THEORETICAL ASPECTS OF THE INTERACTION BETWEEN GRAIN-BOUNDARIES AND IMPURITIES

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THEORETICAL ASPECTS OF THE INTERACTION BETWEEN GRAIN-BOUNDARIES AND

IMPURITIES

P. Guyot, J.-P. Simon

To cite this version:

P. Guyot, J.-P. Simon. THEORETICAL ASPECTS OF THE INTERACTION BETWEEN GRAIN- BOUNDARIES AND IMPURITIES. Journal de Physique Colloques, 1975, 36 (C4), pp.C4-141-C4-149.

�10.1051/jphyscol:1975415�. �jpa-00216320�

THEORETICAL ASPECTS OF THE INTERACTION BETWEEN GRAIN-BOUNDARIES ANW IMPURITIES

P. G U Y O T and J.-P. S I M O N LTPCM-ENSEEG

Institut National Pol ytechnique, Grenoble, France

RCsumC. - On dCcrit dans cet article I'interaction impuretC-joint de grains dans une solution solide B I'Cquilibre. On passe d'abord en revue les diffkrents termes'de I'interaction Clastique, puis on calcule I'interaction Clectronique d'une impurett htttrovalente avec un joint dans un cristal mCtallique normal. On utilise des interactions de paires dCduites de pseudopotentiels pour calculer d'abord I'Cnergie de joints de grains, la mCthode Ctant ensuite Ctendue au calcul de I'interaction avec une impuretC. L'interaction est oscillante et dCcroit avec la distance comme

-

Abstract. - We describe i n this paper the interaction between impurity and grain-boundary in solid solutions at the equilibrium. The different terms of the elastic interaction are first reviewed. One then calculates the electronic interaction of a heterovalent impurity with a boundary in n cryst;~l of normal metal. One use pair interactions deduced from pseudo- potentials to calculate grain-boundary energies, the method being then extended to the calculation of the interaction with an impurity. This interaction is found oscillating and decreases with the distance as r'. The resulting binding energy can be sizeable when compared with the elastic effect for highly misoriented boundaries.

I. Introduction. - As emphasized by Friedel [I], w e must, in t h e theoretical analysis of grain- boundary-impurity interactions, distinguish bet- ween direct and indirect o r thermodynamic interac- tions.

T h e later ones occur f o r instance in the precipita- tion of over-saturated solid solutions, when the precipitation starts a t t h e boundary ; the resulting solute concentration gradient rules thereafter the precipitation kinetics. This is also t h e case when the solute atoms are swept by a moving boundary, like in discontinuous precipitations o r in eutectoid transformations. This kind of interaction will not be herein described.

On the contrary, in solid solutions at the equilibrium, t h e migration of solute atoms towards boundaries is ruled by diffusion processes assisted by a drift flow which results from the interaction gradient. An attractive interaction leads t o an accumulation of impurities a t the boundary referred t o a Cottrell atmosphere type. A repulsive o n e leads t o a depletion around the boundary. T h e value of the interaction energy in the core of the boundary (binding energy) gives t h e relative concentration of the impurities with respect t o the bulk ; its range gives the width of the atmosphere and determines t h e diffusion kinetics in the early stages of the segregation.

T h e direct interactions have t w o origins : - One is elastic. I t takes into account the different atomic volumes of crystal matrix and

impurity atoms, the difference in their elastic constants, a s well a s the matrix and impurity anisotropy. T h e interaction is calculated by using the elastic continuum model.

- T h e second o n e is electronic o r chemical. I t comes f r o m the electronic structures of both boundary a n d impurity ; it can in principle be evaluated by a continuum model of free electrons o r better by a discrete lattice model, with nearly free electrons. Because in metals the charge perturbations a r e screened by the conduction elec- trons t h e electronic interaction is expected t o be small.

Our purpose is t o give first a brief review of the elastic part of the interaction, which is generally the more important. T h e electronic contribution for impurities in normal metals will be next considered.

