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Isotopic-substitution neutron diffraction as a probe of the structural environment of cations in superionic glasses
Solid State Ionics 105 (1998) 39–45
Isotopic-substitution neutron diffraction as a probe of the structural environment of cations in superionic glasses S.R. Elliott Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1 EW, UK
Abstract Isotopic-substitution neutron diffraction of superionic glasses can be used to probe the local and intermediate-range structure around modifier (mobile) cations in superionic glasses. This approach is illustrated by application to two glassy superionic systems: (Ag 2 S)(GeS 2 ) and (Li 2 S)(SiS 2 ). Keywords: Cation environment; Superionic glasses; Neutron diffraction; Isotopic substitution
1. Introduction Determination of the structure of amorphous solids is not a trivial problem. The lack of long-range order associated with the presence of disorder means that structural information can be obtained, if at all, only in a statistical sense, i.e. as distributions of structural parameters (e.g. of nearest-neighbour bond lengths, bond angles, etc.). How can an amorphous structure be characterized? The simplest approach, and one that is amenable to experiment, is in terms of the pair correlation function, r (r). This quantity is related to the radial distribution function (RDF), J(r) 5 4pr 2 r (r), where J(r)dr is the probability of finding a pair of atoms separated by a distance r in an interval dr. Note that this is a very simplistic function. It is spherically symmetric (the spatial variable is a scalar, not a vector), and is consequently a one-dimensional representation of a three-dimensional structure. Moreover, since it considers only pair correlations, ignoring higher-order correlations (involving triplets of atoms, etc.), any orientational structural information is not
included. Despite these shortcomings, the RDF provides as much structural information about an amorphous solid as it is possible to obtain in a single diffraction experiment. The RDF consists of a series of peaks at different values of r. For a crystalline material, these are very sharp and well-separated: each peak then corresponds to a particular coordination shell surrounding a given origin atom, whose area gives the coordination number, the position gives the radius of the shell and the width gives the mean-square displacement of the coordination shell radius (e.g. due to thermal fluctuations). Matters are more complicated for an amorphous solid. Here, generally it is only the first peak in the RDF, corresponding to the first (i.e. nearest-neighbour) coordination shell that is wellseparated from the rest. Thus, it is only for this peak that an unambiguous estimate of the relevant coordination number can generally be made. All other peaks in the RDF of an amorphous solid, even a monatomic material, overlap, increasingly so with increasing distance r from a given origin atom, until the RDF becomes featureless and merges with
0167-2738 / 98 / $19.00 1998 Elsevier Science B.V. All rights reserved. PII S0167-2738( 97 )00447-5
40
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
the average-density parabola, J(r) → 4pr 2 r0 , i.e. r (r) → r0 the average density of the material. Hence, an unambiguous interpretation of higher-lying peaks in the RDF cannot be made from experiment alone, since all peaks higher than the first have contributions from several coordination shells. The widths of the peaks in the RDF of an amorphous solid increase with r since static disorder, as well as thermal (vibrational) disorder, is present, and this is cumulative for higher-lying peaks. The position is even more complicated for multicomponent amorphous systems, where there is even more overlap between peaks in the RDF associated with the partial pair correlation functions, ri j (r), between pairs of different, as well as the same, atoms. For an n-component solid, there are a total of n(n 1 1) / 2 separate partial correlation functions. Obviously, a single diffraction experiment used to obtain an RDF cannot provide all the partial functions: for this purpose, a number of scattering experiments (in each of which the atomic scattering factors are different) equal to the total number of partial correlation functions is necessary. Performing diffraction experiments in which some, or all, of the atomic scattering factors are changed can be achieved in a number of ways. The type of radiation (neutron, X-ray or electron) used can be changed, in which all the atomic scattering factors