Finite symmetry operations

Consider a mirror plane that is perpendicular to the Z-axis and intersects with this axis at the origin (z = 0). This plane will reflect objects leaving their x and y coordinates unchanged but the z coordinate of the initial object would be inverted and will become -z after the reflection operation is performed. Therefore, the symbolic description of this mirror reflection operation is x, y, -z. 

The one-fold rotation converts the object into itself (A A), which results in x, y, z. 

The center of symmetry inverts all three coordinates of the object in the point with coordinates 0, 0, 0 (A D), which results in -x, -y, -z. 

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As a result of symmetry transformation (Eqs. 2.111 and 2.112), both the magnitude of the structure amplitude and its phase may change. Finite symmetry operations (t, tj and tj are all 0) usually affect the phase angle, while infinite operations, i.e. those which have a non-zero translational component, affect both the magnitude and the phase. 

Symmetry axes can only have the multiplicities 1,2,3,4 or 6 when translational symmetry is present in three dimensions. If, for example, fivefold axes were present in one direction, the unit cell would have to be a pentagonal prism space cannot be filled, free of voids, with prisms of this kind. Due to the restriction to certain multiplicities, symmetry operations can only be combined in a finite number of ways in the presence of three-dimensional translational symmetry. The 230 possibilities are called space-group types (often, not quite correctly, called the 230 space groups). 

Only certain symmetry operations are possible in crystals composed of identical unit cells. In three dimensions these are one-, two-, three-, four- and six-fold rotations and each of these axes combined with inversion through a centre to give I, 2 ( = m, mirror plane), 3, 4, and 6 operations. Five-fold rotations and rotations of order 7 and higher, while possible in a finite molecule, are not compatible with a three-dimensional lattice. 

The symmetry operations, G, of the space group acting on an atom placed at an arbitrary point in space will generate a set of mo equivalent atoms in the unit cell. Operation of the lattice translations, R, acting on this set generates an infinite array of such atoms, with the finite set of ma atoms being repeated at each point on the lattice. This is illustrated in Fig. 10.1 in which nia = 4 and each of the rectangles defined by the horizontal and vertical lines represents a unit cell that is identical with the one outlined with heavy lines. 

In order to apply group theory to the physical properties of crystals, we need to study the transformation of tensor components under the symmetry operations of the crystal point group. These tensor components form bases for the irreducible repsensentations (IRs) of the point group, for example x, x2 x3 for 7(1) and the set of infinitesimal rotations Rx Ry Rz for 7(1 )ax. (It should be remarked that although there is no unique way of decomposing a finite rotation R( o n) into the product of three rotations about the coordinate axes, infinitesimal rotations do commute and the vector o n can be resolved uniquely... 

A single crystal, considered as a finite object, may possess a certain combination of point symmetry elements in different directions, and the symmetry operations derived from them constitute a group in the mathematical sense. The self-consistent set of symmetry elements possessed by a crystal is known as a crystal class (or crystallographic point group). Hessel showed in 1830 that there are thirty-two self-consistent combinations of symmetry elements n and n (n = 1,2,3,4, and 6), namely the thirty-two crystal classes, applicable to the description of the external forms of crystalline compounds. This important... 

The number of elements in a group is called its order g. A group may have a finite or infinite order (finite or infinite group). All symmetry operators of a point or a space group fulfill the conditions 1 - 4. 

This family of operators can be regarded as an extension of the family of point symmetry operators. Symmorphy is a particular extension of the point symmetry group concept of finite point sets, such as a collection of atomic nuclei, to the symmorphy group concept of a complete algebraic shape characterization of continua, such as the three-dimensional electron density cloud of a molecule. In fact, this extension can be generalized for fuzzy sets. 

Another classification is based on the presence or absence of translation in a symmetry element or operation. Symmetry elements containing a translational component, such as a simple translation, screw axis or glide plane, produce infinite numbers of symmetrically equivalent objects, and therefore, these are called infinite symmetry elements. For example, the lattice is infinite because of the presence of translations. All other symmetry elements that do not contain translations always produce a finite number of objects and they are called finite symmetry elements. Center of inversion, mirror plane, rotation and roto-inversion axes are all finite symmetry elements. Finite symmetry elements and operations are used to describe the symmetry of finite objects, e.g. molecules, clusters, polyhedra, crystal forms, unit cell shape, and any non-crystallographic finite objects, for example, the human body. Both finite and infinite symmetry elements are necessary to describe the symmetry of infinite or continuous structures, such as a crystal structure, two-dimensional wall patterns, and others. We will begin the detailed analysis of crystallographic symmetry from simpler finite symmetry elements, followed by the consideration of more complex infinite symmetry elements. 

It is practically obvious that simultaneously or separately acting rotations (either proper or improper) and translations, which portray all finite and infinite symmetry elements, i.e. rotation, roto-inversion and screw axes, glide planes or simple translations can be described using the combined transformations of vectors as defined by Eqs. 1.38 and 1.39. When finite symmetry elements intersect with the origin of coordinates the respective translational part in Eqs. 1.38 and 1.39 is 0, 0, 0 and when the symmetry operation is a simple translation, the corresponding rotational part becomes unity, E, where... 

In addition to the symmetry within any particular cell, the points in the neighboring cells are related by symmetry to those in that particular cell. Thus we can add symmetry operations that contain an element of translation as well as the other elements appropriate to the finite figure. The addition of translation to the possible symmetry operations greatly increases the number of possible combinations of the symmetry elements. There are 230 possible combinations (space groups) any atomic arrangement in a crystal must have the symmetry corresponding to one of these 230 combinations of symmetry operations. To determine the space group requires a detailed examination of the crystal by x-rays. 

In the case of symmetry operations the product is the successive performance of these operations. If a group has a finite number of elements it is said to be a discrete group. The point groups, which arise firom the rotations and reflections of bodies such as isolated molecules are discrete groups. 

The space group of a crystal structure can be considered as the set of all the symmetry operations which leave the structure invariant. All the elements (symmetry operations) of this set satisfy the characteristics of a group and their number (order) is infinite. Of course, this definition is only valid for an ideai structure extending to infinity. For practical purpose, however, it can be applied to the finite size of real crystals. Lattice translations, proper or improper rotations with or without screw or gliding components are all examples of symmetry operations. 

Point group is a mathematical term and denotes a set of symmetry operations (mirror planes, screw axes, etc.) that can be done on a finite entity (for example, a geometrical shape). This set of operations when applied to the entity, will leave at least one point unchanged in terms of its position. 

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