Peak intensities and photoelectric cross sections

~oloumaiofElectronSpec~oscopyandRehredPhenomena,5 (1974)895-910 OElsevier Scientific F'ublisbingCompany, AmsterdamPrintedinThe Netherlands PEAK

INTENSITIES

IN THE

c.

s.

SUDDEN

PHOTOELECTRIC

CROSS

SECTIONS

APPROXIMATION

FADLEY

Department (U.

AND

of

Chemistry,

University

of Hawaii,

Honolulu,

Hawaii

96822

S. A.)

SUMMARY The

sudden

approximation

has

predicting

the

origins

and

satellites

due

to both

multiplet

effects. nearly

The

applicable

electron Aberg

[Chem.

exact

In

and

calculated that

a sum

all

of

emission

are

processes

the

peak

peak

is extended peak

shake-up

or

intensities

On

and

this in

associated

other

approximation soft

x-ray

of

hand,

primary

with

over

all

and

this

model

relationship

cross

involving

sections It

represent

multielectron

are

approximate frozen-orbital

to

a

photoelectron

one-electron

to correspond

that

sections

Corrections

absorption

the

and

approximation.

cross

those

such

of

photoelectric

photo-

Manne shown

limitations

with

Several

in x-ray

average

of

and

it most

structure

fine

for

shake-off-

make

a weighted

unrelaxed

shake-off+

and

as

frozen-orbital

including

sections. the

shake-up-

to a consideration

associated

subshell,

to

potential

or

structure

previously

Sources

intensities

intensities

cross

involved

certain

frozen-orbital

discussed. in

all

studies

approximation,

have

is equal

fine

emission

this

(1970)282]

of

approximation

inner-shell

energy

in numerous

and

this

commonly-utilized

a given as

in

Utilizing 7

paper,

it

theoretical

expected

splittings

including

XPS

such

such

in comparing to such

in

from

processes

are

this

individual

is shown

this

binding

energies,

discussed,

between

(XPS).

utilized

intensities

involved

Letters,

Theorem

satellites.

as

Phys.

binding

relative

to high-energy

spectroscopy

Koopmans'

are

assumptions

been

the

coefficient

thus

required

processes methods

for

cross

totality

only doing

sections

of

measurements

well

G.

896 above good

and

threshold, agreement

this

found

observation

in previous

is

consistent

comparisons

with

between

the

FADLEY

s.

relatively

experiment

and

theory.

INTRODUCTION The their

partitioning

individual

several most vided

the

about

the

spatial

basic

essential

forany

core

and

spetitroscopy sections x-ray

process

described

Any

for

in terms

relevant

electron

satellite

structure

15-25X

of

all

14,151

and

solids

[18-203.

in XPS

in what

in may

The

sudden

of

the

are,

as

noble high

for

possible example,

are

found

gases

and

certain

50%

approximation

of all has

of

the

sections

i,s of

photoelectron to cross

such the

as

soft

basic

to be adequately [1,13-173.

1 keV. consider

not

only

multielectron responsible to occur small

events

been

about

intensities

analysis

must

and

as

the

for which

order

for

information

as

attention

appears

pro-

allowed

cross

of

approximation

various

spectra,

the be

1121,

of the

of valence

experiments

threshold

thus

the

as well

in x-ray

our

of photoabsorption also

to obtain

subshell

observed

related

sudden

transitions

events

of

one

properly

into

for

sections,

make-up[6,7]

restrict

above

are

description

Two-electron

shall

be

Q

important

cross

understanding

peaks

closely

simple

but

[l-4],

sections

provides

is possible

measurements

energies

transitions,

tions.

and

hv well

of a

photon

detailed

We

coefficient

can

a knowledge

theoretical

[7-111. to XPS

absorption

it

photoelectron

(XPS)

relevant

absorption

The

detailed

valence

scattering

atomic-orbital

Furthermore,

is

subshell

photoabsorption

[5] and

states.