are altered, but even if this were possible to implement, for the ternary glass compositions considered in this paper where a total of six partial correlation functions are involved (n 5 3) this in itself would be insufficient. Instead, atom-specific diffraction techniques are required, in which the scattering factor of a particular atom can be varied systematically. This can be done in one of two ways. In the case of X-ray scattering, a tunable X-ray source (e.g. from a synchrotron) can be tuned so that the X-ray wavelength is in the vicinity of an X-ray absorption edge of a particular element: the X-ray scattering factor is very different for energies just below and just above the absorption edge. This is the basis of the ‘anomalous X-ray scattering (AXS) technique’. The alternative approach, the subject of this paper, involves neutron diffraction and the replacement of an isotope of a particular element, i, having a neutron-scattering length b i , with another having a very different
neutron scattering length, b i* : this is the basis of the ‘isotopic-substitution neutron-diffraction (ISND) technique.’ The latter technique is potentially more powerful than AXS in that the neutron-scattering factor, b, unlike the corresponding X-ray scattering factor f(Q), is not a function of momentum transfer (scattering vector), Q: f(Q) is a rapidly decreasing function of Q, and this therefore limits the range of Q-values over which X-ray data may be measured, in turn limiting the resolution of real-space correlation functions obtained by Fourier transformation of the diffraction data. However, AXS has the advantage that, in principle, every element can be singled out, although in practice, the X-ray wavelengths corresponding to the absorption edges of lighter elements are so long that their use is precluded. ISND has the disadvantage that not every element has two or more (inexpensive) naturally-occurring isotopes with sufficiently different neutron-scattering lengths. The atom specificity of, for example, the ISND technique can be illustrated for the simplest case of a binary alloy A xA B x B , where element A has two isotopes A and A* with different scattering lengths, bA and bA* , respectively. The total RDF can be written as: J(r) 5 x 2A b 2A JAA (r) 1 x B2 b B2 JBB (r) 1 2x a x b bA b B JAB (r) . (1) (It should be noted, in passing, that another advantage of neutron diffraction over X-ray diffraction is that the RDF, obtained by Fourier transformation of the measured interference function, i(Q), can be written exactly in the form of Eq. (1) when the scattering factors are Q-independent.) An ISND measurement of the same material with isotope A replaced by A* gives: J*(r) 5 x 2A b 2A* JAA (r) 1 x B2 b B2 JBB (r) 1 2xA x B bA* b B JAB (r) .
(2)
Hence, a first difference of these two quantities gives: J9 5 J 2 J* 5 x 2A (b 2A 2 b 2A* )JAA 1 2xA x B b B (bA 2 bA* )JAB , (3) which is a pair-correlation function involving A
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
41
atoms alone as origin. A third ISND experiment involving yet another isotope, A**, then allows, in principle, just the A-related pair-correlation function, JA A (r), to be obtained directly by a double difference (together with the other correlation functions, JA B and JB B , by algebraic manipulation). Note that this technique for obtaining partial pair-correlation functions only works if the differences in scattering lengths (bA 2 bA* ), etc., are sufficiently large that, for example, the matrix-inversion procedure involved in the extraction of JA A in a double-difference experiment is well-conditioned. Fig. 1. Interference functions measured by neutron diffraction for three isotopically-substituted samples of glassy (Ag 2 S)(GeS 2 ).
2. Isotopic-substitution neutron diffraction of superionic glasses In this paper, ISND experiments of two, structurally related, superionic glass systems will be described, namely (Ag 2 S)(GeS 2 ) and (Li 2 S)(SiS 2 ), in which isotopic substitution of the cations, Ag and Li respectively, was carried out. For the silver-containing glass, 1 0 7 Ag (b57.5 fm) and 1 0 9 Ag (b54.2 fm), as well as nat Ag (b55.9 fm), were used. For further details, see Lee et al. [1]. In the case of the Li-containing glass, nat Li (b5 21.9 fm) and 6 Li (b5 1.4 fm) were used. Note that natural Li (mainly 7 Li) has a negative neutron-scattering length, thereby providing a big contrast with 6 Li; however, 6 Li strongly absorbs neutrons, and this makes ISND experiments, and the subsequent data analysis, of this system very difficult (see Ref. [1]).
