Total cross sections for (e--He) ionization

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Total cross sections for (e–He) ionization

K. Mukherjee, Keka Choudhury, P. Mazumdar, S. Brajamani

To cite this version:

K. Mukherjee, Keka Choudhury, P. Mazumdar, S. Brajamani. Total cross sections for (e–He) ion-

ization. Journal de Physique II, EDP Sciences, 1991, 1 (1), pp.11-17. �10.1051/jp2:1991136�. �jpa-

00247497�

J.

Phys.

II1

(1991)

11-17 _JANVIER 1991, _PAGE II

Classification

Physics

Abstracts

34.80D

Total

_cross

sections for (e--He) ionization

K. K.

Mukhe(ee (I),

Keka Basu

Choudhury (2),

P. S. Mazumdar (3) ^and S.

Brajamani

(3)

(1) Novodaya

Vidyalaya, Khumbong, Imphal,

795113

Manipur,

India

(2)

Department

of

Physics, Jadavpur University, Jadavpur,

Calcutta, 700032 West

Bengal,

India (3)

Department

of

Physics, Manipur University, Canchipur, lmphal,

795003

Manipur,

India

(Received13 March 1990, accepted19

September

1990)

Rksumk. On calcule les sections efficaces totales d'ionisation par

impact klectronique

dans le domaine

d'knergie

40-150 eV et dans le moddle de l'onde dkforrrke, off l'on utilise des ondes dkforrrkes pour les Electrons incidents, diffusks et

kjectks.

Nos rksultats sont

comparks'aux

rksultats

expkrimentaux

ainsi

qu'aux

autres

prkdictions thkoriques.

Abstract. Total _cross sections for electron

impact

ionization have been calculated in the energy range 40-150eV in _a distorted _wave model which

employs

distorted _waves for the

incident,

scattered and

ejected

electrons. The present results _are compared with

experimental

results and other theoretical

predictions.

1. InUoducfion.

Ionization of helium atom

by

electron

impact

iS _{a common} process in natural

plasmas.

In

laboratory plasmas

it is

particularly

relevent to controlled thermonuclear fusion devices because helium is

produced

_{as a} result of D-T fusion reaction. Helium is also the

simplest

terrestrial _gas and is therefore _a natural candidate for _{use as} standard atomic collision

measurements. Even also from the theoretical

point

of view after atomic

hydrogen

it is _an ideal

system

^for the

study

of the electron

impact

ionization because of the

availability

of

accurate _wave function and absence of inner shell effects.

There exists _a number of

experimental

results for the total _cross section

(TCS)

for electron

impact

ionization of helium.

Recently

Shah _et al.

[I]

have measured the TCS for

(e--He)

ionization

using

_a

pulsed

crossed beam

technique.

Their results _are in very

good agreement

with the

experimental

results of

Montauge

_et al.

[2].

^{On the} theoretical side most of the recent calculations _on

(e--He)

ionization have been directed towards

obtaininj

the

triple

and double differential _cross sections

[3].

Bell and

Kingston [4]

Economides and McDowell

[5]

and Sloan

[6]

have used the first Born

approximation (FBA)

to obtain the TCS for

(e--He)

ionizations.

But below _an incident energy of 400eV the FBA results overestimate the,

experimental

results. Recent calculations

by

Bartchat and Burke

[7]

based _on the _use of the R-matrix

12 JOURNAL DE

PHYSIQUE

II M

method

provide

_cross sections

larger

than the measured values. The results obtained

by

Peach

[8] by using

the Ochkur

approximation

are in fair

agreement

with the

experimental

values whereas the

Born-Exchange (BE)

results of Peach

[8]

rather underestimate the measured values. The first distorted _wave calculations for TCS of

(e--He)

ionization _were

performed b§

Bransden _et al. [9] but their results overestimate the

experimental

results below _an incident energy of 250 eV.

Recently Campeanu

et al.

[10]

have carried out _an elaborate and consistent distorted _wave calculation for TCS of

(e--He)

ionization. Their results in the model DWE are

in very

good agreement

^{with the measured} values above the incident _energy of 150 eV but below 150 eV there is _some

divergence

between their results and the measured

values,

the

measured values

being

lower.

