original values are those calculated by Asaad xt).

4. Comparison with Previous Work and with Experiment It is of interest to compare the results obtained above, where use has been made of the transition amplitudes calculated by Callan, with the corresponding results

410

w.N..~SAAD

which can be obtained when transition amplitudes computed by other workers are used instead. We first compare the above results with the transition probabilities obtained using t h e amplitudes calculated by Rubenstein and Snyde-" 16) for argon, krypton and silver, the amplitudes calculated by Burhop 9) for silver and those computed by Asaad 11) for mercury. Rubenstein and Snyder, and also Asaad, used wave functions for a self-consistent field. Burhop used screened hydrogenic functions with screening constants given by Slater's rules. In obtaining the new results, we used the values of the mixing parameters obtained before. It was found that, for Z = 18, the values of the transition amplitudes calculated by Callan give rise to considerably larger values for the transition probabilities for the (2s)°(2p) 6 and the (2s)(2p) 5 configurations, and considerably lower values for the (2s) 2(2p) 4 configuration, than those based on Rubenstein and Snyder's amplitudes. For Z = 36 and Z = 47, the two results based on their values for the (2s)(2p) 5 configuration become nearly equal, while the situation remains the same as for argon for the (2s)°(2p) 6 and the (2s)Z(2p) 4 configurations. Callan's values and Burhop's values of the amplitudes for silver give rise to comparable values for the transition probabilities of all lines~ although the intensities of the 3P 2 and the ID 2 lines for the (2s)2(2p) 4 configuration based on Callan's values are still appreciably lower than those based on Burhop's values. The same holds also for Z = 80 when comparison is made with results based on Asaad's amplitudes. The same pattern is again maintained for the configurations (2s) 2, (2s)(2p) and (2p) 2. As the values of the strength of the LILI(1S0) line, calculated by these authors, are different, especially for the lighter elements, the results obtained for the relative intensities (see tables 4 and 4(a)) of the various lines of the spectrum, while still in general agreement as to order, are not as to numerical values. The greatest discrepancy occurs for the most intense line, the 1D2. For argon, Callan's values lead to a relative intensity of this line of about 1.8, whereas Rubenstein and Snyder's values lead to a value of about 7.5. The discrepancy diminishes, however, as Z increses, but is still present. It is therefore of interest to compare the results of the calculations made by the different authors for the lighter elements with the experimental values. Table 5 gives such a comparison for Z = 26 and Z = 29. The experimental values for Z = 26 are those of Moussa and Bellieard 1~). The different lines in the spectrum are not resolved. Only the total transition probability for each of the (2s)(2p) 5 and (2s)2(2p) 4 configurations is given relative to that of the (2s)°(2p) 6 configuration. In this comparison, therefore, the kind of coupling assumed in the theoretical treatments is unimportant, since all three couplings which may be assumed give rise to the same sum for each configuration. From table 5, it is seen that Caiian's values give a ratio of 2.5 : 1 for the transition probabilities of (2~)(2p) 5 relative to (2s)°(2p)6. This Js in good agreement with the experimenal value 2.2 : I. However, the ratio of the (2~)~(2p) 4 to (2s)°(2p) 6 transition probabilitieg'i given by Callan's values is too

INTENSITIES OF THE K-LL AUGER LINES

411

TABLE 4 Comparison of the relative intensities for the (2s)°(2p) 6, (2s)(2p) 5 and (2s)2(2p) 4 configurations 18 Z LtL~(1Pt) L1Lz(sP0) L1Ls(aP1) LxLs(sP~) LIL2(aSo) L~Ls(XDI) L3Ls(3P0) L3 I_~(sP~) a) b) c) ~)

36

47

80

•)

~)

.)

~)

.)

~)

o)

.)

,)

1.71 0,05 0.15 0.25 0.14 1.81 0.001 0.01

2.58 0.07 0.23 0.37 0,58 7.51 0.003 0.06

1.91 0.05 0.54 0.27 0.18 3.04 0.12 0.69

2.44 0.07 0.69 0.36 0.45 7.33 0.29 1.66

1.45 0.05 1.12 0.27 0.15 3.13 0,19 1.11

1.78 0.07 1.36 0.36 0.32 6.59 0.41 2.34

1.80 0.05 1.37 0.24 0.19 4.72 0.24 1.68

1.07 0.05 1.68 0.26 0.14 3.28 0.26 1.54

1.22 0.06 1.92 0.32 0.25 5.40 0.46 2.52

Calculated using the transition amplitudes computed by Callan ~2). Based on the transition amplitudes calculated by Rubenstein and Snyder 7a). Calculated using Burhop's values ') of the transition amplitudes. Based on Asaad's values n).

TABLE 4(a) Comparison of the relative intensities for the (2s) a, (2s) (2p) and (2p) z configurations

18 Z

Ll~a(aP0) L1L2(aPx) L1Ls(sPz) LxLa(1Pa) LzI,~(aPo) LsL~(~P~) L3Ls(XD:~) Lsl-,aeS0)

.

