Brillouin zone volume

Because (k) = (k + G), a knowledge of (k) within a given volume called the Brillouin zone is sufficient to detennine (k) for all k. In one dimension, G = Imld where d is the lattice spacing between atoms. In this case, E k) is known once k is detennined for -%ld < k < %ld. (For example, m the Kronig-Peimey model (fignre Al.3.6). d = a + b and/rwas defined only to within a vector 2nl a + b).) In tlnee dimensions, this subspace can result in complex polyhedrons for the Brillouin zone. 

In Equation (5.58) the outer summation is over the p points q which are used to sample the Brillouin zone, is the fractional weight associated with each point (related to the volume of Brillouin zone space surrounding q) and vi are the phonon frequencies. In addition to the internal energy due to the vibrational modes it is also possible to calculate the vibrational entropy, and hence the free energy. The Helmholtz free energy at a temperature... 

Let us review the calculation of the number of electrons in successive Brillouin zones. The calculation has as its starting point a distribution of free electrons. In a volume V to which the electrons are restricted the most stable pair of electrons occupies the lowest energy levels, with nearly zero kinetic energy, and correspondingly long wave-lengths. As the number of electrons... 

The brittleness of these intermetallic compounds suggests an electronic structure involving a filled Brillouin zone. It was pointed out by Ketelaar (1937) that the strongest reflection, that of form 531, corresponds to a Brillouin polyhedron for which the inscribed sphere has a volume of 217 electrons per unit cube, which agrees well with the value 216 calculated on the assumption that the sodium atom is univalent and the zinc atoms are bivalent that is, calculated in the usual Hume-Rothery way. It has also been... 

In this chapter we will have a closer look at the methods of the reconstruction of the momentum densities and the occupation number densities for the case of CuAl alloys. An analogous reconstruction was successfully performed for LiMg alloys by Stutz etal. in 1995 [3], It was found that the shape of the Fermi surface changed and its included volume grew with Mg concentration. Finally the Fermi surface came into contact with the boundary of the first Brillouin zone in the [110] direction. Similar changes of the shape and the included volume of the Fermi surface can be expected for CuAl [4], although the higher atomic number of Cu compared to that of Li leads to problems with the reconstruction, which will be examined. 

The conclusion to be drawn from Eq. (37) is thfit the volume of the first Brillouin zone is equal to the reciprocal of file volume of the primitive cell. It should be noted that the scalar product... 

A is the volume of the unit cell in the direct lattice of the crystal The range of integration is restricted to the first Brillouin zone of the crystal, and the volume of the zone is (27t)3/A. 

In the Debye model, the Brillouin zone (see section 3.3) is replaced by a sphere of the same volume in the reciprocal space (cf eq. 3.57) ... 

The volume of Brillouin zone Vg corresponds to (IttY times the reciprocal of cell volume 17, commonly adopted in crystallography, and the origin of the coordinates is at the center of the cell (and not at one of the corners) ... 

Thus, the reciprocal lattice of a simple cubic lattice is also simple cubic. It is shown in Fig. 5.7 in the xy plane, where it is clear that the bisectors of the first nearest-neighbour (100) reciprocal lattice vectors from a closed volume about the origin which is not cut by the second or any further near-neighbour bisectors. Hence, the Brillouin zone is a cube of volume (2n/a)2 that from eqn (2.38) contains as many allowed points as there are primitive unit cells in the crystal. The second, third, and fourth zones can... 

For many metals, the "nearly free" electron description corresponds quite closely 10 the physical situation. The Fermi surface remains nearly spherical in shape. However, it may now he intersected by several Brillouin zone boundaries which break the surface into a number of separate sheets. It becomes useful to describe the Fermi surface in terms not only of zones or sheets filled with electrons, but also of zones or sheets of holes, that is. momentum space volumes which are empty of electrons. A conceptually simple method of constructing these successive sheets, often also referred lo as "first zone. "second zone." and so on was demonstrated by Harrison. An example of such construction is shown in Fig. 2. 

The N q vectors allowed by the boundary conditions just fill the first Brillouin zone (BZ) of volume equal to vb, the volume of the primitive unit cell of the reciprocal lattice. Because of this dense, uniform distribution of q vectors it is possible to treat q as a continuous variable and thus replace... 

The total number of transitions per unit volume and time from Eq. (17) is obtained by dividing by t and summing over j (occupied bands) and / (unoccupied bands). Integrating over the Brillouin zone gives ... 

Most materials are two- or three-dimensional, and while one dimension is fun, we must eventually leave it for higher dimensionality. Nothing much new happens, except that we must treat k as a vector, with components in reciprocal space, and the Brillouin zone is now a two- or three-dimensional area or volume.915... 

Fig. 2. Band structure and total DOS of bulk V2O5. The energy bands are shown for characteristic paths connecting high symmetry points of the irreducible part of the orthorhombic Brillouin zone (BZ) which is included at the bottom. All energies e(k) are taken with respect to that of the highest occupied state. The DOS is given in states per unit volume and per eV.

We shall see below that the transport properties above are consistent with the picture of Luttinger chains in (a - b) planes. The exponent a = 0.7 derived from the constant volume transverse data leads to K = 0.22. This value of allows in turn a prediction for the constant volume T-dependence for p . The only scattering process through which electron-electron collisions can contribute to resistivity in this 1-D electron gas occurs when the total momentum transfer is commensurate with a Brillouin zone wave vector. For the situation of a 1/4-filled 1-D band which is likely to apply to (TMTSFljPF as the dimerization can be forgotten in first approximation (A ,[Pg.254]

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