Lattice spacing trends in close-packed hexagonal phases based on the noble metals

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Lattice spacing trends in close-packed hexagonal phases based on the noble metals

T.B. Massalski

To cite this version:

T.B. Massalski. Lattice spacing trends in close-packed hexagonal phases based on the noble metals.

J. Phys. Radium, 1962, 23 (10), pp.647-654. �10.1051/jphysrad:019620023010064701�. �jpa-00236655�

647

and ionic and covalent character become possible.

What happens ^{as the area of} the Fermi surface decreases is that the range of ^{distance over which}

. metallic screening ^occurs increases. Thus in a

semimetal with a very small Fermi surface, ^one might ^{have a net} charge, ^{in a} covalent bond loca- lized on anatomic scale which is surrounded by ^a neutraliziny.cloud ^{with a radius of order a hundred}

atomic separations. ^{Thus in} principle, ^{the theorem}

holds for se’1limetals, but becàuse of ^the long screening ^{radius there is a} greater ^re3e nblance ^to

an insulator than to a _{metal as} regards ^{ionic or}

covalent character.

Aeknowledgement.

^{I am} very grateful ^to

Dr. Walter A. Harrison for searching ^{criticism of}

the Hartree calculations ^{and to Prof. J. C,} Phillips

for general discussions.

n

REFERENCES AND FOOTNOTES -

[1] ^GELL-MANN (M.) and BRUECKNER (K. A.), Phys. Rev., 1957, 106, 364.

[2] ^HUBBARD (J.), ^Proc. Roy. Soc., London, 1957, ^A 240, 539; 1958, ^A 243, 336 ; 1958, ^A 244,199.

[3] ^PHILLIPS (J. C.) and KLEINMAN (L.), Phys. Rev., 1959, 116, 287. ^KLEINMAN (L.) and PHILLIPS (J. C.), Phys.

Rev., 1959, 116, 880; 1960, 117, 460 ; 1960, 118, 1153. PHILLIPS (J. C.), ^J. Phys. ^Chem. Solids, 1959, 11, 226.

[4] ^COHEN (M. H.) ^{and HEINE} (V.), Phys. Rev., 1961,122,

1821.

[5] ^NOZIÈRES (P.) ^{and PINES} (D.), Nuovo Cimento [X], 1958, 9, 470,

[6] EHRENREICH (H.) ^and COHEN (M. H.), Phys. ^Rev.

1959,115, ^786.

[7] ^{Such a} separation ^{is not} possible ^{for the transition} metals, ^{the rare earth} metals, ^{and the} actinides, and

is marginal for the noble metals.

[8] Their work (ref. 3) ^{has been} generalized ^{to include} many-body ^effects by BASSANI, GOODMAN, ROBIN-

SON and SCHRIEFFER, Phys. ^Rev. (in press).

[9] ^COHEN (M. H.) and PHILLIPS (J. C.), Phys. Rev., 1961, 124,1818.

[10] ^PHILLIPS (J. C.), Phys. Rev., 1961, 123, ^420.

[11] ^PHILLIPS (J. C.) ^{and KLEINMAN} (L.), Phys. Rev.

(in press).

LATTICE SPACING TRENDS IN CLOSE-PACKED HEXAGONAL PHASES BASED ON THE NOBLE METALS

By ^{T. B. MASSALSKI.}

Mellon Intitute, Pittsburgh 13, Pennsylvania, ^{U. S. A.}

Résumé.