where T 0 5 4prr0 kbl 2 ,
(5)
˚ 2 1 . (Note that are shown in Fig. 2 for Q m a x 527.8 A T(r) ~ J(r) /r and therefore basically increases linearly with r – cf. Eq. (5).) It can be seen that the first ˚ is the same for all Ag-isotopic peak at r52.23 A compositions. This peak is due to Ge–S correlations, for which ˚ 5 | 2.23 A) | 2x Ge x S b Ge b S T GeS (r) ; T(r 5
(6)
this is independent of the Ag scattering length, as
2.1. (Ag2 S)( GeS2 ) system The corrected neutron-diffraction interference functions, i(Q), for the three isotopically-substituted samples studied are shown in Fig. 1. It can be seen that there are pronounced differences in the curves at low Q-values, particularly in the region of the first sharp diffraction peak. The corresponding pair-correlation functions, T(r), obtained by Fourier transformation of the i(Q) curves: Q max
T(r) 5 T 0
E Qi(Q) sin Qr dQ , 0
(4)
Fig. 2. Pair-correlation functions, T(r), obtained by Fourier transformation of the interference functions shown in Fig. 1 ˚ 2 1 ). The peak at 2.23 A ˚ is due to Ge–S correla(Q m a x 527.8 A tions.
42
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
˚ is due observed.The second peak in T(r) at r.2.55 A to Ag–S correlations.In this case ˚ 5 | 2.55A) | 2xAg x S bAg b S TAgS (r) , T(r 5
(7)
and so the corresponding peak in plots of T(r) /bA g should be independent of bAg , as seen in Fig. 3. A weighted sum of Ag-related pair-correlation functions can be obtained by a first difference of two total correlation functions measured for two different Ag isotopes. Fig. 4 shows such first-difference
curves for various permutations of Ag isotopes, where the correlation function involved is D(r) 5 T(r) 2 T 0 5 kbl 2 G(r) ,
(8)
which therefore oscillates about zero. The first difference can be written as the sum of three terms: DD(r) 5 D(r) 2 D*(r) 5 AGAgS (r) 1 BGAgGe (r) 1 CGAgAg (r) ,
(9)
˚ is Fig. 3. Pair-correlation functions, T(r) /bAg . The peak at 2.5 A due to Ag–S correlations.
where A(B) 5 2xAg x S(Ge) (bAg 2 bAg* ) and C 5 2 x Ag (b 2Ag 2 b 2Ag* ). Ag-related pair correlations can be identified at ˚ (Ag–S), .3 A ˚ (Ag–Ag) and 3.7 A ˚ r52.55 A (Ag–Ge). The small Ag–Ag peak can be revealed more clearly by means of a double difference. Fig. 5 shows GAg Ag (r) obtained by inversion of the matrix equation involving the two first differences D2D* and D2D**. Although noisy, such a plot shows ˚ evidence for a peak in GAg Ag (r) at r.3 A. A weighted sum of the three remaining paircorrelation functions, GGes , GSS and GGe Ge , can be found by further algebraic manipulation of two total correlation functions, D(r) and D*(r) together with GAg Ag (r) found as above (see Ref. [2] for more details). Evidence for a (very short) S–S nearest˚ is found in this neighbour separation at r.2.7 A way. With these peak assignments from different analy-
Fig. 4. First-difference functions DD(r) for the pairs of Ag isotopes indicated.
Fig. 5. GAg Ag (r) obtained by a double-difference procedure.
