Calculation of Kr photoabsorption spectrum fine structure within the KN23 ionization threshold region

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Calculation of Kr photoabsorption spectrum fine structure within the KN23 ionization threshold region

V. Sukhorukov, A. Hopersky, I. Petrov

To cite this version:

V. Sukhorukov, A. Hopersky, I. Petrov. Calculation of Kr photoabsorption spectrum fine structure

within the KN23 ionization threshold region. Journal de Physique II, EDP Sciences, 1991, 1 (5),

pp.501-509. �10.1051/jp2:1991184�. �jpa-00247534�

Classification

Physics

Abstracts 32 80F

Calculation of Kr photoabsorption spectrum ^{fine structure} within the KNz~ ionization.threshold region

V L

Sukhorukov,

A. N

Hopersky

and I D. Petrov

Rostov Railway Engineers Institute, Chair of

Physics,

344017 Rostov-on-Don, USSR

(Received5July

1990, revised 29January 1991,

accepted

31January

1991)

Absbact. The near edge

KN~~-fine

structure of Kr atom is calculated _via the

configuration

interaction method

including

_a monopole rearrangement ^{of electron shells The} theoretical

spectn~m form _{is in} good agreement with the expenmental one The calculated absolute values of the oscillator

strengths

and the

photoabsorption

_cross section are

predictions

1. Introduction.

Inner shell

photoabsorption spectra

^of atoms, molecules and solids _near ionization threshold have

complex

fine structures XANES

(X-ray Absorption

Near

Edge Structure).

XANES may be caused both

by

the

shape

of the

potential

of environment

[1, 2]

and

by many-body

effects

[3-6]. Therefore,

studies of the nature of

many-body

effects and their effect _on

photoabsorp-

tion spectra are of

special importance

It _is fruitful to

perform

these studies _on atoms where effects of environment _on XANES _are excluded

Expenmental

spectra ^{of Ne and Ar}

photoabsorption

within the KX-threshold regions

(X

_is the additional

vacancy)

have been obtained _m reference

[3] ~Ne.

X

=

L)

and reference

[4]

(Ar

X

=

fil~.

A

semi-empirical

_assignment of spectra ^{has been made in} these _papers. A

more accurate

assignment

of

KX-photoabsorption

spectra has been

presented

_m reference

[5],

where it has been shown that

photo-double excitation/ionization

_processes

play

_a

significant

role _in

forrnJng

the fine _structure features. Cross sections of these processes may be calculated with _a

good degree

of accuracy within the approximation ^of

monopole

rearrangement of

electron shells

(MRES)

which accounts for the deformation of outer electron shells because of appearance of inner vacancy.

The

expenmental study

of multi-electron transitions _near the K-threshold of Kr has shown the existence of _a

complex

fine _structure which _is caused

by photo-double excitation/ionization

processes

[6].

^The last conclusion of reference

[6]

is based _on

semi-empincal analysis

^of

optical

data and _on a comparison of

KX-photoabsorption

_spectra of

Ne, Ar,

Kr.

The _{main aim} of this work _{is to} give a more accurate assignment and to

predict

the absolute values of Kr

KN23-Photoabsorption spectrum

^features on the

grounds

of _a

non-empirical

calculation. One addluonal _{aim is} to

give

_a

comparative analysis

of

KX-photoabsorption spectra

m a series

Ne, Ar,

Kr

JOURNAL DE PHYSIQUE II T V MA> 1991 25

502 JOURNAL DE

PHYSIQUE

II M 5

2. Method and the results of calculation.

A method of calculation of

photo-double excitation/ionization

_processes has been

developed

and described in detail _m reference

[5], therefore,

_{we present} here

only

the basic formulae with bnef conunents.

