Diffusion studies of vacancies and impurities

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Diffusion studies of vacancies and impurities

David Lazarus

To cite this version:

David Lazarus. Diffusion studies of vacancies and impurities. J. Phys. Radium, 1962, 23 (10),

pp.772-778. �10.1051/jphysrad:019620023010077201�. �jpa-00236678�

772

[2] Detailed discussions of these methods are presented ⁱⁿ

the National Physical Laboratory Symposium No. 9,

The Physical Chemistry of Metallic Solutions and Intermetallic Compounds. H. M. S. O., London,

1959.

[3] See e.g. PRIGOGINE (I.), ^{The Molecular} Theory ^of Solutions, North Holland Publishing Company, Amsterdam, 1957.

[4] An authoritative discussion of the lattice theories of solution is given ^{in E. A.} Guggenheim’s ^{" Mixtures} ",

Oxford University Press, Oxford, 1952.

[5] ^FRIEDEL (J.), ^{Adv. in} Physics, 1954, 3, ^446.

[6] ^GORDY (W.) ^and THOMAS (W. ^J. O.), ^{J. Chem.} Physics, 1956, 24, ^439.

[7] ^WHITE (J. L.), ^ORR (R. L.) and HULTGREN (R.), ^Acta Met., 1957, 5, ^747.

[8] ^KÖSTER (W.) and RAUSCHER (W.), ^Z. Meta!lk., ^1948, 39, ^111.

[9] ^ZENER (C.), ⁱⁿ Thermodynamics ⁱⁿ Physical ^Metal- lurgy, ^{A. S.} M., Cleveland, ^1950.

[10] ^HULTGREN (R.), ^Private communication.

[11] ^PINES (B. J.), ^J. Physics, ^U. S,B ^S. R., 1940, 3, 309.

[12] ^LAWSON (A. W.), ^{J. Chem.} Physics, 1947, 15, ^831.

[13] ^KLEPPA (O. J.), ^Acta Met., 1955, 3, ^255.

[14] ^KLEPPA (O. J.), ^J. Phys. Chem., 1956, 60, ^846.

[15] ^KLEPPA (O.,J.) ^{and KING} (R. C.), To Acta Met (in press).

[16] ^RAYNOR (G. V.), Progress ^{in Metal} Physics, 1949, 1, 1.

[17] KLEPPA (O. J.), ^KAPLAN (M.) and THALMAYER (C. E.),

J. Phys. Chem., 1961, 65, ^843.

[18] In the dilute range, the differential ^excess quantities

of the solvent are related in a simple ^{manner to the}

curvature of the integral, ^excess quantity. (See ^also

ref. [21].)

[18A] Note added in proof. Very recent theoretical work

by Blandin and Deplante reported during ^{this col-} loquium, represents ^an improvement on the earlier Friedel theory, ^and appears ^to account for this change

in sign.

[19] ^KLEPPA (O J.) ^{and THALMAYER} (C. E.), ^J. Phys.

Chem., 1959, 63, 1953.

[20] ^WILSON (E. G.), Private communication.

[21] ^KLEPPA (O J.), ^Acta Met., 1960, 8, ^435.

[22] ^KLEPPA (O. J.), ^Acta Met., 1960, 8, ^804.

[23] PRATT (J. N.) ^Trans. Faraday Soc. 1960, 56, 975.

[24] ^CHAN (J. P.), ^ANDERSON (P. D.), ^ORR (R. L.) ^and

HULTGREN (R.), ^{4th Tech.} Report, Mineral Research

Laboratory, Berkeley, Calif.,1959.

[25 ^HOARE (F. E.) ^{and YATES} (B.), ^Proc. Roy. Soc., 1957, A 240, 42.

[26] ^HOARE (F. E.), ^MATTHEWS (J. C.) and WALLING (J. C.),

Proc. Roy. Soc., 1953, ^A 216, ^502.

[27] ^ORIANI (R. A.) and MURPHY (W. K.j, ^Acta Met., 1962, 10, ^879.

[28] ^WEISS (R. J.) ^{and TAUER} (K. J.), ^J. Phys. ^Chem.

Solids, 1958, 4, 135.

DIFFUSION STUDIES OF VACANCIES AND IMPURITIES

By ^DAVID LAZARUS, ,

Department ^of Physics, University ^of Illinois, Urbana, Illinois, ^{U. S. A.}

Résumé.