La variation des paramètres cristallins ^{et du} rapport e/a ^{dans les} alliages ^hexa-

gonaux compacts de métaux nobles a été étudiée en détail, ^avec des méthodes de haute precision,

sur tous les exemples ^{connus. On trouve une} corrélation remarquable ^{entre les résultats} experi-

mentaux d’une part, la concentration en électrons et la zone de Brillouin d’autre part. ^Les phases hexagonales examinées forment trois groupes, chacun ayant ^un comportement caractéristique ^en

ce qui ^{concerne le} rapport e/a, ^{le volume} atomique ^et les écarts par rapport ^{à la} linéarité. ^L’appa-

rition d’un contact entre la surface de Fermi et les faces {10.0} et {00.2} de la zone de Brillouin peut être détectée avec précision, ^{et un modèle} simple permet d’estimer les intervalles ^{entre bandes}

dans la zone. Les valeurs obtenues sont comprises ^{entre 0,5 et 1} eV, ^{selon le} rapport e/a, ^{la concen-}

tration en électrons et la nature du solute.

Abstract.

The changes of lattice spacings ^and of the axial ratio in h. c. p. intermediate phases

based on the noble metals constitute an interesting chapter ^{in the} theory ^of alloys. ^{We have recen-} tly completed the survey of the lattice spacings of all known h.c.p. phases ^{which belong to this}

group using high precision methods and the results show a very remarkable correlation with the electron concentration and the Brillouin zone. The investigated h.c.p. phases ^fall naturally

into three distinct groups ^each showing ^a characteristic behaviour in terms of axial ratio, ^volume per atom, and systematic deviations from linearity. ^{Onset of} overlap of electrons across the

{10.0} and {00.2} sets of Brillouin zone faces can be detected with high accuracy in terms of electron concentration and the use of a simple ^electron theory permits ^{one to evaluate} approximate

values of the band gaps ⁱⁿ the zone. The obtained values range between 0.5-1.0 eV depending

on the axial ratio, ^the electron concentration and the nature of the alloying elements. These and other details will be discussed.

LE JOURNAL DE PHYSIQUE ^{ET LE RADIUM}

TOME 23, ^OCTOBRE 1962,

Some thirty years ago Jones [1] interpreted ^the changes ^with composition ^{of lattice} spacings ^of

the E and n phases in the Cu-Zn system ⁱⁿ terms ^of overlaps of electrons across the Brillouin zone. At

that time the available data consisted of only ^a

few points ^{in the Cu-Zn} system ^{and was non-}

existent for other similar syste ns. By ^{now thé}

lattice spacings ^{of all} phases ^which possess ^{the close-}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010064701

packed hexagonal ^{structure in} binary systems

based on the noble metals (Cu, Ag, Au) ^{have been}

measured with a high degree of accuracy and they

reveal a very striking ^and interesting picture. ^The

purpose of this _{paper is} ^to summarize the available data and to discuss the possible interpretations.

The typical phases ^{with thé} close-packed ^hexa-

gonal (h,c.p.) ^structure may be conveniently ^divi-

ded into three natural groups which ^are denoted by

the Greek symbols 03B6, E ^and ~ according ^{to their}

axial ratio (c/a), ^lattice spacing trends, ^electron

concentration (e/a) and solute content. The range of stability ^{in terms of} e/a ^{of all} known 03B6, é and -1 phases ^{are shown} in ^{figure 1. In this figure are}

FiG. 1.

Ranges of maximum stability ^{of the} oc, 03B6, ^e and -1 phases ^{in terms}

of electron concentration (after Massalski and King [2]).

also included the ranges of stability ^{of the corres-} ponding ^a phases ^{which are terminal solid solutions}

with the face-centred cubic (f.c.c.) ^{structure and}

which are of interest in connection with the trends in the lattice spacings ^{of the} h.c.p. phases. ^The changes ^{of the} axial ratio with e ja ^{in the} h.c.p.

phases ^{listed in} figure ^{1 are shown in} figure ^2.

The 03B6 phases follow the ^oc terminal solutions and

occur within the e la range between 1.22 and 1.83.

Their axial ratios lie relatively ^{close to the ideal}

value for the close-packing ^of spheres

and they ^{decrease with increases of} e/a. ^{The e and}

1) phases always contain zinc or cadmium as the

major constituent. The axial ratios of the 03B6 phases

fall below the ideal value, ^{in the} region ^of

decreasing initially ^with e/a ^and showing ^an

increase near zinc-, ^or cadmium-rich boundaries.

