MACROSCOPIC MANIFESTATION OF LOCAL DISORDER IN
MIXED CRYSTALS AND ALLOYS
Yu.Rosenberg(1), V.Sh.Machavariani(2), A.Voronel(2), S.Garber(2), A.Rubstein(2),
A.I.Frenkel(3), E.A.Stern(4)
(1)
Wolfson Center for Materials Research, Tel-Aviv University, 69978, Israel.
(2)
Raymond and Beverly Sackler School of Physics and Astronomy
Tel-Aviv University, 69978, Tel-Aviv, Israel.
(3)
Materials Research Laboratory, University of Illinois at Urbana - Champaign.
(4)
Physics Department, Box 351560, University of Washington,
Seattle, WA 98195-1560, USA.
Abstract: A correlation between precise X-ray diffraction patterns and atomic size mismatch
in disordered mixed crystals (alloys and ionic crystals) is observed. The anisotropy of the
elastic moduli has been taken into account for evaluation of the strain energy density of the
mixed crystals revealed in XRD measurements. The novel improved procedure of microstrain
estimation relevant for metallic alloys is developed. The precursor of the phase
transformation for a quenched disordered Au-Cu alloy is identified.
PACS Numbers: 61.10. i; 61.72.Dd; 61.72.Hh; 78.30.Ly
INTRODUCTION
It is widely accepted that in the first approximation the lattice parameter of a mixed crystal or
alloy varies linearly with the composition (Vegard's law). The real structures deviate from
this oversimplified view considerably. In our extended study of mixed ionic salts with atomic
size mismatch RbBr-KBr [1,2], RbCl-RbBr [1,2,3], and AgCl-AgBr [4] strong deviations of
the local structure obtained by X-ray absorption fine structure (XAFS) technique from the
average structure obtained by X-ray diffract ion (XRD) measurements have been revealed.
The equilibrium atomic positions have been found by the XAFS analysis to be shifted from
the periodic lattice sites, ascertained by diffraction. Our further XAFS investigation of the
disordered metallic alloy AuxCu1-x (Ref. [5]) also reveals considerable deviations of the
interatomic distances from those predicted by XRD. In our present careful XRD investigation
we are trying to find the characteristic features of diffraction patterns of crystals with disorder
ascertained by XAFS. The linewidths as a function of diffraction angle have been carefully
measured for different concentrations of Au-Cu, Au-Ag, RbBr-RbCl and AgBr-AgCl. It has
been found that a clear correlation exists between the built in atomic size disorder and the
XRD line broadening in all the investigated mixed crystals.
EXPERIMENTAL
The Au-Cu alloys were quenched to avoid phase separation. The details of samples'
preparation were presented in Ref. [1,2,3,4,5]. Homogeneity of the alloys and mixtures was
verified by XRD and no trace of phase separation or superstructure were observed for all the
concentrations. The compositions of the mixed samples were established by energy
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dispersive spectoscopy with a scanning electron microscope. All specimens were found to be
homogenious within the accuracy of 1%.
XRD data were collected in the range 30-120 2 range with Cu K-radiation on :
powder diffractometer "Scintag" equipped with a liquid nitrogen cooled Ge solid state
detector. Peak positions and widths of Bragg reflections were determined by self-consistent
profile fitting technique with Pearson VII function [6]. Contributions of K2 radiation were
subtracted from the total profiles; the obtained results correspond to only the K1 component
of the K-doublet. Lattice constant computation was carried out by a reciprocal lattice
parameters refinement. The determined lattice constants are presented in Figures 1a and 2a as
deviations from the Vegard's law.
The Williamson-Hall [7] approach allows us to separate two different causes for a line
broadening. It is assumed usually [8] that the broadening  of Bragg reflection (hkl)
originated from a finite grain size of the polycrystal follows the Sherrer equation:
= /( cos hkl)
(1)
where  is the X-ray wavelength, hkl is a Bragg angle,  is a mean "effective" size of
coherent scattering region normal to the reflecting planes.
If this is the only reason for the broadening then the value ( cos hkl) should be
independent of the diffraction angle. However, the real situation is different because of a
persistence of the additional stress-induced broadening  which is given by the Wilson
formula:
= 4 tan hkl
(2)
Here  is a dimensionless value ("microstrain") which is usually assumed to be proportional
to a square root of density of dislocations. Assuming that the contributions to the Bragg peak
are mutually independent and both have a Cauchy-like profile, one gets for the total reflection
width hkl:
hkl =  +  = /( cos hkl) + 4 tan hkl
(3)
Plotting now the measured value (hkl cos hk) as a function of the argument (4 sin hkl) one
can estimate microstrain  from the slope of the line and  from its intersection with vertical
axis.
RESULTS AND DISCUSSION
The obtained microstrains  are presented in Figures 1b and 2b. It should be pointed out that
our experimental widths are not corrected for an instrumental broadening. Therefore, the
values of microstrains measured for the pure components give us the instrumental zero
points.