Les défauts ponctuels dans les métaux ont d’abord été introduits pour expliquer les phépomènes ^de diffusion, ^{et le} succès des modèles est généralement mesuré par le succès dans la correlation des résultats des mesures de diffusion. Dans cet article, ^on passe en ^{revue l’utilisa-} tion ^{de la} diffusion ^comme instrument d’étude des imperfections, ^{et on} cherche à définir les limites de la validité des modèles théoriques ^à la lumière des études expérimentales ^{de la} variation, ^en fonction de la température, ^{de la} pression ^{et de la masse,} de la diffusion dans un ensemble de métaux purs et de solutions solides.

Abstract.

Point defects in metals were first introduced to explain diffusional phenomena,

and the success of the models is generally ^measured by ^{the success in} correlating results of diffusion measurements. In this paper, the ^use of diffusion as a tool to study imperfections ^{will be} reviewed, and an attempt made to assess the limits of validity of theoretical models in the light ^of experi-

mental studies of the temperature, pressure, and ^mass dependence of diffusion in a variety ^of

pure metals and solid solutions.

LE JOURNAL DE PHYSIQUE ^{ET LE RADIUM TOME} 23, ^OCTOBRE 1962,

1. Introduction.

Diffusional phenomena ^are

basic to reactions in metallic systems. ^Diffusion limits ^the rate of phase transformations, solubility,

creep, grain growth, ^and recrystallization. ^Dif-

fusion rates dictate whether a material will be useful in a given environment, ^{as in} high tempe- rature ^{reactors, under} high ^flux conditions, ^{or com-} pletely useless, ^as in corrosive atmospheres. ^Tech-

nical interest in the field, therefore, ^has always ^been lzigh.

From a purely scientific viewpoint, ^{the most} important problems ^have been associated with deli-

neating specific mechanisms for diffusion which

permit ^{the observed} large ^{flues of matter without} perturbing ^the essentially perfect ^{lattice structure.}

Of the _many mechanisms suggested ^to explain ^dif- fusion, ^the concept ^{of mobile} point defects, parti- cularly interstitials and vacancies, ^has proven ^most viable. Since point ^{defects were} essentially

invented " ^to explain diffusion, ^{it is} perhaps appropriate ^{to consider} how diffusional measu-

rements have been useful _as, a tool for studying point defects in various systems,.

In homogeneous systems, the diffusion coefficient,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010077201

is invâriably ^{f ound to} vary ^with température ^accor- dirig ^{to the} simple ^{Arrhenius relation}

where the frequency factor, Do, ^{and activation}

energy, Q, ^are temperature independent. ^{In order}

to equate measured values of Do and Q ^{with terms}

of fundamental interest, ^some basic model for the process is needed. From the well known theory ^of

random flights, ^the diffusion coefficient for a cubic lattice it easily ^{shown to be} given by

where à is the lattice parameter, pd, the proba- bility ^{that a defect be} present ^at any lattice site,

v the probability per second that ^an atom execute

an elementary diffusional jump, ^and f, ^{the corre-}

lation factor, ^{a measure of the} degree ^{of ran-}

domness of the successive jumps ^{of an} atom,

varying ^{from 0 for} completely correlated jumps ^{te 1}

for completely ^random jumps. ^The correlation factor can be shown to be given by

where cos 0 is the _average value of the cosine between successive jumps ^{of any} given ^atom

If defects are produced ^{in thermal} equilibrium

and randomly distributed through ^the lattice, pd is

simply the fractional concentration of defects,

where A GI is the incrément in Gibbs free _{enorgy of} the crystal ^{on formation ouf a} defect. If the jump

itself is treated by ^the theory ^{of absolute reaction}

rates, ^which despite occasional objections [1], ^has

proven sufficient ^for analysis of all diffusional phe-

nomena, ^the jump fréquéney mày be written as

The pre-exponential ^term, vo, is interpretable ⁱⁿ

terms of.the ratio of the product ^{of the 3N normal,} frequencies ^{of the} crystals ^{with the atom in its} equi-

librium state to the product ^{of the 3N - 1 normal.}

frequencies ^{when the atom is at the} point ^{of maxi-}

mum energy during ^the diffusional jump [2]. ^The exponential ^{term is} simply ^{the ratio of the} parti-

tion functions of the system ^{with the atom in the} excited state and in the equilibrium ^{state. If we} impose ^a single constraint on the system, omitting