0n.e or QnJj

relative

inelastic

of

cross

spectroscopy

measurements,

dynamics

distribution

electronic

the

of

such

photoelectric

Photoelectron

for measuring

effects

From

[2,4,7-111.

atomic

contributions

[l-lo].

methods

that

total

subshell

reasons

direct

of

transifor

in approximately molecules

in certain

utilized

one-

with

[2,4, inorganic

some

SUDDEN

APPROXIMATION

PEAK

.sucoess

in predicting

well

their

as In

for

recent

photon it

321, the

years,

N-electron

identical

orbital

that

or unrelaxed

depending

upon

and

energies three

near

basic

1 keV,

involve

no direct

tions.

30th

body yet

methods been

the

the

final

rise

sudden

are

are

such of

into

used

to

to a significant

between ionic

all

extent

these

the

Such

atoms

effects

categories dipole

[LO,22,23,25, for

resulted

photon

in at

more

rigorous but

calculation

least

they

of multielectron

effects,

to the

frozen-

[27,30,31],

and

such

an assump-

two

agreement

have

[3,21-

initial

state,

multipoles

good

been

describing

essentially

[L,13-17] treat

made

non-relativistic

neutral the

have of

Theorem.

in rather

for

as

orbitals

N-l-electron

calculations

the

majority

unrelaxed

used:

approximation be

vast

one-electron

divide

to allow

can

calculations

including

ofthese

all

the

possible,

[13-203.

to Koopmans'

relativistic

Although

[33-371

In

atom

transitions

section

approximation

attempt

applied

that

sections

spanning

the

cross

to XPS,

and

results

tabulations

of

giving

and

the

intensities

in the

cross

the

[3,21,24,26,28-301 27,31,32]

relative

assumed

state

to

two-electron

relevant

electrons

atomic

of

a number

energies

passive

types

and

is explicitly

N-l

tion

the

energies

897

INTENSITIES

these

transimany

-

have

not

of XPS

cross

sections. Thus, section

questions calculated

as

cular, included usual

precisely transitions

as

the

In

this

to

the

basis

if any,

simplifications

photoelectron

of

on

to what, in it.

arise

of an

effects

paper,

[14-173,

an

the

effects

[38].

The

implications

results

are

that,

sudden

unrelaxed

of alL

then

unrelaxed

show

with

includes

experimental

significance

of

within

discussed.

state,

subshell

result

and,

transitions the

approximation

one-electron this

of a subshell

final

of multielectron

we

associated

spectra

exact

cross and

cross

in parti-

are

framework

of

analyses

of

section

u,,JU

multielectron

in the

the

interpretation

c.

898

s.

FADLEY

DER IvAT ION The tion

basic

assumptions

analysis

(1) The

that

we

excitation

shall

is assumed

trons

not

caused (2) We

have

basically

use

dipole

use

to be time

by

final-state that

elements

the

that

neglected.

The

additional

final-state

set

of

initial-state

set

dipole

operator

between

state

determinant

pf


the

double

the

signed

minor

NXN

matrix

Dfi

final-state are

thus

whose

in general as

the

matrix

elements orbitals

Thus,

an

initial-state by =

approxima-

an

inner

outer

elec-

in the

Hamiltonian

one-electron

orbitals

non-relativistic

between

all

is assumed one-electron

primary

excitation

arise { @;

to be

because

1

is not

element

of

determinant

matrix can

the

yi and

be

relaxed.

equal the

small

to

the

N-electron a final-

C <~;(L)lrl%(L),Dfi(jIk) j,k all

are

orbitals.

N2 matrix

it

[39]:

removing

elements

being

the

change

the

the matrix

is over

one-electron

distinguished

describing

by

from

that

of

with

1.

j,k

rapid

composed

associated

is given

formed

the

sense

is permitted,

: ;.I&N)> i=l 1

sum

photoelectron

coordinates,

one-electron I ak

sudden

coupling.

effects

than

common

screening.

relaxation

net

other

to

angular LS

the

in the

functions

approximation,and

enough

high-energy

to adjust

and

of

[1,13-171:

"sudden"

wave

in radial

(3) Although

are

decreased

determinantal

separable

implications

of a primary

subshell do

and

occupied the

the

jth -

orbitals row

overlaps

That

elements,

is, and

and

and

kth -

between (Dfi).

