Bloch boundaries

Let us consider lithium as an example. In the usual treatment of this metal a set of molecular orbitals is formulated, each of which is a Bloch function built from the 2s orbitals of the atoms, or, in the more refined cell treatment, from 2s orbitals that are slightly perturbed to satisfy the boundary conditions for the cells. These molecular orbitals correspond to electron energies that constitute a Brillouin zone, and the normal state of the metal is that in which half of the orbitals, the more stable ones, are occupied by two electrons apiece, with opposed spins. 

If we assume that the Bloch functions have periodic boundary conditions ... 

In solid-state physics the opening of a gap at the zone boundary is usually studied in the free electron approximation, where the application of e.g., a ID weak periodic potential V, with period a [V x) = V x + a)], opens an energy gap at 7r/a (Madelung, 1978 Zangwill, 1988). E k) splits up at the Brillouin zone boundaries, where Bragg conditions are satished. Let us consider the Bloch function from Eq. (1.28) in ID expressed as a linear combination of plane waves ... 

It is necessary to state now that the rigorous fulfillment of the Bloch theorem needs an infinity lattice. In order to calculate the number of states in a finite crystal, a mathematical requirement named the Bom-Karman cyclic boundary condition is introduced. That is, if we consider that a crystal with dimensions Nxa, N2b, /V3c is cyclic in three dimensions, then [5]... 

The wave function has the same amplitude at equivalent positions in each unit cell. Thus, the full electronic structure problem is reduced to a consideration of just the number of electrons in the unit cell (or half that number if the electronic orbitals are assumed to be doubly occupied) and applying boundary conditions to the cell as dictated by Bloch s theorem (Eq. 4.14). Each unit cell face has a partner face that is found by translating the face over a lattice vector R. The solutions to the Schrodinger equation on both faces are equal up to the phase factor exp(zfe R), determining the solutions inside the cell completely. 

In this equation, N is equal to the number of unit cells in the crystal. Note how the function in Eq. 5.27 is the same as that of Eq. 5.19 for cyclic tt molecules, if a new index is defined ask = liij/Na. Bloch sums are simply symmetry-adapted linear combinations of atomic orbitals. However, whereas the exponential term in Eq. 5.19 is the character of the yth irreducible representation of the cychc group to which the molecule belongs, in Eq. 5.27 the exponential term is related to the character of the Mi irreducible representation of the cychc group of infinite order (Albright, 1985). This, in turn, may be replaced with the infinite linear translation group because of the periodic boundary conditions. It turns out that SALCs for any system with translational symmetry are con-stmcted in this same manner. Thus, as with cychc tt systems, there should never be a need to use the projection operators referred to earher to generate a Bloch sum. 

This result at Eq. (24), given explicitly, for instance, by Harris and Cina [16], when inserted into the appropriate form of the Bloch equation, is readily verified to be an exact solution, as well as satisfying the boundary condition at Eq. (7). The Bloch equation is evidently a powerful tool for treating problems of applied fields of arbitrary strength. 

Using Bloch s theorem and cyclic boundary conditions [8] the atomic displacements can be expressed as a product of a maximum amplitude, ( /), and a phase factor, the wave vector k. Solutions to Eq. (4.43) are ... 

Here Aatom is the number of atoms per unit cell, N ew is the number of unit cells considered (very large), i and j labels include the x, y, z components of different atoms and q is a mass weighted coordinate, after Eq. (4.5). The interaction cannot depend on the absolute positions of the atoms but their relative positions. Again a major simplification to this equation comes from the application of the Bloch theorem and cyclic boundary conditions [8]. We shall represent the positions of all atoms in the solid by reference to a chosen unit cell, the zeroth cell (superscript 0), then for any k ... 

In disordered metals, the electrons couple with the static structure at any. K-value and are heavily damped close to pseudo Brillouin-zone boundaries. A definite dispersion relation does not exist (Fig. 5.3bl) and Bloch s theorem is no longer valid. Electronic states cannot be described as eigenstates of the system and, strictly speaking, cannot be translated to the reduced-zone scheme. 

---