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Structural simulation of an amorphisation reaction
Journal of Non-Crystalline Solids 156-158 (1993) 532-535 North-Holland
jouR HALor ] ~ ' ~ l ~ i ~
Structural simulation of an amorphisation reaction Ji-Chen Li a, N Cowlam
a
and J.E. Evetts b
a Department of Physics, University of Sheffield, SheffieM $3 7RH, UK t, Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, UK
Static relaxation calculations have been made to simulate an amorphisation reaction at an atomic level. In the model, the late transition metal diffuses into the lattice of the early transition metal and occupies both interstitial and substitutional sites, which produces positive and negative stress fields. At sufficiently high concentrations, the accumulation of these stress fields breaks down the crystalline lattice into a structurally disordered array. Trials on clusters of fcc unit cells using Lennard-Jones potentials were extended to a full calculation on a I100 atom binary array. The results are presented in terms of the partial pair distribution functions and the partial structure factors obtained.
1. Introduction
It is generally agreed that a negative heat of mixing, - A H , and a fast diffusion of one alloy constituent into the lattice of the other are necessary for a successful solid state 'amorphisation' reaction. However, there is no indication, as yet, of the relative importance of agencies such as gaseous impurities, composition gradients and interface stress and few models have emerged to describe these reactions on an atomic scale. The early stages of the mechanical alloying of powders are chiefly important for creating lamellar structures having large, clean, interfaces between the constituents. This represents a highly disordered version of the geometry which is already present in a multilayer sample. Also, since alloys of equiatomic concentration can be produced in solid state reactions, this means that large numbers of atoms of one element must diffuse across these interfaces into the lattice of the second, often with characteristic times for the reaction of minutes rather than hours. Although the mechanisms of fast diffusion are still the subject of debate [1,2], we suggest that amorphisation reactions can only proceed at the observed rates if the Correspondence to: Dr N. Cowlam, Department of Physics, University of Sheffield, Sheffield $3 7RH, UK. Tel: + 44-742 768 555, ext. 4295. Telefax: + 44-742 728 079.
large flux of the diffusing species occupies both interstitial and substitutional sites in the host lattice. We postulate, given the size difference of the alloy 'constituents is typically 15%, that the sum of the positive and negative strain fields which this occupation imposes on the host lattice can cause a critical collapse into a structurally disordered state. We believe that there may be a loosely defined reaction front between the parent lattice and the material which has been occupied by the diffusing species. This latter material will have transformed to the amorphous state if the concentration of the introduced atoms is high enough. We decided, therefore, to investigate whether any insight into the amorphisation reactions could be gained from static relaxation calculations on model systems, as an extension of our previous work on structural relaxation in metallic alloy glasses [3,4]. The process involved is one of seeding high concentrations of interstitial and substitutional defects into a crystalline array and of using the relaxation method to remove the atomic overlaps and density defects so produced. While these calculations can provide no actual information on the diffusion of the moving species into the host lattice, they demonstrate, nevertheless, that the presence of a high concentration of an introduced species can lead to a structural collapse.
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
J.-C Li et al. / Structural simulation of an amorphisation reaction
533
2. Structural relaxation calculations
Debye equation [7],
Structural relaxation calculations [5] have been described many times in the literature. The total force on an atom n at r n in a structure is given by the sum of pair forces between the atoms,
Sij(Q ) = ENij( p p
Fn(rn) = Y'~ Fnm(r.m ).
3. Structural models considered
(1)
m ~:n
If the atom n is displaced a distance, ~rn, while all the other atoms are kept fixed, then the change in the force is given by Fn(r n +
6r.)
=Fn(rn)
+
(~r." ~7)Fnm(rnm).
(2)
The force-free position of atom n is given by
e.(r. + ~rn) =
0,
(3)
or
-Fn(rn) = (~r. V)F.m(r.m ).
(4)
The displacements, Brn, can be determined from eq. (4) and the 3r, for all the atoms calculated in turn. The process can be done most conveniently by computer. Usually fractions of ~r n are added to an atom in turn, to bring the force to zero, before moving on to the next atom. In the program already developed by us [6] and used in the previous work [3,4], the size of the step is chosen automatically according to the magnitude of the force. If the starting (or the finishing) point of the relaxation calculation is a structurally disordered array, then it may be specified in terms of the partial pair distribution functions. N~j(r) counts the average of i-j pairs in radial shells r to r + A r around an i-type atom: 1 EN, •
Nij(r)=-~j~n ~m~(r-lrn-rml ).