The q phases ^are terminal solid solutions based on

zinc or cadmium. Their initial axial ratios lie

considerably ^{above the ideal} value of 1.633 and

they increase with increasing e ja towards the values _{of pure} zinc (cla

1.8563) ^or pure cad- mium (cla

1.8859).

In order to emphasize ^certain characteristic trends in the lattice spacings in each group of

h.c.p. phases it is convenient to compare the

03B6 phases ^{with the oc} terminal solid solutions which

precede them in the respective phase diagrams

and to compare the e phases ^with the q ^terminal

solid solutions which follow them in the respective phase diagrammes. This is illustrated diagrama- tically ⁱⁿ figure 3 in which the phase ^{fields are}

drawn with the characteristic shapes usually

observed.

To a first approximation ^{the structure of the a} and 03B6 phases ^are very similar, differing only ^{in the} stacking ^{of the} elosest-packed planes. Therefore,

it can be assumed that the contributions to the

changes of lattice spacings ^{from such factors as} the

différence in atomic sizes, electrochemical inter-

actions, etc., ^should vary in a smooth and conti-

nuous manner across through ^{a and 03B6} phases.

Any specific deviation may be attributed to the

change ^{of structure} between the two phases ^and

hence to the change in the Brillouin zone and, ^rela-

ted to it, ^a change ^{in the} possible interactions between the Fermi surface and the Brillouin zone.

The characteristic features in the general ^trends

with composition ^{in the a} spacings ^{of the 03B6} phases

649

FIG. 2.

Changes of axial ratio (c/a) with electron concentration (e/a) ^{in the} 03B6, ^E and 7] phases (after Massalski and King [7]).

FIG. 3.

Characteristic positions ^{of the} phase diagramme a) ^{in the oc - 03B6} region ^of systems ^{based on the noble}

metals and b) ^{in the e} - ui region ^of systems ^{based on}

zinc or cadmium.

FIG. 4.

The Da’ and Aa deviations in the ( phases

based on the noble metals (after Massalski and King [9]).

are illustrated in figure ^{4. These trends are} parti- cularly clear if the lattice spacings within the ter- minal solid solutions (1) ^are plotted together (marked a’) ^{with the} corresponding a spacings ⁱⁿ the 03B6 phases. Ag-based systems ^are represented , by ^the Ag-Sn system, ^Cu-based systems by ^the

Cu-Ge system and Au-based systems by ^{the Au-Sn} system. ^{An initial} contraction, ^1Ya’ may be obser- ved in the closest packed spacings ^on passing

from f.c.c. to the h.c.p. structure. This contrac- tion increases depending ^on the solvent in the order Ag ^{Cu Au. For each} group ôf systems

based on one solvent the 0394a’ contraction is found to be nearly ^{the same.}

By contrast the values of mean volume _per atom change smoothly ^{from the oc to the 03B6} phases, ^{the values in the 03B6} phases falling ^almost exactly ^{on the values} extrapolated ^{from the corres-} ponding ^{a terminal solid} solutions. ^A typical

trend may be illustrated by ^{the data for the Au-Sn} system ^{as shown in} figure ^5. It may be seen that

FIG. ^5.

Lattice spacing ^{trands in a and 03B6} phases ^{in the} system ^Au-Sn (after ^{Massalski and} King [5]).

the 1Ya’ contraction ^occurs at nearly ^constant

volume per atom ; ^{and hence a} complementary expansion, Ac’, may be expected ^{to exist in the}

c spacings. This is revealed (see fig. 5) by ^com- paring ^{the c} spacings ⁱⁿ the 03B6 phases ^with hypo-

thetical c’ spacings ^{ïn a} alloys ^derived by ^multi-

(1) For this purpose the cubic ^a spacings ^{are divided} by B/2B ^{so that a direct} comparison ^can be made between the closest distance of approach of atoms in the {111}

planes of the f.c.c. structure - ^{and the} equivalent ^closest

distance of approach of atoms in the {oo. 2 } ^{planes of}

the h.c.p. structure, a’ = 1 IV2- - ai. 0,. c. c"

2/V3 .af:c.c..

plying ^{the. a’} spacings by the ideal axial ratio.

