The atomic arrangement in disordered alloys is classified as an intermediate structure between the non-crystalline glass and crystalline structures, because different metal atoms randomly occupy the lattice accompanied by small displacements from the periodic positions. Therefore, it has been difficult to investigate the atomic arrangement of disordered alloy. However, reverse Monte Carlo (RMC) calculation with complementary data sets of extended X-ray absorption fine structure (EXAFS) and X-ray diffraction (XRD) is a useful method to visualize the atomic arrangement in disordered alloys.

Ferromagnetic Fe-Ni disordered alloy is a suitable material to investigate the distortion of atomic arrangements by the magnetovolume effect (Yokoyama and Eguchi, 2011; Kousa et al., 2020; Yokoyama, 2021). The large magnetovolume effect of Fe-Ni alloys compresses the lattice with increasing temperature, which offsets normal thermal expansion and leads to the zero thermal expansion. This is known as the Invar effect. Disordered Fe₆₅Ni₃₅ is a well-known Invar alloy that exhibits zero thermal expansion in a wide temperature range up to the Curie temperature, T _C ≈ 505 K, at ambient pressure (Guillaume and Seances, 1897). We have previously examined the compressibility of Fe-Fe, Fe-Ni, and Ni-Ni atomic pairs in the Invar alloy using RMC calculations, revealing elongation of Fe-Fe atomic pairs compared with the other two atomic pairs (Ishimatsu et al., 2021). Because the elongation disappears with the pressure-induced transition to the paramagnetic phase, we concluded that the magnetovolume effect is due to the elongation of Fe-Fe atomic pairs.

To systematically discuss the relationship between Fe-Fe atomic pairs and the Invar effect, herein we performed a similar analysis on a Ni-rich alloy of Fe₅₅Ni₄₅, known as a pressure-induced Invar alloy, and compared the results with those of the Fe₆₅Ni₃₅ Invar alloy. Fe-Ni alloys recover normal thermal expansion with a slight increase in the Ni content from 35 at.%, but the pressure-induced Invar effect occurs at a characteristic pressure. According to a report by Dubrovinsky et al. (Dubrovinsky et al., 2001), Fe₅₅Ni₄₅ undergoes the pressure-induced Invar effect at ≈ 7.7 GPa, although this alloy has a 7.5 times larger thermal expansion coefficient (α ≈ 9 × 10^–6/K) than the Invar alloy (α ≈ 1.19 × 10^–6/K) at ambient pressure (Hayase et al., 1973). Therefore, this study focuses on the atomic arrangement at the vicinity of the pressure-induced Invar effect of the Fe₅₅Ni₄₅ alloy. Consequently, our RMC calculation with complementary data sets of EXAFS and XRD demonstrate that the atomic arrangement in disordered Fe-Ni alloys is not completely random and that the Fe₅₅Ni₄₅ alloy also exhibit elongation of the Fe-Fe atomic pairs, depending on the magnetovolume effect under pressure.

Polycrystalline Fe₅₅Ni₄₅ non-Invar alloys were grained from ingots and pelletized for the EXAFS and XRD measurements. The pressure dependences of EXAFS and XRD were independently measured on beamline 39XU at SPring-8 (Kawamura et al., 2009) using a fresh (uncompressed) pellet. We utilized diamond anvil cells to apply pressure to the sample. The applied pressure was monitored using the conventional ruby fluorescence method. Nano-polycrystalline diamond anvils were utilized for the EXAFS measurement to prevent Bragg diffraction from the anvil crystals superimposed on the absorption profile (Irifune et al., 2003; Ishimatsu et al., 2012, 2016). A Fluorinert 70 + 77 solution was used as a pressure transmitting medium. All the measurements were performed at room temperature.

