Structure of rare-earth chalcogenide glasses by neutron and x-ray diffraction

Many of the trivalent rare-earth ions exhibit intra-4f shell luminescence that can be exploited to make photonic glasses for, e.g. fibre amplifiers and lasers that operate at wavelengths compatible with fibre communications technology [1–3]. Here, the luminescence of the material will depend, e.g. on the solubility of rare-earth ions in the glass, and on the tendency of rare-earth ions to cluster within their matrix material. The latter leads to a non-uniform distribution of rare-earth ions that can limit the fluorescence lifetime, or lead to the formation of rare-earth compounds that are optically inactive. The luminescence can also be affected by multiphonon relaxation, wherein the probability of non-radiative channels is increased if the phonon energies of the host glass are close to gaps in energy between the 4f electron states of a rare-earth ion. It is therefore desirable to develop glassy materials in which the rare-earth ions are uniformly distributed within a glass matrix having appropriately small phonon energies and low transmission losses at the optical wavelengths of interest.

Chalcogenide glasses, containing one or more of the chalcogen elements (S, Se, Te), have photonic applications that range from digital memory storage to thermal imaging and infrared transmitting media [4–6]. They are also ideal hosts for luminescent rare-earth ions because their maximum phonon energies (e.g.  ∼460 cm⁻¹ for Ge–Ga–S versus  ∼340 cm⁻¹ for Ge–Ga–Se glasses in which the rare-earth solubility is high [7, 8]) are generally small as compared to silica (∼1100 cm⁻¹) and fluoride glasses (∼400–650 cm⁻¹) [3, 4, 9], although it is important to avoid an introduction of the vibrational bands associated with impurities to the phonon density of states [9–13]. The high refractive index of these materials (>2.1) helps to increase the cross section for stimulated emission [14, 15]. Even though rare-earth ions are barely soluble in the binary Ge–S system [10], germanium sulphide based glasses have attracted much attention because of their transparency in the visible spectrum. Here, the rare-earth solubility can be enhanced by making multicomponent glasses such as R₂S₃–Ga₂S₃–GeS₂ [7, 16–21], in which the ability of Ga to form negatively charged GaS₄ tetrahedra helps to balance the positive charge on rare-earth ions. The incorporation of Ga also has a tendency to reduce rare-earth clustering [22–24]. This reduction will either (i) increase the fluorescence lifetime if all of the rare-earth ions initially form a single population of clustered species, or (ii) increase the fluorescence intensity corresponding to the longer fluorescence lifetime if the rare-earth ions initially form two populations of clustered versus dispersed species. Accordingly, there have been many investigations to characterise the physico-chemical, optical and luminescent properties of R₂S₃–Ga₂S₃–GeS₂ glasses to explore their potential for use in, e.g. optoelectronic devices [7, 11, 13, 15–18, 20, 21, 23, 25–42]. There have been comparatively few investigations of the corresponding selenide or mixed sulphide/selenide glasses [9, 40–48].

In this paper, the method of neutron diffraction with isomorphic substitution (NDIS) of the rare-earth element [49–54] was employed to measure the structure of the rare-earth glasses (${{R}_{2}}{{X}_{3}}$)_0.07(Ga₂X₃)_0.33(GeX₂)_0.60, with $R=\text{La}$ or Ce and $X=\text{S}$ or Se. X-ray diffraction was also used to measure the structure of the sulphide glass. The aim is to provide a point of comparison for the effect of different chalcogens on the glass structure. The regions of glass formation in ternary ${{R}_{2}}{{X}_{3}}$–Ga₂X₃–GeX₂ systems containing large rare-earth ions are described in [16, 18, 20, 21]. The composition chosen for study contains a relatively large atomic fraction of the rare earth species c_R  =  0.0368 which is necessary for an application of the NDIS method. Here, the assumption is made that the 4f electrons of La³⁺ and Ce³⁺ , which characterise the difference between their electronic configurations, are sufficiently removed from the valence shell that the cations share a common structural chemistry and form isostructural compounds [1, 55]. This premise is supported by the similarity between the ionic radii of these trivalent cations (1.16 versus 1.14 Å for eightfold coordinated cations) [56] and the similarity between the Pettifor chemical parameters for La and Ce (0.705 versus 0.7025) [57]. There is a significant difference, however, between the neutron scattering lengths of La and Ce that will lead to a contrast between the measured neutron diffraction patterns for isostructural glasses containing one or other of these elements. In this way, it is possible to measure difference functions that separate, essentially, the R–R and R-μ pair-correlation functions from the μ−μ' pair-correlation functions, where μ (or μ') denotes a matrix atom, i.e. Ge, Ga or X. This simplification in the structural complexity associated with a single diffraction pattern enables the NDIS method to provide the type of site-specific structural information that is required in the development of realistic structural models, a prerequisite for establishing the structure-function relationships for rare-earth chalcogenide glasses.

The paper is organised as follows. The essential diffraction theory is given in section 2, and the experimental methodology is described in section 3. The results for the sulphide and selenide glasses are presented in section 4, and are discussed in section 5 by reference to (i) the structural models that have been proposed and (ii) the presence of rare-earth clustering. Conclusions are drawn in section 6.

