Positron trapping in Au

~aierials Chemistry and Physics, 25 (1990) 523-529

POSITRON

TRAPPING

523

IN Au

A. BELAIDI Ecole

normale

supirieure

H. AOURAG* Laboratoire

E. N- s. E- T- d’oran

de physique,

and 8. KHELIFA

, Departement

d’optique

3 1 100 Es-senia Received

, Departement

( Algeria )

3 1000 Oran

de physique , Universite

d’Oran

Es-s&a

( Algeria )

February

2,

1990 ; accepted

March

12,

1990

ABSTRACT Trapping both

of positrons

the density

three-state of Au.

each

state

model

An attempt and their

divacancy

defects

and the structure

trapping

case

by lattice

i.e.

dependence

contribution

of defects.

free

is made

has proved

to be a powerful

In this work,

( bulk I, monovacancy

to calculate on both

tool in probing

we are concerned

and divacancy

the fractions

of positrons

in the

annihilating

time

and sample

temperature.

rate

is observed

for temperatures

to the annihilation

with a

states

in

A significant near

melting.

INTRODUCTION Positron

physics

The existence more

than

is concerned

of the positron

50 years

during

the last

unique

information

of defects

ago.

were

and momentum

of particular

time

from

1Opm to lmm

(5 1 ps f . After

rapidly

a further

medium

useful

about

information

The application of solids

* permanent

address

technique

density

that

in condensed

/ [ 11-131

with matter.

by Anderson

[21

has known a rapid growth

lies in observations

of problems

positrons

and verified

by means

positrons

matter

can provide

physics.

of positron

is injected

it penetrates

all of its ioitiai

longer

(??r 100 ps I, it annihilates

period

with the emission

the electronic

system

annihilation

into a standard

: Institut

into a medium,

losing almost

of the positron

has developed

[l]

Studies

annihilation

interest.

a source

of the surrounding

of low-energy

by Dirac

annihilation

This growth

on a wide variety

When a positron of order

The positron

two decades.

[3-IO]

technique

with the interaction was predicted

of high energy to the outside method

Universite

within

a very short

with an electron

gamma

rays which

convery

world.

to investigate

tool of solid state

d’electronique,

energy

to a depth

science,

defect capable

de Be1 Abbes,

0 Elsevier Sequoia/Printed

properties of yielding

22000 SBA ( Algeria )

in The Netheriands

524

unique

information

condensed

The sensitivity

a firm experimental as follows: positive

ions because

becomes

void,

the positron

can relax

state.

will be reduced

. The tendency charge.

therein

Around

a lattice

The localization

of positron

of positrons

1) The concentration

energy.

energy

from

when some hole or

seen

by the

of the core

annihilation

rate.

electrons

In solids,

by the presence

has two important

can be deduced

Thus, or other

density

density

affected

by the

into intersti-

advantage.

the positrons

in the core

are strongly

can be understood

to interstitials

dislocation,

the electron

defect,

at the defects

of defects

reflects

decrease

states

state

in

in solids has

of the positron

vacancy,

with a considerable

in any state

with a corresponding

and variety

repelled

to the ground

of defects

of defects

of such trapping

This squeezing

of a lattice

structure

types

is strongly

contribution

rate

and internal

to various

in a metal

in the form

annihilation

in that

number

a positive

available

The positron positron

positron

of its positive

provides

space

of positrons

[ 3-10,141

basis

a thermalized

tial regions

configuration

of concentration,

matter.

the

of defects.

consequences:

the ratio

of trapped

and free

.

positron

2) The annihilation

characteristics

giving

information

then

unique

Positron peratures

annihilation close

different,

in divacancies

positrons

internal is usually

(Tm ) , However,

to melting

wherein

of trapped on their

the contribution

role at high temperatures

(~0.9

reflect

local properties

electronic

structure.

neglected

in most

for Au the situation

of divacancies

metals

even

seems,

to the annihilation

of defects,

for tem-

to some

rate

extent,

may play a

Tm ).

CALCULATIONS When some a chemical form.

deviation

impurity,

particule

density

assume

that

problem

the positrons

different

states

t (t>O)

then,

i at time clearly

and if the positrons transitions write

rates

:

dni ( t 1 -+ dt

e i=l

wavefunction

Thus in highly

of a variety

theoretical

defected

t = 0. n.(O) ’

into a system

=- no.

can make transitions

between

Kij ( Kij + K.. ; Kii t 0 ) which J’

r j=i

from

and rapidly

systems

there

annihilation

the various

are comparable

states

is

We

into N of

in state

in the i-th

the

rates,

terms.

thermalize

population rate

or even

a delocalized

phenomenological

the positron

ff the annihilation

dislocation

or in the extreme

and associated

in purely

If ni( t ) denotes

strongly

the impurity or disordered

states

can be formulated

are injected

such a vacancy,

may depart around

of positron

N (q+

is introduced

may be enhanced

may be localized. the possibility

The basic

periodicity

the positron

The positron

always

from

state

i at time is 4

at time-independent

with the

4

then

we may

N Kij ) ni (t ) = r j=i

Kji ni (t )

(1)

At tirne

n(t)

t the positron

=?

population

still

is :

surviving

ni(t)

(2)

i and the lifetime

is given

_

by :

N -n(t) where

= I(t)

=

both Ii and

Ti are complicated

from eqn. (1) .An analytical Hence , some of ( 1 ) arise trapping

model

( labelled

to other

to be negligible.

states

to make

and some

difficult

are assumed in which

at rates

K

‘1 becomes:

from

they which

the escape

when N is small.

tractable.

