Direct radiative recombination and photoeffect cross sections for arbitrary atomic Coulomb bound states

(1)

HAL Id: jpa-00247899

https://hal.archives-ouvertes.fr/jpa-00247899

Submitted on 1 Jan 1993

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Direct radiative recombination and photoeffect cross sections for arbitrary atomic Coulomb bound states

F. Aaron, A. Costescu, C. Dinu

To cite this version:

F. Aaron, A. Costescu, C. Dinu. Direct radiative recombination and photoeffect cross sections for

arbitrary atomic Coulomb bound states. Journal de Physique II, EDP Sciences, 1993, 3 (8), pp.1227-

1239. �10.1051/jp2:1993193�. �jpa-00247899�

(2)

Classification P/>ysics Abstracts

32.80F 52.20F

Direct radiative recombination and photoeffect

_cross

^sections

for arbitrary ^atomic Coulomb bound states

F. D.

Aaron,

A. Costescu and C. Dinu

Department

of Theoretical

Physics, University

of Bucharest, 76900

Bucharest-Magurele,

Romania

(Received 28 December J992,

accepted

in

final fat-m

14 April 1993)

Abstract.

Simple

exact

analytical

_cross sections for direct radiative recombination (DRR) and atomic

photoeffect

for

arbitrary

(n,

f,

m states are obtained in the nonrelativistic Coulomb

dipole

approximation.

These may also be used in the _case of DRR _or

photoeffect

_on the

Rydberg

states of

highly

ionized _atoms with N electrons by

considering

an effective ionic

charge Z~rr=

Z N.

Asymptotic

formulae in the energy of the

incoming panicle

_are derived. The numerical values of the DRR _cross sections obtained with _our formulae _are in very good agreement with the

existing

numerical relativistic results. This is true for high Z values and large

energies

of the incoming electron.

1. Introduction.

The interest for the atomic processes

involving highly

excited states, due to the

development

^of nuclear fusion

techniques

and

Rydberg

spectroscopy, ^has sustained constant

growth

in the last few years. The direct radiative recombination

(DRR),

the dielectronic recombination and the collisional ionization determine the ionization

equilibria

ⁱⁿ

plasma. Also,

for _a

fairly large

temperature range, the DRR dominates both the dielectronic recombination and bremss-

trahlung, being

the _most

important

mechanism of energy loss in hot

plasmas.

As is well-known in fusion

plasma physics,

the radiation of

high

and intermediate Z atomic number

impurity ions, having only

a few electrons in the L

shell,

is

mainly

due to DRR. For such

highly

ionized _atoms the Coulomb

approach

with _an effective

charge

_Z~~~

=

Z

N,

where

N is the number of electrons of the ion, is

excellent, especially

for the DRR _on

highly

excited states. Further considerations _on

screening

effects in the _case of low levels _or target ^ions

having

more residual electrons are

given

in the works of Hahn and Rule

Ii,

Kim and Pratt

[2]

and

Burgess

and Summers

[3].

Kramers obtained the first

analytical

result for the DRR _cross section in the semiclassical

approach [4]. Using

the

pertinent

matrix elements obtained

by

Gordon

[5],

Stobbe calculated

the _exact

analytical

_cross section in the quantum nonrelativistic _case for the

photoeffect

and the

DRR on the

ground

and first excited states

[6].

Karzas and Latter derived _an exact

analytical

(3)

formula for the

photoeffect

_cross section from

arbitrary

states in the nonrelativistic

dipole approximation,

but their formula is rather

cumbersome, involving

_a number of

~F

j

(a,

b _; _c _z

hypergeometric

Gauss functions

[7].

A few years ago exact

analytical

formulae _were

given

for

the DRR _cross sections _on the nS, nP and nD subshells in the nonrelativistic

dipole

approximation (DA) [8].

In section 2 of this paper we obtain

simple analytical

formulae for the DRR _cross sections for

arbitrary in, f, m)

atomic _states.

Indeed,

_our method leads _to considerable mathematical

simplifications.