In an x-ray or neutron diffraction experiment on (${{R}_{2}}{{X}_{3}}$)_0.07(Ga₂X₃)_0.33(GeX₂)_0.60 glass, the desired structural information is described by the total structure factor [58]

$$\begin{eqnarray}&&F(Q)=\underset{\alpha =1}{\overset{n}{\sum}}\,\underset{\beta =1}{\overset{n}{\sum}}\,{{c}_{\alpha}}{{c}_{\beta}}\,{{f}_{\alpha}}(Q)f_{\beta}^{\ast}(Q)\left[{{S}_{\alpha \beta}}(Q)-1\right],\end{eqnarray} \tag{ 1 }$$

where α and β denote the chemical species, n  =  4 is the number of different chemical species, ${{c}_{\alpha}}$ represents the atomic fraction of chemical species α, ${{f}_{\alpha}}(Q)$ and $f_{\alpha}^{\ast}(Q)$ are the scattering length (or atomic form factor) and its complex conjugate for chemical species α, respectively, ${{S}_{\alpha \beta}}(Q)$ is a Faber–Ziman [59] partial structure factor, and Q is the magnitude of the scattering vector. ${{S}_{\alpha \beta}}(Q)$ is related to the partial pair-distribution function ${{g}_{\alpha \beta}}(r)$ by the Fourier transform relation

$$\begin{eqnarray}&&{{g}_{\alpha \beta}}(r)-1=\frac{1}{2{{\pi}^{2}}\,{{n}_{0}}\,r}{\int}_{0}^{\infty}\text{d}Q\,Q\,\left[{{S}_{\alpha \beta}}(Q)-1\right]\,\text{sin}(Qr)\end{eqnarray} \tag{ 2 }$$

where n₀ is the atomic number density and r is a distance in real space. The mean coordination number of atoms of type β, contained in a volume defined by two concentric spheres of radii r_i and r_j centred on an atom of type α, is given by

$$\begin{eqnarray}&&\bar{n}_{\alpha}^{\beta}=4\pi \,{{n}_{0}}\,{{c}_{\beta}}{\int}_{{{r}_{i}}}^{{{r}_{j}}}\text{d}r\,{{r}^{2}}{{g}_{\alpha \beta}}(r).\end{eqnarray} \tag{ 3 }$$

The scattering lengths are independent of Q for the case of neutron diffraction, but not for the case of x-ray diffraction. In order to compensate for this Q dependence, the total structure factor can be rewritten as

$$\begin{eqnarray}&&S(Q)=1+\frac{F(Q)}{|\langle \,f(Q)\rangle {{|}^{2}}}\end{eqnarray} \tag{ 4 }$$

where $\langle \,f(Q)\rangle ={\sum}_{\alpha}{{c}_{\alpha}}{{f}_{\alpha}}(Q)$ is the mean scattering length. The corresponding real-space information is contained in the total pair-distribution function

$$\begin{eqnarray}&&G(r)=1+\frac{1}{2{{\pi}^{2}}\,{{n}_{0}}\,r}{\int}_{0}^{\infty}\text{d}Q\,Q\,\left[S(Q)-1\right]\,\text{sin}(Qr).\end{eqnarray} \tag{ 5 }$$

In x-ray diffraction experiments, the normalisation defined by equation (4) has the advantage that it allows for a better resolution of the peaks in G(r).

Consider a neutron diffraction experiment performed on two samples of R–Ge–Ga–X glass that are identical in every respect, except that $R\,\text{represents}\ \text{La}$, a 50:50 mixture of La and Ce, or Ce. Let the corresponding total structure factors be denoted by ${{}^{\text{La}}}F(Q)$, ${{}^{\text{Mix}}}F(Q)$ and ${{}^{\text{Ce}}}F(Q)$, respectively, where ${{b}_{\text{La}}}>{{b}_{\text{Mix}}}>{{b}_{\text{Ce}}}$. Then, if the rare-earth elements can be regarded as isomorphic, the contribution to a total structure factor from the μ-${{\mu}^{\prime}}$ pair-correlation functions can be eliminated by taking a difference function such as

$$\begin{eqnarray}\begin{array}{*{35}{l}} \Delta F_{R}^{(1)}(Q) & \equiv & \,{{}^{\text{La}}}F(Q){{-}^{\text{Ce}}}F(Q) \\ {} & = & 2{{c}_{R}}{{c}_{X}}{{b}_{X}}\left({{b}_{\text{La}}}-{{b}_{\text{Ce}}}\right)\left[{{S}_{RX}}(Q)-1\right] \\ {} & {} & +\,2{{c}_{R}}{{c}_{\text{Ge}}}{{b}_{\text{Ge}}}\left({{b}_{\text{La}}}-{{b}_{\text{Ce}}}\right)\left[{{S}_{R\text{Ge}}}(Q)-1\right] \\ {} & {} & +\,2{{c}_{R}}{{c}_{\text{Ga}}}{{b}_{\text{Ga}}}\left({{b}_{\text{La}}}-{{b}_{\text{Ce}}}\right)\left[{{S}_{R\text{Ga}}}(Q)-1\right] \\ {} & {} & +\,c_{R}^{2}\left(b_{\text{La}}^{2}-b_{\text{Ce}}^{2}\right)\left[{{S}_{RR}}(Q)-1\right]. \\\end{array}\end{eqnarray} \tag{ 6 }$$

Related difference functions are defined by $\Delta F_{R}^{(2)}(Q)\equiv \,$$\,{{}^{\text{La}}}F(Q){{-}^{\text{Mix}}}F(Q)$ and $\Delta F_{R}^{(3)}(Q)\equiv \,{{}^{\text{Mix}}}F(Q){{-}^{\text{Ce}}}F(Q)$. Similarly, it is possible to eliminate the R-μ pair-correlations by taking a difference function such as

$$\begin{eqnarray}\begin{array}{*{35}{l}} \Delta {{F}^{(1)}}(Q) & \equiv & \,{{}^{\text{La}}}F(Q)-\left[{{b}_{\text{La}}}\,/\,\left({{b}_{\text{La}}}-{{b}_{\text{Ce}}}\right)\right] \Delta F_{R}^{(1)}(Q) \\ {} & = & \left[{{b}_{\text{La}}}\,{{}^{\text{Ce}}}F(Q)-{{b}_{\text{Ce}}}\,{{}^{\text{La}}}F(Q)\right]\,/\,\left({{b}_{\text{La}}}-{{b}_{\text{Ce}}}\right) \\ {} & = & \Delta {{F}_{\mu {{\mu}^{\prime}}}}(Q)-c_{R}^{2}{{b}_{\text{La}}}{{b}_{\text{Ce}}}\left[{{S}_{RR}}(Q)-1\right], \\\end{array}\end{eqnarray} \tag{ 7 }$$

