Crystalline structures. Basic concepts Crystals are solids in which the atoms are regularly arranged with respect to one another. This regularity of arrangement can be described in terms of symmetry elements. These symmetry elements determine the symmetry of the physical properties of a crystal. Lattice. A lattice is a set of point in space such that the surroundings of the one point are identical with those of all the others. The type of symmetry described by the lattice is referred to as translational symmetry. A crystal is a body composed of atoms arranged in a lattice such that there exists three fundamental translation vectors a, b and c. The atomic arrangement in a crystal appears the same in every respect when viewed from a point r as from a point r if: r = r + n 1 a + n 2 b + n 3 c Where n 1, n 2 and n 3 are arbitrary integers. Many choices of translation vectors a, b and c are possible. However, they are called primitive translation vectors, if starting from any lattice point, all the other lattice points can be reached with a suitable choice of n 1, n 2 and n 3. It is essential to distinguish the lattice formed by a set of points from the crystal structure. Page 1 of 19.
A crystal structure is formed by associating with every lattice point a basis of one or several atoms. These atoms are identical in composition, arrangement and orientation. An atom may or may not actually "sit" directly on the lattice point. Page 2 of 19.
Symmetry elements. The symmetrical arrangement of atoms in crystals is described formally in terms of the elements of symmetry. The symmetry arises because an atom or group of atoms is repeated in a regular way to form a pattern. Any operation of repetition can be described in terms of one of the following three different types of pure symmetry elements or operators. Translation This describes the fact that similar atoms in identical surroundings are repeated at different points within the crystal and any one of these points can be brought into coincidence with any other by an operation of translational symmetry. Such an operation is defined by: Rotation A crystal possesses an n-fold axis of rotational symmetry if it coincides with itself upon rotation about the axis of 360 o /n = 2π/n radians. In crystals, axes of rotational symmetry with values of n = 1, 2, 3, 4 and 6 are the only ones found. These correspond to repetition every 360, 180, 120, 90 and 60 o and are called monad, diad, triad, tetrad and hexad axes, respectively. Reflection This operation is that of reflection in a mirror. Objects that are mirror images cannot be made to superimpose on each other. Page 3 of 19.
Restrictions on symmetry elements. All crystals show translational symmetry. A given crystal may or may not possess other symmetry elements. All of the symmetry elements in a crystal must be mutually consistent. For example, there are no 5-fold axes of rotational symmetry because such axes are not consistent with the translational symmetry of the lattice. Crystal systems. A crystal system contains all those crystals that possess certain axes of rotational symmetry. In any crystal there is a necessary connection between the possession of an axis of rotational symmetry and the geometry of the lattice of that crystal. Because of this connection, a certain convenient conventional unit cell can always be chosen in each crystal system. A unit cell can be thought of as the fundamental building block from which the whole crystal can be generated without gaps when the unit cells are stacked face to face. The symmetry characteristics of the seven basic crystal systems into which all crystalline materials can be grouped are shown below: Page 4 of 19.
The crystal systems These conventional unit cells are in some cases non-primitive i.e. they contain more than one lattice point (there are lattice points at locations other than the cell corners). Space Lattices (Bravais Lattices) Let us consider the types of space lattice (i.e., the regular arrangement of points in three dimensions), which are consistent with the various combinations of rotational axes particular to each of the seven crystal systems. Investigation will show that more than one arrangement of points is consistent with a given set of rotational symmetry elements. However, the number of essentially different arrangements of points is limited to 14. These are the 14 Bravais lattices whose description is given below: Page 5 of 19.
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Selection of the 14 Bravais lattices Some of the unit cells of the Bravais lattices are primitive while others are non-primitive. Why should there not be a space lattice with a base-centered tetragonal unit cell? The following figure shows that such a lattice would be equally well described by a primitive tetragonal unit cell with the same c parameter but with an a parameter equal to the a parameter of the body-centered tetragonal lattice divided by 2. Page 7 of 19.
Therefore, no real difference exists between the base-centered and primitive tetragonal lattices and so the base-centered tetragonal lattice is not a truly separate lattice. For each lattice not described by a primitive unit cell, one could have chose a primitive cell instead, but it would not have indicated the obvious symmetries inherent in the lattice as effectively. For example, the face-centered cubic lattice can be described by a primitive rhombohedral (trigonal) unit cell with an angle of 60 o. Page 8 of 19.
