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Photoelectric cross sections and multi-electron transitions in the sudden approximation
Volume
25,
number
CHEMICAL
2
PHOTOELECTRIC
1.5 March 1974
PHYSICS LETTERS
CROSS SECTIONS AND MULTI-ELECTRON IN THE SUDDEN APPROXIMATION
TRANSITIONS
C.S. FADLEY Departmer~t
of Ckmistry,
University
of Hawaii.
Hor~olulu.
Received 5 July 1973 Revised manuscript received 7-S December
Hawaii
96822.
USA
1973
Within the usual assumptions associated with sudden-approximation analyses of photoelectron spectm. it is shown that unrelaxed subshell cross sections calculated on the basis of unit overlap between initial- and final-state-passive electrons implicitly include the effects of both one-electron and multi-electron transitions. Such cross sections therefore need not be directly related to the intensities of the one-electron transition peaks discernible in X-ray photoelectron spectroscopy (XPS), but should describe the processes associated with soft-X-ray absorption coefficient measure: ments well above threshold. The latter prediction is consistent with available comparisons between experiment and theory. The relationship of XPS peak intensities and their ratios to unrelaxed cross sections is also discussed within this approximation.
The partitioning of total atomic photoelectric cross sections into their subshell contributions is important for several reasons, among them the interpretation of photoelectron spectra and absorption coefficients. We shall restrict our attention to cross sections relevant to X-ray photoelectron spectroscopy (XPS or ESCA) and closely related experiments such as the measurement of soft-X-ray absorption-coefficients, for which photon energies’are of the order of 1 keV_ Furthermore, only inner-shell absorption processes well away from threshold will be considered, making plausible the use of the sudden approximation in treating multi-electron transitions [l ,2] . Al& though in recent years several calculations of such cross sections have beenmade [3-13, and references therein], a direct inclusion of the effects of multi-electron transitions has been attempted for relatively few cases (for example, refs. [ 14-161). Furthermore in the vast majority of these calculations_ it is assumed-that the one-electron orbitals describing the N-l passive electrons are unrelaxed in the final state, a Koopmans’-theorem-like approximation. Thus, questions arise as to the precise significance of such unrelaxed cross sections, particularly with respect to their seeming neglect of multi-electron processes that are found in XPS spectra to be involved in from 10 to 50% of all inner-shell absorption events 12, 17-221. The sudden approximation has proven useful in analyzing such processes [ 1,2, 17-221, and we apply it here to an analysis of unrelaxed cross sections; Our derivation is. ,. analogous to that of Manne and &berg [l] , who were cbncerned with the relationship of unrelaxed (Koopmans’ theorem) binding energies to the various experimental binding energies associated with one-electron and multi-election transitions. Their final result is quoted for comparison below. In the sudden-approximation analysis we shall use, it is assumed that: (1) the primary inner-shell excitation is.sudden relative to the adjustment of the outer electrons to. change in screen: ing; ~. (2) although final-state relaxation is permitted, the net effects associated with all one-electron-matrix elements ‘: _.‘other than that describing the primary excitatiorrare neglected (more’accurate methods do exist for,dealing _.:‘:~‘. with the additional matrix elements and overlap factors arising ‘with relaxed final states [4, 141; but these have-:. 5.. ., :. ._, .: not yet been widely applied); . ,... .. -. : :-‘:‘. _,. .T _. : -_-... : 225: :-. ..
CHEMICAL PHYSICS LETTERS
Volume 25. number 2
(3) the one-electron matrix elements range of final states involved;
describing
the primary
excitation
(4) the dipole approximation is used; (5) all initial and final states are-described by linear combinations figuration with L and S as good quantum numbers.
The primary photoelectron excitation is separated anti-symmetric products of the form [l] :
I5 March 1974
are approximately
of Slater determinants
out by writing
constant
over the energy
belonging to a single con-
the initial- and final-state
wavefunctions
as
(1) (3 where r~$,~~,,~,( 1) is the initial one-electron orbital involved in the primary excitation, @(N-l) represents the N-l passive electrons in the initial state, QG,p_my’.ms s,,( 1) represents a photoelectron with kinetic energy E and angular I”’ = I + 1, and @(N-l)
momentum
is a final-state wavefunction
for the N-i electrons
of the ionic core.
We shall only deal explicitly with one-electron and two-electron transitions, as the derivation an identical way with the inclusion of higher-order effects. There are thus three basic absorption be considered [2], and these are indicated schematically below.