where $\Delta {{F}_{\mu {{\mu}^{\prime}}}}(Q)\equiv {\sum}_{\mu =1}^{m}{\sum}_{{{\mu}^{\prime}}=1}^{m}{{c}_{\mu}}{{c}_{{{\mu}^{\prime}}}}{{b}_{\mu}}{{b}_{{{\mu}^{\prime}}}}\left[{{S}_{\mu {{\mu}^{\prime}}}}(Q)-1\right]$ contains only the μ-${{\mu}^{\prime}}$ pair-correlation functions and m  =  3 is the total number of matrix species. Related difference functions are defined by $\Delta {{F}^{(2)}}(Q)\equiv \,{{}^{\text{La}}}F(Q)-\left[{{b}_{\text{La}}}\,/\,\left({{b}_{\text{La}}}-{{b}_{\text{Mix}}}\right)\right] \Delta F_{R}^{(2)}(Q)$ and $\Delta {{F}^{(3)}}(Q)\equiv \,{{}^{\text{Mix}}}F(Q)-\left[{{b}_{\text{Mix}}}\,/\,\left({{b}_{\text{Mix}}}-{{b}_{\text{Ce}}}\right)\right] \Delta F_{R}^{(3)}(Q)$. The coherent neutron scattering lengths are ${{b}_{\text{La}}}=8.24$(4) fm, ${{b}_{\text{Mix}}}=6.54$(4) fm, ${{b}_{\text{Ce}}}=4.84$(2) fm, ${{b}_{\text{Ga}}}=7.288$(2) fm, ${{b}_{\text{Ge}}}=8.185$(20) fm, ${{b}_{\text{S}}}=2.847$(1) fm and ${{b}_{\text{Se}}}=7.970$(9) fm [60]. The ratio of the R–R weighting factor $c_{R}^{2}{{b}_{\text{La}}}{{b}_{\text{Ce}}}$ to the weighting factors for the μ-${{\mu}^{\prime}}$ pair-correlations is small (e.g. in the case of $\Delta {{F}^{(1)}}(Q)$, 0.285% for the sulphide glass versus 0.087% for the selenide glass), i.e. the $\Delta {{F}^{(i)}}(Q)$ functions are dominated by the μ-${{\mu}^{\prime}}$ pair-correlation functions.

The real-space functions corresponding to $\Delta F_{R}^{(i)}(Q)$ and $\Delta {{F}^{(i)}}(Q)$ ($i=1,2\,\text{or}\,3$) are obtained by Fourier transformation, and are denoted by $\Delta G_{R}^{(i)}(r)$ and $\Delta {{G}^{(i)}}(r)$, respectively. The equations for, e.g. $\Delta G_{R}^{(1)}(r)$ and $\Delta {{G}^{(1)}}(r)$ are obtained from equations (6) and (7), respectively, by replacing each ${{S}_{\alpha \beta}}(Q)$ function by the corresponding partial pair-distribution function ${{g}_{\alpha \beta}}(r)$.

3.1. Glass synthesis and characterisation

3.2. High energy x-ray diffraction experiments

3.3. Neutron diffraction experiments

4.1. Sulphide glasses

The x-ray total structure factors ${{S}_{\text{X}}}(Q)$ measured for the rare earth sulphide glasses (figure 1(a)) have a first sharp diffraction peak (FSDP) at 1.08(2) Å⁻¹ (table 2) that is indicative of ordering on an intermediate length scale [71]. The atomic form factors for the rare-earth ions are similar because the atomic numbers for La and Ce are ${{Z}_{\text{La}}}=57$ and ${{Z}_{\text{Ce}}}=58$, respectively. Thus, the close similarity between the measured ${{S}_{\text{X}}}(Q)$ functions (figure 1(b)) is indicative of structural isomorphism.

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Table 2. First three peak positions Q₁–Q₃ in the reciprocal-space functions measured for glassy (${{R}_{2}}{{X}_{3}}$)_0.07(Ga₂X₃)_0.33(GeX₂)_0.60, where R represents La, Ce or Mix, and X represents S or Se. The peak positions r₁–r₄ in the corresponding real-space functions are also listed, together with the coordination number $\bar{n}_{\text{Ge}}^{X}$ obtained from the area under the peak at r₁ by assuming that $\bar{n}_{\text{Ga}}^{X}=4$, and the coordination number $\bar{n}_{R}^{X}$ obtained from the area under the peak at r₂.

