A new way of using positron-lifetime measurements to study lattice defects

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A new way of using positron-lifetime measurements to study lattice defects

Ch. Janot, B. George, M. Boidron

To cite this version:

Ch. Janot, B. George, M. Boidron. A new way of using positron-lifetime measurements to study lattice defects. Journal de Physique, 1979, 40 (1), pp.39-45. �10.1051/jphys:0197900400103900�. �jpa- 00208882�

A

new way

of using positron-lifetime measurements

to

study lattice defects

Ch. Janot, ^B.

George

Laboratoire de Physique ^du Solide, Faculté des Sciences, C.O. n° 140, ⁵⁴⁰³⁷ Nancy Cedex, ^France

and M. Boidron

Centre d’Etudes Nucléaires, ^{B.P. n°} 6, ⁹²²⁶⁰ Fontenay ^aux Roses, ^France (Reçu ^{le 26} juin ^1978, accepté ^{le 21} septembre 1978)

Résumé. ²⁰¹⁴ Dans le but d’étendre les mesures de temps de vie des positons ^vers les hautes températures ^{et d’éviter}

les complications ^{dues aux} annihilations dans la source et à la surface de l’échantillon, ^des experiences ^{ont été}

réalisées à partir ^{de sources} intégrées ^{où les} isotopes émetteurs de positons ^{sont à l’état} d’impureté isolée dans le materiau étudié. Par ailleurs, ^{on montre} qu’il ^est possible, ^à partir ^{des mesures de} temps ^de vie, ^d’obtenir l’énergie

de formation des lacunes

EF1V,

^sans être tributaire des imperfections du modèle du piégeage. ^On explique ^aussi

les correlations observées précédemment ^entre

EF1V

^{et la} température Ti ^du point d’inflexion dans la courbe

03C4(1/T).

Abstract. ²⁰¹⁴ With the aim of extending positron-lifetime measurements to high temperature ^and avoiding ^surface

and source contributions, experiments ^{have been} performed ^with positron emitting isotopes ^{in a state of diluted}

solid solution in the material investigated. ^Further, it has been shown that the positron-lifetime ^{data can be used}

in a new way ^to obtain the vacancy formation energy

EF1V

without the many uncertainties of the trapping ^model.

The previously observed linear correlation between

EF1V

and the inflection temperature Ti ^{in the} 03C4(1/T) ^curve

has also been explained.

Classification Phvsics Abstracts

61.70 ^{- 78.70B}

1. Introduction. ^- Since the

pioneering

^{work of}

MacKenzie et al. [1] positron annihilation has become

a widely ^{used tool for} investigating ^the

properties

^of

vacancies in metals in thermal

equilibrium.

^The many

experimental

results that have been obtained and the many ways of

analysing

^{them have been} extensively

presented

^{in a number of review papers [2]-[7].}

1. 1. ^- Positron sources for lifetime measurements

were often

prepared by depositing

^{the source}

isotope

(commonly ^22Na in the form of sodium chloride)

either directly ^{on a surface, or} between thin

organic

(or metallic) foils sandwiched between two plates ^of

the material to be

investigated.

Measuring tempe-

ratures are then limited either by ^the

melting

point

or by ^the

evaporation

^{of the source} material. The method has the extra disadvantage ^{of a}

relatively

large fraction of

positrons

annihilating ^{either in the}

source-supporting

^{material or at the}

specimen

^surface,

giving

^{rise to serious} background

problems [8],

[9], [ 10]

and drastic limitations in the total activity ^{that can be} really ^used.

Integrated source-specimen

systems ^for

high temperature positron annihilation

experiments

have been recently

designed

[11]-[13]. ^In this so-called sealed-source

technique,

^{the source material} ( N ¹⁰

uCi

of carrier-free

22NaCl)

^is

hermetically

sealed between two discs of the material to be

investigated

^{which are}

electron-beam welded under

high

^vacuum.

Obviously,

this method has

proved

^{to be}

quite

^{effective at}

high

temperature ^{and has} brought ^some

improvements regarding

^the parasitic

intensity

^{of the} long-lived

source component. Limitations _are, the still present surface component ^{and a}

possible

break of the weld under the vapour pressure of the source material [11], [14].