In the

present

_{paper we}

plan

to calculate the TCS for

(e--He)

ionization in the energy range 40-150 eV.

Campeanu

et al.

[10]

have

employed

_a full

screening

model. But in the low energy

region (40-150 eV)

under consideration the system

gets

^{ionized before the scattered}

(faster)

electron _can leave and therefore this electron like the

ejected (slower)

electron _sees the final state of the target

[I I].

So _we have assumed that both the

outgoing

electrons _move in the field of the residual He+ nucleus. We have also taken into account the effects of the final channel and target distortions which have been found to be

important

in electron-atom

scattering [12, 13].

2.

Theory.

The TCS for the ionization of _a helium atom

by

_an

unp,olarized

beam of electrons is

given

in terms of

singlet/triplet

direct and

exchange scattering iriplitudes by

Q

= ~

ii

^~~

^{ff g°}

^~ + 3

g'

^~

di~ dk~ (2.1)

(2

_{gr k~}

where _k~ is the momentum of the incident electron and k~ and k~ are the momenta of the scattered

(faster)

and the

ejected (slower)

electrons

respectivily.

_{The two}

oitgoing

electrons

obey,

the energy c6nservation relation

E,

=

E~

₊

E~

+

E~ (2.2)

where

Ei, E~

and

E~

are

respectively

the

energies

of the

incident,

scattered and

ejected

electrons. E~ ^is the _energy for the

single

ionization of helium.

f°

and

f'

_are

respectively

the

singlet

and

triplet

direct

scattering hmplitudes

and

g' is

_the

triplet exchange scattering hmplitude.

Following

Geltman

[14]

_{we can} write

f~

"

£ _Xk<(Zf,

~

) #il(~2,

_~

₃₎ (

+

) _#i1(~l,

~2,

~3)1(2.3)

>2

g~

= x~ (z~, r,

) ~/(r~/

_r

_~)

₊ ^~~

~i(r,,

_r~, r

)) (2.4)

2 'T ^~ h2 ~>3

,

where S

= 0 and S

=

I

stan( respectively

for the

singlet

and

triplet scattering amplitudes.

The

subscript

I refers to the incident electron and 2 and 3 to the helium electrons.

x~~(z~,

^r

j)

is the _wave function of the scattered electron. z~ is the effective

charge

_{as seen}

by

the scattered electron. As _we have

-already

mentioned

in

_{the scattered electron}

moves in the

field of the residual He+ nucleus _so that z~ = 1.

M I (e--He) IONIZATION CROSS SECTIONS 13

~i(r,,

_r2>

r~)

is the _wave function of the initial state of the electron-helium system.

Taking

into _account the effect of the

target

distortion it _can be written _as

[15]

#i1(~>, ~2,

~3)

"

l~(~>)(@ (~2)

_@

(~3)

_{+ X(~>,}

r2)

_@

(~3)

_{+ X(~>,}

~3)

_@

(~2)) (2.5)

where

F(r,)

is the _wave function of the incident electron.

X(r,, r~)

is the

dipole

_component of the distorted

part

^{of the} target wave function. It is

given by [15]

jj~

t(r,, r2)

_@>s

~p(~2)

~~ ~y ^y

(2.6)

X

(r,, r2)

" '~

~)

r2

~ ~

where

t(r,, r~)

=

~

_{~' ~ ~~}

(2.7)

0 r, < r~

and

R,~

~

~(r)

=

z('~

₂

z~ r~

+

r~)

^exp

^{(- z~ r) (2.8)}

with

z~ ^=1.598.

We have

ignored

the

target

^distortion in the calculation of the

exchange

contribution to the ionization

amplitude.

The

exchange

has been introduced _as if the atomic

charge

cloud _were not distorted whereas target distortion is included _as if

exchange

did not take

place.

Since the

distortion of the atomic

charge

cloud is

negligible

at small

distances,

where

exchange

is the most

important [16]

this choice is not

expected

to effect the results

significantly.

4~o(r~,

r~) is the helium

ground

state _wave function. We have used for it the

analytic

Hartree- Fock

(HF)

_wave function of

Byron

and Joachain

[17]

*o(r2, r3)

= R

(r2)

_R

(r3) (2.9)

with

iL

(r)

=

Ris(r) Yoo(?) R,~(r)

=

Njexp(- ar)

_{+ exp}

(- fir)1 (2.10)

where

N

=

2.6049754,

_a

= I.4I

,

p

= 2.61

, ⁼

0.799.