(5)
Here N/ and Nj are the numbers of i- and j-type atoms, respectively and r = p Ar, where p is an integer. An interval of A r = 0.05 A was used in the calculation and the functions are presented in histogram form. The partial structure factors (PSFs), Sij(Q), have also been derived in order to monitor the results before and after relaxation and to make comparison with experimental data obtained by diffraction methods. The Sij(Q) may be obtained from the Nij(r) using a form of the
sin(Qp Ar) ar)
(Qp
(6)
Ar)
Here Q is the scattering vector Q = 4wsin0/A.
The amorphisation reaction in metallic multilayer 'diffusion couples' has been investigated using relaxation calculations in the following way. For simplicity, we have taken the cases of small atoms (S) inserted both interstitially and substitutionally into an fcc lattice of a larger (L) atom. We have chosen the size of the atoms to imititate cobalt (d = 2.50 ~,) inserted within a fictitious fcc zirconium lattice (d = 3.20 .~). Although fast diffusion is thought to proceed by interstitial occupation, note that in this lattice the (½, ½, 1) octahedral interstice can contain a sphere only 1.32 .~ in diameter and theo( 1, ~, 1 z) 1 tetrahedral interstice a sphere 0.72 A in diameter. Clearly substantial distortion of the lattice must accompany any such occupation and occupation of the small tetrahedral site is likely to require interaction with an adjacent defect. Lennard-Jones (6-12) potentials, ~bLL(r), ~bLs(r) and ~bss(r), have been used throughout these calculations with minima at the distances given above and the well-depths, %, taken from our work on Ni63.TZr36 metallic glass [4] as specified in table 1. At first, a series of tests was made in order to confirm that the program was working satisfactorily for these crystalline cases and also to help decide which defects or combination of defects to introduce into the L-lattice. These tests included first seeding a small S-atom into either the (½, ½, ½) or the (1, 1, 1) interstice which locally extends the L lattice and second placing either a small S-atom or a vacancy substitutionally, which contracts the L-lattice. These tests were done for Table 1 Details are given of the interatomic pair potentials ~bij(r) used
~0 (eV): r 1 (,~):
~bLL(r)
~bLs(r)
q~ss(r)
- 0.07
- 0.10
- 0.08
3.20
2.85
2.50
J.-C. Li et al. / Structural simulation of an amorphisation reaction
534
one unit cell (14L atoms), 2 x 2 x 2 unit cells (64L atoms) and 3 x 3 x 3 unit cells (127L atoms). In these latter cases, the forces on the atoms were calculated out to r--9.60 ~ which corresponds to third neighbour distances of the L atoms. Space does not permit us to describe the resuits of these trials in detail, but they showed that a high density of defects placed randomly with respect to one another in the L-lattice, with their strain fields overlapping in an irregular way, would be an appropriate model to use. A larger array of 6 x 6 x 6 unit cells (ll00L atoms) was seeded with one S-atom in the large (1, ½, ½) interstice, and one randomly in a smaller (¼, ¼, ¼) type interstice and one S-atom randomly substituted for an L-atom. This array has a Ls0Ss0 composition. The same 6-12 potentials were used again truncated at 9.6 .&. 250 iterative cycles of relaxation were made with examination after each 50 cycles to monitor the results. Energy minima were reached slowly and the change in energy between cycles was less than than 2%. In all, about 1 week of cpu time on an IBM 3083 was used. As an illustration of the results obtained, the NLL(r) pair distribution function is shown in fig. 1, for the parent L-lattice and after 50 cycles and 250 cycles of iteration. These distribution functions demonstrate the gradual loss of crystalline character of the array as the relaxation proceeds. Figure l(b) shows that after 50 cycles the interatomic distances of the parent lattice can still be easily discerned although the function is continuous over all r values. It may be noted at this point, that the structure after 50 cycles perhaps corresponds better to an intuitive view of the parent disordered lattice, than the actual starting
Z
0
1
2
t,
5
6
7
8
point of the calculations. This starting point is physically artificial because there are large atomic overlaps which must be removed. By contrast with the NLL(r) function after 50 cycles, fig. l(c) shows that after 250 cycles the second neighbour distance r(1, 1, 0)= 1.414r I has completely disappeared. Further, the second peak and its shoulder (at 5.5 and 6.3 ,~, respectively) can be seen to
Atomic
Pair distribution
Partial structure
pair
function (,~)
factor (-&-1)
rt 3.15 2.90 2.40 3.20
r2 5.45 4.65 4.60 [ 4.53 l, 5.50
10
Fig. 1. The NLL(R) pair density function is shown for the Ls0Ss0 parent array (a) and after 50 (b) and 250 (c) cycles of iteration as an example of the atomic scale changes which accompany the structural relaxation.