A technique

using

^metal

specimens containing positron

^source

isotopes

^{as bulk}

impurities

^is

presented

in this _paper.

1.2. - Data on positron annihilation near ther-

mally generated vacancies in metals are conven-

tionally

analysed by ^a trapping ^{model which is}

generally recognized

^{to contain}

questionable approxi-

mations and too many

adjustable

parameters [ 15], [16] ;

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197900400103900

40

factors such as thermal expansion, temperature

depen-

dence of the

trapping

^rate,

detrapping

mechanisms,

self-trapping

^or

phonon-assisted trapping

[17], pre-

sence of other defects

(impurities,

dislocations...)

have influences that cannot be taken into account as

they should be. This usually

produces

^{values of the}

monovacancy formation _energy

E ;,

^{which are some-}

what low when

compared

^with results from established

techniques, ^{or are not} intemally consistent [18], [19], [3]. There is additional concern arising from the fact that for instance, ^{data on Al can be}

analysed

equally

well [20] ^{in terms of a}

temperature-independent

^{or a} strongly

temperature-dependent Eiv

^{of the} type

suggested

by Popovic et ^al. [21] ; ^{moreover, when}

the model is used with metals

of higher

melting

points,

such as the noble metals [22], [13] ^the agreement ^with

other methods is far from

satisfactory. Consequently,

it has often been

suggested

[23], [6],

[24], [12]

^that,

until the _many uncertainties of the

trapping

^model

have been clarified, ^it may be

preferable

^to analyse

positron

^data through ^{a more}

phenomenological approach,

using ^the

empirical

^linear correlations between

Eiv

^{and some} characteristic temperatures :

1.1.1. ^-

T;

which is the temperature of the inflec- tion

point

^{in the} plot ^{of i} (mean lifetime of

positrons)

against 1/T :

E’

(eV) ^{= 1.13 x 10-3} Ti (K) [6].

1.1. 2. ^- The threshold temperatures

Tt

^{at which} thermally

generated

^vacancies

produce

^measurable

positron trapping.

^{This has} the additional

advantage

of

giving

^a

precise

determination of

Eiv

^without

approaching

^the melting

point

^{- a}

major

^{one in}

dealing

with refractories [12], ^{with metals}

having high

vapour pressure in the solid state [24], [13], ^{or with} alloys [6].

It will be the main _purposes of this _paper to show that :

a)

positron

annihilation data can be fruitfully analysed ^{without the} need of the

trapping

^model formalism,

b) the linear correlation between

E F

^and Ti ^{is not}

fortuitious but derived from the trapping ^{model and}

must be used carefully because of a

possible

crystallo- graphic ^structure

dependence

^{of the}

proportionality

factor,

c) computer ^simulated

r(I/T)

^{curves as derived}

from the

trapping

^{model are} strongly ^affected by

changes

ⁱⁿ

trapping

rates,

detrapping

process, ^etc.

except for the value

of E1v

^{that can} be calculated from the characteristics of the inflection

point

Both the new source technique ^{and the new method}

of analysis ^{will be tested on} aluminium metal.

2.

Expérimental

procédure ^{and results. - The} electronic apparatus ^{used to measure the}

positron-

annihilation lifetimes is basically ^a high-resolution timing ^{device with fast and slow} coincidence circuits, plastic scintillators, ^a

time-to-amplitude

^conversion capacitor ^{and an} energy discriminator. The data are

accumulated in a multichannel analyser ^{in the form}

of a

histogram

^{of the number of events as a function}

of time. In a metal this is the sum of exponential components ^folded with the resolution of the instru- ment and a constant

background

^{due to random}

coincidence. The mean lifetime as a function of temperature ^{is obtained} by ^a computer ^fit

analysis

as described elsewhere [25], [14]. ^A

good

estimate of the resolution _may be obtained

by

substituting ^{a 6°Co}

source in

place

^{of the}

sample.

^{Since 6°Co emits two}

y-rays within - _{5 ps} of each other, ^the measured time spectrum ^{will be} essentially the resolution of the instrument. Such a prompt coincidence curve has been found to have a half width of 300 ps. In the lifetime

region ^{studied here} ( 150 ⁱ

300 ps) the uncertainty

will be of about _{2 ps.}

A new type ^of

positron-lifetime

^{source has been} tested,

using

^{5 sCo}

positrons emitting isotope

^{in a}

state of diluted solid solution in the material to be

investigated.

