Reciprocal Lattice Unit Cell

The three reciprocal lattice vectors are a, b, and c defined by the nine relations below 

Note A condensed notation used by crystallographers is as follows a..b.= S., where S. is the Kronecker operator (i.e., for i = j, 5i =1 and for S. =0). On the other hand, a slightly different notation is used in solid state physics aj.bj= InS... 

Window material Long (IR) and short (UV) cutt-offs Refractive Index (Hd) Comments 

CaFj (fluorite Irtran-3) 0.130 9.01 77000 1110 22315 322 1.434 Resists most acids and alkalis withstands high pressure insoluble in water. 

(barium fluoride) 0.149 13.51 67000 740 19417 214 1.46 Brittle crystal insoluble in water good resistance to fluorine and fluorides. 

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The direct lattice and reciprocal lattice unit cells are marked on the crystal pattern of a planar hexagonal net in Figure 16.10, using eqs. (13), (14), and (18). The scales chosen for... 

It is also a corollary of Eqs. (1) through (3) that the volume v of the reciprocal-lattice unit cell is the reciprocal of the volume V of the crystal unit cell. Since there is one reciprocal-lattice point per cell of the reciprocal lattice, the number of reciprocal-lattice points within the limiting sphere is given by... 

Pack and Monkhorst °° have suggested that a commensurate grid of points is a suitable option for this purpose. In their method, the grid size depends on a parameter, the shrinking factor s, that specifies how many equidistant k points must be taken along each direction of bi, hx, and bs inside one reciprocal lattice unit cell so that the total number of points in the grid, s, is equal to s , with n denoting the order of periodicity ( = 3 for three-dimensional crystals). 

Figure Bl.21.3. Direct lattices (at left) and corresponding reciprocal lattices (at right) of a series of connnonly occurring two-dimensional superlattices. Black circles correspond to the ideal (1 x 1) surface structure, while grey circles represent adatoms in the direct lattice (arbitrarily placed in hollow positions) and open diamonds represent fractional-order beams m the reciprocal space. Unit cells in direct space and in reciprocal space are outlined.

In this diagram, a series of hexagon-shaped planes are shown which are orthogonal, or 90 degrees, to each of the corners of the cubic cell. Each plane connects to cuiother plane (here shown as a rectangle) on each fiace of the unit-cell. Thus, the faces of the lattice unit-cell and those of the reciprocal unit-cell can be seen to lie on the same pltme while those at the corners lie at right angles to the corners. 

The direction of these vectors is perpendicular to the end faces of the direct lattice unit cell. The lengths of the basis vectors of the reciprocal lattice are the inverse of the perpendicular distance from the lattice origin to the end faces of the direct lattice unit cell. For the square and rectan-... 

Figure 3.2.1.7 Real-space unit cells of the five two-dimensional Bravais lattices and the corresponding reciprocal-space unit cells.

The amplitude and therefore the intensity, of the scattered radiation is detennined by extending the Fourier transfomi of equation (B 1.8.11 over the entire crystal and Bragg s law expresses die fact that this transfomi has values significantly different from zero only at the nodes of the reciprocal lattice. The amplitude varies, however, from node to node, depending on the transfomi of the contents of the unit cell. This leads to an expression for the structure amplitude, denoted by F(hld), of the fomi... 

Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

Figure BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

Note that the denominator in each case is equal to the volume of the unit cell. The fact that a, b and c have the units of 1/length gives rise to the terms reciprocal space and reciprocal latlice. It turns out to be convenient for our computations to work with an expanded reciprocal space that is defined by three closely related vectors a , b and c, which are multiples by 2tt. of the X-ray crystallographic reciprocal lattice vectors ... 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

The prindple of a LEED experiment is shown schematically in Fig. 4.26. The primary electron beam impinges on a crystal with a unit cell described by vectors ai and Uj. The (00) beam is reflected direcdy back into the electron gun and can not be observed unless the crystal is tilted. The LEED image is congruent with the reciprocal lattice described by two vectors, and 02". The kinematic theory of scattering relates the redprocal lattice vectors to the real-space lattice through the following relations... 

Figure 4.26. Schematic drawing of the LEED experiment on a single costal with a unit cell given by vectors a-i and O2. The LEED pattern corresponds to the reciprocal lattice described by vectors and 02. ...

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