Table 2 Peak positions in the pair distribution functions Nq(r) and in the PSFs Siy(Q) for the 1100 atom relaxed Ls0Ss0 model
L-L L-S S-S fcc L lattice
g
Radial distance r in
r2s
Q1
Q2
Qzs
AQ1
6.20 6.40
2.50 2.85 3.10
4.43 5.00 5.60
4.80 -
0.61 0.80 0.93
J.-C. Li et aL / Structural simulation of an amorphisation reaction I
I
I
I
I
I
I
I
I
535
4. Conclusions
t.-L
u_
i_
g_
l
I
I
I
I
I
l
I
I
2
~
S
6
7
8
9
10
Scattering Vector O. in A~
Fig. 2. The partial structure factors, Sij(Q), are shown derived from the L50550 array after 250 cycles of iteration.
occur at the classic ratios r 2 = 1.72r I and r2~ = 1.96r I which are familiar from structural investigations of metallic alloy glasses [9]. Peak positions from the three partial distribution functions are given in table 2. The PSFs derived after 250 cycles of iteration are shown in fig. 2. They exhibit all the characteristics usually associated with the amorphous state. SLL(Q) has a well defined first peak and a second peak with a shoulder on the high r side. The oscillations in Sss(Q) are well developed, while those in SLs(Q) appear to grow throughout the relaxation process as if chemical order between the L- and S-atoms was being favoured by the slightly deeper well-depth of the ~bLs(r) potential. The widths, AQ, of the first peaks in the PSFs lie between about 0.6 and 0.93 ,~-1, giving a range of structural disorder A r = 2"rr/AQ of between 7 and 10 ~,. This implies that there is a characteristic range of structural reorganisation on this length scale which is smaller than the extent of the cluster, which is 27 ,~ in diameter.
The present investigation has been undertaken with two aims. The first was to determine whether static relaxation calculations could be used to model an amorphisation reaction. The second was to investigate whether the presence of a large concentration of randomly arranged defects in a crystalline array would lead to a structural collapse into an amorphous state. The characteristic results shown in figs. 1 and 2 indicate that this is possible, although the process is not yet optimised. The model used is open to criticism, because it does not provide any information about the introduction of the diffusing species into the host lattice. However, we believe it was worthwhile to investigate first whether a structural collapse is predicted to occur in a static relaxation calculation (using our existing computer programs) before attempting a more complex model. These preliminary calculations can be extended in at least three different ways in order to improve their effectiveness. First, oscillatory potentials [3,4] derived from structural data on metallic alloy glasses can be used since they consistently outperform 6-12 potentials in relaxation calculations on these materials [6]. Second, the starting array can be built by joining together disordered crystalline sub-units each of which contained one or more defects, like those used in our initial trials. Third, appropriate sub-units could be used both statically as here and also translated through different (random) multiples of the lattice dimension in order to imitate the diffusion process more faithfully. References [1] Y. Nakamura, H. Nakajima, S. Ishioka and M. Koiwa, Acta Metal. 36 (1988) 2787. [2] W. Petry, G. Vogl, T. Flottman and A. Heidemann, in: Springer Proceedings in Physics, Vol. 10 (Springer, Berlin, 1986) p. 134. [3] Ji-Chen Li and N. Cowlam, Phys. Chem. Liq. 17 (1987) 29. [4] J.-C. Li, N. Cowlam and F. He, J. Non-Cryst Solids 112 (1989) 101. [5] L. von Heimendahl, J. Phys. F9 (1979) 161. [6] Ji-Chen Li, PhD thesis, University of Sheffield (1987). [7] P. Debye, Ann. Phys. 46 (1915) 809. [8] B.E. Warren, X-ray Diffraction (Addison Wesley, Reading, MA, 1969). [9] G.S. Cargill III, Solid State Phys. 30 (1975) 227.
