The symmetry properties of the energy bands of α- and β-quartz

Chirality-asymmetry force between α-quartz and copper block

“…We can take the energy split due to the spin-orbit interaction as the value of the matrix element Ψ 0 | µ k • B(r k ) |Ψ n approximately [11] whose value can be obtained from published theoretical calculations that Ψ 0 | µ k • B(r k ) |Ψ n ≈0.01 eV. [12] So, we have the estimation…”

“…(8) The lattice constants were determined experimentally as a 0 = 4.89 Å, and c 0 = 5.375 Å (1 Å=0.1 nm). [12] In potential (6), the detailed expression of the sum of the products of matrix elements for valence electrons in a primitive cell is…”

“…module | ψ kn | r ika |ψ k0 | of the kth valence electron can be taken approximately as the distance between the centres of the tetrahedron and the ith primitive cell, where a = x, y, z. For valence electrons in the tetrahedron whose centre is located on the boundary of primitive cell, we can take| ψ kn | r ikx |ψ k0 | = a 0 /2, | ψ kn | r iky |ψ k0 | = 0 and | ψ kn | r ikz |ψ k0 | = c 0 /2 [12]. Modules of matrices of valence electrons in the tetrahedron centred at interface can be taken as| ψ kn | r ikx |ψ k0 | = 0, | ψ kn |r iky |ψ k0 | = a 0 /2 and | ψ kn | r ikz |ψ k0 | = c 0 /6.…”

On Energy Bands of Crystals with Lowered Point Symmetry

“…Several approximate conclusions (e.g. with u-quartz, approximate mirror symmetry of E ( b ) in 2, small finite slopes in certain points k [14]) follow from the only crystallographical possibility of existence of a more symmetrical crystal. It seems therefore advisable in general to ask whether a more symmetrical structure is possible, whenever an anisotropic crystal is studied.…”