Stopping power of protons in He at low energies

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Stopping power of protons in He at low energies

Y. Boudouma, A. Chami, H. Beaumevieille, M. Boudjema

To cite this version:

Y. Boudouma, A. Chami, H. Beaumevieille, M. Boudjema. Stopping power of protons in He at low energies. Journal de Physique I, EDP Sciences, 1992, 2 (9), pp.1803-1810. �10.1051/jp1:1992246�.

�jpa-00246661�

Classification

Physics

Abstracts

61.80M 79.20R

Stopping _power of protons ^{in He at low} energies

Y.

Boudouma,

A. C.

Chami,

H. Beaumevieille and M.

Boudjema

Institut de

Physique,

U.S.T.H.B., B-P. 32, El-Alia, Bab Ezzouar,

Alger, Algeria

(Received 30 December 1991, revised 8

April

1992, accepted 1 June J992)

Abstract.

Using

the

Scattering Approach

for

Stopping

in _an Elecdon Gas, the Local

Density Approximation

(LDA), _an adiabatic criterion, ^and capture ^{and loss} condibutions to the elecdonic energy loss we have calculated the elecdonic

stopping

_power of protons ⁱⁿ helium and

hydrogen

gases at low

energies.

The combined model used

give

a

significant

deviation from the well known

velocity proportionality

of the

stopping

_power at very low energy in the two cases. Our calculations

explain

quite ^{well this} striking deviation

reported

in recent papers. A good agreement ^{is obtained} with

experimental

values in _case of He.

Introduction.

A variation of the

stopping

_cross section

proportional

to the

projectile velocity

in the

region

of low velocities _was

predicted by

theories

conceming

the electronic energy loss of

heavy

ions _on

atoms

developed by

Firsov

[I]

and Lindhard and Scharff

[2].

Such _a

dependence

is also obtained when the target ^is a

degenerate

free electron gas

[3-5].

This behavior is also used in

empirical

formulae deduced from the

compilation

of several

experimental

data

[6-8].

However,

in this energy range

(4-20keV

for

protons),

the _cross sections _cannot be considered _to be final. This is due _to the

large

uncertainties associated with the target

preparation

and to the energy measurement of scattered or transmitted

particles.

Some authors

[9]

^noted

that,

in _{some cases,} the

stopping

_cross section values

brought

in

by

Andersen and

Ziegler [6]

_{seem too}

high

at low energy. This _was confirmed

by

the recent work of Golser and Semrad

[10]

who observed _a

Significant

deviation from

velocity proportionality

in the

Stopping

cross sections of protons ^with energy lower than 20 kev in helium gas.

In the

following,

_we

study

the proton

slowing

down in _a free electron gas. The

generalization

_to the atomic system ^is obtained

through

the local

density approximation

LDA. An

application

is afterwards

given

for the protons

slowing

down in helium target ^and

compared

to Golser and Semrad results

[10].

Likewise _{a test} for H ₊ H system ^is

presented.

We _use, unless otherwise noted, the Hartree atomic units

(e

=

fi

= m ₌

1).

1804 JOURNAL DE PHYSIQUE I N° 9

Theoretical

approach.

The linear response

theory,

based

mainly

on the Bom

approximation,

underestimates the

increase of the

density

around _a

positive

Static

point charge placed

in _an electron gas

[I I].

This

limiting

_case leads _{uS to} think that when the

particle, undergoing

the

Slowing,

_moves with low

velocity,

the

Screening

effect iS underestimated which leads to incorrect

Stopping

_power.

Therefore the

hypothesis

of linear response iS _not valid. It becomes necessary ^to use a non

linear

approach

of the response of electron gas.

Moreover, complied

to _a low

perturbation

from _an ion of low

velocity,

_Some electrons of the electron gas

adapt

their motion _to this

perturbation.

An alternative to these _two

approaches

is constituted

by

the Ferrell and Ritchie

theory [12].

Studies in this _sense have been conducted

by

different

authors,

either in the framework of the elastic

scattering

of the electrons _on the screened

potentiel

induced

by

the host

particle (Binary

Encounter

Approximation BEA) [12-13],

and in the framework of the

Density

Functional

approach

DFA

[14].