Jk

the

Dfi(j ]k) is

column

from

initial-

and

= <@?I ak>. 3

primary

ak = @nR = a given

(1)

core

excitation orbital

the

There is

and

O! = azn(" = a high-energy photoelectron state with angular momentum J 1.V A. The terms involving all matrix elements other than nil. are

thus

neglected.

This

assumption

is at

least

partly

justified

by

SUDDEN

APPROXIMATION

the

fact

that

all

matrixelements

D fi(jlk).

some

these

elements

will

matrix

be

determinant, <9

by an

one-electron electron

Limit

typical

XPS

dealing

with

utilizing

the

the

effects

final

an

appropriate and

electron Linear

in all

states

so

Configurations,

is convenient the of q&N)

separate

initial-and

the =

form &

elements

latter

to

are

one

for

if multi-

has

considered

infinite-excitationNo quantitative .appear to exist for

neglected. terms

more

accurate This

[L,33].

anaLyses

that

of XPS

methods

assumption peak

for has

in

intensities

[14-16,18-201. describing

very

over

nearly

all

that

out

final-state

the

be

wave

belonging the

over

photoelectron the

adequately

energy

range

represented

by

transitions. to be

each

primary

constant

it can

assumed

of determinants to

be

although

so that

are

The

[40]

tn an

additional

discussed

element

value

can

small

equal

zero

overlap

determinants

overlaps

they

previous

to be

average final

and

these

generally

Aberg

[14,17].

or

transi-

non-zero

excitation.

smaller

small

bound-bound

small

to the

by

inconceivable

approximately

approximation

involved,

combination

writing

been

matrix

states

Initial

of

have

is assumed

of

that

for a

XPS

by much

energies,

sudden

one-electron

excitation

products

concludes

made

high-energy

elements

is not

example,

compared

involved

multipLied

it

to compensate

but

and

them

been

It

matrix

(for

determinant

are

excitation

case

(4) The

overlap

transitions

estimates'of

any

for

are

Howev&r,

when

transitions,

additional

energy

enough

B expected

&l~l@n&

such

large

particularly

multiplied

by

other

determinants

of

899

INTENSITIES

overlap

tions)

(5)

PEAK

primary

wave

adequately function

described will

to a single photoelectron

functions

as

by

at most

pure

be a

codfiguration. excitation

anti-symmetric

1161:

QnRm

m (1) R

@(N--l)

(PaI

s

(a)

c.

900

where

em

Qnh

(1)

is the

initial

one-electron

orbital

involved

passive

electrons

s‘.

FADLEY

in the

s primary

excitation,

initial

state,

energy

e and

function the

for

primary

@(N-l)

represents

Qs, a"zm~"',m,"' (I)

angular

momentum

the

electronsof

N-l

excitation. H(N)‘?

the

N-l

is a photoelectron

a"

= R & the

1, and ionic

orbital

with

is a

final

were

not

Yf(N-1)

core

in the

that

kinetic state

wave

involved

in

By. definition,

$f)

=

Ei(N>

(3)

‘i+(N)

and H(N-1) in which initial and

H(N)

is the

initial

state

energy,

H(N-1)

Ef(N-1)

final

state

is a

total

wave

function.

hv We

shall

sitions, order

= Ef(N-1)

Yf(N-1)

as

only the

deal

considered

and

the

state

- El(N)

final

are

are

state

energy.

+

E =

N-l-electron

thus

three

indicated

i-

is a total

Hamiltonian, not

of course,

a valid

requires

that

E

(5)

one-electron

proceed

Ei(N)

o (.N-1) is thus

Eb(ng)

with

would

Hamiltonian,

conservation,

explicitly

There

these

is

(4)

N-electron

Energy

derivation

transitions.

state

final

= Ef(N-1)

Yf(N-1)

and

two-electron

in an

identical

basic

absorption

schematically

way

for

tran-

higher-

mechanisms

to be

below:

One-electron: (n'Qi)P

. . . (nQIq... &N)

.a

,

Two-electron . . . (nQIq...