jouR HALor ] ~ ' ~ l ~ i ~
Structural simulation of an amorphisation reaction Ji-Chen Li a, N Cowlam
a
and J.E. Evetts b
a Department of Physics, University of Sheffield, SheffieM $3 7RH, UK t, Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, UK
Static relaxation calculations have been made to simulate an amorphisation reaction at an atomic level. In the model, the late transition metal diffuses into the lattice of the early transition metal and occupies both interstitial and substitutional sites, which produces positive and negative stress fields. At sufficiently high concentrations, the accumulation of these stress fields breaks down the crystalline lattice into a structurally disordered array. Trials on clusters of fcc unit cells using Lennard-Jones potentials were extended to a full calculation on a I100 atom binary array. The results are presented in terms of the partial pair distribution functions and the partial structure factors obtained.
1. Introduction
It is generally agreed that a negative heat of mixing, - A H , and a fast diffusion of one alloy constituent into the lattice of the other are necessary for a successful solid state 'amorphisation' reaction. However, there is no indication, as yet, of the relative importance of agencies such as gaseous impurities, composition gradients and interface stress and few models have emerged to describe these reactions on an atomic scale. The early stages of the mechanical alloying of powders are chiefly important for creating lamellar structures having large, clean, interfaces between the constituents. This represents a highly disordered version of the geometry which is already present in a multilayer sample. Also, since alloys of equiatomic concentration can be produced in solid state reactions, this means that large numbers of atoms of one element must diffuse across these interfaces into the lattice of the second, often with characteristic times for the reaction of minutes rather than hours. Although the mechanisms of fast diffusion are still the subject of debate [1,2], we suggest that amorphisation reactions can only proceed at the observed rates if the Correspondence to: Dr N. Cowlam, Department of Physics, University of Sheffield, Sheffield $3 7RH, UK. Tel: + 44-742 768 555, ext. 4295. Telefax: + 44-742 728 079.
large flux of the diffusing species occupies both interstitial and substitutional sites in the host lattice. We postulate, given the size difference of the alloy 'constituents is typically 15%, that the sum of the positive and negative strain fields which this occupation imposes on the host lattice can cause a critical collapse into a structurally disordered state. We believe that there may be a loosely defined reaction front between the parent lattice and the material which has been occupied by the diffusing species. This latter material will have transformed to the amorphous state if the concentration of the introduced atoms is high enough. We decided, therefore, to investigate whether any insight into the amorphisation reactions could be gained from static relaxation calculations on model systems, as an extension of our previous work on structural relaxation in metallic alloy glasses [3,4]. The process involved is one of seeding high concentrations of interstitial and substitutional defects into a crystalline array and of using the relaxation method to remove the atomic overlaps and density defects so produced. While these calculations can provide no actual information on the diffusion of the moving species into the host lattice, they demonstrate, nevertheless, that the presence of a high concentration of an introduced species can lead to a structural collapse.
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
J.-C Li et al. / Structural simulation of an amorphisation reaction
533
2. Structural relaxation calculations
Debye equation [7],
Structural relaxation calculations [5] have been described many times in the literature. The total force on an atom n at r n in a structure is given by the sum of pair forces between the atoms,
Sij(Q ) = ENij( p p
Fn(rn) = Y'~ Fnm(r.m ).
3. Structural models considered
(1)
m ~:n
If the atom n is displaced a distance, ~rn, while all the other atoms are kept fixed, then the change in the force is given by Fn(r n +
6r.)
=Fn(rn)
+
(~r." ~7)Fnm(rnm).
(2)
The force-free position of atom n is given by
e.(r. + ~rn) =
0,
(3)
or
-Fn(rn) = (~r. V)F.m(r.m ).
(4)
The displacements, Brn, can be determined from eq. (4) and the 3r, for all the atoms calculated in turn. The process can be done most conveniently by computer. Usually fractions of ~r n are added to an atom in turn, to bring the force to zero, before moving on to the next atom. In the program already developed by us [6] and used in the previous work [3,4], the size of the step is chosen automatically according to the magnitude of the force. If the starting (or the finishing) point of the relaxation calculation is a structurally disordered array, then it may be specified in terms of the partial pair distribution functions. N~j(r) counts the average of i-j pairs in radial shells r to r + A r around an i-type atom: 1 EN, •
Nij(r)=-~j~n ~m~(r-lrn-rml ).