Accompanying ^{the 1Yc’ increase and} the 1Ya’ de-

crease in the 03B6 phases’are ^{the initial} higher-than-

ideal values ^{of the} axial ratio. The detailed trends in the axial ratio in all phases ^{are shown in} figure ^6.

Corresponding ^{to the 1Yc’ and 1Ya’} deviations the

FIG. 6.

Changes of axial ratio with electron concen-

tration in 03B6 phases ^{based on the} noble metals (after King and Massalski [6]).

initial c ja ^is highest ^{in Au-based} systems ^and

decreases in the order Au > Cu > Ag.

In addition ^to the a’ decrease and the c’ increase described above an accelerated increase (marked ^0394a

in figures ^{2 and} 5) ^{occurs in the a} spacings ^{in each}

group of03B6 phases ^{based on a common solvent at a} surprizingly ^narrow range of ela. The actual values of ela appear ^to be characteristic of each noble métal falling ^{in the} range e/a =: 1. 415-1.43 ; 1.39-1.40 ^and 1.35-1.36 ^for alloys ^{based on} Cu, Ag ^{and Au} respectively [2]. ^{In the} range ^{of the Aa} deviation the volume per ^atom trends indicate a

slight departure ^towards higher ^{values from the}

smooth curves extrapolated from the terminal solid solutions. This may be ^seen in figure ^{7 in which}

the lattice spacings ^{in all ot} and t phases ^{based on}

silver are plotted together arranged ^{to follow the} plan ^{of the} periodic ^{table for the} B-subgroup ^cle-

ments which follow silver. The inclusion of both a

and t ^lattice spacings ^{trends in} figure ⁷ clearly

shows that the Aa deviation occurs in all 03B6 phases irrespective of whether the genera.l trend of the lattice spacings ^{is to} decrease with e/a ^{or to increase}

with e ja.

The slight increase of the volume per atom in the

region ^{of the da déviation is} insufficient to compen- sate the total change ^{in the a} spacings ^{and hence}

a compensating decrease in the c spacing ^{may be}

observed..

. The characteristic trend in the behavior of the

, .lattice spacings ^{in the e} and -1 phases may ^{be illus-}

651

FIG. 7.

Lattice spacing trands ^{in oc} and y phases ^{based on silver} (after King and ^Massalski [6]).

trated with reference to the système Ag-Zn ^and

Au-Zn as shown in figure ^8. Thé data for the oc

phases ^is again included for comparison. ^The exis-

tence of the Aa deviation, ^{similar to the Aa devia-}

tion observed in the 03B6 phases, ^is clearly ^evident

within the entire range of thé e phases ^{when com-} parison is made between the lattice spacings ^{of a} and 03B6 phases. The volume per ^atom deviation,

from a line extrapolated ^{from the a} phases, is

however now quite pronounced. ^Another striking

feature of the s phases ^{is an} accelerated increase in the c spacings (marked Aci ⁱⁿ figure 8) ^observed

near the Zn-rich boundaries of. thé s phases.

These increases in the ^c spacings ^{cause a reversal}

in the trend of the axial ratio in thé s phases ^{from a}

decrease to an increase. This is illustrated in fi- gure 9 for the four known s phases. Judging by ^the

data in figure ^{9 the initial values of} e/a ^{at which a}

deviation of the axial ratio (resulting ^{from the} Ace expansion) becomes noticeable occurs slightly

above e/a ^{~ 1.8 and increases in the order}

It is of interest to note that within the r phases ^the

volume trends indicate a further positive ^déviation

from the original ^line extrapolated ^{from the ce} phases. ^{At the same time the increase in thé c} spa-

cings ⁱⁿ thé 7) phases, and the decrease in the a

spacings, ^{occur at} higher ^rateû than those indicated

by similar trends ^near the Zn-rich boundaries of the c phases (see fig. 8).

