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Shinya Hosokawa, Jens Rüdiger Stellhorn, Nathalie Boudet, Nils Blanc, Eisuke Magome, László Pusztai, [Shinji Kohara](https://orcid.org/0000-0001-9596-2680), Kazutaka Ikeda, Toshiya Otomo

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©2024 The Physical Society of Japan 

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Shinya Hosokawa,Jens Rüdiger Stellhorn,Nathalie Boudet ,Nils Blanc,Eisuke Magome, László Pusztai ,Shinji Kohara,Kazutaka Ikeda,Toshiya Otomo.(2024).
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Local- and Intermediate-range Partial Structure Study of As–Se Glasses.
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Physical Society of Japan,93,014601.https://doi.org/10.7566/JPSJ.93.014601.

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[Local- and Intermediate-range Partial Structure Study of As–Se Glasses](https://mdr.nims.go.jp/datasets/67b6e0a7-06d0-4a77-a12d-50b1484e6aa4)

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71038.dviJ. Phys. Soc. Jpn.Local- and intermediate-range partial structure study of As–Se glassesShinya Hosokawa1 ∗, Jens Rüdiger Stellhorn2, Nathalie Boudet3, Nils Blanc3, EisukeMagome4, László Pusztai5,6, Shinji Kohara7, Kazutaka Ikeda8, and Toshiya Otomo91Institute of Industrial Nanomaterials, Kumamoto University, Kumamoto 860-8555, Japan2Department of Physics, Nagoya University, Nagoya 464-8602, Japan3Universitité Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble Cedex 9,France4Kyushu Synchrotron Light Research Center, Tosu 841-0005, Japan5Wigner Research Centre for Physics, 1525 Budapest, Hungary6International Research Organization for Advanced Science and Technology (IROAST),Kumamoto University, Kumamoto 860-8555, Japan7National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan8Neutron Industrial Application Promotion Center, Comprehensive Research Organizationfor Science and Society (CROSS), Tokai 319-1106 Japan9Institute of Materials Structure Science, High Energy Accelerator Research Organization(KEK), Tsukuba 305-0801, Japan1/27Journal of the Physical Society of JapanTo investigate the relationship between the partial structures and the stiffness transitionin AsxSe1−x glasses, anomalous X-ray scattering (AXS) and X-ray and neutron diffraction(XRD and ND) experiments were carried out. For the AXS experiments, anomalous termsnear the absorption edges were experimentally obtained instead of the theoretical valueswith large ambiguities. The results were analyzed by reverse Monte Carlo (RMC) model-ing to obtain partial structure factors, S i j(Q), partial pair distribution functions, gi j(r), andthree-dimensional atomic configurations. The S i j(Q) and gi j(r) functions gradually vary withx; however, an important change was observed in the intermediate-range element-selectiveatomic structures (the so-called hyper-ordered structures) near the stiffness transition com-position. With decreasing x across the so-called intermediate phase compositions, a rapiddecrease of the As–As wrong bonds is visualized. However, the other anomalies found inGe–Se glasses are not clearly observed, such as a rapid decrease in pre-shoulder position inS SeSe(Q), a rapid decrease in the number of edge-sharing connections, and an exclusion ten-dency of the connections between the As(Ge) atoms sharing two Se atoms, which may berelated to the anisotropic pyramidal atomic configurations around the As atoms in the As-Seglasses in contrast to the isotropic tetrahedral ones around the Ge atoms in the Ge-Se glasses.1. IntroductionA mean-field theory1, 2) is a simple idea for understanding various experimental anomaliesaround a specific composition of the rigidity percolation threshold at the average coordinationnumber 〈r〉 = 2.4, where the number of configurational constraints per atom is equal tothe degrees of freedom in three dimensions. The character of a network glass undergoes atransition from rigid at 〈r〉 > 2.4 to easily deformable (floppy) at 〈r〉 < 2.4. In the case of theAsxSe1−x glasses handled in this paper, the threshold composition corresponds to x = 0.40, theAs2Se3 stoichiometric alloy, if the coordination numbers of the As and Se atoms are exactly3 and 2, respectively. At the beginning of more than 15 years of researches, the findingswere limited to the relationship between the rigidity percolation and physical properties ofbinary3, 4) and ternary5, 6) chalcogenide glasses and liquids.Remarkable experimental progress concerning the stiffness transition has been achievedby Boolchand and coworkers on GexSe1−x glasses.7, 8) They demonstrated that the results ofRaman scattering, modulated scanning calorimetry (MDSC), and Mössbauer spectroscopyon a fine composition grid provide evidence that the transition occurs over a well-defined∗Corresponding author: shhosokawa@kumamoto-u.ac.jp2/27J. Phys. Soc. Jpn.composition range between an onset point at 〈r〉 = 2.40 (x = 0.20) and a composition point at〈r〉 = 2.52 (x = 0.26), where the glasses are characterized to be in an unstressed-rigid regioncalled an intermediate phase (IP).To explore the relationship between atomic structures in short and intermediate ranges inGexSe1−x glasses, Hosokawa et al. carried out anomalous X-ray scattering (AXS) and neutrondiffraction (ND) experiments, and analyzed the obtained experimental data by reverse MonteCarlo (RMC) modeling to obtain partial structure factors, S i j(Q), partial pair distributionfunctions, gi j(r), and the corresponding three-dimensional (3D) atomic configurations.9, 10)From these experiments, important indications of the stiffness transition were observed onthe basis of the intermediate-range element-selective atomic structures (hyper-ordered struc-tures). Namely, a sudden decrease in prepeak intensity in S GeGe(Q), an abrupt disappearanceof Ge–Ge wrong bonds, anomalies in the connection ratio of edge- and corner-sharing GeSe4tetrahedra, and a characteristic change in the features of tetrahedral connections were real-ized with decreasing x across the IP concentration range. During the analytical process, weconfirmed that AXS is sufficiently sensitive to obtain intermediate-range structures,9) whileND has an excellent capability to correctly determine the nearest neighboring structures.10)Concerning the As–Se glasses handled in this study, Georgiev et al.11) carried out a