Specification of crystallographic planes and directions. Coordinates of points within the unit cell. Use the edges of the unit cell as coordinate directions. Express the position of a point in dimensionless coordinates that are fractions of the edge lengths of the unit cell, For example, the body-centered position in the cubic, tetragonal or orthorhombic lattices has the coordinates: 1/2, 1/2, 1/2. The six face centered positions have the coordinates: 1/2, 1/2, 0 0, 1/2, 1/2 1/2, 0, 1/2. Lattice planes. 1/2, 1/2, 1 1, 1/2, 1/2 1/2, 1, 1/2. Any three lattice points determine a lattice plane on which many more lattice points will lie. Occasionally it is necessary to consider planes, which do not contain any lattice points. Both are specified in the same way. The procedure is as follows: An arbitrary lattice point is selected as the origin (however, the plane should not pass through the origin). Identify the points at which the plane intercepts the unit cell directions (defined by the cell edges) so that the intercepts are measured in units of cell edges. Define these intercept lengths as a, b and c. Page 9 of 19.
Instead of these intercepts, the Miller indices h, k and l are used to define the plane. The Miller indices are the reciprocals of the axial intercepts multiplied by a suitable factor n such that the indices are only whole numbers (reduction to lowest integers depends upon the crystal structure). Miller indices are enclosed in round brackets without commas (hkl). If an intercept is negative, the corresponding index has a bar placed above the number. Page 10 of 19.
Additional Considerations If the plane to be described does not intercept an axis, it will have a Miller index of 0, i.e. the intercept is at co. Planes and their negative are identical. Planes and their multiples are not, in general, identical. Planar density represents the number of atoms per unit area (i.e. fraction of plane occupied by atoms) counting only those atoms whose centers lie on the plane of interest. Planes and their multiples having the same planar density are identical. Planes which contain the same combination of indices (i.e., indices given in different order and with positive and negative values) and have the same planar density belong to a family of equivalent planes. This is indicated by enclosure in braces {hkl} Page 11 of 19.
Lattice directions. A line parallel to the desired direction is drawn through any lattice point which is then taken as the origin. The coordinates of any point along the line are determined in units of cell edge (lattice) parameters in that direction. The indices u, v, w are taken as the smallest set of integers with the same ratio as the above parameters. If any of the coordinates are negative, the corresponding index is denoted with a bar above it. Direction indices are enclosed in square brackets [uvw]. Page 12 of 19.
Additional considerations. A direction and its negative are not identical. A direction and its multiple are identical. Linear density represents the number of atoms per unit length. For the special case where atoms are uniformly spaced along a given direction, determine the repeat distance between adjacent atoms counting only those atoms whose centers lie on the direction of interest. The linear density is then the inverse of the repeat distance. Non-parallel directions, which contain the same combination of indices (i.e., indices given in different order and with positive and negative values) and have the same linear density belong to a family of equivalent directions. This is indicated by enclosure in angle brackets <hkl>. In the cubic system, a direction that has the same indices as a plane is perpendicular to that plane. Page 13 of 19.
Special case of the hexagonal system. In order that crystallographic equivalent directions and planes have the same set of indices, a four-axis or Miller-Bravais coordinate system based upon the full hexagonal prism as the unit cell is used. The unit cell is defined by three basal coplanar vectors 120 o apart and a fourth vector perpendicular to the base. Specification of directions: [uvtw] where t = -(u+v) Specification of planes: (hkil) where i = -(h+i) Page 14 of 19.
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Planar spacings and interplanar angles. The various sets of planes in a lattice have various values of interplanar spacing (d hkl ) where d hkl represents the perpendicular distance between the planes. The planes of large spacing have low indices and pass through a high density of lattice points whereas the reverse is true of small spacing. The interplanar spacing (d hkl ) is a function of both the plane indices and the particular crystal system. A similar dependence exists for calculations of cell volumes and the interplanar angle (φ). The angle φ between the plane (h 1 k 1 l 1 ) of spacing d 1 and plane (h 2 k 2 l 2 ) of spacing d 2 and the unit cell volumes may be obtained from the following equations: Page 17 of 19.
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