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
ses, a given T(r) curve in Fig. 2 can be (with gaussian functions) to obtain coordination numbers of the respective coordinations. However, this procedure is only really secure for the first (split) peak ˚ where there is minimal overlap from at r|2.4 A higher-lying peaks. In this way, nearest-neighbour coordination numbers for S atoms around Ge (3.73) and Ag (2.89) are found (Fig. 6), consistent with tetrahedral coordination of Ge, as expected, and three-fold coordination of Ag atoms. With this structural information, a structural model for the short- and intermediate-range order in glassy (Ag 2 S)(GeS 2 ) can be proposed [2], as shown in Fig. 7. Elongated GeS 4 tetrahedra (the very short S–S ˚ identified corresponds to an S–Ge– distance of 2.7 A S bond angle of u 575.78, rather than the average tetrahedral angle u .1098) are proposed to be connected to two others through their apices, thereby forming chain-like intermediate-range ordering. The Ag 1 ions are presumed to sit between neighbouring chains, where they are coordinated by three nonbridging S atoms. Thus, on the basis of this structural model that is consistent with the ISND data, it can be speculated that the conduction pathway for the Ag 1 ions in this superionic glass is along the inter-chain space parallel to the chains, bordered by non-bridging S 2 ions, although, of course, this study of the static structure can give no direct information about the ionic dynamics.
Fig. 6. Fitting of peaks corresponding to coordination shells of Ge–S, Ag–S, Ag–Ag, and Ag–Ge and Ge–Ge combined, to the T(r) curve for the nat Ag sample.
43
Fig. 7. Model for the short- and intermediate-range glassy structure in glassy (Ag 2 S)(GeS 2 ).
2.2. ( Li2 S)( SiS2 ) system The corrected neutron-diffraction interference functions for the two isotopically-substituted samples studied are shown in Fig. 8; large differences in the low-Q region are seen, as in Fig. 1. The corresponding total correlation functions obtained by Fourier transforming the i(Q) data (Eq. (4)) are shown in Fig. 9. ˚ is due to Si–O correlations, First peak at r.1.6 A resulting from inadvertent oxygen contamination of this very water-sensitive sample. The second peak at ˚ is due to Si–S nearest-neighbour correlar.2.2 A tions. The large, negative-going peak in the curve in Fig. ˚ is indicative of 9(b) for the nat Li sample at r.2.5 A a correlation involving Li (since b( 7 Li),0); it corresponds to Li–S nearest-neighbour correlations. Curve fitting to both T(r) curves for the two isotopically-substituted samples (not shown – see Lee et al. [1]) indicates that the Li 1 ions are coordinated by approximately three sulphur atoms. Thus, since the two systems under study are both
44
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
Fig. 8. Interference functions measured by neutron diffraction for two isotopically-substituted samples of glassy (Li 2 S)(SiS 2 ).
based on tetrahedrally-coordinated network-forming systems and, in both, the anion coordination of the network-modifying cations is the same, it can be presumed that very similar short- and intermediaterange order (as shown in Fig. 7) characterizes both glassy systems.
3. Conclusions
Fig. 9. Pair-correlation functions, T(r), for the two Li-isotope samples studied, obtained by Fourier transformation of the interference functions shown in Fig. 8: (a) 6 Li; (b) nat Li. The dotted lines are the fits, and the solid thick lines the residuals, resulting from subtraction, of three gaussian peaks.
The aim of this paper has been to demonstrate that isotopic-substitution neutron-diffraction studies can provide, in favourable circumstances, much structural information on complex glasses, such as glassy superionic materials. Not only can the average (static) nearest-neighbour structural environment of the (mobile) cations be determined, but also information on the short- and intermediate-range order characterizing the network-forming matrix can be obtained. Although only structural information pertaining to the first (and maybe second) coordination shell can be obtained directly and unambiguously from the diffraction data, as shown here, nevertheless structural information relating to larger interatomic separations and intermediate-range order remains encoded in the data, and some insight into structural order at such larger distances can be gleaned by combining the diffraction data with simulation tech-
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
niques, such as (reverse) Monte Carlo and moleculardynamics simulations.
Acknowledgements The author is grateful to his colleagues, Professor M. Ribes and Dr. A. Pradel, and particularly Dr. J.H. Lee, for their contributions to this work.
45
References [1] J.H. Lee, A. Pradel, M. Ribes, S.R. Elliott, Phys. Rev. B. 56 (1997) 10934. [2] J.H. Lee, A.P. Owens, A. Pradel, A.C. Hannon, M. Ribes, S.R. Elliott, Phys. Rev. B 54 (1996) 3895.
