Theoretical study of low-energy positron scattering on alkaline-earth atoms in the relativistic polarized orbital approximation

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Theoretical study of low-energy positron scattering on alkaline-earth atoms in the relativistic polarized orbital

approximation

Radoslaw Szmytkowski

To cite this version:

Radoslaw Szmytkowski. Theoretical study of low-energy positron scattering on alkaline-earth atoms

in the relativistic polarized orbital approximation. Journal de Physique II, EDP Sciences, 1993, 3 (2),

pp.183-189. �10.1051/jp2:1993120�. �jpa-00247821�

Classification Physics Abstracts

34.808-34.90

Short Communication

Theoretical study of low-ener~y ^{positron scattering}

on

alkaline-earth atoms in the relativistic polarized orbital approximation

Radoslaw

Szmytkowski

Institute of Theoretical Physics and Astrophysics, University of Gda4sk, Wita Stwosza 57, 80-952 Gdailsk, Poland

(Received

6 October1992, accepted in final form 2

December1992)

Abstract. The elastic low-energy positron scattering _on Be, Mg, Ca, Sr, Ba and Ra atoms has been investigated at energies below 100 eV. The ab initio calculated polarization potentials applied in these calculations _were obtained by solving the coupled Dirac-Hartree-Fock equations.

Comparison between present results and those of previous model calculations shows serious discrepancies. Particularly, _we do _not predict existence of the Ramsauer-Townsend minima in total elastic _cross sections.

The

low-energy

elastic

scattering

of electrons from

light (Be, Mg

^and

Ca)

and

heavy (Sr,

Ba and

Ra)

alkaline-earth metals has attracted considerable interest in the past ^few years

[1-8].

However, only

_a few

investigations

have been made for the

scattering

of slow

positrons

_on

these targets. ^{Kurtz and Jordan} [9] carried _out model calculations _on elastic

e+-Be, _Mg

and Ca

scattering

for

positron energies varying

from 0 to eV. In turn Khare and co-workers

employed

the

optical

model

potential approach

in their studies _on

e+-Mg

_[10] and e+-Ca

[III scattering

in the

impact

_{energy ranges} 10-500 eV and 10-75

eV, respectively.

For

heavy

targets the

only

calculation _was

performed by

Jask61ski [12] ^{who obtained} s and _p-wave

phase

shifts for e+-Ra

scattering

in the static field

approximation.

So

far,

_no

experimental

data _on e+- alkaline-earth atom collisions have been

reported.

In this paper, we present results of the first ah initio

study

of elastic

positron scattering

from Be,

Mg, Ca, Sr,

Ba and Ra atoms in the energy

region

below 100 eV. The

approach employed

has been the relativistic

polarized

orbital

theory

formulated _some time ago [13] ^and

applied

to the elastic

scattering

of slow

positrons

from

Hg

_[14] and Zn and Cd

jib]

atoms.

In the relativistic

polarized

orbital

approximation

the

positron

scattered

elastically

_{on a} closed-shell _atom is treated _as the Dirac

particle moving

in

a central field and

a system ^of two

184 JOURNAL DE PHYSIQUE II N°2

radial differential

equations describing

the

projectile

has _a form CA

~

()~~

^{+ ch}

)P(r)

+

(-mc~

_{E +}

v(r)) Q(r)

= 0,

~~~

CA

~()~~

^{+ ch}

)Q(r)

⁺

^(-mc~

^{+ E}

^{v(r)) P(r)}

= 0, where E is the total _energy of the

positron (including

its rest energy

mc~)

and _~t

=

(2j +1) (L j),

while

j

and L have their usual

meanings

[16]. ^The

spherically symmetric potential

V

consists of static

(Vs)

and

polarization (VP)

parts

V(r)

₌

Vs(r)

+

VP(r). (2)

In _our calculations the static

potential

Vs has been obtained in _a standard _way [13]

using

the radial atomic orbitals

generated

ⁱⁿ a

single-configuration approximation by

the MCDF code of Grant _et al. [17]. ^{The feature which}

distinguishes

the relativistic

polarized

orbital method from many other

approaches

is the fact that in this

approximation

the

polarization potential

VP is

calculated in _a

fully

ah initio _manner

by solving

the

coupled

Dirac-Hartree-Fock

equations

[13, 14]. ^This very laborious work has been done with the aid of _a program POLAR

[18].

Only

the

dipole

term in the

polar12ation potential

has been used in present calculations. The

dropping

of the other

multipole

terms is

justified by

the fact that in the relativistic

polar12ed

orbital

approximation

the non-adiabatic

dynamic

distortion effects _are not taken into account.

Far from the atom the

scattering potential

should be of the form

where the coefficients _ak and

6flk-1

_are both

positive

and

comparable

in

magnitude. Therefore, neglecting

the

dynamic

distortion effects

(I.e. putting flk-1

"

0)

we were forced to

drop

out

other than

dipole

contributions _to the

polar12ation potential.

Once the

scattering potential

V is

known,

the

scattering phase

shifts b~ can be calculated and the relativistic version of the variable

phase

method has been used for this _purpose. In this

method _one introduces _a so-called

phase

function

b~(r)

which satisfies _a first-order nonlinear differential

equation

~~~~~~

=

-J~~ ~(~ _(jL ^(fir)

cos

b~(r) fiL(I(r)

_{sin b~}

(r)j

~

dr _c

(4)

~f( ²

-1. _ch^~

(jL+i(I<r)

^cos

^{b~(r) fiL+i (fir)}

^sin

b~(r)j

with _an initial condition

b~(0)

₌ 0.