2. Elastic interactions. - W e shall successively examine t h e interactions with boundaries which can be analysed in terms of well individualized disloca- tions (typical case of sub-grain boundaries), and then with high angle grain-boundaries.

2.1 SUB-GRAIN BOUNDARIES. - F o r a misorien- tation 0 less than

-

At a distance r > x from t h e boundary, the stresses exerted by all boundary types (edge, mixed edge, screws, crossed grids of screw dislocation arrays) decrease with r like exp(- crrlx) [2] ; there

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975415

C4-142 P. GUYOT A N D J.-P. SIMON

are nevertheless two cases where a boundary has long range stresses : a finite array of edge disloca- tions and an infinite but asymmetrical tilt boundary, as shown in figure 1 ; this last boundary has at large r non vanishing stress components and a hydrostn- tic pressure

giving rise t o a non-zero long range interaction by size effect ; but being proportional to V , p, i. e. in exp(-4 r r l x ) , a strong impurity drift is not expected. A finite boundary is similar at long distance to an isolated dislocation with burgers vector n b ; its long range interaction cart-then -be.

analysed as described below for an infinite boun- dary at short distance.

FIG. I . - Asyrnrnetric;~l tilt hound:rry.

For r < x, the elastic interaction is practically deduced from the stress field of an isolated dislocation. We shall therefore distinguish three origines to the interaction W(r) :

a ) a size effect :

stress field ^a,j, we have Wzb(r) a v l r 2 , for edge or screw dislocations. The interaction is repulsive for impurities harder than the matrix, attractive for softer impurities ;

c ) an anharmonic effect, due to deviation from simple harmonic elastic behaviour of the crystal.

Like the elastic constant effect it also leads to a second order interaction W;'h a Avlr2 [4].

Hence for non-pure screw dislocation arrays in an isotropic medium, the size effect in Ilrdomina- tes. For a screw array, the elastic constant effect and the anharmonic effect are superimposed to give an interaction in l l r 2 , but in an anisotropic medium the interaction is also in Ilr.

2 . 2 H I G H ANGLE GRAIN BOUNDARIES. - For a misorientation larger than 20°, the core dislocations are no more individualized. Therefore the stresses exerted by the boundary decrease exponentially with r, practically from the outset of the boundary.

One expects an additional interaction which decrea- ses less rapidly with r, due to an image force concept :

a ) In an anisotropic crystal, the impurity sees the misoriented half-crystal in which it does not stay with average elastic constants different from those of the host half-crystal. It leads to an interaction by image force given by [ S ] :

Depending upon the misorientation of the boun- dary the shear modulus y ' of the half-crystal not containing the impurity is larger or smaller than p

for the host half-crystal, implying that the impurity is repelled or attracted by the boundary.

b) At high enough temperature for the stresses created by the impurity to be relaxed (by shear or where p is the hydrostatic pressure of the disloca- vacancy emission or absorption in the boundary) in tion and A v the difference in atomic volume the grain boundary, the boundary can be considered as a free surface. Putting p' = 0 in (4) we get an between matrix and impurity atoms. Therefore

attractive interaction Wib a -

1

(,L, 1s

-

binding energy WSb ^{' p}

1

1

.

the shear modulus of the crystal. For pure screw 4 71. r3 ^{' ( 5 )} dislocations p = 0 in an isotropic crystal ; but if one

considers the crystal anisotropy a s well as the impurity shape anisotropy, one is left with an interaction Wahu also in -

1

b ) a n elastic constant effect, due t o the diffe- rence in elastic constants between impurity and matrix.The interaction is then the elastic energy stored by the boundary in the impurity atom of volume v :

where e{ is the induced stress free strain in the impurity. e/i being linearly related to the dislocation

An uppei limit of the binding energy can be estimated assuming that the stored elastic energy by the impurity is completely relaxed in the boundary core [6] :

4 AvZ

w5b

9 p - 7

(9

Table I summarizes the different elastic interac- tion energies.