Useful

For example,

to thermalize can either

may be determined

even

the problem

of the Kij are zero.

I , say)

Then eqn. (1)

of b; and K.. which ‘I

to ( 1 ) is always

is necessary

[ 141 all the positrons

state

transitions

functions

solution

simplification

when N is small

delocalized

(3)

$exp(-Tit)

‘L i=l

versions

in the simple

at t = 0 into a common

annihilate

at rate

probability

(K.

1’

6, or make

) is assumed

N dn, (t) +

‘q+

Klj)

t

n,(t)

= 0

j-1

dt

(4) dnj (t)

-

+ bjnj(t)

= 0

dt The boundary

conditions

del (N = 3))

eqn. (4)

are now nj ( 0 ) = qj

no.

In the case

of three-state

trapping

mo-

yields:

dn, (t)

-

+ ‘q

+ K12

+ K,3)

n,(t)

= 0

dt dn2 ( t ) -

+ S2n2(t)

= K,2n,(t)

+ 63n3(t)

= K,3n,(t)

(5)

dt dn3 ( t 1 dt where

the subscripts

states,

respectively.

I , 2 and 3 refer The solution

of the decoupled

by :

n,(t)

= no exp [-(a,

+ K,2

to the bulk ( matrix ) , monovacancy

+ K,3)tj

set of differential

and divacancy

equations

is given

526 “0 K12 n2(t)

= -

exp[-(6,

+ K,2

+ K,3)tl

- expW2t)I

(6)

62 “0 Ki3 n,(t)

= -

exp [- (6,

+ K ,2 = K,3)tl

- exp(U3t)]

?3 and the positron

population

at time

t :

(7)

where

b, , e2, Q3 are the annihilation

cancy

states,

vacancy

divacancy

the concentration K K

12 = Klv

,

of traps

matrix,

respectively.

monovacancy

rates

The are usually

from

assumed

and diva-

the bulk to mono-

to be proportional

to

ii-e;

= fllvClv

13 = K2v = p2v C2v

Such an assumption traps

would seem

Czv
(Clv,

and divacancy C

in the metal

K,2 and K, 3 are the transition

respectively.

and to a

rates

). Here

to be realistic

for low concentration

ulv and u2v are the specific

concentration,

respectively.

C

Iv and C2v

of small

trapping

rate

can be written

volume

per unit vacancy [ 151 as :

exp(-E,,,/kT) Iv = Blv

(9)

C2v = ( Z/2 1 B2v exp ( - E2JkT where

B

IV

)

and B2v are the usual pre-exponential

exp ( S2v/k ), respectively t2max

corresponding

t2max

= Log (

entropy

and Z is the coordination

to maximum

7 + Klv

monovacancy

factors

number

fraction

exp ( S,,,/k ) and

of the lattice.

is given

The time

by :

+ Ktv I /

(a, + Ktv + KZv)

used

in this paper are taken from [ 161 . In I . Figure l(a) to l(d) show the fraction

(IO)

62

RESULTS The experimental lations

we take,

annihilating

in each

the positrons cancy

monovacancy ficant

state

for Au

from

T = 0.6 Tm ,

For temperatures trapping,

,

no =

-vs; time,

annihilate

trapping,

observable.

data

for simplicity

a delocalized the

effects

around

the number

but the contribution

for four different state

temperatures.

( note

of positron

of divacancies

trapped

At T = 0.4 Tm all

by monovacancies

the onset rate

of va-

become

and the saturation

in monovacancies

to the annihilation

calcu-

of positrons

the scale 1. At the onset

trapping

0.7 Tm , in between of positrons

our

is still

become

signi-

unobservable.

of

2

4

6

8

10

2

x 100~s

4

6

8

IO

x loops

n.

Fig.

Numbers

I . Plot of n, , n2 and n3 VS. time.

cy and divacancy

scale 2

3

to scale

x 2 10’

x3109

-b

to scale

x 20

x2 IO4

80

-c -d

to scale

x 1

x3 IO2

20

to scale

x 10-2

x2

However,

for temperatures

divacancy

contribution

0.14

approching

the fractions

in the metal

in Fig. 2 (b ) where

100

of positrons

still

matrix,

positron

therefore

trapping

to positron

annihilating

deduced

from

increasing the curve

3 lo-2

80

0.5

96

3.5

point

Le.

T

is observed. --&

surviving the decay

whose

in monovacancy a saturation

temperature.

3 lo-7

20

in divacancies

referring

In Fig. 2 ( c 1, t2max shows

3

q

0.9Tm

in the sample is a simple seems

intensities

with time

exponential.

