- Email: [email protected]
Extended-range structural correlations in amorphous solids
]OURNA/, O F
ELSEVIER
Journal of Non-CrystallineSolids 192& 193 (1995) 98-101
Extended-range structural correlations in amorphous solids A. Uhlherr
a
S.R. Elliott b,,
a CSIRODivision of Chemicals and Polymers, Clayton, Australia b Departmentof Chemistry, University of Cambridge, Lersfield Road, Cambridge CB2 1EW, UK
Abstract
Extended-range order (ERO), extending beyond 35 A, has been discovered in very large (> 10o000 atom) structural models of a-Si. These weak, but quasi-periodic, atomic-density fluctuations have a period of R = 3.4 A, which corresponds to the position of the first sharp diffraction peak (FSDP) in the structure factor of a-Si at Q = 1.9 ~,-~. In fact, more than half of the FSDP intensity is associated with the ERO fluctuations for r > 10 ,~. Atomic-void-based correlations are as well defined as atom correlations at large distances: the FSDP is equivalently found to be a chemical-order prepeak in Scc(Q) in the Bhatia-Thornton formalism calculated for the atom-void packing, thereby supporting the void model for the origin of the FSDP.
I. Introduction
The nature, and extent, of medium-range order (MRO) in (covalent) amorphous solids remains one of the outstanding problems in glass science [1]. The so-called 'first sharp diffraction peak' (FSDP) in the structure factor of such materials has often been taken as a signature of MRO, although its origin remains controversial [2]. Although every diffraction pattern obviously must have a first peak, the FSDP for covalent network glasses is unique in that it exhibits an anomalous behaviour in its temperature, pressure and composition (modifier content) dependence compared with all other peaks in S(Q), and as such would seem to have a qualitatively different origin. The FSDP has in the past often been assumed to arise from a particular distance in real space in the medium-order range ( 5 - 1 0 A), but this is extremely unlikely in view of the anomalously narrow width of
the peak. It is more likely that the FSDP can be regarded as a pseudo-Bragg peak associated with some type of quasi-periodicity in the real-space structure of the amorphous material, with a period given by R = 2 7 r i O 1 = 3-6~, (where Q1 is the position of the FSDP), extending over a correlation range given by D = 2 x r / A Q 1 (where AQI is the half-width of the FSDP), with correlation lengths of D = 20 commonly inferred in this way [3]. However, little discussion seems to have been given as to the structural origin of such quasi-periodicity and why it should persist to such great distances in, ostensibly, a random structure. We report here the first direct observation of such quasi-periodic, real-space density fluctuations in a very large continuous random network of a-Si. Further details are given elsewhere [4,5].
2. The structural model
* Corresponding author. Tel: +44-1223 336 525. Telefax: +44-1223 336 362.
The random network model of a-Si analysed in this study, containing 13 824 atoms, was constructed
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0022-3093(95)00334-7
A. Uhlherr, S.R. Elliott /Journal of Non-Crystalline Solids 192&193 (1995) 98-101
by Holender and Morgan [6] by molecular dynamics, together with a cell-doublingo procedure. The length of the cubic cell is L = 66.5 A; thus a radial distribution function (RDF) can be computed up to a length of 33.3 .~.
99
,
1
!
3. R e s u l t s
r
The radial density function calculated from the coordinates of the model is shown in Fig. 1 and, as expected, when plotted on a normal scale seems featureless for r > 10 A, implying that the structure is homogeneous beyond such a distance. However, on an expanded scale, atomic density fluctuations are clearly evident up to the maximum calculable distance [4]. Moreover, these extended-range fluctuations are quasi-periodic, with a period R = 3.4 .~ (although the period decreases slightly with increasing r). A clue to the origin of the ERO can be obtained by reference to the neighbour-specific partial RDFs, g,(r), where n is the 'topological' distance [7] between pairs of atoms, defined as the number of bonds in the shortest connecting percolation path. These partial correlation fluctuations are also shown for comparison in Fig. 1 for 1 < n < 13, whence it can be seen that the ERO period, R---3.4 .~ in the
2.5
•
...............
_J ,ooi
x
(a) ~"~,|~ o.,%,,.-,~ ~
#,,"~'11~.J,~ _ ~ J ~" ,~'~.
Fig. 2. Projection of a 12 A thick slice of the model containing 2476 atoms: (a) atoms shaded according to their value of n with respect to an origin atom at the centre (grey, odd n; black, even n); (b) straightforward projection with atoms shaded according to depth (dark, top; light, bottom).