The interpretation ^{of some of} the ^above data supplelnents ^{and extends the} original suggestion’by

Jones [1] ^that overlaps of électrons across the Brillouin ^zone discontinuities exist in the e and 1) phases ^{and that} they ^are responsible ^{for the}

behavior of the lattice spacings. ^The presently summarized ^data permits ^{one to locate the onset of} the overlaps in ^{terms o.f} e/a and it throws addi-

tional light upon the relationship between the lat- tice spacings and ^{the band} structure..

A vertical and horizontal ^section through ^the

energy ^zone suggested by ^Jones [1] is. shown in

figure ^{10. This zone is bounded} by ^{20 faces : 6 of}

the 110.61 type (the ^A faces), 2 of the JOO.21

FIG. 8.

Lattice spacing trends in oc, ^e and 7) phases (after Massalski and King [7]).

type (the ^B faces) and 12 of the f 10.1 } ^type

(the ^C faces). ^{The number of states} per atom, n, enclosed within the inner zone (without ^{the domes} produced by the intersection of the C fa,ces) ^is given by

The values of n for 03B6 phases with ideal axial ratio and for thé e phasmes ^with cla ^{= 1.55 are 1.745}

and 1.721 respectively, [2] but since the ^zone is

incomplete [3] overlaps may ^{occur at} much lower values of ela. Following Jones it may be assumed that overlaps produce ^{a net electron stress} tending

to distort the Brillouin zone and _{that any} change

in the zone shape should be reflected by ^{a corres-} ponding change ^{in the lattice} spacings ^{in real} space in directions opposite ^{to the} displacement ^{of the}

zone faces. If the volume remains constant, ^a

positive ^deviation say in the a direction [11.0]

must be compensated by ^a negative deviation in the c direction [Ô0.2].

The Da deviation in both the t ^{and s} phases may

be taken as evidence of the f 10. O} ^{(across the}

A faces) overlap [4-6]. ^{The available data} suggests

that this overlap ^{first occurs in the} region ^of the t phases ^at relatively low values of e/a ^{between 1. 36}

and 1.4 [2]. ^{This is not} suprizing ^{since the con-}

tact between a spherical Fermi surface and the

10 . 0 } f aces ^{of the zone would occur at e Ja ~ 1.14.}

The trends in the lattice spacings ^{in the} Ag-Cd system, ^{where both} the 03B6 ^{and s} phases ^are stable, suggests that, ^once started, the influence of the

( 10 .0 ) } ^overlap is about the same in both ( ^and

e phases. However the rate at which the a spa-

cings expand ^{in différent} systems (as ^measured by Aal(a -Lla) [2}) changes ^{from one} system ^{to ano-}

ther and appears ^to be dependent ^{on additional}

factors such as the magnitude ^{of band} gaps in the Brillouin _zone, elastic constants, ^etc. [2].

The reason for the existence of a two-phase region between the e and -1 phases ^{in the Cu-Zn} system, both of which posses the h.c.p. ^structure,

has been attributed by ^Jones [1] ^{to the lack of}

tao.21 ^{overlap in tbe E phases. The} interpreta-

tion of the Ac, ^deviation suggests however that

653

FIG. 9.

Changes ^of axial ratio with electron concen-

tration in e phases ^{based on} zinc and cadmium (after Massafski and King [7]).

this overlap ^{does occur in the s} phases ^{near the} zinc-, ^{or cadmium -rich} boundar ies. In terms of the shape ^{of the e} phase ^fields (shown ⁱⁿ figure 3)

the overlapped range corresponds ^to alloys quen- ched from the extended corners of the e phase

fields (marked X ⁱⁿ figure 3) suggesting ^{that it is}

enhanced by the température. Nevertheless the lack of a continuous smooth transition between the s and -n phases deserves further study [7].