In fact _we _use the momentum

representation

and also express the _wave function of _an

arbitrary (n, f,

_m

₎

_state in terms of _a differential

operator acting

_upon the

ground

state wave function. This _was _not done

by previous

authors except ^for ^Brandus et al. in

[81.

Due to the

crossing

symmetry ^of ^the ^matrix ^element

describing

the

one-photon

_processes, we

also obtained the

photoeffect

_cross section. Thus, well-known textbook results _are

generalized.

Section 3 is devoted to the discussion of the results.

Asymptotic

formulae in the energy of the

incoming particle

_are also derived. In section 4 _we

give

numerical results obtained with _our

analytical

formulae. An excellent agreement ^with ^the relativistic numerical results of Kim and Pratt

[2]

is found. In fact, it _was

recently proved analytically,

for

Rayleigh scattering

and

photoeffect,

that the

multipole

effects are cancelled

by

the relativistic kinematical corrections

[17].

This

_processes, ⁱⁿ ^the nonrelativistic DA, is

(s P)~,

₌

a/(q)(s q) a,(q)d3q, ₍₁₎

where P is the _momentum operator ^for ^the ^electron ^and s is the

polarization

of the

photon.

For DRR,

f

= in,

f,

_m and I

= p,

(n, f,

_m ₎

_being

the quantum ^numbers

corresponding

to the atomic state into which the electron is

captured

and _p

being

the initial momentum of the

electron. We notice that the _same matrix element also describes the

photoeffect.

The wavefunction a;

(q)

= a

(q p)

of the

incoming

electron with momentum p, normalized in the energy and solid

angle

scale has the

integral representation

a(q;P)= ~~~~j~°'~~~ l'~l _j" ^di

_~

~~

"~

_~~~°> ^'

