Fine Structure of Zr₈₀Pt₂₀ Amorphous Alloy Determined from Anomalous X-ray Scattering (AXS) Data by Applying Reverse Monte-Carlo (RMC) Simulation Method

Abstract

The fine structure of a Zr₈₀Pt₂₀ amorphous alloy was evaluated by anomalous X-ray scattering (AXS) coupled with reverse Monte Carlo (RMC) simulation. Featureless short-range ordered structures were preferentially formed by the Zr component, similar to those described by dense random packing (DRP), while the Pt atoms adopted a somewhat different structure. The Pt atoms adopted Zr-rich coordination and an icosahedral local atomic arrangement, in unique chemical short-range ordering (CSRO) as well as topological short-range ordering (TSRO). The present AXS-RMC analysis also suggested that the pre-peak signal at 17 nm⁻¹ in the X-ray diffraction profile was largely attributed to the correlation between the Pt–Pt pair with a separation of approximately 0.4–0.5 nm. The geometrical and chemical features of the common neighbors of the middle-range Pt–Pt pairs indicate the unique CSRO and TSRO, where both chemical and density fluctuations were observed in two distinct regions: a Zr-rich region with high packing density and a Pt-rich region with low packing density.

Fig. 1 Experimental interference functions, QΔi_Pt(Q), QΔi_Zr(Q), and Qi(Q), and corresponding values obtained from RMC simulation.

1. Introduction

Atomic ordering in both the short-range region and the middle-range region in amorphous alloys is known to be closely correlated with unique material properties.^1–4) Nevertheless, atomic ordering in amorphous alloys is poorly understood and suitable methods of describing the subtle differences in the ordering are still far from complete.

As an example, the icosahedral (and icosahedral-like) local atomic arrangement is suggested to be one of the specific topological short range orderings (TSROs) in Zr-based amorphous alloys.⁶⁾ Researchers propose that such TSRO is associated with solute-centered, solvent-shell type chemical short-range ordering (CSRO).⁷⁾ In contrast with short-range ordering (SRO), the so-called middle-range ordering (MRO), which involves atomic ordering reaching the next shell around the first nearest neighbor region, is yet to be fully elucidated and visualized.

Experimental evidence of MRO has been reported based on the occasional observation of a pre-peak signal in the X-ray diffraction (XRD) and neutron diffraction (ND) patterns of several amorphous alloys.^8–11) Recent studies of Zr–Pt binary amorphous alloys^12–14) strongly suggest that the pre-peak signal may provide information about the characteristic atomic ordering of amorphous alloys through a relatively less complex chemical approach. Structural studies of Zr₈₀Pt₂₀ amorphous alloys also suggest that the icosahedral (-like) local atomic arrangement is the dominant TSRO and the pre-peak arises from the enhanced middle range Pt–Pt correlation.^7,14–17) However, no quantitative information has yet been obtained with respect to the relationship between the icosahedral TSRO and MRO or the specific atomic arrangements that contribute to enhancing the middle-range Pt–Pt correlation.

The main purpose of this work is to report the fine structure of a Zr₈₀Pt₂₀ amorphous alloy, determined from anomalous X-ray scattering (AXS) data¹⁷⁾ by applying the reverse Monte-Carlo (RMC) simulation method, from which a method for describing the atomic ordering in the middle-range region is proposed by introducing a new approach for distribution of the Bernal polyhedral units.

2. Methodology of AXS-RMC Analysis

The AXS data for the Zr₈₀Pt₂₀ amorphous alloy were taken from our previous paper,¹⁷⁾ and the procedure for the RMC simulation employed in this work is identical to that described in our previous works.^5,18,19) Only the essential points are given below for convenience of discussion. When the incident energy for AXS is set close below the absorption edge, E_abs, using Pt as an example in this case, anomalous dispersion becomes significant. The resulting variation in the signal intensity, Δi_Pt(Q, E₁, E₂), is attributed to a change in the real part of the anomalous dispersion terms of Pt.   

