e/a Determination for the transition metal element TM in Al –– Cu –– TM –– Si (TM ¼ Fe and Ru) approximants and B2-compounds by means of the FLAPW-Fourier method Uichiro Mizutani * ,I , Ryoji Asahi II , Tsunehiro Takeuchi III , Hirokazu Sato IV , Oleg Y. Kontsevoi V and Arthur J. Freeman V I Toyota Physical & Chemical Research Institute, Nagakute, Aichi 480-1192, Japan II Toyota Central R & D Laboratories, Inc., Nagakute, Aichi 480-1192, Japan III Ecotopia Science Institute, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan IV Department of Physics, Aichi University of Education, Kariya-shi, Aichi 448-8542, Japan V Department of Physics and Astronomy, Northwestern Univ., Evanston, IL 60208, U.S.A. Received June 14, 2008; accepted July 15, 2008 Phase stability / Electron concentration / Hume-Rothery electron concentration rule Abstract. By using the FLAPW-Fourier method, we could determine the e/a value for Al–– Cu––TM ––Si (TM ¼ Fe and Ru) 1/1 1/1 1/1 approximants and sev- eral B2 compounds including CuZn, NiZn, NiAl, MnZn, and AlMg. The NiAl, NiZn and MnZn B2-compounds involving the transition metal element as a partner element are found to be no longer regarded as the 3/2 electron compounds. Moreover, we found that the e/a value is not uniquely assigned to a given transition metal element but depends on its surrounding environments. Hence, it is dif- ficult to use it as a universal parameter in an alloy design. 1. Introduction There exist two different definitions for the electron con- centration, which plays a key role in the discussion of phase stability: one is the e/a, defined as an averaged va- lency of constituent elements in an alloy, and the other VEC or the number of valence electrons per atom filled into the valence band. Obviously, the e/a is the parameter appearing in the Hume-Rothery rule, whereas the VEC is essential in determining the Fermi level in the valence band. One must cautiously select either e/a or VEC as an electronic structure-controlling parameter, depending on the stability mechanism involved [1–3]. The phrase of the Hume-Rothery stabilization mechan- ism has been referred to by many researchers, when the density of states (DOS) of a complex alloy phase is char- acterized by the pseudogap at the Fermi level. This is be- cause many quasicrystals were discovered by using the Hume-Rothery e/a rule as a guide [4]. This has encour- aged us to use e/a as a parameter in stabilizing a complex phase. However, this is quite misleading, since there are two different mechanisms for the formation of the pseudo- gap: one the Fermi surface-Brillouin zone (FsBz) interac- tion and the other orbital hybridization. Mizutani et al. [1–3] applied the FLAPW-Fourier method to a series of gamma-brasses to identify the origin of the pseudogap at the Fermi level and concluded that only those, in which the resonance of electrons at the Fermi level with a parti- cular set of lattice planes yields the pseudogap, can be claimed to obey the Hume-Rothery stabilization mechan- ism and are entitled to use the e/a as a critical parameter. Instead, the VEC must be chosen as an appropriate para- meter for alloys, in which the Hume-Rothery stabilization mechanism fails [2, 3]. The empirical Hume-Rothery e/a rule is not limited to complex alloy phases but also has been applied to simple phases like fcc and bcc. However, one can hardly judge from the first-principles band calculations whether or not the Hume-Rothery stabilization mechanism works in these simple phases, since the FsBz interaction involved is too weak to produce a sizable pseudogap at the Fermi level. Another important issue remains concerning the e/a deter- mination for the transition metal element involved. The e/a rule originally proposed by Hume-Rothery in 1926 pointed to the regularity that, in spite of little connection in composition, all three compounds Cu 5 Sn, Cu 3 Al and CuZn crystallize into a common structure of the bcc phase with the possession of e/a ¼ 3/2 [5]. Given that NiAl B2- compound had been regarded as a 3/2 electron compound with a zero valency for Ni, Hume-Rothery [6] addressed a naı ¨ve question as to if mono-valency should be assigned to Ni in NiZn, which also possesses the B2 structure. We have developed the FLAPW-Fourier method to reli- ably determine the e/a value for the transition metal ele- ment in various gamma-brasses by extracting the electro- nic state with the largest Fourier component of the FLAPW wave function outside the muffin-tin sphere with subsequent construction of a single-branch energy disper- sion relation for such mobile electrons [1, 2]. The e/a value can be decisively determined from the Fermi dia- Z. Kristallogr. 224 (2009) 17–20 / DOI 10.1524/zkri.2009.1078 17 # by Oldenbourg Wissenschaftsverlag, Mu ¨nchen * Correspondence author (e-mail: riken-mizutani@mosk.tytlabs.co.jp) Properties / Applications