l(q Pf)~

₊

_IS

₊

_ipji I)j2j2

where N

=

~~~ ~" _"

'~~ §)/

^~ ^"

_~pm~)'~~ ₍₃₎

(2

_gr

E

=

p~/1~

is the electron energy in

Ry*Z~

_units.

a is the fine-structure constant and m~ is the rest mass of the electron. Natural units h

= c =

I _are used.

For the final

in, f, _m)

_state, _we _use the wavefunction written in the form

[9,

10]

u,inn,lP)

₌

5"lv A?~lv, x)

~ ~ ~ ^,

(5)

llP~X)

+ l~

~i>/n

(4)

with

f+~

i

C~t

n

Yim ( ) =

3

Ai~lll, x)

a,,

a~

(~c, x)

=

~

^~

Ii 0)

' %x~~

~c

%~c

i

By

convention,

[(n~

I

)(n~ 4) (n~ f~)]

^~ _is _taken _to _be

one for

f

=

0. The coefficients

Ci)

~,, ~t

appearing

in the

spherical

harmonics

(8)

_are the

components

^of a

totally symmetric

and traceless tensor of rank

f [Il, 12].

Combining

the above

equations

_one obtains

(s

P

)~i~

~ ⁼

~

P(~~

^5~^(Hi

)(~ sin'(p,

~

i

Ii 1)

(2

_gr

n

with

8~~(p,

_~

)

=

(

s~

Aj~)(~, x)a~(~,x)K(p

_~,

x)j~~~ ⁽¹²⁾

and

K(p;/~,x)=

=

j'~dfl'j _)"~ j

~ ~

~~)

~ ~

(13)

(e~o) R3

(q

₊ _/~

)((q Pi +x) ~p(I ()- ie] )

Using

the residue theorem, the

integral (13)

becomes

~

3

(f~ 1)"

_i

K(p

_~c,

x)

=

~ ~ ^,

(14)

~

f2

_P _~lP~ _-P°X

where

~2

^~

~°'~

_~ _~ _'~

II i'~~]~.'i~ii)i'i~ ^~~

^~ ⁺ ^~~ ^~^~~

⁽¹⁵⁾

Taking

into _account how the

operator Aj~~a~

^acts on

K(p

_; _~c,

x)

and also

using

the above- mentioned

properties

of the coefficients C

if

~~, ~~,

after _a

lengthy,

but rather

elementary calculation,

_we obtain the

following

result

Is

P

)~i~,

~ ⁼ ~V~i

iv £~io

Iv )(S P)

_Ytm ^~ + £mu

Iv )

IS x P

) Lynn,

^~

,

Ii 6)

(5)

where

~V,,t

iv

= I ⁺ ^' 2~ + ~

'~~

_~~~ _~~~ _~ _~~~ _~~ ~

" ^~ ~

x

1

1-exp(-2 «(v() t~5

~ 2

exp

(I

_arg N 2 _v _p,,

x

[v

+ I

]_t

~

i (17)

~

'"'

^~~

n~

ia

i~ ^m

(a

₊ I

)ia

₊

2).. (a

₊

j ),

iai_~

m

(a )(a 2).. (a j), (18)

iaiom1,

w~ =

tan~'

^~

,

(v( (19)

£n101"1~

~

~ ( _ij f v)I$~l"

f _i

j

~

jV

₊

f

₊ _{i j~}

~ X

1+

~~/

^'"° ~"°~"°

n

x

(ijvj 3~,,.~j

+

-2j _jvj2+ _2i(I

+

I)jvj +2i(I

₊

I)j 3~,,j

» i I

v ^~

Sj°~(v

~~ i~

~~

,i~ ^i~

^~~~

jr

₊ 2

I

₊ _I

)! in I

r

)!

^'

12°)

£mu

iv )

₌

=~~P(~(i-*'i)j~~~-~-~~~ _j(j ~-~~~ir+2i+1)iiJ~-I-r-i)! i~~~

vi~i~ i~+~i ^n'~ i~~c~t _CIC~

ii j _is _)i +j j-s

^~22~

Sj°~(v)

=

f

^~'~ ^~~~ ^~'~ ^~ ^~

^~'~~

2.2 THE DRR cRoss SECTION. For this _case _we choose _a coordinate frame in which the

Ox~

axis is taken

along

the momentum of the

incoming

electron I-e- _p

= pe~. The matrix element

given by (16)

becomes

(s P)~t~

~~ ⁼ ~V~t

(v)

_p

/

~

j£~t~(v)

_s~

3~

o +

£~tj iv )

x

4 _"

x

~

[(sj

+

is~) 3~

j

(sj is~) 3~ )

(25)

2 I ^'~

(6)

The _cross section for DRR _on _a bound

(n, f,

_m _state is

given

ⁱⁿ the nonrelativistic DA

by

~#

"

(~

_" _{)~ ~e}

~C Ii

(S ^P)nim,_p ^~

dflk

'

(26)

P

s

where r~ =

a~c

^is ^the classical electron

radius, ~c

is the reduced

Compton ~avelength

^of ^the

electron and k is the energy of the emitted

photon. Using

the relation k

=

) ^E

+

~

and

me _n

carrying

out the _sum _over the

polarizations

and the

integral

_over all the directions of the

momentum of the emitted

photon,

k, _we

finally

have

~ f _~ _i

(n~

I

)(n~

4

) (n~ i~)

~

DRR

~4

^I+ 6

~ 2

r ~

f ^~

»im ~

3 ~ ~

n~ ^+~

~

~j2i+4 exp(-4(V( Wn)

X

[" +11-I(

j~j2 ^2i+1~

l

-exp(-2 ar(V()~

~_

n~

x

£nioi

_v

_)1~

_3m,

o +

iii

₊ i

)

_£nip

iv )1~

13m, _j ⁺ 3m, _j

)) ¹²⁷⁾

The DRR cross section for _an

arbitrary in, fl'subshell

_is

"fl~

I

~

i "# ₍₂₈₎

m I

From

(27)

we get

"fl~

~

2~~ " ~/

~~

~C

^~~~

~n25

