4 Fe-Cr alloy. Although, the sigma phase is a very complex object due to a high number of atoms per unit cell, five different sublattices with high coordination numbers (12- 15), each showing chemical disorder, which altogether results in a huge number of possible atomic configura- tions, it was described reasonable well in terms of only one adequately chosen configuration. From the calcula- tions, it is also evident that the dynamics in particular sublattices is different. The method was also successfully used to calculate the dynamics of the disordered alloy in the alpha-phase. Although its crystallographic structure is much simpler, but, due to the chemical disorder, the number of possible atomic configurations is high, making the calculations not trivial. We have also obtained rele- vant thermodynamic quantities without necessity of us- ing empirical parameters. Such a complex alloy has been studied for the first time within the combined NRIXS and theoretical ab initio approach, and it may provide understanding of lattice dynamics in a wide variety of disordered systems. fact that the center shift measured by M¨ ossbauer spec- troscopy constitutes a sum of the chemical isomer shift, which is independent on the atomic motion, and the second-order Doppler shift, which is a relativistic cor- rection to the atomic energy levels purely due to mo- tion. The second-order Doppler shift is proportional to the vibrational kinetic energy of the $^{57}$Fe atom [15], i.e. SOD[mm/s] = -0.00565 × E k [meV]. Only under the as- sumption that the chemical isomer shift is temperature independent, the M¨ ossbauer measurement can provide the correct Debye temperature. This does not seem to be the case here, and our NRIXS data present a notable im- provement in the understanding of the role of vibrations in the Fe-Cr system. The value of the vibrational entropy measured at 298 K for the α -phase agrees quite well with inelastic neutron scattering results on samples of similar composition [14]. Also the difference in the entropy val- ues, Δ S = S σ − S α = 0 . 095 ± 0 . 009 k B as determined in the present experiment from the Fe-partial DOS agrees well with the corresponding difference calculated from the equation Δ S = 3 k B ln(Θ Dσ / Θ Dα ) = 0 . 07 k B , where k B is the Boltzmann constant and Θ Di is the Debye tem- perature as determined from the second-order Doppler shift for the α ( i = α ) and the σ ( i = σ ) phase, respec- tively [3]. The corresponding theoretical value of Δ S is equal to 0.058 k B . Taking into account the approxima- tions used in the calculations, the agreement seems to be quite satisfactory. In summary, we have revealed, both experimentally and theoretically, significant differences in the partial-Fe phonon DOS of the α and σ phases of a quasi-equiatomic TABLE II: Physical quantities derived from the measured and calculated Fe-partial DOS. The units are meV/atom for vi- brational kinetic energy, E k ; k B /atom for vibrational entropy, S , and specific heat C V ; N/m for mean force constant, D ; K for Debye temperatures, Θ D . experiment theory quantity α σ α σ f @298K 0.782 ± 0 . 001 0.768 ± 0 . 002 E k @298K 42.4 ± 0 . 1 41.4 ± 0 . 2 D @298K 156 ± 1 157 ± 2 149.41 150.59 C V @298K 2.747 ± 0 . 006 2.748 ± 0 . 007 2.746 2.752 S @298K 3.252 ± 0 . 006 3.347 ± 0 . 007 3.354 3.412 f @20K 0.9150 ± 0 . 0002 E k @20K 18.93 ± 0 . 06 D @20K 155.1 ± 0 . 7 f @0K 0.9194 ± 0 . 0001 0.9149 ± 0 . 0003 E k @0K 19.32 ± 0 . 07 18.95 ± 0 . 06 Θ D ( f @298K) 417 398 Θ D ( C V ) 399 398 Θ D ( f @20K) 387 Θ D ( f @0K) 403 385 ∗ Corresponding author: dubiel@novell.ftj.agh.edu.pl [1] K. H. Lo, C. H. Shek and J. K. L. Lai, Materials Sci. Eng. R: Reports 65 , 39 (2009). [2] J. Cieslak, M. Reissner, W. Steiner and S. M. Dubiel, Phys. Stat. Sol. 205 , 1794 (2008). [3] J. Cieslak, B. F. O. Costa and S. M. Dubiel, J. Phys.: Condens. Matter 18 , 10899 (2006). [4] S. I. Simdyankin, S. N. Taraskin, M. Dzugutov, and S. R. Elliott, Phys. Rev. B 62 , 3223 (2000). [5] W. Sturhahn, T. S. Toellner, E. E. Alp, X. Zhang, M. Ando, Y. Yoda, S. Kikuta, M. Seto, C. W. Kimball and B. Dabrowski, Phys. Rev. Lett. 74 , 3832 (1995). [6] W. Sturhahn, J. Phys.: Condens. Matt. 16 , S497 (2004). [7] W. Sturhahn, Hyperfine Interact. 125 , 149 (2000). [8] G. Kresse, and J. Furtm¨ uller, Phys. Rev. B 54 , 11169 (1996). [9] G. Kresse, and J. Furtm¨ uller, Comput. Mater. Sci. 6 , 15 (1996). [10] P. E. Bl¨ ochl, Phys. Rev. B 50 , 17953 (1994). [11] G. Kresse and D. Joubert, Phys. Rev. B 59 , 1758 (1999). [12] K. Parlinski, Z. Q. Li and Y. Kawazoe, Phys. Rev. Lett. 78 , 4063 (1997). [13] K. Parlinski, Software PHONON, Krakow, 2007. [14] M. S. Lucas, M. Kresch, R. Stevens and B. Fultz, Phys. Rev. B 77 , 184303 (2008). [15] W. Sturhahn, and A. I. Chumakov, Hyperfine Interact. 123/124 , 809 (1999). The results reported in this study were partly obtained within the project supported by the Ministry of Sci- ence and Higher Education, Warsaw (grant No. N N202 228837) and the Project no. 44/N-COST/2007/0. Use of the Advanced Photon Source was supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02- 06CH11357.