Bloch INDEX

The orbitals <]) j(k r) are Bloch functions labeled by a wave vector k in the first Brillouin zone (BZ), a band index p, and a subscript i indicating the spinor component. The combination of k and p. can be thought of as a label of an irreducible representation of the space group of the crystal. Thequantity n (k)is the occupation function which measures... 

Singh, H.S. and T.V. Reddy. 1990. Effect of copper sulfate on hematology, blood chemistry, and hepato-somatic index of an Indian catfish, Heteropneustes fossilis (Bloch), and its recovery. Ecotoxicol. Environ. Safety 20 30-35. 

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. 

Electron Microdiffraction. Spence and Zuo, 1992. Contains well-dociunented Fortran listings for programs to simulate CBED patterns by Bloch Wave method, and multislice. Indexed patterns shown with HOLZ to speed indexing. 

I. Flamant et al. successfully applied the FGSO basis set in Fourier Space Restricted Hartree Fock (FS-RHF) in a study of identification of conformational signatures in valence band of polyethylene. In 1998, they used a distributed basis set of s-type Gaussian function (DSGF) in FS-RHF. The method briefly is to use RHF-Bloch states (p (k,r), which are doubly occupied up to the Fermi energy Ep and orthonormalized. k and n the wave number and the band index, respectively. 

It is a basic consequence of the translational symmetry of a solid that its Kohn-Sham eigenfunctions can be uniquely labeled by four quantum numbers, the band index n and a wavevector k, as in xj/ y.. The diagram s n,k) that represents the n, k dependence of the corresponding eigenenergies is called the band structure. The Bloch theorem asserts that the t>e written in the form of a Fourier... 

It is convenient to describe the state of each atom in terms of Wannier functions Xj) [Wannier 1937] that are localized at lattice sites labeled by index j. The Wannier functions are superpositions of the delocalized Bloch eigenfunctions k) of the same band,... 

Rp is the vector which translates a point in one unit cell to the corresponding point in another. The index p identifies the unit cell, or the atom, ip is an atomic orbital, which is the same for all atoms. The (k)s are called Bloch functions and are simply symmetry-adapted linear combinations of the orbitals -0-... 

Using Equation 6.D.21, we can obtain the rate of propagation of the planar waves of the Bloch electrons given by the band index n and the wave vector k. 

In addition to the Bloch vector, a complete description of the electronic states in a crystal requires a band index j, which may be defined such that... 

A typical example where such a procedure is needed is in the application of the LMTO method to molecules. Furthermore, in this situation Bloch s theorem does not apply and k is therefore not a good quantum number. Instead, the k dependence should be substituted by a Q , Q dependence, where Q is a site index. Formally, the LMTO matrix for molecules may be obtained by substituting... 

The wavevector is a good quantum number e.g., the orbitals of the Kohn-Sham equations [21] can be rigorously labelled by k and spin. In three dimensions, four quantum numbers are required to characterize an eigenstate. In spherically symmetric atoms, the numbers correspond to n, I, m, s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent quantum numbers in a crystal are k, k, k and spin. The spin index is usually dropped for non-magnetic materials. 

Bloch JM (1985) Angle and index calculations for a z-axis X-ray diffractometer. J Appl Cryst 18 33-36... 

The eigenvectors of the lattice periodic Hamiltonian of the Kohn-Sham-Dirac equation (28) are Bloch states fen) with crystal momentum k and band index n. They are expressed by the ansatz... 

M = 2K). Then, all Bloch functions have the periodicity of the Bom von Karman zone, i.e., of the lengfli 2K a, with a being the length of one unit cell. With n being the band index, any function with this periodicity can be expanded in flie Bloch functions. 

In (6.5) the subscript n indicates the band index and fe is a continuous wave vector that is confined to the first Brillouin zone of the reciprocal lattice. The index n appears in the Bloch theorem because for a given k there are many solutions to the Schrodinger equation. Because the eigenvalue problem is set in a fixed finite volume, we generally expect to find an infinite family of solutions with discretely-spaced eigenvalues which we label with the band index n. The wave vector k can always be confined to the first Brillouin zone. The vector k takes on values within the Brillouin zone corresponding to the crystal lattice, and particular directions like r,A,A,Z (see Figures 4.13-4.15). 

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