These studies lead _{to cross} sections which _are

slightly

different from those obtained

by

the

linear

theory. However,

to our

knowledge,

these calculations have _not been extended _to

atomic Systems.

In this

work,

we use the BEA collisional

approach.

The basic formula to evaluate

Stopping

cross sections can be deduced

Simply

from kinematical considerations

[15].

We

briefly

recall

the

grounds.

Let _a proton ^Of maSS

m~

^and

velocity

_v makes _a collision with _an electron Of

velocity

v' which

belongs

to an electron gas of

density

_n.

In the reference frame connected to the

moving _proton,

which is here the center Of the _maSS System, w and w' _are the velocities Of _an electron before and after the collision

respectively.

The proton ^{and electron} masses

being

_very

different,

_we have

(w(

_m

iw'i

For _a

Scattering angle

o and

taking

into account the axial Symmetry ^{around the direction} defined

by

_w, the value of the momentum transfer

averaged

over the azimuth is

(6p)

=

w(I

_cos

o). (1)

Let

f(v'),

be the

velocity density

in

phase

_space ^{Of the} electron gas. We have

If (v')

^dv'

= n

(2)

v>

The number of scattered electrons

during

_a time interval At is

(W( f(V')dV'dW(W, o)

At

where d« is the collisional differential _cross section.

The momentum transfer for these electrons will be :

(6p)

₌

f (v') dv'(

_w At w

ld« (w,

o

) (

I _cos o

) (3)

v, e

Following

the

collisions, only

the electrons that _can go

beyond

the Fermi level _can be excited

their velocities must then

satisfy,

_{at zero}

temperature,

v~-v<

(w(

_<v~+v, where v~ ^{is the Fermi}

velocity

of the electrons gas. The _mean

slowing

force _or the energy loss per

unit

length undergone by

the

heavy projectile

is _:

~

lf(

')

~

'j

^'

(

^' ~

@)(j

@)

(~)

$

^{~ ~ ~ ~ ~ ~ ~ ~' ~~~}

~ V'

~

using

relation

(2),

this

expression

becomes

~

= vv~ n«~

(v~)

in the _{case v « v}

~

(5. I)

dr

~

=

v~ _n«~

(v )

in the _{case v} » _v

~

(5.2)

dr

where «~ =

ld« (I

_cos

o)

is the

transport

cross section.

With _a

good approximation,

these _two relations _can be unified into _one which presents ^the

advantage

to be

applicable

for any

velocity [15-16].

~

= vv~

n«~~(v~) (6)

dr

where v~ is the _mean relative

velocity

between the proton ^{and the} target electrons of the gas

[17].

3 ~ _~ ² _~

1

4

~~~

~ 3 v~ 15 v~

~ ~ ~~

~~'~~

V~(V, VF) =

11

+

~~

j

v m v~.

(7.2)

5 _V

The transport cross section «~ can be Obtained in the framework of _a non-linear response

by studying

the

scattering

of electron gas

by

a

potential

V

(r).

«~(v~)

=

[ ( _(I

+

I) sin~ 18j(v~) 8j~ j(v~)1. (8)

k~ j=o

Where

8j

is the

phase

shift of _wave I

computed

at

velocity

_v~.

A low energy

hydrogen

beam

traversing

_any

stopping

material _can be considered _{as a} three

charge

states system constituted

by H+, H°

and H- with constant fractions

(at equilibrium) 4+, 4°

_and

_4~.

These

equilibrium

fractions _were

already

examined in detail

[18]

and it appears, that in the

case of helium gas,

4

is very small

(less

than I fb in _our incident energy

range).

Furthermore

the energy loss

involving

the three

components

^have a same Order of

magnitude.

This

consideration leads _{us to}

neglect

the H- contribution.

The energy loss

including

_capture and loss processes

(CL)

_can then be written _:

dW ₌ 4 ⁺

[(1-

_{« ~o)}

dEjj

+

«jodejo]

+

~1°[(1 _~xoj) dEoo

+ «oj

deo~]. (9)

Where _:

dEj~ (resp. dEoo)

is the energy loss in helium gas

computed

^for H+

(resp. H°)

without C.L. _processes.

1806 JOURNAL DE

PHYSIQUE

I N° 9

deja (resp. deoj)

_represents the energy loss

occurring during

_capture

by

H+

(resp.

ionization of

H°).

From the ionization

potentials

of H and He _{we assume} that

dejo (resp, deoj)

is very close to lo eV

(resp.

13.6

eV).

The

equilibrium

fractions 4 ⁺ and

4

^° _are calculated

by using

the capture ^{and ionization}

cross sections

(«jo

and

«oj) reported by

Bamett

[18].

«jo

(resp,

_~xoj) is the capture

(resp. loss) probability.

All these

quantities

_are normalized _to the _same target thickness

lfl~at/cm~

_{in order to}

obtain total electronic

stopping

_cross section