\begin{align} &\Delta i_{\textit{Pt}}(Q,E_{1},E_{2}) \\ &\ \equiv \frac{\{I(Q,E_{1}) - \langle f^{2}(Q,E_{1}) \rangle\} - \{I(Q,E_{2}) - \langle f^{2}(Q,E_{2}) \rangle\}}{c_{\textit{Pt}}\{f'{}_{\textit{Pt}}(E_{1}) - f'{}_{\textit{Pt}}(E_{2})\}W(Q,E_{1},E_{2})}\\ &\ = \frac{c_{\textit{Pt}}\Re \{f_{\textit{Pt}}(Q,E_{1}) + f_{\textit{Pt}}(Q,E_{2})\}}{W(Q,E_{1},E_{2})}(a_{\textit{PtPt}}(Q) - 1)\\ &\ \quad + \frac{c_{\textit{Zr}}\Re \{f_{\textit{Zr}}(Q,E_{1}) + f_{\textit{Zr}}(Q,E_{2})\}}{W(Q,E_{1},E_{2})}(a_{\textit{PtZr}}(Q) - 1) \end{align} (1)

  

\begin{equation} W(Q,E_{1},E_{2}) = \sum_{k = 1}^{2}c_{k}\Re \{f_{k}(Q,E_{1}) + f_{k}(Q,E_{2})\} \end{equation} (2)

Here, Q and E are the wave vector and the incident X-ray energy, respectively, and c_i is the atomic fraction. The atomic scattering factor of the i-th element is given by the equation: $f_{i}(Q.E) = f_{i}^{0}(Q) + f'_{i}(E) + if''_{i}(E)$, where f⁰(Q) is the scattering factor of the atom at an energy sufficiently away from the absorption edge, and f′(E) and f′′(E) are the real and imaginary parts of the anomalous dispersion terms,²⁰⁾ respectively. Equation (2) includes two functions: I(Q.E) and W(Q.E₁, E₂). The function I(Q.E) is the coherent X-ray scattering intensity, obtained by ordinary analysis coupled with the generalized Krough-Moe-Norman method.²¹⁾ The function W(Q, E₁, E₂) is calculated using eq. (2), where the symbol $\Re \{ \} $ denotes the real part of the values in the brackets. The value of Δi_Pt(Q, E₁, E₂) is a function of the two partial structure factors, a_PtPt(Q) and a_PtZr(Q). Similarly, Δi_Zr(Q, E₁, E₂) measured at the Zr K absorption edge contains contributions from a_ZrZr(Q) and a_PtZr(Q). Fourier-transforms of the functions Δi_Pt(Q, E₁, E₂) and Δi_Zr(Q, E₁, E₂) using the average number density (ρ₀) of the system readily yields the environmental radial distribution functions (RDFs) 4πr²ρ_Pt(r) and 4πr²ρ_Zr(r), respectively.

The detailed parameters for the RMC simulation²²⁾ are summarized as follows. The simulation was initiated with a model comprising 25000 atoms (Zr: 20000 and Pt: 5000) for the dense random packing (DRP) structure in a cubic hyper-cell with a unit size of L = 8.102 nm. The number of atoms and cell size were selected for the model to meet the chemical composition as well as the measured density (8.74 Mg/m³) of the Zr₈₀Pt₂₀ sample. The atomic positions in the DRP structure were obtained using the algorithm proposed by Clarke et al.²³⁾ by employing the Goldschmidt radii (Zr: 1.60 nm, Pt: 1.38 nm).²⁴⁾ Through the present RMC simulation, the partial distribution functions for each pair of constituents and the corresponding structural factors could be readily calculated, where this structural information is very useful for discussing the SRO and MRO observed for amorphous Zr₈₀Pt₂₀.

3. Results

Figure 1 shows the experimental interference functions, Qi(Q), QΔi_Pt(Q), and QΔi_Zr(Q), as well as those determined from the AXS data using the RMC simulation method (hereafter referred to as AXS-RMC). The AXS-RMC simulation results reproduced the experimental structural functions well, and the atomic positions obtained from the present AXS-RMC simulation seem to be much closer to the atomic arrangements of amorphous Zr₈₀Pt₂₀.

Fig. 1

Experimental interference functions, QΔi_Pt(Q), QΔi_Zr(Q), and Qi(Q), and corresponding values obtained from RMC simulation.

