Study of local order in Au75Cu25 and Au70Cu 30. Results of diffuse scattering measurements

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Study of local order in Au75Cu25 and Au70Cu 30.

Results of diffuse scattering measurements

M. Bessière, Y. Calvayrac, S. Lefebvre, D. Gratias, P. Cénédèse

To cite this version:

M. Bessière, Y. Calvayrac, S. Lefebvre, D. Gratias, P. Cénédèse. Study of local order in Au75Cu25

and Au70Cu 30. Results of diffuse scattering measurements. Journal de Physique, 1986, 47 (11),

pp.1961-1976. �10.1051/jphys:0198600470110196100�. �jpa-00210392�

Study of local order in Au75Cu25 ^and Au70Cu30. ^{Results of diffuse scattering}

measurements

M. Bessière, ^Y. Calvayrac, ^S. Lefebvre, ^{D. Gratias and P. Cénédèse}

CECM, 15, ^rue Georges Urbain, ⁹⁴⁴⁰⁰ Vitry ^{sur Seine Cédex, France} (Requ ^{le 15 mai 1986,} accepté ^{le ler} juillet 1986)

Résumé.

L’ordre local dans Au75Cu25 ^et Au70Cu30 ^{a été étudié à partir de mesures} d’intensité diffuse sur des monocristaux trempés depuis ^une température ^T telle que T/Tc ^{~1,03 et 1,2 (Tc:} température de transition entre la phase ^{ordonnée à} grande ^{distance et la} phase désordonnée). Les données ont été analysées par la méthode de Borie-Sparks, (Acta Cryst. ^{A 27} (1971) 198-201). ^Les paramètres ^{d’ordre à courte distance et de} déplacement ^du premier ^{ordre sont} déterminés. Une structure fine des maxima d’intensité diffuse, ^en quatre lobes, ^{est observée à la} position spéciale ^{100 dans} l’espace réciproque pour ^tous les cas étudiés. Les cartes d’intensité recalculée d’ordre à courte distance montrent que les fonctions de corrélation de paires (paramètres

d’ordre à courte distance) lointaines sont essentielles pour reconstruire la ^structure fine de l’intensité

expérimentale. ^La détermination des potentiels d’interactions de paires ^à l’aide de deux approximations thermodynamiques (Bragg-Williams ^et la méthode variationnelle des amas avec le quadruple ^tétraèdre-

octaèdre pour ^amas maximum) ^{est donnée. Les} probabilités ^des configurations atomiques ^locales (non

mesurables par des techniques expérimentales) ^sont calculées par deux techniques : a) ^{Une méthode de}

simulation conduisant à une configuration ^{où les} paramètres ^{d’ordre à courte distance ont des valeurs aussi} près que possible ^{de celles} déterminées expérimentalement (Gehlen ^{et Cohen,} Phys. ^{Rev. A} 139, 844-855).

b) ^Une approche analytique ^{basée sur} la méthode variationnelle des amas avec le quadruple ^tétraèdre-

octaèdre pour ^amas maximum (Cénédèse et al., Progress ⁱⁿ X-ray ^Studies by Synchrotron radiation, Strasbourg, 1985). L’intensité diffuse dans Au75C25 ^{a été} aussi mesurée ^en dessous de Tc (T/Tc ^{~ 0,89) pour}

deux temps ^{de recuit} (1/2 ^{h et 24} h). L’intensité diffuse pour le traitement thermique ^{le plus court est similaire}

à celle obtenue après ^le traitement à T/Tc ~ ^{1,03. Le} traitement thermique ^le plus long ^{conduit à un fort}

renforcement de l’intensité diffuse correspondant ^{à une} augmentation ^du degré ^d’ordre.

Abstract.

Local order in Au75Cu25 ^and Au70Cu30 ^{has been} investigated by X-ray ^diffuse scattering ^from single crystals quenched ^from T/Tc ~ ^{1.03 and 1.2 (Tc:} transition temperature ^between long range ordered and disordered phases). ^Data analysis has been made by ^the Borie-Sparks ^method (Acta Cryst. ^{A 27} (1971) 198-201). Short range order and first order displacement parameters ^are determined. A fourfold splitting ^{of the}

diffuse intensity ^contours is observed at 100 special positions ⁱⁿ reciprocal space in all studied cases. The recalculated short-range-order intensity maps show that long-range two-site correlation functions (SRO parameters) ^are necessary ^for recovering ^{the fine structure of} experimental intensity. ^The determination of

pair-interaction potentials ^{with two} thermodynamic approximations (Bragg-Williams and Cluster variation Method with quadruple tetrahedron-octahedron for maximum cluster) ^is given. ^The probabilities ^{of local}