MDSCexperiment on AsxSe1−x glasses and determined the IP region between 〈r〉 = 2.29 and 2.37(x = 0.29 and 0.37, respectively). Two differences are seen in the features of the IP. The firstone is that the x range of IP is a wide value of 0.08 compared with 0.06 for GexSe1−x glasses.In the sense of 〈r〉, however, the AsxSe1−x glasses have a smaller ∆〈r〉 of 0.08 than 0.12 for theGexSe1−x glasses. The second one is that the IP range shifts to the smaller 〈r〉 values from theoriginal value of 2.40 in the AsxSe1−x glasses but toward the opposite direction in the GexSe1−xglasses. For this reason, Georgiev et al.11) predicted the existence of fourfold-coordinatedquasi-tetrahedral Se=AsSe3 configurations instead of threefold-coordinated AsSe3 pyramidsby about 30% in number to increase the 〈r〉 values. This results in breaking the “8−N bonding(octet) rule” that is usually accepted for network glasses.The above idea motivated experimental studies of partial structures, and Hosokawa etal.12) carried out AXS experiments on As2Se3 in combination with RMC modeling to exam-ine whether the 8 − N bonding rule is broken around the As atoms or not. Note that ND withisotope substitution (NDIS) experiments is usually very effective for obtaining partial struc-tures. Since As has only one stable 75As isotope, however, this technique is limited to the useof Se isotopes. In fact, recent NDIS studies of AsxSe1−x glasses at x = 0.40, 0.35, and 0.30by Polidori et al.13) result in only difference functions around the As and Se elements. An ab3/27J. Phys. Soc. Jpn.initio molecular dynamics (MD) simulation was also conducted to confirm the experimentalresults. The results are rather ambiguous, i.e., the experimental AXS gave the coordinationnumber around As, NAs, of 3.26(2), which is in good agreement with Georgiev et al.’s predic-tion, while the theoretical data resulted in NAs = 3.07(3), which could not confirm the validityof the experimental results.Subsequent AXS experiments were carried out also in the IP region.14) The obtained NAsvalues were 3.29 and 3.69 at x = 0.33 and 0.29, respectively, showing good coincidence atx = 0.33 but apparent overestimation at x = 0.29. In addition, the spectral changes in theS AsAs(Q) partials were not systematic, particularly on the first peaks at about Q = 20 nm−1.To investigate the relationship between the local- and intermediate-range atomic struc-tures and the stiffness transition in AsxSe1−x glasses, four technical improvements that en-able more systematical and reliable structural data than the above previous results to be ob-tained were achieved in the present study. 1) Experiments were performed in a wide range of0.20 ≤ x ≤ 0.40 including the floppy, unstressed-rigid (IP), and rigid regions. 2) As K AXSmeasurements were conducted using a new setup at Kyushu Synchrotron Light Research Cen-ter (Saga-LS). 3) Experimental anomalous terms of atomic form factors were obtained andused for the analysis. 4) ND data were introduced to obtain more accurate first-neighboringinformation. As a result, different conclusions were obtained for the NAs values from those inthe previous papers.In this article, we explain the experimental technique in Sec. 2, present full sets of S i j(Q)s,gi j(r)s, and the 3D atomic configurations in Sec. 3, and discuss the specific features of theglass structures, such as prepeak heights and positions, partial coordination numbers, andconnections of AsSe3 pyramids, in relation to the stiffness transition or IP compositions inSec. 4. Finally, we summarize the present results in Sec. 5.2. Experimental and analytical proceduresBulk AsxSe1−x glass samples with x = 0.20, 0.25, 0.29, 0.33, 0.37, and 0.40 were preparedby quenching the melts in silica ampoules containing proper compositions of As and Se withpurities of 99.999 at.%. The melts were homogenized at 600◦C for at least 48 h with thefrequent stirring of the ampoule and quenched in air. The temperature was chosen by takingthe boiling point of As element at about 613◦C and of Se at about 685◦C into account.AXS measurements near the As K absorption edge (11.868 keV) were carried out atBL15 of the Saga-LS at Tosu, Japan. The voltage and current of this ring are 1.7 GeV and100–300 mA, respectively. To compensate relatively weak X-ray intensities from such a small4/27J. Phys. Soc. Jpn.synchrotron ring, we used two Si drift detectors (SDDs) to sensitively detect the elastic scat-tering X-rays and correctly estimate fluorescent X-ray contributions in the detected scatteringsignals. The elastic scattering signals reach 0.6–0.8 million counts at the S (Q) maximum ina reasonable beamtime at Saga-LS; these are sufficient for the analysis to obtain differentialstructure factors, ∆kS (Q). The details of this AXS detecting system are given elsewhere.15)AXS experiments near the Se K edge (12.658 keV) were performed at BM02 of theEuropean Synchrotron Radiation Facility (ESRF) at Grenoble, France. To obtain sufficientelastic signals and exclude the fluorescent X-rays, we used a bent graphite crystal analyzerand a scintillation counter on a 1-m-long arm. The details of this detecting system are givenin Ref. 9 and in a review article.16) Both the AXS experiments were carried out in reflectionmode for rectangle samples with a width of about 5 mm, a length of about 10 mm, and athickness of about 2 mm flattened with an emery paper.The improvement of these AXS experiments is made to enable us to experimentally de-termine anomalous terms of atomic form factors near an absorption edge. For this, X-rayabsorption spectra, µ, were measured in the fluorescence mode near an absorption edge ofthe k element. The energy ranges for the absorption measurements were about 11.8-12.0 and12.6-12.8 keV for the As and Se edges, respectively. The obtained µ spectra were smoothlyconnected with the imaginary part f ′′ of the anomalous term, since µ is proportional to f ′′and is expressed asµ = −2λre∑ini f ′′i ,where λ is the wavelength of X-rays, re is the classical electron radius, and ni is the number ofelectrons in the ith element. For this, Sasaki’s theoretical data17) were used for f ′′ outside themeasured E range. The upper three spectra of Fig. 1 show the f ′′ spectra near the (a) As and(b) Se K absorption edges at selected x values of 0.20, 0.29, and 0.40. As seen in the panels,the f ′′ spectra as well as those at other x values are mostly independent of x.The real part f ′ of the anomalous term can be calculated using the Krammars–Kronigrelationf ′(E) =2πP∫ ∞0E′ f ′′(E′)E′2 − E2dE′,where P denotes the principal value. The results are given in the lower spectra in Fig. 1. Asseen in the figures, the f ′ spectra are mostly independent of x.The dashed lines in Fig. 1 indicate the energy positions at 20 and 200 eV below thecorresponding K absorption edges where the present AXS measurements were carried out5/27J. Phys. Soc. Jpn.