Basic _atomic orbitals

(AO)

It has been shown _in reference

[5]

that when

calculating

KX-

ionizationlexcitation

_cross sections _one

has,

_in the first

place,

to take MRES into account, and to achieve tbJs it suffices _{to use}

non-orthogonal

AO obtained

separately

for the1nltial and the final states Thus AO radial parts of the imual state were obtained

by solving

Hartree-Fock

(RF) _equations

of Kr

ground

state The final state AO _were obtained

by solving

HF

equations averaged

_over the excited

configuration.

To obtain AO of discrete excited states, tile HF equations for the

configuration

ls~

4p~ nin'i'

_{were solved. To obtain AO of continuum}

(ep

_a frozen _core

approxbnation

of the

configurations

ls~

4p~ 5p(6p)

for ls

4p-5p(6p)

_ep

process and of the

configuration ls~~ 4p~~

for

ls4p-epe'p

_process had been

exploited

Within the

calculation,

the _core

AO,

discrete and continuous excited _state

energies

and the

energies

of

ls4p-nin'i'

excitations _were obtained _as the differences of total

energies

of

corresponding configurations.

Thus Is

4p-spot excitationlionlzation

_{energy at} the threshold

(e =0)

_is 14

344.7eV,

that of Is

4p-6pei

_is 143482eV and the threshold _energy Is

4p-eie'i

is 14 351.9 eV. A relativistic correction to the energy of Is level of reference

[7]

was taken into account, it is 224.0 eV.

To calculate the

KN~~ photoabsorption

_spectrum of Kr the

following

excitation had been considered

~~~~ ~5p~~ ~4d~~ _~~~~~~ ~5s~~

~~~

The choice of the excitations

(I)

_is based _on the expenence

gained

_in the calculation of KX-

spectra

of

Ne,

Ar

[5]

and _on using the smallness parameter

characterinng

the

mixing

of the wavefunction

QVF)

of

corresponding configuration

with the WF of ls~

4p~ 5p~ configura- tion,

because the transitions from the

ground

state into the states of tins

configuration

have the

greatest

^oscillator

strengths.