However,

Tc then

may

96% and 3.5%

sites , respectively

and divacancy

up to a temperature

to

the decay

are approximately

with the value

at

are supposeo

to play a role,

The value of the threshold

is in good agreement

a significant

Figs. 2 f a ) and 2 ( b f

In Fig. 2 ( a ) all the positrons

as the sum of two exponentials

with

rate

respectively.

be taken

rapidly

monovacan-

(%1

1O-2

the melting

to the annihilation

T = 0.4 Tm and T = 0.9Tm,

fraction 2

1

-a

annihilate

to bulk,

respectively.

1

are depicted

1, 2 and 3 refers

.

decreases

temperature

Tc = 763K

Tc = 755 + 20 K obtained

52%

n -a

I

T = 0.4 Tm .5

0

*.

*. ..*. X. .‘,..,.

i

2

4

6

t X

IO x lOUps

2

46

I(

IO x loops

% area

50

0 +

.7

.9

.5

.7

.9

Tc a. n ( t ) vs; time plot for T = 0.4 Tm. 6. n ( t ) vs; time plot for T = 0.9 Tm . 5. tzmax vs; temperature plot. -d. Fractions (in % ) of positrons annihilating 1. bulk ; 2. monovacancy ; 3. divacancy

Fig. 2.

by Seeger

rate

[ 181 . Furthermore,

t2max

may

in each

have

some

in each

state

state

relations

vs. temperature

with the transition

K,v. The fractions

of positron

Fig. 2 (d ). The proportion 0.6 Tm then

it decreases

ped in monovacancies. lation

rate

trapped

from

close

divacancy

annihilating

with a corresponding But as T approches

monovacancies

strongly

may come is appreciable.

indicative

in at lower

are

temperature

curves

of divacancy temperatures

effects.

of positron

metals

Seeger

if the positron

trap-

to the annihi-

in the number

for the noble

in

up to

of positrons

contribution

decrease

shown

dominant

in the fraction

Tm the divacancy

The S and H parameter

to melting effects

with

in the bulk remains

increase

to play a role with a corresponding

in monovacancies.

a behavior that

starts

annihilating of positron

[

[ 171 show 181 noted

detrapping

CONCLUSION Positron

trapping

In our calculations is negligible. procedure

we assumed

However

at temperatures

that

the positron

this assumption

at high temperatures

near

Tm may be significant

thermal

is less clear

for the S or H curves -in Au ought

of divacancies cancies

in divacancies

detrapping

for noble

to take

and eventually

by monovacancies

metals.

into account

of positron

in Au.

A proper

fitting

of the contribution

detrapping

from

monova-

.

Sirnilar

work

noble

on

metals

and low melting

point

metals

(In,

Sn , Pb ) is in

progress.

REFERENCES

I

P.A.M.

2

C.D. Anderson,

Dirac , Proc;

Camb.

Phys.

Rev.,

3

R.N. West , Positron’

4

A. Belaidi , H.P. Leighly

5

A. Belaidi

and R.N. West,

A. Belaidi

, H.P.

6

-IO2 ( 19X/)

Studies

Phil.

Sot.

Sci. , Jfj ( 1930 ) 361 .

Math.

( 1932 ) 405 .

2

of Condensed

Matter,

and R.N. West , J: Phys. Phys.

Taylor

and Francis,

London.

1973

F : Met. Phys. , 12 ( 19X2 ) 813 .

Sol. (a ), 102 ( 1987 ) IO/ .

Stat.

Leighly , P.G. Coleman

and R.N. West,

Phys:

Stat.

SOL (a),

127 . and R.N. West , 3. Phys; F : Met. Phys. , E

( 19% ) 1001 .

7

A. Belaidi

8

A. Belaidi , H.P. Leighly , P.G. Coleman and R.N. West, Studies of Defects March I987 , Wernigerode , G. D. R.

Y

A. Belaidi , Ph. D thesis , University

IO

A. Belaidi , H. Aourag and B. Khelifa , 2e Rencontre Internationale sur les Sciences des surfaces des Materiaux , IO-12 December 1988, 0. P. U. Oran ( Algeria ), 1989 , p.223.

II

H. Aourag , M. Phii. thesis,

I2

P.E. Minarends IV79 .

I3

H.

Aourag

,

University

in P. Hautojarvi

I4

D.C. Connors

I5

A.C. Damask and C.J. New York,1971. ,

and R.N. W-St,

Phys.

Dienes , Point

of East

Defects

16

D.

Dlubeck

17

0.

Sueko ,

IX

A. Seeger , Appl. Phys; , 4 ( 1974 ) I83 .

0. Brummer

and N. Menyendorf

3. Phys; Sot. Jpn.,

Anglia

, U. K.

on Positron

, 1987 .

in Solids ,Springer

, R.N. West and B. Khelifa Letts.

Meeting

Anglia , U. K. , 1980 .

(ed. ) , Positrons

A. Belaidi , T. Kobayasi 191 .

,

-I55 (1989)

of East

European

Verlag , Berlin

, Phys.

Stat.

A, 30 ( l96Y ) 24 . in Metals, , Appl. Phys.,

-36 ( 1974 ) 464 .

Gordon

and Breach,

13 ( 1977 ) 67 .

Sol. ( b )