Bloch wavevector

Here the Bloch wavevector k is 2% times the crystallographic reciprocal lattice vector ... 

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 tln-ee dimensions, four quantum numbers are required to characterize an eigenstate. In spherically syimnetric atoms, the numbers correspond to n, /, m., s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent... 

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. 

Expressed formally, the wave must be matched in amplitude at the surface and in phase velocity parallel to the crystal strrface. This implies that the tangential components of D and H must be continuous across the strrface, and the components in the crystal strrface of the wavevectors inside and outside the strrface must be the same. If n is a rmit vector normal to the crystal strrface, whatever the values of k o or the resttlting Bloch wave inside the crystal, then... 

Figure 4.18 The effect of spherical incident waves on the excitation of Bloch waves, (a) Reciprocal space the divergent incident beam has wavevectors ranging from P j O to P 2 O. (b) Real space energy is distributed throughout the Borrmann fan ABC. The beams generated outside the crystal are indicated...

Equation (8.4.2) suggests that a wavefunction uk(r) needs to be found by standard quantum-chemical means for only the atoms or molecules in the one direct-lattice primitive unit cell. For each of the Avogadro s number s worth of fermions in a solid, the factor exp(ik R) in Eq. (8.4.2) provides a new quantum "number," the wavevector k, that guarantees the fermion requirement of a unique set of quantum numbers. The Bloch waves were conceived to explain the behavior of conduction electrons in a metal. 

Band Structure for the Free-Electron Case. If the electron is free, then the Bloch functions are simple plane waves, because the wavef unctions nk(r) used for the expansion Eq. (8.4.2) are themselves plane waves. For an electron gas with no lattice and no imposed symmetry, Fermi-Dirac statistics apply At 0 K all electrons pair up (spin-up and spin-down), with an occupancy of 2 for every k value from k 0 to the Fermi wavevector kF =1.92/rs = 3.63 a0/rs, and from zero energy up to the Fermi energy eF = h2kF2 /rn 50.1 eV rs/a0) 2, where rs is the radius per conduction electron and a0 is the Bohr radius, and the energy levels are spherically symmetric in k-space. The Fermi surface is a sphere of radius kF. Note that the ratio (rs/a0) varies from 0.2 to 1.0 nm for metals (Table 8.3). 

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... 

Even though in similarity to free particle wavefunctions the Bloch wavefunctions are characterized by the wavevector k, and even though Eq. (4.80) is reminiscent of free particle behavior, the functions V nkfr) are not eigenfunctions of the momentum operator. Indeed for the Bloch function (Eqs (4.78) and (4.79)) we have... 

In the BCS theory, the normal (non-superconducting) state of the metal is described by Bloch single particle eigenstates (8) k> labelled by a wavevector k,... 

Sk is the overlap matrix for the Bloch functions for the wavevector k, with E] being the energy matrix and A the matrix of coefficients. Fk is the Fock matrix, which consists of a sum of one- and two-electron terms. The values of k are typically selected to sample from the first Brillouin zone according to a special scheme as described in Section 3.8.6. When these terms are expanded they involve infinite sums over the nuclei and electrons in the lattice. As is usual in a Hartree-Fock approach the one-electron terms involve the sum of a kinetic energy term and one due to the Coulomb interaction between the nuclei and the... 

An analogous expression is obtained in three dimensions. We now need to consider periodic systems. As we have discussed, the wavefunction for a particle on a periodic lattice must satisfy Bloch s theorem. Equation (3.85). The wavevector k in Bloch s theorem plays the same role in the study of periodic systems as the vector k does for a free particle. One important difference is that whereas the wavevector is directly related to the momentum for a free particle (i.e. k = p/h) this is not the case for the Bloch particle due to the presence of the external potential (i.e. the nuclei). However, it is very convenient to consider Hk as analogous to the momentum and it is often referred to as the crystal momentum for this reason. The possible values that k can adopt are given by ... 

The operator within the parentheses remains the same if (r - - na) is substituted for p. The Bloch theorem (231, 437) states that because of this periodicity, iy(r -f a) must be the same as to within a phase factor, that is (r - - a) = (r). Each eigenfunction that satisfies Eq. (2) has a wavevector associated with it such that translation by a lattice vector na is equivalent to multiplying the eigenfunction, v (r), by the phase factor exp(/ifc a). 

For crystaUine solids the translation symmetry of the Hamiltonian is taken into account in any electronic-structure calculations as it allows calculations to be made for the basis connected only with the primitive unit cell. In the translation-symmetry-adapted basis the matrix JI has a qnasidiagonal structure with identical blocks related to an irrep k of the translation symmetry group T. As the latter is Abelian its irreps are one-dimensional. The translation s3Tnmetry adapted functions are known as Bloch functions and numbered by wavevector k. Use of the point symmetry of a crystal allows the number of Bloch functions calculated to be decreased and further block-diagonaUzation of Hamiltonian of a crystal to be made. 

In Sect. 3.1 we have considered the space-group irreps D( )(g) defined by stars k of wavevectors k from the first Brillouin zone and by irreps of little groups Gfc. Let us construct in the space of induced rep (q,(3) new basis functions that span the space of irreps of the translation group T (Bloch functions)... 

This procedure gives the labels of the induced rep q, / ) in the k basis corresponding to those in the q basis, ie. the results of the reduction of the induced rep over irreps of the group Gfc. All the information obtained can be specified by listing the symmetry (the labels of irreps) of the Bloch states with wavevectors k corresponding only to a relatively small number of k points in the BriUouin zone forming a set K. The set K contains the inequivalent symmetry points of the BriUouin zone and one representative point from each inequivalent symmetry element (symmetry line or symmetry plane) if the latter does not contain the points of higher symmetry. 

The character of the locahzation of Wannier functions depends on the analytical properties of Bloch states (as a function of the wavevector) that are essentially determined by the nature of the system under consideration. One can arbitrary change only the form of an unitary transformation of Bloch functions. It is just this arbitrariness that is used in the variational approach [42] to assure the best localization of Wannier functions. The accuracy of the Wannier functions obtained by the proposed method is determined solely by the accuracy of the Bloch functions and the size of the supercell used. As the calculations have shown, the proposed method is reUable and useful in the problem of generation of the locaUzed Wannier functions. In the two examples... 

This condition is satisfied for both the infinite crystal and the basic domain only the sets of values of the wavevector k for which (4.101) is satisfied are different in these two cases. By applying Bloch s theorem (4.101) to the wavefunctions in (4.97) for the one-electron DM of the basic domain, we obtain... 

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