-10-8-6-4-202468f’   (electron units)12.8012.7012.6012.5012.40E  (keV)AsxSe1-x  glassesSe K edgex = 0.200.290.400.400.290.20f”  (electron units)-10-8-6-4-202468f’   (electron units)12.0011.9011.8011.7011.60E  (keV)AsxSe1-x  glassesAs K edgex = 0.200.290.400.400.290.20f”  (electron units)(a)(b)Fig. 1. Experimental anomalous terms f ′ (real) and f ” (imaginary) near the (a) As and (b) Se K absorptionedges of selected AsxSe1−x glasses. For clarity, the correct curves are at x = 0.40 and others are displacedupward by 1. Dashed lines indicate the energy positions at 20 and 200 eV below the corresponding edges wherethe present AXS measurements were carried out. (Color online)(Enear and Efar, respectively). Since the f ′ and f ′′ values at six compositions are almost thesame, we employed the average values in the following data analyses, as tabulated in Table I.The errors were estimated to be 0.02 for f ′ at Enear and 0.01 for other values. For clarity, thecorrect curves are at x = 0.40 and others are displaced upward by 1.6/27J. Phys. Soc. Jpn.Table I. Experimentally obtained f ′ and f ′′ values of As and Se in electron units at energies measured andthe theoretical values17) in parentheses.Element E (keV) f ′Asf ′′Asf ′Sef ′′SeAs 11.668 –3.673 0.507(–3.704) (0.512) (–2.241) (0.580)11.848 –6.039 0.602(–6.054) (0.497) (–2.419) (0.564)Se 12.458 –3.627 0.497(–2.236) (3.494) (–3.739) (0.514)12.638 –5.683 0.532(–1.960) (3.413) (–5.625) (0.501)In Table I, values in parentheses indicate the theoretical results calculated by Sasaki.17)The differences in f ′ obtained in the present experiments are larger than theoretical values byabout 0.7 and 9.0% near the As and Se edges, respectively, which may affect the validity of thecoordination numbers. In the previous AXS experiments on GexSe1−x glasses,9) where Enearwas selected at the nearer energy of 15 eV below the corresponding K edges, corrections fromtheoretical f ′ and f ′′ values were achieved by comparing the S i j(Q) data of GeSe2 obtainedfrom AXS results with reliable NDIS results obtained by Petri et al.,18) and the differences ofabout 14 and 3% were estimated for f ′ of Ge and Se, respectively. Hence, to determine localglass structures, it is important to experimentally obtain f ′ and f ′′ values. For the followinganalysis, theoretical values are used at energies where the experimental values are missing.To obtain element-selective structural information, one can obtain the differences betweentwo scattering data values at Enear and Efar. The differential structure factors, ∆kS (Q), can berelated to the differential intensity, ∆kI, asαk∆kI(Q, Efar, Enear) = ∆k[〈 f2〉 − 〈 f 〉2] + ∆k[〈 f 〉2]∆kS (Q),where αk is a normalization constant and ∆k[ ] indicates the difference between the valuesin brackets at Efar and Enear close to an absorption edge of the kth element.The ∆kS (Q) functions are separated into three S i j(Q)s as∆kS (Q) =∑i=As,Se∑j=As,SeWki jS i j(Q),where the weighting factors, Wki j, are given byWki j(Q, Efar, Enear) = xix j∆k[ fi f j]∆k[〈 f 〉2],7/27J. Phys. Soc. Jpn.1.00.50.00.50.00.50.00.50.0Wij0.400.300.20xAs(a) ΔAsS(Q)(b) ΔSeS(Q)(c) SX(Q)(d) SN(Q)As-AsAs-SeSe-SekFig. 2. Wki jvalues of As–As (©), As–Se (�), and Se–Se (△) correlations for AsxSe1−x glasses as a function ofx for (a) ∆AsS (Q), (b) ∆SeS (Q), (c) S X(Q), and (d) S N(Q). Wki jvalues for X-ray data are shown at Q = 22 nm−1near the first maximum of S X(Q). (Color online)The Wki jvalues were calculated by using the f ′ and f ′′ values together with theoretical valuesof the usual energy-independent term, f0(Q), taken from the literature.19)Circles, squares, and triangles in Fig. 2 show the Wki jvalues of the As–As, As–Se, and Se–Se correlations, respectively, as a function of x for (a) ∆AsS (Q), (b) ∆SeS (Q), and (c) total X-ray structure factors, S X(Q), at Q = 22 nm−1 near the first maximum of S X(Q). They all varyslightly with Q. Since WkAsSeand WkSeAsare the same as easily understood by the definition,and the twice of WkAsSeis generally used as the weighting factor of S AsSe(Q). According tosuch a common practice, twice the values are drawn for WkAsSein Fig. 2. As expected fromthe principle of AXS, the As–As and Se–Se contributions are highly suppressed in ∆SeS (Q)and ∆AsS (Q), respectively. Concerning the As composition x, the As–As contribution termsincrease with x, while the Se–Se terms decrease. The As–Se terms change with x, dependingon the respective structure factors.ND measurements were carried out over a wide Q range from 1.6 to more than 1000 nm−1,using the neutron total scattering spectrometer (NOVA)20) installed at BL21 of the Material8/27J. Phys. Soc. Jpn.and Life Experimental Facility (MLF) of the Japan Proton Accelerator Research Complex(J-PARC), Tokai, Japan. The incident neutron beam was generated by the proton acceleratorwith an output power of 400 kW. Measurements were performed in the time-of-flight mode,with neutron energies between 0.0013 and 5.7 eV, and a pulse repetition rate of 25 Hz. Detailsof the neutron detecting system are given elsewhere.10)The same glassy samples of about 5×10×2 mm3 were also used for the ND measurementsto measure the structures with the same compositional and thermal conditions. The sampleswere contained in standard pure V sample containers with an outer diameter of 10.0 mmand a thickness of 0.1 mm. Experiments took 4 h each. Detected ND intensities from thesamples were corrected for instrumental background, absorption by samples and cells,21) andmultiple22) and incoherent scattering. The scattering lengths and absorption cross-sectionsfor the constituent nuclei were taken from the literature.23) These corrections were carried outwith the mvaSq program coded by the NOVA group.24)The Wki jvalues for total neutron structure factors, S N(Q), obtained from ND experimentsare shown in Fig. 2(d); they were calculated from scattering