Notably, a pre-peak signal was clearly observed at approximately Q = 17 nm⁻¹ for the experimental functions Qi(Q) and QΔi_Pt(Q), but not for the function QΔi_Zr(Q). This observation suggests that the pre-peak signal is associated with the Pt–Pt interaction in the middle-range ordering.¹⁷⁾

Figure 2 shows three partial structure factors, S_ij(Q)s, of the Zr₈₀Pt₂₀ amorphous alloy in the so-called Faber-Ziman (FZ) form obtained from the present AXS-RMC simulation and those of the DRP structure model. Because the pre-peak signal at 17 nm⁻¹ was clearly observed in S_PtPt(Q)_AXS-RMC, the middle range ordering structure of the Zr₈₀Pt₂₀ amorphous alloy is attributed to the Pt–Pt pair correlation and the corresponding chemical fluctuation.

Fig. 2

Partial structure factors, S_PtPt(Q), S_ZrZr(Q), and S_PtZr(Q) calculated from the RMC simulation and using DRP structural model.

Figure 3 shows the partial pair distribution functions, g_ij(r)s, determined from the AXS-RMC results, as well as those determined from the DRP structure model. The vertical dotted lines indicate the nearest neighbor atomic distances for the Pt–Pt, Zr–Pt, and Zr–Zr pairs (Pt–Pt: 0.276 nm, Zr–Pt: 0.298 nm, Zr–Zr: 0.320 nm), estimated from the Goldschmidt radii.²⁴⁾ The first peaks of the partial pair distribution functions, g_PtPt(r)_AXS-RMC and g_ZrZr(r)_AXS-RMC, were found at a distance that is just twice the Goldschmidt radii. On the other hand, the first peak corresponding to g_ZrPt(r)_AXS-RMC had an apparent maximum at a distance shorter than that estimated using the Goldschmidt radii. The upper limit of nearest neighbor region for each atomic pairs used in calculation of SRO parameters and common neighbor analysis (CNA) were set as the near of the minimum g(r) as follows, Zr–Zr: 0.420 nm, Zr–Pt: 0.400 nm and Pt–Pt: 0.380 nm.

Fig. 3

Partial pair distribution functions, g_ij(Q)s, calculated from 3-D structural model obtained by RMC simulation. The corresponding functions for the DRP structure are also illustrated for comparison.

For all three g_ij(r)_AXS-RMC partial pair distribution functions, a second correlation peak consisting of two humps was observed at a distance of 0.45–0.65 nm. Importantly, the partial pair distribution function g_PtPt(r)_AXS-RMC indicates significant enlargement at the shorter side of the second correlation peak. This unique correlation provides key information on the MRO developed in the present Zr₈₀Pt₂₀ amorphous alloy.

4. Discussion

4.1 Short range ordering

Table 1 lists the structural parameters, including the average atomic distance and the coordination numbers for the nearest-neighbor region, calculated from the three-dimensional atomic models obtained from the AXS-RMC simulation results. The structural parameters determined from the DRP model are also presented in Table 1 in the round brackets.

Table 1 Average atomic distance: r_ij, first peak maximum for g_ij(r): p_ij, average coordination number: Z_ij, chemical occupancy of coordinated atoms, average atomic size ratio, R^*, from RMC simulation. The R^* values for the DRP structure are also presented in round brackets for comparison.

Examination of the listed parameters from AXS-RMC readily suggests chemical affinity between the first-nearest Pt–Zr pair, as well as inhomogeneous distribution of Pt. As an example, the maximum of the first peak of the Pt–Zr pair was found at 0.282 nm, where the distance is shorter than that of the DRP structure. This is likely to arise from the chemically strong bond associated with the small mixing enthalpy of Pt and Zr.²⁵⁾ On the other hand, the average atomic distance, r_ij, and the average coordination number, Z_ij, of Zr were not significantly different from those of the DRP structure. This again implies that the averaged local structure of Zr surrounded by the solvent is well approximated by the DRP structure.

The Voronoi tessellation²⁶⁾ is a well-known and useful method of evaluating the fine topological features around specific components of amorphous structural models. We analyzed the distribution of the Voronoi polyhedral units of the AXS-RMC and DRP structural models in a previous report.¹⁷⁾ The distribution of the Voronoi polyhedral units around the Zr atoms agrees well with the DRP structure, while some differences can be found in the distribution of the Pt atoms. This contrast suggests that the TSRO around the Pt atoms is influenced by chemical interactions. The icosahedral and icosahedron-like coordination indicated by the Voronoi indices of (0 0 12 0 0) or (0 2 8 2 0) were the dominant TSRO in the Voronoi polyhedra distribution for the Pt atoms. This result indicates that the Pt center adopts icosahedral short-range ordering in the Zr₈₀Pt₂₀ amorphous structure as one of the most important TSROs.