(5)

Here

~~~

~

+

~~

~~~

and

1<~

=

(E _mc~) (E

₊

_mc~) /(ch)~, (7)

while

jL (fir)

and fiL

(fir)

_are the Riccati-Bessel functions. In the

equation (4)

^the _upper

sign (-)

should be taken for _{~t >} 0 and the lower _one

(+)

for _{K <} 0. Then the

phase

shift b~ can

be obtained from the relation

b~ ₌ lim

b~(r). (8)

r-co

and the total elastic _cross section

QT

_can be calculated via

expression

The variable

phase approach

offers also _a method of calculation of _a

scattering length

_a which is defined _as

~

iii ^~~~

~~~~

In this method the

scattering length

is calculated

by solving

the

equation

_[14]

with _an initial condition

b(0)

₌ ⁰

(12)

and then

using

_a formula

a ₌ ao tan

b(cc), (13)

where _ao denotes the Bohr radius.

The calculations of the

polarization potentials yield

_{as a}

by-product

the electric

polariz-

abilities of the targets. ^{In table I} we present ^the

dipole polarizabilities

obtained both in the relativistic and nonrelativistic _cases. As could be

expected,

the relativistic effects in target

structures are almost

negligible

for the three

lighter

atoms while it is evident that

they play

a

significant

role for heavier targets. ^{In table} I _we also list relativistic results obtained in the _same

approximation by

Kolb et al. [19]. ^{An excellent} agreement between the two sets of

relativistic data confirms correctness and _accuracy of the _program POLAR.

Table I.- Relativistic and nonrelativistic

dipole polarizabilities

for alkaline-earth atoms

(in a().

nonrelativistic relativistic

Atom present present Kolb et al [191

Be 45. 62 45. 59 45. 6

Mg ^{81.59 81. 16 81. 2}

Ca 185.5 182.8 182.8

Sr 246.0 232. 6 232. 6

Ba 368.2 324.0 324.0

Ra 441. 5 296. 2

In table II _we present ^{calculated values} of the

positron scattering lengths.

Their

large positive

values

signify

that

scattering potentials

_can support

weakly

bound states with

binding energies

Eb

fi2 Eb _cr

~~~~

(14)

18fi JOURNAL DE PHYSIQUE II N°2

Table II. Positron

scattering lengths

a

(in ao)

and

binding energies

Eb

(in eV)

for alkaline- earth atoms.

Present Kurtz and Jordan [91

Atom

a E a E

Be 67.6 0. 003 26. 1 0.02

Mg ^{30. 1} 0.02 58. 3 0. 004

Ca 14.3 0.07 82. 5 0.002

Sr 12.7 0.08

Ba 8.6 0.2

Ra lo. 8 0.1

Ol O

l$

©

.I e

~

o oJ 0J

0J O

~

Q _.

O

.~

4~

~j0J

~-

Q~ .

~- f3

4~

E-O o.oi

D1

td~

©

,fl

$ e~/Mg

t~

0J 0J O

~ _.

Q .

O

.~ ~~

4~

0J . .

n~ . . .

~-

tJ

~-ol

+J O ~

a u

E- io~

o,oi o-i i io ioo

Energy (eV)

Fig. 2. Total _cross section for elastic scattering of positrons _on magnesium:

(-),

present;

(.),

Kurtz and Jordan [9]; (Q), Khare et al. [10].

"

o

~

io~

e~/Ca

$

.~

t~

0~

0~ _.

O .. ., .

b

~

. D

~

.~

0~4J 1$

$~

~_ ^D

1$ ~

dJ

O E- 10~

0.Ol

Energy (eV)

Fig.

3. Total _cross section for elastic scattering of positrons _on calcium:

(-),

present;

(.),

Kurtz and Jordan [9];

(o),

Khare et al. [I

Ii.

188 JOURNAL DE PHYSIQUE II N°2

/~~

"O

~/~',

gJ ,

~~

©

O

3 ,

I "',

"

. ,

g~ ,

g~ _,

o

.,,

~ ',

°

fl

^~'

',~

j

__-_~~

T

".

~- ".

l~ ",

+J O

E-

10'

Energy (eV)

Fig. 4. Total _cross section for elastic scattering of positrons _on strontium, barium and radium:

(-),

Sr, present; (-

-),

Ba, _present;

(...

), Ra, present.

calculations of Kurtz and Jordan

predict

the existence of the very shallow Ramsauer-type minima for

Mg

^{and Ca} at

energies

0.6 and I-b

eV, respectively,

while _{our curves} decrease

monotonically

in this energy

region.

At

higher energies

differences between present ^results and those of Khare et al. _are not _so

large

but still

significant.

Our results for

Sr,

the Ba and Ra

are

plotted

in

figure

4. For radium the results obtained

using

the

phase

shifts of Jask61ski [12]

are not shown because of the

unreliability

of the

approximation

used in his calculations. No other

investigations

are available for

comparison

for these three targets.

We also carried _out

a number of calculations of differential elastic _cross sections at various

energies

but in view of the lack of relevant

experimental

data _we do not present ^{results here. The} numerical values of the

phase shifts, differential,

total elastic and elastic momentum transfer

cross sections _are available from the author _upon request.

Acknowledgements.

am indebted to Professor C2.

Szmytkowski

for critical

reading

of the

manuscript.

This work

was

supported by

the Komitet Badafi

Naukowych

under

grant PB/916/P3/92/03.

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