would proceed in mechanisms to
Olte-electron:
(3) Two-eI&ron
drake-up/shake-o_fy
---(llI)~---(Iil’)p
hV
L*S -_,
-..(,,1)4-l
...(,I’~‘)P-l
($/“)I
sqlv-l),,Ef(fv-I),
wv,,ev
L’ &?’ + Q ’
E 1kl 7’
(4)
in these transitions, we have specified electron configurations, L and S values, wavefunctions, and total energies for initial and final states. The notation (no* refers to a filled inner subshell, whereas (n’Z’)p can represent either a filled or partially-filled outer subshell. The final state of the second electron involved in a two-electron transition is indicated by (~“1”)~ ~where it is understood that in a shake-off process, 12” is replaced by a kinetic energy E;_ The initial state we assume to have the same L.S. and f?(N) for all transitions, but various final state energies will be possible, including perhaps extra structure induced by L’,S’ multiplet splittings [2, 231. We have suppressed the magnetic quantum numbers and the individual subshell coupling schemes necessary to completely specify each initial and final state, but they are implicitly included in the subscripts (Y,B, and y_ The final-state subscripts 0 and y thus include I”’ , L’, S’, all magnetic quantum numbers, and a further index which may be required to delineate a given state if more than one coupling scheme within the same configuration gives rise to the same L’,S’ The cross section for a transition from an initial state Q to a final state fl or -y will be given by [I] : a
&4(Y)
=
CI~QscYr’“:“~,~~IrlQ,~~,S~I~Itu’(N-i~,,,,l~~N-l~,~l’,~
[2, 231.
(5)
in which C is a constant for a given hv. The overlap between \kf(Ar-l)p(,j and @(IV-I), will be non-zero only if both represent the same L.S.hI,, andMs values and if the one-electron orbitals comprising both contain the-same set of N-1 values of I. These conditions. give rise to the well-known monopole
selection rules, including N = I” - I’ = 0
]2,4,LS]
It is now convenient
over all of the degenerate
to sum over all transitions
associated
with a given final-state’corrfiguration
initial states involved, as is done in determining
subshell cross secti&.
.. 226
‘.
and to average This will yield a
Votume 75, number 2
1.5 March
CHEWCAL PHYSICS LETTERS
1974
true one-eIectron cross section or11and various two-electron cross sections a,,I(,l,~_,l,~~~~),in which the parenthetical subscripts indicate the initial and final states of the second electron. This summing and averaging wiIl proceed in an identical way for all cross sections due to the monopole character of transitions within thcN-1 passive electrons. The resulting cross sections will then be:
in which C’ is a constant. C,, , and Rz,l,, are the usual coefficients and radial dipole matrix elements appearing cross section calculations, P is an average kinetic energy to be defined more precisely below, the subscript o refers to the one-electron-t~nsition final co~~~g~imtion~ and the subscript r indicates that only radial overlap integrals remain to be calculated. Eqs. (6) are generaiizations of expressions utilized in previous sudd&-approximation analyses of XPS spectra [2] _ In relating these cross sections to an unrelaxed cross section r+, it is useful to expand the initial-state passive etectrons as represented by @(N-l), in terms of the complete set of orthono~al final-state ion-core wavefunctions representing all possible transitions [ 11 : in
@(lV-l)a = Cie$vI)pqNI>abPf(lv- i)i3 f Jj-J\Irf(fv- I)pqNI)pf(lVI), + **-. (7) P Y As we have not explicitly distinguished shake-up and shake-off transitions, certain portions of the sum over r should be interpreted as integrals over the shake-off portion of the spectrum. If the @(lV-I), are normalized, then one consequence of this expansion is that the radial overlaps must satisfy
Ic~f(~-l)ol~~(N-I)~ri~ -I-
c
It~r(~-1),1.,._,*,,,,l~(.~-i)~r12 + *a- =
It
(8)
n’l’. n”1”
where the sum extends over all alIowed two-electron-transition final configurations. Utilizing an expansion such as that in eq. (7). Manne and Aberg [ 11 have shown that a Koopmans’ theorem binding energy is equal to a weighted average binding energy over all possibfc final states. That is, if this binding energy is denoted by Eb(nl)KT zz Eb(~zl)u, it is given by Eb(“l)u
= 5
It*‘(N-l),
I@$V-I)~l%-b(~l/)6,
(91
6’0 in which the index 6 is used to distinguish over degenerate final states and averaging thus to the cross section for in approximation as F = hv By contrast, unrelaxed section electron transition, and can be calculated G,cX-,
=C,(~f(~~,~r.IY’ion,lZ .r=1 J u = W”P,,,~,mi”,m;
cp(N-1)
all final states of different energies and the overlap implies summing over degenerate initial states. The weighting of each binding energy js a to that state. the average kinetic energy is correctly &,(“l)“. a specific is based upon what appears to be a pure oneas:
I5 +s%+~&#~mr~(iV-l)(I$ a j=* : (10)
= clc@~*,l’~t,mi”.m;” lYl@,~,Q2The superscript u in all cases refers to an un&laxed final state. These cross sections as before, yielding the unrelaxed subshell cross section as
can be smnnied
.,
:
and averaged’. : ‘. ,. .(;I) : _ : 227 ._.-: ., :‘-:
.Volume 2.5, number 2
CHEMICAL
PHYSICS
15
LETTERS
March 1974
The constants and radial dipole matrix elements in this equation are the same as those in eqs. (6). because the monopole selection rules dictate that states of the same symmetries are involved in the summing and averaging, theaverage value of E, for unrelaxed photoelectron states will be equal to E, and essentially the same radial wavefunctions will be involved_ Combining eqs. (6). (S), and (1 1) thus means that U-
a Ill
-
=,,,-I-
c
liI’.li’l”
@nl(,iI’-n”l”)
+..--
(12)
That is. the unrelaxed cross section represents a total cross section for all one-electron and multi-electron processes. The above derivation can also be generalized to encompass simple multiplct splittings in the final states by appro1231 _A more general priately partitioning a given configuration cross secti.on amon, 0 different couplingschemes expression of our final result is thus a cross-section analogue of eq. (9): (13) in which 6 has the same meaning and a,,1 6 represents a summed and averaged cross section to a set of final states in a given configuration with the same L’, S’, and energy. As #, in this approximation somewhat accidentally includes all possible photoabsorption events, it may be misleading to compare such cross sections directly to experimental results in which the different events are distinguished on the basis of final-state energy, as is the case in photoelectron spectroscopy. For example, the intense one-electron peak (or peaks if multiplet splittings are present) observed in XPS spectra should be proportional in total area to arlr in eq. (6a). From cqs. (6a) and (1 I), it is clear that a:[ could be multiplied by the appropriate overlap squared as a general procedure for obtaining a first-order estimate of CI,~~_ At the level of this approximation, it might also be adequate to use a single diagonal product of one-electron radial overlaps squared, and it is this product that has been used in most shake-up- and shake-o’ff-intensity analyses [2,4] _The final-state one-electron orbitals mi&t be determined by a hole-state Hartree-Fock calculation_ A much simpler procedure would be to use the equivalent-core approximation and thus assume that the outer subshells in the final state are represented by radial functions characteristic of the atom with next liigher atomic number [2], and that furthermore the inner zubshells show negligible relaxation (very nearly unit overlap). This approximation is discussed further below. By contrast, oil should be directly comparable with an experimental situation such as that prevailing in softX-ray absorption coefficient measurements wel! above threshold. In such experiments, the photon energy can be presumed high enough to participate in all photoabsorption events of significant intensity, and in a manner describable within the sudden approximation_ The measurement is thus not sensitive to which absorption events are taking place, but simply to the sum of the probabilities for all events, whichis precisely given by 0% in eq. (13). Therefore, good agreement might be expected between experimental X-ray absorption coefficients and theoretical values based on unrelaxed total cross sections, provided that the dominant contributors are inner-shell absorptions well above threshold_ This prediction is consistent with previous comparisons of experiment and theory, in which typical agreement well away from threshold is in the approximate range of 5%lO%, with no consistent under- or over-estimation of u by theory [3, 5.6, I@, 24]_ By contrast, if ail did not somehow include the effects of multielectron transitions, it would be expected to be consistently too low by approximately 20% or more in predicting such total cross sections [2. l7--22]_ It has. in fact, been noted by McGuire [6] that the good agreement in such comparisons appears to indicate that no significant correction of such cross sf?ctioris for final state relaxation is necessary, although no explanation for this observation was given. As one example of such a comparison, fig. 1 shows experimental absorption coefficient data $ie to Henke et al. [24] and .tieore,tical curves based on unrelaxed calculations by Veigele et al. [ 10, I I]. For-shorter.wavelengths and/or iegibns we11 away from threshold, the agreement is remarkably good. It should be noted, howevdr, that this is piobably some&hat.forttiitous, as ihe use of slight& different approximations in calculating the same crosqsections by Kennedy and Manson [9] yields theoretical values differing by at least 10%.even in regiiqns.corresponding to-veq_good agreement in fig. 1 [S-l I] _ As a final point, we consider one-electron-transition subshell c&s section iatios, den0te.d by IJ~,~../u,;_ Such -228
_’ .,
.,
,.
Voiume 25, number 2
CHEMICAL
PHYSiCS
I5 March 1974
LETTERS
IO’ KRYPTON
‘O
X(A)
loo
Fig. I. Comparison of experimental 124 j and theoretical f 10, 111 X-my absorption coefficients values are based on unrelased subshell cross sections. (h = 10 A corresponds to kv = 1250 eV.)