~~~'.

N dR

The

computation

of

~~~

^and

~°°

is made

through

the formula

(6),

the main

problem

is

r r

therefore the choice of

scattering potential.

As

long

_as the incident proton ^have a

velocity

^low

compared

to target

electrons,

_{we can} treat its interaction with the

stopping

medium

through

a

screened

potential. Thus,

we choose the elastic

scattering potential

derived

by Apagyi

and

Nagy [13]

:

Vsc(r)

₌ e~ ^~~ +

"

(10)

r 2

~x

(r~)

= no + ~xj

e~~"

_{+ ~x~}

e~~~'(I

< r~

<15) (11)

where _am

~x(r~)

is the

screening

_parameter which

depends

on the electronic

spacing

parameter, r~ =

(3/4 wn)~'~,

introduced

by

the authors to _assure the

self-consistency,

_or in other words _to

satisfy

the Friedel _sum rule.

We _use the _same

potential

with _«

=

2 in

equation (9)

in order to obtain the

scattering potential V~

relative to

hydrogen projectile.

We

adopt

the

Calogero

method

[19]

for the

computation

of

phase shifts,

_put in

practice by

Ferrell and Ritchie

[12],

the

generalization

to

an atomic system constituted

by

_a

slowing

atom is made

through

the local

density

approximation

_:

~

=

14 ^{wr~~ (r)}

^dr

⁽¹²⁾

N dR dr

where ~

(r)

^is function of _n

(r),

the electronic

density,

derived from the _wave functions _as dr

classified

by

Herman and Skillman

[20].

Lindhard and Scharff

[21]

have shown that the LDA

gives practically

the _same results when

applied

to _an

independent particle

model of the _{atom as} well _{as to} its classical orbital

picture

with _resonance

frequencies.

Within the classical treatment of the atom, in the 4-20 kev energy range, the

projectile velocity

is lower than the _mean orbital

velocity

of the helium _atom electrons which is

approximately

1.6 _a.u. in the Thomas-Fermi model. Hence the

target

^electrons can

adjust

their motion to that of the

projectile

and

give

rise to a

polarization

force that

compensates

^the force exerted

by

the

projectile

and thus _a

negligible

_energy loss _: that is the so-called Bohr criterion

[22].

Then _we consider that the

projectile

acts

mainly

_{as a} screened proton ^{rather than} bare ion. In

fact,

the LDA allows to take into _account of the energy loss due to the outermost

electrons in the

following

_{way : we} divide the electronic

density

n

(r),

into

elementary portions

and _we

study

the gas response in each of them. We

postulate

_a cut-off criterion

analogous

to the Bohr's _one and decide that those

portions

with v~ greater ^{than the}

projectile velocity

_v do not contribute to the

slowing

down process. Such _a condition is

equivalent

to a fair _« shell correction _» introduced in _a

gradual

_way. The introduction of this criterion in

equation (6)

has

as consequence ^the non consideration of

large impact

_parameter collisions. The collective excitations _are then

neglected.

This

approximation

is

justified

because _our

computations

consider isolated atoms;

therefore the energy transfer

phenomena by

collective collisions is less

important

than in condensed _matter.

To test this

method,

_we have also carried _out

computations

of

stopping

_power for

hydrogen

in

hydrogen

_up to 15 kev. This is

highly justified

since in this energy range the capture process is dominant and protons ^behave

mainly

_as

hydrogen

atoms.

Results and discussion.

The protons

stopping

_power in He has been

computed

from 4 to 40 kev

by treating

the target

as electron gas and

by using

^{the formalism}

developed previously,

^{for the} different

potentials Vsc

and

V~.

Our theoretical

predictions (cf. Fig. I)

show _a

good

agreement ^{with the}

experimental

data of Golser and Semrad

[10]

and

Phillips [23],

_as well as the theoretical calculations of Schiwietz

[24]