It should be noted that the topological preference of the Pt center for icosahedral(-like) SRO can be explained by using the ratio, r^*, of the first-nearest Pt–Zr and Zr–Zr distances, which corresponds to the ratio of the distance from the center to the vertex and that to the edge in the Pt-centered icosahedrons. The calculated ratio r^* was 0.951 = 0.311/0.327, which clearly coincides with r_ico^* = 0.951 for the ideal icosahedron.

4.2 Common neighbor analysis and middle-range ordering with structural inhomogeneity

Analysis of the SRO in the first nearest-neighbor region is useful for understanding the fundamental structural features of amorphous alloys. However, it is sometimes necessary to conduct further analysis with respect to the second nearest-neighbor region. The present Zr₈₀Pt₂₀ amorphous alloy could be included in this category because the partial pair distribution function of the Pt–Pt pair indicates a significant enlargement at approximately 0.45 nm in the second correlation region, as shown in Fig. 3. To extend the analysis to the second nearest-neighbor region, the CNA^27,28) was employed.

An interesting pair correlation was selected for CNA, where the paired atoms were specified as the root-pair atoms. The CNA procedure ranks the atomic configuration for the root-pair atoms by using CNA indices of three numbers, [n₁ n₂ n₃]_CN. Here, n₁ is the number of neighbors common to the root-pair atoms, n₂ is the number of connections between the common neighbors, and n₃ is the longest continuous number of connections between common neighbors.

Figure 4 shows the results of CNA for the Pt–Pt correlation associated with the second peak region of the partial g_PtPt(r)_AXS-RMC and g_PtPt(r)_DRP. Schematic diagrams of the typical common neighbor connections found in this region, i.e., [333]_CN, [322]_CN, and [211]_CN, are illustrated in Fig. 5. The numbers, the average distance of the root-pairs, and the elemental fraction of the common neighbors for each connection are summarized in Table 2. The root-pairs in the second nearest-neighbor region are ranked into four CNA indices, i.e., [333]_CN, [322]_CN, [211]_CN, and [100]_CN. The middle-range Pt–Pt correlation around 0.45 nm mainly consists of the connection defined by the three indices [333]_CN, [322]_CN, and [211]_CN. This connection is characterized by a short distance and a remarkably higher population of Zr in the common neighbors. This result readily suggests that the second-nearest Pt–Pt pairs, which are closely associated with the pre-peak signal, accommodate Zr as common neighbors to produce a Zr-rich CSRO around Pt. At the same time, there is a high preference for the [322]_CN and [211]_CN connections at 0.45 nm as another interesting feature observed in the present AXS-RMC results. This is discussed in the next paragraph by using a new approach employing the distribution of the Bernal polyhedral units.

Fig. 4

Results of CNA for middle-range Pt–Pt pair from RMC simulation (a), corresponding data for DRP structure (b), and the difference between both sets of values (c).

Fig. 5

Schematic diagrams of bonding between root pair atoms and common neighbor atoms with CNA index of (a) [333], (b) [322], and (c) [211].

Table 2 Number, average atomic distance of Pt–Pt root pair, r, occupancy of element of the common neighbor atoms of [333]_CN, [322]_CN, and [211]_CN from RMC simulation. The corresponding values for the DRP structure are also presented in round brackets for comparison.