The smallness paranJeter was determined with the

help

of the

expression

(5p~ V[ nin'i')

~~~~'~'

AE(5p~ nin' I'

^'

where

(5p~ V( nin'i')

_is the _mean interaction between the

configurations ^ls~~ 4p~~ 5p~

and ls~

4p~ nin' I'

_which

was estimated _as

(f~i f~,1, )~ (f~i being

the _mean radius of the

correspondJng AO),

and

AE(5p~ nin'f')

_{is the difference of energies of}

corresponding

excitations. Thus the calculation of

K~2/K~ii

^had

given

I

=

) ^~~~~~ ~~

=

~~ ~'~~

~'~'~ ^~'~

^~~

= l10

(2)

K4F

f~ AE(5p

4d

(4.65

_a-u

0,37

^~V

Expression (2)

^allows one to assume that the effect of is

4p-4P

_{exaltation on the spectrum is}

two orders less than that of

ls4p-4d2

excitation and not include it _in calculation. An

analogous

selection had been made for all the excitations included _in the calculation.

Together

with the excitations

(I)

the

following

channels of continuous spectrum were taken

into account in the calculation: Is

4p-(spep,6pep, epe'p).

The channels of

diffenng

symmetnes

were not included because the transition

amplitudes

for them _are small _as

compared

with the

amplitudes

of the transiuons into the above mentioned channels.

Energies

and

wavefunctions of

the excited _states To construct the full WF of the atom in the final state, basic vectors of

LS-coupling

_were used

(spin-orbital

interacuon for

4p, ni

and

n'i'

_electrons

being weak)

a

LS)

₌ ls~

4p~ (Lo So) ^{nfn' f'} (L£) LS)

Here the

ls-4p

Coulomb interaction had been

neglected

because the

splitting

of ~P and

~P _terms of the

configuration

ls~

4p~

_is 0.90 eV which _is less than the natural width of Kr Is level amounting to 2.69eV

[8].

^After

calculating

_via the method

[9]

^{the matrix elements}

(a

^' LS V _a

LS)

of the

operator

V

=

H~

V~~(K),

where H~ _{is a} Coulomb operator ^and

V~~(K)

is the average over the

configuration

K

Table I

Spectral features of

Kr

KN~~-XANES

_m the region

of

double Is 4 p-n _j

ii n~f~

photoexcitation (see Fig. I).

x x io4, Mb «iP) (2)

2 41 0 11 0 30 035p21'D) ₊ 0 14d2('D) ₊

+ 5s 4d[0 9('D) ₊ 0 2(3D)]

2 0 16 0 52 38 5p2[0 2('D) 0 5(iS) 0 5(3P)] ₊ 4d2[061'S) 0 21'D)]

+ 5s 5d[0 3('D) 0 1(3D)]

3 0 00 2 09 5 51 5p2[0 1('S) 0 3('D) 08(3P)] 0 25s 5d('D) ₊

+ 4d~10 5('D) o I('s)j

4 0 78 0 63 66 5p2[0 2('S) ₊ 0 7('D)] 054d2(3P)

0.14d 5d(~D) 0 35s

4d('D)

5 81 0 46 22 5p2[0 2('D) 0 2('S) ₊ 0 1(3P)] + 4d2[0 3('D) ₊ 04(3P)]

+ 5p 6p[0 1('S) ₊ 0 2('P) 02(35) ₊ 03(3P) + 0 5(3D)]

4d 5d[0 1('D) ₊ 02(3P) ₊ 0 3(3D)]

6 2.31 0 18 0 46 0 25p2('S) 4d 5d[0 4('S) ₊ 0 4(D) 06(3P) 0 3(3D)]

+ o24d2j( is) ₊ (3P)j 5p 6pjo 2('P) ₊ o i('D) o4(3P)j

7 2 46 0 78 2 05 0.25p2~'S) 5p 6p[0 1(lP) 0 7('D) ₊ 02(35) 04~3P) ₊ 0 4(3D)]

4d2[0 2('S) 0 1('D)] ₊ 0 24d 5d[('S) ('D)]

8 2 72 0 66 75 045p2('S) ₊ 0 15s2(lS) ₊ 4d2[04(lS) ₊ 02(3P)] ₊

+ 5p 6p[04('S) ₊ 0 3(lD) ₊ 0 1(3P) 02(3D)] ₊

+ 4d 5d[- 0 2('S) ₊ 0 2('P) ₊ 0 5(lD) 02(3P)]

N _is the number of component in figure I

(I) Zero of the photoexcitation _energy corresponds to the pos1tlon of the line N

= 3, _w

=

14 342 3 eV (2) Final _state of photoexcltation (only the components ^with (a~ i ~~t~j~s~ m 01 _are listed)

~~~~~

~j

^~l ii n2t2(LS) ^~~ ~~ ~~~~~~~)fll ~l~2

~2(£S),

_~P)

~l 1n2 2 LS

504 JOURNAL DE

PHYSIQUE

II M 5

potential,

^and

solving

secular equation, ^the energies

(li~

and WF of the final states were

obtained

(ELS)

=

£

_a~~

aLS) (3)

The

energies

and wavefunctions of the final states for the transitions

having

the

greatest

oscillator

strengths

_are listed _m table1.

Cross sections

of excitation/ionization

_processes The oscillator

strength

of the transition from the

ground

state into the state of

(3)

has been calculated via the formula _:

fi~~~(w ⁾

₌ ^~ w

£ a~~(aLS[[D

^[

0)

,

(4)

3

~

"

~

where the

photon

_{energy w}

=

E

Eo

is in a-u-,

Eo

is the HF energy of the

ground

state, ^and

the

amplitudes

A

=

(aLS[[ Q

^[

0)

are determined

according

to

[5] by

the expressions.

