e/a classification of Hume–Rothery Rhombic Triacontahedron-type approximants based on all-electron density functional theory calculations

Abstract

There are three key electronic parameters in elucidating the physics behind the Hume–Rothery electron concentration rule: the square of the Fermi diameter (2k_F)², the square of the critical reciprocal lattice vector and the electron concentration parameter or the number of itinerant electrons per atom e/a. We have reliably determined these three parameters for 10 Rhombic Triacontahedron-type 2/1–2/1–2/1 (N = 680) and 1/1–1/1–1/1 (N = 160–162) approximants by making full use of the full-potential linearized augmented plane wave-Fourier band calculations based on all-electron density-functional theory. We revealed that the 2/1–2/1–2/1 approximants Al₁₃Mg₂₇Zn₄₅ and Na₂₇Au₂₇Ga₃₁ belong to two different sub-groups classified in terms of equal to 126 and 109 and could explain why they take different e/a values of 2.13 and 1.76, respectively. Among eight 1/1–1/1–1/1 approximants Al₃Mg₄Zn₃, Al₉Mg₈Ag₃, Al₂₁Li₁₃Cu₆, Ga₂₁Li₁₃Cu₆, Na₂₆Au₂₄Ga₃₀, Na₂₆Au₃₇Ge₁₈, Na₂₆Au₃₇Sn₁₈ and Na₂₆Cd₄₀Pb₆, the first two, the second two and the last four compounds were classified into three sub-groups with  = 50, 46 and 42; and were claimed to obey the e/a = 2.30, 2.10–2.15 and 1.70–1.80 rules, respectively.

Keywords:

• Hume-Rothery electron concentration rule
• FLAPW-Fourier method
• 2/1-2/1-2/1 approximants
• RT-type approximants
• interference condition
• Hume-Rothery stabilization mechanism

Acknowledgements

One of the authors (UM) is grateful for the financial support of the Grant-in-Aid for Scientific Research (Contract No. 23560793) from the Japan Society for the Promotion of Science. Q.L. was supported by the Office of the Basic Energy Sciences, Materials Sciences Division, US Department of Energy (DOE). Ames Laboratory is operated for DOE by Iowa State University under contract No. DE-AC02-07CH11358. We would like to thank Prof. Tsunehiro Takeuchi, Toyota Technological Institute, Nagoya, Japan for providing us atomic structure data for the 1/1–1/1–1/1 approximant Al₉Mg₈Ag₃.

Notes

1. The value of (e/a)_X for the element X in the periodic table has been determined using the FLAPW-Fourier band calculations and in most cases reported elsewhere [1,5–11].

2. In the present work, both Al₄₈Mg₆₄Zn₄₈ and Al₈₄Li₅₂Cu₂₄ are referred to as Al₃Mg₄Zn₃ and Al₂₁Li₁₃Cu₆, respectively.

3. The atomic positional data for the approximant Al₄₅Mg₄₀Ag₁₅ or Al₉Mg₈Ag₃ were not included in [17]. The data were kindly communicated from Prof. T. Takeuchi, Toyota Technological Institute, Nagoya. See [17] for the definition of sites B and C.

4. The e/a values for skutterudites TMX₃ (X = Co, Rh, Ir and Ni, X = P, As and Sb) and Zintl compounds AB (A = Li and Na, B = Al, Ga, In and Tl) were deduced from the FLAPW-Fourier method to be centred at 4.34 ± 0.20 and 2.09 ± 0.06, respectively [10]. This is nothing but the validity of the Hume–Rothery electron concentration rule for these families of compounds.