from the partition ^{function of the} ground ^{state the}

one vibrational degree ^{of freedom} associàtéd with the direction of motion, ^{then AGm can be iden-}

tified with the incrément of Gibbs free _{energy of} thé contrained system ^{when the} diffusing ^{atom is}

at its position ^of maximum energy. Since deletion of one out of some 1023 degrees ^{of freedom is not} expected ^{to create} any significant differences ^{in the}

analysais, ^it is reasonable to associate A G. simply

with the isothermal, isobaric work required ^to

move the diffusing ^{atom through the saddle} point

in the elementary jump [3]. ^{Thus one} may write

for the diffusion coefficient,

where As and AH are the changes ⁱⁿ entropy ^and enthalpy. ^The frequency factor, Do, ^{is then seen} to be simply,

and the activation energy

From ^{f ormal} thermodynamic arguments, ^the

third and fourth terms on the right ⁱⁿ (8) ^cancel identically, ^{so that}

Thus if Do and Q ^{are round to be} temperature independent, ^{it follows that AS and AH must be} temperature independent. ^For self-diffusion, f is ^a geometrical ^{constant which} depends only ^{on the}

diffusion mechanism and the lattice. Therefore,

thé third term on the right ⁱⁿ (9) depends ônly on

thé temperature ^{variation of a and} ),, ^for this case, and is expected ^{to be small.} The measured acti- vation energy ^for self-diffusion iq, théréforé, _équà.

table to the sum of the enthalpy changes ôjssociated with lormation and motion of defects.

Since experimental values for Q ^are obtainable to

high precision, these values may be compared ^with

values of AHI and AH. determined by ^{other means,}

as by measurements of resistivity ^of quenched

wires [4] ^or by comparison of x,ray and dilato- metric thermal expansion coefficients [5]. ^The

fact that the sum of the energies ^obtained by ^elec-

trical measurements on quenchèd ^wires agrees ^exac-

tly ^{with the} diffusional activation énergies ^{is the} strongest support ^{for the} validity ^{of the} interpre-

tation of the quenching experiments.

Expérimental ^values for Q may also be com-

pared, ^of course, with calculated values of AH

and AHm. This has served ^as much more of a

check on the validity ^{of a} theoretical model than on

the experiment itself, ^since thooretical énergies ^are

calculated as differences between large ^{terms of} opposite sign [6], ^{and are} seldotn known to any

precision. Small changes ⁱⁿ the theoretical ^models

can cause large changes in the calculated energies

for defects. As recently ^shown [7], introduction of non-sphericity in the Fermi surface of copper

causes a factor two increase in the calculated value for the formation _energy of a vacancy.

Theoretical calculations have given strong sup-

port ^to the conclusion that vacancies, rather than

interstitials, ^{or other more} complex mechanisms,

are predominantly responsible for diffusion in metals under equilibrium conditions. However,

because of uncertainties in the calculations, ^an

exact identification ^cannot be made on this basis.

The final delineation of the mechanism rests, ^even- tually; ^{on a} relatively ^sound experimental ^basis.

Two straightforwarct ^diffusion experiments ^dic-

tate the acceptance ^on the vacancy mechanism.

The first of these is the well-known Kirkendall effect [8], ^which gives direct evidence for a current of imperfections ^{to balance the net current of}

atoms in chemical diffusion, ^{as shown} by ^the macroscopic ^{motion of markers} placed ^{at the initial}

interface. This could be explained by ^{either a}

current of vacancies toward the side of the couple containing ^{an excess of the} fast-diffusing ^cons-

tituent or by ^a flow of interstitialcies in the oppo- site direction. The effect cannot be attributed to any exchange ^of sub-doundary mechanisms. The obvious evidence of porosity ^on the side of the

couple containing ^{an excess of} fast-diffusing atoms, under most experimental conditions, ^{can be most} simply explained ^{in terms of the} condensation of

excess vacancies about impurities.

The final distinction between vacancy and inters-

titialcy mechanism s can be made on the basis of measurements of the mass dependence of the diffu-

sivity [9]. Apart ^from predicable correlation

effects, ^the diffusivity ^should vary ^as 1/meff,

where the effective mass of the diffusional complex ’

is given by [2]

where ml is the ^mass of the tracer, m2 the mass of the host atom, ^{and the term} C2 , ^a direction cosine

squared, ^{measures the} fraction of the kinetic _energy carried by ^{the tracer in the} elementary jump. ^If

the kinetic _energy ^were equally ^shared by n ^atoms

in the jump process,

Agreement ^with experiment ^{is obtained} only ^for

n ⁼ 1. This excludes the interstitialcy ^mecha-

nism where n ⁼ 2, ^{as well as} exchange, ring, ^and

crowdion mechanisms.