The mechanism responsible ^{for the initial Aa’}

and Ac’ deviations observed ia the ( phases ^{is not}

clear. In this low region ^of ela overlaps ^{are unli-} kely ^{to take} place ^{and the} possible ^mechanisms

may be expected ^to be related to interactions between the Fermi surface with thé Brillouin ^zone

before contact and after contact with a set of zone

faces [2, 8]. ^{If the Fermi} surface is assumed to be

nearly spherical ^{it is possible to} associate thé range of e ja near 1.36 with ^an approach ^{of the Fermi}

surface towards the {OO. 2B ^{faces [5, 8]. However}

following the récent studies of the Fermi surface

topology in the pure noble metals [9] ^{it is} likely

that within the h. c. p. phases ^the Fermi surface will be also, ^{at least} initially, considerably ^dis-

torted from a sphère having " necks " in contact with the 100.2 ^{faces, since} these faces are equi-

valent and lie at the same distance from the orjgin

of k-space ^{as do} the 11111 faces in the zone ^{of the}

f.c.c. structure. Thus the Ac’ deviation might ^be

related ^to the details of the " neck " topology ^and

hence also to the magnitude ^of the energy gaps [2].

Finally ^the knowledge ^of the e ja values at which

FIG. 10.

The Jones’ zone for an idéal close-packed ^hexa- gonal ^structure (after Massalski and King [2]).

overlaps ^first appear ^to take place in various

h.c.p. phases, ^{as evicenced} by thé ^{onset of Aa and} Aci deviations, permits ^{one to} Make ^{an ocder of} magnitude estimate of the energy _baud gaps in the Brillouin. _zone, uxing a free électron approxi-

mation. This is perhaps ^not unreasonable since there are several indications that although ^{the ban’d}

gaps ^are relatively large in pure Soble ^metals they

may be much smaller in thé alloys [2].

Taking the energy of free électrons

thé band gap ^{across a zone} face may be expressed

as

where khkl ^and Àjly, correspond ^{to thé radii of the}

Fermi surface at overlap ^{and at contact with} respect to a (h, k, l) zone face.

Thé values of k ^are related to the électron concen-

tration by ^the expression

where Q is the volume _{per atom.} The value of ko

can be calculated for the condition of contact between a spherical ^Fermi surface ^{and a} (hkl) face

and the value. ^of k’ _may be derived from the expéri-

mental data for the électron concentration ^{at which}

initial overlap appears ^to take place ^{across the}

Rame face. Such calculations yield several inte-

resting points. ^{The band} gaps ^across the ’( 10 . 0 )

faces in the 03B6 phases ^{in the} region ^of the ( 10. 0) } overlap appear ^to be less than 1 eV (AE(10.c) ^{- 0.73}

eV for the Ag-Sn system [6]) ^{and the band} gaps

across the 100. 21 ^{faces in the e phases in’the région} of 100*.21 overlapappear ^to be about ^{1 eV}

eV for the Ag-Zn system [7]). ^{If a} slight change

in the slope ^{of the c} spacings ^{in the} Ag-Sn system is interpreted to ^{be a} result of the {OO. 2} ^overlap

in that system (for ^{details see ref.} [7]). ^{The cor-} responding ^band gap is, equal ^to

.

0. 41 eV at ela == 1. 51.

The same band gap ^at the electron concentration value in the region ^{of the} (10.0) overlap ^{has been}

estimated to be AE(oo.2) ^{=1.32 eV at} e ja ^{= 2 .4} [6]. Therefore the lattice spacings ^{data indicate}

that the band gaps in the Brillouin ^zone of the

hexagonal phases ^{decrease with the increase of}

the solute content.