Let us try to probe deeper. Eqs. (2) and (3) assumes that the microstrain  is uniform
in all crystallographic directions. This assumption is doubtful for a crystalline material. It is
more physical to consider an anisotropic magnitude hkl. In Ref. [9] a uniform stress
(deformation pressure)  was assumed. In this case  in Eqs. (2) and (3) has to be substituted
by an anisotropic microstrain hkl=/Ehkl, where Ehkl is the Young's modulus in direction hkl.
The Eq. (3) transforms into the Eq. (4):
hkl = /( cos hkl) + 4 tan hkl /Ehkl
(4)
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Figure 1. Part a: The lattice parameter for
AuxCu1-x and AuxAg1-x metallic alloys
presented as a deviation from the Vegard's
law a-aVegard. Part b: The microstrain  as
determined by the XRD for AuxCu1-x and
AuxAg1-x metallic alloys.
Figure 2. Part a: The lattice parameter for
AgClxBr1-x and RbClxBr1-x ionic crystals
presented as a deviation from the Vegard's
law a-aVegard. Part b: The microstrain  as
determined by the XRD for AgClxBr1-x and
RbClxBr1-x ionic crystals.
However, it may be more reasonable to expect the real parameter of deformation to be a
density of a deformation energy u. Thus assuming this density being uniform u=(hkl)2Ehkl/2
(Hooke's law), one has to substitute  in Eqs. (2) and (3) by the anisotropic microstrain
hkl= 2u / Ehkl . As a result one obtains the following equation:
hkl = /( cos hkl) + 4 2 / Ehkl tan hkl u
(5)
Figures 3 and 4 present the comparison of these three evaluation procedures for the
Au0.20Cu0.80 and Au0.56Ag0.44 alloys, respectively. Figures 3a and 4a correspond to Eq. (3),
Figures 3b and 4b correspond to Eq. (4), Figures 3c and 4c correspond to Eq. (5). Young's
modulus Ehkl for cubic crystals [10] were calculated using a linear interpolation of data from
Ref. [11]. The scattering of the points from the linear expression is much less for the parts c.
All the other concentrations exhibit an analogous behavior. Therefore, the assumption of the
uniform deformation energy is found to be better for the metallic samples. As for the mixed
salts the broadening effect is an order of magnitude smaller and is too small to see a
significant difference between the approaches.
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Figure 3. The analysis of the angular dependence of the Bragg peaks' widths in assumption of
the uniform deformation  (part a), uniform stress  (part b) and uniform deformation energy
u (part c) for Au0.20Cu0.80 alloy.
Figure 5 presents the calculated density of the deformation energy for the metallic
(part a) and non-metallic (part b) samples. One can see from both figures that the maximal
deformation energy density roughly corresponds to the 50-50 concentration range. In
substitutionaly disordered crystal each cite may be occupied either by atom of size d 1 or by
that of size d2 with probabilities (1-x) and x respectively. The relative size dispersion  can be
written as [12]:

d
2

(1  x)d12  xd22  ((1  x)d1  xd2 ) 2
(d1  d 2 ) 2

x
(
1

x
)
2
((1  x)d1  xd2 ) 2
d
(6)
2
d
Indeed, this dispersion has a maximum at 50-50 concentration independent of the
disparity of size but its maximal value is greater for the greater disparity. In the case of ionic
crystal a distance between anions is a corresponding relevant parameter. Figure 5
demonstrates the correlation between the density of deformation energy and the size
dispersion. In spite of the fact that the relative size dispersion (and, simultaneously, the
deformation energy) within the ionic crystals is about an order of magnitude less than in the
metallic alloys and is within the uncertainty of our measurements, the results do not
contradict a positive correlation between u and the size dispersion.
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Figure 4. The analysis of the angular depandence of the Bragg peaks' widths in assumption of
the uniform deformation  (part a), uniform stress  (part b) and uniform deformation energy
u (part c) for Au0.35Ag0.65 alloy.
The anisotropy of the microstrains is compatible with our results presented in Figure 6
for lattice parameters ahkl of the Au-Cu alloys evaluated from the positions of the different
individual Bragg reflections. The set of ahkl has been calculated for every sample from lattice
spacing of corresponding Bragg reflections. The deviations of lattice parameters for different
planes from the average depend systematically on concentration. The difference grows
rapidly in the midst of th e concentration range while this effect disappears for pure Au and
Cu. It means that the effect exceeds considerably the uncertainty of the measurement. Let us
stress here the difference in the experimental origins of Figures 5 and 6. While the
microstrains are determined by the XRD peaks' widths analysis the lattice parameters ahkl for
different crystallographic planes are determined from the peaks' positions. Therefore, the
Figures 5 and 6 present the results of two independent experimental techniques.
The result presented in Figure 6 originates, probably, in different macroscopic
deformations within the grains in different crystallographic directions. This discrepancy is
maximal between (111) and (200) planes. It is compatible with the greater stiffness of the
direction [111] (space diagonal) in the Au-Cu alloys [10,11]. Thus, the deformation
(expansion) in the direction [200] is easier. Such a difference between the lattice parameters
for the different crystallographic planes may be considered as a precursor of a transformation
of the Au-Cu alloy from the metastable disordered fcc phase to other phases.
