ABSTRACT
Crystal: An ideal crystal consists of a regular repetition in space of atoms or groups of atoms. Lattice: The group of points from which the observed atomic environment is the same as at the origin. This identity concerns both the chemical nature of the atoms and their orientation. Basis: The atoms or group of atoms which constitute the structural unit applied to each points of the lattice. Primitive vectors of the lattice: To go from one point of the crystal to an equivalent point, in which the arrangement and orientation appear exactly the same, it is sufficient to effectuate a translation of the form T = + +m n p a b c in which a b c, , are the primitive vectors of the matrix and m, n, p are integers (positive, zero, or negative).Crystal structure = Lattice + Basis
2. Simple and Multiple LatticesThe choice of vectors a b c, , is not unique. The vectors a b c, , are said to be primitive if the translations allow a complete description of all the points of the crystal lattice. The parallelepiped a b c)( ¥ corresponds to a simple (or primitive unit) cell of the lattice. A multiple cell is constructed from the non-fundamental vectors
a b c¢ ¢ ¢, , , where the parallelepiped contains n equivalent points (n > 1): the corresponding non-primitive cells, therefore, consists of n primitive cells and the order of the lattice cell will be n, where n = 2 or n = 4. 3. Lattice Rows and Miller Indices
To describe the crystal structure it is sufficient to state the choice of lattice vectors, and the nature and position of atoms that make up the basis. These positions are expressed with the help of the vectors
a b c, , , considered as the unit vectors: r a b cj j j ju v w= + + . The lattice points are arranged along various rows and planes. When the rows and the planes are parallel and equidistant to each other they are equivalent and they are represented with the same symbols. A series of parallel rows is represented by (m, n, p) where r
= + +m n pa b c when the row is parallel to the line that connects the origin to the lattice point m, n, p. A series of parallel planes can be represented all by Miller indices (h, k, l), which describe the equation of the form hx + ky + lz = 1 of plane nearest the origin and using a, b, c units (see Ex. 8). This definition means that the intersections of the plane (h, k, l) with axes x, y, and z are 1/h, 1/k, 1/l, respectively. The atoms chosen to define a plane must not be collinear. 4. Point SymmetryIn nearly all crystals one or several directions are equivalent. This orientation or point symmetry of a crystal can be represented by the symmetry of the figure formed by the group of half-lines which, emanating from the same point 0, are parallel to the directions from which all the properties of the crystal are identical. The point symmetries that are encountered include the rotations of order n
around an axis (the angle of rotation is 2p/n with n = 1, 2, 3, 4, 6) and the rotation-inversions, written as n (1 ∫ inversion with respect to 0, 2 ∫�m: mirror symmetry, 3 4 6, , ). Several symmetry elements can be associated around a point but the number of distinct combinations and possibilities is limited to 32. That is, there are 32 symmetry point groups which result in a classification of 32 crystal classes.