It appears therefore that the interaction for high angle boundaries is at shorter range ( l l r 3 ) than for low angle boundaries ( l l r ) .

The binding energy takes its principal part from the size effect, except for twist sub-boundaries in

Elastic interaction for sub-boundaries and high angle boundaries

size effect

Sub-boundaries elastic constant effect W,sb a k 2

r < x = b/B ^r

(3 < 200 anharmonicity effect Au Wiba

*

High angle image force (anisotropy) W , B ~ a f Au2 -

grain boundaries r3

8 > 20, image force (free surface) W f b a - Av2 -

r3 isotropic medium. It is therefore always negative.

Its value is given for various impurities in alumi- nium in table 11, for pure tilt sub-boundaries (size plus modulus effect) and high angle boundaries. It is typically of a few tenths of eV.

The binding energy is less for twist boundaries : as an example a twist sub-boundary in copper has :I

binding energy by anisotropic size effect of - . 7 eV with an aluminium atom.

Similarly the anisotropic term is small a s com- pared with the relaxation one for high angle boun- daries - .04 eV for aluminiuni with a tilt boun- dary misoriented of 7r/4 around { 100 ', in copper.

3 . Electronic interactions. - Grain-boundary and impurity introduce local perturbations in the elec- tronic structure of the crystal. Their interaction results from the interference between the two perturbations, and can first be solved in the free electron approximation. But taking into account the discrete nature of the ions, nearly free electrons must be used. One is then led to utilize the total energy of a metal which includes a band structure term expressed in terms of ion pair interactions. Of course the analysis is then limited to normal metals.

3.1 FREE EI>ECTRONS. - For an impurity in a metal the defect or excess of nuclear charge A Z is screened by the conduction electrons, producing long range oscillations in cos(2 k~ r)/r3 of both electron charge density and electrostatic poten- tial cp.

They are the so-called Friedel oscillations 171.

The interaction between two impurities proceeds

from the overlapping of their diffraction shells and can be approximated by [8]

Similarly the local density change in high angle boundary core and in edge dislocation core (then in a low angle tilt boundary) should induce Friedel oscillations with an asymptotical form in sin (2 kF r)lr2 for a grain boundary (plane oscillations), and in sin (2 k F r)ly512 for a dislocation (cylindrical oscillations). In the dislocation case oscillations coming out from the long range elastic dilatation, also in sin (2 k F r)1r512, must be superimposed [93.

As previously shown the boundary-impurity interaction can be obtained by .multiplying the screened electrostatic potential of the boundary by the impurity charge difference AZ. Such an estima- tion is necessarily rough, because both screens are not handled self-consistently. Furthermore the boundary core perturbation is probably too large to be treated by first order perturbation theory therefore modifying the amplitude of the oscilla- tions and shifting their phase.

To our knowledge no calculation of cp(r) has.been done, except for the dilatation of an edge disloca- tion, which contribution is small, leading to a binding energy of a few eV for AZ = 1 in Cu.

The nearly free electron approach given in the next section allows in principle a more realistic estimation of the interaction, for it takes into account the position of the lattice ions on both sides of the boundary and of the vacant sites in the boundary core.

3 . 2 N E A R I . Y FREE E1,ECTRONS. - Using a per- turbation theory up t o second order, the total energy E of a metal is classically given by

E, represente the kinetic and electrostatic energies of the electrons corrected for exchange and correla- tion, and is constant at constant volume. Ei is the interaction energy of an i ion with its own screen.

E(rij), pair interaction between i and j ions, is the energy of the bare j ion in the screened potential of the i ion. Ei and E(ri,) also vary with the crystal volume through the dielectric constant, which is function of exchange and correlation, and through the Fermi vector k , .