L,S -?2+

yf(N-+

&N)

shake-up (nVQq)p

or

L,S

Ef(N-1)

A_

values, electron whereas subshell.

transitions, wave

we

functions,

states. (n'a')p The

The can final

+

Ed,

J?21 (6)

B

B

Qfl

EB'

shake-off: . ..(IIQ)~-~...(~~~')~-~~~'~Q")~ Yf(N-l+

In these

L’,S’

...(nQ)q-l...(nyQl)P

have and

represent state

‘specdfied

of

(n&)4 either

electron

the

for

refers a

second

Q-Cl (7)

Ef(N-l)y

total:energias

notation

+ EB,

L',S'

filled

configurat.ions, initial

to a filled

and

electron

final many-

inner

or partially involved

L and

subshell,

filled in a

outer two-

S

SUDDEN

APPROXIMATION

electron

transition

a shake-off state

we

tions, gies

PEAK

process,

assume

but, for

is

indicated

the

in general,

different

L',S'

1411.

I-WZ ,=kt ML,

and

the

completely .included

specify in the

labelling

of

final

a"l, L',S',

which

may

scheme

all

The

the

cross

necessary

section

and

have

final

magnetic

to specify

for

any

state,

the

S and

a given

onk

in which

C

is a

gives

and

Equations

The

overlap

(1) and

between

represents

the

comprising

both

give

rise

_f,?_.g”

= 0

It given

fixed

this

and

L,S,ML,

MS

the

the well-known

and

hv.

cross

yf(N-l)S(+)

is :now convenient state

states

will

for

(2),

contain

ener-

if more

a

transitions

numbers to

f3

two-

specification further

than

the

to

implicitly

ylabelling

and

due

subscripts

a

in

one

index

coupling

same

L',S'

will

be

[14,41].

given

by

2

(8)

same

With section

@(N-l), values

set

assumptions reduces

will

and

of N-l

if

(l),

selection

sum

all

(3)

to:

be

non-zero

the

only

one-electron These

a values.

monopole

(2) and

rules,

if both orbitals

conditions Aa

including

=

[1,14,17}*

final

initial This

to

same

transi-

necessary

are

include

to

all

quantum

the

numbers,

rise

of these

they

subscripts

state

initial

configuration

states;

y thus

The

different

schemes

but

that

CII i-l

=

constant.

for

magnetic

N

an~,a-B(y)

L,S

coupling

quantum

configuration

1

state

the

initial and

s 'I.

energy

final

subshell

states,

is understood

to distinguish

surpressed

and

it

therefore

possible

The subscr-ipts

same

a kinetic

o labelling

final

be necessary

within

We

subscripts

where

in a given

initial

states.

by

indiv-ldual

each

one-electron

electron

(n"A")l,

energy

be

values

splittings and

same

it may

multiplet

MS

by

nt' is replaced

to have

901

INTENSITIES

yield

configuration

involved, a

to

true

as

over and

is done

one-electron

transitions

to average

over

in determining cross

section

associated all

of

subshell ona

and

the

with

a.

degenerate_

cross various

sections. two-

c.

902

electron

cross

sections

scripts

indicate

the

summing

and

cross

as

passive

electrons

tum numbers

will

to assumptions a

=

na.

subscript

and

the

matrix

subscript

remain

given

in Equation

to be

[14,17].

the

to expand in terms

way

values

state

resulting

of

within and

these

the

N-L

magnetic

sections

quanSubject

configurations.

cross

This

electron.

for all

of L',S'

sub-

are:

(loa)

to the

energy

of

the

usual

that

to be defined

only

The

radial

overlaps

(10)

are

previous

sections

the

initial-state

the

complete

are

sudden

more

passive

set

precisely

final

below,

configuration, overlap

inte-

determinants

of the

generalizations

of

approximation

unrelaxed

cross

electron-s

(lob)

and

calculations

one-electron

thus

to an

coefficients

section

one-electron-transition

Equations

cross

are

the

analyses

section

as

type

of

ulLeul it

represented

of orthonormal

final-state

shake-up

shake-off

by

ion-core

[17]:

not

explicitly

portions

shake-off

order

second

transitions

% I a*L appearing in cross

in certain

these

functions

certain

of

final

the

of the

identical

and

kinetic

(1).