lengths, bi, of 6.58 fm for As and7.970 fm for Se,23) instead of f for the X-ray data. bSe is larger than bAs by 21%, while fSeexceeds fAs by only about 3%. Thus, WSeSe values for S N(Q) are slightly larger than those forS X(Q), whereas the others are the opposite.RMC modeling25, 26) was applied to obtain atomic configurations from the present ex-perimental data of two ∆kS (Q)s, S X(Q), S N(Q), and the corresponding neutron total pairdistribution functions, gN(r). Calculation cubic boxes contained 10,000 atoms in total withedge lengths determined using density data.27) Initial atomic configurations were generatedby hard-sphere Monte Carlo simulations. Three constraints were applied to the RMC calcu-lations: 1) shortest atomic distance of 0.220 nm, 2) weak 8 − N connectivities, and 3) weakbond angle constraint of 100◦ around As atoms. RMC simulations were carried out using theRMC++ program package.28)3. ResultsThe circles in Fig. 3 show experimental results of (a) ∆AsS (Q), (b) ∆SeS (Q), (c) S X(Q),(d) S N(Q), and (e) gN(r) of AsxSe1−x glasses. The S X(Q) results seem to be consistent withthe pioneering XRD data obtained in 1973 by Renninger and Averbach,29) and the S N(Q) andgN(r) results are in good agreement with those of a recent ND work with natural isotopicabundances by Polidori et al.13) All of the spectra gradually vary with changing x, and seemto have no characteristic features in the structures across the IP region of x = 0.29 − 0.37.9/27J. Phys. Soc. Jpn.Fig. 3. Circles indicate experimental results of (a) ∆AsS (Q), (b) ∆SeS (Q), (c) S X(Q), (d) S N(Q), and (e) gN(r)of AsxSe1−x glasses. Solid curves represent the best fits of RMC modeling. For clarity, the data are displacedupward by 1 for (a)–(d) and 5 for (e). (Color online)The spectral features of structural factors with different Wki jvalues are very different fromeach other; ∆AsS (Q)s have large and sharp prepeaks at about Q = 12 nm−1, and small firstpeaks at about 20 nm−1. On the other hand, ∆SeS (Q)s exhibit only shoulders at the largerQ value of about 15 nm−1 and large first peaks. Both the S X(Q) and S N(Q) spectra showintermediate features between those of ∆AsS (Q)s and ∆SeS (Q)s. The S X(Q) and S N(Q) spec-10/27J. Phys. Soc. Jpn.tra look similar; however, the prepeaks in S X(Q)s are slightly more prominent than those inS N(Q)s. These differences are, of course, due to the differences in Wki jgiven in Fig. 2. Whenthe Q values exceed 30 nm−1, all the spectra in Figs. 3(a)–(d) look very similar to each other.The gN(r) functions in Fig. 3(e) have prominent peaks at about 0.23 nm and a second peak atabout 0.37 nm.The solid curves in Fig. 3 indicate the RMC fits for the corresponding spectra. Althoughthe coincidences between the results of experiments and the RMC fits are very good, smallinconsistencies between the experimental data and the RMC fits are seen in the AXS data of∆kS (Q) in Figs. 3(a) and (b). This is because the contrasts in the AXS data are only severalpercent of the scattering data of S X(Q)s. With decreasing x, the prepeak heights in ∆AsS (Q)sdo not show systematic changes with x, while the shoulders in ∆SeS (Q)s at about 15 nm−1seem to slowly rise. The prepeaks or shoulders in S X(Q)s and S N(Q)s become broadened.The heights of prominent peaks in gN(r)s become alightly lower, indicating that the As–Seand Se–Se bond lengths are almost identical, and the average coordination numbers decreasewith decreasing x.Figure 4 shows (a) S AsAs(Q), (b) S AsSe(Q), and (c) S SeSe(Q) partial structure factors ofAsxSe1−x glasses obtained by the present RMC modeling. For the stoichiometric As0.40Se0.60glass, a sharp peak is observed at the prepeak position of total structures of about 12 nm−1shown in Figs. 3(c) and (d). The first peak of S AsAs(Q) is clearly observed at about 20 nm−1,which is different from the previous AXS reports,12, 14) but looks reasonable upon comparisonwith those at different x values. This would be a result of adding the ND data to the presentRMC analysis. The S AsSe(Q) spectrum also has a prepeak at a similar position in S AsAs(Q).A sharp minimum is seen at about 18 nm−1, which is a slightly lower Q position than thefirst peak position in S X(Q) and S N(Q) of about 22 nm−1. A small shoulder is observed atthe sharp first peak in S X(Q) and S N(Q). Beyond 30 nm−1, the spectral features are similarto those in S N(Q), indicating that the As–Se bonding is the main contribution in As0.40Se0.60glass. The S SeSe(Q) spectrum has a shoulder (not a peak) at about 15 nm−1, a sharp and largefirst peak at about 22 nm−1, and a second peak at about 37 nm−1 (similar features to S X(Q)and S N(Q)). These features were also observed for the GeSe2 glasses.9, 10)With decreasing x, all of the S i j(Q) spectra gradually change. The prepeak in S AsAs(Q)exists over the entire x range measured, while the prepeak height does not change system-atically with x. The first peak in S AsAs(Q) does not change distinctly but looks not to besystematic. The S AsAs(Q) spectrum remains unchanged approximately beyond 30 nm−1, andthe errors become more conspicuous with decreasing x. These unstable spectral features in11/27J. Phys. Soc. Jpn.Fig. 4. (a) S AsAs(Q), (b) S AsSe(Q), and (c) S SeSe(Q) of AsxSe1−x glasses obtained by the present RMC mod-eling. For clarity, the data are displaced upward by 1. (Color online)the As–As correlations are due to the small WkAsAsvalues even for the ∆AsS (Q) data shownin Fig. 2. The prepeak in S AsSe(Q) remains unchanged, while the minimum at about 18 nm−1is highly buried with decreasing x. The remaining spectra beyond 30 nm−1 are almost thesame, indicating that the As–Se bonding features are unchanged over the entire x range mea-sured. The shoulder in S SeSe(Q) at about 15 nm−1 becomes prominent, and the height of thefirst peak decreases with decreasing x. On the other hand, the oscillation beyond 50 nm−1becomes larger, indicating that the number of Se–Se bonds increases.Here, it should be noted that detailed atomic configurations, particularly intermediate-range order cannot be correctly obtained without including the AXS data. In fact, a recentpaper on the structure of As-Se glasses reported the S i j(Q) functions by using a RMC model-ing with only total XRD data.30) However, they highly disagree with the present results in thelow Q region up to about 18 nm−1, i.e., the S AsAs(Q) and S AsSe(Q) data