X	Function	Q1 (Å−1)	Q2 (Å−1)	Q3 (Å−1)	r1 (Å)	$\bar{n}_{\text{Ge}}^{X}$	r2 (Å)	$\bar{n}_{R}^{X}$	r3(Å)	r4(Å)
S	${{}^{\text{La}}}{{S}_{\text{X}}}(Q)$	1.08(2)	2.19(2)	3.71(2)	2.25(2)	4.0(1)	3.03(3)	8.5(2)	3.66(5)	—
S	${{}^{\text{Mix}}}{{S}_{\text{X}}}(Q)$	1.08(2)	2.19(2)	3.71(2)	2.25(2)	4.1(1)	3.03(3)	8.4(2)	3.66(5)	—
S	${{}^{\text{Ce}}}{{S}_{\text{X}}}(Q)$	1.08(2)	2.19(2)	3.71(2)	2.25(2)	4.1(1)	3.03(3)	8.4(2)	3.66(5)	—
S	${{}^{\text{La}}}{{S}_{\text{N}}}(Q)$	1.05(2)	2.18(2)	3.68(2)	2.25(2)	3.9(1)	3.01(3)	8.8(9)	3.63(5)	—
S	${{}^{\text{Mix}}}{{S}_{\text{N}}}(Q)$	1.07(2)	2.19(2)	3.68(2)	2.25(2)	3.9(1)	2.97(3)	11.8(9)	3.62(5)	—
S	${{}^{\text{Ce}}}{{S}_{\text{N}}}(Q)$	1.05(2)	2.19(2)	3.68(2)	2.24(2)	3.9(1)	2.94(3)	10.2(9)	3.60(5)	—
S	$\Delta {{F}^{(1)}}(Q)$	1.05(2)	2.22(2)	3.67(2)	2.24(2)	4.0(1)	—	—	3.59(5)	—
S	$\Delta {{F}^{(2)}}(Q)$	1.05(2)	2.23(2)	3.67(2)	2.25(2)	4.1(1)	—	—	3.61(5)	—
S	$\Delta {{F}^{(3)}}(Q)$	1.05(2)	2.21(2)	3.67(2)	2.24(2)	4.1(1)	—	—	3.57(5)	—
S	$\Delta F_{R}^{(1)}(Q)$	1.05(2)	1.95(2)	—	—	—	3.05(3)	8.0(2)	3.89(5)	4.69(5)
S	$\Delta F_{R}^{(2)}(Q)$	1.02(2)	1.92(2)	—	—	—	3.16(3)	7.9(2)	3.84(5)	4.56(5)
S	$\Delta F_{R}^{(3)}(Q)$	1.06(2)	1.97(2)	—	—	—	3.07(3)	8.0(2)	3.85(5)	4.70(5)
Se	${{}^{\text{La}}}{{S}_{\text{N}}}(Q)$	1.03(2)	2.08(2)	3.50(2)	2.38(2)	3.9(1)	3.09(3)	8.1(9)	3.82(5)	—
Se	${{}^{\text{Ce}}}{{S}_{\text{N}}}(Q)$	1.03(2)	2.06(2)	3.50(2)	2.38(2)	3.9(1)	3.14(3)	9.8(9)	3.85(5)	—
Se	$\Delta {{F}^{(1)}}(Q)$	1.03(2)	2.05(2)	3.50(2)	2.37(2)	4.0(1)	—	—	3.87(5)	—
Se	$\Delta F_{R}^{(1)}(Q)$	1.09(2)	2.46(2)	—	—	—	3.11(3)	8.5(4)	3.84(5)	4.75(5)

The first peak in ${{G}_{\text{X}}}(r)$ at 2.25(2) Å (figure 2) is attributed to a superposition of nearest neighbour Ga–S and Ge–S pair-correlations, and its position compares with Ga–S and Ge–S intra-tetrahedral bond distances of $2.19\leqslant {{r}_{\text{GaS}}}$(Å) $\leqslant 2.37$ in crystalline Ga₂S₃ [72–74] versus $2.17\leqslant {{r}_{\text{GeS}}}$(Å) $\leqslant 2.28$ in crystalline GeS₂ [75–77]. The atomic form factors for Ga and Ge are similar because their atomic numbers are similar at ${{Z}_{\text{Ga}}}=31$ and ${{Z}_{\text{Ge}}}=32$. Hence, by converting ${{S}_{\text{X}}}(Q)$ to ${{F}_{\text{X}}}(Q)$ and dividing by ${{c}_{\text{Ge}}}{{c}_{\text{S}}}\left[\,f_{\text{Ge}}^{\ast}(Q){{f}_{\text{S}}}(Q)+{{f}_{\text{Ge}}}(Q)f_{\text{S}}^{\ast}(Q)\right]$, the Q-dependent weighting factors on the Ga–S and Ge–S partial structure factors are more-or-less removed. In this way [78], the Fourier transform of the resultant Q-space function enables the Ga–S and Ge–S coordination numbers to be extracted by direct integration of the first peak. If it is assumed that $\bar{n}_{\text{Ga}}^{\text{S}}=4$, then $\bar{n}_{\text{Ge}}^{\text{S}}=4.1(1)$ is obtained by integrating over the range 1.90 $\leqslant r\left({\mathring{\rm A}}\right)\leqslant 2.64$ (table 2). The results are therefore consistent with the formation of both GaS₄ and GeS₄ tetrahedra. The second peak in ${{G}_{\text{X}}}(r)$ at 3.03(3) Å is attributed to nearest-neighbour R-S pair-correlations by comparison with the bond lengths found in crystalline R–Ga–Ge–S, R–Ga–S and R–Ge–S materials with $R=\text{La}$ or Ce (table 3). By converting ${{S}_{\text{X}}}(Q)$ to ${{F}_{\text{X}}}(Q)$, dividing by ${{c}_{R}}{{c}_{\text{S}}}\left[\,f_{R}^{\ast}(Q){{f}_{\text{S}}}(Q)+{{f}_{R}}(Q)f_{\text{S}}^{\ast}(Q)\right]$, and Fourier transforming into real-space [78], a mean coordination number $\bar{n}_{R}^{\text{S}}=8.4(2)$ is obtained by integrating over the range $2.70\leqslant r\left({\mathring{\rm A}}\right)~\leqslant 3.19$ (table 2).

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Table 3. The Ga–X, Ge–X and R–X coordination numbers $\bar{n}_{\text{Ga}}^{X}$, $\bar{n}_{\text{Ge}}^{X}$ and $\bar{n}_{R}^{X}$, and corresponding bond distances ${{r}_{\text{Ga}X}}$, ${{r}_{\text{Ge}X}}$ and r_RX, respectively, in crystalline R–Ga–Ge–S, R–Ga–X or R–Ge–X materials, where $R=\text{La}$ or Ce and $X=\text{S}$ or Se. In several of these materials, Ga or Ge have both tetrahedral and octahedral coordination environments. Also given are the minimum nearest-neighbour R–Ga, R–Ge and R–R distances $r_{R\text{Ga}}^{\text{min}}$, $r_{R\text{Ge}}^{\text{min}}$ and $r_{RR}^{\text{min}}$, respectively. More information is available on the crystal structures of the sulphides as compared to the selenides.