~~~ ~~~

~~ ~ ~~~ ~

~

2i

/~ ^( ₎ ~nf°~~~

^~

~~~

_~

nfl~~'

^~~~~

(i

^~

^'"(

^~~ ^~~~ ^" ^~

n~

Obviously

we can write

I-1 .~

Iv

+

il-il~

=

lvl~~ fl

1+ ~

~

j=o

[Ml

and the convention

[a low I,

mentioned in

ii 8), implies

that the

product

is

equal

_to I for

f

=

0.

2.3 THE PHOTOEFFECT cRoss sEcTioN. Now it is useful _to choose _a coordinate frame in

which the

Ox~

^axis ^is ^taken

along

the momentum of the

incoming photon.

The

photoeffect

cross section for _a transition from _a bound state to continuum is

given

in the DA

by

«I'm

=

2 _« ^~re

~c Z

_IS

_P)p,

_ni»>1~

drip 13°1

~

JOURNAL DE PHYSIOUE>I T 3, N' 8, AUGUST 1993 48

(7)

Using

the

expression (16)

we obtain

~ ~ ~ ~ l

(n~ )(n~

4

(n~ i~)

_eXP

(-

4

" W,>

"~~'

~ ~

" '~

~~

(aZ)~ _{(2 f} _{)(2 f}

₊ 3 n~ + ~ l ~XP

(-

2

+

2(3 m~ i~

I Re

£,pow

£,,tj iv

exp

(- j

,

j31)

where K

= k

IA

^~/2

m~)~

^' ^is ^the energy of the

incoming photon

in

Ry

_* ^Z~ units.

Obviously, according

to the conservation of energy we have

(v(

= ,

K~

~.

j ~~/

^I n

(32)

£,,to iv ^and

£~tj (v

are

given by (20)

and

(21)

where

v should be

replaced according

to

(32).

The

photoeffect

_cross section for the entire

(n, ii

subshell is

f

ph

£

^ph (~~)

~»i " ~<lim

'

<n -1

so that we obtain

~ ~~ ~

~2 f

+ i I

(n~

I

)(n~ 4)... (n~ i~) eXP(- 4(

_" _W«)

"~~

^~ ^~ "

3 ^~~

~~

(aZ)~

n~ +~ l ~XP (- 2 _" _" ^~

x

~~~~ ~jj i

+

j~~K- ((£,,io(v)(~

+

f(f

₊

i)(£,,ij(v)(~) (34)

K ~~o n

Considering equations (29)

^and

(34)

_one _can

easily

_see that the well-known

relationship pertaining

to the detailed balance

principle

is checked

~~~~

=

~~. ₍₃₅₎

~

"$/ _P~

According

_to

equations (27), (29), (31)

and

(34)

_one _can

immediately

_see that the DRR and the

photoeffect

_cross sections

depend only

_on the dimensionless variable _v and the quantum

numbers _n,

f

and _m. Such

scaling properties

are well-known and _were

pointed

out

by

^Hahn ^and

Rule

[I]

and for _more

sophisticated

_cases

by

Kim and Pratt

[2].

3. Discussion of results.

Putting

^f

=

0,

and 2 in

equation (29)

_we obttain the results

previously given by

^Brandus

et al.

[8]. Also,

for _n ₌ 1, 2 and 3

equations (31)

and

(34) give

for the

photoeffect

_cross

sections the Stobbe results. Our formulae

give

_very

simple expressions

~2

and

">,nDRR "

~4

+ 2

j~2 (~2

~

) j~2 j~ )2)

~ 2_n ₊ 2

~

3