Isotopic-substitution neutron diffraction as a probe of the structural environment of cations in superionic glasses S.R. Elliott Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1 EW, UK
Abstract Isotopic-substitution neutron diffraction of superionic glasses can be used to probe the local and intermediate-range structure around modifier (mobile) cations in superionic glasses. This approach is illustrated by application to two glassy superionic systems: (Ag 2 S)(GeS 2 ) and (Li 2 S)(SiS 2 ). Keywords: Cation environment; Superionic glasses; Neutron diffraction; Isotopic substitution
1. Introduction Determination of the structure of amorphous solids is not a trivial problem. The lack of long-range order associated with the presence of disorder means that structural information can be obtained, if at all, only in a statistical sense, i.e. as distributions of structural parameters (e.g. of nearest-neighbour bond lengths, bond angles, etc.). How can an amorphous structure be characterized? The simplest approach, and one that is amenable to experiment, is in terms of the pair correlation function, r (r). This quantity is related to the radial distribution function (RDF), J(r) 5 4pr 2 r (r), where J(r)dr is the probability of finding a pair of atoms separated by a distance r in an interval dr. Note that this is a very simplistic function. It is spherically symmetric (the spatial variable is a scalar, not a vector), and is consequently a one-dimensional representation of a three-dimensional structure. Moreover, since it considers only pair correlations, ignoring higher-order correlations (involving triplets of atoms, etc.), any orientational structural information is not
included. Despite these shortcomings, the RDF provides as much structural information about an amorphous solid as it is possible to obtain in a single diffraction experiment. The RDF consists of a series of peaks at different values of r. For a crystalline material, these are very sharp and well-separated: each peak then corresponds to a particular coordination shell surrounding a given origin atom, whose area gives the coordination number, the position gives the radius of the shell and the width gives the mean-square displacement of the coordination shell radius (e.g. due to thermal fluctuations). Matters are more complicated for an amorphous solid. Here, generally it is only the first peak in the RDF, corresponding to the first (i.e. nearest-neighbour) coordination shell that is wellseparated from the rest. Thus, it is only for this peak that an unambiguous estimate of the relevant coordination number can generally be made. All other peaks in the RDF of an amorphous solid, even a monatomic material, overlap, increasingly so with increasing distance r from a given origin atom, until the RDF becomes featureless and merges with
0167-2738 / 98 / $19.00 1998 Elsevier Science B.V. All rights reserved. PII S0167-2738( 97 )00447-5
40
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
the average-density parabola, J(r) → 4pr 2 r0 , i.e. r (r) → r0 the average density of the material. Hence, an unambiguous interpretation of higher-lying peaks in the RDF cannot be made from experiment alone, since all peaks higher than the first have contributions from several coordination shells. The widths of the peaks in the RDF of an amorphous solid increase with r since static disorder, as well as thermal (vibrational) disorder, is present, and this is cumulative for higher-lying peaks. The position is even more complicated for multicomponent amorphous systems, where there is even more overlap between peaks in the RDF associated with the partial pair correlation functions, ri j (r), between pairs of different, as well as the same, atoms. For an n-component solid, there are a total of n(n 1 1) / 2 separate partial correlation functions. Obviously, a single diffraction experiment used to obtain an RDF cannot provide all the partial functions: for this purpose, a number of scattering experiments (in each of which the atomic scattering factors are different) equal to the total number of partial correlation functions is necessary. Performing diffraction experiments in which some, or all, of the atomic scattering factors are changed can be achieved in a number of ways. The type of radiation (neutron, X-ray or electron) used can be changed, in which all the atomic scattering factors are altered, but even if this were possible to implement, for the ternary glass compositions considered