~4~~))~~)$i~

"

(~l

~~^{~~ ~} ~2 ~~^~~~

(LS)

n'

12 ~~2(l~' ~), ~l'

⁽

~/

⁽

^°, ~~)

"

"

(~

i

)~

N

imax

~~ ^~ ~

~ l') ^~~

^{~ ~ ~~ ~~~}

^{~) ~'~~)} _(D12

+

(~

i

)~'~~ _C12) (5)

In

(5) 0, ~5j

_is the

ground

state WF

,

p =

ii

₊

i~

₊

S,

N _{is a}

product

^of

overlap integrals

of the AO not involved _in the transition,

i~~~

₌ max

(ij, i~)

_;

1~12 "

(n~2~4~

_~l

~l) (n'~2~~2~2),

ci~

=

(n~ i~l1411

ni

ii) (n~ i~Ini~),

the first order _expression for the matnx element of the operator is

(F

is a Ferml

level) [10]

(n~i~[[d[[

_ni

ill

"

(n2 f2(~(

_~i

ii) i ^~~~~~~~~( )i~~~

^{~~ ~~~ ~~~}

nj<n<F ~~ ₂ ~ ~~

(ni in' I')

=

)~ P~i (r) P~,

_I,

(r)

dr

,

(nf _jr

n'

I' ) oo

=

P~i (r) P~,

_I,

(r)

_r dr

o

If _one of the

electrons,

_{i-e np or}

n'p,

is in _a continuum, one may

neglect

the electrostatic

splitting

of the _core and _{sum up}

squared amplitudes (5)

In tins case the expression of the

excitation/ionization

cross section is

~npep(~

4

~ 2

~

~2

~j~ npep

(~)

IS4p

j

0 0 is4p ,

l~~())~~

ii ~

~ ~max

(~

~2 +

1) (D12

+

C12) ~1~12 _f~12)

,

(~)

where

ao15

_a fine _structure constant, _ao is Bohr's radius If both electrons go to continuunl, double ionJzation cross sections _are calculated with the expression

w -JPj~~~

"ls4p(W )

_"

"~i~~(W ) dE, (9)

0

Table II.

ls-ep

and Is

4p- (~) ^pe'p photoabsorption

_cross sections _in Kr _m the region

of

e

KN~~-threshold (see Fig. I).

«(f((

_x 10~, Mb

w

('),

eV

«[f

_x 10~

(2),

Mb

n = 5 _n

=

6 _n

=

e'

14 338.0 154 12 0 50 0 12 0.00

14 342.3 145 44 2 88 0 25 0.00

14 347.3 137 24 16 50 50 0.00

14 354.0 129 55 19 25 4 38 75

14 358.0 125.05 18 75 3 75 4.25

(1)

Kr K-level relativistic _{correction Am}

~~j ⁼

224.0 eV

[7]

is taken into account.

(2)

Results of references

[11, 12].

ax ~o~/tl6

~

s

is4j ntej

6

,----,

,'

--,-,_

/" ~'sly-sjtj

/ , i i

<4.a5 k~liev i i ,

I i I

/~I

I

l~

I

j

j

,

'~

,'

,'

/

,

~,' ,,

"

'~_~ _,,is4j -iii)

fi.55

~ J) )jL(~

i 3 ~ 5

Fig

I K-XANES of Kr atom

(-)

this work's theory

,

(-)

double

photoexcitation

spectrum (- -) excitation/ionization and double

photoionization

channels Insert expenmental _curve of reference ,

[6] o~absorbed

photon

_energy Lorentzian

amplitudes

_are calculated _via

(10)

with

rj~

= 2 69 eV [8]

The notation of the final states labelled here with numbers _is given in table1.

506 JOURNAL DE

PHYSIQUE

II M 5

where the integration is _over the surface

e + e'

= w IP

i~~~, ^e and e' _are the

energies

of the electrons in continuum and

IPj~~~

is ls

4p

ionization

potential

of Kr.