TABLE TI

Binding energies in eV of different impurities in A1 with : tilt sub-boundaries (size plus modulus effects) WsP,

+

N a t u r e o f t h e

impurity Mg Si Zr Cr Mn F e Cu Zn

w,r)r

wz$

w2

C4-144 P. GUYOT AND 3.-P. SIMON

The pair interaction is

with

In (9) the direct ion-ion first term has been added t o the ion-electron-ion interaction. v(y) is the form factor of the bare ion screened by the dielectric constant ~ ( q ) . T h e logarithmic singularity at q = 2 k~ of the Hartree dielectric function ~ ( q ) produces long range Friedel type oscillations of E ( r ) in cos (2 k~ r)/r3.

Writing E in the form of the series (8) allows to calculate the energy of a crystal transformation, a t constant volume and at 0 K, in which the only variation of t h e s t r u c t u r e d e p e n d e n t t e r m I/2 Zi, E(rii) is involved : prediction of phase stability [10], atomic configurations and energy of crystalline defects not modifying the crystal volume, such as stacking faults [lo] and screw dislocations [ I 11. The extension of the method t o the calculation of the energy of boundaries others than perfect twins must necessarily consider the energy variation due to local density changes. It is unfortunately not the case for calculations made by Morse o r similar type potentials when the cohesive energy of a crystal is uniquely expressed by a sum of pair interactions. Before trying t o calculate the boundary energies and boundary-impurity interac- tions, let's first briefly review the available pair interactions.

3 . 2 . 1 Pair interactions :

Different E(r) have been proposed in pure metals depending upon v(q) and ~ ( q ) .

i. point like ions (Hartree's model) :

ii. non local pseudo-ions : the pseudo-potentials are calculated from first principles, orthogonalizing the conduction electron waves to the core electron ones by an O.P.W. method (Harrison [l2], Pick [13]).

111. ... local pseudo-ions (Heine-Abarenkov-Ani-

malu [14])

v(r) = - VO for r < rm

- Ze2

- - - for r > rm r

where V, is fitted on the core electron energy levels. Shaw [IS] optimized such a potential by varying r,,, with the quantum number I and electron energy ^E ; his pair interaction, calculated with a dielectric constant corrected for exchange and correlation has no minimum near the first neighbour position. The comparison between Shaw and Pick pair interactions in Al can be made on figure 2.

FIG 2. - Pair interaction in Al. - Pick [ I R ] - - Shaw [IS].

In all these models the core electron wave functions in the metal are assumed to be the same a s in the free ion. Due to their d orbital overlapping in the metal, noble and transition metals are excluded from the previous description. Aluminium with its high ionic radius is also at its limit of validity.

In an alloy assumed to be infinitely dilute, the pair potential between matrix atom M and impurity I is formally similar to (9) ^:

with (12)

The form factor u ~ ( q ) of the bare pseudo-ion I is generally adjusted on the elastic constants of the impurity metal, as for instance homovalent impuri- ties in Li, Na, K f161. Corrections for the shifts of kF and conduction band bottom have been per- formed by Gupta 1171.

We have used in this work the pair potential for point like ions : in such a case v(q) = - 4 n-Ze2/0o q 2 and therefore

where EMM stands for the pair potential in the pure matrix. It is easy to show that potential (13) results also from Ashcroft type pseudo-ions if r , ~ = r , ~ .

This naive potential is certainly wrong at short distance ; but it is actually hard to say whether it is

worse than those described above for heterovalent impurities, which failed in explaining the order- disorder phenomena in Mg-Li alloys (*). Further- more the coefficient C Y M M ZIIZM of its asymptotical form

Z, cos (2 k , r )

E M I ( r ) @ M M

-

2, (2 k , rI3 ⁽¹⁴⁾

is for dilute alloys of Mg or Zn in A1 (which will be presently studied) in good agreement with the value deduced from quadrupolar effect measu- rements [IS].

In any case the applications should be limited to

I

I

3 . 2 . 2 . Grain boundary energy. - The following analysis, limited to pure symmetrical tilt grain- boundaries will be published in detail else where.