-

36

~

0.05 0.15 0.25 1,71 0.001 0.02 1.81 0.14

.~--~"

0.07 0.23 0~37 2.58 0.003 0,08 7.49 0.58

- -"--x

0.05 0.33 0.27 2.11 0.05 1.67 2.06 0.25

0.07 0.43 0.36 2.70 OA2 4,03 4.96 0.61

47 /"

-

80

-

0.05 0.55 0.27 2.02 0.08 2.46 1.78 0.26

0.07 0.69 0.36 2.46 0.18 5,18 3.74 0.55

0.05 0.65 0.24 2.52 0.10 3.72 2.68 0.32

0.05 0.88 0.26 1.87 0.13 3.14 1.68 0,27

0.06 1.01 0.33 2.14 0.22 5.17 2.76 0.49

See table 4 for explanation of the notes ~-~)

"FABLE 5 Comparison with experiment for Z =: 26 and Z :: 29

Z:~29

Z=26 (~nf.

Exp.

Ref. aT) (2s)°(2p) e ~,~)(2p) b (2s)t(2p) *

1 2.2 6.8

Theoretical

I)

b)

i 2.5 3.1

1 3.5 9.3

Experimental Ref. 7)

Ref. xs) 1 2.8 9,9

1 4.4 t8

,)

Tbeoretleal ~)

1 2.6 3.4

*) Theoretical values obtained from Callan's ealculations la). o) Theoretical values estimated by interpolation o f Rubenstxin and Snyd~r's results 1.).

1 3.5 9~5

412

w . N . ASAAD

low, being 3.1 : 1, as compared to the experimental 6.8 : 1. Rubenstein and Snyder's values, when interpolated, give for Z = 26, too large a value for this ratio (9.3 : 1). Again, for Z = 29, Callan's value of 2.6 : 1 for the ratio of the transition probabilities of the (2s)(2p) 5 and (2s)°(2p) 6 configurations is in good agreement with the experimental value of 2 . 8 : 1 , measured by Bellicard, Moussa and Haynes 18). However, Callan's value of 3.4 : 1 for the ratio of the transition probabilities for the (2s)Z(2p) 4 and (2s)°(2p) 6 configurations is too low compared with their experimental value of 9.9 : 1. The corresponding value of this ratio deduced by interpolation from Rubenstein and Snyder's calculations is 9.5 : 1, in remarkably good agreement with experiment. The experimental values of Sokolowski and Nordling 7) for Z = 29 are also listed in table 5. It is at once seen that their values are larger than both the theoretical values and the previously quoted experimental values of Bellicard, Moussa and Haynes. However, it is quite remarkable that Sokolowski and Nordling managed to resolve the different lines in the spectrum for such light elements as Cu and Ge. TABLE 6 Comparison with experiment for Z = 47

Experimental 14) L1L1 L:Lz LIL 3 L2L, L2La LaL~

1 1.3±0.2 t 1.3~0.2] 2.6~0.4 0.5:k0.2~ 3.2~0.4} 5.5:k0.8 1.8~0.21

,) 1 1.51 1.39 0.15 3.13 1.30

Theoretical b) 1

(0.61) t (2.29)] 2.9 (0.08) 4.6 (2.03)

(2.46)j

3.6 9.6

a) The present work. The results given in brackets are based on the assumption of a two-electron system. b) Theoretical values deduced from Rubenstein and Snyder's calcu!ations 16).

Table 6 gives a comparison of our calculations for the relative line intensities in the K-LL Auger spectrum of silver with the experimental values of Johnson and Foster ~4). It is at once seen that the correct assumption of almost closed L shell gives rise to satisfactory agreement with experiment for all lines, with the exception of the weakest line L2 L2 and the line L 3 t 3. Table 6 shows also that the sum of the line intensities for the (2s)(2p) 5 and for the (2s)Z(2p) 4 configurations, relative to the line strength for the (2s)°(2p) 6 configuration, calculated using the transition amplitudes computed by Callan are in quite good agreement with the corresponding experimental values. Those calculated using Rubenstein and Snyder's values are, however, too high compared with experiment. Experimental results for the K-LL Auger spectrum of elements with Z in the range from Z = 52 to Z = 66 have been accumulating. In table 7, comparison is made of the theoretical results obtained with the experimental values of Erman and Suj-

INTENSITIES OF THE K-LL AUGER LINES

413

kowski 19) and of Graham, Brown, Ewan and Uhler 6) for Z --- 55. The experimental results due to Marguin and Moussa 20) for Z = 56 are also given in this table for comparison. It can readily be seen that the agreement is excellent for all the lines between our calculated values and the experimental results of Erman and Sujkowski. The experimental values of Marguin and Moussa for Z = 56 are nearer to their values than to those of Graham et al. TABLE 7