‘0
X(&
‘00
for the noble gases. The theoretical
ratios can be directly determined for many systems from XPS, provided that the effects of inelastic scattering are adequately allowed for [24]‘. If such a ratio is compared to theory, it may be ~&~/a~~ that is calculrtted, as in the first detailed studies of this kind on Kr by Krause f 171 and Manson and Cooper [7] and on Ne by Wuilleumier and Kwuse f20] and Kennedy and Manson [9] _ In such a comparison, experimental and theoretical ratios will agree only if the net effect of multi-eiectron~transitions is to reduce the one-electron cross section relative to the total subshell cross section by the same fraction for both subshells. This appears to have occurred in the study on Kr, as has been noted by Krause [ 171 in explaining the relatively good agreement between experiment and theory for the ratios ass Jajd and 03p/03d, and the fair agreement for c&/o&t and a4pfo3, [7] _A possible explanation for such a cancellation of errors due to multi-electron transitions lies in the equivalent-core model, provided that it is assumed that the major fraction of relaxation (non-unit overlap) between the initial state and one-electron : final state is associated with the outermost subshells. Because such outer subshells should experience very nearly the same core interactions for B hole in any inner subshell, the overlap correction of equation 63 should be approximately equal for all inner-subshell holes. Therefore, inner-subshell cross section ratios crn~~/un~ might be expected to be rather close to the theoretical ratios ~$1, /air. This would be particularly true for ratios within a given shell. a,&~,+ as the net change in outer electron screening should be more nearly equal for !11’and ni holes. This prediction is thus consistent with the good agreement between experiment and theory for o&osd and u~~/cQ~ in Kr 17. 171. However, the preceding argument would not be expected to hold for a ratio of an outer-subshell cross section and an inner-subshell cross section, as the equivalent-core model cannot be applied in an identical way to both final hole states, and, more importantly, it is welt known that strong intmshell correlation effects can cause. significant devietions of multi-electron transition probabilities from those predicted by the sudden approximation for the case of primary photoelectron excitation from an outer subshell [Z, 17,201. Such effects could be at least partially responsible for the larger deviations between experiment and theory for a4Jusd and ~~/a& in Kr 17, 17). Outer-subshell cross section ratios should therefore also be affected by correlation in a non-simple way, although it has been found that the uzs/02p ratio for Ne is in rather good agreement with theoretical &/a& .yfues [9,20]. This Iast observation is not explained within the context of the simple sudden-approximation analysis presented here. ‘_ T&e authors &press~gmtitude to B.L. He&, P.S. &&IS, and J.W.‘C%oper for helpful comments re]evam to : ,: this work. The support of the National Science-Foundation &also’gratefully acknowledged~. -,. ._ “. _ ‘. .. : -. :_ .: I ., .,,._ : :: zgj -; .. . . . .,‘. ” ‘. :. :. -1.:
CHEMICAL PHYSICS LETTERS
Volume 25. number 2
: 15 hfarch 1974.
References R. hfanne and T. Aberg, Chem. Phys. Letters 7 (1970) 282. [2] T.A. Carlson, M.O. Krause and W.E. hfoddeman, J. Phys. (Paris) C2 (1971) 102, and references
f I]
therein.
[3] G. Rakavy and A. Ron, Phys. Rev. 159 (1967) 50. 141 U_ Fano and 3-W. Cooper. Rev. Mod. Phys. 40 (1968) 441.
[S] ST, hfanson and J.W. Cooper, Phys. Rev. 165 (1968) 126_ [6] E.J. McGuire, Phys. Rev. 175 (1968) 20. f7f ST. Manson and f-W_ Cooper, Phys. Rev, 177 (1969) X.57. [8] E. Storm and Hf. Israel, Nuclear Data Tables A7 (1970) 565. [9] D.J. Kennedy and S.T. Manson, Phys. Rev. AS (1972) 227. [lo] E.M. Henry, CL. Bates and W.J- Veigele, Phys. Rev. A6 (1972) 2131. [I 1 f W.J. Veigefe, qtornic Data Tables 5 (1973) 51. [12] J.H. Scofield. Lawrence Livermore Laboratory, Report No. UCRL-51326 (1973). f13] R.H. Pratt, A. Ron and H-K. Tseng, Rev. Mod. Phys. 45 (1973) 273. [ 14 1 R.J.W. Henry and L. Lipsky, Phys. Rev. 153 (I 937) 5 1. [IS] A.F. Starace, Phys. Rev. A2 (1970) I 18. [ 161 M.Ya. Amusya, N.A. Cherepkov and L.V. Chemysheva, Zh..Eksperim. i Teor, Fiz. 60 (1971) 160 [English transl. Soviet Phys. JETP 33 (1971) 901. [ 17 J M.O. Krause. Phys. Rev. 177 (1969) 1.51. [ 181 R. Siegbahn, C. Nordling. G. Johansson. J. Hedman, P-F, Heddn, K. Hamrin. U, Celius, T. Bergmark, L-0. Werme, R. hfanne and Y. Baer, ESCA appiied to free molecules (North-Hohand, Amsterdam. 1969). [I91 T. Novakov, Phys. Rev. B3 (1971) 2693; T. Novakov and R. Prins. in: Electron spectroscopy. ed. D.A. Shirley (North-Holland. Amsterdam. 1972) p. 821. 1201 F- WuiBeumier and hf.0. Krause, in: Electron spectroscopy, ed. D.A. Shirley (North-Holland, Amsterdam, 1972) p. 259. f2 I] G.K. Wertheim. R.L. Cohen, A. Rosencwaig and H-J. Guggenheim, in: Etectron spectroscopy, ed. D.A. Shirley (North-Holland, Amsterdam. 1972) p. 813. [22] K.S_ Kim and R.E. Davis, J. Electron Spectry. 1 (1972/73) 251; Ji.S_ Kim, private communication. [23] C.S. Fadley, D.A. Shirley, A.J. Freeman, P.S. Bagus and J.V. Mallow, Phys. Rev. Letters 23 (1969) 1397; C.S. FadIey and D.A. Shirley, Phys. Rev. A2 (1970) 1109; C.S. Fadley, in: Efectron spectroscopy, ed. D.A. Shirley (North-~oli~nd* Amsterdam, 1972) p. 781. [24] B.L. Henke and R.L. Et@, in: Advances in X-ray analysis, Vol. 13 (Plenum Press, New York, 1970) p_ 781. B.L. Henke and E. Ebisu, in: Advances in X-ray anaIysis. Vol. 16 (Plenum Press, New York), to be published.
