Lattice translation vector

For the reconstruction of the occupation number density (k) in the repeated zone scheme one uses the reciprocal form factor at lattice translation vectors R, as (k) can be written as [9]... 

Calculate the volume of a unit cell from the lattice translation vectors. 

A cell direction is designated by the vector r, which is a combination of the lattice translation vectors a, b, and c ... 

Negative directions are indicated by an overbar [1-1] and are called the one negative one one direction. All directions are relative to the origin where the three lattice translation vectors originate (see Figure 1.19). 

The cell volume, V, can be calculated using the lattice translation vectors ... 

Mathematically, this is a triple scalar product and can be used to calculate the volume of any cell, with only a knowledge of the lattice translation vectors. If the lattice parameters and interaxial angles are known, the following expression for V can be derived from the vector expression ... 

Let w be the non-lattice translation vector that belongs to Ru in the coset expansion, eq. (2). Then... 

Symmetry plane or symmetry line Graphic symbol Glide vector in units of lattice translation vectors parallel and normal to the projection plane Printed symbol... 

Here x a refers to the ath atomic orbital in the unit cell of the crystal described by the lattice translation vector, t. The CRYSTAL code is also capable of calculating charge density in a solid using the density functional theory (DFT) at local density approximation (LDA) or at generalized gradient approximation (GGA). 

The simplest model of plastic flow was historically founded on the idea of a shearing process in which entire crystal planes underwent motion with respect to one another in a uniform fashion. This model envisions a certain energy cost for relative motion of adjacent crystal planes. The picture that emerges is that of uniform translation of the upper half of the crystal with respect to the lower half by multiples of the appropriate lattice translation vector. For example, if we imagine a perfect crystal as indicated schematically in the left hand panel of fig. 8.7, the... 

The latter two are the 2-D direct and reciprocal lattice translation vectors respectively. In contrast, the vectors r, A, B, C, are 3D vectors throughout. 

Let us suppose that we are mainly interested iu electrons that inhabit the crystal. As we know, electrons glue particles of a solid. As far as electrons are concerned, it is convenient to describe the lattice by the primitive lattice translation vectors. A primitive unit cell, which can fiU up all space, is important in this case. Such a unit ceU in the real space is called the A gner-Seitz ceU. 

A unit cell in a Bravais lattice is described by a set of vectors U], ai, 03. The position of any atom in a real (direct) crystal lattice can be determined by the lattice translation vectors r. 

Let us suppose that a real crystal lattice consists of a set of points R that are determined by vectors r. Any property p of the crystal lattice is periodic, p r) = p r + R), where R is the lattice translation vector. 

The A V(r) value is substantial in the interatomic area where eigenfunction is small and on the contrary potential correction vanishes in the lattice site. is quite a good approximation to a stationary state of the crystal for all R in the Bravais lattice. So will also be for tpn r — R), where R is the lattice translation vector, n is the energetic band level and collectively represents a full set of atomic quantum numbers. 

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