Integer lattice points

Consider the set of integer lattice points in three-dimensional space, i.e., points of the form (p,ij,r>, where p, q, and r are integers. Show that this set is countable. 

This means that — (n + n +n ), since kxL = 2-kux, kyL = 27rnj and kzL = 2-kUz for the propagation vector k = kx,ky,kz), where n = fix,riy,nz) defines a set of integer lattice points in three dimensions with each point or triplet of integers corresponding to a photon state for a fixed polarization. The number of lattice points (states) per polarization in a frequency range lj to w+dw can be figured as follows. Note that k" cP — i.e., n =... 

In general, for a box which is positioned at a cubic lattice point n (= rij.L, riyL, n L) with n, Tiy, being integers) ... 

Because the electrons do not penetrate into the crystal bulk far enough to experience its three-dimensional periodicity, the diffraction pattern is determined by the two-dimensional surface periodicity described by the lattice vectors ai and ai, which are parallel to the surface plane. A general lattice point within the surface is an integer multiple of these lattice vectors ... 

There are an even number of lattice points equispaced on a ring, which are represented by the integers 1,2,..., A. The restriction U — v = 1 limits the summation to nearest neighbor contributions, and since A is periodic the identity A + 1 = 1 holds. Along a bond the pairs interact through an electrostatic force the magnitude and sign determined by the coefficient oce- The... 

They change Px into Py, but the difference is so minor that they are often considered the same distribution, and are denoted by the same name. The transformation can be used to transform the distribution to a standard form, for instance one with zero average and unit variance. In the case of lattice distributions one employs (5.4) to make the lattice points coincide with integers. In fact, the use of (5.4) to reduce the distribution to a simple form is often done tacitly, or in the guise of choosing the zero and the unit on the scale. 

If one examines the series in Equation (2.9) it will be seen that the terms will tend to reinforce whenever Q=2vl/d, where / denotes an integer. These values of Q define a one-dimensional reciprocal lattice and whenever Q takes on one of these values the diffracted waves will reinforce. These values of Q correspond to the familiar Bragg reflections given by 2d sin 0 =/A. It should be noted that for finite values of N the reciprocal lattice points have a finite width. 

Let the primitive translations in the real lattice be the vectors a, b, c, then these quantities multiplied by the integers u, v, w, respectively define the lattice points. Thus... 

It is not always possible to choose a unit cell which makes every pattern point translationally equivalent, that is, accessible from O by a translation a . The maximum set of translationally equivalent points constitutes the Bravais lattice of the crystal. For example, the cubic unit cells shown in Figure 16.2 are the repeat units of Bravais lattices. Because nt, n2, and w3 are integers, the inversion operator simply exchanges lattice points, and the Bravais lattice appears the same after inversion as it did before. Hence every Bravais lattice has inversion symmetry. The metric M = [a, a ] is invariant under the congruent transformation... 

It is frequently formd that it is not possible to find a primitive rrrrit cell with edges parallel to crystal axes chosen on the basis of symmetry. In such a case the crystal axes, chosen on the basis of symmetry, are proportiorral to the edges of a unit of stracture that is larger than a primitive unit cell. Such a rrrrit is called a nonprimitive unit cell, and there is more than one lattice point per nonprimitive unit cell. If the nonprimitive unit cell is chosen as small as possible consistent with the symmetry desired, it is found that the extra lattice points (those other than the comer points) lie in the center of the unit cell or at the centers of some or all of the faces of the imit cell. The coordinates of the lattice points in such a case are therefore either integers or half-integers. 

If the coordirrate system is not chosen in corresporrdence to a primitive unit cell, not all the lattice points have coordinates that are integers and all lattice points will lie on planes in the set orrly if certain restrictiorrs are placed on the combinations of values that the Miller indices hkl may asstrme. 

For a body-centered lattice I, some of the lattice points have coordinates that are expressible as integers mnp and some have coordinates that mrrst be expressed as half integers m +, n +, p +. For the latter,... 

Crystallographic direction Vector between two lattice points where the direction is indicated by [u V w], where u, v, and w do not contain a common integer. The integers u, v, and w are called the indices of the crystallographic direction and specify an infinite set of parallel vectors. 

Any reciprocal lattice vector, or reciprocal lattice point is uniquely specified by the set of three integers, hkl, which are the Miller indexes of the family of planes it represents in the crystal. Thus there is a one-to-one correspondence between reciprocal lattice points and families of planes in a crystal. It will be seen shortly that the reciprocal lattice is the Fourier transform of the real lattice, and vice versa. This was in fact demonstrated experimentally in Figure 1.7 of Chapter 1 by optical diffraction. As such, reciprocal space is intimately related to the distribution of diffracted rays and the positions at which they can be observed. Reciprocal space, in a sense, is the coordinate system of diffraction space. 

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