atomic configurations (not measurable by experimental techniques) ^are calculated by ^two techniques : a) ^A

simulation method leading ^{to a} configuration where short range parameters ^{are as close as} possible ^{to the} experimental ^ones (Gehlen ^and Cohen, Phys. ^{Rev. A} 139, 844-855). b) ^An analytical ^CVM approach ^with quadruple tetrahedron-octahedron for maximum cluster (Cénédèse et al. , Progress ⁱⁿ X-ray ^Studies by Synchrotron radiation, Strasbourg, 1985). ^{The diffuse} intensity ⁱⁿ Au75Cu25 ^{has been} also measured below Tc

(TlTc ^~ 0.89 ^{for two times of annealing (1/2 h and 24 h). The diffuse intensity for the short} heat-treatment is similar to that obtained after the treatment at T/Tc ~1.03. The longer heat-treatment results in a

sharpening ^of the diffuse intensity distribution corresponding ^{to an increase in the} degree ^{of order.}

Classification

Physics ^Abstracts

61.55H

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470110196100

1. Introduction.

The short-range ^order (SRO) ^diffuse scattering ^from

the disordered Au3Cu ^alloy has been studied by

means of electron [1] ^and X-ray diffraction methods

[2]. ^{The SRO diffuse} intensity ^{of the} X-ray study

showed ^an _egg shapped intensity distribution located around the superlattice position, ^{with the} major ^axis along the line between the two nearest fundamental reflections, ^{whereas a more} precise picture ^was

obtained by ^electron microscopy: ^a splitting ^into

four diffuse spots around the ( 110 ) superlattice position ^was observed. This discrepancy ^{between X-}

ray and electron diffraction results was due to the lack of resolution of the X-ray spectrometers.

Advances both in the resolution of the X-ray spectro-

meters and in data analysis ^{allow the} splitting ^{to be}

seen by X-ray diffraction, ^{as we} have shown in a

previous paper [3].

In the ^same study ^{the SRO} and size effect parameters have been determined and the pair

interaction potentials have been calculated by ^the Clapp-Moss ^method [4].

The variation of local order with temperature ^and concentration is of considerable interest. The data

can be used to model important physical properties

such as thermodynamic (pair interaction potentials),

mechanical and electrical properties. ^{There are a}

few set of experiments exhibiting ^this variation of local order with temperature : ^CuAu [5], Cu3Au [6]

and Ni3Fe ^{[7], and with} temperature ^{and concentra-}

tion : Ni-Mo (10.7 ^{and 20 at % of} Mo) [8] ^{and Cu-Al} (9.13 ^{and 14 at % of} Al) [9].

The first publication [3] presents ^the result of SRO determination in AU75Cu25 ^{annealed at 485 K}

(T/Tc = 1.03 , ^{this one reports the} results of the diffuse X-ray scattering ⁱⁿ Au75Cu25 ^{annealed at}

573 K (T/Tc = 1.2 ^{and Au7oCU30 annealed at}

507 K (r/Tc-1.03) ^{and 573 K} (r/Tc-1.2).

These two alloys ^{form a} solid solution with a face centred cubic structure above the ordering tempera-

ture (To = 473 K ^for Au75Cu2, ^{and 493 K for} Au70CU30). ^Below To, D. Gratias et al. [10] ^have

shown for AU75CU25 ^that, in addition to the usual ordered structure (type L12), one-dimensional perio-

dic antiphase ^structure (PAP) ^is present ^{in the} upper part ^{of the ordered} region ^{in the} phase diagram. ^For Au7oCu3o ^{we report a short} study by electron micros- copy showing the existence of a two-phase ^domain (L12 ^and PAP).

2. Experimental.

2.1 ALLOY PREPARATION. - The alloys ^are prepa- red by melting ^{of 99.999 % Au and ASARCO}

99.999 % Cu under secundary ^{vacuum. The} single crystals ^{are grown in a} graphite ^{crucible under}

vacuum by ^a horizontal Bridgman technique. They

are then homogenized by annealing ^{for 2 h at}

1 258 K and slowly ^{cooled to room} temperature.

Slabs (14 mm in diameter and 2 mm in thickness),

obtained by spark machining, ^are disordered by annealing ^{for 1 h at 773 K,} slowly ^cooled (2 h) ^{to the} annealing temperature, annealed for 1 h and rapidly

cooled either by ^water quenching, ^or by pulling ^out

the furnace.

