Crystallography and Crystal Defects. Anthony Kelly. Читать онлайн. Newlib. NEWLIB.NET

Автор: Anthony Kelly
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: Физика
Год издания: 0
isbn: 9781119420163
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target="_blank" rel="nofollow" href="#fb3_img_img_f90509bb-d6bf-557e-b346-40c45865cad4.png" alt="images"/>, where 0 < ε1, ε2 ≪ 1. Under these circumstances the centres of the atoms at X, A and Y lie outside the unit cell, so that the atom count within the unit cell is simply two. We note that the minimum number of atoms which a unit parallelogram could contain in a sheet of graphene is two, since the atoms at O and O′ have different environments.

      To describe the atomic positions in Figure 1.1a we chose OXAY as one unit parallelogram. We could equally well have chosen OXTA. The choice of a particular unit parallelogram or unit cell is arbitrary, subject to the constraint that the unit cell tessellates, that is, it repeats periodically, in this case in two dimensions. Therefore, NQPM is not a permissible choice – although NQPM is a parallelogram, it cannot be repeated to produce the graphene structure because P and M have environments which differ from those at N and Q.

      A formal definition of the lattice is as follows: A lattice is a set of points in space such that the surroundings of one point are identical with those of all the others. The type of symmetry described by the lattice is referred to as translational symmetry. The lattice of the graphene crystal structure drawn in Figure 1.1a is shown in Figure 1.1b. It consists of a set of points with identical surroundings. Just a set of points: no atoms are involved. Various primitive unit cells are marked in Figure 1.1c. A primitive unit cell is defined as a unit parallelogram which contains just one lattice point. The conventional unit cell for graphene corresponding to OXAY in Figure 1.1a is outlined in Figure 1.1c in heavy lines and the corresponding x‐ and y‐axes are marked.

      In general, the conventional primitive unit cell of a two‐dimensional net which shows no obvious symmetry is taken with its sides as short and as nearly equal as possible, with γ, the angle between the x‐ and y‐axes, taken to be obtuse, if it is not equal to 90°. However, the symmetry of the pattern must always be taken into account. The net shown in Figure 1.1b is very symmetric and in this case we can take the sides to be equal, so that a = b and γ = 120°.

      Comparisons of Figures 1.1a and b emphasize that the choice of the origin for the lattice is arbitrary. If we had chosen O′ in Figure 1.1a as the origin instead of O and then marked all corresponding points in Figure 1.1a, we would have obtained the identical lattice with only a change of origin. The lattice then represents an essential aspect of the translational symmetry of the crystal however we choose the origin.

Definition of the smallest separations a, b, and c of the lattice points along the x-, y-, and z-axes respectively, together with the angles α, β, and γ between the axes for a lattice in three dimensions. The numbers give the elevations of the centres of the atoms, along the z-axis, taking the lattice parameter c as the unit of length.

      It is apparent from this drawing of the crystal structure of caesium chloride that the coordination number for each caesium ion and each chlorine ion is eight, each ion having eight of the other kind of ion as neighbours. The separation of these nearest neighbours, d, is easily seen to be given by:

      (1.1)equation

      since a = b = c.

      The number of units of the formula CsCl per unit cell is clearly 1.