2. Impurity-vacancy interactions.

Precise values of Do and Q ^{can be obtained for} self-diffusion in a pure material, and also for impurity ^diffusion.

Experimentally, ^{the diffusion rates of solvent and}

solute atoms are found frequently ^{to be} quite ^diffe-

rent. Solutes may diffuse faster ^or slower than solvent atoms, ^{and with} différent activation ^ener-

gies ^and frequency ^factors. Diffusivities ^{can also}

change radically ^on formation of even dilute alloys.

Since the vacancy has been shown to be the funda- mental defect which permits diffusion, ^{the diffe-}

rences in diffusion rates must be attributable ^to interactions between impurity ^atoms and vacancies From the form of eq. (7), such interactions may affect Do mostly through ^{the terms} f ^and AS, ^since changes ⁱⁿ jump ^{distance and} vo ^are expected ^{to be}

small. From (8), the activation energy is expected

to change, ^for impurities, through ^a change ^{in AHt} equal ^{to the} binding energy between the vacancy and impurity, ^{and a} change ^{in OHm} equal ^{to the} change ^{in barrier} height ^for jump ^{of the} impurity.

The correlation factor f, ^whose temperature depen-

dence could be ignored ^for self-diffusion, ^{now must}

be considered as an explicit function of tempe-

rature. We may calculate ^cos 0 and f, ^{to a} good approximation [10], by considering ^three separate jump frequencies, vl, the rate of interchange ^{of the}

tracer with a neighboring vacancy, ’J2, the rate of

jump of solvent atoms which are mutual nearest

neighbors ^{of both tracer} and vacancy, and ^v3, the

jump ^rate of solvent atoms which are not nearest

neighbors ^{of the} tracer, and whose motion thereby

dissociates the tracer and vacancy. By ^a straight-

forward procedure, ^{it can be shown that}

Since the various frequency ^{terms will have diffe-}

rent teinperature dependences ^{for cases other than} self-diffusion, f may be strongly temperature ^de- pendent ^for impurity diffusion, and the measured activation _energy, Q, ^{cannot be} simply equated ^to

the sum of the enthalpy changes ^for formation and motion of the vacancy ^{next to} the impurity ^atom.

Indeed, ^{as evident} from the form of (12), ^{if AHm}

should be very small for the impurity, ^{such that} 2v1» 2v2 + ^5.15 V3, the diffusion coefficient for the tracer would depend ^on AHf, ^{for the} tracer, ^but only ^on AH-2 ^and l1Hma ^for jumps of the solvent atoms. The change ⁱⁿ activation energy for diffu- sion of the impurity ^{relative to that} for self-diffu- sion of the solvent in the pure crystal, ^{will be}

where the subscript ^{1 refers to formation and}

motion of vacancies in the pure solvent. The term C includes effects from the temperature ^de- pendence ^{of the} correlation factor, ^{and can be de-}

termined from (7), (8), ^and (11), by setting

v1 ^- V,, e-,AHM,/k.T, ^{etc... The} pre-exponential

and enthalpie ^{terms can be} evaluated from the

measured Dos ^and the correlation factor t8 ^{for the} solvent, whereby, ^{C is} given hy

From (13) ^and (14), AQ ^{is seen to} depend only ^on differences in energy for formation and motion of vacancies adjacent ^{to solute atoms. Since these}

differences are generally small, fairly ^crude appro- ximations may be used ^to calculate the changes ⁱⁿ

activation _energy.

Considerable success has been achieved [11]

using ^a simple Thomas-Fermi model where the interaction is attributed entirely ^to screening ^{of the}

excess valence of the impurity ^atom by ^{the Fermi}

electrons of the solvent, with the interaction ^ener-

gies ^ascribed entirely ^{to the} potential energy of the screened impurity ^{atom and the} vacancy, thelatter being considered simply ^{as a} charge

^{e. Accor-} ding ^{to this} model, ^the change in activation energy is

where _y is a constant of order unity which varies

slowly ^with Z, ^{Z is the} valence différence between solute and solvent, ro is the interatomic distance, and q is the classical screening ^radius given by q2 .= ^25/8 EJl2f1t, EF being the ^Fermi energy of the.

solvent. The model is compared ^with experiment

in Table 1.

