- A general combined symmetry operation, can be symbolized by β t. (SEITZ operator)

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Sylvia Moody
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1 SPACE GROUP THEORY (cont) It is possible to represent combined rotational and translational symmetry operations in a single matrix, for example the C6z operation and translation by a in D 6h is represented by the 4x4 matrix (Seitz operator) which takes x, y, z x-y+1, x, z, The combined operation can be symbolized C 6z a - A general combined symmetry operation, can be symbolized by β t. (SEITZ operator) - The β part represents the 3x3 submatrix in the upper left of the 4x4 matrix the rotational (proper or improper) operator - The t part represents the 3 x l submatrix on the top of the last column the translational part of the symmetry operation. - The last row of the 4x4 matrix is always 0001 ( dummy line) device which makes the matrix multiplication work out to appropriately transform the positional parameters, when they are in a 4xl column matrix with a 1 in the bottom row. Note: The translation represented by symbol t need not be a lattice translation. (e.g. when the operation is a screw or glide operation)

2 RECIPROCAL LATTICE To obtain basis functions (irreducible representations) which may be used to investigate the properties (diffraction, electronic structure, vibrational (phonon) modes) of 3D structures (crystalline solids) with a regular internal structure space group theory. 1. First, seek a function which transforms into itself under the translational symmetry operations (i.e., a function which has the periodicity of the lattice). a basis function for the "totally symmetric" representation 2. For the purpose of finding a basis function for the totally symmetric representation of the pure translations it is useful to know a function of that is incremented by an integer when r is incremented by T. a vector K such that for with such a vector, we can express the periodicity of the lattice by For the expression to be true: K T = integer

3 We know that T = is an integer. The definitions mean that a* is perpendicular to b and c (proportional to b x c). Therfore; a* = α (b x c) Since a* a = a a* = 1 α (a b x c) = α V cell = 1 or α = [V cell ] -1 then a* = [V cell ] -1 (b x c) Similarly, With K = m*a* + n*b* + p*c* sin 2πK r; cos 2πK r; and exp 2πiK r have the periodicity of the lattice.

4 Remember: K r is dimensionless. Therefore K has the dimension of reciprocal length called a reciprocal vector or, since m*, n* and p* can be allowed all integral values thereby generating a lattice (a reciprocal lattice), K is called a reciprocal lattice vector. The essential feature used in the definition of the reciprocal lattice vector is K T = integer (or an integral multiple of 2π) Hence if β t = a symmetry operation of a crystalline solid then K βt = is an integer if K T = integer. However if K βt is an integer then β 1 K T is also an integer. (true for a given T and all of the reciprocal vectors generated by the rotational part of the symmetry operations of the space group). Thus if K is a reciprocal lattice vector so too is βk the reciprocal lattice has the same rotational symmetry as the real lattice. Note: in reciprocal space the unit length coordinates are given as h, k and l

5 Reflection conditions Within the Crystallographic tables: - The Reflection conditions are listed in the right-hand column of each Wyckoff position. These conditions were called Ausloschungen (German), missing spectra (English), and extinctions (French) and 'Conditions limiting possible reflections'; They are also referred to as 'Systematic absences'. There are two types of reflection conditions: (1) General conditions. They apply to all Wyckoff positions of a space group, i.e. they are always obeyed, irrespective of which Wyckoff positions are occupied by atoms in a particular crystal structure. (2) Special conditions ('extra' conditions). They apply only to special Wyckoff positions and occur always in addition to the general conditions of the space group. Note that each extra condition is valid only for the scattering contribution of those atoms which are located in the relevant special Wyckoff position. E.g.: If the special position is occupied by atoms whose scattering power is high, in comparison with the other atoms in the structure, reflections violating the extra condition will be weak. The General reflection conditions are due to one of three effects: (i) Centered cells. - The conditions apply to the whole three-dimensional set of reflections hkl. - They are called integral reflection conditions. - These conditions result from the centering vectors of centered cells. They disappear if a primitive cell is chosen instead of a centered cell. - Note that the centring symbol and the corresponding integral reflection condition may change with a change of the basis vectors (e.g. monoclinic: C A I). (ii) Glide planes. - The resulting conditions apply only to two-dimensional sets of reflections, i.e. to reciprocal lattice nets containing the origin (such as hk0, h0l, 0kl, hhl). - For this reason, they are called zonal reflection conditions. - The indices hkl of these 'zonal reflections' obey the relation hu + kv + lw = 0, where [uvw] (the direction of the zone axis), is normal to the reciprocal-lattice net. - Note that the symbol of a glide plane and the corresponding zonal reflection condition may change with a change of the basis vectors (e.g. monoclinic: c n a).

6 (iii) Screw axes. - The resulting conditions apply only to one-dimensional sets of reflections, i.e. reciprocal lattice rows containing the origin (such as h00, 0k0, 00l). - Called serial reflections conditions. In real (direct) space, the reflection conditions of type (ii) and (iii) correspond to projections of a crystal structure onto a plane or onto a line.

The structure of liquids and glasses. The lattice and unit cell in 1D. The structure of crystalline materials. Describing condensed phase structures

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