The software RMCProfile was used to carry out the RMC calculations (Krayzman et al., 2008). A cluster structure, which reproduces the experimental results of both EXAFS and XRD, was found by RMC calculation. By using conventional EXAFS analysis, element-dependent differences in the local structure have been reported (Yokoyama and Eguchi, 2011; Kousa et al., 2020; Yokoyama, 2021). However, it is typically difficult to distinguish the neighboring atoms to the X-ray absorbing atoms by conventional EXAFS analysis, because metal elements with close atomic numbers, such as Fe and Ni, give comparable scattering amplitudes of the photoelectrons. In contrast, the RMC calculation constructs and optimizes the best model of the atomic cluster that matches the experimental EXAFS profiles at multiple absorption edges via trial-and-error iterations. RMCProfile is suitable for this purpose, because it allows for the input of a dataset of EXAFS spectra at the Fe and Ni K-edges together with XRD patterns. A cluster of randomly distributed 2201 Fe and 1799 Ni atoms to a 10 × 10 × 10 fcc unit cell was used as the starting structure. In our calculation, atom swapping was not allowed. Therefore, long-range disorder was unchanged during the calculation. The minimum atomic separation was set to 1.8 Å. Each RMC calculation was performed for 720 min. Although the calculation time depends on the computer power, we judged that this time was sufficient to achieve convergence by monitoring the residual of the fitting. All single scattering paths calculated by the FEFF85L code were used in the EXAFS part of the RMC optimization. Furthermore, chainlike multiple scattering paths, which give significant intensities comparable to those of the single scattering paths, were also included in our model. The RMC calculation was performed three times, and the averaged result was used for the discussion.

The calculated profiles are in good agreement with both EXAFS and XRD experimental data, as shown in Figure 1. The good reproducibility allows us to investigate how the distorted local structure coexists with the long-range periodicity in the calculated atomic cluster. The absolute values of Fourier transformed (FT) profiles (|FT [k ² χ(k)]|) increase with increasing pressure at both the Fe and Ni K-edges, indicating that Debye-Waller factor decreases with the volume compression (Figures 1A, B). The prominent peak at radial distance R = 2.1–2.2 Å, which corresponds to the first nearest neighboring (1NN) 12 Fe/Ni atoms of the X-ray absorbing atom, shows significant enhancement of the peak intensity with increasing pressure. This trend was well reproduced by the RMC calculation in a wide range of R (up to 6 Å).

Experimental XRD patterns at selected pressures are compared with the calculated profiles in Figure 1C. The pressure dependence of the 111 and 200 Bragg peaks were chosen for our XRD measurements using a flat panel with a limited window size of 50 × 50 mm². The RMC calculations successfully reproduced the appearance of 111 and 200 Bragg peaks. As shown in Figure 1D, the calculated atomic cluster conserves the sequential stacking of the atomic planes, although the atoms are displaced from the ideal positions of the fcc lattice. Therefore, the RMC calculations found a structure, in which the local structure was largely distorted; however, the disordered atomic cluster maintained the long-range periodicity to satisfy the Bragg law owing to pseudo fcc symmetry.

Figure 2A shows the partial pair distribution functions g _ij (R), where i, j = Fe or Ni, in the range of R = 1.8 Å to 6 Å  corresponding to the histogram of the first to fifth nearest neighboring (1NN–5NN) Fe-Fe, Fe-Ni and Ni-Ni atomic pairs at selected pressures. The g _ij (R) function is normalized to approach unity with increasing R (Keen, 2001). In Figure 2A, the cross marks represent the lengths of atomic pairs expected from the lattice constant of a cluster with fcc symmetry, revealing small deviations. Moreover, the broad g _ij (R) profile around the expected peak positions indicates the displacements of Fe/Ni atoms from the fcc lattice positions.