~~~~~~

2 5 2

~~~~~~~~+'~

_~

~ ^~~

(2~ l)~ (2~~2)~ ~

~

'"' ~~~~

n~

x

~~ *

[(2n~+n+ I)(v(~+2n~(2n~-n+ I)(v(~+n~(n-1)(2n- _Iii.

I

-exp(-2 ar(v()

(37)

As

usual,

we define the Gaunt factor for the _n shell _as

~DRR

~n

~~Krl

^~~~~

n

where

trf~~

is the _cross section for the entire _n shell and is

given by

n

DRR

£

^DRR

(~~)

"n " "<,I

f o

and

tr,)~'~

is the Kramers _cross section

[4]

«i~~~ H

re

~c ^~ "j lj

~

Ho)

~

Considering

the low energy limit _we find lim g~ =

0.924.

Comparing

with the _same limit for

E _~ 0

the

K,

L and M shells

[13]

I-e- lim gj =

0.797,

lim g~ =

0.876,

lim g~ =

0.908,

_we

point

out

E~0 E~0 E~0

that,

_as

expected

with

increasing

n, the semi-classical Kramers

approach gives

better

predictions

for the _cross section values.

In the

high

_energy limit

(E

~ co, v ~ 0

)

we obtained the

following asymptotic

^formulae

Hi)

for _n

= 1,

2,

and

f

=

0 and

(9)

~$RRaS ~

~4i+6'T~,.

i~

^~~~

^~~~~~ /~

^~~~ ^~~~ ^~

^~~~~~

_X

~ 3 ^~

i(2i);12 n2

⁺³

j~

^~

lvl~ ~~+~

~2

j i~+I

₊ _I _I

I(I

₊

_I)

jj ^~j eXP(-41"1

_Wn)

~

Ii"

_~

~i~~'

~~

iii +1)12i +1)~i~n~12i

+1)

^~ ^l

-exp(-2 «(v(1'

(42)

for _n

= 2, 3, and

f

=

1, 2,

., n I.

For very

high energies

the ratios of the main

asymptotic

terms are

"$f~~~

2~ ^~

If

I

)i f f2

iF$l~~~ [(2 f)!

_]~ n~ n~

E~' ^~'~'

^' ^~ ^' ^~~~~

~()~

=

~

l ^~~

~~

_~~

,

f

= 1,

2,

,

n 2.

(44)

&~i

^~~

(f

₊

₁₎₍₂

f ₊ _I

)

_n

One _can

easily

_see that the recombination on S _states is dominant _at

high energies.

Using

the detailed balance

principle, equation (35),

we obtained the main

asymptotic

term for the

photoeffect

_cross section in the

high

_energy

limit,

_as follows

~~~~~ ~~

i

~

z2 ~ ³ l12

'

~~~~

for n ₌

1, 2,

and

f

=

0 and

iii

~~

=

2~~

^~~ ^~"^~~ ^~" _l _l ^~~

~ ail

_~~,

(46)

(2

f

)!

]~ n~ n~ _K

for _n

=

2,

3, and

f

=

1, 2,

...,

n I.

Obviously, according

to the above-mentioned detailed balance

principle,

the ratios of the main

asymptotic

terms are the _same _as in the DRR _case,

equations (43)

and

(44),

^where

E ₌ K

~

WI-

n

4. Numerical results and conclusions.

Numerical values for DRR _cross sections

(Eq. (29))

on some

in, ii subshells,

for different atomic numbers Z and for different

incoming

electron

energies E~ (in kev)

are

given

in table I.

The corrected Gaunt factor,

g[°",

is defined

by tl$~~

~~~~ ₌

(47)

«jKr)corn

^' where

3$~~

N

=

£ tr$~~,

N

=

Min

(n I,

II

(48)

I=o

and

tr)~r)~°",

called the corrected Kramers _cross section

[14],

is

tr)~r)C°"

=

tr)~r)[

I a,,

(E) b~(E

+ 2

a~(E) b~(E )]

,

(49)

(10)

Table I. Values

for

DRR _cross sections

tr$~~ (in

10~ ^~~

cm~) for

various atomic numbers Z and

incoming

elecn.on

energies E~.