2.2 INTENSITY MEASUREMENTS. - The diffraction

experiments have been performed using ^cobalt Ka radiation focused at the receiving ^slits by ^a doubly ^bent pyrolitic graphite monochromator. The second and higher order harmonic wavelengths ^are

eliminated using ^{a Si} (Li) solid-state detector. The beam divergencies ^are adjusted ^{to allow measure-}

ments of the diffuse intensity ^at points ⁱⁿ reciprocal

space separated by ^Ah

^0.1 reciprocal-lattice ^unit (r.l.u.). ^{Due to the} large instrumental width of

Bragg peaks, ^the measurements were stopped ^close

to 0.25 r.l.u. around these peaks. ^The intensity ^of

the direct beam is normalized from the integrated

intensities of the Bragg peaks ^{of a standard} Ni3Fe briquet. ^The crystal is mounted ^{on a} specially designed cryostat, allowing measurements at 80 K.

The detailed description of the diffractometer has been given ^elsewhere [11].

The total diffuse intensity results from the simulta-

neous effect of the short-range ordering ISRO ^and

the static and thermal displacements ^{of atoms. This}

last contribution, very weak ^at 80 K, ^{has been}

eliminated using the method of Walker and Chipman [12]. ^The experiments ^{have been} performed ^{at 80 K.}

The remaining ^diffuse intensity, ID may then be written as :

where N cAe B ( fA - f B ) 2 designates ^{the usual}

Laue monotonic scattering ; ^N is the total number of atoms in the X-ray beam, CA ^and CB ^{are the}

concentrations of A and B chemical species, ^the

atomic scattering factors of which are respectively fA ^and fB.

Since these terms obey ^different symmetries they

can be separated following ^{the «} Borie-Sparks approach » developed by Gragg ^{and Cohen} [13, 14]

(the ^{volume of} measurement was sampled ⁱⁿ recipro-

cal space ^at 1 500 points ^{at interval of o.1} r.l. u. ). ^The Warren-Cowley parameters ^a, the first and second orders displacement parameters, respectively ^{y and} 8, s, ^are obtained by Fourier inversion of their

corresponding intensities [15].

The statistical counting ^{error attached to the}

diffraction data induces a subsequent ^{error on the}

SRO parameters which has been estimated using ^the analysis developed by ^{Wu et al.} [16].

3. Analysis of the diffraction data.

3.1 PRELIMINARY RESULTS TO THE SRO DETERMI- NATION.

As already mentioned, ^{the measure-}

ments have been carried out at low temperature ⁱⁿ

order to minimize the thermal diffuse scattering

background. Therefore, both the kinetics of establis- hment of SRO at the annealing temperature ^{and the} subsequent quench ^{to room} temperature ^{have been} first studied.

Au75CU25 ^{samples, first} disordered at 773 K have been annealed at 485 K during ^various annealing

times (1/2 ^{h, 1 h and 20} h) and then air quenched.

The state of local order has been roughly ^estimated by comparing the diffuse scattering distribution around the 100 special point (Fig. 1). ^The intensity

distributions for 1 h and 20 h are essentially ^{the same}

with a slight re-enforcement for the latter case, the maximum intensity attaining ^{12 Laue units instead of}

11 Laue units for the former case. No effect of the conditions of the quench (air/water) ^has been detec- ted at this temperature.

For higher annealing temperature (573 K) ^{a diffe-}

rence in the intensity map is observed. For example

the Au70cu3O ^air quenched sample ^{exhibit a far more} developed ^{state of order} (Fig. 2) analogous ^{to the}

one obtained after an annealing treatment at 507 K.

A complementary study of the order-disorder transition in Au75CU25 ^and Au70cu3O ^by differential

scanning calorimeter [17, 18] confirms the X-ray

results : figure ³ shows, ^for example, ^{that the} equili-

brium state of SRO is almost obtained after annea-

ling ^{1 h at 507 K for} Au7Ocu30.

The final procedure ^{has been set} up ^as follows : all

samples have been disordered at 773 K and slowly (2 K/min) ^{cooled to the} annealing temperature where they ^have been maintained for 1 h before either a water quench (from ⁵⁷³ K) ^{or an air} quench (from ^lower temperatures).

Fig. ^1.

^Diffuse intensity around 100 for AU75CUL25

annealed. 1/2 h at 485 K : : -.-.- ; 1 h at 485 K :

; 20 h at 485 K : - - -.

Fig. ^2.

^Diffuse intensity ^{around 100 for} Au7oCu3o

annealed at 573 K and cooled: a) by removing ^the furnace ; b) by ^water quenching.

Fig. ^3.