~/(r~,r~) _represents

_the final _state of the helium

subsystem. Following Phillips

and McDowell

[18]

_we have taken it _as

4rf(r2> r3)

=

) _{lx~(ze> r2) v(r3)}

+

(- ')~x~(ze> r3) v(r2)1 (2.")

where

V(r)

is the _wave function of the residual He+ ion

8 >j2

V(r)

=

exp(-

2

r) (2.12)

x~~(z~,

^r

)

is the _wave function of the

ejected

electron. z~ is the effective

charge

as seen

by

the slower electron.

JOURNAL DE PHYSIQUE II T ', M ', JAMVIER lW' 2

14 JOURNAL DE

PHYSIQUE

II lK°

As

already

^stated we have taken z~ ₌ I. It is to be noted the choice of the _wave function of

the helium

subsystem

in

~the singlet

state is not

orthogonal

to

~Po(r~,r~). We,

have

orthogonalised

the final state wave function in the

singlet

state

by replacing _x~~(z~,

_r in

equation (2.

^{I I}

) by

k~(Ze,~)"X~(Ze,~)~R(~)( jXi(Ze,r')X _JL(r')dr') ^(2.13)

The _wave function

F(r,)

of the incident electron is

decomposed

into

partial

_{waves as}

[15]

F(r, )