This treatment obviously ignores ^several impor-

tant factors ; ^{the screen} of the vacancy, the change

in kinetic energy of the electrons, ^{and the oscil-} latory ^terms necessary ^to satisfy ^{the Friedel sum}

TABLE 1

TOMIZUKA (C.) and SLIFKIN (L.), Phys. Rev., 1954, 96, ^610.

HOFFMAN (R.) ^and al., ^Acta Met., 1955, 3, ^417.

SAWATSKY (A.) and JAUMOT (F.), Phys. Rev., 1955, 100, ^1627.

HOFFMAN (R.), ^Acta Met., 1958, 6, ^95.

PIERCE (C.) and LAZARUS (D.), Phys. Rev., 1959, 114, ^868.

HINO (J.) ^and al., ^Acta Met., 1957, 5, ^41.

TOMIZUKA (C.), Private communication.

HIRONE (T.) ^and al., Private communication.

INMAN (M.) ^{and BARR} (L.), ^Acta Met., 1960, 8, ^112.

MACKLIET (C.), Phys. ^Rev., 1958,109,1964.

rule. It appears [12] ^{that the first two terms} may cancel identically, ^{and thé third be a small effect at} nearest-neighbor ^distances.

Since only ^{valence electrons are} considered, ^this

model cannot explain impurity-vacancy ^interac-

tions for impurities ^{of the same valence as the}

solvent. It may be possible ^{to include such} effects,

as well as effects of the size of the solute by ^a straightforward ^extension of the model [13].

Results of the model give ^excellent agreement

for solutes of higher valence, ^but notably poor agreement ^for electronegative ^solutes (e.g., Fe, Co,

Ni in Cu). This is to ’ be expected, ^{since such} impurities, particularly transition metals, ^would probably ^{not be screened} by ^free electrons, ^but

more probably by ^{bound states. This lack of} agreement ^is consistent with magnetic ^measu-

rements of the effects of these paramaghetic impu-

rities dissolved in noble metals.

Since faster diffusing constituents increase the number of vacancies ⁱⁿ solution by ^{the term}

AH t.-I:1Htl’ ^{addition of a} fast-diffusing ^solute

as a finite constituent in a dilute binary alloy ^is expected ^{to cause} large increases in the rate of diffusion of the solvent. Opposite ^{effects are} expected ^for alloys containing ^a slow-diffusing ^im- purity. ^These conclusions are in excellent agree- ment with precise measurements of diffusion in dilute alloys.

The present treatment cannot be readily ^exten-

ded to alloys ^of high ^solute concentration, ^because

of difficulty ⁱⁿ accounting for all the dïfferent _pos- sible jumps and solute-solvent configurations.

However, in ordered alloys, striking ^{effects due to}

enforced correlation of diffusional jumps ^{to main-}

tain order are readily determinable. In an orde- red structure, séquences of vacancy jumps will, ^on

the _average, cause differences in the rate of dif- fusion o£ the two constituents, depending ^{on the}

initial position of the vacancy in the ordered ^struc- ture. The initial position of the vacancy will

depend ^{to a} considerable extent on the degree ^of non-stoichiometry ^{of the} alloy. Such conside- rations give ^a full rationale for the differences in rates of diffusion of Au and Cd in ordered Au Cd

alloys ^near equiatomic composition [14].

3. Diûusion at high pressure.

Just as measu-

rement of the temperature dependence ^{of diffusion}

rates permits determination of the enthalpy ^and entropy changes associated with the formation and motion of defects, measurement of the variation of diffusion rates with pressure gives information rela-

ting ^to the volume changes associated with their formation and motion. Since, according ^to (6), ^the diffusivity depends exponentially ^{on the increment}

in Gibbs free _energy, we may ^{use the} identity

Then the volume changes resulting ^{from for-}

mation and motion of defects can be determined

TABLE 2

TOMIZUKA (C.) ^{et al.,} Bull. Amer. Phys. Soc., 1960, ^II 5,181.

EMRICK (R.), Phys. Rev., 1961, 122, ^1720.

TICHELAAR (G.) and LAZARUS (D.), Phys. Rev., 1959, 113, ^438.

BARNES (R.) ^and al., Phys. ^Rev. Lettres, 1959, 2, ^202.

NACHTRIEB (N.) ^and al., ^{J. Chem.} Physics, 1952, 20, ^1189.