^Kinetic measurements by differential scanning calorimetry ^on Au70CU30- ^{AQ is the enthalpic variation}

measured on continuous heating, ^after preannealing ^at

507 K for various times.

3.2 X-RAY DIFFUSE SCATTERING RESULTS.

The

investigation ^of X-ray ^diffuse scattering ^on AU7SCU25

near T/Tc = 1.03 ^{has been reported in a previous}

paper 3]. Besides the quantitative determination of SRO parameters ^{the most} important ^{results are the}

existence of :

a) ^a strong size-effect modulation,

b) ^{a fourfold} splitting ^{of SRO} intensity ^at 100,110

and equivalent positions ⁱⁿ reciprocal space,

c) long-range pair correlations which are neces-

sary to reconstruct the SRO intensity map.

The present study enlarges ^{the set} of data charac-

terizing ^{the SRO} by ^a scanning ^{in both} temperature

and concentration.

Figure 4 shows the SRO intensity distribution in the h2, ^h29 0) reciprocal plane ^{for a} AU75Cu25 alloy

annealed at 573 K and water quenched. Figures 5a,

b show the SRO intensity ^{in the same} plane ^{for a} Au.¡OCU30 alloy ^{annealed at 507 K and 573 K} respecti- vely. The main diffuse intensity is located around the 100 special point ^{and a} tendency ^{of fourfold} splitting

is clearly ^{observed as in} Cu3Pd [19] ^and Cu3Au [6].

For comparable ^low annealing temperature ^the

state of SRO is ^more developed ^for Au70CU30 ^than

for Au3Cu. This agrees with the experimental phase diagram which does not show any maximum ^at the

stoichiometry Au3Cu [3].

Also an elongation of the 100 peak along ^the [100]

direction is observed which decreases with increasing

copper ^content and lowering temperature. ^This elongation ^is perpendicular ^{to the one} observed in

Cu3Au and has been attributed in the past ^{to a} strong value of the pair potential V3 [20].

The list of SRO parameters ^is given ^{in tables I, II,}

III with the corresponding ^errors determined ^as described in paragraph 2.2. The absolute value of these parameters ^becomes rapidly ^small (less ^than 0.005) ^{with a few} exceptions typically ^{at the llth}

shell.

The SRO intensity ^{has been} resynthesized by performing the inverse Fourier transform of previous

SRO parameters (a lmn ). ^{Figure 6 shows an exam-}

ple of the recalculated intensity map with a Imn up ^to the 15th, ^the 45th, ^{and the} 68th shell. In all cases, ^a

qualitatively good ^fit requires a Imn up ^to the 68th shell (Fig. 7). ^{The SRO} intensity recalculated with a lmn only up ^to the 15th shell gives ^{some of the} shape

of the SRO intensity distribution. The characteristic

splitting of the fine structure of SRO intensity ^is

induced by ^the high ^order almn, although ^their

absolute values are small compared ^{with those of the}

lower order parameters.

High ^{harmonics are} required ^for recreating ^the

fine structure intensity ^{at non rational} positions ⁱⁿ reciprocal space. These long ^range pair correlations, observed for both compositions ^at temperatures above Tc, ^are consistent with diffraction patterns ^of the antiphase structures (PAP) ^{which are known to} develop ⁱⁿ Au3Cu [10]. ^These structures are also found in Au70cu30 ^{as shown in} figure ^{8. A} two-phase region ^{made of PAP and} Ll2 structures appears for

specimen annealed under vacuum for five days ^at 480, 485 and 491 K, ^{and air cooled. In all cases, the}

diffraction patterns show four satellites around the superstructure spot ^{and the} images ^{show a clear} tendency ^{for PAP} structures.

For the Au75CU25 single crystal the diffuse intensity

has also been measured for an annealing temperature

Fig. ^4.

Experimental isointensity map (in ^{Laue monoto-}

nic units) of SRO for the h3

⁰ reciprocal ^lattice plane ⁱⁿ AU75Cu25 ^{annealed at 573 K.}

Fig. ^5.

Experimental isointensity map (in ^{Laue monoto-}

nic units) of SRO for the h3

⁰ reciprocal ^lattice plane ⁱⁿ

AU70CU30: ^{a) annealed at 507 K ; b) annealed at 573 K.}

Table I.

SRO parameters with their errors for AU75Cu25 ^{annealed at 573 K.}

Table II.

SRO parameters with their errors for AU7oCu3o ^{annealed at 507 K.}

below Tc (1/2 h and 24 h at 423 K) ^{in the} L12 ^{zone of}

the phase diagram.

After annealing for 30 min we obtain the same

parameters ^{and the same} distribution of SRO inten-

sity ^{as for the} equilibrium ^{SRO at 485 K} (Fig. 9).