2.0
x20
1.5
1.0
0.5
00 0
5
1~
r/A
15
20
25
Fig. 1. Radial density function (solid curve) for the a-Si model with, inset, a 20-fold magnification of the large-r region clearly exhibiting quasi-periodic extended-range order. The lower curves are the neighbour-specific partial RDFs, g,(r), where n ( = 1-13) is the number of neighbours from an origin atom along the shortest connecting percolation path.
total RDF, corresponds to the spacing of alternate (even-even or odd-odd) peaks in the partial g,(r). The quasi-periodicity associated with the ERO oscillations is not associated with nano/micro-crystallinity, even though, confusingly, the ERO period averaged over the full range of the oscillations, namely R---3.4 .~, is very close to the (111) interplanar spacing of the diamond-cubic crystal structure, d l l 1 -~ 3.2 ~.. However, the ERO fluctuations exhibit radial symmetry about any atom taken as origin; no evidence of crystallite domains is apparent. The radial symmetry is most clearly visible in a projection of a slice of the model (Fig. 2(a)), in which atoms are shaded differently depending on
100
A. Uhlherr, S.R. Elliott/Journal of Non-Crystalline Solids 192&193 (1995) 98-101
whether their topological distance from an arbitrary origin atom at the centre of the slice is even (black: n = 2-26) or odd (grey: n = 1-25). For comparison, no such radial order is apparent in a normal projection of the same slice in which atoms are not distinguished by virtue of their topological separation (Fig. 2(b)).
4. Discussion We believe that the ERO evident in Figs. 1 and 2(a) is due to propagated order (short-range in the case of a-Si) associated with the structural frustration inherent in maintaining network connectivity at large distances from a given origin atom. The propagated nature of the ERO oscillations cannot be understood in terms of a normal pair correlation function (RDF), but instead requires a conditional probability function based on triplet correlations. Specifically, what is required is the probability pa,(6), that, given an atom O at the origin a n d another atom i at a distance r i from the origin which is an nth neighbour of O, there is a third atom j at a distance = rj - r i from i which is an (n + An)th neighbour of O. Evaluation of p ~ ( 6 ) for the model, with r i taken arbitrarily to be 14 ___0.05 A, shows that the dominant distribution function is for A n = 2 and this peaks at 8 = 3.4 .& [4]. Thus, any density fluctuation (positive or negative) at a distance r o from an origin atom is pre~rentially propagated with a periodicity of R = 3.4 A. Hence, it is short-range order (specifically next-nearest-neighbour correlations) that are propagated. It should be noted that the peak in p 2 ( 6 ) does n o t occur at the average second-neighbour separation in a-Si, namely 3.83 A, but instead at an appreciably smaller value than this [4]. The reason for this propagation of short-range order, lies, we believe, in the structural frustration associated with the maintenance of network connectivity at large distances. Essentially, the existence of an atom i at a separation ri from an origin atom O, and which is an nth neighbour of O, severely limits the positioning of the neighbours of i ((n + An)th neighbours of O), since atoms with a separation from O smaller than r i have already been counted as ( n - An)th neighbours of O. This, combined with the tetrahedral symmetry about each atom and the o
fact that shortest percolation paths defining topological distances become roughly linear with increasing n, imposes a greater directional constraint (and hence a greater contraint on d) for An = 2 pairs than for An = 1 pairs [4]. Finally, mention should be made of the likelihood of ERO occurring in compound materials, such as the AXz-type oxide and chalcogenide glasses (A = Si,Ge;X = O,S,Se). The structure of AX2-type materials can be regarded as being a decoration by X atoms of the mid-points of the nearest-neighbour bonds between tetravalent A atoms. Thus, by analogy with the previous results, it is expected instead that An = 4 correlations ( A - X - A - X - A , or even X - A - X - A - X ) will be the dominant contribution to the production of ERO; i.e., it is now MRO that is propagated.