4.3 Geometrical features of MRO around Pt

To discuss the topological features in the connection for the specific middle-range Pt–Pt ordering associated with the pre-peak signal, we introduce the distribution of the Bernal polyhedral units.²⁹⁾

The connection specified by the CNA index [333]_CN indicates a structure composed of two Pt atoms as a root-pair and three atoms as common neighbors. This local structure can be considered as a unit of face-sharing tetrahedra (FST), as shown in Fig. 5. Because the tetrahedron is one of the dense packing structural units among Bernal polyhedra, the connection with the [333]_CN index corresponds to a relatively more dense packing structure. The [333]_CN connection could be further ranked by the number of Zr atoms among the three neighbors (Zr₃, Zr₂Pt, Zr₁Pt₂, and Pt₃), as shown in Fig. 6 and Table 3. The present AXS-RMC results readily demonstrate that for the [333]_CN connection, the Pt–Zr₃–Pt linkage is preferred among the common neighbors. These structural features could be attributed to the strong chemical affinity between Pt and Zr. This is also attributed to the saturation of the quasicrystalline phase in the annealed sample.¹⁷⁾

Fig. 6

Distribution of Pt–Pt root pairs with [333]_CN index for each compositional pattern of common neighbor atoms (Zr₃, Zr₂Pt, Zr₁Pt₂, and Pt₃) from RMC simulation (a), corresponding values for DRP structure (b), and difference between values (c).

Table 3 Number of [333]_CN indexed Pt–Pt root pairs with each compositional pattern of common neighbors, and average atomic distance, r, of Pt–Pt root pairs from RMC simulations. The corresponding values for the DRP structure are also presented in round brackets for comparison.

The [211]_CN connection can be expressed as edge sharing between two triangles, as shown in Fig. 5. Because the [211]_CN connection does not directly lead to a three-dimensional packing scheme of atoms, a supplemental bonding chain (SBC) associated with the root-pair atoms is introduced here to discuss the topological image for this connection. The atomic arrangement of common neighbors and SBCs for the [211]_CN connection can be classified as either a linkage of several Bernal polyhedra or an independent polyhedron. As shown in Fig. 7 and Fig. 8, the [211]_CN connections are classified as face-sharing tri-tetrahedra (FSTT), capped half-octahedron (CHO), partial tetragonal dodecahedra (PTD), partial tri-capped trigonal prism (PTTP), or partial bi-capped tetragonal anti-prism (PBTA), and the independent polyhedron is ranked as a ‘non-Bernal polyhedron’.

Fig. 7

Schematic diagrams of Bernal polyhedra corresponding to [211]_CN, (a) face shared tri-tetrahedra (FSTT), (b) capped half-octahedron (CHO), (c) partial tetragonal dodecahedron (PTD), (d-1,2) partial tri-capped trigonal prism (PTTP), and (e-1,2) partial bi-capped tetragonal anti-prism (PBTA).

Fig. 8

Distribution of Pt–Pt root pairs with [211]_CN in middle-range with the corresponding Bernal polyhedra from RMC simulation (a), corresponding values for DRP structure (b), and difference between both values (c). The number of “Bernal polyhedra” is sum the of FSTT, CHO, PTD, PTTP, and PBTA.

The AXS-RMC data in Table 4 suggest that the FSTT and CHO arrangements are preferred in the range of 0.4–0.5 nm, as well as the Zr-rich common neighbors and SBCs. These features are similar to those found for the [333]_CN connection, and are attributed to the Pt–Pt middle range ordering associated with the strong bonds in the nearest-neighbor Pt–Zr correlation. On the other hand, the PTD, PTTP, PBTA, and non-Bernal polyhedron arrangements predicted from the AXS-RMC results indicate a slight preference for Pt compared to the case of the DRP structure. These structural units with relatively lower packing fractions may differ from the [333]_CN arrangement having a higher packing fraction with Zr-rich common neighbors.

Table 4 Numbers, fractions, average Pt–Pt root pair distances, r, average composition of common neighbor atoms and SBC atoms for each Bernal polyhedra and non-Bernal polyhedra with [211]_CN indexed Pt–Pt pair.

The [322]_CN index corresponds to a structural unit composed of two atoms as the root-pair and three atoms as common neighbors. This [322]_CN structural unit can be expressed as a half octahedron. In order to discuss the three-dimensional packing structure associated with Bernal polyhedra, structural analysis with SBC was also employed. The results are illustrated in Figs. 9 and 10 and Table 5. The [322]_CN connections indicate a preference for the PTD, PTTP, PBTA, and non-Bernal polyhedra. Similar to the [211]_CN structural unit, the total amount and fraction of Pt in the SBCs of these structural units is higher than those of the DRP model.