After annealing for 24 h the measured diffuse

intensity ^is high : the maximum intensity is 100 times the Laue monotonic intensity instead of 7 times the Laue monotonic intensity obtained after annealing

for 30 min. Since the effects of displacements ^have

been neglected, only the minimum volume has been measured to obtain ISRO. The value of the diffuse

intensity and those of the SRO parameters

(Table IV) ^{show that the} alloy evolves towards its

long range order equilibrium ^{state. We note on the}

diffuse intensity map (Fig. 10) that the fourfold

splitting is still detected but ^an additional strong

central peak ^has appeared ^{and the} elongation ^of spots along ^the (100) directions has vanished : this

shape may be due ^to small cuboid domains and should vanish once the size of the domains is large enough.

The intensity has been also recalculated with a lmn parameters : ^{for the two} annealing treatments, a lmn up ^to the 68th shell ^are necessary ^{to have a}

good fit with the experimental ISRO, ^{as for the}

previous studies in Au75Cu25 (Fig. 11).

Table III.

SRO parameters with their errors for AU7oCu3o ^{annealed at 573 K.}

Fig. ^6.

Isointensity map synthesized ^from experimental

almn parameters, ⁱⁿ AU75 Cu2 .5 ^{annealed at 573 K : a) up to}

the 15th shell ; b) ^{up to the 45th} shell ; c) ^{up to the 68th}

shell.

Fig. ^7.

Isointensity map synthesized ^from experimental

almn parameters up ^to the 68th shell in Au7oCu3o ^{annealed :} a) ^{at 507 K ;} b) ^{at 573 K.}

Table IV.

SRO parameters for AU75Cu25 ^annealed

below Tc : 1/2 ^{h and 24 h at 423 K.}

Fig. ^8.

Au70cu3O ^{annealed at 491 K: a) electron}

diffraction ; b) ^{dark field} image with g = 100; ^{the foil is} oriented close to [001].

Fig. ^9.

Experimental isointensity map (in ^{Laue monoto-}

nic units) ^{of SRO for} the h3

⁰ reciprocal ^lattice plane ⁱⁿ

AU75Cu25 annealed 1/2 h at 423 K.

Fig. ^10.

Experimental isointensity map (in ^{Laue mono-}

tonic units) of SRO for the h3 = 0 reciprocal ^lattice plane

in Au75Cu25 annealed 24 h at 423 K.

3.3 EFFECT OF STATIC ATOMIC DISPLACEMENT. - The analysis of the diffuse intensity measurements shows the importance of the effect of static atomic

displacement ^{in Au-Cu} alloys. ^{This result is} expected owing ^{to the} difference in size between the two

atoms, ^{as indicated} by ^the important ^{variation of}

lattice parameter ^versus concentration (Aa ⁼

5 x 10-3 A per atomic percent). ^An insufficient correction of this effect does not permit observation of the fine structure of the SRO intensity : ^{this can} explain why Batterman’s results [2] ^{failed to show}

the fourfold splitting of the SRO intensity.

The effect of static displacements ^{of the atoms}

from the nodes of the average lattice may be observed directly ^{on the} diffuse 100 peak ⁱⁿ figure ^{1 :}

the maximum is displaced from its theoretical posi-

tion. The same effect was noted by ^Batterman [2]

with AU75Cu25 and Roberts [21] ^with CU75Au25.

The values of the Fourier coefficients

( ’Yimn, ^’Y{mn’ ’Yimn) of the static displacement ^inten- sity (I1) ^{are given} in table V for Au75Cu25 ^and Au70Cu30 ^{annealed at} T Tc = 1.03 and 1.2. These

is ^values depend ^{on the structure} factors and the ratio of concentrations, ^{so a more detailed} analysis ^is

necessary (prior any comparison with other studies)

Fig. 11.

Isointensity map synthesized ^from experimental

a Imn parameters up ^to the 68th shell in AU75Cu25 ^{annealed :} a) ^{1/2h at} 423K; b) ^{24 h at 423 K.}

in order to obtain the average atomic displacement

of atomic pair ((Xlmn)’ (Ylmn’ (Zlmn)’

^{It must}

be emphasize ^{that this} quantity is ^the only ^{one which}

permits comparison ^{of the} displacement ^{effects in}

different alloys.

The relation fixing 1’Îmn may be written [22] :

where CAu ^and C Cu ^are concentrations, f Au ^and f Cu ^are scattering factors and (Xr:,cu) ^and (xf’:;u) ^{are the}

average displacement ^{of atomic} pair along ^{the x axis at the} position ^lmn

Table V.