^{In the case of} interest, ^the specimen ^was

a

cylinder

^{of 99.999}

%

purity aluminium, ^{3 mm thick} and 10 mm in diameter in which about 50

yci

^{of S8Co}

were diffused by

annealing

^{near the}

melting

tempe- rature, ^after

electrodeposition

^{of 58CO from} chloride ;

undiffused cobalt and

possible

^corrosion

products

were carefully removed from the

sample

^surface by

polishing. Although

^{the two} y-photons ^{emitted at}

birth and annihilation of

positrons

^{are less} easily

discriminated here than with a 22Na source (0.511,

1.28 MeV for 22Na and 0.511, 0.81 ^{MeV for}

58Co),

the

improvement regarding

^the

intensity

^{of the}

long-

lived source and surface components is obvious

(Fig.

1). ^In addition such a

diffused

^source is very suitable for

high

temperature work and lifetime measurements after thermal or mechanical treatments.

Aluminium has been chosen as a test material because of the abundance of good data available

regarding

^the vacancy formation _energy

(see

^for

instance [6] ^table

II).

^{To obtain}

reproducible

^{data in}

the low temperature part ^{of the} 7(T) curve, the _spe- cimen has to be

carefully

^{annealed near the}

melting

temperature ^{in order to} stabilize the dislocations in the material.

Table 1

gives

^some

typical

^{values of the mean}

lifetime i of the

positrons

^{as a function}

of temperature.

Table I.

Fig. ^{1. -} Positron-lifetime spectra in aluminium : (a) ^with 22Na conventional source; (b) ^{with 58CO} diffused ^source.

Fig. ^{2. -} Temperature dependence ^{of the mean lifetime in alumi-}

nium : (a) i( T) ^{full line :} best fit from a trapping ^model (upper scale) ; (b) 77(l / T)

(lower

^scale).

(c)- dr-/d(I/T)1

This is

pictured

ⁱⁿ figure ²

(curve (a))

along ^{with two}

other curves : the variation of ? with the

reciprocal

temperature

(curve (b))

and the derivative curve

d’f/d(I/T) (curve (c))

^{whose maximum}

^gives

^the

coordinates of the inflection

point

ⁱⁿ 7(1/T), ^{that is :}

Let us remark that

using T;

⁼ 653.6 K in the

empirical

^formula

proposed

by Doyama [6]

(Efv

^{= 1.13 x 10 - 3}

rj

^{results in}

E F

^{= 0.73 eV}

which is in

good

agreement with the other available data [26].

3. An

original

^{method to calculate}

Eiv

^from 1(T)

data. ^- In this section the

trapping

model will be

fully criticized. In particular, ^{it will be} shown that a

direct

fitting procedure

^{of the} 7(T) ^{curve on the}

trapping

^{model formula cannot lead to the} proper value of the _vacancy formation _energy

Efv.

^The

drastic influence of factors such as the

positron

trapping rate, the

detrapping

occurrence, the thermal

expansion

^{of the} lattice, the presence of other defects,

a proper choice for the asymptotic ^{lifetime in the}

lattice iL

^and in the vacancy trap i 1 v will be systema- tically analysed. Incidentally, ^the empirical ^formula

E’(Ti)

^{will be shown to be} contained in the

trapping

model.

Eventually, ^{we will} demonstrate that the many uncertainties of the

trapping

^{model can be} easily

bypassed

^if

E’

^is calculated from the coordinates of the inflection

point

^{in the} i(1/T) ^curve.

3.1 THE TRAPPING MODEL

(see

for instance Ref. [6], [19],

[27]).

^{- Let the} number of free positrons ^{in the}

lattice and those

trapped

by ^vacancies

be nL

^and n 1 v

respectively.

^{Then the rate} equations ^are given by :

where :

. Jll v’ the

specific

positron trapping ^{rate is} usually

written

concentration at température ^T [26],

is the

detrapp-

ing ^rate expressed ^{in terms} ’of a

vacancy-positron

binding

energy eB, the positron ^{mass m and the radius a}

of the

trapping

^volume.