Fig. 9

Schematic diagrams of Bernal polyhedra corresponding to [322]_CN, (a) partial tetragonal dodecahedra (PTD), (b) partial tri-capped trigonal prism (PTTP), and (c) partial bi-capped tetragonal anti-prism (PBTA).

Fig. 10

Pt–Pt root pair distribution with [322]_CN in middle-range with corresponding Bernal polyhedra from RMC simulation (a), corresponding values for DRP structure (b), and difference between both values (c). The number of “Bernal polyhedra” is the sum of PTD, PTTP, and PBTA.

Table 5 Numbers, fractions, average Pt–Pt root pair distances, r, average composition of common neighbor atoms and SBC atoms of each Bernal polyhedra and non-Bernal polyhedra with [322]_CN indexed Pt–Pt pair.

Based on these structural features, the Pt–Pt middle range correlation associated with the pre-peak can be classified into two types of structures. One structural type is representative of FST, FSTT, and CHO, associated with a higher packing density of atoms and common neighbors. This case also includes SBCs with a strong preference for Zr. The other is represented by PTD, PTTP, PBTA, and non-Bernal polyhedra, having a lower packing density of atoms with common neighbors and SBCs, with slight preference for Pt.

5. Concluding Remarks

By using the reverse Monte-Carlo (RMC) simulation method, anomalous X-ray scattering (AXS) data for a Zr₈₀Pt₂₀ amorphous alloy were analyzed in order to characterize the particular structural features of the alloy, including those associated with the pre-peak signal observed in the X-ray diffraction pattern. The AXS-RMC analysis also provides the g_ij(r)s and S_ij(Q)s partial functions for the Zr₈₀Pt₂₀ amorphous alloy, which are necessary for discussing the structural features in the middle range ordering. The resultant atomic positions were used to determine the common neighbors, middle range ordering with structural inhomogeneity, and the geometrical features for MRO around a specific atom such as Pt. The main results are summarized as follows:

1. (1)    The present AXS-RMC analysis for the short-range ordering region clearly indicates that the Pt atom accommodates the coordination ligands of Zr in the first nearest-neighbor region to form an icosahedral-like local structure.
2. (2)    The information on the partial S_PtPt(Q) and the corresponding g_PtPt(r) clearly suggests that the pre-peak signal at 17 nm⁻¹ in S(Q) is mainly attributed to the Pt–Pt middle range correlation at approximately 0.4–0.5 nm. This particular middle range ordering of Pt–Pt correlations corresponds mainly to the polyhedral connections described by the [333]_CN, [211]_CN, and [322]_CN indices.
3. (3)    The geometrical and chemical features of the common neighbors of the middle-range Pt–Pt pairs indicate unique CSRO and TSRO with both chemical and density fluctuations in two distinct regions: the Zr-rich region with high packing density and the Pt-rich region with low packing density.
4. (4)    Notably, in this particular middle range ordering of Pt–Pt correlations, chemical and density fluctuations were both clearly identified by using the unique geometrical linkages based on the Bernal polyhedral units. This new approach is expected to be utilized for characterizing the middle range ordering of a variety of amorphous alloy systems.

Acknowledgment

The authors express their gratitude to Dr. H. Nitani, Institute of Materials Structure Science, High Energy Accelerator Research Organization, Tsukuba, Japan, for assistance with the AXS measurements (Proposal No. 2019G646).

We are also grateful to Professor Y. Waseda, Tohoku University for his valuable comments and discussions. A part of this work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers JP16H04210 and JP19K15653.