Results

of

the calculation and comparison with the experiment Oscillator

strengths

of the transitions from the

ground

state into the states of

(3)

_were calculated _using

(4)

and _are

presented

in table I. We also present ^the

amplitudes

of Lorentzian _curves calculated _via the

expression

1(W )

=

4 7r«o

aifis4p(W )/ris, (lo)

where

ri~

is the Kr ls level width. This _was done to make the theoretical spectrum convenient

for companng with

expenment.

Each component of the

spectrum

^is

presented

with the

Lorentzian curve, and the _{curves are} summed _up. The _cross sections of the transitions Is

4p-

npep and Is

4p-epe'p

_were calculated _via the formulae

(5-9), they

_{are given in} table II. The theoretical

single

ionization _{cross sections} for Kr

[I1, 12]

are also

presented

in the _sanJe table to make it

possible

to compare with the

expenment

the _sum of

single

and double

ionizationlexcitation

_cross sections.

In

figure

I the theoretical

KrKt§~

spectrum obtained _in the way descnbed above is shown.

The measured spectrum of reference

[6]

is also

given.

One may see a

good qualitative

agreement of the spectra. One _may,

therefore,

state that the assignment of the spectrum which _is obvious

(see

Tab. I and

Fig I)

_is reliable

enough Generally,

_our

assignment

is the

same as the

qualitative

_one of reference

[6],

but _one should note on the

bnghtest peak

A

(the peak

at _w

=

14 343 eV in

Fig. I)

that _its ongin is connected not

only

with the excitation of electrons into discrete states but also with

photoiomzation

of _atom.

3.

Comparison

of calculated KX

photoabsorpfion

_spectra of

Kr,

Ar and Ne.

The calculation of the

photoabsorption _spectra

_in the

regions

of KX-ionization for Ne

(X= L),

Ar

(x= M)

and Kr

(X= N)

had made it

possible

to understand the main mechanisms

forming spectral

_structures

Spectral

intensities The main mechanism

determining integral

intensities of spectra is a

monopole

rearrangement of the electron shells at the appearance of ls- and X-vacancies. To charactenze the relative value of excitation and ionJzation processes we list _in table III several values _:

f(A)

=

zfisn~p(W ⁽ⁱ¹⁾

Table III

Changes

_m relative value

of

excitation and ionization processes m the _row

A

=Ne, Ar,

Kr.

A Ne Ar Kr

f(A

_x 10~

(1)

^{2 37 0 44 0.06}

« j

(A

(2) ^{2 75 0 53 0 07}

x(A )

_x

10~~

₍₃₎ 0.29 0.30 0 31

(ii, (2), (3)

^{See the formulae}

(11)-(13)

of this work.

Ii

"I(A)

"

z "I$§ d~, (12)

n

o x

(A )

=

«i(A )/(2 ^7r~

_«o

aif(A)) ₍₁₃₎

where the oscillator

strengths

and ionization _cross sections _are determined

by (4)

and

(7)

_;

A =

Ne, Ar, Kr,

_nj

=

2, 3, 4, Integration

_in

(12)

is made in _a-u One may see from table III

that, although f

and «j values decrease with the

growth

of the atomic

number,

their ratio

charactenzing

the relative value of the exaltation and ionization _cross section

changes weakly Among

the

pecuhanties

of the calculation _one should note the

importance

of the inclusion of additional terms _in

(6)

and the necessity to demand the WF of

higher lying (in energy)

configurations

to be

orthogonal

to the lower

lying

WF of the _same symmetry

Thus, calculating

ls

4p-5p2

exaltation oscillator

strengths,

the inclusion of additional _terms led to

increase the value of

f by

47

9b,

and inclusion of

orthogonahty

condition for the WF of

configurations

ls~

4p~ 5p2

and ls~

5p

led to _a 15 ifi decrease of the value

off-

In

all, taking

additional _terms in

(6)

and

orthogonahty

conditions into account increased the _cross sections

by

30 9b This value _is somewhat less than those for

f/f~~(Ne )-72

ifi and

f(f~~(Ar )-43

ifi. This allows _us to conclude that the role of additional _terms and

orthogonahty

conditions _m the calculation of excitation and ionization

probabilities

decreases with

growth

of atomic

number,

and this conclusion may be useful _in calculations of

photo-double excitation/ionization

processes in multi-atomic systems ^{where inclusion of the} effects under discussion is difficult

Table IV.