XPS

In relating

' a*1

calculated.

utilized

spectra

C

r tidicates

expressions

have

same

in all

elements

o refers

grals

we

the

parenthetical

2 -f-C a-1Rr;2e_l] .Irj2 L+lRS,IL+l 3

[c

is a constant

the

As

that

(5) above,

r1,211, 3 is an average

wave

in an

character

be accessible

= C’

dipole

*(N-L),

states

the

FADLEY

].Ir12

C'

is useful

final

proceed

the monopole

(4) and

anQ(n@%t-Il"L")

radial

and

rc

C’

in which

will

ensures

in which

nJ(n'R 1_nlralr),

initial

averaging

sections,

5

s.

of

portion

excitations

the of

could

distinguished sum the also

over

y-should

spectrum. appear

be

interpreted

Final

in this

and

states

expansion,

as

integrals

resulting as

transitions,

from

indicated.

over higherIf

SUDDEN the

APPROXIMATION are

@(N-l)a

the

where

sum

PEAK

normalized,

extends

903

INTENSITIES one

then

over

all

consequence

allowed

of

this

expansion

two-electron-transition

is that

final

con-

figurationsUtilizing 1173

have

expansion

shown

weighted if we

an

that

average this

Eb(n#

=

index

energies

and

the

8 is used

overlap

degenerate

energy

thus

proportional

Therefore, of an

-E

nQ ,a-a

The

,“=

forms

c

all

energy

possible

Eb(nfi)KT

final

z Rb(nA)U,

summing

all over

final

degenerate

The

weighting

to

cross

section

for

kinetic the

na

to a

states.

That

states

states.

from

Aberg

is equal

it

initial the

and

is given

is,

by

Eb (nQ)*

to distinguish

average

(nk)

cross

energy

final of

different states

each

is correctLy

to

and

binding

a transition

corresponding,

subshell

of

to

that

_

primary

defined

in this

of

u

(14)

section

for

one-electron

Equations

(cY~(N)~~I

:

a

specific

transition,

(?a),

(2b),

and

transition

is based

and

calculated

(8)

can

be

upon

what

from

as:

?;lYi(N)$12

i=l

superscript

cross

b

to be a pure

modified

CJ

the

electron

unrelaxed

appears

binding

(111, Manne

as

E=hv

An

by

implies

over

approximation

over

energy

averaging

excitation

in Equation

I12 C 6=0

the

state.

that

Theorem

energy

binding

in which

is

as

a Koopmans'

binding

denote

such

sections

u

can

in all now

be

cases

refers

summed

and

to an averaged

unrelaxed as

final

before,

state.

yielding

These the

c.

904

unrelaxed

subshell u

The

constants

same

as

dictate

given

states

the

of

good

(16)

means

matrix

same

value

act for and

(Lk),

approximation

be

in this

the monopole

symmetries

of

(16)

elements

because

(lo),

in Equation

to a very

as

dipole

the

average

by 2

thus

radial

in Equations

that

averaging, be

and

those

section

FADLEY

r c R-l-lRE, pIL+1 + CL_1R;yQ-I 1

UC C’

nR

cross

s.

are

same

involved.

in the

wave

Combining

the

rules summing

photoelectron

radial

are

selection

involved

unrelaxed

the

Equation

and

states

functions

will.

will

Equations(lO),

(12),and

that

(17) That

is,

all

the

unrelaxed

one-electron

also

be

states

generalized by

different

final

result

section

multielectron

schemes

is thus analogous co 11 cr =z (5 nR nR.6 6=0

in which cross

6, has

section L',S',

the

same

to a set and

simple

partitioning

coupling

energy.

a total The

multiplet

a given

1413.