REFERENCES

• 1)   M.F.  Ashby and  A.L.  Greer: Scr. Mater. 54 (2006) 321–326.
• 2)   C.A.  Schuh,  T.C.  Hufnagel and  U.  Ramamurty: Acta Mater. 55 (2007) 4067–4109.
• 3)   M.M.  Trexler and  N.N.  Thadhani: Prog. Mater. Sci. 55 (2010) 759–839.
• 4)   A.  Inoue and  A.  Takeuchi: Acta Mater. 59 (2011) 2243–2267.
• 5)   T.  Kawamata,  Y.  Yokoyama,  M.  Saito,  K.  Sugiyama and  Y.  Waseda: Mater. Trans. 51 (2010) 1796–1801.
• 6)   T.  Fukunaga,  K.  Itoh,  T.  Otomo,  K.  Mori,  M.  Sugiyama,  H.  Kato,  M.  Hasegawa,  A.  Hirata,  Y.  Hirotsu and  A.C.  Hannon: Mater. Trans. 48 (2007) 1698–1702.
• 7)   H.W.  Sheng,  W.K.  Luo,  F.M.  Alamgir,  J.M.  Bai and  E.  Ma: Nature 439 (2006) 419–425.
• 8)   K.  Saksl,  P.  Jóvári,  H.  Franz and  J.Z.  Jiang: J. Appl. Phys. 97 (2005) 113507.
• 9)   K.  Saksl,  P.  Jóvári,  H.  Franz,  Q.S.  Zeng,  J.F.  Liu and  J.Z.  Jiang: J. Phys. Condens. Matter 18 (2006) 7579–7592.
• 10)   S.Y.  Wang,  C.Z.  Wang,  M.Z.  Li,  L.  Huang,  R.T.  Ott,  M.J.  Kramer,  D.J.  Sordelet and  K.M.  Ho: Phys. Rev. B 78 (2008) 184204.
• 11)   L.  Yang,  J.Z.  Jiang,  K.  Saksl and  H.  Franz: J. Phys. Condens. Matter 19 (2007) 476217.
• 12)   J.  Saida,  M.  Matsushita and  A.  Inoue: J. Appl. Phys. 90 (2001) 4717–4724.
• 13)   B.S.  Murty,  D.H.  Ping,  M.  Ohnuma and  K.  Hono: Acta Mater. 49 (2001) 3453–3462.
• 14)   D.J.  Sordelet,  R.T.  Ott,  M.Z.  Li,  S.Y.  Wang,  C.Z.  Wang,  M.F.  Besser,  A.C.Y.  Liu and  M.J.  Kramer: Metall. Mater. Trans. A 39 (2008) 1908–1916.
• 15)   N.A.  Mauro,  V.  Wessels,  J.C.  Bendert,  S.  Klein,  A.K.  Gangopadhyay,  M.J.  Kramer,  S.G.  Hao,  G.E.  Rustan,  A.  Kreyssig,  A.I.  Goldman and  K.F.  Kelton: Phys. Rev. B 83 (2011) 184109.
• 16)   A.  Hirata,  L.J.  Kang,  T.  Fujita,  B.  Klumov,  K.  Matsue,  M.  Kotani,  A.R.  Yavari and  M.W.  Chen: Science 341 (2013) 376–379.
• 17)   K.  Sugiyama,  T.  Kawamata and  T.  Muto: J. Phys. Conf. Ser. 809 (2017) 012005.
• 18)   Y.  Waseda,  E.  Matsubara and  K.  Sugiyama: Sci. Rep. Res. Inst. Tohoku Univ. A 34 (1988) 1–14.
• 19)   M.  Saito,  C.  Park,  K.  Omote,  K.  Sugiyama and  Y.  Waseda: J. Phys. Soc. Jpn. 66 (1997) 633–640.
• 20)   D.T.  Cromer and  D.  Liberman: J. Chem. Phys. 53 (1970) 1891–1898.
• 21)  Y. Waseda: The Structure of Non-Crystalline Materials, (McGraw-Hill International, New York, 1980) pp. 27–51.
• 22)   R.L.  McGreevy and  L.  Pusztai: Mol. Simul. 1 (1988) 359–367.
• 23)   A.S.  Clarke and  J.D.  Wiley: Phys. Rev. B 35 (1987) 7350–7356.
• 24)  E.A. Brandes and G.B. Brook: Smithells Metals Reference Book, 7th ed., (Elsevier Butterworth-Heinemann Ltd., London, 1992) pp. 4–41.
• 25)   A.  Takeuchi and  A.  Inoue: Mater. Trans. 46 (2005) 2817–2829.
• 26)   J.D.  Bernal: Nature 183 (1959) 141–147.
• 27)   A.S.  Clarke and  H.  Josson: Phys. Rev. E 47 (1993) 3975–3984.
• 28)   J.D.  Honeycutt and  H.C.  Andersen: J. Phys. Chem. 91 (1987) 4950–4963.
• 29)   J.D.  Bernal: Nature 185 (1960) 68–70.