.
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-,
:
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-.230'-::.
._
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.
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.,
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25,
number
CHEMICAL
2
PHOTOELECTRIC
1.5 March 1974
PHYSICS LETTERS
CROSS SECTIONS AND MULTI-ELECTRON IN THE SUDDEN APPROXIMATION
TRANSITIONS
C.S. FADLEY Departmer~t
of Ckmistry,
University
of Hawaii.
Hor~olulu.
Received 5 July 1973 Revised manuscript received 7-S December
Hawaii
96822.
USA
1973
Within the usual assumptions associated with sudden-approximation analyses of photoelectron spectm. it is shown that unrelaxed subshell cross sections calculated on the basis of unit overlap between initial- and final-state-passive electrons implicitly include the effects of both one-electron and multi-electron transitions. Such cross sections therefore need not be directly related to the intensities of the one-electron transition peaks discernible in X-ray photoelectron spectroscopy (XPS), but should describe the processes associated with soft-X-ray absorption coefficient measure: ments well above threshold. The latter prediction is consistent with available comparisons between experiment and theory. The relationship of XPS peak intensities and their ratios to unrelaxed cross sections is also discussed within this approximation.
The partitioning of total atomic photoelectric cross sections into their subshell contributions is important for several reasons, among them the interpretation of photoelectron spectra and absorption coefficients. We shall restrict our attention to cross sections relevant to X-ray photoelectron spectroscopy (XPS or ESCA) and closely related experiments such as the measurement of soft-X-ray absorption-coefficients, for which photon energies’are of the order of 1 keV_ Furthermore, only inner-shell absorption processes well away from threshold will be considered, making plausible the use of the sudden approximation in treating multi-electron transitions [l ,2] . Al& though in recent years several calculations of such cross sections have beenmade [3-13, and references therein], a direct inclusion of the effects of multi-electron transitions has been attempted for relatively few cases (for example, refs. [ 14-161). Furthermore in the vast majority of these calculations_ it is assumed-that the one-electron orbitals describing the N-l passive electrons are unrelaxed in the final state, a Koopmans’-theorem-like approximation. Thus, questions arise as to the precise significance of such unrelaxed cross sections, particularly with respect to their seeming neglect of multi-electron processes that are found in XPS spectra to be involved in from 10 to 50% of all inner-shell absorption events 12, 17-221. The sudden approximation has proven useful in analyzing such processes [ 1,2, 17-221, and we apply it here to an analysis of unrelaxed cross sections; Our derivation is. ,. analogous to that of Manne and &berg [l] , who were cbncerned with the relationship of unrelaxed (Koopmans’ theorem) binding energies to the various experimental binding energies associated with one-electron and multi-election transitions. Their final result is quoted for comparison below. In the sudden-approximation analysis we shall use, it is assumed that: (1) the primary inner-shell excitation is.sudden relative to the adjustment of the outer electrons to. change in screen: ing; ~. (2) although final-state relaxation is permitted, the net effects associated with all one-electron-matrix elements ‘: _.‘other than that describing the primary excitatiorrare neglected (more’accurate methods do exist for,dealing _.:‘:~‘. with the additional matrix elements and overlap factors arising ‘with relaxed final states [4, 141; but these have-:. 5.. ., :. ._, .: not yet been widely applied); . ,... .. -. : :-‘:‘. _,. .T _. : -_-... : 225: :-. ..
CHEMICAL PHYSICS LETTERS
Volume 25. number 2
(3) the one-electron matrix elements range of final states involved;
describing
the primary
excitation
(4) the dipole approximation is used; (5) all initial and final states are-described by linear combinations figuration with L and S as good quantum numbers.
The primary photoelectron excitation is separated anti-symmetric products of the form [l] :
I5 March 1974
are approximately
of Slater determinants
out by writing
constant
over the energy
belonging to a single con-
the initial- and final-state
wavefunctions
as
(1) (3 where r~$,~~,,~,( 1) is the initial one-electron orbital involved in the primary excitation, @(N-l) represents the N-l passive electrons in the initial state, QG,p_my’.ms s,,( 1) represents a photoelectron with kinetic energy E and angular I”’ = I + 1, and @(N-l)
momentum
is a final-state wavefunction
for the N-i electrons
of the ionic core.
We shall only deal explicitly with one-electron and two-electron transitions, as the derivation an identical way with the inclusion of higher-order effects. There are thus three basic absorption be considered [2], and these are indicated schematically below.