^The ^number ⁱⁿ

parentheses

is minus the power

often multiplying

the entry. The last column

gives

the corrected Gaunt

factor g[°"

Z=26,

E~=50keV

l 5

908(3)

0.9862

2 7

679(4)

9

997(5)

0.9789

3 2

294(4)

3

538(5)

7

946(7)

0.9779

4 9

705(5) 579(5)

4

781(7)

4

059(9)

0.9775

5 4

976(5)

8

288(6)

2

810(7)

3

489(9) 404(1

0.9774

10 6

7 5

349(16)

0.9772

100 6

235(9) 082(9)

⁴

360(1II

⁸

447(13)

9

426(15)

6

798(17)

0.9772

Z

= 42

,

E~

= 100 kev

l 9

711(3)

0.9912

2

274(3)

2

129(4)

0.9829

3 3

815(4)

7

551(5)

2

204(6)

0.9818

4

616(4)

3

372(5) 327(6) 465(8)

0.9814

5 8

287(5) 771(5)

7

800(7) 259(8)

6

598(1ii

0.9813

10

038(5)

2

0.9810 100

039(8)

2

313(9) 211(10)

³

050(12)

4

430(14)

4

162(16)

0.9810

Z

=

74, E~

=

50 kev

l

714(1)

1.0008

2 2

475(2) 915(2)

1.0013

3 7

698(3)

7

003(3) 142(3)

1.0000

4 3

313(3)

3

170(3)

6

941(4)

4

423(5)

0.9996

5

714(3) 676(3)

4

100(4)

3

816(5) 183(6)

0.9995

10 2

172(4)

2

188(4)

6

091(5)

8

041(6)

5

807(7)

2

461(8)

183(7)

2

219(7)

6

424(8)

9

304(9)

7

983(10)

4

497(1ii

0.9992

(11)

Table I

(continued).

Z

=

74

,

E~

=

100 kev

~ DRR

~DRR ~DRR

DRR DRR

~DRR

con

"n0 nl n2 "n3 "n4 n5 ~n

6

447(2)

^1.0076

2 8

961(3)

4

071(3)

0.9986

3 2

732(3) 466(3) 276(4)

0.9972

4

165(3)

6

588(4)

7

715(5)

2

566(6)

0.9968

5 5

997(4)

3

0.9964

100 7

570(8)

4

558(8)

7

082(9)

5

365(10)

2

370(1ii

6

815(13)

0.9964

where

a~(E)

=

0.172825 K

~

E

,

~

(50)

b~(E)

=

0.033060 K ^~

(E~

E _x K ₊ 1.5 _x

K~).

We _are reminded that the corrected Kramers _cross section, that

gives

the contribution of the entire _n shell _to the DRR,

significantly improves

the semiclassical result. As _one _can _see, for

high

_n, the first twelve subshells

give

98 fb _or _even _more of the recombination _on the entire shell.

Table II

gives

the

comparison

of _our numerical results with the relativistic _ones of Kim and Pratt

[2],

for the total atomic DRR _cross sections. Our numerical

results,

for the _cross sections for the entire atom, were evaluated

by summing

_up the contributions of the first _one hundred shells. Let _us observe the excellent agreement ^between our nonrelativistic DA results and the

relativistic _ones. This _was

expected because,

_as _was

pointed

out

[14],

the corrected Kramers

cross

section,

for rather

high

Z and electron

energies

_up to loo kev, is in very

good

agreement

with the relativistic results of Kim and Pratt.

Except

for the _case of E~ =

loo kev, when the nonrelativistic

approach

is not

expected

to

give

_very

good predictions,

_our numerical results

are

significantly

closer to the relativistic results than those

predicted by

Hahn and Rule I

(see

Tab. I of Kim and Pratt

[2]).

^We ^believe ^that our numbers _are _correct because the

simplicity

of

our formulae allows _an easy control of the numerical accuracy.

Certainly,

the nonrelativistic

approach

is

good

at small

energies,

I-e- when the

scaling

_parameter _v is

large

for _a

given

Z.

For 100 kev

significant discrepancies

are to be

expected,

as our numbers show.

N

=DRRaS

jj

^DRRaS

(~ )

"« " "ni

0

The

corresponding

relative _errors _are

always negative. They

_are also

given,

in

fl,

but their

(12)

Table II.