315
Figure 5. Part a: The density of the
deformation energy u for AuxCu1-x and
AuxAg1-x metallic alloys. Dotted (Au-Cu) and
dashed (Au-Ag) curves show the relative size
dispersion  (right scale). Part b: The density
of the deformation energy u for AgClxBr1-x
and RbClxBr1-x ionic crystals. Dotted (AgClAgBr) and dashed (RbCl-RbBr) curves show
the relative size dispersion  (right scale).
Figure 6. The lattice parameter ahkl evaluated
from the positions of the different individual
Bragg reflections (presented as a deviation
from Vegard's law ahkl-aVegard for AuxCu1-x
metallic alloy.
The difference in ahkl-aVegard between the (200) and the other (hkl) diffraction peaks is
in apparent disagreement with macroscopic cubic symmetry. However, the broadening of the
diffraction lines indicates that there is no perfect long range order due to the limited mean
"effective" size of the coherent scattering region normal to the reflecting planes and an
anisotropic microstrain hkl. Both of these effects broaden the diffraction peaks destroying the
long-range cubic symmetry, allowing the microstrains to violate cubic symmetry locally.
Another aspect of the results presented here to be noted is that the Au-Cu alloys were
rapidly quenched to avoid the thermal equilibrium ordering and phase separation while the
Au-Ag alloy and the mixed salts were cooled more slowly, since their disordered phases are
thermally stable. Thus, one expects that the quenched alloy would have higher strains than
the others, as observed. The microstrains are an inherent feature of the non-thermal
equilibrium quenched state and any attempt to anneal them would ca use the alloy to change
phase. These microstrains are an inherent feature of the disordered alloy at the lower
temperatures and their large magnitude is related to the large relative atomic size disparity in
the disordered Au-Cu alloys. The energy associated with the microstrains is the important
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factor in making the disordered phase of Au-Cu alloys thermally unstable at lower
temperatures.
CONCLUSIONS
1.
2.
3.
A positive correlation exists between both the line positions and the line widths of the
XRD pattern and the relative atomic size disparity of the disordered mixed crystals AuCu, Au-Ag, AgCl-AgBr and RbCl-RbBr.
Different ways to account the elastic anisotropy of the crystals are compared. It is found
that the suggested concept of the uniform density of strain energy u gives the most
adequate presentation of our experimental XRD data on the alloys. This strain energy
density depends on concentration the same way as the relative atomic size dispersion.
The difference between the lattice parameters of Au-Cu alloy evaluated from the
different Braggs' reflections (200) and (111) depends on concentration in a systematic
way. This fact is interpreted as a precursor of the phase transformation.
ACKNOWLEDGEMENTS
This work was partially supported by The Aaron Gutwirth Foundation, Allied Investments
Ltd.(Israel), by DOE Grant No. DE-FG02-96ER45439 through the Materials Research
Laboratory at the University of Illinois at Urbana-Champaign (AIF), and by DOE Grant No.
DE-FG03-98ER45681 through the University of Washington, Seattle (AIF and EAS).
REFERENCES:
1.
A.I.Frenkel, E.A.Stern, A.Voronel, M. Qian, M.Newville, Buckled crystalline structure
in mixed ionic salts// Phys. Rev. Lett. 71, 3485 (1993).
2. A.I.Frenkel, E.A.Stern, A.Voronel, M. Qian, M.Newville, Solving the structure of
disordered mixed salts// Phys. Rev. B 49, 11662 (1994).
3. A.I.Frenkel, E.A.Stern, A.Voronel, S. Heald, Solid State Commun. 99, 67 (1996).
4. A.I.Frenkel, A.Voronel, A.Katzir, M. Newville, E.A.Stern, Physica B 208&209, 334
(1995).
5. A.I.Frenkel, E.A.Stern, A.Rubshtein, A. Voronel, Yu.Rosenberg, J. Phys. IV, 7, C21005 (1997); A.I.Frenkel, V.Sh.Machavariani, A.Rubshtein, Yu. Rosenberg, A.Voronel,
E.A.Stern, Local structure of disordered Au-Cu alloys, in preparation.
6. J.Langford, D.Louer, Rep. Prog. Phys. 59, 131 (1996).
7. G.K.Williamson, W.H.Hall, Acta Metallurgica 1, 22 (1953).
8. H.P.Klug, L.E.Alexander, X-Ray Diffraction Procedures for Polycrystalline and
Amorphous Materials (NY, Wiley, 1974).
9. K.Reimann and R.Wurschum, J. Appl. Phys. 81, 7186 (1997).
10. A.E.H.Love, A Treatise on the Mathemetical Theory of Elasticity (NY, Dover, 1944).
11. Landolt-Bornstein, Numerical Data and Functional Relationships in Science and
Technology, New Series, Volume 11 (Springer-Verlag, Berlin, Heidelberg, NY, 1979).
12. A.Voronel, S.Rabinovich, A.Kisliuk, V. Steinberg, T.Sverbilova, Phys. Rev. Lett. 60,
2402 (1988).
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