in this paper where a total of six partial correlation functions are involved (n 5 3) this in itself would be insufficient. Instead, atom-specific diffraction techniques are required, in which the scattering factor of a particular atom can be varied systematically. This can be done in one of two ways. In the case of X-ray scattering, a tunable X-ray source (e.g. from a synchrotron) can be tuned so that the X-ray wavelength is in the vicinity of an X-ray absorption edge of a particular element: the X-ray scattering factor is very different for energies just below and just above the absorption edge. This is the basis of the ‘anomalous X-ray scattering (AXS) technique’. The alternative approach, the subject of this paper, involves neutron diffraction and the replacement of an isotope of a particular element, i, having a neutron-scattering length b i , with another having a very different
neutron scattering length, b i* : this is the basis of the ‘isotopic-substitution neutron-diffraction (ISND) technique.’ The latter technique is potentially more powerful than AXS in that the neutron-scattering factor, b, unlike the corresponding X-ray scattering factor f(Q), is not a function of momentum transfer (scattering vector), Q: f(Q) is a rapidly decreasing function of Q, and this therefore limits the range of Q-values over which X-ray data may be measured, in turn limiting the resolution of real-space correlation functions obtained by Fourier transformation of the diffraction data. However, AXS has the advantage that, in principle, every element can be singled out, although in practice, the X-ray wavelengths corresponding to the absorption edges of lighter elements are so long that their use is precluded. ISND has the disadvantage that not every element has two or more (inexpensive) naturally-occurring isotopes with sufficiently different neutron-scattering lengths. The atom specificity of, for example, the ISND technique can be illustrated for the simplest case of a binary alloy A xA B x B , where element A has two isotopes A and A* with different scattering lengths, bA and bA* , respectively. The total RDF can be written as: J(r) 5 x 2A b 2A JAA (r) 1 x B2 b B2 JBB (r) 1 2x a x b bA b B JAB (r) . (1) (It should be noted, in passing, that another advantage of neutron diffraction over X-ray diffraction is that the RDF, obtained by Fourier transformation of the measured interference function, i(Q), can be written exactly in the form of Eq. (1) when the scattering factors are Q-independent.) An ISND measurement of the same material with isotope A replaced by A* gives: J*(r) 5 x 2A b 2A* JAA (r) 1 x B2 b B2 JBB (r) 1 2xA x B bA* b B JAB (r) .
(2)
Hence, a first difference of these two quantities gives: J9 5 J 2 J* 5 x 2A (b 2A 2 b 2A* )JAA 1 2xA x B b B (bA 2 bA* )JAB , (3) which is a pair-correlation function involving A
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
41
atoms alone as origin. A third ISND experiment involving yet another isotope, A**, then allows, in principle, just the A-related pair-correlation function, JA A (r), to be obtained directly by a double difference (together with the other correlation functions, JA B and JB B , by algebraic manipulation). Note that this technique for obtaining partial pair-correlation functions only works if the differences in scattering lengths (bA 2 bA* ), etc., are sufficiently large that, for example, the matrix-inversion procedure involved in the extraction of JA A in a double-difference experiment is well-conditioned. Fig. 1. Interference functions measured by neutron diffraction for three isotopically-substituted samples of glassy (Ag 2 S)(GeS 2 ).
2. Isotopic-substitution neutron diffraction of superionic glasses In this paper, ISND experiments of two, structurally related, superionic glass systems will be described, namely (Ag 2 S)(GeS 2 ) and (Li 2 S)(SiS 2 ), in which isotopic substitution of the cations, Ag and Li respectively, was carried out. For the silver-containing glass, 1 0 7 Ag (b57.5 fm) and 1 0 9 Ag (b54.2 fm), as well as nat Ag (b55.9 fm), were used. For further details, see Lee et al. [1]. In the case of the Li-containing glass, nat Li (b5 21.9 fm) and 6 Li (b5 1.4 fm) were used. Note that natural Li (mainly 7 Li) has a negative neutron-scattering length, thereby providing a big contrast with 6 Li; however, 6 Li strongly absorbs neutrons, and this makes ISND experiments, and the subsequent data analysis, of this system very difficult (see Ref. [1]).
