Wigner-Seitz primitive cell

Fig. B.2. First Brillouin zone of the fee BL, showing the critical points. Its geometry is the same as that of the Wigner-Seitz primitive cell of the bcc BL...

The Wigner-Seitz primitive cell of the lattice is the smallest region, of volume... 

The range of k-values between — Ti/a < k < n/siisknownsiSthQ rstBrillouinzone (BZ). The first BZ is also defined as the Wigner-Seitz primitive cell of the reciprocal lattice, whose construction is illustrated in Figure 2.75. First, an arbitrary point in the reciprocal lattice is chosen and vectors are drawn to all nearest-neighbor points. Perpendicular bisector lines are then drawn to each of these vectors the enclosed area corresponds to the primitive unit cell, which is also referred to as the first Brillouin zone. 

One has to take into account, however, that the unit cell which is relevant for spectroscopy is the primitive (or Wigner-Seitz) unit cell. It is a parallelepiped from which the entire lattice may be generated by applying multiples of elementary translations. Face- and body-centered cells are multiple unit cells. The content of such a cell has to be divided by a factor m to obtain the content of a primitive unit cell. This factor m is implicitly given by the international symbol for a space group P and R denote primitive cells (m = 1), face-centered cells are denoted A, B, C (m = 2), and F m = 4), and body-centered cells are represented by I m = 2). Examples are described by Turrell (1972). 

The basis vectors Aj determine an LUC and a new Bravais superlattice. The LUC thus constructed has volume V), = LVa and consists of L primitive cells. The superlattice vectors A are linear combinations (with integral-valued coefficients) of the basis vectors Aj. The matrix 1 in (4.141) is chosen such that the point symmetry of the new superlattice is identical to that of the original lattice (the corresponding transformation (4.141) is called a symmetric transformation, see Sect. 4.2.1). The type of direct lattice can be changed if there are several types of lattice with the given point symmetry. The LUC is conveniently chosen in the form of a Wigner-Seitz (WS) cell, which possesses the point symmetry of the lattice. 

Fig. 40 Calculated constant-intensity surface in 4-beam laser interference patterns. The primitive units (contents of Wigner-Seitz unit cell) is shown inset in each case, a Scheme-1, high-index beam vectors interference, producing pattern of 922-nm lattice constant, b Scheme-2, low-index beam vectors interference, producing FCC pattern with 397-nm lattice constant. In both case, the use of 355-nm YAG laser was assumed. Scale bars 500 nm...

We previously introduced the concept of a primitive cell as being the supercell that contains the minimum number of atoms necessary to fully define a periodic material with infinite extent. A more general way of thinking about the primitive cell is that it is a cell that is minimal in terms of volume but still contains all the information we need. This concept can be made more precise by considering the so-called Wigner-Seitz cell. We will not go into... 

Figure 16.3. (a) Construction of the Wigner-Seitz cell in a 2-D hexagonal close-packed (hep) lattice, (b) Primitive unit cell of the hep lattice. 

It is possible, as well, to define the primitive unit cell, by surrounding the lattice points, by planes perpendicularly intersecting the translation vectors between the enclosed lattice point and its nearest neighbors [2,3], In this case, the lattice point will be included in a primitive unit cell type, which is named the Wigner-Seitz cell (see Figure 1.2). 

A construction in reciprocal space identical to that used to delineate the Wigner-Seitz cell in direct space gives a cell known as the first Brillouin zone, (Figure 2.7). The first Brillouin zone of a lattice is thus a primitive cell. 

The Wigner-Seitz cell is the smallest cell of the crystal lattice. While the unit cell is chosen for reasons of symmetry, the Wigner-Seitz cell represents the full periodicity of the electron shell. The Wigner-Seitz cell is an example of the construction of a primitive elementary cell. The characteristic feature of the primitive cell is that it contains only one lattice point. The rules for constructing a Wigner-Seitz cell are ... 

Figure 2.75. The Wigner-Seitz construction of a primitive unit cell for a 2-D lattice.

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