A noted result of g _ij (R) is the elongation of 1NN Fe-Fe pairs in comparison with 1NN Fe-Ni and Ni-Ni pairs. To highlight the elongation, an enlarged plot of g _ij (R) profiles around the 1NN peak is shown in Figure 2B. Compared to the profiles of g _Fe-Ni(R) and g _Ni-Ni(R) at 1.0 GPa, the peak and whole profile of g _Fe-Fe(R) around the 1NN Fe-Fe pairs shift to the longer R direction. As the pressure increases, the shift of the g _Fe-Fe(R) gradually decreases, and the g _ij (R) profiles of Fe-Fe, Fe-Ni and Ni-Ni pairs are located at the same positions at pressures above 18.9 GPa. The elongation and pressure-induced shrinkage of the Fe-Fe pairs have also been observed in the g _ij (R) profiles of Invar alloys (Ishimatsu et al., 2021).

To investigate the elongation of the Fe-Fe atomic pairs quantitatively, average lengths of the 1NN atomic pairs, R _Fe-Fe, R _Fe-Ni, and R _Ni-Ni were evaluated. Gaussian fitting was performed within a range of 2.2 Å ≤ R ≤ 2.8 Å, which encompasses the 1NN peaks in the asymmetric real space function. The fitted values of R _ij are plotted in Figure 3A as a function of pressure. We confirmed that a similar result was obtained by Gaussian fitting of the converted profile to the radial distribution function n _ij (R) (Keen, 2001). Figures 3A–C summarize the pressure dependences of R _ij , the integrated intensity of X-ray magnetic circular dichroism (XMCD), and the compression curve of Fe₅₅Ni₄₅ alloy. The compression curve was derived from the XRD patterns. Our previously reported results for the Fe₆₅Ni₃₅ alloy are shown in Figures 3D–F for comparison.

Figure 3A shows that R _Fe-Fe is longer than R _Fe-Ni and R _Ni-Ni in the low-pressure region. The evaluated bond lengths are 2.536, 2.513, and 2.515 Å for Fe-Fe, Fe-Ni, and Ni-Ni pairs, respectively, at 1.0 GPa. The magnitude of the elongation for the R _Fe-Fe is estimated to be approximately 0.021–0.023 Å, which is comparable to that of Invar alloy ( ≈ 0.02 Å).

The integrated XMCD at the K-edge corresponds to the 4p orbital moment (Igarashi and Hirai, 1994; Torchio et al., 2011). Because the 4p orbital moment is induced by the net magnetization, the integrated XMCD allows for the investigation of the pressure dependence of the magnetization. As shown in Figures 3B, E, both alloys undergo a phase transition from the ferromagnetic to the paramagnetic phase with increasing pressure. The vertical lines indicate the characteristic pressures labeled P _I and P _c , where the magnetization starts to decrease and becomes nearly zero, respectively. At pressures above P _c , the magnetic structure enters a paramagnetic phase. Fe₅₅Ni₄₅ alloy exhibits higher pressures (P _I ≈ 10.6 GPa and P _c ≈ 18.7 GPa) than the Fe₆₅Ni₃₅ alloy (P _I ≈ 1 GPa and P _c ≈ 7 GPa), indicating that the ferromagnetic order in Fe₅₅Ni₄₅ is more stable under pressure. This trend is consistent with the ≈ 215 K higher T _C of Fe₅₅Ni₄₅ than that of Fe₆₅Ni₃₅ (Hayase et al., 1973). The softening behavior appears as a hump in the compression curves near the onset pressure P _I as shown in Figures 3C, F. According to previous reports (van Schilfgaarde et al., 1999; Dubrovinsky et al., 2001), the Invar effect occurs at pressures of the softening behavior and is accompanied by the destabilization of the ferromagnetic phase at P _I .

In Invar alloys, the destabilized ferromagnetic structure leads to a rapid shrinkage of R _Fe-Fe due to the pressure-induced loss of the magnetovolume effect above P _I ≈ 1 GPa. Similar elastic anomalies of softening and shrinkage of R _Fe-Fe have also been observed for Fe₅₅Ni₄₅. Fe₅₅Ni₄₅ is characterized by the wide pressure range of the pressure-independent ferromagnetic state below P _I . The broader pressure range of the stable ferromagnetic state enables us to understand the correlation between the magnetic instability, elongation of Fe-Fe atomic pairs and the softening behavior.

