Solid-state systems translational vector

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

When it comes to crystals, it is clear that the system under study is trans-lationally invariant in all three spatial directions, and Bloch s theorem utilizes the translational s)mimetry to generate the crystal s wave function, composed of crystal orbitals which are also called electronic bands. We therefore imagine an idealized solid-state material whose electronic potential V possesses the periodicity of the lattice, expressed by a lattice vector T, that is... 

All atoms are equivalent by translation a. Although this is only a simplified ideal situation, it is used as a first example in many introductory textbooks on physical chemistry and solid state physics, because it is a problem simple enough to be treated analytically, especially if an easy approximation such as Hiickel s model Flamiltonian is applied. Several important simplifications in Hiickel s model make the calculation very easy, while preserving the main topological characteristics of the system. In this simple model, only one pz AO is considered for each atom. The different orbitals will be identified by the g lattice vector of the cell in which they are centered and denoted as p. Hiickel s approximation prescribes simple rules for the determination of the overlap and the Hamiltonian matrices with two parameters, a and p ... 

The periodie boundary eondition (PBC) has to be included to perform solid state ealeulations and surfaees. The potential of a periodic system has the property deseribed by eq. (9), where r is an arbitrary position defined within the unit eell and R is the translational vector defined in Fig. 1. 

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