Comparison with experiment for Z = 55 Theory (present work) L1L1 LxL~ L1La{ 'P*

1 1.32 0"27/ 1.65

L2L2 L2L3 L3L3

0.14 3.19 1.51

Experiment Ref. 10) 1 1.414-0.10

Exp. 20) Z = 56

Ref. 6) 1 1.71i0.07 0'42!0"19t

1

1.4 1.99±0.07

3.394-0.25 4.23±0.07 1o44±0.10 1.91±0.06

1.8

2.9 1.5

Graham, Bergstr6m and Brown s) have pointed out that previous non-relativistic calculations using intermediate Coupling theory gave rise to too small values for the intensity of the satellite lines expressed relative to the total intensity of the line doublets. The most prominant discrepancy between theory a-nd experiment is for the fractional intensity aP0/(XP 1 + 3Po) of the L 1 L2 line. This discrepancy is still present in our work, pointing to the need of taking into account relativistic effects which have been ignored in the above calculations. As Z increases, the coupling approaches extreme j-j coupling and relativistic effects become more and more prominent. It is therefore expected that the results of this paper for the more heavy elements will be in poor agreement with experiment. It is to be noted that inj-j coupling the L t L 3 line intensity is always double that of the L1 L2 line, as can immediately be seen from the expressions in sect. 3, in contrast with experimental results. For Z = 47, for example, the two lines were found by Johnson and Foster x4) to have nearly equa ! values; for the heavier elements, the L 1L 2 line is always more intense than the L~ L3 line (see for example the experimental results o f Bergstr6m and Hill is) for mercury). It is evident from the present work that intermediate" coupling theory has gueeessfully removed this descrepaney for intermediate atomic numbers. Also, it has been shown by Asaad It) that for Z = 80, relativistic calculations for the intensities of the lines in the K-LL Auger spectrum are in remarkably good agreement with experiment. (See also Listengarten 21)). However, such calculations assume extreme j-j coupling.

414

w . N . ASAAD 5. C o n c l u s i o n

It is n o w clear that, whereas L-S coupling can be safely used in calculating the intensities o f the various lines in the K - L L A u g e r spectrum of elements with a t o m i c n u m b e r up to 25, j-j coupling is n o t fully approached even at Z = 80. I n t e r m e d i a t e coupling theory should therefore be applied for the calculation of the K - L L A u g e r spectrum of atoms with intermediate a n d large atomic n u m b e r . Non-relativistic calculations carried out using intermediate coupling lead to good agreement with experiment for intermediate atomic n u m b e r s when the coupling is assumed to b e between the holes. It successfully accounts for the variation with the atomic n u m b e r Z of the line strengths of the K - L L A u g e r spectrum. It remains to apply intermediate coupling theory properly in the relativistic calculations of the K - L L Auger spectrum o f the heavier elements. The a u t h o r w o u l d like to t h a n k Dr. E. J. Callan a n d Dr. R. L. G r a h a m for comm u n i c a t i n g their results to him in a d v a n c e o f publication a n d Professor J. N. Snyder for sending h i m a copy of Dr. R u b e n s t e i n ' s thesis~ He also wishes to t h a n k Professor E. H. S. B u r h o p for his interest in the work a n d for his valuable suggestions a n d criticism which helped the a u t h o r in p u t t i n g the paper in its present form.

References

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

W. N. Asaad and E. H. S. Burhop, Proc. Phys. Soc. 71 (1958) 369 G.T. Ewan, J. S. Geiger, R. L. Graham and D. R. MacKenzie, Can. J. Phys. 37 (1959) 174 G. T. Ewan, R. L. Graham and L. Grodzins, Can. J. Phys. 38 (1960) 163 G. T. Ewan and J. S. Merrit, Can. J. Phys. 38 (1960) 324 R. L. Graham and J. S. Merrit, Can. J. Phys. 39 (1961) 1058 R. L. Graham, F. Brown, G. T. Ewan and J. Uhler, Can. J. Phys. 39 (1961) 1086 E. Sokolowski and C. Nofdling, Ark. Fys. 14 (1959) 557 R. L. Graham, I. Bergstr6m and F. Brown, Nuclear Physics 39 (1962) 107 E. H. S. Burhop, Proc. Roy. Soc. 148 (1935) 272 J. C. Slater, Phys. Rev. 36 (1930) 57 W. N. Asaad, Proc. Roy. Soc. 249 (1959) 555 E. J. Callan, Phys. Rev. 124 (1961) 793 C. Froese, Proc. Roy. Soc. 239 (1957) 311; 244 (1958) 390; 251 (1959) 534 F. A. Johnson and J. S. Foster, Can. J. Phys. 31 (1953) 469 I. Bergstr/~m and R. D. Hill, Ark. Fys. 8 (1954) 21 R. A. Rubenstein and J. N. Snyder, Phys. Rev. 97 (1955) 1653; R. A. Rubenstein, Ph.D. Thesis (University of Illinois, 1953) A. Monssa and J. B. Bellicard, J. Phys. Radium lSa (1954) 85 J. B. Bellicard, A. Moussa and S. K. Haynes, Nuclear Physics 3 (1957) 307 P. Erman and Z. Sujkowski, Ark. Fys. 20 (1961) 209 P. Marguin and M. A. Moussa, J. Phys. Radium 21 (1960) 149 M. A. Listengarten, Izvestia Akad. Nauk USSR (Phys. Ser.) 25 (1961) 792