0

TL( T) and r, 1v(T)

^are usually

naively

supposed

to vary only through the lattice thermal expansion,

that is :

and

iL

^and

To

^are merely deduced from the extrapolated

values of the 7(T) experimental ^{curve at low and} high temperature (a ^{= 8.4 x} 10- 5 K-1 in Al).

Solution of eq. (1) ⁱⁿ steady ^state conditions gives

the mean lifetime of positrons i(T) :

42

Table II.

When there are many kinds of trapping centres, M1v

Ci v

^{must be}

replaced

in eq. (1) by ^a

sum 03A3

^mi

Ci

i over all the different

trapping

^{centres. In metals for} instance, ^{there is} always ^a contribution from the so-

called Frank lattice dislocations which is

equivalent

^to

a concentration of

Cd

^vacancies whatever the tempe-

rature.

By

using

^{a standard} least-square

procedure,

eq. (2)

has been fitted as usual to the present

experimental

data on the assumption ^{that no}

detrapping

^of

positrons

occurs. The values for best fit are given ^{in table II}

and have been used to draw the full line in figure ^2a.

In addition to a too low value of

E’V,

the results are

unrealistic in

particular

because of an overestimate of the

trapping

^{rate. Indeed, from} eq. (1) ^the

expected

order of magnitude ^for

lilv is

^{about 200}

ps-1,

^10’

smaller than the fitted value. We will see later that the

discrepancy

^comes mainly ^{from the} opposite ^influences

of

detrapping

^{and thermal} expansion ^{which are}

difficult to take into account with their true values in the

trapping

^model.

3.2 THE ORIGINAL « INFLECTION POINT » METHOD.

-

If detrapping

might ^be neglected, eq. (2) ^{could be}

rewritten :

where t(T),

yl,(T)

^and

C1v(T)

^are supposed ^{to be} temperature

dependent

^{as described in 3.1. In the}

following,

7(T) ^as given by eq. (3) ^will be referred to

as the

simple

trapping ^{model. A}

straightforward

^calcu- lation,

stating

^that

d2t/d(I/T)2

^{= 0 for the} particular temperature Ti, shows that at the inflection point ^of the curve t(I/T), ^the fraction of

positrons annihilating

in a

perfect region

of the lattice is equal ^{to the fraction}

annihilating

^at vacancies, ^and consequently :

or

which is

just

^an expression ^of

Efv

equivalent ^{to the}

so-called empirical formula of Doyama [6].

The reason

why

^Ln

[.LI v 1:L ex p slv k ]

^{i s a cons-}

tant, ^{is not} very clear and might vary from ^a f.c.c.

structure, ^to b.c.c. or h.c.p.... because of the influence of the

crystallographic

^{order on}

sBfv

and of the

influence of the packing ^{fraction on}

iL(T). In

^any

case, eq. (4) ^{cannot be used} directly ^{to calculate}

Efv

since

J1v(Ti}

^and

SF1v

^{are not} given by

positron

annihilation experiments.

However,

calculating 1

^{dI at the} inflection point

dT

temperature

Ti,

^and using eq. (4) ^with

results in a new

expression

^for

El,

^{that is :}

It is easy ^to show by numerical estimates that the two first terms in the right ^{hand side}

of eq.

(5) ^account

for about 10-’ in

Eiy

^of vacancies in metals; ^this contribution will be neglected ^{in the}

following

^because

of its irrelevance to experimental accuracy in

E ;,

measurements. The x term contribution is

relatively

small as well (about ^{4 x}

10- 2)

^{but will be} kept ^{so far.}

Then,

Eiy

^{can be}

expressed

^from eq. (5) :

Tl, î5(T;)

^and

dT/d(I/T)i

^are directly obtained from the derivative curve

of i(1/T)

(Fig. 2c) ;

iL(T;)

^{has to}

be calculated by ^{a thermal}

expansion

formula from the 7(T) ^curve

extrapolated

^{at low} temperature

(iL

^{= 165 ps at 20 °C). There is no need to know}

,rlv(Ti)

and this is _very fortunate as it will be shown in a

moment.

Although

^{the true value of x does not} really

matter for the

E F

calculation, ^{it is} sometimes of interest to determine the temperature

dependence

^of

the

trapping

^{rate. It will be seen} further that x ⁼ 0.5,

as found by ^the

trapping

^model fitting

procedure

(Table II), ^{is the more} reasonable.

