Crystal Field expanded on renormalized spherical Harmonics Ckm
- Slides: 8
Crystal Field expanded on renormalized spherical Harmonics Ckm Maurits W. Haverkort Institute for theoretical physics – Heidelberg University M. W. Haverkort@thphys. uni-heidelberg. de
From potentials to matrix elements of the Hamiltonian Crystal-field theory Approximate the solid by a single atom in an effective potential And matrix elements of the Hamiltonian: !!!! WARNING !!!! Crystal-field potentials do not exist. They are effective Hamiltonians introduced to mimic covalent bonding. (crystals bind due to a gain in kinetic energy and a smaller loss in potential energy, i. e. the contribution of the crystalfield potential to binding is negative) Covalent bonds are stronger than ionic bonds
From potentials to matrix elements of the Hamiltonian Atomic basis sets: (also Gaussian, muffin-tin orbitals, …) Expand potential on spherical harmonics.
From potentials to matrix elements of the Hamiltonian Atomic basis sets: (also Gaussian, muffin-tin orbitals, …)
Not all values of Akm are allowed (many are zero) Selection rules on angular momentum conservation Hamiltonian is Hermitian
Not all values of Akm are allowed (many are zero) Symmetry related: For all “rotations” of the point-group Ci we have Cubic: (Oh) eg d t 2 g
Not all values of Akm are allowed (many are zero) Tetragonal: (D 4 h) eg d a 1 g in d rte b 1 g o ps p n u x o l e ted _gro a r e en int v e eg s um /po e t 2 g l b doc stry a l i va and hemi a pbs 2 g files s_c u ro elp ysic g int ica H g/ph o r p mat y. or e th the ant o l Al Ma w. qu ww / / p: t t h
