Site Crystallographic positions NN A B C D E Total A 2i (0, 0, 0 ) - 4 - 4 4 12 B 4f (0.4, 0.4, 0 ) 2 1 2 4 6 15 C 8i (0.74, 0.66, 0 ) - 1 5 4 4 14 D 8i (0.464, 0.131, 0 ) 1 2 4 1 4 12 E 8j (0.183, 0.183, 0.252) 1 3 4 4 2 14 TABLE I: Atomic crystallographic positions and numbers of NN atoms for the five lattice sites of the Fe-Cr σ -phase. The phonon DOS’s measured on $^{57}$Fe for both phases of the Fe-Cr alloy are presented in Figure 1. The dif- ferences in the energy range covered by the spectrum and the discrete structure of the spectrum are signifi- cant. The spectrum obtained for the α -phase is found to be similar to that of pure Fe [14] exibiting a distinct peak For the optimized structures the phonon dispersions and density of states were calculated using the direct method [12, 13]. The dynamical matrix of the crystal is constructed from the H-F forces generated while dis- placing atoms from their equilibrium positions. For con- sidered structures each atom must be displaced in three directions. For the α and σ -phases, a complete set of H- F forces is obtained from 48 and 90 independent atomic displacements, respectively. The amplitude of the dis- placements equals 0.03 ˚ A. To minimize systematic errors we applied displacements in positive and negative direc- tions. Finally, the phonon frequencies are obtained by the diagonalization of the dynamical matrix for each wave vector. The phonon DOS is calculated by the random sampling on the k-point grid in the reciprocal space, and then the thermodynamic functions are obtained within the harmonic approximation. sidered. However, for the sake of computer and time ca- pacity, we have chosen at random only five to be included in our calculations. The real σ -Fe 52 . 5 Cr 47 . 5 sample was approximated by a σ -Fe 53 . 3 Cr 46 . 7 one. The latter was studied in the 1 × 1 × 1 tetragonal supercell with 30 atoms (16 Fe and 14 Cr atoms). The structure optimization was done using the spin-polarized density functional to- tal energy calculations performed within the generalized gradient approximation (GGA) using the VASP pack- age [8, 9]. The valence electrons for each atom (electron configuration: d $^{5}$s1 and d $^{6}$s2 for the Cr and Fe atoms, respectively) are represented by plane wave expansions. The wave functions in the core region are evaluated us- ing the full-potential projector augmented-wave (PAW) method [10, 11]. The integrations in the reciprocal space were performed on the 8 × 8 × 8 and 4 × 4 × 4 grid for α - and σ -phase, respectively. During the optimization, the Hellmann-Feynman (H-F) forces and the stress ten- sor were calculated and the structure optimization was performed in two steps. First the lattice constants were determined assuming the appropriate symmetry, then the atomic positions were found in a fixed unit cell. The crystal structure optimization was finished when resid- ual forces were less than 10 − 5 eV/ ˚ Aand stresses were less than 0.1 kbar. The calculated lattice constants are a = 5 . 64 ˚ Aand a = 8 . 60 ˚ A, c = 4 . 74 ˚ Afor the cubic and tetragonal symmetry, respectively. The optimized mag- netic moments on the Fe atoms are ordered ferromag- netically with the average values 2.03 and 0.96 µ B , in the α and σ phase, respectively. On the Cr atoms, the antiparallel arrangement occurs with the negative mean magnetic moments -0.17 and -0.28 µ B in the respective phases. the Advanced Photon Source. The vibrational properties such as the Fe-partial phonon density-of-states were stud- ied using the method of nuclear resonant inelastic X-ray scattering (NRIXS) [5, 6]. Synchrotron radiation X-rays were monochromatized to a bandwidth of 1.2 meV and tuned in energy ranges of +/- 80 meV (room temperature measurements) and -20 meV to +80 meV (measurement at 20 K) around the $^{57}$Fe nuclear transition energy of 14.4125 keV. The X-ray flux and beam size at the sam- ple position were 4 × 10 9 photons/s and 0.3 × 2 mm $^{2}$, re- spectively. Data collection times were about 3 hours for the room temperature measurement of each sample and about 12 hours for the low temperature measurement of the σ -phase sample. We followed previously described evaluation procedures [5, 6] using the publicly available PHOENIX software [7]. The following quantities were derived directly from the data: Lamb-M¨ ossbauer factor, f , kinetic energy per atom, E k , and the mean force con- stant, D . No specific assumptions about the character of the vibrations had to be made to obtain these values. The Fe-partial DOS was derived by direct data inversion using the Fourier-Log method under the assumption of quasi- harmonic vibrations. The consistency of this pro- cedure was verified by independent calculation of Lamb- M¨ ossbauer factor, kinetic energy per atom, and mean force constant from the DOS and by agreement of these values with same quantities obtained directly from the data. Then the following quantities were calculated using the Fe-partial DOS: vibrational specific heat at constant volume, C V , and vibrational entropy, S . The assignment of Debye temperatures, Θ D , is based on the Debye model, i.e., the DOS is proportional to energy squared, and they are widely used in the literature. With the determina- tion of the Fe-partial DOS, we have surpassed the Debye model but find it useful to provide Debye temperatures for comparison. Using the values of f and C V , we deter- mined commonly presented values for Θ D . In calculations, both phases of Fe-Cr alloy were mod- eled by the appropriate atomic configurations placed in a supercell with the periodic boundary conditions. The disordered α -Fe 52 . 5 Cr 47 . 5 alloy was approximated by the α -Fe 50 Cr 50 one, for which we used the 2 × 2 × 2 bcc cell with 16 atoms. For random distribution of atoms, there are about 500 different atomic configurations to be con- 2