5. The first sharp diffraction peak The FSDP of a-Si occurs at Q1 = 1.9 ,~-1 [8]. The Holender-Morgan [6] model of a-Si accurately reproduces the experimental structure factor, including the FSDP, when correlations at all length scales in the model are taken into account. However, Fourier transformation of the ERO oscillations for r > 10 shows that they contribute more than half of the intensity of the FSDP [5]. Thus, it can be concluded that small structural models, containing only° of order 1000 atoms and having diameters of ~ 10A, cannot be expected to reproduce satisfactorily the intensity of the FSDP. Good agreement with the FSDP intensity obtained with such models can only be achieved therefore by the imposition of an otherwise unacceptable increase in local order (e.g., reduction in bond-angle fluctuations). An equivalent explanation for the origin of the FSDP in network glasses has been given in terms of the chemical ordering of cation-centred clusters and interstitial voids [2,9]. For the case of monatomic materials, e.g., a-Si, this picture reduces to a simple packing of atoms and voids, as envisaged originally by Bl&ry [10]. A void analysis [11] of the model for a-Si reveals that atom-void and void-void correlations are as well defined as atom-atom correlations, and they also exhibit ERO extending to similarly large distances [5]. For r > 10 A, the oscillations in
A. Uhlherr, S,R. Elliott /Journal of Non-Crystalline Solids 192&193 (1995) 98-101 25
.
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,
.
,
,
,
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.
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.
10
12
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Fig. 3. Bhatia-Thornton partial structure factors, S ~ ( Q ) (a,fl = N, C), calculated from the coordinates of the atoms and interstitial voids in the model, together with the total structure factor.
101
the voids are rather regular in shape and the distribution of void sizes is correspondingly rather narrow and symmetric [5]. As a result, one would expect ERO to be optimal and to extend to large distances. For the case of AX2-type materials, on the other hand, the distribution of void sizes (in, for example, a-SiO 2) is much broader and is very asymmetric [11]. Thus, this extra degree of disorder, coupled with the enhanced structural flexibility associated with the inclusion of two-fold-coordinated X atoms, might be expected to reduce appreciably the magnitude and extent of the ERO oscillations. Analysis of ultralarge models of AX2-type amorphous systems is required to confirm this point.
6. Conclusions the atom-atom and void-void partial RDFs are in phase with each other, but are in antiphase with those for the atom-void partial RDF [5]. The void-based model [2,9] for the origin of the FSDP ascribes the FSDP to a chemical-order prepeak occurring in the concentration-concentration structure factor (in the Bhatia-Thornton formalism) that is associated with the ordering of atoms and voids. That the atoms and voids are strongly correlated in the model of a-Si is attested by the existence of quasi-periodic oscillations in the atom-void partial RDF to beyond 30 A [5]. The three BhatiaThornton partial structure factors, S~13( Q) ( a, fl = N, number density; C, concentration) can be readily calculated from the coordinates of atoms and voids in the model, and are shown in Fig. 3. It can be seen that only See(Q) exhibits a pronounced peak at Q1 = o 1 1.9 A - , the position of the FSDP, lending support to the void-based picture. Finally, it should be remarked that perhaps the monatomic, tetravalent system a-Si (or Ge) may be the best in which to observe ERO at large distances. This is because fluctuations in atomic density can be equivalently regarded as arising from interstitial voids interspersed between the atoms. For the case of a-Si,
Extended-range order (ERO), in the form of weak, but quasi-periodic, fluctuations in atomic density, has been identified in a large (13 824 atom) model of o a-Si. The ERO oscillations, extending beyond 35 A, arise from propagated short-range order (secondneighbour correlations) and contribute significantly to the production of the first sharp diffraction peak in the structure factor.
References [1] S.R. Elliott, Nature 354 (1991) 445. [2] S.R. Elliott, J. Phys. Condens. Matter 4 (1992) 7661. [3] A.P. Sokolov, A. Kisliuk, M. Soltwisch and D. Ouitmann, Phys. Rev. Lett. 69 (1992) 1540. [4] A. Uhlherr and S.R. Elliott, J. Phys. Condens. Matter 6 (1994) L99. [5] A. Uhlherr and S.R. Elliott, Philos. Mag. B71 (1995) 611. [6] J.M. Holender and G.J. Morgan, J. Phys.: Condens. Matter 3 (1991) 1947; 7241. [7] R.J. Temkin, J. Non-Cryst. Solids 28 (1978) 23. [8] J. Fortner and J.S. Lannin, Phys. Rev. B39 (1989) 5527. [9] S.R. Elliott, Phys. Rev. Lett. 67 (1991) 711. [10] J. Bl&ry, Philos. Mag. B62 (1990) 469. [11] S.L Chan and S.R. Elliott, Phys. Rev. B43 (1991) 4423.