(5)
Here N/ and Nj are the numbers of i- and j-type atoms, respectively and r = p Ar, where p is an integer. An interval of A r = 0.05 A was used in the calculation and the functions are presented in histogram form. The partial structure factors (PSFs), Sij(Q), have also been derived in order to monitor the results before and after relaxation and to make comparison with experimental data obtained by diffraction methods. The Sij(Q) may be obtained from the Nij(r) using a form of the
sin(Qp Ar) ar)
(Qp
(6)
Ar)
Here Q is the scattering vector Q = 4wsin0/A.
The amorphisation reaction in metallic multilayer 'diffusion couples' has been investigated using relaxation calculations in the following way. For simplicity, we have taken the cases of small atoms (S) inserted both interstitially and substitutionally into an fcc lattice of a larger (L) atom. We have chosen the size of the atoms to imititate cobalt (d = 2.50 ~,) inserted within a fictitious fcc zirconium lattice (d = 3.20 .~). Although fast diffusion is thought to proceed by interstitial occupation, note that in this lattice the (½, ½, 1) octahedral interstice can contain a sphere only 1.32 .~ in diameter and theo( 1, ~, 1 z) 1 tetrahedral interstice a sphere 0.72 A in diameter. Clearly substantial distortion of the lattice must accompany any such occupation and occupation of the small tetrahedral site is likely to require interaction with an adjacent defect. Lennard-Jones (6-12) potentials, ~bLL(r), ~bLs(r) and ~bss(r), have been used throughout these calculations with minima at the distances given above and the well-depths, %, taken from our work on Ni63.TZr36 metallic glass [4] as specified in table 1. At first, a series of tests was made in order to confirm that the program was working satisfactorily for these crystalline cases and also to help decide which defects or combination of defects to introduce into the L-lattice. These tests included first seeding a small S-atom into either the (½, ½, ½) or the (1, 1, 1) interstice which locally extends the L lattice and second placing either a small S-atom or a vacancy substitutionally, which contracts the L-lattice. These tests were done for Table 1 Details are given of the interatomic pair potentials ~bij(r) used
~0 (eV): r 1 (,~):
~bLL(r)
~bLs(r)
q~ss(r)
- 0.07
- 0.10
- 0.08
3.20
2.85
2.50
J.-C. Li et al. / Structural simulation of an amorphisation reaction
534
one unit cell (14L atoms), 2 x 2 x 2 unit cells (64L atoms) and 3 x 3 x 3 unit cells (127L atoms). In these latter cases, the forces on the atoms were calculated out to r--9.60 ~ which corresponds to third neighbour distances of the L atoms. Space does not permit us to describe the resuits of these trials in detail, but they showed that a high density of defects placed randomly with respect to one another in the L-lattice, with their strain fields overlapping in an irregular way, would be an appropriate model to use. A larger array of 6 x 6 x 6 unit cells (ll00L atoms) was seeded with one S-atom in the large (1, ½, ½) interstice, and one randomly in a smaller (¼, ¼, ¼) type interstice and one S-atom randomly substituted for an L-atom. This array has a Ls0Ss0 composition. The same 6-12 potentials were used again truncated at 9.6 .&. 250 iterative cycles of relaxation were made with examination after each 50 cycles to monitor the results. Energy minima were reached slowly and the change in energy between cycles was less than than 2%. In all, about 1 week of cpu time on an IBM 3083 was used. As an illustration of the results obtained, the NLL(r) pair distribution function is shown in fig. 1, for the parent L-lattice and after 50 cycles and 250 cycles of iteration. These distribution functions demonstrate the gradual loss of crystalline character of the array as the relaxation proceeds. Figure l(b) shows that after 50 cycles the interatomic distances of the parent lattice can still be easily discerned although the function is continuous over all r values. It may be noted at this point, that the structure after 50 cycles perhaps corresponds better to an intuitive view of the parent disordered lattice, than the actual starting
Z
0
1
2
t,
5
6
7
8
point of the calculations. This starting point is physically artificial because there are large atomic overlaps which must be removed. By contrast with the NLL(r) function after 50 cycles, fig. l(c) shows that after 250 cycles the second neighbour distance r(1, 1, 0)= 1.414r I has completely disappeared. Further, the second peak and its shoulder (at 5.5 and 6.3 ,~, respectively) can be seen to
Atomic
Pair distribution
Partial structure
pair
function (,~)
factor (-&-1)
rt 3.15 2.90 2.40 3.20
r2 5.45 4.65 4.60 [ 4.53 l, 5.50
10
Fig. 1. The NLL(R) pair density function is shown for the Ls0Ss0 parent array (a) and after 50 (b) and 250 (c) cycles of iteration as an example of the atomic scale changes which accompany the structural relaxation.
