Space groups notations

We can now complete our answer to the question, What information is conveyed when we read that the crystal structure of a substance is monodime P2JC7" The structure belongs to the monoclinic crystal system and has a primitive Bravais lattice. It also possesses a two-fold screw axis and a glide plane perpendicular to it. The existence of these two elements of symmetry requires that there also be a center of inversion. The latter is not specifically included in the space group notation as it would be redundant. 

These were largely due to inappropriate use of the method being applied. For the Rietveld based techniques, these errors included (i) entry of incorrect crystal structure information, including atom coordinates and thermal vibration parameters, (ii) incorrect space group notation, (iii) incorrect atom site occupation parameters, (iv) allowing structure parameters (especially thermal parameters) to refine to physically unrealistic values, and (v) not completing the refinement. 

Note. The Hermann-Mauguin space group notation for any particular crystal comprises two parts. The first part identifies the Bravais lattice type into which the crystal belongs and the second part identifies the total symmetry of the array of atoms in the crystal and therefore also the crystal system. In the second part that identifies the symmetry, only those symmetry elements are included in the symbol that are necessary to describe the space group uniquely. The remainders are being omitted since they follow, as a necessary consequence. 

In reading a space group notation it is important to remember the following ... 

A conqtlication is that rectangular columnar phases have been found with three different two-dimensional lattices, having the space group notations P2 /a, T 2/a ... 

The traditional eharaeterisation of an electron density in a crystal amounts to a statement that the density is invariant under all operations of the space group of the crystal. The standard notation for sueh an operation is (Rim), where R stands for the point group part (rotations, reflections, inversion and combinations of these) and the direct lattice vector m denotes the translational part. When such an operation works on a vector r we get... 

Note 3 The smectic C structure corresponds to monoclinic symmetry characterised by the symbol C2h, in the Schoenflies notation and the space group t Hm in the International System. 

Note 4 The point-group symmetry is C2h (2/m) in the Schoenflies notation, and the space group, 121m in the International System. 

Note 3 The relevant space group of a Colh mesophase is P 6lmmm (equivalent to P 6/m 2 m in the International System and point group Dhh in the Schoenflies notation). 

We cannot present here the complete derivation, as it is very lengthy. However, we shall discuss some representative cases in detail and these should serve to convey the essential ideas. For each illustrative case, we shall give the space group symbol and the conventional diagrams and tables used by X-ray crystallographers. On the basis of these specific examples the general rules for notation and diagrams will be relatively easy to appreciate. 

The space group G of a crystal is the set of all symmetry operators that leave the appearance of the crystal pattern unchanged from what it was before the operation. The most general kind of space-group operator (called a Seitz operator) consists of a point operator R (that is, a proper or improper rotation that leaves at least one point invariant) followed by a translation v. For historical reasons the Seitz operator is usually written R v. However, we shall write it as (R ) to simplify the notation for sets of space-group operators. When a space-group operator acts on a position vector r, the vector is transformed into... 

Note that in eq. (3), as in Chapter 16, no special symbol is used to signify when (R w +1) is a space-group function operator since this will always be clear from the context. It will often be convenient (following Venkataraman et al. (1975)) to shorten the notation for a space-group operator to... 

Obviously, much of the development of crystallography predates the discovery of diffraction of X-rays by crystals. Early studies of crystal structures were concerned with external features of crystals and the angles between faces. Descriptions and notations used were based on these external features of crystals. Crystallographers using X-ray diffraction are concerned with the unit cells and use the notation based on the symmetry of the 230 space groups established earlier. 

We write creation and annihilation operators for a state 1/1) as a and aA, so that ) = a lO). We use the spin-orbital 2jm symbols of the relevant spin-orbital group G as the metric components to raise and lower indices gAA = (AA) and gAA = (/Li)3. If the group G is the symmetry group of an ion whose levels are split by ligand fields, the relevant irrep A of G (the main label within A) will contain precisely the states in the subshell, the degenerate set of partners. For example, in Ref. [10] G = O and A = f2. In the triple tensor notation X of Judd our notation corresponds to X = x( )k if G is a product spin-space group if spin-orbit interaction is included to couple these spaces, A will be an irrep appearing in the appropriate Kronecker decomposition of x( )k. 

The Wyckoff notation for a set of equivalent positions consists of two parts (i) the multiplicity M, which is the number of equivalent positions per unit cell, and (ii) an italicized small letter a starting at the bottom of the list and moving upward in alphabetical order. For a primitive unit cell, M is equal to the order of the point group from which the space group is derived for centered cells, M is the product of the order of the point group and the number of lattice points per unit cell. 

Table 1-4 lists the point symmetry elements and the corresponding symmetry operations. The notation used by spectroscopists and chemists, and used here, is the so-called Schoenflies system, which deals only with point groups. Crystallographers generally use the Hermann-Mauguin system, which applies to both point and space groups. 

The Hermann- Mauguin notation is generally used by crystallographers to describe the space group. Tables exist to convert this notation to the Schoen-flies notation. The first symbol is a capital letter and indicates whether the lattice is primitive. The next symbol refers to the principal axis, whether it is rotation, inversion, or screw, e.g.,... 

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