NACHTRIEB (N.) ^and al., ^{J. Chem.} Physics, 1959, 31, ^135.

BUTCHER (B.) ^{and RUOFF} (A.), Bull. Amer. Phys. Soc., 1961, ^II 6, ^243.

BOSMAN (A.) ^and al., ^J. Physique Rad., 1959, 20, 241.

BASS (J.) and LAZARUS (D.), ^{To be} published.

TICHELAAR (G.) ^and al., Phys. Rev., 1961, 121, ^748.

from measurements of the pressure dependence ^of

D at constant T,

The second term on the right ⁱⁿ (17) depends only slightly ^on pressure, and ^can be evaluated from the compressibility and the pressure depen-

dence of the elastic constants.

The major restrictions on studying ^diffusion

under high pressure ^are entirely technical. Large hydrostatic pressures, in the _{range of} thousands of

atmospheres, ^are required, ^{and these must be main-} tained, frequently ^{in a} high temperature ^environ-

ment, for extensive periods of time. It is also

important that the pressures be purely hydrostatic,

since even small plastic strains may make large

differences in the equilibrium ^defect concentration.

Because of these problems, only ^{a few data are} available, and in many of these cases, resort ^was taken ^to indirect determination of diffusion rates, by NMR, ^anelastic relaxation, ^and magnetic ^rela-

xation techniques. The results of currently ^avai-

lable measurements for metals are shown in Table ^2.

From these data it is apparent ^{that in} both

b. c. c. and f. c. c. metals, ^the activation volume for substitutional diffusion is considerably ^{less than}

one atomic volume. For the one case studied, AVm is found to be extremely small, ^{but the}

vacancy volume itself, A Vf, ^is clearly ^{small com-} pared ^{to an atomic} volume, implying considerable relaxation of neighboring ^{atoms inward about the}

vacancy. These results are inconsistent with most calculations of vacancy ^volumes [14], which pre- dicted essentially ^no relaxation in close-packed metals, ^{but are in} agreement ^{with the recent calcu-}

lations of Tewordt and coworkers [16]. ^If large

relaxations are indeed present, ^this fact should be taken into account in theoretical calculations of the formation energy of the vacancies, despite ^the

obvious complication ^that relaxation lowers the

symmetry ^{of the} lattice about the defect.

The case of AV. for interstitials is, ^at best, ^con- fusing. Substantial activation volumes are obser- ved for interstitials in vanadium, ^{but none for}

interstitials in iron, ^{this last fact} being ^confirmed by ^two separate measurements. It thus appears that in irominterstitial atoms may be highly ^char- ged positive ions. This conclusion is consistent with the large ^{observed rate of} migration ^{of carbon}

atoms to the negative electrode when Fe-C alloys

are heated under an electric field [17].

4. Non-equilibrium diffusion,.

Diffusion mea-

surements may also be performed ⁱⁿ specimens

which are not equilibrated, ^either by virtue of the presence ^of electrical, thermal, ^{or chemical} gra-

dients, ^{or in which an excess of} defects is present

through plastic deformation or irradiation. Diffu- sion measurements under such conditions are rarely straightforward. Complications ^{arise in} satisfying

the boundary conditions necessary for analytic ^solu-

tion of the diffusion equation, ^{since the} diffusivity

is generally ^{a function of both} position ^{and time.}

At low temperatures, ^{radiation or} plastic ^defor-

mation cause large enhancements in diffusion rates,

presumably ^{due to the formation of an excess of}

vacaniecs and interstitials. However, ^{the diffusion}

rates do not follow first-order kinetics, ^{and are}

thus not simply analysed.

At high temperatures, ^excess vacancies and interstitials introduced athermally apparently equi-

librate rapidly ^and give ^{little effect on} the diffusion

rate [18]. ^At intermediatetemperatures, ^enhanced diffusivity ^is observed, ^but this may be traceable to enhanced boundary. ^diffusion resulting ^from large

increases in dislocation density.