do not show distinctand systematic prepeaks and the positions of the shoulders in S SeSe(Q) largely shift towardsthe shorter Q values forming a small peaks.Figure 5 shows (a) gAsAs(r), (b) gAsSe(r), and (c) gSeSe(r) of AsxSe1−x glasses obtainedfrom the present RMC fits. On the gAsAs(r) function of the As0.40Se0.60 glass, a large peakis observed at about 0.23 nm with a height of about 2.6, as shown in Fig. 5(a), indicating alarge number of As–As homogeneous wrong bonds, which were also reported in the resultsof the NDIS experiment.13) With decreasing x, this homopolar peak once decreases in heightand becomes broadened in the IP composition range, and again recovers in the floppy region.However, the errors markedly increase in this region owing to the small Wki jvalues, and thus,12/27J. Phys. Soc. Jpn.Fig. 5. (a) gAsAs(r), (b) gAsSe(r), and (c) gSeSe(r) of AsxSe1−x glasses obtained by the present RMC modeling.For clarity, the data are displaced upward by 2 for (a), 8 for (b), and 3 for (c). (Color online)the recovery in the low x region is not a decisive evidence of the prepeak features.The second peak of gAsAs(r) of the As0.40Se0.60 glass is also very high, about 2, repre-senting the correlations between the AsSe3 pyramids with corner- or edge-sharing connec-tions. Since the shoulder at the shorter side of the second peak in not clearly seen, unlikethe Ge–Se results,9, 10) the fraction of edge-sharing AsSe3 pyramids would be rather small.With decreasing x, the height of the second peak seems to be almost unchanged, although thegAsAs(r) functions are highly scattered. Beyond the second peak, the statistics are very poor,particularly in the low x region, owing to the small WAsAs values even for the ∆AsS (Q) data,as given in Fig. 2.For the gAsSe(r) function at x = 0.40 shown in Fig. 5(b), a large and sharp first peak isobserved at r ∼ 0.240 nm with a height of about 9. The second peak indicates strong As–Secovalent bonds. The second broad peak is located at about 0.37 nm with a height of about1.8, exhibiting a Se atom attached to an AsSe3 pyramid, i.e., As–(Se)–Se correlation. Suchfeatures in the first and second peaks do not vary with decreasing x. Only a small differenceis observed in the third peak detected at about 0.57 nm; it is smeared out with decreasing x.In Fig. 5(c), when x = 0.40, a small peak is found at about 0.237 nm with a heightof about 1.5, which again represents the presence of the so-called homopolar wrong bonds.With decreasing x, the first Se–Se peak grows gradually and systematically up to a heightof about 4 with x = 0.20, owing to the extra Se atoms in the glasses. The second maximumof gSeSe(r) when x = 0.40 is located at r ∼ 0.36 nm with a height of about 2.8, originatingfrom Se–(As)–Se correlations inside the AsSe3 pyramid. With decreasing x, the second peak13/27J. Phys. Soc. Jpn.decreases very gradually in height with a mostly unchanged interatomic length. The changewould originate from the decreasing number of pyramids and a wider distribution of the Se–(Se)–Se interatomic distance for the compensating Se chains in the As–Se glasses.The left panels of Fig. 6 show the 3D atomic configurations obtained by the present RMCmodeling at x = (a) 0.40, (b) 0.33, and (c) 0.25. Red tetrahedra indicate the AsSe3 pyramids.The size of each picture in Fig. 6 is about 3.25 nm, being one-half of the RMC simulationboxes. At (a) x = 0.40, the AsSe3 pyramids are mainly connected by corner sharing, whilea certain number of edge-sharing connections are also observed. As mentioned above, thereare certain numbers of homopolar wrong bonds. To emphasize them, only the As–As andSe–Se bonds are shown in the right panels of Fig. 6 as solid and dashed lines, respectively,where the atomic configuration is the same as in the left panel. Bonds are defined as lengthsbelow 0.28 nm in the first peak regions in Figs. 5(a) and (c). Although a certain fraction ofSe–Se short bonds are formed by large distortions of pyramids, a large number of individualand topologically wrong bonds are confirmed in the As0.40Se0.60 glass. With decreasing x, thenumber of Se–Se bonds increases, as expected. On the other hand, a small number of As–Aswrong bonds remain even at x = 0.25 (or at the lowest As fraction of x = 0.20, as shown inFig. 5(a)).4. Discussion4.1 Local structuresTo discuss partial coordination numbers and bond angles, atomic bonds are again definedas interatomic distances below 0.28 nm, where the first and second peaks in the gi j(r) func-tions are well separated, as shown in Fig. 5. The partial coordination number, Ni j, is definedas the mean number of type j atoms around type i atoms. The previous ri j and Ni j results arewell compiled by Polidori et al. in Table III of Ref. 13. Table II shows structural parametersof the partial interatomic distance, ri j, and Ni j obtained in previous experimental12–14) and the-oretical31, 32) works at selected x values of 0.20, 0.30, and 0.40, as well as the present resultsat x = 0.20, 0.29, and 0.40. The ri j values from the NDIS experiments are not given in thetable because the gi j(r) functions were not evaluated owing to the limited number of isotope-enriched samples. The typical values estimated from their analysis are given in parenthesesunder the rAsSe column.By comparing the overall features of these parameters, we found a common tendency thatthe ri j values in the results of AIMD calculations are larger than in those of the experiments by0.05–0.15 nm. The important advantage of the RMC modeling is that this inverse technique14/27J. Phys. Soc. Jpn.