Comparison

betvveen the relativistic results

of

Kim and Piatt,

trfl(~p,

and _oat- nonrelativistic

i-esu/ts, tr(@~~, for

the total atomic DRR _cross sections

(in 10~~~ cm~).

Various _cases

of

atomic numbers Z and

incoming

electron

energies

_E~

(in kev)

at-e

considered. Relative _errors _are

given

in the last column.

E~

«i~fP _~~@&D

^~~~~~~~~ ~~~°~

Z

= 26 _x 103 2.944 _x 1.21fb

5 41 _x 10~ 3.396 _x 0.41 fl

10 1.21 _x 10~ 1.210 _x 0.00

50 7.21 7.333 ₊ 1.70 fl

100 1.75 1.825 ₊ 4.28 fl

Z

= 42 9.99 _x 103 9.766 _x 2.24 fl

5 1.28 _x 103 1.266 _x 1.09 fl

10 5.02 _x 10~ 4.944 _x 1.51 fl

50 .18 _x 10' 4.200 _x 101 ₊ 0.48 fl

100 1.20 _x 10' 1.216 _x 101 ₊ 1.33 fl

Z

= 74 4.0 _x 104 3.827 _x 4.32 fl

5 5.63 _x 103 5.413 _x 3.85 fl

lo 2.35 _x 103 2.247 _x 4.38 fl

50 2.65 _x 10~ 2.504 _x 5.51 fl

Table III.

Comparison

betvveen _our _exact,

tl$~~,

and _our

asymptotic, 4f~~~~,

_cross

sections

(in

10~ ^~~

cm~), for

the _cases

ofatomic

numbers Z

=

26 and 42 and

incoming

^election

energies E~

₌ ⁵⁰ ^and ^100keV. ^The ^number in

paiantheses

is minus the power

often multiplying

the _em.y. Relative _errors _al-e

gii,en

in the

fourth

and seventh columns,

respectively.

Z

=

26

E~

₌ 50 kev

E~

₌ ^loo ^kev

n

tl$ 4f~~

^Rel. er.

$f 4$~~

^Rel. er.

4

133(4) 131(4)

0.18fl 2

601(5)

2

599(5)

0.07fl

5 5

833(5)

5

806(5)

0.46 fl

335(5) 334(5)

0.07 fl

6 3

385(5)

3

362(5)

0.68 fl 7

738(6)

7

724(6)

0.18 fl

7 2

136(5)

2

117(5)

0.89 fl 4

877(6)

4

866(6)

0.23 fl

8

432(5) 418(5)

0.98 fl 3

269(6)

3

261(6)

0.24 fl

9

007(5)

9

952(6)

1.17 fl 2

297(6)

2

290(6)

0.30 fl

lo 7

343(6)

7

251(6)

1.25 fl

675(6) 670(6)

0.30 fl

50 5

888(8)

5

758(8)

2.21fl

341(8) 334(8)

0.58 fl

loo 7

361(9)

7

188(9)

2.35 fl

677(9) 667(9)

0.62 fl

(13)

Table III

(continued).

Z =42

E~

₌ ⁵⁰ ^kev

E~

= loo kev

~

jDRR

_~DRRaS _Rei,

er.