obtained above lo kev within the frame of _a

coupled-channel theory.

They

are

drastically

different from the

semi-empirical

values classified

by

Andersen and

Ziegler [6]

and _more

recently by Ziegler,

Biersack and Littmark

[7].

The

following

table shows the values of the

p

_{exponent as}

reported by

different authors

using e(v )

~

VP which links here the

velocity dependence

of the

stopping

_power between 4

and lo kev.

ml ⁰

fi

^{, , ,} ,

~ ^A

'

u A

'

-J 4

tl ~

~ ^''

ul

l'~'

_O

b

~

+w

,

^{TRIM 90}

~

^OJIIC~ COLSER-SEJfRA

~ ANBERSEN-ZIECLEII

~

i

AA AA A L#'

... pHILLIPS

Q$ O

OUR CALCULATIONS

'tl ' iZ

'tl

4 6 8 z 4 6 8

0 00

ENERGY (key)

Fig.

I.

Stopping

_cross section of protons ⁱⁿ He _are shown _some

experimental

results (o) [lo], (.) [23], recommended

empirical

values (- -) [6], (- -) [7], L-W- theoretical

predictions

(A ) [5]. The full line, inferred from

polynomial

fit, is obtained in

our combined model

through

the

use of

Vsc,

VH, and (CL) _processes (see text).

1808 JOURNAL DE PHYSIQUE I N° 9

References

p

Andersen-Ziegler [6]

TRIM 90

[7]

0.5

Golser-Semrad

(experimental) [10]

3.34-2.23

Semrad

(theory) [25]

4

Our

predictions (theory)

2.20

The

experimental

data of Golser and Semrad include the energy loss

by

nuclear collisions. As

evaluated

by

the LSS

theory [26],

the nuclear

stopping

_cross section is

equal

to

0, I _x 10~ ^~~

eV.cm~

_{for protons of 5 kev and becomes}

rapidly negligible beyond

this energy.

In

figure

I the results _are

reported.

The dashed lines represent ^{the well known semi-}

empirical

fits of Andersen and

Ziegler

and

Ziegler,

Biersack and Littmark.

We _note

large discrepancies

between _our

predictions

and those of the Lindhard and Winther

theory (LW) [5]

_as well _as with the fits

[6-7].

In the _case of

hydrogen

_gas target, our calculations

(cf. Fig. 2)

show _a linear

dependence

with

hydrogen projectile

_energy whatever the

potential used,

this is

roughly

the case of the

recommended values of

Phelps [27].

The results _are somewhat different from

Phillips

values

[23] (the

values in

Fig.

2 _are

reported

under the

assumption

of linear

additivity

of

stopping

cross

sections)

the

Phillips

values _are, in _our

knowledge,

the

only existing

_ones derived

from classical energy loss

experiments

in the low energy

region.

However, the electronic

density

used here is that of free atom, while the

experimental

measurements deal with

H~

target gas. Hence, a

quite

different electronic distribution is needed at this stage. ^Moreover we

j

~

~

~

~ ^~ ,,'''''''

'~ ,,'''

~ _.,,"

-

-x'

Q ^.

,,'

-

~ ^'

++ . ,'

_

~ ^'

~ ,,''

/ '

~ ,,''

U _

~_,~,

^{,, -}

0$ fi ^'~

^~'

_{~~~ ~(~~~~} ^{~~~~~ ~c}

73

^' _{... PIIELP.S'}

~

_xxxxx pjIILLII'S'

$z

ANDER,ih'N-ZIECLeR

7j

z 4 6 ~ z

10 ENERGY Ike()

Fig.

2.

-Stopping

cross section of

hydrogen

in

hydrogen.

Test of the combined model with two

scattering potentials

_VH and

Vsc

are shown

experimental

results (x) of protons [23] and

hydrogen

(.) [27] in H2 (linear

additivity

of

stopping

is assumed here), and recommended

empirical

fit (- -) [6].

think that _a refinement of the

potential

and

probably

the _manner that electrons enter the

slowing

down process, could enable _us to take _account of the

experimental

results.

The

potentials

used _to

compute

^the electronical energy loss ask for _some commentaries.

The

Vsc potential

is

derived,

under the

assumption

that the Friedel _sum rule must be

verified.

However,

since _{we are} concemed

by

outermost electrons

through

the

LDA,

the

electron response ^is no

longer

adiabatic in this _case. The constraint linked to sum rule is then lifted.

We have,

through

the choice of

V~, implicitly neglected