The

to Equation

cross

above

derivation

splittings

configuration clearest

section

can

in the cross

expression

for

final

section of

the

(13):

(18)

meaning of

represents processes.

to encompass

appropriately

among

same

and

cross

and

final

Qna

states

Equation

(13)

t

8 represents

in a given can

thus

a

summed

and

configuration

alternatively

averaged

with

the

be written

as

m

E+d”

E (nQ16 gf?oanQ,6 b

=

u

DISCUSSION As

on,"

in this

approximation

possible

photoabsorption

sections

directly

distinguished

on

events,

to experimental the

basis

of

(19)

u

na

thus

somewhat

it may

be misleading

results

final

accidentally

state

in which energy,

includes

to compare the

as

different

is the

case

such events in

all cross are

SUDDEN

APPROXIMATION

photoelectron peaks

if multiplet

ciated

in

with

intensity

This

to o&u_

For

splittings

primary

905

INTENSITIES

spectroscopy.

proportional

al

PEAK

are

present

to Ong.

n$ excitation situation

is

the

example,

The

intense

observed

[41])

total

should,

in XPS

photoelectron

on

illustrated

one-electron

the

other

in Figure

peak(or should

intensity

hand,

be

be asso-

proportion-

1.

l- empeak(s) a cnl

Shake-up

-

--Shake-oft

-

-

Relationship between unrelaxed Fig. 1. electron cross sections one and observed photoelectron spectrum free of inelastic increases to the right. unrelaxed

Thus,

cross

peak

intensities,

that

may

tudes

have

from

From by

the

appropriate

At

the

to use

level only

level (10)

of the

(N-1) R > <@' I@ niL n’R

and

this

dependence

(16), overlap

of ona

or any

approximation,

square in which

of

the

it

directly

has

different

been

is clear

as

related

to

one"

in

for

multielectron-transition

for all

factors factors

in the except

be

be

overlap those

multiplied

obtaining

cross

probably

magni-

solids[lB-201.

could

procedure

also

transitlons

relative

observed

that

a general

it would

one-electron

of multielectron

and/or

diagonal

n = n'

be

presence

to another,as

squared

estimate

not

in the

chemical

core

Equations

first-order

need

particularly

strong

one

sections

cross sections on~~ and oneintensities in a hypothetical Kinetic energy scattering.

a

section.

reasonable determinant representing

a

c.

906

shake-up rule

or

Aa

= 0.

It

of

shake-up

analyses An

even

simpler

tion

1143

could

be

next

shake-off

and

can

be

Such

Cooper

[3]

and

Manson

[28]

present

of

Lfght

if

one-electron

good

elements

same

net

cross

section

fraction

on Kr, agreement

as

for has

been

between

outer

[8,10,11], n'a'

thus

monopole

sudden been

carried

out

core in the

characteristic the

selection

approximation

equivalent

subshells

of

have by

Ne,

mean

of

approximafinal

the

for n*R

such and

provided (non-unit

and

the

fair

relative

and

na

thus

been

show

path

variation

with

by

made

to

the

subshells.

and

directly

related from

ratios

by

121 and

Krause

In

[4] and

such

subshell

appears

theory

to

the

agreement

for

of errors

nJI both

within

the

that

is

over lap}

is associated

with

between the

same

due atom

assumed the

outermost

o,s/a3d

One

to multielectron lies

that

initial

[3].

in the

the

levelsin

state

subshells.

and

cross

section

have

occurred the

ratios

/o 3d

by

in the

and

explanation

transitions

fraction

the

relatively

0

possible

such

will

reduce

core of

one-electron

Because

the

is to

equivalent

major

Manspn

Kennedy

ratios

in explaining for

with

comparisons,

theoretical

total

to

XPS

ans~t/o~~

transitions

[Z]

one-

al. [lO],forcore

and

This

by Krause

of

Krause

Nefedov,et

compounds.

intensFties

determined

and

experimental

experiment

further

with

subshells

be

experimental

of multielectron

noted

can be

Wuilleunier

and

a cancellation

it

state

atom

3s Q3p'=3dJ

[14,l.7].,

inner

free

the

in principle

relative

both

the

of electron

that

effect

of

have

furthermore

in solid

indicates

the

the

the

FADLEY

overlap).