&f Si~

~~~' _~~'

4 7

997(4)

7

883(4)

1.43 fl

966(4) 959(4)

0.36 fl

5 4

150(4)

4

028(4)

2.94fl

014(4) 006(4)

0.79fl

6 2

419(4)

2

318(4)

4.18 fl 5

888(5)

5

820(5)

1.15 fb

7

530(4) 450(4)

5.23 fl 3

717(5)

3

662(5)

1.48 fl

8

028(4)

9

659(5)

6.04 fl 2

493(5)

²

451(5)

1.68 fl

9 7

234(5)

6

748(5

6.72 fl

753(5) 719(5)

1.94 fl

10 5

281(5)

4

896(5)

7.29 fl

279(5) 252(5)

2. ii fl

50 4

250(7)

3

738(7)

12.05 fl

026(7)

9

889(8)

3.62 fl

100 5

313(8)

4

638(8)

12.70fl

283(8) 233(8)

3.90 fl

minus

sign

is omitted. The chosen Z and E~ ^values

correspond

to

iv

~ 0.70. For such

iv

values the relative _errors do not exceed 13 fl when _n increases _to 100.

Considering higher energies (k

_s _m~ the values of the

photoeffect

cross section from the K

shell, predicted by

^the nonrelativistic DA, _agrees within 30 fl with the relativistic _ones

[15, 16]. Recently,

the

following formula,

which takes into account both the relativistic kinematical

corrections,

for all intermediate states, and the

multipole

terms was derived

[17]

x +

l~ Iii

⁺

«~z~)li

⁺ ²

l~ ⁽ⁱ ^~z~i~/~l ^«~z~ l~ l~i

,

for k» me

(521

where

xo(k)

=

~m

^m ² ^1/2

w 2 _tan ^' aZ

~ ~ ~

~~ ~

~

~~

~~~~ ~

~

~~

,

k _m 2 EK 11 _«

2z2)'/2

^'~~

«

2z2

k

m m 2 1/2

w + 2 tan~ aZ

~ ~ ~

~~ ~

~~~~~~~ ~

~

~~

, SK ~ k _w 2 EK

~e

~

2

z2

_(~

~ 2

z2)1/2

k

e~

being

the K shell ionization _energy.

(14)

This formula proves the cancellation of the

multipole

effects

by

the relativistic kinematical corrections.

Comparing

_our

asymptotic

formula

(45)

with formula

(52),

_one _may _see that for k _» m~

only

_an extra factor of (m~/k)~~~ occurs in the nonrelativistic _case.

Roughly speaking,

this factor is of the order of

unity

for the

energies

considered.

Acknowledgment.

Two of _us

(F.

D. Aaron and A.

Costescu)

_are

grateful

to Professor R. H. Pratt from the

University

of

Pittsburgh

for

stimulating

discussions.

References

[ii Hahn Y. and Rule D. W., J.

Phys.

B _: At. Mol. Phys. 10 (1977) 2689.

[2] Kim Y. S. and Pratt R. H.,

Phys.

Rev. A 27 (1983) 2913.

[3]

Burgess

A. and Summers H. P., Mon. Not. R. Astion. Sac. 226 (1987) 257.

[4] Kramers H. A., Philos. Mag. 46 (1923) 836.

[51 Gordon W., Ann. Phys. 2 (1929) 1031.

[6] Stobbe M., Ann. Phys. 7

(1930)

661.

[7] Karzas W. J. and Latter R.,

Astrophys.

J.

Supp/.

55 (1961) 167.

[8] Brandus I., Costescu A. and Manolescu A., J. Phys. B _: At. Mol. Phys. 20 (1987) 4615.

[9] Costescu A., Brandus I. and Mezincescu N., J.

Phys.

B

: At. Mol. Phys. 18 (1985) L II.

[10I Costescu A., Aaron F. D., Schneider I. and Mihailescu I. N., Opt. Lea. ii (1986) 449.

[I II Whittaker E. T. and Watson G. N., A Course of Modern

Analysis (Cambridge University

Press,

Cambridge,

1950).

[12]

Applequist

J., J. Phys. A _: Math. Gen. 22 (1989) 4303.

[13] Costescu A., Manolescu A. and Mezincescu N., Rev. Roum.

Phys.

30 (1985) 65.

[14] Costescu A. and Mezincescu N., Phys. Lett. 105A (1984) 359.

[15] Kissel L., Pratt R. H. and

Roy

S. C., Phys. Rev. A22

(1980)

1970.

[16] Roy S. C. and Pratt R. H.,

Phj,s.

Rev. A26 (1982) 651.

[17] Costescu A.,

Bergstrom

P. M. Jr., Dinu C. and Pratt R. H., to be