(a) x = 0.40(c) x = 0.25(b) x = 0.33Fig. 6. Atomic configurations of glassy AsxSe1−x obtained by the present RMC modeling at selected x valuesof (a) 0.40, (b) 0.33, and (c) 0.25. Left panels: AsSe3 pyramidal units. Right panels: As–As and Se–Se homopolarbonds indicated by solid and dashed lines, respectively. (Color online)can complement the lack of missing data of AXS results in the large Q region, as was clearlygiven by Waseda et al.33) The inclusion of ND data over the wide Q range in this study,moreover, made justified our results in the first shell regime. In addition, the total scatteringg(r) data on As2Se3 glass are dominated by the As–Se heteropolar partial in both XRD and15/27J. Phys. Soc. Jpn.Table II. ri j (nm), and Ni j (atoms) values obtained from present experiments and previous experimental andtheoretical works at selected x values.x rAsAs rAsSe rSeSe NAsAs NAsSe NAs NSeAs NSeSe NSe Ref.0.40 0.240 0.241 0.238 0.48 2.54 3.02 1.69 0.35 2.04 Present0.241 0.237 0.225 0.73 2.53 3.26 1.69 0.32 2.01 AXS12)(0.241) 0.63 2.37 3.00 1.58 0.42 2.00 NDIS13)0.255 0.245 0.237 0.65 2.40 3.05 1.60 0.42 2.02 AIMD31)0.258 0.248 0.242 0.70 2.31 3.01 1.54 0.45 1.99 AIMD32)0.255 0.245 0.240 0.53 2.54 3.07 1.69 0.32 2.01 AIMD12)0.30 (0.240) 0.01 2.99 3.00 1.28 0.72 2.00 NDIS13)0.257 0.247 0.239 0.07 2.94 3.01 1.26 0.74 2.00 AIMD31)0.29 0.244 0.240 0.237 0.18 2.85 3.03 1.17 0.85 2.02 Present0.241 0.235 0.229 0.12 3.57 3.69 1.46 0.54 2.00 AXS14)0.20 0.240 0.240 0.235 0.19 2.77 2.96 0.69 1.35 2.04 Present0.259 0.247 0.239 0.05 2.96 3.01 0.74 1.27 2.01 AIMD31)ND results, and the obtained first peak position has been at about 0.241 nm consistentlysince the 1970s29, 34) to now.13, 35) Furthermore, the rSeSe value should approach the interatomicdistance of Se glass of about 0.234 nm29, 36) with decreasing x. Therefore, the experimentalri j values are correct and the theoretical ones have systematic deviations to larger values.Other large differences are seen in the coordination numbers obtained from previous AXSexperiments.12, 14) In particular, the NAs values are much larger than the 8 − N rule of three.There may be two reasons why the results are different. First, we used the experimentallyobtained anomalous terms, f ′ and f ′′, for the atomic form factors f , as shown in Fig. 1 andTable I, which improved the qualities of the ∆kS (Q) functions and the RMC analysis, ratherthan the theoretical values in the previous papers.12, 14) Second, the S N(Q) and gN(r) data wereincluded in the present RMC analysis, as shown in Figs. 3(d) and (e). As mentioned above,owing to a wide Q range measured by ND, the validity of the information on the first nearestneighbors is highly improved compared with the results of the RMC analysis without NDdata. By comparing with the NDIS results by Polidori et al.,13) the NAsAs value of 0.01 atx = 0.30 is much smaller than the present value of o.18 at x = 0.29. Such a discrepancy mayresult from the fact that no good As isotopes exists in the NDIS measurement. The existenceof ∆AsS (Q) data in our AXS measurements may give correct S AsAs(Q), gAsAs(r), and NAsAsvalues.Figure 7 shows the x dependence of the Ni j values obtained from the present RMC anal-16/27J. Phys. Soc. Jpn.IPFig. 7. Averaged partial coordination numbers Ni j. Solid curves represent total coordination numbers aroundAs (upper) and Se (lower) atoms. Dashed lines indicate the ideal Ni j values assuming the chemically orderedcontinuous-random-network model.37) (Color online)ysis. For the annotations in the figure, As–As and As–Se indicate Ni j values for neighboringAs and Se atoms around As, and similarly, those around Se are given as Se–As and Se–Se. Dashed lines indicate the ideal Ni j values predicted by Zachariasen using a chemicallyordered continuous-random-network model.37) The solid curves represent the total coordina-tion numbers around As and Se (NAs and NSe), which are almost 3 and 2, respectively, in the0.20 ≤ x ≤ 0.40 range. This result indicates that the 8 − N bonding rule is applicable toAsxSe1−x glasses for all the concentrations measured.At x = 0.40, the majority of the As and Se atoms are surrounded by three Se and twoAs atoms, respectively. The average numbers of homopolar As–As and Se–Se wrong bondsare 0.48 ± 0.10 and 0.35 ± 0.10, respectively. The NAsAs value of 0.48 is slightly smallerthan others as shown in Table II, but acceptable within the range of errors of the present andprevious studies. The NAsAs value is in agreement with those obtained from other experimentaland theoretical works. As mentioned above, NAs ∼ 3.0 and NSe ∼ 2.0 are concluded from theresults of the present experiments, which contradicts the previous AXS result of a breakingdown of the 8−N rule around As by Hosokawa et al.12) and the prediction of an additional As–Se double bond on the basis of the mean-field theory.11) Owing to the improvements of thepresent experiments and analyses by using the experimental f ′ and f ′′ values and includingND data, the conclusion in the previous paper regarding As2Se3 glass, i.e., the breakdown of17/27J. Phys. Soc. Jpn.the 8 − N rule, is doubtful.With decreasing x, NSeAs decreases and NSeSe increases, which is mostly in line with theresults of the chemically ordered continuous-random-network model37) shown by the dashedlines in Fig. 7, except for the presence of homopolar wrong bonds. Note that NAsAs largelydecreases from about 0.5 to about 0.2 when crossing the IP concentration range indicatedin yellow color in Fig. 7. This phenomenon would be reasonable because the As–As wrongbonds reflect the stressed-rigid nature of the AsxSe1−x glasses at x > 0.36, and the stress in theglasses is released in the IP region, as was found in the Ge–Se glasses.10) In the floppy regionof x < 0.26, there are a certain number of As–As bonds, i.e., the wrong bonds intrinsicallyremain even in the flexible As–Se networks.Figure 8 shows the (a) As–Se–As and (b) Se–As–Se bond angle distributions of AsxSe1−xglasses as a function of cos θ. The θ values are indicated at the top of the figures. At x = 0.40,the As–Se–As and Se–As–Se distributions have broad peaks centered at about 97◦ and 100◦,respectively. The As–Se–As and Se–As–Se bond angles of crystalline As2Se3 with a mono-clinic space group have three different values in the ranges of 85.6–101.5◦ and 90.6–104.3◦,38)respectively, and the glass phase has slightly larger angles. The bond angle distributions atx = 0.40 were examined experimentally with the usual combination of XRD, ND, and RMCanalysis by Fábián et al.,35) and the peak positions of 97 ± 2◦ and 99 ± 2◦ were obtained forthe As–Se–As and Se–As–Se configurations, respectively, in good agreement with the presentresults although they did not utilize the AXS or NDIS data of greater element sensitivity.To our knowledge, there have been two AIMD studies of the bond angle distributionsof As2Se3 glass,31, 39) and both studies indicated broad distributions around either As or Secentered at about 100◦. A specific result was given for the As–Se–As bond angle distributionsby Bauchy et al.31) where there is an additional shoulder at about 90◦. They did not identifythe origin of the shoulder, and it is