Crystal	$\bar{n}_{\text{Ga}}^{X}$	${{r}_{\text{Ga}X}}$ (Å)	$\bar{n}_{\text{Ge}}^{X}$	${{r}_{\text{Ge}X}}$ (Å)	$\bar{n}_{R}^{X}$	rRX (Å)	$r_{R\text{Ga}}^{\text{min}}$ (Å)	$r_{R\text{Ge}}^{\text{min}}$ (Å)	$r_{RR}^{\text{min}}$ (Å)	Reference
La2Ga2GeS8	4	2.24–2.29	4	2.19–2.26	8	2.93–3.12	3.75	4.45	4.87	[79]
La3Ga0.5(Ge0.5/Ga0.5)S7	4	2.20–2.25	4	2.20–2.25	8	2.89–3.28	3.57	3.68	4.42	[80]
	6	2.59–2.63
LaGaS3	4	2.09–2.53	—	—	8.3	2.82–3.48	3.50	—	4.33	[81]
LaGaS3	4	2.20–2.33	—	—	8	2.81–3.59	3.69	—	4.33	[82]
La6Ga3.33S14	4	2.23–2.28	—	—	8	2.85–3.39	3.56	—	4.44	[83]
	6	2.66–2.86
Ce6Ga3.33S14	4	2.23–2.25	—	—	8	2.82–3.38	3.53	—	4.41	[83]
	6	2.63–2.84
La4Ge3S12	—	—	4	2.19–2.24	9	2.86–3.74	—	3.71	4.05	[84]
La2GeS5	—	—	4	2.80–2.98	8	2.61–3.89	—	3.82	4.06	[85]
La2GeS5	—	—	4	2.18–2.25	8.5	2.83–3.32	—	3.76	4.30	[86]
La2GeS5	—	—	4	2.18–2.26	8.5	2.84–3.33	—	3.77	4.32	[87]
LaGe1.25S7	—	—	4	2.17–2.22	8	2.85–3.17	—	3.51	4.34	[88]
	—	—	6	2.63–2.64
La4Ge3S12	—	—	4	2.19–2.25	9	2.86–3.73	—	3.71	4.05	[89]
La6Ge3S14	—	—	4	2.00–2.34	8	2.85–3.46	—	3.64	4.47	[90]
	—	—	6	2.45–2.66
Ce4Ge3S12	—	—	4	2.19–2.24	9	2.85–3.72	—	3.70	4.03	[89]
Ce4Ge3S12	—	—	4	2.20–2.24	9	2.86–3.72	—	3.69	4.02	[91]
Ce6Ge3S14	—	—	4	2.12–2.29	8	2.86–3.25	—	3.58	4.38	[90]
	—	—	6	2.65–2.85
Ce6Ge2.5S14	—	—	4	2.13–2.27	8	2.86–3.26	—	3.56	4.37	[87]
	—	—	6	2.64–2.84
La6Ga3.33Se14	4	2.33–2.34	—	—	8	2.92–3.36	3.66	—	4.57	[83]
	6	2.79–2.80
Ce6Ga3.33Se14	4	2.27–2.31	—	—	8	2.94–3.38	3.63	—	4.54	[83]
	6	2.72–2.73
La3Ge1.48Se7	—	—	4	2.31–2.37	8	3.02–3.27	—	3.69	4.55	[92]
	—	—	6	2.84–2.88
La6Ge3Se14	—	—	4	2.27–2.33	8	3.02–3.35	—	3.72	4.59	[93]
	—	—	6	2.77–2.78
Ce3Ge1.47Se7	—	—	4	2.32–2.37	8	3.00–3.27	—	3.66	4.52	[92]
	—	—	6	2.82–2.86
Ce6Ge3Se14	—	—	4	2.26–2.31	8	2.99–3.33	—	3.69	4.56	[93]
	—	—	6	2.74–2.76

The neutron total structure factors ${{S}_{\text{N}}}(Q)$ measured for the rare-earth sulphide glasses are shown in figure 3, and the corresponding total pair-distribution functions ${{G}_{\text{N}}}(r)$ are shown in figure 4. The positions of the leading peaks in both reciprocal and real space are listed in table 2. There is a small but measurable contrast between the reciprocal-space functions that is particularly marked in the region of the FSDP at ≃1.05 Å⁻¹, as emphasised by the difference functions $\Delta F_{R}^{(i)}(Q)$ shown in figure 5(a). As for the x-ray results, the first peak in ${{G}_{\text{N}}}(r)$ at 2.25(2) Å is attributed to a superposition of nearest neighbour Ga–S and Ge–S pair-correlations. On the assumption that $\bar{n}_{\text{Ga}}^{\text{S}}=4$, integration of this peak over the range $2.02\leqslant r$(Å) $\leqslant 2.64$ gives $\bar{n}_{\text{Ge}}^{\text{S}}=3.9(1)$. The second peak in ${{G}_{\text{N}}}(r)$ at ≃2.94–3.01 Å is attributed to nearest neighbour R–S pair-correlations. The $\bar{n}_{R}^{\text{S}}$ values obtained by integrating this peak over the range $2.76\leqslant r$(Å) $\leqslant 3.19$ are not, however, the same (table 2), which indicates a contribution to the second peak from μ-${{\mu}^{\prime}}$ pair-correlations. The height of the nearest-neighbour R–S peak is more pronounced in ${{G}_{\text{X}}}(r)$ (figure 2) as compared to ${{G}_{\text{N}}}(r)$ (figure 4), because the large atomic number of R gives larger weighting factors to the R-β partial pair-distribution functions.