=

kj

^"~

~j (2 ij

_{+ I I}

~'exp (isf)

~~~~'~'~

x

Pi (cos (it, _{f, )) (2.14)}

f, 0

' ~> ^'

where

it

_{is the orbital}

_angular

_momentum

_quantum

_number of the incident electron. The radial

part

_Hi

(ki, r,)

of

F(r,)

satisfies the

intregro

differential

equation

,

d2

~ f~

(f~

+ I

m ⁺

ki

2 2 V>s

>s(~l)

2

Vpol(~l)

X ~LI

(ki, ~l)

~

Xi (~l)

_~>

R>s(~l) (2.15)

dry

_r, ' '

with

, ioJ

0

rj

f ri r, r'

= ~ 2.17

,( ⁾

,

+1

)

r~

Equation (2.15)

is to be solved

subject

to the usual

bounda~y conditions,

Hi

(ki, 0)

=

0,

_{~z I (k~,}

j) k/ "~sin k~

r

i,

_{gr +}

I

(2.18)

at,

is the

phase

shift. The

polarization

induced

by

the

incoming

electron is

only

taken into account

by

_means of the direct

polarization potential V~~j(r)

so that

(2.15) produces

elastic

scattering

solutions in the adiabatic

exchange approximation,

the effect of

exchange polarization

_terms

having

been

neglected.

This is consistent with the fact that in

calculating

the

exchange amplitude

_we have not considered the effect of

target

distortion. The direct

static

potential

_Vj~

&(r)

is

given by

'

Vis,_is

(r)

= + R

~s

(t

~

R_is

(t )) ^{(2 .19)}

and

V~~j(r) by

v~~j(r)

₌

(~j~(t)

~

x

(r,

t

~

(2.20)

lK°

(e--He)

IONIZATION _CROSS SECTIONS ₁₅

We have

decomposed _Xk~(z~,

_r

) (A

=

f _or

e)

into

partial

_{waves as}

4gr ^{~ ~~} _f

Gf~(k~,Z~,r)

Xk~

(Zl,

~

)

"

f £ £

^~

~

X Yf~ _~~

(f)

_Yi~

~~

(k

i eXp

(-

_I

lJ f~

(2.21)

1 f~ 0 m~ f~

where l~i~ is the

corresponding phase

shift.

i~

^and

i~

are

respectively

the orbital

atigular

momentum- quantum ^numbers of -the scattered and

ejected

electrons. The radial _wave functions

Gi~(kj, z~,r)

for both the

outgoing

electrons _are the

exchange polarization

functions for the

scattering

of electrons

by

He+ ion

(Sloan [19]):

3. Results and discussions.

The

major

trouble in the evaluation of

scattering amplitudes f~

_and

_{g~ (S}

=

0,1)

is

encountered in the evaluation of radial

integrals occurring

in them. The number of

partial

waves

required

to obtain convergence increases with energy

resulting

in _severe cancellation in

the _{sums over} the

angular

momenta. We have evaluated the radial

integrals

_{upto a} radial distance of 80 _a-u-

using

16

point Gauss-Legendre Quadrature by partitioning

the entire

integral

into _a number of

integrals (Sil [20]).

The radial parts of the _wave functions of the

incidint elictron

_{and two}

outgoing

electrons have been obtained

by solving thi corresponding integrodifferential equations by

Numerov method

(Sloan [19])

_{upto a} radial distance of

r =

30 _a-u- with _a

step

^size of 0.01a-u- For _r

~ 30 _a-u- the radial _wave functions have been

replaced by

their wellknown

asymptotic'forms (McDowell

_et al.

[21]).

The

integral

over

k~ in ^the

expression

for

Q

^has been carried _out

by using

8

fioint Gauss-Legendre Quidrqture.

Special

attention has been

given

_on the convergence of

Q

with respect to the

angular

momentum quantum number

it, _i~

and i~. ^The maximum value of

i~

has been taken _us 5. For

E~ =40eV the maximum value

(i~)~~~ of-it

is taken _as 10 whereas for

E;

=.150eV

(I,)~~~

+ 20.

i~

^{has been} obtained from the

triangle

rule

involving

_i~,

i~.

and

i~.

^{It has been} observed that

for

a

particular

value of _{i~ and}

i~

contributions to

Q

for

i~

_~ 5

correspond nearly

to the terms of "a

geometric

series. Therefore _the contributions of

partial

_waves

i~

_~ 5 _are

approximated by

the series. We have noted that at the lowest energy considered

(40 ev~

the

error incured due to this.

approximation-

is less than I fb. '

Table I presents ^the total _cross section

Q

for

(e--He)

ionization calculated

by using

the

preseit

_method

_together

_{with the FBA results of Bell and}

_{Kingston [4]}

_{and DWE results of}

Campeanu

et al.

[10].

In

figure

I _we have

plotted

the present ^results

together

with the FBA

[4]

and DWE results

[10]

and

the,experimental

results of Shah _et al.

[Ii.

It is evident from

figure

I that

althiugh

_the

prisent

method

provides improved

_agreement between the

Table I. Total _cross sections

(in

units

of'gra) for (e--He)

ionization.

Energy (ev~

FBA

[4]

DWE

[10]

Present

40 0.3210~ ^~ 0.2217 0.2164

60 0.5509 0.4189 0.3782

10 6.6159

_{0.4767 0.4445}

100

0.(158

_0.4750

I.4491

120 0.6050 0.4648 0.4393

150 0.5529 0.4408 0.4127

16 JOURNAL DE

PHYSIQUE

II M

A

i I _i

i ¹

i ii

OJ i

(

_I

I

I I 'I

I o

30 40 50 60 70 loo f30 f40 f50

E;(ev)

Fig.

1. Total _cross section

(Q)

for (e--He) ^ionization (in units of grao~. ^{A FBA} [4] ^B : DWE [10]

C _: Present results _; I

Experiment ii].

theoretical results and the,measured values there is still _some

discrepancy

between them. The situation is _some what similar to the electron

impact

ionization of atomic

hydrogen [22].

Further

improvement

_may await the

development

of methods which allow for _correct

asymptotic

form of the final _{state wave} function _as done

by

Brauner _et al.

[23, 24]

who have derived _an exact form of the

three-body

Coulomb _wave function in the

asymptotic region

for the electron

impact

ionization of atomic

hydrogen by using

a modification of the method of

Pluvinage [25, 26].

Acknowledgements.

The authors thank Prof. Dr. N. C. Sil

fir

his keen

iiterest

_{in the}

_problem.

_Thanks

are due to

Dr.

Devraj

for his

help.

One of

the_authors (KBC)

is thankful to Prof. Dr. R. M. Dreizler for valuable discussions with him

during

her stay ^{in the Institut fur} Theoretische

Physik,

Universitht Frankfurt and to Council of Scientific and Industrial Research

(CSIR),

New

Delhi,

India for _a financial assistance.

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