Kr,

gaseous

number

to use

that

ratios

gaseous

for

analysis only

study

for

be

Comparisons

anlR,u/anaU

and

a number

can

included

processes

subshells

ratios

a

functions

and

for

have

that

that

effects

two

[2,4,7,9,10].

theoretical

would

(unit

the

from

level

radial

corrected

peaks

0n'_61/onl.

the

by

we

shake-off

number

that

and

assume

relaxation

electron

agree

and

thereby

Provided

this

procedure

atomic

negligible

spectra

is at

represented

higher

energy

process

s.

outer

for model,

relaxation final

state

subsheLls

SUDDEN

APPROXIMATION

should

experience

inner

equal

section

within

ratios

a given

thus

By

such

be

as

well

high

significant

events all

are

taking

events,

which

good or

The

agreement

x-ray

cross

shell

absorptions previous

ment

well

consistent

By

contrast,

tions, 20% has,

it

or more

be

been

appears

simply

to

not

expected

noted to

be

approxi-

cross to

the

true

for

ratios

screen-

prediction and

given

by

between and that

is

theory

threshold.

is

in the

include

for

to be such

od"

total

that

of

no

Therefore, sections

on unrelaxed are

inner-

is consistent

in which range

typical

of f5-lO%,

too

the

significant

low

agreewith

[23,24,26,29,42].

of multielectron

sections

that

for

cross

based

prediction

effects

sudden

absorption

(18).

total

theory

events

the

contributors

theory,

energy

probabilities

values

approximate

cross [26]

the

dominant

and

photon

to which

experimental

This

coefficient the

in Equation

consistently

by McGuire indicate

sum

experimentaL

within

sensitive

of Q by the

an

photoabsorption

describable

theoretical the

with

absorption

in all

the

of experiment

in predicting

in fact,

comparisons

above

did

This

experiments,

not

or over-estimation

if QnaU

in any

electron

experiment

x-ray

In such

is thus

threshold

under-

would

in soft

expected

comparisons from

in outer

comparable

in a manner

provided

well

close

n6 holes.

between

directly

coefficients

sections,

away

no

and

but

be

rather

change

and

should

particularly

be

to participate

is precisely

might

total

be

measurement

absorption

with

enough

place,

n&'

agreement

threshold.

intensity,

approximation.

for

net

a hole

inner-subshell

to be

would

for

{lOa)

Therefore,

the

prevailing

above

Equation

expected

as

interactions

[2,3].

should

that

of

This

good

in Kr

onau

presumed

be

equal

the

core

subshells.

/ang,

nearly

u3p/o3d

measurements

nA'

same

correction

on,A,u/ondu.

with

contrast,

situation

the

msght n'A' /o nX

be more

and

nearly

inner

shell,U

consistent

o3s/o3d

of

u

907

INTENSITIES

overlap

all

ratios

should

can

the

for

theoretical

ing

vary

subshell,

mately

PEAK

by

transi-

approximately

It

[4,i4,15,18-201. good

agreement

correction

in such of

such

c. s.

908

cross

sections

nation

for

this

,final state

observation

in several

Thus, transitions mation

for

analysis

experimental

was

respects

in unrelaxed could

relaxation

although

no

expla-

given.

the

cross

be an

is necessary,

FADLEY

implicit-

sections

important

inclusion

predicted factor

in

of multielectron

by the

this

sudden

approxi-

interpretation

of

data.

ACKNOWLEDGMENTS The Cooper

author and

support

of

expresses

T. Ab erg the

gratitude

for

National

to B.

L.

helpful

comments

Science

Foundation

Henke,

P. S.

relevant

to

is also

Bagus,

J. W.

work.

The

this

gratefully

acknowledged.

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