Thus, using eqs. (6) ^and (4) along ^{with the data of}

the present ^work (see ^section 2) ^{and the} entropy coefficient

SF Ik

^{= 1.8 in} aluminium obtained else- where [26], gives :

E F

^{= 0.70 + 0.03 eV}

,Mu°v

^{= 164}

ps -1

which are indeed reasonable values.

3.3 ADVANTAGES ^{OF THE «} INFLECTION POINT » METHOD COMPARED TO THE FITTING PROCEDURE. ^-

As eq. (6) giving

Eiv

^{in the} inflection point ^{method is}

deduced from a trapping ^model formula, ^{it could be}

thought

^{that the same} uncertainties could still influence the result. To show that it is not the _case, i(1/T)

behaviours have been simulated by computer ^calcu- lations and the influence of the various factors has been

analysed.

^{The main} conclusion is that the

inflection

point keeps ^on

giving

^{the same}

Efv

^value

(within

experimental

accuracy) ^even when drastic

changes ^{are observed in the}

i( 1 / T)

^curve.

3.3.1 Specific trapping ^rate influence. ^- Using

eqs. (3) ^and (4), ^sets

of 7(T)

^{values can} be calculated with arbitrary though reasonable parameters

(Efv

^{= 0.70}

eV, TO

^{= 170 ps,}

T1v

^{= 260}

ps)

^and

different temperature

dependence

^{of the}

specific trapping

^rate [x ^{= -} 0.5, 0, ^{0.5 and 1 in a}

J1.1v(T) = MIVO T x law .

^Then,

^T-,,(I/T)

^{curves are}

Mlv(T ) iv T 0 _vo/

^law

J

^Then

^(I/T)

^{curvcs are}

drawn and

compared

^{to each other} (see

Fig.

3).

Fig. ^{3. -} Computer simulation for different specific trapping ^rate

functions _{Jl1V =} u’v(TITO)’ : ^{: (a) x = - 0.5 ; (b) x = 0 ;} (c) x=0.5;(d) x= 1.

The main features are the

following :

e all the calculated curves are

quite

^similar except in the

high

temperature

region

e the recalculated value

E F (X)

^{obtained from the}

inflection

points

^{of all the}

ix( 1 / T)

^{curves and with}

eq. (6) ^{are all the same} within reasonable _accuracy range, ^as evidenced in table III

Table III.

e in addition x ⁼ 0.5 seems to be the best value to fit the experimental ^{data at}

high

temperature ;

Tx> 0.5(I/T) slopes

^are

slightly

^{too low}

and Tx 0.5(I/T)

slopes ^are

slightly

^too large ^at

high

temperature.

3.3.2 Influence

of

^other

defects.

^{- The simulation}

will be limited here to the influence of dislocations, that will be considered as equivalent ^{to an extra}

concentration

Cd

^of vacancies, kept ^{constant with}

changing

temperature. ^The

TD(1/T)

^{curves have to be}

calculated from _eq. (3) ^with

C,v(T)

substituted by

Clv(T)

⁺

Cd. Again,

^all calculated curves are very similar within realistic values of

Cd ( 104

^{to 106 cm - 2}

in well annealed aluminium) and recalculated

E’(Cd)

values by the inflection point ^{method are all the same} (Table IV). Incidentally, ^{the rather}

unexpected

large

values of

7,,.P(I/T)

^{below 280 OC appear as not due to}

dislocations (within reasonable concentrations) ^and

cannot so far be interpreted

by

^{a coherent}

trapping

model (Fig. 2a).

Table IV. ^- (same units as in table III).

3.3.3 Influence of the thermal expansion. ^{- In}

eq. (3),

Ta(T)

^can be calculated with the parameter

set as given ^{in section} 3.3.1, with rlv and _iL either

kept

^{constant or} being temperature

dependent

The

-=taCT),

^{for a} ranging ^{from 0 to the thermal} expan- sion coefficient in aluminium (a ^{= 8.4 x 10- 5}

K-1),

are quite ^{similar at low} temperature, ^but

diverge drastically

^at

high

temperature. ^But

again,

^{the recal-}

culated values of

Eiv(rx)

given by ^the inflection

point

method (eq. 6) ^{are not} really

changed.