too large to be composed of edge-sharing AsSe3 pyramids,in contrast to those of GeSe4 tetrahedra in GeSe2 glass.10)With decreasing x, the As–Se–As distributions become rather broad and scattered ow-ing to the decrease in the number of the pyramid connections with only one Se atom (Se1),as shown later. The As–Se–As bond angle distributions shown in Fig. 8(a) do not seem tochange systematically from the broad peak centered at about 97◦ at x = 0.40. However, itis clear that the broad peak is suddenly smeared out below x = 0.25, which may be relatedto that the glass system enters the floppy regime. The Se–As–Se distributions remain mostlyunchanged, whereby the local atomic configurations of AsSe3 pyramids do not change with x.Only Bauchy et al.31) discussed the x dependence of the bond angle distributions by obtaining18/27J. Phys. Soc. Jpn.Bond Angle Distribution  (Arb. units)-1.0 -0.5 0.0cos(b) Se-As-Sex = 0.200.250.290.330.370.40  (°)180 120 90 70Bond Angle Distribution  (Arb. units)-1.0 -0.5 0.0cos(a) As-Se-Asx = 0.200.250.290.330.370.40  (°)180 120 90 70Fig. 8. Distributions of (a) As–Se–As and (b) Se–As–Se bond angles. Curves are displaced upward for clarity.(Color online)four types of bond angle (Se–Se–As and Se–Se–Se in addition to the above), and rather com-plex x dependences were reported. In addition, they obtained the x dependence of the averagedihedral angle and the standard deviation, which indicate rather characteristic features in theIP concentration region. On the other hand, the experimentally obtained results in the presentwork show simple and gradual x variations, and cannot compare with the above theoreticalpredictions.4.2 Intermediate-range orderA prepeak in the S (Q) spectra in covalent glasses suggests the existence of intermediate-range order (IRO) in the atomic configurations of glasses. Large changes were observed inthe total S (Q) of GexSe1−x glasses,9, 10, 40) where with decreasing x, the position largely shiftstowards larger Q values and the height rapidly decreases, as shown in Figs. 2(c) and (d) ofRef. 10 for the XRD and ND data, respectively. By separating them into the partial S i j(Q)s, itwas found that with decreasing x, the prepeaks in the Ge–Ge and Ge–Se partials show slightincreases in position, while the shoulder position in S SeSe(Q) rapidly decreases. The prepeakheight in S GeGe(Q) rapidly decreases in the IP concentration range of 0.26 ≥ x ≥ 0.20. Thesepartial behaviors in the prepeak/shoulder induce the interesting changes in the prepeak of thetotal S (Q) of GexSe1−x glasses.In the present AsxSe1−x case, the x dependence of the prepeak in total S (Q) is very small,as shown in Figs. 3(c) and (d), with only peak heights decreasing with decreasing x. Figure19/27J. Phys. Soc. Jpn.IP(a)(b)Fig. 9. (a) Qp and (b) S i j(Qp) in the As–As (©), As–Se (�), and Se–Se (△) partials of S i j(Q)s. (Color online)9(a) shows the x dependence of the prepeak/shoulder positions, Qp, in the As–As, As–Se, andSe–Se partials of S i j(Q) indicated by circles, squares, and triangles, respectively, which areobtained from Fig. 4. The peak positions of the As–As and As–Se correlations show mostlythe same values of about 12 nm−1, and that of Se–Se gradually decreases with decreasingx. In the hatched IP concentration region, the decreasing rate is slightly higher, but not soprominent as that of the Ge–Se glasses shown in Fig. 7(a) of Ref. 10.Figure 9(b) shows the heights of the prepeaks for the partials, S i j(Qp). The prepeakheights for As–Se are almost a constant value of about 0.9, and those of Se–Se show a gradualincrease from 0.4 to 0.6 with decreasing x. On the other hand, those of As–As have a ratherscattered x dependence owing to the small WAsAs values, as shown in Fig. 2. If anything, adecrease is observed in the IP concentration region, while it is not clear when compared withthe Ge–Ge prepeak height in GexSe1−x glasses in the IP region. The x dependence in the fea-tures of prepeaks in the As–Se glasses have been taken up only for total S (Q) functions,41, 42)and the relationship to the stiffness transition has not been discussed clearly. Golovchak et al.reported that the prepeak width increases at about x = 0.3 in both the XRD and ND data.42)On the other hand, the present data cannot clarify such a broadening as anything other than20/27J. Phys. Soc. Jpn.Table III. x dependence of the fractions of corner-, edge-, and face-sharing connections between the AsSe3pyramids.x Fcorner (%) Fedge (%) Fface (%)0.40 92.5 7.4 0.10.37 91.8 8.0 0.20.33 90.8 8.6 0.60.29 92.3 7.0 0.80.25 92.4 7.5 0.10.20 92.4 7.5 0.1experimental error.Next, we discuss the connections of the AsSe3 pyramids. In crystalline monoclinicAs2Se3, the pyramids are connected with each other by fully corner-sharing configurations.38)Table III shows the x dependence of the corner-, edge-, and face-sharing connections whentwo As atoms approach each other as second neighbors by sharing a Se atom. As for theAs2Se3 glass at x = 0.40, only about 7.5% of pyramids are connected accidentally by edgesharing, and corner sharing dominates the pyramid connections. This value is much smallerthan the edge-sharing fractions of GeSe4 tetrahedra (20–25%) in the stressed-rigid region ofthe Ge–Se glasses, as shown in Fig. 11(b) of Ref. 10. The face-sharing connection is negligi-ble. With decreasing x, all fractions remain basically unchanged, which is unlike the Ge–Seglass case, where the edge-sharing fraction rapidly decreases, as shown in Fig. 11(b) of Ref.10. These differences would be due to the fact that the AsSe3 pyramid clusters do not includethe stresses caused by the overconstraint of the averaged coordination numbers.Then, we discuss the connections between two pyramids. As mentioned above, the ma-jority of the connections between the AsSe3 pyramids in the As2Se3 glass are corner sharing,and there are a small fraction of edge-sharing connections. For both connections, the Asatoms share only one Se atom between them; this was referred to as the S1 connection inthe previous discussion. At the As2Se3 composition, the glass system is considered to be inthe stressed-rigid region.11) Figure 10 shows the x dependence of the fraction of the numberof Se atoms belonging to Sen chains between two As atoms, i.e., the fraction of Se atomsintervening between the As atoms.At x = 0.40, about 72% of Se atoms are connected with two As atoms by corner-, edge- orface-sharing connections (Se1). About 18% of Se atoms form the As–Se–Se–As connections(Se2). The remaining 10% of Se atoms are located in longer Se chains between two As atoms.21/27J. Phys. Soc. Jpn.0.80.60.40.20.0Fraction of Sen chains0.400.350.300.250.20xAsSe1Se2Se3Se4Se5+IPFig. 