The energy y of the boundary being the diffe- rence between the total energy of a bicrystal B and a single crystal S, the procedure is as follows : i. calculation of y when B and S have the same atomic density as shown on figure 3 for n (013) boundary in a f .c.c. structure. y is then given by the variation of the structure term of (8). As in reference [lo] we group the atoms in A, planes parallel to the boundary. If &'J(z) is the interaction energy between two planes Ai and Aj, distant of z on each side of the virtual boundary plane in S, and if this pair becomes a pair A, A,. in B, with an interaction energy

FIG. 3. - a ) f . c . c . single crystal seen along [100]. h ) non- relaxed bicrystal. The tilt boundary plane is (013), with a

misorient;rtion 0 = 37" a r o ~ ~ n d [1001.

an exhaustive counting of the pair variations gives the boundary energy :

At short z, ~ 'is calculated by summation over the j

atoms of both planes in the real space. A summa- tion in the reciprocal space of the Ai plane gives the asymptotical form of ri' :

k i sin

(JZ$XGj

g i j ( z ) CL

- 2

z 2

( * ) Beauchamp, P . , private communication

for A < 2 k ~

for A > 2 k ~ ,

where A are the reciprocal lattice vectors of A,.

As k ~ is proportional to ZIi3, low dense A' planes keep a preponderant oscillating interaction even for alkali metals.

An example of q$i' (2) for a (013) boundary in A1 is shown .. ^- in figure 4.

11. for boundaries others than (111) or (113) the structure has to be relaxed. In a first approximation this relaxation has indeed been performed roughly, with a very limited number of steps and only normally t o the boundary plane. Empty cores, like for (013) shown on figure 5, are occasionally obtained. The energy variation with volume has been estimated either in using a theoretical expres- sion f o r the term Eo of (S), or through the experimental value of the formation energy of a vacancy in the case of empty cores. The volume variation of the pair interaction E(rij) has not been taken into account.

FIG. 4. - Variation of the interaction energy rpO, when a pair of (013) planes AoAs in Al at a distance nd is replaced by a pair AoAa (calculated from the Pick potential). The arrow indicates where the asymptotical form (16) coincides with the exact

calculation.

FIG. 5. - Full core (a) and relaxed empty core (b and c) for a (017) houndary The minimum boundary energy corresponds to

the C) case.

C4-146 P. GUYOT AND J.-P. SIMON

Tilt grain boundary energy has been calculated Fortunately this time, the energy variations with different pair potentials for the following performed in the two brackets of (17) need no more boundaries in Al and f.c.c. Li : (013) and (012) volume corrections for being defined between around

[loo],

results indicated in table I11 for Al, are obtained Using the I-M pair interaction (12) one then with the Pick potential ; they are in reasonable obtains the following interaction :

agreement with experience and expresses well the lower values for the twins (1 1 1 ) and ( 1 13). Besides,

the ( 1 1 1 ) twin energy appears t o be the half of the W I ( r ) = 52

(z'

+

intrinsic stacking fault energy [ l o ] . Z M ^j,zl

where 0 is the area of the unit cell in the plane TABLE 111 parallel to the boundary and z' the distance between Grain holrndary energy at 0 K in A1 calculated from the boundary and the AJ (or A") plane in the 2 part Pick potential [ I 31. of the crystal. The impurity position r fixes .i and z'

fixes j'.

Boundary type ( 1 1 1 ) ( 1 1 3 ) (013) (012) From equations (16) and (18) it is clear that W l ( r )

- - - - - oscillates at large r like

3 . 2 . 3 Grain boundary-impurity interaction. - Similarly to a work of Nourtier and Saada [19] on stacking faults, the grain-boundary-impurity inte- raction can easily be deduced from the previous analysis.

An impurity I at a distance r from a boundary has with it an interaction energy W,(r) given by the difference between the dissolution energy of I in the bicrystal B at an infinite distance of the boundary, i. e. in a single crystal S, and the dissolution energy in the bicrystal R at the dis-

tance I- Whence

sin

(Jm .