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The difference functions $\Delta F_{R}^{(i)}(Q)$ and $\Delta {{F}^{(i)}}(Q)$ are shown in figure 5, and the corresponding real-space functions $\Delta G_{R}^{(i)}(r)$ and $\Delta {{G}^{(i)}}(r)$ are shown in figure 6. The first peak in $\Delta G_{R}^{(i)}(r)$ at ≃3.09 Å is attributed to R–S pair-correlations (table 3) and its integration over the range $2.76\leqslant r$(Å) $\leqslant 3.50$ gives a mean coordination number $\bar{n}_{R}^{\text{S}}=8.0(2)$. This value is smaller than obtained from the second peak in ${{G}_{\text{X}}}(r)$ (table 2), which is consistent with a contribution to the latter from μ-${{\mu}^{\prime}}$ pair-correlations. The second peak in $\Delta G_{R}^{(i)}(r)$ at ≃3.86 Å is attributed to a superposition of R–Ge and R–Ga pair-correlations (table 3). The third peak in $\Delta G_{R}^{(i)}(r)$ at ≃4.65 Å is expected to have a contribution from nearest neighbour R–R pair-correlations on the basis that the minimum R–R distance is $r_{RR}^{\text{min}}\simeq$4.02–4.87 Å for the crystalline sulphides reported in table 3. As in the case of ${{G}_{\text{N}}}(r)$, the first peak in $\Delta {{G}^{(i)}}(r)$ at 2.24(2) Å is attributed to a superposition of Ga–S and Ge–S pair-correlations and, if it is assumed that $\bar{n}_{\text{Ga}}^{\text{S}}=4$, a coordination number $\bar{n}_{\text{Ge}}^{\text{S}}=4.1(1)$ is obtained by integrating over the range $2.02\leqslant r$(Å) $\leqslant 2.64$. The second peak in $\Delta {{G}^{(i)}}(r)$ at ≃3.59 Å overlaps with the first peak in $\Delta G_{R}^{(i)}(r)$ at ≃3.09 Å, which confirms a contribution from μ-${{\mu}^{\prime}}$ pair-correlations to the second peak in ${{G}_{\text{N}}}(r)$.

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It was not possible to extract reliably the R–R partial structure factor from the measured data sets by using difference function methods [51, 68, 94]. This situation is likely to have originated from the small atomic fraction of the rare-earth ions (c_R  =  0.0368), and the possibility of some residual μ-${{\mu}^{\prime}}$ pair- correlations in $\Delta G_{R}^{(i)}(r)$, i.e. the sample compositions may not have matched exactly.

4.2. Selenide glasses

The neutron total structure factors ${{S}_{\text{N}}}(Q)$ measured for the rare-earth selenide glasses are shown in figure 7, and the corresponding total pair-distribution functions ${{G}_{\text{N}}}(r)$ are shown in figure 8. The positions of the leading peaks in both reciprocal and real space are listed in table 2. The first peak in ${{G}_{\text{N}}}(r)$ at 2.38(2) Å is attributed to a superposition of nearest neighbour Ga–Se and Ge–Se pair-correlations, and its position compares with intra-tetrahedral bond distances of $2.32\leqslant {{r}_{\text{GaSe}}}$(Å) $\leqslant 2.48$ in crystalline Ga₂Se₃ [95, 96] versus $2.34\leqslant {{r}_{\text{GeSe}}}$(Å) $\leqslant 2.37$ in crystalline GeSe₂ [97]. On the assumption that $\bar{n}_{\text{Ga}}^{\text{Se}}=4$, integration of this peak over the range $2.15\leqslant r$(Å) $\leqslant 2.58$ gives $\bar{n}_{\text{Ge}}^{\text{Se}}=3.9(1)$. The results are therefore consistent with the formation of a network containing both GaSe₄ and GeSe₄ tetrahedra. The second peak in ${{G}_{\text{N}}}(r)$ at ≃3.09–3.14 Å is attributed to nearest-neighbour R–Se pair-correlations by comparison with the bond lengths found in crystalline R–Ga–Se and R–Ge–Se materials with $R=\text{La}$ or Ce (table 3). The $\bar{n}_{R}^{\text{Se}}$ values obtained by integrating this peak over the range $2.82\leqslant r$(Å) $\leqslant 3.19$ are not, however, identical (table 2), which indicates a contribution to the second peak from μ-${{\mu}^{\prime}}$ pair-correlations.

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The difference function $\Delta F_{R}^{(1)}(Q)$ and its Fourier transform $\Delta G_{R}^{(1)}(r)$ are shown in figures 9(a) and 10(a), respectively. The $\Delta F_{R}^{(1)}(Q)$ function is relatively noisy, so the data set was truncated at ${{Q}_{\text{max}}}=10$ Å⁻¹ to reduce Fourier transform artifacts in $\Delta G_{R}^{(1)}(r)$. The first and second peaks in $\Delta G_{R}^{(1)}(r)$ at 3.11(3) and 3.84(5) Å are attributed to the nearest-neighbour R–Se pair-correlations and to a superposition of the R–Ga and R–Ge pair-correlations, respectively. As compared to $\Delta G_{R}^{(1)}(r)$ for the corresponding sulphide glass (figure 6(a)), the first peak is higher than the second peak because ${{b}_{\text{Se}}}/{{b}_{\text{S}}}=2.799$. The first peak gives a coordination number $\bar{n}_{R}^{\text{Se}}=8.5(4)$ by integrating over the range $2.76\leqslant r$(Å) $\leqslant 3.56$ to the first minimum, which compares to $\bar{n}_{R}^{\text{Se}}=8$–9 for large rare earth ions in related crystalline materials (table 3). This coordination number is different to those found from the ${{G}_{\text{N}}}(r)$ functions (table 2), which points to an overlap of the the R−Se and μ−μ' pair-correlation functions in G_N (r), and to the difficulty in extracting $\bar{n}_{R}^{\text{Se}}$ from the total pair-distribution functions. The third peak in $\Delta G_{R}^{(1)}(r)$ at 4.75(5) Å is likely to have a contribution from R–R pair-correlations on the basis that $r_{RR}^{\text{min}}\simeq 4.52$–4.59 Å for the crystalline selenides reported in table 2.