ELSEVIER
Journal of Non-CrystallineSolids 192& 193 (1995) 98-101
Extended-range structural correlations in amorphous solids A. Uhlherr
a
S.R. Elliott b,,
a CSIRODivision of Chemicals and Polymers, Clayton, Australia b Departmentof Chemistry, University of Cambridge, Lersfield Road, Cambridge CB2 1EW, UK
Abstract
Extended-range order (ERO), extending beyond 35 A, has been discovered in very large (> 10o000 atom) structural models of a-Si. These weak, but quasi-periodic, atomic-density fluctuations have a period of R = 3.4 A, which corresponds to the position of the first sharp diffraction peak (FSDP) in the structure factor of a-Si at Q = 1.9 ~,-~. In fact, more than half of the FSDP intensity is associated with the ERO fluctuations for r > 10 ,~. Atomic-void-based correlations are as well defined as atom correlations at large distances: the FSDP is equivalently found to be a chemical-order prepeak in Scc(Q) in the Bhatia-Thornton formalism calculated for the atom-void packing, thereby supporting the void model for the origin of the FSDP.
I. Introduction
The nature, and extent, of medium-range order (MRO) in (covalent) amorphous solids remains one of the outstanding problems in glass science [1]. The so-called 'first sharp diffraction peak' (FSDP) in the structure factor of such materials has often been taken as a signature of MRO, although its origin remains controversial [2]. Although every diffraction pattern obviously must have a first peak, the FSDP for covalent network glasses is unique in that it exhibits an anomalous behaviour in its temperature, pressure and composition (modifier content) dependence compared with all other peaks in S(Q), and as such would seem to have a qualitatively different origin. The FSDP has in the past often been assumed to arise from a particular distance in real space in the medium-order range ( 5 - 1 0 A), but this is extremely unlikely in view of the anomalously narrow width of
the peak. It is more likely that the FSDP can be regarded as a pseudo-Bragg peak associated with some type of quasi-periodicity in the real-space structure of the amorphous material, with a period given by R = 2 7 r i O 1 = 3-6~, (where Q1 is the position of the FSDP), extending over a correlation range given by D = 2 x r / A Q 1 (where AQI is the half-width of the FSDP), with correlation lengths of D = 20 commonly inferred in this way [3]. However, little discussion seems to have been given as to the structural origin of such quasi-periodicity and why it should persist to such great distances in, ostensibly, a random structure. We report here the first direct observation of such quasi-periodic, real-space density fluctuations in a very large continuous random network of a-Si. Further details are given elsewhere [4,5].
2. The structural model
* Corresponding author. Tel: +44-1223 336 525. Telefax: +44-1223 336 362.
The random network model of a-Si analysed in this study, containing 13 824 atoms, was constructed
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0022-3093(95)00334-7
A. Uhlherr, S.R. Elliott /Journal of Non-Crystalline Solids 192&193 (1995) 98-101
by Holender and Morgan [6] by molecular dynamics, together with a cell-doublingo procedure. The length of the cubic cell is L = 66.5 A; thus a radial distribution function (RDF) can be computed up to a length of 33.3 .~.
99
,
1
!
3. R e s u l t s
r
The radial density function calculated from the coordinates of the model is shown in Fig. 1 and, as expected, when plotted on a normal scale seems featureless for r > 10 A, implying that the structure is homogeneous beyond such a distance. However, on an expanded scale, atomic density fluctuations are clearly evident up to the maximum calculable distance [4]. Moreover, these extended-range fluctuations are quasi-periodic, with a period R = 3.4 .~ (although the period decreases slightly with increasing r). A clue to the origin of the ERO can be obtained by reference to the neighbour-specific partial RDFs, g,(r), where n is the 'topological' distance [7] between pairs of atoms, defined as the number of bonds in the shortest connecting percolation path. These partial correlation fluctuations are also shown for comparison in Fig. 1 for 1 < n < 13, whence it can be seen that the ERO period, R---3.4 .~ in the
2.5
•
...............
_J ,ooi
x
(a) ~"~,|~ o.,%,,.-,~ ~
#,,"~'11~.J,~ _ ~ J ~" ,~'~.
Fig. 2. Projection of a 12 A thick slice of the model containing 2476 atoms: (a) atoms shaded according to their value of n with respect to an origin atom at the centre (grey, odd n; black, even n); (b) straightforward projection with atoms shaded according to depth (dark, top; light, bottom).