+

C

A,Z' (z'

+

The variations of W l ( r ) calculated with the Pick potential for f.c.c. Li and Al, with a valency difference of

+

The sensitivity to the pair potential EMM has been tested for ( I 1 I ) and ( I 13) twins in Al and is represented on figure 9 for the ( 1 1 1 ) twin. Unless the W I ( r ) shape stays unchanged, its magnitude varies significantly with the potential ; table TV

gives the binding energy for a AZ = - 1 impurity in AI, taken a s the first minimum value of W l ( r ) .

The analysis of these results shows that : where, according to figure 6 , E I Z B (or E M Z B ) is the

sum of the pair interactions of I (or a matrix atom i. the wave length of the long range oscillations is M a t the same position as I ) with the 2 B part of the larger f o r dense boundary planes, as expected bicrystal ; E1zS (or EMZS) has the same meaning but from (19) : 9

hi

hi

A

ii. at the close proximity of the boundary, the interaction energy is about 0.1 eV for Al and Li, and its sign varies either with the boundary type or t h e 42 = 2, - Z,cl sign. For :I dense boundary like ( I 1 I ) the binding energy is one order of magnitude lower,

-

Frc, 6 . - Definition o f the interaction between boundary and iii. beyond

-

impurity. tion energy is of a few eV.

Binding energy (in e V ) with ( 1 1 1 ) and ( I 13) twins for a AZ ⁼ - 1 impurity in A / .

Potential Pick [13] Shaw [15] Animalu [20] Ashcroft [21] Shyu-Gaspari [22]

- - - - - -

WB (111) - 0.017 ^- 0.005

-

WB (113) - 0.11 ^- 0.03 - 0.128 - 0.055 - 0.08

FIG. 7. - Interaction energy W , ( r ) for high angle boundaries (a and b) and twins (c and d) in f . c . c . Li and Al. The distance requals n times the unit length indicated on each profile.

P. GUYOT AND J.-P. SIMON

F I G . 8. - (013) bicrystal in Al (h case of fig 5) T h e n u n ~ h e r s give in eV the interaction energy of 21 AZ = - I impurity, when

substituted a t the corresponding lattice positions.

4 . Conclusion. - In this work we have estimated the theoretical interaction between impurity and grain boundary from both elastic and electronic points of view. The electronic analysis is only meaningful for normal impurities in normal metals.

Unless the used pair potentials lead to reasonable grain-boundary energies, the obtained electronic interactions are certainly semi-quantitative at least at short distance.

On this basis it appears that :

i. for low angle boundaries the elastic interaction dominates in magnitude and range the electronic one ;

ii. for high angle boundaries the electronic interac- tion can be relatively important ; it should decrease

( a t u 5!

F I G . 9. - Interaction energy W , ( r ) for the (I I I ) twin in Al, c a l c u l a t e d f r o m t h e p a i r i n t e r a c t i o n of : P i c k [13], t S h a w [15], O A n i m a l u [201, 0 A s h c r o f t [21],

*

Gaspari [22].

less rapidly with the distance (r2) than the elastic interaction (r-'). However an impurity drag can not result from its sinusoidal nature.

Finally it is well known that transition impurities are of practical interest for metallurgical purposes ; one expects a stronger effect taking its origin in their virtual d bound states. But at the moment no evaluation of the interaction has been made.

Aknowledgments. - We thank the P.U.K. Alumi- nium Branch and the Direction GCnCrale d e la Recherche Scientifique e t Technique (contrat nb.

73-7- 1559) for financial support.

References

[ I ] FRIEDEL J. : 4e Colloque d e Mttallurgie, Saclay-PUF, (1960), p. 95.

[2] LI J . C. M. : Electron microscopy clnd Strength o f crystals, (J. Wiley et Sons) 1963. 713.

[3] HIRTH J . P., L.OTHE P. : Theory of disloccrtions (McGr;rw Hill) 1968.

[4] BUL.L.OUCH R., NEWMAN R. C. : Harwell report AERE.

R6215 (1969).

[S] BACON D. J . : Phys. Stat. Sol. b 50 (1972) 607.