Experimental ^values of ^atomic displacement parameters yx., Yfmn’ Yfmn.

Here, ^the majority species (Au) ^{is the atom of} higher scattering ^factor and XIC-U-C U ) ^and

1. ^{are of the same order of} magnitude. ^So

the second term of relation (2) ^{is small} compared ^to

the first term. Neglecting ^this term, ^{values of}

1. ^{were obtained (Table} VI) using, ^{for the} scattering factors, the values at sin 0/A ⁼ 0.2.

For the first nearest-neighbour ^atomic displace-

ment, it ^is interesting ^to compare these results with the values obtained form a hard-sphere ^{model in}

which the atoms in the alloy ^{are assumed to retain}

their size in the _pure state. With the distance of closest approach, d110 ^{= 2.884 A for Au} [23], ^and

the cell parameter, a ^{= 3.9853} A for Au75Cu25 ^and

a ⁼ 3.9618 A for Au70Cu30 [24], ^the displacements ⁱⁿ

a hard-sphere ^{model are estimated to be}

Xl oA° ⁼ 0.023 for Au75Cu25 ^{and 0.030 for} AoCu30. These values are considerably larger ^than

the experimental values. The ^same result has been

given ^for Cu3Pd by ^{Oshima et ale} [19]. Although ^the hard-sphere ^{model is a crude} approximation, ^{we can}

conclude from this comparison ^{that the atoms in the} alloy ^{do not retain} their proper sizes and the Au atoms are compressed.

Table VI.

Average displacement of ^atomic pair AuAu along ^{the X axis at the} position ^{lmn :} XA,l >,.

The intensity ^{of diffuse} scattering ^{due to the}

second order effects in the atomic displacements

(I2) ^is considered only ^{as a} correction to I SRO

since, ^{due to the lack of} information near the

positions ^{of the} Bragg peaks, ^{we are unable to}

obtain reliable Fourier coefficients of 12. ^However

this correction remains an important ^one, becoming

more significant ^{for weak} SRO intensities. For

instance, ^an analysis ^of the diffuse scattering ^inten- sity ^with only ^the correction of the first order effects

yields ^a positive ^value of a 110 ^for Au75Cu2, ^annealed

at r/Tc ^{= 1. 2.}

4. Discussion.

4.1 STATE OF ORDER. - Kinematical diffraction

experiments ^lead only ^{to the} determination of the

pair correlation functions. In fact, ^a given ^{state of}

SRO would require ^the complete ^set of correlation functions associated with all the different kinds of clusters. The quality ^{of a} given description depends

on the order of the correlation functions which ^are taken into account. If only ^the point correlation functions are known (as ^{in the} Bragg-Williams model) ^{the state of order is described} by ^the

concentration and, ^for long range order (LRO), by

the LRO parameter ^{which is a} point correlation function. The second level of approximation ^{is the} pair correlation function description ^{obtained from}

diffuse intensity measurements. No experimental techniques ^are presently ^{available which} give ^accu-

rate estimates of triplets, quadruplets, ^{etc. correla-}

tion functions ; ^{therefore a} theoretical model is

required.

Two different approaches ^are possible:

a) ^A simulation method leading ^{to a} configuration

whose SRO parameters ^{are as close as} possible ^to

the experimental ^ones [25]. This method makes

implicit ^{use of the} geometrical frustrations induced

by the lattice : it gives ^one possible configuration

consistent with the experimental ^{set of} pair ^correla-

tion functions with no particular thermodynamical

properties. ^The multiplet correlation functions issued

from this kind of model are such that all the linear constraints due ^to positive probabilities requirements

are satisfied. The quality of this method is better for

higher connected lattices.

b) By thermodynamical requirements : ^{choice is}

made of an Ising-Hamiltonian. ^The multiplet ^correla-

tion functions are obtained from a maximization of

an entropy functional. This method has been develo-

ped by Clapp [26] using ^the ignorance ^{function as} entropy functional and more recently by Cenedese et al. [27] using ^{a CVM} entropy ^as suggested by ^De

Fontaine [28].

