NFE approximation for the e/a determination for 3d-transition metal elements and their intermetallic compounds with Al and Zn

Abstract

First-principles full-potential linearized augmented plane wave (FLAPW) band calculations with subsequent FLAPW-Fourier analyses have been performed for elements from K to Cu in period 4 of the periodic table to determine the effective electrons per atom ratio (e/a). For the series of 3d-transition metals (TM), the determination of the square of the Fermi diameter , from which e/a is derived, has been recognized not to be straightforward because of the presence of a huge anomaly associated with the TM-d states across the Fermi level in the energy dispersion relation for electrons outside the muffin-tin sphere. The nearly free electron (NFE) approximation is newly devised to circumvent this difficulty. The centre of gravity energy is calculated from the energy distribution of the square of the Fourier coefficients for the FLAPW state . The NFE dispersion relation is constructed for the set of and in combination with the tetrahedron method. The resulting e/a values are distributed over positive numbers in the vicinity of unity for elements from Ti to Co. Instead, the e/a values for the early elements K, Ca and Sc and the late TM elements Ni and Cu were determined to be close to one, two, three, 0.50 and unity, respectively, using our previously designed local reading method. In addition, the composition dependence of e/a values for intermetallic compounds in X-TM (X = Al and Zn) alloy systems was studied to justify an appropriate choice between the local reading and NFE methods for respective elements.

Keywords:

• FLAPW band calculations
• electrons per atom e/a
• FLAPW-Fourier method
• transition metals

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. For the sake of convention, at particular symmetry points is simply expressed as .

2. The summation is taken under the condition and, hence, it counts the degeneracy of the state , which is generally more than two at high symmetry points in the Brillouin zone.

3. The FLAPW wave function is normalized with respect to the volume of a system. Hence, strictly speaking, is not normalized to unity, since the wave function outside the MT spheres does not cover the volume inside it. It is generally slightly higher or lower than unity, depending on the amount of charges inside the sphere.

4. The summation is taken under the condition (see note 2).

5. The value of for the set of lattice planes for hexagonal crystal with lattice constants a and c is given as , where is the volume per atom. Since the relation holds, we omit the third index . For the sake of clarity, the set of lattice planes for hcp lattice is expressed as in the present work.