Primitive translation vectors

Zones bounded by planes defined by equation 38 are consistent with the reduced-vector zones. These reciprocal-lattice planes are simply the planes bisecting the vectors K and normal to them. It is noteworthy that lattices with the same type of translational symmetry have equivalent zone patterns since zone structure is determined by the K vectors, and these are determined by the primitive translation vectors. 

A crystal is a physical object - it can be touched. However, an abstract construction in Euclidean space may be envisioned, known as a direct space lattice (also referred to as the real space lattice, space lattice, or just lattice for short), which is comprised of equidistant lattice points representing the geometric centers of the structural motifs. Any two of these lattice points are connected by a primitive translation vector, r, given by ... 

Fixed-Point-Free Motions. These include translations, screw rotations, and glide reflections. Because the primitive translation vector, Eq. 1.2, joins any two lattice points, an equivalent statement is that Eq. 1.2 represents the operation of translational symmetry bringing one lattice point into coincidence with another. However, we must choose the basis vectors (a, b, c) so as to include all lattice points, thus defining a... 

It has just been stated that a band stracture diagram is a plot of the energies of the various bands in a periodic solid versus the value of the reciprocal-space wave vector k. It is now necessary to discuss the concept of the reciprocal-space lattice and its relation to the real-space lattice. The crystal structure of a solid is ordinarily presented in terms of the real-space lattice comprised of lattice points, which have an associated atom or group of atoms whose positions can be referred to them. Two real-space lattice points are connected by a primitive translation vector, R ... 

TABLE 4.1. Primitive Translation Vectors of the Real-Space Cubic Lattices R = ua- + v32 + was... 

Taking the primitive translation vectors for one of the real-space cubic lattices from Table 4.1, Eqs. 4.25-4.27 can be used to obtain the primitive translation vectors for the corresponding reciprocal lattice, which are given in Table 4.2. By comparing Tables 4.1 and 4.2, it is seen that the primitive vectors of the reciprocal lattice for the real-space FCC lattice, for example, are the primitive vectors for a BCC lattice. In other words, the ECC real-space lattice has a BCC reciprocal lattice. 

What are the primitive translation vectors of the reciprocal lattice ... 

Like the diamond stracmre discussed earlier, the honeycomb stracture is not itself a Bravais lattice. If the lattice is translated by one nearest-neighbor distance, the lattice does not go into itself. There are two nonequivalent, or distinct types of sites per unit cell, atoms a and b, separated by a distance Uq, as shown later in Figure 4.6. However, a Bravais lattice can be created by taking this pair of distinct atoms to serve as the basis. Doing so, shows that the vectors of the two-dimensional hexagonal lattice, a and U2, are primitive translation vectors. A given site on one sublattice with coordinates (0, 0), has three nearest neighbors on the other sublattice. They are located at (0, U2), (fli, 0), and (- , 0). 

Now if a, b, c are the primitive translation vectors that define the unit cell of the three-dimensional array of scattering points, then we have, using Eq. (3.6), the following conditions for diffraction maxima ... 

We consider here a few particular space groups. The fee BL is generated by three primitive translation vectors making equal angles with one another. 

In simulations assuming a crystalline structure, the collision sequence is deterministic once the impact point of the projectile at the surface and its direction into the solid are given. A list of target atom positions is then required this can be constructed as in standard solid-state theory with the help of three primitive translation vectors, starting from a basis of one or more atoms. A major and elaborate task in such simulations is, hence, the calculation of a list of next neighbors for each collision sequence and the search procedure to find the next collision partner. 

The space groups are listed and described in the International Tables for X-Ray Crystallography (1962). For our purposes, we are interested only in the primitive unit cell, or the smallest unit cell that can be used to reproduce the crystal by means of translations only. The primitive unit cell often has lower symmetry than the conventional unit cell (Kittel, 1968). According to the international nomenclature for space groups, the letter P denotes a primitive lattice. If we deal with such a crystal, the conventional unit cell is the primitive cell. If, however, the group designation begins with another letter (e.g., I, f, C), then the primitive cell has to be determined and this is smaller and less symmetric than the conventional um t cell. The unit cell vectors of the primitive cell are called primitive translation vectors. 

In the case of plane-wave basis sets the scaling proceeds on the reciprocal-space vectors as G- (1+ ) G, which is seen by the definition aj b< = 6 4, where and b are real- and reciprocal-lattice primitive translation vectors, respectively. Thus one finds the derivative of reciprocal-space vectors given by... 

The group Ta is an invariant subgroup of Ta (translation group of the infinite lattice with primitive translation vectors a ) so that the cosets E a )TA i = 1,2,...,L) in the decomposition... 

This is the first time that the name unit cell has been used, a name that seems innocuous enough. In fact, some care is needed in its use because there is no unique definition of unit cell for any crystal structure. For any crystal structure there is an infinity of acceptable choices of unit cell—the only requirement on it is that it contains one primitive structural unit and that it generates the entire crystal by translation operations alone. A unit cell need not have six faces and its faces need not be planar (just as tiles with curved edges can cover a surface). It is often convenient to choose as unit cell a volume defined by the primitive translation vectors themselves but, equally, for other purposes this can be an inconvenient choice. 

In addition to the above symmetry operations concerning the unit cell, the crystalline state is characterized by a triple periodicity along the three crystallographic axes Si, aa, and a3- This periodicity is expressed by a primitive translation vector in the direct space given by... 

Unfortunately, at this point, the actual complexities of UPtj became apparent. UPtj is a non-symmorphic lattice (two atoms per unit cell, separated by a nonprimitive translation vector). Because of this, the observed dynamic susceptibility is not invariant under reciprocal lattice translations (ignoring form factors, it has a periodicity of two reciprocal lattice vectors in the c direction and three in the basal direction, due to the non-primitive translation vector). Using this susceptibility in a gap equation, then, gives gap functions which are not properly lattice periodic. Thus both the solutions of Norman and of Putikka and Joynt are invalid. 

These are the Cartesian coordinates of a set of primitive translation vectors for an fee lattice. 

Thus the primitive translation vector in reciprocal space, G = h A + fc B + C for a bcc direct lattice (fee reciprocal lattice) is... 

We started our discussion in this chapter by assuming an infinitely extended crystal. Now we shall introduce the Born-von Karman or periodic boundary conditions as we did for the linear chain in Chap.2. For this purpose we subdivide the infinitely large crystal into "macrocrystals". Each macrocrystal is a parallelepiped defined by the vectors N a, N a, N a, where 1 2 3 primitive translation vectors and N, N2, are large... 

Equation (3.34) specifies the possible values of q. In order to express these conditions in a simple manner, we introduce the reciprocal lattice. The primitive translation vectors of the reciprocal lattice are the three vectors 2, definded by [3.1]... 

Fig.3.4. a) Unit cell and primitive translation vectors of the CsCl structure. b) Brillouin zone of the CsCl structure [3.5]... 

In Figure 3.5, the body-centered-cubic (bcc) lattice and the corresponding Brillouin zone are illustrated. The primitive translation vectors are... 

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