where T 0 5 4prr0 kbl 2 ,
(5)
˚ 2 1 . (Note that are shown in Fig. 2 for Q m a x 527.8 A T(r) ~ J(r) /r and therefore basically increases linearly with r – cf. Eq. (5).) It can be seen that the first ˚ is the same for all Ag-isotopic peak at r52.23 A compositions. This peak is due to Ge–S correlations, for which ˚ 5 | 2.23 A) | 2x Ge x S b Ge b S T GeS (r) ; T(r 5
(6)
this is independent of the Ag scattering length, as
2.1. (Ag2 S)( GeS2 ) system The corrected neutron-diffraction interference functions, i(Q), for the three isotopically-substituted samples studied are shown in Fig. 1. It can be seen that there are pronounced differences in the curves at low Q-values, particularly in the region of the first sharp diffraction peak. The corresponding pair-correlation functions, T(r), obtained by Fourier transformation of the i(Q) curves: Q max
T(r) 5 T 0
E Qi(Q) sin Qr dQ , 0
(4)
Fig. 2. Pair-correlation functions, T(r), obtained by Fourier transformation of the interference functions shown in Fig. 1 ˚ 2 1 ). The peak at 2.23 A ˚ is due to Ge–S correla(Q m a x 527.8 A tions.
42
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
˚ is due observed.The second peak in T(r) at r.2.55 A to Ag–S correlations.In this case ˚ 5 | 2.55A) | 2xAg x S bAg b S TAgS (r) , T(r 5
(7)
and so the corresponding peak in plots of T(r) /bA g should be independent of bAg , as seen in Fig. 3. A weighted sum of Ag-related pair-correlation functions can be obtained by a first difference of two total correlation functions measured for two different Ag isotopes. Fig. 4 shows such first-difference
curves for various permutations of Ag isotopes, where the correlation function involved is D(r) 5 T(r) 2 T 0 5 kbl 2 G(r) ,
(8)
which therefore oscillates about zero. The first difference can be written as the sum of three terms: DD(r) 5 D(r) 2 D*(r) 5 AGAgS (r) 1 BGAgGe (r) 1 CGAgAg (r) ,
(9)
˚ is Fig. 3. Pair-correlation functions, T(r) /bAg . The peak at 2.5 A due to Ag–S correlations.
where A(B) 5 2xAg x S(Ge) (bAg 2 bAg* ) and C 5 2 x Ag (b 2Ag 2 b 2Ag* ). Ag-related pair correlations can be identified at ˚ (Ag–S), .3 A ˚ (Ag–Ag) and 3.7 A ˚ r52.55 A (Ag–Ge). The small Ag–Ag peak can be revealed more clearly by means of a double difference. Fig. 5 shows GAg Ag (r) obtained by inversion of the matrix equation involving the two first differences D2D* and D2D**. Although noisy, such a plot shows ˚ evidence for a peak in GAg Ag (r) at r.3 A. A weighted sum of the three remaining paircorrelation functions, GGes , GSS and GGe Ge , can be found by further algebraic manipulation of two total correlation functions, D(r) and D*(r) together with GAg Ag (r) found as above (see Ref. [2] for more details). Evidence for a (very short) S–S nearest˚ is found in this neighbour separation at r.2.7 A way. With these peak assignments from different analy-
Fig. 4. First-difference functions DD(r) for the pairs of Ag isotopes indicated.
Fig. 5. GAg Ag (r) obtained by a double-difference procedure.
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
ses, a given T(r) curve in Fig. 2 can be (with gaussian functions) to obtain coordination numbers of the respective coordinations. However, this procedure is only really secure for the first (split) peak ˚ where there is minimal overlap from at r|2.4 A higher-lying peaks. In this way, nearest-neighbour coordination numbers for S atoms around Ge (3.73) and Ag (2.89) are found (Fig. 6), consistent with tetrahedral coordination of Ge, as expected, and three-fold coordination of Ag atoms. With this structural information, a structural model for the short- and intermediate-range order in glassy (Ag 2 S)(GeS 2 ) can be proposed [2], as shown in Fig. 7. Elongated GeS 4 tetrahedra (the very short S–S ˚ identified corresponds to an S–Ge– distance of 2.7 A S bond angle of u 575.78, rather than the average tetrahedral angle u .1098) are proposed to be connected to two others through their apices, thereby forming chain-like intermediate-range ordering. The Ag 1 ions are presumed to sit between neighbouring chains, where they are coordinated by three nonbridging S atoms. Thus, on the basis of this structural model that is consistent with the ISND data, it can be speculated that the conduction pathway for the Ag 1 ions in this superionic glass is along the inter-chain space parallel to the chains, bordered by non-bridging S 2 ions, although, of course, this study of the static structure can give no direct information about the ionic dynamics.
