Reciprocal lattice primitive unit cell

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... 

Example 16.3-2 Find the reciprocal lattice of the planar hexagonal net which has the primitive unit cell shown in Figure 16.10. 

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... 

FIGURE 6.8 The lattice of asymmetric units, represented by the symbol for a question ( ), is the same in both (a) and (b). The corresponding reciprocal lattice is accordingly the same for both (c) and (d). If the unit cell is chosen as in (a) to be a primitive unit cell, then every lattice point in (c) is occupied. If, however, the real unit cell is chosen to be centered as in (h), then half of the reciprocal lattice points, indexed according to this centered cell, are systematically absent, and a checkerboard pattern of diffraction intensities is observed. 

Problem 4.3. Show that in a three-dimensional lattice the number of distinctly different k vectors is N1N2N3. Since these vectors can all be mapped into the first Brillouin zone whose volume is bi (b2 x bj) = (27r)fyvv where vv = ai (32 X 33) is the volume of the primitive unit cell of the direct lattice, we can infer that per unit volume of the reciprocal lattice there are M N2N3 / [ (2n ) /w] = wNiN2N3/(2n ) = Q/(27r) states, where Q = L L2L3 is the system volume. Show that this implies that the density (in Z -space) of allowed k states is 1 / (27r ) per unit system volume, same result as for free particle. 

The reciprocal lattice of a BL whose primitive unit cell is defined by three vectors ai, a2 and a3 is generated by three primitive vectors... 

Sometimes it can be advantageous to construct the reciprocal lattice of the centred rectangular oc-lattice using the primitive unit cell. In this way it will be found that the primitive reciprocal lattice so formed can also be described as a centred rectangular lattice. This is a general feature of reciprocal lattices. Each direct lattice generates a reciprocal lattice of the same type, i.e. mp — mp, oc —> oc, etc. In addition, the reciprocal lattice of a reciprocal lattice is the direct lattice. 

The electronic motion is extraordinarily related to the reciprocal lattice. Instead of dealing with just one electron dispersion relationship E k) there is an infinite number of equivalent dispersion relationships (see Figure 4.12) such that enei E(k) = E(k g) for all g. However, because of the k-space periodicity it is enough to only study the primitive unit cell of the reciprocal lattice, known as the first Brillouin zone, all other cells have the same characteristics. 

Coincidence-IA (POL) p,q,r, and s are all rational numbers. A superceU, which in the case of Fig. 7(b) is a 2 x 2 array of primitive cells, defines the phase-coherent registry with the substrate. By convention, the supercell is defined by comers that coincide with substrate lattice points. If these sites are considered energetically preferred, this condition implies that the other overlayer lattice points on or within the perimeter of the supercell are less favorable. Consequently, if only the overlayer-substrate interface is considered, coincidence is less preferred than commensurism. Two alternative primitive unit cells are depicted here, constructed from different primitive lattice vectors. Though the matrix elements differ, the determinants and, therefore, the areas are identical. Note that the description of the unit cell with coinciding with [0,1] illustrates the reciprocal space criterion = ma [m = 1). 

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. 

We turn our attention next to specific examples of real crystal surfaces. An ideal crystal surface is characterized by two lattice vectors on the surface plane. Hi = aix + aiyS, and H2 = 02xX -I- a2yy. These vectors are multiples of lattice vectors of the three-dimensional crystal. The corresponding reciprocal space is also two dimensional, with vectors bi, b2 such that b aj = IrrStj. Surfaces are identified by the bulk plane to which they correspond. The standard notation for this is the Miller indices of the conventional lattice. For example, the (001) surface of a simple cubic crystal corresponds to a plane perpendicular to the z axis of the cube. Since FCC and BCC crystals are part of the cubic system, surfaces of these lattices are denoted with respect to the conventional cubic cell, rather than the primitive unit cell which has shorter vectors but not along cubic directions (see chapter 3). Surfaces of lattices with more complex structure (such as the diamond or zincblende lattices which are FCC lattices with a two-atom basis), are also described by the Miller indices of the cubic lattice. For example, the (001) surface of the diamond lattice corresponds to a plane perpendicular to the z axis of the cube, which is a multiple of the PUC. The cube actually contains four PUCs of the diamond lattice and eight atoms. Similarly, the (111) surface of the diamond lattice corresponds to a plane perpendicular to the x -I- y -I- z direction, that is, one of the main diagonals of the cube. 

Note that we could have represented the bcc and fee structures by their primitive lattices and since these primitive unit cells contain single atoms, there is no interference and all h k reflections would occur. Now if we take the primitive indices for the fee reciprocal lattice shown in Figure 6.1 and map these primitive indices into the nonprimitive fee reciprocal lattice as shown in Figure 6.10a, we generate points at (200), (110), (220), etc. in the first layer ( = 0) and (101), (211), (301), etc. in the second layer = ) layer. These points correspond to the allowed reflections of a bcc direct lattice in accordance with the structure factor calculations (Equation 6.34), which requires hkl to be net even. Similarly, one can see in Figure 6.10b that the primitive vectors in the bcc reciprocal lattice will map out (200), (220), (222), (400), etc. in the even-numbered planes and (111), (311), (331), etc. 

It has been shown previously that B twins have a reticular character for both type I and type II twins. This means that not all the nodes of the reciprocal lattice have homologues in the twin operation (one over three for all the twins of B). This is due to the fact that the unit cell of the primitive lattice of B is three times the unit cell of the lattice of A, so that the reciprocal lattice of B is three times as dense. All B twins are related to the structure of A, so that only the nodes related to A have an homologue in the twin operation. For instance nodes 111, 202, 020, 313, 310,... of the reciprocal lattice that can be related to nodes of A have homologues in each of the twin operation, while nodes Til, 112, 201, 402,... that are not related to nodes of A have no homologues in a twin operation, as shown on fig. 10. So the reticular character of the B twins, nearly superlattice conserving twins (NSLCT), can be related to the complete description of the relationship existing between the A and B structures. 

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. 

The nonprimitive reciprocal lattice vectors A, B, and C are given by Equation 6.9 using the nonprimitive lattice translation vectors a, b, c. These nonprimitive reciprocal lattice vectors are along the x, y, z axes and form a cubic unit cell with length lit/a and a basis of (000), (110), (101), (Oil) as may be seen in Figure 6.1. Note that the primitive reciprocal lattice vector A = - /2 A. ... 

To decide which of these many vectors to use, it is usual to specify the points at which the plane intersects the three axes of the material s primitive cell or the conventional cell (either may be used). The reciprocals of these intercepts are then multiplied by a scaling factor that makes each reciprocal an integer and also makes each integer as small as possible. The resulting set of numbers is called the Miller index of the surface. For the example in Fig. 4.4, the plane intersects the z axis of the conventional cell at 1 (in units of the lattice constant) and does not intersect the x and y axes at all. The reciprocals of these intercepts are(l/oo,l/oo,l/l), and thus the surface is denoted (001). No scaling is needed for this set of indices, so the surface shown in the figure is called the (001) surface. 

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