[6] ESHEI.BY J . D. - Solicl State Phys. 3 (1956) 79.

[7] FRIEDEI. J . : A d v . Phys. 3 (1954) 446.

181 B L A N D I N A. : 1. Physiqtte R a d . 22 (1961) 507.

[9] PERRIER J., PEUTZ M. : Int. Report. Orsay (1965).

[lo] BLANDIN A., FRIEDEL J., SAADA G. : J. Physique Colloq. 27 (1966) C3-128.

(1 11 R A R I E R J., GRII.HE J . : J Phys. C h e n ~ . Solids 34 (1973) 103 1.

[I21 HARRISSON W. A. : Pseudo-potentials in the Theory of

Metals (Benjamin, N.Y.) 1966.

[I31 PICK R. : I. Physique 28 (1967) 539.

[I41 ARARENKOV I V., HEINE V. : Phil. Mag. 12 (1965) 529.

I151 SHAW Jr. : J. Phys. C 2 (1969) 2335.

[I61 TORRENS 1. M., GERL M. : Phys. Rev. 187 (1969) 912.

1171 GUPTA 0. P. : Phys. Rev. 174 (1968) 668.

[I81 MINIER M . : ProprihtPs Electroniqlies d e s MPtalr.~ e t Allinges (Masson) 1973, p. 174.

[I91 NOURTIER C., S A A D A G. : Fundamental aspects of Disloca- tion theory, NBS (U.S.) Spec. Pub[. (1970) 317, 11, p. 1259.

I201 ANIMALU A. 0 . E . : Phil. Mag. 11 (1965) 379.

[211 S H Y U W. M.. GASPARI G . D. : Phys. Rev. 163 (1967) 667 [221 SHYU W. M . , WEHLING J. M., CORDES M. R., GASPARI G .

D . : Phys. Rev. B4 (1971) 1802.

DISCUSSION

P. LESBATS : Dans le modMe gComCtrique que ment ressortir une variation de l'interaction a trbs vous avez expost, les plans rkticulaires sont courte distance, mais elle reste ailleurs pratique- conservCs et ne sont pas dCformCs. Dans les joints ment identique.

autres que la macle (1 11) il y a en fait une relaxation Je peux donner une autre explication ^: une qui va faire que les oscillations vont se brouiller ou analyse en plans plus fine, pour tenir compte de la se superposer et plus ou moins disparaitre. relaxation, revient B considkrer plus de plans moins

P. GUYOT : E n ce qui concerne les joints pCriodiques que j'ai trait&, une relaxation des atomes au voisinage du joint n'emp2che pas une analyse par plans atomiques parallaes au plan du joint. La relaxation se traduit donc simplement par une variation relative de la distance entre quelques plans au voisinage du joint. Ceci peut changer 1'Cnergie d'interaction de I'impuretC B trbs courte distance du joint, mais ne peut remettre en cause la prCsence des oscillations. Nous avons par exemple calculC l'interaction B une impuretC avec un joint (013) non relax&, en ce sens que nous n'avons pas enlev6 d'atomes dans le cceur du joint (structure du joint de coi'ncidence). La comparaison avec le joint (013) relax&, prCsente dans I'exposC, fait effective-

denses en atomes. IndCpendamment de leur dis- tance, ces plans vont avoir des interactions compor- tant davantage d'harmoniques car ils auront plus de A < 2 kt.. La superposition conduit par suite B des longueurs d'onde d'oscillation plus courtes, et ceci uniquement B trks courte distance du joint.

G. SAADA : Les calculs que vous d6veloppez pour la sCgrCgation d'impuretCs sont valables B 0 K.

Avez-vous tent6 d'analyser l'effet de la tempCra- ture ?

P. GUYOT : Non. En ce qui concerne la modula- tion de sCgrCgation, si jamais elle est dCcernable B 0 K et tout prks du joint, elle devrait disparaitre rapidement B tempCrature croissante, compte tenu des petites amplitudes des oscillations d'interaction.