To date, very little attention has been directed toward study of diffusion under large electrical and thermal gradients, despite ^{the obvious} importance

of such effects in the design of materials for reactors and direct conversion devices. Available data are

fragmentary ^and frequently inconsistent, ^{and do}

not permit any definite conclusions about the effects of such gradients ^{on defects.}

Discussion

A. SEEGER.

1 should like to remark that for the noble metals the agreement between the mena-

sured activation volumes as given in Lazarus’ paper and the theoretical results are as good ^{as one} might expect. ^{If we assume} that it is justified ^{to com- ,}

bine the experimental ^{data for} Ag and Au with each other, subtracting the activation volume for vacancy migration ^{in Au from the activation}

volume for self-diffusion in Ag gives ^{us 7.7} cm3 j

mole. This corresponds ^{to a} vacancy volume of 75 percent ^{of the} atomic volume. No detailed theoretical calculations for the relaxation of the atoms surrounding ^a vacancy have been published

for Ag ^{or Au.} However, ^the results should not be too différent from those on Cu. Seeger ^and

Mann (1) employed ^{three différent} Born-Mayer- potentials for copper and found for the vacancy volume o.92 %,0.81 % ^{and 0.71} % of ^{the atomic} volume, respectively. ^{The values} just quoted ^are

increased only slightly, ^{if we allow for the aniso-} tropy ^{of the} strain-field surrounding the vacancy

(G. Schmid, unpublished). ^{As a} good theoretical estimate we might ^{consider that} the vacancy volume in copper is about 0.8 atomic volumes (2),

which is in accord with the experimental ^{data on}

noble metals.

(1) ^SEEGER (A.) ^{and MANN} (E.), ^J. Phys. ^Chem. Solids, . 1960, 12, ^326.

(2) ^SEEGER (A.), ^MANN (E.) ^{and v. JAN} (R.), ^J. Phys.

Çhem. Solids, 1962, 28, ^639,

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INTERACTIONS ENTRE LACUNES TREMPÉES ET IMPURETÉS DANS LES MÉTAUX ET ALLIAGES

Par Y. QU.ÉRÉ,

Centre d’études nucléaires de Fontenay-aux-Roses

Résumé.

Les données expérimentales concernant les interactions entre lacunes trempées ^et impuretés dans les métaux sont encore rares.

Lomer et Cottrell ont interprété des résultats de Roswell et Nowick sur le frottement interne dans un alliage Ag-Zn trempé, par ^une interaction forte lacune-zinc. Depuis, différentes techniques (microscopie électronique, dureté, résistivité) ^ont permis ^d’évaluer quelques énergies .de ^liaison lacune-impureté, ^en particulier ^{dans des} alliages ^{à base} d’aluminium, ^{d’or et de} platine. ^Nous

faisons le point ^{sur ces résultats et décrivons une} expérience qui ^{met en évidence une} forte inter- action entre lacune et oxygène ^dans l’argent.

Abstract.

Expérimental ^{data on} interactions between quenched-in vacancies and impurities

in metals are still rare.

Lomer and Cottrell proposed ^an interpretation of results by Roswell and Nowick on internal fric- tion in Ag-Zn, ^{based on a} strong interaction between a vacancy ^{and a} zinc atom. Since then, ^dif-

ferent experimental techniques (electron microscopy, hardness, resistivity) have onablede an

évaluation to be made of some vacancy-impurity binding energies especially ⁱⁿ aluminium, gold

and platinum ^based alloys. ^We present these results and describe an experiment which pro- vide évidence of a strong interaction between a vacancy and ^an oxygen atom in silver.

LE JOURNAL DE PHYSIQUE ^{ET LE RADIUM TOME} 23, ^OCTOBRE 1962,

Depuis ^{bientôt une dizaine d’année, un} gros effort expérimental ^{a été porté sur l’étude des}

lacunes dans les métaux très purs. Une partie

notable de cet effort consiste d’abord à obtenir un

métal exempt ^{de toute} impureté. ^ce qui ^{contribue à} expliquer ^les progrès ^{de nos} connaissances sur les lacunes dans les métaux ^{nobles pu dans l’alumi-}

nium _{que l’on} peut tous obtenir correctement

purifiés.

Parallèlement, ^mais plus timidement, ^{on com-}

mençait ^à s’intéresser aux interactions entre lacunes et impuretés ^{et l’on est} obligé ^{de constater}

que dans ce domaine, les données expérimentales

sont encore rares, peu précises ^et sujettes ^{à caution.}

Si l’énergie ^{de formation ou} l’énergie ^{de mobilité}

des lacunes sont maintenant connues avec une

excellente précision pour ^un certain nombre de métaux, ^{on n’a} encore sur l’énergie ^{de liaiso de}

deux lacunes par exemple, que des appréciations

d’ailleurs assez divergentes.