10. Fraction of the number of Se atoms belonging to Sen chains. The yellow region represents the IPconcentration region. (Color online)With decreasing x, the fraction of S1 connections gradually decreases owing to the increasein the numbers of Se atoms in the glasses. The fraction of Se2 connections increases untilthe end of the IP region at x = 0.29, and then starts to decrease rapidly. The S3 connectionsexhibit a similar x dependence to the Se2 connections with a smaller fraction of about 3/5.On the other hand, the fraction of Se atoms belonging to longer chains of more than five Seatoms rapidly increases in the floppy region of x < 0.29.A similar analysis was carried out on GexSe1−x glasses, and the following results wereobtained in relation to the stiffness transition shown in Fig. 12 of Ref. 10. Firstly, the fractionof Se1 atoms exhibits a dip in the IP region in Ref. 10, while the present result does not.Secondly, the fraction of Se2 connections slightly decreases in this region, which contradictsthe present result. Thirdly, the fraction of Se4+ connections (Se atoms belonging to chainswith more than 4 atoms) exceeds that of S1 connections at the central composition of the IPregion, whereas the S1 connections are dominant there in the present As–Se glasses, and therapid increase in S4+ (= S4 + S5+) starts at the end of the IP region. In the previous paperon the Ge–Se glasses, it was discussed that these behaviors of the connections of GeSe4tetrahedra originate from a large configurational stress in the Se2 connections and a relevantphase separation tendency between the S1 and S4+ chain connections. In the present As–Se glasses, on the other hand, such presence of stresses is hardly expected because AsSe3pyramids have good constraints to form glasses in the sense of rigidity percolation theory and22/27J. Phys. Soc. Jpn.do not form a specified configuration nearby. Accordingly, the differences in the pyramidalconnections may be related to the anisotropic pyramidal atomic configurations around the Asatoms in the As-Se glasses in contrast to the isotropic tetrahedral ones around the Ge atomsin the Ge-Se glasses.5. ConclusionsAXS, XRD, and ND experiments were carried out on AsxSe1−x chalcogenide glasses andthe results were analyzed by RMC modeling to obtain the partial atomic structural informa-tion in S i j(Q)s, gi j(r)s, and 3D atomic configurations and to find the relationship betweenthem and the stiffness transition. By using the experimentally obtained anomalous terms forthe atomic form factors and including the ND data for RMC analysis, the quality of the par-tial structural results was highly improved, particularly for the coordination numbers. TheS i j(Q) and gi j(r) functions seem to gradually change with x; however, an important changeis detected, i.e., a rapid decrease of As–As wrong bonds is visualized in the IP region, asin the Ge–Se glasses.10) However, the other anomalies found in the Ge–Se glasses, such asa rapid decrease in the pre-shoulder position in S SeSe(Q), a rapid decrease in the number ofedge-sharing connections, and an exclusion tendency of the connections between the As(Ge)atoms sharing two Se atoms, were not clearly observed. These differences may be due to thefact that the AsSe3 pyramids in the As–Se glass do not have any structural stress in contrastto the GeSe4 tetrahedra in the Ge–Se glasses.Finally, we note that the qualities of S AsAs(Q) and gAsAs(r) are not sufficient probablybecause of the small WAsAs values even in the ∆AsS (Q) spectra of the AXS data. In particular,the prepeaks seem not to change systematically in height with x and are not in good agreementwith the existing AIMD data.31) A possible way of improving the data quality with partialinformation is to include the NDIS data, as shown in Ref. 13, in the RMC analysis; this isnow being planned.Another way to improve the structural information is to ameliorate the statistics of theAXS data by increasing the incident X-ray flux. Since the beamline at the ESRF used for thepresent study is a bending magnet one, it is possible to replace it with an undulator insertiondevice for the X-ray source. At the SPring-8, new AXS equipments have been developed atpreviously BL13XU and currently BL47XU, and an intenser X-ray beam (about two ordersof magnitude) and a better energy resolution using a LiF analyzer crystal (about 30 eV at10 keV43)) were achieved.44, 45) In fact, much reasonable results in the pre- and first peakregion of ∆GaS (Q) were obtained on a Ga2Ge3Se9 glass at the SPring-846) compared with that23/27J. Phys. Soc. Jpn.measured at the ESRF,47) and the similar improvements is highly expected for the presentAs-Se glasses.AcknowledgmentThe AXS experiments were performed at BM02 of the ESRF (No. HS1860) and at BL15of the Saga-LS (Nos. 1802007A and 2005048F). The supporting AXS experiments werecarried out at BL13XU (Nos. 2021A1181 and 2021B1110), and BL47XU (No. 2022A1306).The ND measurements were carried at BL21 of MLF in the J-PARC (No. 2017B0047). SHacknowledges the Japanese Society for Promotion of Science (JSPS) for Grant-in-Aids forTransformative Research Areas (A) ‘Hyper-Ordered Structures Science’ (Nos. 21H05569and 23H04117) and for Scientific Research (C) (No. 22K12662), and Japan Science andTechnology Agency (JST) for CREST (No. JPMJCR1861) for financial supports.24/27J. Phys. Soc. Jpn.References1) J. C. Phillips, J. Non-Cryst. Solids 34, 153 (1979).2) M. F. Thorpe, J. Non-Cryst. Solids 57, 355 (1983).3) S. S. Yun, H. Li, R. L. Cappelletti, R. N. Enzweiler, and P. Boolchand, Phys. Rev. B 39,8702 (1989).4) Y. Wang, M. Nakamura, O. Matsuda, and K. Murase, Physica B 263–264, 313 (1999).5) M. Tatsumisago, B. L. Halfpap, J. L. Green, S. M. Lindsay, and C. A. Angell, Phys. Rev.Lett. 64, 1549 (1990).6) R. Böhmer and C. A. Angell, Phys. Rev. B 45, 10091 (1992).7) X. Feng, W. J. Bresser, and P. Boolchand, Phys. Rev. Lett. 78, 4422 (1997).8) P. Boolchand, X. Feng, and W. J. Bresser, J. Non-Cryst. Solids 293–295, 348 (2001).9) S. 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