e/a determination for 4d- and 5d-transition metal elements and their intermetallic compounds with Mg, Al, Zn, Cd and In

Abstract

The present work is devoted to the determination of the effective electrons per atom ratio e/a by means of first-principles full-potential linearized augmented plane wave-Fourier method for elements from Rb to Ag in Period 5 and from Cs to Au in Period 6 of the periodic table and is regarded as a continuation of the preceding work done for elements from K to Cu in Period 4. The value of e/a was determined by reading off the square of the Fermi diameter, from the dispersion relation for electrons outside the Muffin-Tin spheres. A straightforward reading of the ordinate at the Fermi level, i.e. local reading method was validated for Rb and Cs in Group 1, Sr in Group 2, Y in Group 3, Pd and Pt in Group 10 and Ag and Au in Group 11. Instead, the nearly free electron (NFE) method was found to be indispensable for TM elements from Zr to Rh in Period 5 and those from Ba to Ir in Period 6. The composition dependence of e/a values for intermetallic compounds in X–TM (X = Mg, Al, Zn, Cd and In) alloy systems was also studied. The new Hume–Rothery electron concentration rule was established by constructing e/uc, the number of electrons per unit cell, vs. square of critical reciprocal lattice vector, , diagram for structurally complex metallic alloys having a pseudogap at the Fermi level. A proper use of either the local reading- or the NFE-e/a for the elements as indicated above is found to be essential.

Keywords:

• FLAPW band calculations
• electrons per atom e/a
• FLAPW-Fourier method
• transition metals
• new Hume–Rothery electron concentration rule

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.

Notes

1. There is an exception to this behavior. The Ni concentration dependence of e/a was found to be non-linear for the Al–Ni alloy system [19].

2. Note that the Fermi sphere obtained from the Hume-Rothery plot is not a true Fermi surface determined by first-principles band calculations or experiments.

3. The NFE method has been employed for the 3d-TM elements under the condition L = 20 [19]. As mentioned in Section 2.2, we will apply the same condition for the 4d- and 5d-TM elements. However, its application to Ba resulted in e/a  = 0.93, which is apparently too small. Hence, the NFE method with L = 1, i.e. the use of only the maximum Fourier coefficient in the Fourier series was applied to Ba and also to La.

4. In hexagonal crystal with lattice constant a and c, is expressed as in units of , where is the volume per atom and is the set of Miller indices for hexagonal crystal [19]. In the case of symmetry points L or planes, for example, the value of is reduced to or 2.13 for Y with a = 3.647 A and c = 5.731 A.

5. Note that the value of e/a = 0.51 for Nb is surprisingly small. Indeed, it is the smallest among 3d, 4d and 5d TM elements except for Group 10 TM metals Ni, Pd and Pt. We will discuss more details in Section 4.1.

6. Among CMAs collected from literature, FLAPW band calculations have not yet been performed for β-Al₃Mg₂ and 2/1–2/1–2/1 approximant Al₁₅Mg₄₃Zn₄₂. The e/a value is estimated by taking a composition average of nominal valences of constituent elements, while critical from measured X-ray diffraction spectrum (see more details in [15]).