Fig. 6. Fitting of peaks corresponding to coordination shells of Ge–S, Ag–S, Ag–Ag, and Ag–Ge and Ge–Ge combined, to the T(r) curve for the nat Ag sample.
43
Fig. 7. Model for the short- and intermediate-range glassy structure in glassy (Ag 2 S)(GeS 2 ).
2.2. ( Li2 S)( SiS2 ) system The corrected neutron-diffraction interference functions for the two isotopically-substituted samples studied are shown in Fig. 8; large differences in the low-Q region are seen, as in Fig. 1. The corresponding total correlation functions obtained by Fourier transforming the i(Q) data (Eq. (4)) are shown in Fig. 9. ˚ is due to Si–O correlations, First peak at r.1.6 A resulting from inadvertent oxygen contamination of this very water-sensitive sample. The second peak at ˚ is due to Si–S nearest-neighbour correlar.2.2 A tions. The large, negative-going peak in the curve in Fig. ˚ is indicative of 9(b) for the nat Li sample at r.2.5 A a correlation involving Li (since b( 7 Li),0); it corresponds to Li–S nearest-neighbour correlations. Curve fitting to both T(r) curves for the two isotopically-substituted samples (not shown – see Lee et al. [1]) indicates that the Li 1 ions are coordinated by approximately three sulphur atoms. Thus, since the two systems under study are both
44
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
Fig. 8. Interference functions measured by neutron diffraction for two isotopically-substituted samples of glassy (Li 2 S)(SiS 2 ).
based on tetrahedrally-coordinated network-forming systems and, in both, the anion coordination of the network-modifying cations is the same, it can be presumed that very similar short- and intermediaterange order (as shown in Fig. 7) characterizes both glassy systems.
3. Conclusions
Fig. 9. Pair-correlation functions, T(r), for the two Li-isotope samples studied, obtained by Fourier transformation of the interference functions shown in Fig. 8: (a) 6 Li; (b) nat Li. The dotted lines are the fits, and the solid thick lines the residuals, resulting from subtraction, of three gaussian peaks.
The aim of this paper has been to demonstrate that isotopic-substitution neutron-diffraction studies can provide, in favourable circumstances, much structural information on complex glasses, such as glassy superionic materials. Not only can the average (static) nearest-neighbour structural environment of the (mobile) cations be determined, but also information on the short- and intermediate-range order characterizing the network-forming matrix can be obtained. Although only structural information pertaining to the first (and maybe second) coordination shell can be obtained directly and unambiguously from the diffraction data, as shown here, nevertheless structural information relating to larger interatomic separations and intermediate-range order remains encoded in the data, and some insight into structural order at such larger distances can be gleaned by combining the diffraction data with simulation tech-
S.R. Elliott / Solid State Ionics 105 (1998) 39 – 45
niques, such as (reverse) Monte Carlo and moleculardynamics simulations.
Acknowledgements The author is grateful to his colleagues, Professor M. Ribes and Dr. A. Pradel, and particularly Dr. J.H. Lee, for their contributions to this work.
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References [1] J.H. Lee, A. Pradel, M. Ribes, S.R. Elliott, Phys. Rev. B. 56 (1997) 10934. [2] J.H. Lee, A.P. Owens, A. Pradel, A.C. Hannon, M. Ribes, S.R. Elliott, Phys. Rev. B 54 (1996) 3895.