^{Table V} presents the main features of the

rL(T) and rlv(T)

influences.

Table V.

3.3.4 Influence of ^the

detrapping

occurrence. -

Going

^{back now to the full} eq. (2) ^{we will now simu-}

late the influence of

positron detrapping

^before

annihilation. In this calculation,

rL and

’t 1 v will be

kept

^{constant in a first} stage. ^{The data} presented ⁱⁿ

table VI show that, opposite ^to the influence of thermal

44

expansion,

detrapping

results in

lowering

^the high temperature ^{values of}

t"b(T).

^Again, the recalculated values of

Eiv(h)

^{are not} really ^af’ected by

changes

ⁱⁿ

the binding energy of the vacancy-positron pair.

Just for the sake of it, ^an attempt ^{has been made to} compensate the thermal expansion ^influence by ^allow- ing detrapping ^{to occur so that}

1/2(ia(T)

⁺

:t"b(T»)

would be equal ^{to the}

experimental

^data

Texp(T)’

This resulted in a surprisingly ^{low value of}

f.b=0.67

eV, compared with the theoretical estimate of Hodges [28].

This is

pictured

ⁱⁿ

figure

^4.

Fig. ^{4. -} Influence of thermal expansion ^and detrapping ^occurrence (a) computer simulation of the thermal expansion influence ; (b) computer simulation of the detrapping influence ; (c) : experimental data, full line : computer simulation of a compensation

between thermal expansion ^and detrapping.

Table VI.

4. Discussion and conclusion. ^- This _paper has

reported

^new

experimental

and theoretical approaches

to positron-lifetime

experiments

^{for the} measurement

of the _vacancy formation _{energy in} metals.

It has been shown that

diffused

sources, with the

positron active material as impurities in the investi-

gated metal, ^are very suitable for high temperature work, ^{and that} they ^reduce quite

obviously

^the long

lived source and surface components ^{in lifetime} spectra and permit ^{direct thermal or} mechanical treatment of the specimen.

A direct analysis ^{of the lifetime data} by ^a trapping

model formula has been shown to be quite ^hazardous

without a perfect

knowledge of many physical

^{factors :}

e temperature

dependence

^{of the} specific trapping

rate,

e

binding

energy of the

vacancy-positron

pair ^which might ^{be less tied to each} other than previously

expected,

. presence of other defects,

. and last but not least, ^{the true} temperature

dependence

^{of the}

positron-lifetime

^{in the} perfect

lattice and at vacancy trap. The lattice thermal

expansion approximation

that has been commonly

used until now

might

^{be a} very crude _one,

especially

because of a

possible phonon-assisted trapping

^of

positrons

[17] ^{before the} occurrence of a vacancy.

This

point

^is

supported

by ^the observation of a syste- matic difference in the lifetime obtained in the tran-

sition region ( T ³⁰⁰ OC) ^when measurements carried out at

equilibrium

temperature ^are

compared

^{to data}

obtained in

cooled-samples

containing quenched-in

vacancies [29].

Although derived from the

trapping

model, ^the inflection point method which allows one to calculate

Efy

^{from the} coordinates of the inflection

point

^{in the} T( 1 /T) curve, does not suffer the same uncertainties and

gives

always ^{the same value}

(expected

^{to be}

accurate !) ^of the vacancy formation energy. In fact,

this is not very

surprising.

^{First of all, this} method does not need to know

’t"ly(T)

^{which, on the} contrary, interfers in the

trapping

^model formula and is

impos-

sible to determine from

experimental

data because of the influence of both thermal expansion ^and

detrapping

processes.

Secondly,

^the inflection point

of the :r(I/T) ^{curve is} equally far from the

high

^and

low temperature

regions

and, ^{as a} consequence, is

expected

^to be insensitive to high temperature detrapp- ing ^{and low} temperature

phonon-assisted trapping.

Finally,

^this

inflection

point method has one

impor-

tant advantage

because T;

^{occurs at}

approximately

0.7

T. ( Tm

⁼

melting temperature).

^{In the case of}

refractories or metals with

high

vapour pressure ⁱⁿ the solid state, ^{it will} be

possible

^{to evaluate}

E F

without measurements near the melting

point.

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