Changes

_m the

dkgree of effect ~((l'~ of

the _excitation

Y=np~-md~,

np~-mj

_sm~ d _on K-XANES _m the _row A

=

Ne, Ar,

Kr

A Ne Ar Kr

n, ^m j 3 4 5

m,m~ 3 3 4

E~(md~)

_(~

),

eV 6.50 1.17 0 37

E~(mi

_sm~

d),

eV 0 75 2.20 2 20

R[(md~),

eV 2 51 2.44 2 16

R((md~),

eV 46 1.57 37

R/(m

_{sm~ d}

),

eV 2 89 2.32 1.23

~/(md~) ₍₂₎

0.39 2.09 5.84

~

((m

_sm~ d

)

3.85 1.05 0.56

(~)

E~( Y)

"

E~~(np~j E~~( _Y).

(2) See the formula

(14)

^of this work

508 JOURNAL DE

PHYSIQUE

II M 5

Structure

ofspectra.

The _structures of KX

photoabsorption

_spectra are dominated

mainly by multiplet splitting

of the

configurations

ls" nj

p~ npn'p

and

by

their

polanzation by

npn'

_p-m

dm~

d and

npn'

_p-m

j sm~ d excitations. Coulomb

ls-nj

_p interaction _is

significant only

_m

Ne,

the role of

np-n'p

Coulomb interaction is

approximately equal (the splitting

of

l-2

eV),

and the role of

npn'

_p-m

j

dm~

d and

npn'

_p-m

j sm~ d is shown _m table IV. Listed _m this table _are Slater

integrals R((Y) charactenzing

the value of

mixing

for the excitations Y

=

np~-md~, np~-mj

_sm~ d

(n

=

3, 4,

5 for

Ne, Ar, Kr),

and the energies of these excitations

E~(

Y~

(that

_is the differences between centres of gravities of

corresponding configurations).

The values

~i(Y)

=

iRj( Y~/En(Y) (14)

give the

degree

of Y excitation effect _on spectrum. ^It can be _seen from table IV that the effect

on spectra ^of

np~-md~

excitations grows because of

decreasing

_energy denonfInator and that of

np~-mj sm~d sharply

decrease because of _energy denominator's _increase and

decreasing integrals R((npnp,

_m

i sm~

d).

4. Conclusion.

This

study

allowed _us to give an

assignment

^of Kr KX

photoabsorption spectrum

^based on a

non-empirical

calculation. Our

assignment

is

basically

the _{same as} that of reference

[6]

It _was shown that the _main fine structure features _are determined

by mu1tlplet splitting

of the

discrete excitations and

by

_opening of channels of

single

and double ionization. These

processes' probabilities

_may be described within the MRES

approximation and,

to calculate the fine structure, one has to consider

multiplet splitting

of the

configuration

ls~

4p~ npn'

_p

and its

polanzation by npn'p-m

j

dm~d

excitations

A companson of the results of KX spectra calculations _m

Ne, Ar,

Kr has shown that MRES

is the main mechanism

allowing

one to descnbe

photo-double excitation/ionization probabili-

ties near the KX

threshold,

but relative

significance

of the _main fine structure

forming

mechanisms _vary somewhat _m the _row

Ne, Ar,

Kr Thls

study

_may be useful _in the

analysis

of K-XANES _m 10 _« Z « 36 elements which _are

being widely enough

used in

investigation

of matter's electronic structure

Acknowledgments.

The authors _are

grateful

to Dr. A G Kochur for

helpful

discussions _{on some} of the above

reported

results.

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