Table 2 Peak positions in the pair distribution functions Nq(r) and in the PSFs Siy(Q) for the 1100 atom relaxed Ls0Ss0 model
L-L L-S S-S fcc L lattice
g
Radial distance r in
r2s
Q1
Q2
Qzs
AQ1
6.20 6.40
2.50 2.85 3.10
4.43 5.00 5.60
4.80 -
0.61 0.80 0.93
J.-C. Li et aL / Structural simulation of an amorphisation reaction I
I
I
I
I
I
I
I
I
535
4. Conclusions
t.-L
u_
i_
g_
l
I
I
I
I
I
l
I
I
2
~
S
6
7
8
9
10
Scattering Vector O. in A~
Fig. 2. The partial structure factors, Sij(Q), are shown derived from the L50550 array after 250 cycles of iteration.
occur at the classic ratios r 2 = 1.72r I and r2~ = 1.96r I which are familiar from structural investigations of metallic alloy glasses [9]. Peak positions from the three partial distribution functions are given in table 2. The PSFs derived after 250 cycles of iteration are shown in fig. 2. They exhibit all the characteristics usually associated with the amorphous state. SLL(Q) has a well defined first peak and a second peak with a shoulder on the high r side. The oscillations in Sss(Q) are well developed, while those in SLs(Q) appear to grow throughout the relaxation process as if chemical order between the L- and S-atoms was being favoured by the slightly deeper well-depth of the ~bLs(r) potential. The widths, AQ, of the first peaks in the PSFs lie between about 0.6 and 0.93 ,~-1, giving a range of structural disorder A r = 2"rr/AQ of between 7 and 10 ~,. This implies that there is a characteristic range of structural reorganisation on this length scale which is smaller than the extent of the cluster, which is 27 ,~ in diameter.
The present investigation has been undertaken with two aims. The first was to determine whether static relaxation calculations could be used to model an amorphisation reaction. The second was to investigate whether the presence of a large concentration of randomly arranged defects in a crystalline array would lead to a structural collapse into an amorphous state. The characteristic results shown in figs. 1 and 2 indicate that this is possible, although the process is not yet optimised. The model used is open to criticism, because it does not provide any information about the introduction of the diffusing species into the host lattice. However, we believe it was worthwhile to investigate first whether a structural collapse is predicted to occur in a static relaxation calculation (using our existing computer programs) before attempting a more complex model. These preliminary calculations can be extended in at least three different ways in order to improve their effectiveness. First, oscillatory potentials [3,4] derived from structural data on metallic alloy glasses can be used since they consistently outperform 6-12 potentials in relaxation calculations on these materials [6]. Second, the starting array can be built by joining together disordered crystalline sub-units each of which contained one or more defects, like those used in our initial trials. Third, appropriate sub-units could be used both statically as here and also translated through different (random) multiples of the lattice dimension in order to imitate the diffusion process more faithfully. References [1] Y. Nakamura, H. Nakajima, S. Ishioka and M. Koiwa, Acta Metal. 36 (1988) 2787. [2] W. Petry, G. Vogl, T. Flottman and A. Heidemann, in: Springer Proceedings in Physics, Vol. 10 (Springer, Berlin, 1986) p. 134. [3] Ji-Chen Li and N. Cowlam, Phys. Chem. Liq. 17 (1987) 29. [4] J.-C. Li, N. Cowlam and F. He, J. Non-Cryst Solids 112 (1989) 101. [5] L. von Heimendahl, J. Phys. F9 (1979) 161. [6] Ji-Chen Li, PhD thesis, University of Sheffield (1987). [7] P. Debye, Ann. Phys. 46 (1915) 809. [8] B.E. Warren, X-ray Diffraction (Addison Wesley, Reading, MA, 1969). [9] G.S. Cargill III, Solid State Phys. 30 (1975) 227.