Interestingly, the elongation rate of R _Fe-Fe remains unchanged in the wide pressure region below P _I , and then a rapid contraction of the R _Fe-Fe occurs accompanied by the destabilization of the ferromagnetic state in a pressure range of P _I ≤ P ≤ P _c . As mentioned above, the maximum elongation of R _Fe-Fe is comparable to that exhibited by the Invar alloy. Thus, a large magnetovolume effect commonly works in the ferromagnetic phase of Fe-Ni alloys below P _I , and a similar interpretation of the elongation of R _Fe-Fe can be given to this alloy as well as Fe₆₅Ni₃₅ Invar alloy. Therefore, although we did not measure temperature dependences, we expect that Fe₅₅Ni₄₅ alloy exhibits a pressure-induced Invar effect at pressures near P _I , where a rapid contraction of R _Fe-Fe occurs. It is noted that the softening behavior of Fe₅₅Ni₄₅ is observed at a pressure that is ≈ 3 GPa higher than that reported by Dubrovinsky et al. (Dubrovinsky et al., 2001), where they observed a small hump in the compression curve at P _I ≈ 7.5 GPa. This deviation is attributed to a different nonhydrostatic condition in the sample chamber that strongly depends on the pressure transmitting medium and sample shape. We also note that the lattice constant of the sample recovered the original value, as the pressure was released to ambient pressure. Therefore, it is confirmed that the plastic deformation was not introduced to the sample during the compression.

Finally, to confirm the validity of the element-dependent elongation, the expected value, ⟨R _ij ⟩, was evaluated by using the equation: ⟨ R i j ⟩ = ∫ R 1 R 2 R × 4 π R 2 g i j R ρ j d R ∫ R 1 R 2 4 π R 2 g i j R ρ j d R , (1) where parameters R ₁ and R ₂ define the lower and upper limits of integration, respectively, and ρ _j is the number of j atoms in a unit volume. An advantage of the expected value is to minimize the effects from the asymmetric real space function. We used a wide integration range from R ₁ = 1.8 Å to R ₂ ≈ 3.1 Å  ensuring inclusion of all 12 Fe/Ni atoms in the first shell of the fcc lattice. This range is wide enough to avoid ambiguities due to the integration range. The obtained pressure dependence of ⟨R _ij ⟩ is plotted in Figure 4. Although the elongation of the Fe-Fe bond is reproduced, it is a factor of one quarter smaller than the result of the Gaussian fit (Figure 3A). This is because the integration range includes not only main 1NN peak but also small humps and a moderate slope at both sides of the peak. However, we note that a similar trend is observed in the pressure dependence of the elongation: the elongation starts to decrease at pressures above P _I , and the lengths of three bond become comparable above P _c .

We successfully visualized the atomic arrangement of the disordered Fe₅₅Ni₄₅ alloy using the RMC calculation with the datasets of EXAFS and XRD measurements. The g _ij (R) profile was evaluated from the visualized atomic cluster, which showed that Fe and Ni atoms were displaced from a perfect fcc lattice, while the element-dependent distribution of the atomic bond length revealed an elongation of Fe-Fe pairs in the ferromagnetic phase at low pressures. Similar to the interpretation proposed for the Fe₆₅Ni₃₅ Invar alloy, the elongation of Fe-Fe pairs is the atomic scale origin of the volume expansion due to a large magnetovolume effect. The elongation becomes negligible upon the pressure-induced destabilization of the ferromagnetic state and softening behavior in the compression curve. Therefore, a strong correlation between the elongation of Fe-Fe atomic pairs, magnetization, and elastic anomalies has been clearly demonstrated. We note that the visualization procedure using RMC calculations can be employed to solve various problems of alloys such as the composition dependence of elastic anomalies, magnetism, and hydrogen embrittlement.