2.0
x20
1.5
1.0
0.5
00 0
5
1~
r/A
15
20
25
Fig. 1. Radial density function (solid curve) for the a-Si model with, inset, a 20-fold magnification of the large-r region clearly exhibiting quasi-periodic extended-range order. The lower curves are the neighbour-specific partial RDFs, g,(r), where n ( = 1-13) is the number of neighbours from an origin atom along the shortest connecting percolation path.
total RDF, corresponds to the spacing of alternate (even-even or odd-odd) peaks in the partial g,(r). The quasi-periodicity associated with the ERO oscillations is not associated with nano/micro-crystallinity, even though, confusingly, the ERO period averaged over the full range of the oscillations, namely R---3.4 .~, is very close to the (111) interplanar spacing of the diamond-cubic crystal structure, d l l 1 -~ 3.2 ~.. However, the ERO fluctuations exhibit radial symmetry about any atom taken as origin; no evidence of crystallite domains is apparent. The radial symmetry is most clearly visible in a projection of a slice of the model (Fig. 2(a)), in which atoms are shaded differently depending on
100
A. Uhlherr, S.R. Elliott/Journal of Non-Crystalline Solids 192&193 (1995) 98-101
whether their topological distance from an arbitrary origin atom at the centre of the slice is even (black: n = 2-26) or odd (grey: n = 1-25). For comparison, no such radial order is apparent in a normal projection of the same slice in which atoms are not distinguished by virtue of their topological separation (Fig. 2(b)).
4. Discussion We believe that the ERO evident in Figs. 1 and 2(a) is due to propagated order (short-range in the case of a-Si) associated with the structural frustration inherent in maintaining network connectivity at large distances from a given origin atom. The propagated nature of the ERO oscillations cannot be understood in terms of a normal pair correlation function (RDF), but instead requires a conditional probability function based on triplet correlations. Specifically, what is required is the probability pa,(6), that, given an atom O at the origin a n d another atom i at a distance r i from the origin which is an nth neighbour of O, there is a third atom j at a distance = rj - r i from i which is an (n + An)th neighbour of O. Evaluation of p ~ ( 6 ) for the model, with r i taken arbitrarily to be 14 ___0.05 A, shows that the dominant distribution function is for A n = 2 and this peaks at 8 = 3.4 .& [4]. Thus, any density fluctuation (positive or negative) at a distance r o from an origin atom is pre~rentially propagated with a periodicity of R = 3.4 A. Hence, it is short-range order (specifically next-nearest-neighbour correlations) that are propagated. It should be noted that the peak in p 2 ( 6 ) does n o t occur at the average second-neighbour separation in a-Si, namely 3.83 A, but instead at an appreciably smaller value than this [4]. The reason for this propagation of short-range order, lies, we believe, in the structural frustration associated with the maintenance of network connectivity at large distances. Essentially, the existence of an atom i at a separation ri from an origin atom O, and which is an nth neighbour of O, severely limits the positioning of the neighbours of i ((n + An)th neighbours of O), since atoms with a separation from O smaller than r i have already been counted as ( n - An)th neighbours of O. This, combined with the tetrahedral symmetry about each atom and the o
fact that shortest percolation paths defining topological distances become roughly linear with increasing n, imposes a greater directional constraint (and hence a greater contraint on d) for An = 2 pairs than for An = 1 pairs [4]. Finally, mention should be made of the likelihood of ERO occurring in compound materials, such as the AXz-type oxide and chalcogenide glasses (A = Si,Ge;X = O,S,Se). The structure of AX2-type materials can be regarded as being a decoration by X atoms of the mid-points of the nearest-neighbour bonds between tetravalent A atoms. Thus, by analogy with the previous results, it is expected instead that An = 4 correlations ( A - X - A - X - A , or even X - A - X - A - X ) will be the dominant contribution to the production of ERO; i.e., it is now MRO that is propagated.