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The difference function $\Delta {{F}^{(1)}}(Q)$ and its Fourier transform $\Delta {{G}^{(1)}}(r)$ are shown in figures 9(b) and 10(b), respectively. As for the case of ${{G}_{\text{N}}}(r)$, the first peak in $\Delta {{G}^{(1)}}(r)$ at 2.37(2) Å is attributed to a superposition of Ga–Se and Ge–Se pair-distribution functions and, if it is assumed that $\bar{n}_{\text{Ga}}^{\text{Se}}=4$, a coordination number $\bar{n}_{\text{Ge}}^{\text{Se}}=4.0(1)$ is obtained by integrating this peak over the range $2.15\leqslant r$(Å) $\leqslant 2.64$. The glass composition can be re-written as (R₂Se₃)_0.07[(Ga₂Se₃)_0.3548(GeSe₂)_0.6452]_0.93, i.e. the base glass is rich in GeSe₂. Hence, by comparison with the structure of glassy GeSe₂ [98], the shoulder in $\Delta {{G}^{(1)}}(r)$ at ≃3.1 Å may have a contribution from the Ge–Ge distances in edge-sharing GeSe₄ tetrahedra, the peak at 3.87(5) Å may have a contribution from Se–Se pair-correlations, and the low-r shoulder on this peak at ≃3.6 Å may have a contribution from the Ge–Ge pair-correlations in corner-sharing tetrahedra. In comparison, Raman and multinuclear (⁷¹Ga, ⁷⁷Se) solid state nuclear magnetic resonance experiments on the base glass-forming system (Ga₂Se₃)_x(GeSe₂)_1−x with $0.063\leqslant x\leqslant 0.30$ support the formation of a network based primarily on corner-sharing GaSe₄ and GeSe₄ tetrahedra in which there are also some edge-sharing GeSe₄ tetrahedra and Ge–Ge homopolar bonds that appear in ethane-like Se₃Ge-GeSe₃ units [8]. With increasing x, the glass becomes increasingly Se deficient because the Ga:Se ratio is 1:3/2, i.e. less of the Ga and Ge atoms are able to take coordination environments in which they are bound exclusively to four Se atoms. In response, the fraction of homopolar bonds in ethane-like units increases, and Se atoms can also increase their coordination number from two to three by the formation of triclusters in which each higher coordinated Se atom is shared between three (Ga/Ge)Se₄ tetrahedra. In this way the Ga and Ge atoms can retain fourfold coordination environments.

The local structure in R₂S₃–Ga₂S₃–GeS₂ glasses containing La and Ce has been investigated by extended x-ray absorption fine structure (EXAFS) spectroscopy [30, 31]. In the case of glassy (La₂S₃)_0.167(Er₂S₃)_0.083(Ga₂S₃)_0.417(GeS₂)_0.333, EXAFS investigations at the Ga and Ge K-edges indicate the formation of GaS₄ and GeS₄ tetrahedra with bond distances of ${{r}_{\text{GaS}}}\simeq 2.31$ Å and ${{r}_{\text{GeS}}}\simeq 2.21$ Å [30]. Similar observations were made for the Ga and Ge coordination environments for glasses in the La₂S₃–Er₂S₃–Ga₂S₃–GeS₂ system from other EXAFS investigations at the Ga and Ge K-edges in which the ratio of Ga₂S₃ to GeS₂ was varied [31]. In the case of glassy (R₂S₃)_0.25(Ga₂S₃)_0.417(GeS₂)_0.333 ($R=\text{La}\,\text{or}\,\text{Ce}$), EXAFS investigations at the R L_III-edge gave a coordination number $\bar{n}_{\text{La}}^{\text{S}}\simeq 10.5$ with a bond distance ${{r}_{\text{LaS}}}\simeq 2.99$ Å or a coordination number $\bar{n}_{\text{Ce}}^{\text{S}}\simeq 7.6$ with a bond distance ${{r}_{\text{CeS}}}\simeq 2.96$ Å [30]. In comparison, EXAFS investigations of glasses in the La₂S₃–Ga₂S₃ system at the Ga K-edge and La L_III-edge indicate a network based on corner-sharing GaS₄ tetrahedra with a bond distance ${{r}_{\text{GaS}}}=2.26$–2.27 Å, and a coordination number $\bar{n}_{\text{La}}^{\text{S}}\simeq 7$ with a bond distance ${{r}_{\text{LaS}}}=2.91$–2.93 Å [99].

Raman spectroscopy was used to measure the structure of glassy (La₂S₃)_x[(Ga₂S₃)_0.4(GeS₂)_0.6]_(1−x) with $0\leqslant x\leqslant 0.35$ [7]. For x  =  0, the results were interpreted in terms of a network made from corner-sharing GaS₄ and GeS₄ tetrahedra, in which the S deficiency associated with the Ga:S ratio of 1:3/2 is compensated by the formation of Ga–Ga or Ge–Ge homopolar bonds. It was argued that Ge–Ge bonds predominate on the basis that Ge–S bonds are more easily ruptured than Ga–S bonds, for which the electronegativity difference between the atomic species is greater. Edge-sharing tetrahedral connections are allowed, especially in respect of Ga centered units. The negative charge on units such as GaS$_{4}^{-}$ is compensated by the positive charge on units such as (S₃Ge–GeS₃)²⁺ , where these charges are calculated on the basis of bridging S atoms. When La₂S₃ is introduced, the provision of additional S atoms breaks homopolar bonds to enable all of the Ge and Ga atoms to form corner-sharing GaS₄ or GeS₄ tetrahedra in which some of the S atoms are non-bridging. In addition, edge-sharing tetrahedra are converted to corner-sharing tetrahedra, and the resultant motifs may also contain non-bridging S atoms. Hence, negatively charged tetrahedral units are formed that can balance the positive charge on the added La³⁺ ions.