would proceed in mechanisms to
Olte-electron:
(3) Two-eI&ron
drake-up/shake-o_fy
---(llI)~---(Iil’)p
hV
L*S -_,
-..(,,1)4-l
...(,I’~‘)P-l
($/“)I
sqlv-l),,Ef(fv-I),
wv,,ev
L’ &?’ + Q ’
E 1kl 7’
(4)
in these transitions, we have specified electron configurations, L and S values, wavefunctions, and total energies for initial and final states. The notation (no* refers to a filled inner subshell, whereas (n’Z’)p can represent either a filled or partially-filled outer subshell. The final state of the second electron involved in a two-electron transition is indicated by (~“1”)~ ~where it is understood that in a shake-off process, 12” is replaced by a kinetic energy E;_ The initial state we assume to have the same L.S. and f?(N) for all transitions, but various final state energies will be possible, including perhaps extra structure induced by L’,S’ multiplet splittings [2, 231. We have suppressed the magnetic quantum numbers and the individual subshell coupling schemes necessary to completely specify each initial and final state, but they are implicitly included in the subscripts (Y,B, and y_ The final-state subscripts 0 and y thus include I”’ , L’, S’, all magnetic quantum numbers, and a further index which may be required to delineate a given state if more than one coupling scheme within the same configuration gives rise to the same L’,S’ The cross section for a transition from an initial state Q to a final state fl or -y will be given by [I] : a
&4(Y)
=
CI~QscYr’“:“~,~~IrlQ,~~,S~I~Itu’(N-i~,,,,l~~N-l~,~l’,~
[2, 231.
(5)
in which C is a constant for a given hv. The overlap between \kf(Ar-l)p(,j and @(IV-I), will be non-zero only if both represent the same L.S.hI,, andMs values and if the one-electron orbitals comprising both contain the-same set of N-1 values of I. These conditions. give rise to the well-known monopole
selection rules, including N = I” - I’ = 0
]2,4,LS]
It is now convenient
over all of the degenerate
to sum over all transitions
associated
with a given final-state’corrfiguration
initial states involved, as is done in determining
subshell cross secti&.
.. 226
‘.
and to average This will yield a
Votume 75, number 2
1.5 March
CHEWCAL PHYSICS LETTERS
1974
true one-eIectron cross section or11and various two-electron cross sections a,,I(,l,~_,l,~~~~),in which the parenthetical subscripts indicate the initial and final states of the second electron. This summing and averaging wiIl proceed in an identical way for all cross sections due to the monopole character of transitions within thcN-1 passive electrons. The resulting cross sections will then be:
in which C’ is a constant. C,, , and Rz,l,, are the usual coefficients and radial dipole matrix elements appearing cross section calculations, P is an average kinetic energy to be defined more precisely below, the subscript o refers to the one-electron-t~nsition final co~~~g~imtion~ and the subscript r indicates that only radial overlap integrals remain to be calculated. Eqs. (6) are generaiizations of expressions utilized in previous sudd&-approximation analyses of XPS spectra [2] _ In relating these cross sections to an unrelaxed cross section r+, it is useful to expand the initial-state passive etectrons as represented by @(N-l), in terms of the complete set of orthono~al final-state ion-core wavefunctions representing all possible transitions [ 11 : in
@(lV-l)a = Cie$vI)pqNI>abPf(lv- i)i3 f Jj-J\Irf(fv- I)pqNI)pf(lVI), + **-. (7) P Y As we have not explicitly distinguished shake-up and shake-off transitions, certain portions of the sum over r should be interpreted as integrals over the shake-off portion of the spectrum. If the @(lV-I), are normalized, then one consequence of this expansion is that the radial overlaps must satisfy
Ic~f(~-l)ol~~(N-I)~ri~ -I-
c
It~r(~-1),1.,._,*,,,,l~(.~-i)~r12 + *a- =
It
(8)
n’l’. n”1”
where the sum extends over all alIowed two-electron-transition final configurations. Utilizing an expansion such as that in eq. (7). Manne and Aberg [ 11 have shown that a Koopmans’ theorem binding energy is equal to a weighted average binding energy over all possibfc final states. That is, if this binding energy is denoted by Eb(nl)KT zz Eb(~zl)u, it is given by Eb(“l)u
= 5
It*‘(N-l),
I@$V-I)~l%-b(~l/)6,
(91
6’0 in which the index 6 is used to distinguish over degenerate final states and averaging thus to the cross section for in approximation as F = hv By contrast, unrelaxed section electron transition, and can be calculated G,cX-,
=C,(~f(~~,~r.IY’ion,lZ .r=1 J u = W”P,,,~,mi”,m;
cp(N-1)
all final states of different energies and the overlap implies summing over degenerate initial states. The weighting of each binding energy js a to that state. the average kinetic energy is correctly &,(“l)“. a specific is based upon what appears to be a pure oneas:
I5 +s%+~&#~mr~(iV-l)(I$ a j=* : (10)
= clc@~*,l’~t,mi”.m;” lYl@,~,Q2The superscript u in all cases refers to an un&laxed final state. These cross sections as before, yielding the unrelaxed subshell cross section as
can be smnnied
.,
:
and averaged’. : ‘. ,. .(;I) : _ : 227 ._.-: ., :‘-:
.Volume 2.5, number 2
CHEMICAL
PHYSICS
15
LETTERS
March 1974
The constants and radial dipole matrix elements in this equation are the same as those in eqs. (6). because the monopole selection rules dictate that states of the same symmetries are involved in the summing and averaging, theaverage value of E, for unrelaxed photoelectron states will be equal to E, and essentially the same radial wavefunctions will be involved_ Combining eqs. (6). (S), and (1 1) thus means that U-
a Ill
-
=,,,-I-
c
liI’.li’l”
@nl(,iI’-n”l”)
+..--
(12)
That is. the unrelaxed cross section represents a total cross section for all one-electron and multi-electron processes. The above derivation can also be generalized to encompass simple multiplct splittings in the final states by appro1231 _A more general priately partitioning a given configuration cross secti.on amon, 0 different couplingschemes expression of our final result is thus a cross-section analogue of eq. (9): (13) in which 6 has the same meaning and a,,1 6 represents a summed and averaged cross section to a set of final states in a given configuration with the same L’, S’, and energy. As #, in this approximation somewhat accidentally includes all possible photoabsorption events, it may be misleading to compare such cross sections directly to experimental results in which the different events are distinguished on the basis of final-state energy, as is the case in photoelectron spectroscopy. For example, the intense one-electron peak (or peaks if multiplet splittings are present) observed in XPS spectra should be proportional in total area to arlr in eq. (6a). From cqs. (6a) and (1 I), it is clear that a:[ could be multiplied by the appropriate overlap squared as a general procedure for obtaining a first-order estimate of CI,~~_ At the level of this approximation, it might also be adequate to use a single diagonal product of one-electron radial overlaps squared, and it is this product that has been used in most shake-up- and shake-o’ff-intensity analyses [2,4] _The final-state one-electron orbitals mi&t be determined by a hole-state Hartree-Fock calculation_ A much simpler procedure would be to use the equivalent-core approximation and thus assume that the outer subshells in the final state are represented by radial functions characteristic of the atom with next liigher atomic number [2], and that furthermore the inner zubshells show negligible relaxation (very nearly unit overlap). This approximation is discussed further below. By contrast, oil should be directly comparable with an experimental situation such as that prevailing in softX-ray absorption coefficient measurements wel! above threshold. In such experiments, the photon energy can be presumed high enough to participate in all photoabsorption events of significant intensity, and in a manner describable within the sudden approximation_ The measurement is thus not sensitive to which absorption events are taking place, but simply to the sum of the probabilities for all events, whichis precisely given by 0% in eq. (13). Therefore, good agreement might be expected between experimental X-ray absorption coefficients and theoretical values based on unrelaxed total cross sections, provided that the dominant contributors are inner-shell absorptions well above threshold_ This prediction is consistent with previous comparisons of experiment and theory, in which typical agreement well away from threshold is in the approximate range of 5%lO%, with no consistent under- or over-estimation of u by theory [3, 5.6, I@, 24]_ By contrast, if ail did not somehow include the effects of multielectron transitions, it would be expected to be consistently too low by approximately 20% or more in predicting such total cross sections [2. l7--22]_ It has. in fact, been noted by McGuire [6] that the good agreement in such comparisons appears to indicate that no significant correction of such cross sf?ctioris for final state relaxation is necessary, although no explanation for this observation was given. As one example of such a comparison, fig. 1 shows experimental absorption coefficient data $ie to Henke et al. [24] and .tieore,tical curves based on unrelaxed calculations by Veigele et al. [ 10, I I]. For-shorter.wavelengths and/or iegibns we11 away from threshold, the agreement is remarkably good. It should be noted, howevdr, that this is piobably some&hat.forttiitous, as ihe use of slight& different approximations in calculating the same crosqsections by Kennedy and Manson [9] yields theoretical values differing by at least 10%.even in regiiqns.corresponding to-veq_good agreement in fig. 1 [S-l I] _ As a final point, we consider one-electron-transition subshell c&s section iatios, den0te.d by IJ~,~../u,;_ Such -228
_’ .,
.,
,.
Voiume 25, number 2
CHEMICAL
PHYSiCS
I5 March 1974
LETTERS
IO’ KRYPTON
‘O
X(A)
loo
Fig. I. Comparison of experimental 124 j and theoretical f 10, 111 X-my absorption coefficients values are based on unrelased subshell cross sections. (h = 10 A corresponds to kv = 1250 eV.)