This work has been supported ⁱⁿ part by ^the

United States Atomic Energy Commission, Washington 25, ^{D. C.}

REFERENCES

[1] ^JONES (H.), ^Proc. Roy. Soc.,1934, ^{A 147, 396.}

[2] ^MASSALSKI (T. B.) ^{and KING} (H. W.), Prog. ^{Mat. Sci.,} 1961, 10, 1.

[3] ^JONES (H.), ^The Theory of Brillouin Zones and Elec- tronic States in Crystals, North Holland Publishing Co., Amsterdam, 1960.

[4] ^MASSALSKI (T. B.) and COCKAYNE (B.), ^{Acta Met.,} 1959, 7, 762.

[5] ^MASSALSKI (T. B.) ^{and KING} (H. W.), Acta Met., 1960, 8, ⁶⁷⁷ and 684.

[6] ^KING (H. W.) and MASSALSKI (T. B.), ^Phil. Mag., 1961, 6, 669.

[7] ^MASSALSKI (T. B.) ^{and KING} (H. W.), ^Acta Met., 1962, 10, in _press.

[8] GOODENOUGH (J. B.), Phys. Rev., 1953, 89, 282.

[9] ^HARRISON (W. A.) ^{and WEBB} (M. B.), The Fermi Sur- face, ^John Wiley ^and Sons, ^Inc.

THE RESIDUAL RESISTIVITY DUE TO DEFECT PAIRS (1)

By ^{C. P.} FLYNN,

Department ^of Physics, University ^of Illinois, Urbana, ^Illinois.

Résumé. 2014 Des calculs ont été effectués dans le but de déterminer les contributions des phéno-

mènes de diffusions multiples cohérentes à la résistivité résiduelle des paires ^de défauts dans un

réseau métallique. ^{La théorie a été} appliquée ^aux paires de lacunes et d’impuretés d’étain ^dans

l’or où l’on trouve _que les échanges isotropes ^sont petits, de l’ordre de 5 % ^de la résistivité totale due aux défauts isolés. On trouve une énorme anisotropie ^{dans le cas des} paires de lacunes en

position ^de premiers et seconds voisins, ^{en accord avec} les résultats trouvés _{par Bross} ^et Seeger

avec un modèle différent. Les variations de résistivité isotrope ^et anisotrope ^{oscillent avec la} période kF 03B1/03C0 ^{avec une} décroissance en 03B120142, lorsque la distance a entre les défauts augmente.

Abstract.

Calculations have been performed to determine the contributions of coherent and

multiple scattering phenomena to the residual resistivity ^{of defect} pairs ^{in a} metal lattice. The

theory ^{has been} applied to vacancy and tin impurity pairs ⁱⁿ gold ^{where it} is found that the iso-

tropic changes ^are small, being ^{of the} order of five percent of the total resistivity ^{due to} the isolated defects. The anisotropy ^{in the case} of vacancy pairs at first and second neighbouring sites is found to be considerable, ⁱⁿ agreement with the results of calculations by ^{Bross and} Seeger using ^{a diffe-}

rent model. Both isotropic ^and anisotropic resistivity increments oscillate with period ^kF 03B1/03C0

and decays ^as 03B120142, ^with increasing separation a of the defects.

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

Introduction.

In the theory ^of resistivity ^due

to point ^{defects in a} metal lattice, ^{it is} commonly

assumed that the various .deviations of the lattice from petfect periodicity ^scatter electrons inde-

pendently. ^This being ^the case, the observed linear dependence ^{of residual} resistance on defect (1) This work has been supported ⁱⁿ part by ^{the U. S.}

Atomic Energy Commission.

concentration is automatically ^{built into the theo-}

retical description ^{of the} expriments. Conversely,

the linearity ^{of the} resistivity-concentration ^rela-

tionship up ^to concentrations _{of many} atomic peur-

cent of defects, ^{ànd the} reasonable agreement

between the rate of change ^of resistivity ^{with con-}

centration observed and that predicted ^{on theore-}

tical grounds, may be taken ^to imply ^that point