The present neutron and x-ray diffraction work on the structure of glassy (R₂S₃)_0.07(Ga₂S₃)_0.33(GeS₂)_0.60, where the composition can be re-written as (R₂S₃)_0.07[(Ga₂S₃)_0.3548(GeS₂)_0.6452]_0.93, is consistent with the formation of GaS₄ and GeS₄ tetrahedra as network forming motifs. By contrast, in the crystalline structures of R–Ga–Ge–S, R–Ga–S and R–Ge–S materials with $R=\text{La}\,\text{or}\,\text{Ce}$, the Ga and Ge atoms can form either tetrahedral or octahedral motifs with S atoms, depending on the composition (table 3). The first peak in the measured ${{G}_{\text{N}}}(r)$ functions (figure 4) has a small high-r shoulder at ≃2.55 Å, a distance that is longer than typical intra-tetrahedral Ga–S and Ge–S distances (table 3). It may therefore originate from the longer bond lengths found in higher coordinated Ge or Ga centred polyhedra, or from Ga–Ga homopolar bonds, where the Ga–Ga nearest-neighbour distance is 2.45 Å in crystalline GaS [100], 2.43–2.48 Å in crystalline GaSe [101], or 2.48–2.69 Å in the different polymorphs of crystalline Ga [102–105]. A shoulder at ≃2.55 Å does not, however, manifest itself as a notable feature in $\Delta {{G}^{(i)}}(r)$ (figure 6), i.e. this feature in ${{G}_{\text{N}}}(r)$ may have a contribution from Fourier transform artifacts. The coordination number $\bar{n}_{R}^{\text{S}}=8.0$(2) found from the measured $\Delta G_{\text{R}}^{(i)}(r)$ functions for glassy (R₂S₃)_0.07(Ga₂S₃)_0.33(GeS₂)_0.60 (table 2) compares to $\bar{n}_{R}^{\text{S}}=8$–9 for large rare earth ions in related crystalline materials (table 3).

For glassy (R₂S₃)_0.07(Ga₂S₃)_0.33(GeS₂)_0.60, a nearest-neighbour R–R distance of ≃4.65 Å, as estimated from the position of the third peak in $\Delta {{G}_{\text{R}}}(r)$ (table 2), compares to a minimum nearest-neighbour R–R distance $r_{RR}^{\text{min}}=$4.02–4.87 Å in crystalline R–Ga–Ge–S, R–Ga–S and R–Ge–S materials with $R=\text{La}$ or Ce (table 3). In these crystalline materials, $r_{RR}^{\text{min}}$ corresponds to R–S–R connections, i.e. two rare-earth ions share a common sulphur atom. By contrast, if the rare-earth ions were uniformly distributed within the glass matrix on a simple cubic lattice in order to optimise their separation, the R–R distance $r_{RR}^{\text{opt}}=\left({{c}_{R}}{{n}_{0}}\right){{}^{-1/3}}=8.79$ Å. Hence, there appears to be a clustering of rare-earth ions in the sulphide glass, and a distance $r_{RR}^{\text{min}}\simeq 4.0$–4.9 Å is likely to be representative of R₂S₃–Ga₂S₃–GeS₂ glasses in which this clustering occurs. For glassy (R₂Se₃)_0.07(Ga₂Se₃)_0.33(GeSe₂)_0.60, a nearest-neighbour R–R distance of ≃4.75 Å can be estimated from the position of the third peak in $\Delta {{G}_{\text{R}}}(r)$ (table 2), which compares to $r_{RR}^{\text{opt}}=9.02$ Å for the glass and $r_{RR}^{\text{min}}\simeq 4.52$–4.59 Å in crystalline R–Ga–Se and R–Ge–Se with $R=\text{La}$ or Ce (table 3). For the latter, $r_{RR}^{\text{min}}$ corresponds to R–Se–R connections. In rare-earth glassy materials, a distance $r_{RR}^{\text{min}}\simeq r_{RR}^{\text{opt}}$ is desirable because it will minimise the non-radiative energy transfer between rare-earth ions, thereby helping to optimise the radiative quantum efficiency [14, 36].

The structure of (${{R}_{2}}{{X}_{3}}$)_0.07(Ga₂X₃)_0.33(GeX₂)_0.60 glasses, where R denotes La or Ce and X denotes S or Se, was investigated by using the method of neutron diffraction with isomorphic substitution. X-ray diffraction was also employed to measure the structure of the sulphide glass. The experiments on both the sulphide and selenide glasses reveal structures that are based on networks built from GaX₄ and GeX₄ tetrahedra in which the rare-earth ions reside with an R-X coordination number $\bar{n}_{R}^{\text{S}}=8.0(2)$ or $\bar{n}_{R}^{\text{Se}}=8.5(4)$. The results for the Ga and Ge centered network-forming motifs are consistent with the findings from spectroscopic investigations of glasses in the R₂S₃–Ga₂S₃–GeS₂ system, and show that these conformations are also present in glasses from the R₂Se₃–Ga₂Se₃–GeSe₂ system. For sulphide and selenide ${{R}_{2}}{{X}_{3}}$–Ga₂X₃–GeX₂ glasses in which rare-earth clustering occurs, representative minimum nearest-neighbour R–R distances are likely to be 4.0–4.9 Å and 4.5–4.8 Å, respectively, and correspond to R–X–R connections.

We are grateful to Lawrie Skinner, Joan Siewenie and Veijo Honkimäki for their help with the experiments. PSS would like to thank Bruce Aitken (Corning Inc.) and Andrey Tverjanovich (Saint-Petersburg State University) for useful discussions. The Bath group thanks the EPSRC for support via Grant No. EP/C003594/1, and acknowledges use of the EPSRC funded National Chemical Database Service hosted by the Royal Society of Chemistry. PSS and AZ are grateful to Corning Inc. for the award of Gordon S Fulcher Distinguished Scholarships during which this work was completed. AZ is supported by a Royal Society—EPSRC Dorothy Hodgkin Research Fellowship. The data sets created during this research are openly available from the University of Bath data archive at https://doi.org/10.15125/BATH-00343.