‘0
X(&
‘00
for the noble gases. The theoretical
ratios can be directly determined for many systems from XPS, provided that the effects of inelastic scattering are adequately allowed for [24]‘. If such a ratio is compared to theory, it may be ~&~/a~~ that is calculrtted, as in the first detailed studies of this kind on Kr by Krause f 171 and Manson and Cooper [7] and on Ne by Wuilleumier and Kwuse f20] and Kennedy and Manson [9] _ In such a comparison, experimental and theoretical ratios will agree only if the net effect of multi-eiectron~transitions is to reduce the one-electron cross section relative to the total subshell cross section by the same fraction for both subshells. This appears to have occurred in the study on Kr, as has been noted by Krause [ 171 in explaining the relatively good agreement between experiment and theory for the ratios ass Jajd and 03p/03d, and the fair agreement for c&/o&t and a4pfo3, [7] _A possible explanation for such a cancellation of errors due to multi-electron transitions lies in the equivalent-core model, provided that it is assumed that the major fraction of relaxation (non-unit overlap) between the initial state and one-electron : final state is associated with the outermost subshells. Because such outer subshells should experience very nearly the same core interactions for B hole in any inner subshell, the overlap correction of equation 63 should be approximately equal for all inner-subshell holes. Therefore, inner-subshell cross section ratios crn~~/un~ might be expected to be rather close to the theoretical ratios ~$1, /air. This would be particularly true for ratios within a given shell. a,&~,+ as the net change in outer electron screening should be more nearly equal for !11’and ni holes. This prediction is thus consistent with the good agreement between experiment and theory for o&osd and u~~/cQ~ in Kr 17. 171. However, the preceding argument would not be expected to hold for a ratio of an outer-subshell cross section and an inner-subshell cross section, as the equivalent-core model cannot be applied in an identical way to both final hole states, and, more importantly, it is welt known that strong intmshell correlation effects can cause. significant devietions of multi-electron transition probabilities from those predicted by the sudden approximation for the case of primary photoelectron excitation from an outer subshell [Z, 17,201. Such effects could be at least partially responsible for the larger deviations between experiment and theory for a4Jusd and ~~/a& in Kr 17, 17). Outer-subshell cross section ratios should therefore also be affected by correlation in a non-simple way, although it has been found that the uzs/02p ratio for Ne is in rather good agreement with theoretical &/a& .yfues [9,20]. This Iast observation is not explained within the context of the simple sudden-approximation analysis presented here. ‘_ T&e authors &press~gmtitude to B.L. He&, P.S. &&IS, and J.W.‘C%oper for helpful comments re]evam to : ,: this work. The support of the National Science-Foundation &also’gratefully acknowledged~. -,. ._ “. _ ‘. .. : -. :_ .: I ., .,,._ : :: zgj -; .. . . . .,‘. ” ‘. :. :. -1.:
CHEMICAL PHYSICS LETTERS
Volume 25. number 2
: 15 hfarch 1974.
References R. hfanne and T. Aberg, Chem. Phys. Letters 7 (1970) 282. [2] T.A. Carlson, M.O. Krause and W.E. hfoddeman, J. Phys. (Paris) C2 (1971) 102, and references
f I]
therein.
[3] G. Rakavy and A. Ron, Phys. Rev. 159 (1967) 50. 141 U_ Fano and 3-W. Cooper. Rev. Mod. Phys. 40 (1968) 441.
[S] ST, hfanson and J.W. Cooper, Phys. Rev. 165 (1968) 126_ [6] E.J. McGuire, Phys. Rev. 175 (1968) 20. f7f ST. Manson and f-W_ Cooper, Phys. Rev, 177 (1969) X.57. [8] E. Storm and Hf. Israel, Nuclear Data Tables A7 (1970) 565. [9] D.J. Kennedy and S.T. Manson, Phys. Rev. AS (1972) 227. [lo] E.M. Henry, CL. Bates and W.J- Veigele, Phys. Rev. A6 (1972) 2131. [I 1 f W.J. Veigefe, qtornic Data Tables 5 (1973) 51. [12] J.H. Scofield. Lawrence Livermore Laboratory, Report No. UCRL-51326 (1973). f13] R.H. Pratt, A. Ron and H-K. Tseng, Rev. Mod. Phys. 45 (1973) 273. [ 14 1 R.J.W. Henry and L. Lipsky, Phys. Rev. 153 (I 937) 5 1. [IS] A.F. Starace, Phys. Rev. A2 (1970) I 18. [ 161 M.Ya. Amusya, N.A. Cherepkov and L.V. Chemysheva, Zh..Eksperim. i Teor, Fiz. 60 (1971) 160 [English transl. Soviet Phys. JETP 33 (1971) 901. [ 17 J M.O. Krause. Phys. Rev. 177 (1969) 1.51. [ 181 R. Siegbahn, C. Nordling. G. Johansson. J. Hedman, P-F, Heddn, K. Hamrin. U, Celius, T. Bergmark, L-0. Werme, R. hfanne and Y. Baer, ESCA appiied to free molecules (North-Hohand, Amsterdam. 1969). [I91 T. Novakov, Phys. Rev. B3 (1971) 2693; T. Novakov and R. Prins. in: Electron spectroscopy. ed. D.A. Shirley (North-Holland. Amsterdam. 1972) p. 821. 1201 F- WuiBeumier and hf.0. Krause, in: Electron spectroscopy, ed. D.A. Shirley (North-Holland, Amsterdam, 1972) p. 259. f2 I] G.K. Wertheim. R.L. Cohen, A. Rosencwaig and H-J. Guggenheim, in: Etectron spectroscopy, ed. D.A. Shirley (North-Holland, Amsterdam. 1972) p. 813. [22] K.S_ Kim and R.E. Davis, J. Electron Spectry. 1 (1972/73) 251; Ji.S_ Kim, private communication. [23] C.S. Fadley, D.A. Shirley, A.J. Freeman, P.S. Bagus and J.V. Mallow, Phys. Rev. Letters 23 (1969) 1397; C.S. FadIey and D.A. Shirley, Phys. Rev. A2 (1970) 1109; C.S. Fadley, in: Efectron spectroscopy, ed. D.A. Shirley (North-~oli~nd* Amsterdam, 1972) p. 781. [24] B.L. Henke and R.L. Et@, in: Advances in X-ray analysis, Vol. 13 (Plenum Press, New York, 1970) p_ 781. B.L. Henke and E. Ebisu, in: Advances in X-ray anaIysis. Vol. 16 (Plenum Press, New York), to be published.
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