Both approaches have been used in the present study. ^{In the} geometric simulation, ^{the first} eight ^a’s

were used. Tables VII and VIII present ^{the exhaus-}

tive list of all the non equivalent ^{local atomic} configurations ^of respectively the cubo-octahedron used by Clapp [26] ^{and the} quadruple tetrahedron- octahedron (QTO) used in the CVM minimization of Cdndd6se et al. [27]. Tables IX and X give ^the

results obtained by ^{the two methods,} respectively geometric simulation and CVM minimization. The ratio AP/P represents the variation of the local atomic configuration probabilities ^with respect ^to the fully disordered state. The probabilities ^{for the}

two approaches ^{can be} compared only ^{for the}

Fig. ^12.

a) Labelling the atomic sites in a cubo-

octahedron ; b) Labelling of 116 different clusters genera- ted by ^{the QTO.}

tetrahedra ; ^the triplets considered in the CVM minimization are linear triplets whereas the geome- tric simulation considers all the triplets. ^{In both} approaches, ^the probabilities corresponding ^{to the}

local atomic configuration ^{of the} L12 ^{ordered structu-}

res are higher than those expected ^{for the} fully

disordered state : the 1000 (A3B) tetrahedron

configuration value reaches 0.482 in the geometric

simulation and 0.496 in the CVM minimization for

AU75CU25 ^{annealed at 573 K} (see ^{also 100} triplets,

Table IX ; triplets ^8, octahedron 21 and QTO 106,

Table X). ^Also the octahedron 22 (Table X) ^{which is}

one of the expected clusters in the D022 ^structure (see also the increase in the cubo-octahedron 16,

Table IX, ^{and QTO} 72, ^Table X) ^{has a} positive ^and significative ^deviation (AP/P) ^{which suggests a}

tendency of the Au-Cu alloys ^to order with an

important ^{amount of} antiphase boundaries. This fact may be connected ^to the observed fine structure of SRO intensity.

The two methods lead to the same kind of

equilibrium ^{states. The} major advantage ^{of the}

CVM analytical approach is its short computation

time compared ^{to the} simulation routine, ^{and no}

introduction of periodic boundary conditions in the numerical process [17]. Moreover the multilinear

constrains, ^induced by the cluster probabilities ^{to be} positive, ^{are used to check the} consistency ^{of the} input experimental ^SRO parameters ^with respect ^to the FCC lattice frustration: a linear programming

routine is called at the beginning of the calculation for obeying ^the probabilities constrains. If the rou-

tine failed to find ^a possible solution, ^the experimen-

tal SRO parameters ^are rejected ^as being ^inconsis-

tent with the FCC geometric constrains.

4.2 PAIR POTENTIAL DETERMINATION. - Two dif- ferent approaches ^{have been} used : the second order

expansion ^{of the} Bragg-Williams approximation (also ^{known as random} phase approximation, RPA)

and an inverse CVM approximation.

The RPA leads to the susceptibility ^formula

derived by Clapp ^{and Moss} [4] :

where f3 = l/kT (k : Boltzman’s constant, ^{T : tem-} perature ⁱⁿ Kelvin) ; ^V (q) : Fourier transform of

pair potential

The pair potentials obtained from this expression

are listed in table XI. A good estimate of the quality

of the method is given by the theoretical integrated intensity (Table XI) ^{which is} equal ^{to one for the}

exact free energy functional [29]. ^As expected, ^the

RPA formula gives best results at high temperatures.

Table VII.

List of ¹⁴⁴ different configurations of the cubo-octahedron [26] (see Fig. 12 ^a for labelling of ^the

atomic sites in cubo-octahedron).

For T/Tc =ae 1.2 where the RPA is accurate

enough, ^the pair potentials ^{show a} long range oscillation like in Cu3Au alloys. They ^{do not} vary

drastically with concentration : both signs ^{and order}

of magnitude of the different terms are comparable

for Au75Cu25 ^and Au70Cu30 alloys.

A two dimensional (100) mapping ^of V ( q),

reconstructed with the four first potentials, ^{is shown}

in figure ^{13. This} mapping differs from the experi-

mental isointensity ^{maps : no} splitting ^or elongation

around the 100 special positions ^{are observed for this}

restricted range of pair potentials.

QTO).

Table IX.

Probabilities (P) ^and deviations from ^total

disorder (f1PI P) of ^the principal ^clusters configuration

obtained by ^numerical simulation.

Fig. ^13.

^Potential isointensity map reconstructed with the four first potentials ^obtained by the RPA formula.

The inverse CVM approach ^makes explicit ^{use of}

the pair potentials ^as Lagrange multipliers : they ^are directly determined once the CVM entropy ^{has been} maximized [27]. ^{The modified QTO} approximation

allows the determination of the four first potentials (Table XII).

Table X.

Probabilities (P) ^and deviations from ^total

disorder (AP/P) of ^the principal ^clusters configuration

obtained by ^CVM minimization in the QTO appro- ximation.

Table XI.

Values ofpair interaction potential ^{in meV}

calculated with RPA formula (I ⁼ integrated intensity).

Table XII.