the

eventuality

of interaction of

hydrogen projectile

in excited state, this _seem reasonable since in gases of low

density

the excited states can

decay

to the

ground

_one between consecutive collisions

[24].

We have also tested instead of

Vsc (Eqs. (10)

and

(11)),

_a Yukawa type

potential

~- ^xr V

y

(r)

=

with « =

1.563/(r~)~'~

^and

Vsc

^with _{a =}

I,

the results _are

roughly equivalent

r

with _{a more}

pronounced slope

for

stopping

cross section in the very low energy range. Since the present ^{work deals with the}

possibility

to describe _a deviation from allowable

linearity

of

stopping

with _v, it is _not very

important

at this stage to

reproduce

the _exact

experimental results,

but if Golser's and Semrad's _{ones are} to be confirmed in the

future,

it is

always possible by

_means of

improved screening parameter

^{of used}

potentials

to take into

consideration

experimental

measurements.

Conclusion.

We have used the kinetic

theory

of

stopping,

the

scattering

of electrons _waves and the local

density approximation

to

compute

^{the electronic}

stopping

_cross sections of protons ^{in helium} and

hydrogen

in

hydrogen.

The capture ^{and loss} processes are taken into _account

through

semi

empirical corresponding

_cross sections.

The

slowing

atom is

computed

in the LDA which allowed _us to introduce _an adiabatic criteria similar to Bohr's _one. Thus _we exclude from

slowing,

the electrons which _are able to

react

adiabatically

_to the

acting projectile

force. This appears as a convenient method _to

generalize

the orbital _as well _{as «} shell correction _»

concepts.

The

potentials

involved in

phases

shift calculations _{seem to}

correctly

describe the _mean interaction between the

projectile

and the

target

atom but _are

likely

to be

improved,

thus

giving

results much closer to

experimental

ones.

In the _case of

He,

_our results _are in

good

agreement ^{with the data of Golser and Semrad and} those of

Phillips.

The

striking

deviation from

velocity proportionality

showed

by

the _measures in the lower energy

region

is

correctly explained.

As for the

hydrogen target

^several

problems

are interelated

(validity

of

Bragg

rule, real electronic

density

of molecules and absence of recent

stopping measurements),

however in

spite

of

discrepancies,

our

predictions

show _a behavior very similar _to

Phelps's

data obtained in indirect _manner.

The electronic

charge

distribution of

slowing

atoms is of

importance

in _our model, thus in this later _{case a more} realistic

description

of it is needed.

It is

important

to note that whatever the

scattering potential used,

_our model exhibits a

non-proportionality

of the

stopping

_power to the

velocity

_v of the incident ion. This is in

good

agreement ^{with the} recent measurements

performed directly

_or

indirectly

at low

energies.

The results in _our combined model offer _a

challenging problem

for further

investigations.

Acknowledgments.

The authors _are indebted _to Dr. A.

Bordenave-Montesquieu

for his

appreciable help.

1810 JOURNAL DE PHYSIQUE I N° 9

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Phys.

Rev. 72 (1947) 399.

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Stopping

and

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Ziegler

Eds. 3

(Pergamon

Press, New York, 1977).

[7] _« The

Stopping

and

Ranges

of Ions in Solids _», J. F.

Ziegler,

J. P. Biersack and U. Littmark Eds. I

(Pergamon

Press, Oxford, 1985) and earlier versions of TRIM.

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Phys.

Rev. Lett. 66 (1991) 1831.

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Colloq.

France 38 (1987) C 20, 1465.

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