5. The first sharp diffraction peak The FSDP of a-Si occurs at Q1 = 1.9 ,~-1 [8]. The Holender-Morgan [6] model of a-Si accurately reproduces the experimental structure factor, including the FSDP, when correlations at all length scales in the model are taken into account. However, Fourier transformation of the ERO oscillations for r > 10 shows that they contribute more than half of the intensity of the FSDP [5]. Thus, it can be concluded that small structural models, containing only° of order 1000 atoms and having diameters of ~ 10A, cannot be expected to reproduce satisfactorily the intensity of the FSDP. Good agreement with the FSDP intensity obtained with such models can only be achieved therefore by the imposition of an otherwise unacceptable increase in local order (e.g., reduction in bond-angle fluctuations). An equivalent explanation for the origin of the FSDP in network glasses has been given in terms of the chemical ordering of cation-centred clusters and interstitial voids [2,9]. For the case of monatomic materials, e.g., a-Si, this picture reduces to a simple packing of atoms and voids, as envisaged originally by Bl&ry [10]. A void analysis [11] of the model for a-Si reveals that atom-void and void-void correlations are as well defined as atom-atom correlations, and they also exhibit ERO extending to similarly large distances [5]. For r > 10 A, the oscillations in
A. Uhlherr, S,R. Elliott /Journal of Non-Crystalline Solids 192&193 (1995) 98-101 25
.
'
,
.
,
,
,
i
.
,
,
,
,
.
,
20 i
315
i
,
.
,
,
- -
total
.........
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.....
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.
,
,
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- 0 5
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,
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,
I
4
i
f~
,
,
i
6
.
,
.
i
8
,
,
.
10
12
Q/A
Fig. 3. Bhatia-Thornton partial structure factors, S ~ ( Q ) (a,fl = N, C), calculated from the coordinates of the atoms and interstitial voids in the model, together with the total structure factor.
101
the voids are rather regular in shape and the distribution of void sizes is correspondingly rather narrow and symmetric [5]. As a result, one would expect ERO to be optimal and to extend to large distances. For the case of AX2-type materials, on the other hand, the distribution of void sizes (in, for example, a-SiO 2) is much broader and is very asymmetric [11]. Thus, this extra degree of disorder, coupled with the enhanced structural flexibility associated with the inclusion of two-fold-coordinated X atoms, might be expected to reduce appreciably the magnitude and extent of the ERO oscillations. Analysis of ultralarge models of AX2-type amorphous systems is required to confirm this point.
6. Conclusions the atom-atom and void-void partial RDFs are in phase with each other, but are in antiphase with those for the atom-void partial RDF [5]. The void-based model [2,9] for the origin of the FSDP ascribes the FSDP to a chemical-order prepeak occurring in the concentration-concentration structure factor (in the Bhatia-Thornton formalism) that is associated with the ordering of atoms and voids. That the atoms and voids are strongly correlated in the model of a-Si is attested by the existence of quasi-periodic oscillations in the atom-void partial RDF to beyond 30 A [5]. The three BhatiaThornton partial structure factors, S~13( Q) ( a, fl = N, number density; C, concentration) can be readily calculated from the coordinates of atoms and voids in the model, and are shown in Fig. 3. It can be seen that only See(Q) exhibits a pronounced peak at Q1 = o 1 1.9 A - , the position of the FSDP, lending support to the void-based picture. Finally, it should be remarked that perhaps the monatomic, tetravalent system a-Si (or Ge) may be the best in which to observe ERO at large distances. This is because fluctuations in atomic density can be equivalently regarded as arising from interstitial voids interspersed between the atoms. For the case of a-Si,
Extended-range order (ERO), in the form of weak, but quasi-periodic, fluctuations in atomic density, has been identified in a large (13 824 atom) model of o a-Si. The ERO oscillations, extending beyond 35 A, arise from propagated short-range order (secondneighbour correlations) and contribute significantly to the production of the first sharp diffraction peak in the structure factor.
References [1] S.R. Elliott, Nature 354 (1991) 445. [2] S.R. Elliott, J. Phys. Condens. Matter 4 (1992) 7661. [3] A.P. Sokolov, A. Kisliuk, M. Soltwisch and D. Ouitmann, Phys. Rev. Lett. 69 (1992) 1540. [4] A. Uhlherr and S.R. Elliott, J. Phys. Condens. Matter 6 (1994) L99. [5] A. Uhlherr and S.R. Elliott, Philos. Mag. B71 (1995) 611. [6] J.M. Holender and G.J. Morgan, J. Phys.: Condens. Matter 3 (1991) 1947; 7241. [7] R.J. Temkin, J. Non-Cryst. Solids 28 (1978) 23. [8] J. Fortner and J.S. Lannin, Phys. Rev. B39 (1989) 5527. [9] S.R. Elliott, Phys. Rev. Lett. 67 (1991) 711. [10] J. Bl&ry, Philos. Mag. B62 (1990) 469. [11] S.L Chan and S.R. Elliott, Phys. Rev. B43 (1991) 4423.