Values of pair interaction potential ⁱⁿ

meV calculated with inverse CVM approach ^{in the} QTO approximation.

The signs and order of magnitude between RPA and CVM calculation are in good agreement. ^Howe-

ver the RPA results systematically underestimate the potential values : RPA is a fluctuation analysis ^of

the standard Mean Field Approximation ^{which is}

known to overestimate the ordering. The oscillation of the pair potential ^with respect ^to interatomic distance is clearer in the CVM than in the RPA calculation (Fig. 14). ^{As a} result the third pair potential ^{is far} stronger in CVM than in RPA. A reconstruction of V(q) (Fig. 15) clearly ^shows elongation ^{at the} special position ( 100 ) ^{on the} potential’maps ^as already ^{seen in the} experimental I SRO. It is interesting ^to notice that an important

third pair potential is also necessary for fitting ^the experimental lsRO with the RPA relation, ^{if one}

considers only pair interactions up ^to third

neighbours [20].

Fig. ^14.

Pair interaction potentials vs interatomic distance for Au75Cu25 ^and Au7oCu3o ^{annealed at}

T/Tc = 1.2, ^{(.) by the RPA} formula and (+) by ^the

inverse CVM approach.

Fig. ^15.

^Potential isointensity map reconstructed ^with the four first potentials ^obtained by ^{inverse CVM} approach.

In order to have an estimate of the SRO parame- ters at lower temperatures, ^a direct CVM calculation has been performed starting from the determined

potentials ^at high temperatures. ^The predicted ^four

first SRO parameters ^{are seen} in table XIII. The differences between observed and calculated SRO parameters (al = 10 %, a2, a3, a4

30 %) ^are

consistent with the intrinsic error of the chosen CVM approximation ^{near the} transition tempera-

ture [30].

5. Conclusion.

In the present study, ^the SRO diffuse scattering ^with

characteristic fourfold splitting ^{was measured} by ^X-

ray diffraction from single crystals ^of AU75Cu ^and Au70cu3O alloys. ^{The SRO} ( almn) ^and displacement

Ylmn ) parameters ^were determined.

These results can be judged by several crite- ria [16] :

1) ^From the definition of a Imn’ the value of aooo is 1. Here the different values of ag ^are close to 1 :

Table XIII.

Comparison of ^SRO parameters for Au75CU25 ^{annealed at 485 K} (3) : experimental ^and

calculated in CVM approach ^{with the} first four poten-

tials.

2) a 110 values ^are smaller than their maximum

possible negative ^value (1 - 1/ C A) = - ^{0.333 for}

Au75CU25 and - 0.428 for Au70cu30. ^More generally,

all the experimental values of a Imn have been checked with respect ^to their allowed range. They ^all

fit within the convex polyhedron associated with QTO ^and therefore all the probabilities ^{of the}

different local atomic configurations ^{in the QTO are} positive.

3) The reconstructed diffuse intensities do not differ significantly from the measured values (cf. ^the intensity map established from the experimental

data or recalculated from the almn).

4) ^{The values of static} displacements

( XAuAu are physically acceptable (for ^ins-

tance, compared with those calculated from a hard-

sphere model).

The pair interaction potentials determined by

either method RPA or CVM do not depend ^drasti- cally ^on the concentration. They ^{exhibit a} long ^range

oscillation which is a common characteristic of noble metal alloys ^as compared ^to the transition-transition

alloys (see ^for instance, Fig. 16, Ni3Fe ^{where the} pair potentials slow down rapidly after the first coordination shell).

The accuracy of the pair potential ^{values is} strongly dependent ^{on the} thermodynamical ^treat-

ment specially ^at temperatures ^{close to} the transition temperature. ^This certainly explains ^the important

variation of these values with respect ^to tempera-

ture : the RPA is clearly inadequate ^{for low} tempera-

tures and the modified QTO ^{CVM could be} replaced advantageously by other clusters approximations.

A major difficulty is the cluster size which must be greater ^{than the} largest considered pair interaction distance. For long range interactions, the inverse CVM would be untractable. A k space formula-

Fig. ^16.

Comparaison ^of pair interaction potential ^ratio

Vlmn/Vllo (calculated by inverse CVM approach) ^vs.

interatomic distance for: 0 N76.5Fe23.5 ^{annealed at}

TI Tc ^{= 1.2 [7] ; + AU15CU35 annealed at} T / Tc = 1.2.

tion [29] ^{based on a small cluster} size would be

probably the better way for solving this inverse

problem.

Acknowledgments.

The authors thank Dr M. Harmelin for the measure- ments by DSC and for ^a thorough discussion of their kinetic interpretation.

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