1/2, 1/2,1/2, is the center of a cube. Induces of lattice directions and crystal planes (a) Directions in a crystal Directions in a crystal are

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1 Crystallography Many materials in nature occur as crystals. Examples include the metallic elements gold, copper and silver, ionic compounds such as salt (e.s. NaCl); ceramics, rutile TiO2; and nonmetallic elements, possibly the best known being carbon in the form of diamonds. The science of the crystalline state is called Crystallography. A crystal is best defined as having a periodic structure. That is a crystal is the repetition of a specific arrangement of some known feature. From this comes the concept of a lattice. A lattice is a framework built up from repetition in three dimensions of a series of lattice points. The property of a lattice point is that each lattice point has identical surroundings. It is the feature which when repeated a large number of times creates a crystal. Figure 1 shows a lattice. The dots at line intersections represent lattice points. Only 14 different combinations of lattice points are possible such that the lattice itself is not repeated. These 14 lattices are known as the Bravais lattices and are shown in Fig.2 They are also known as 'translation' groups. Translation vectors a, b, c, (Fig.l) describe the lattice group and serve as reference axes. In many crystals, lattice points consist of groups of atoms. Thus although only 14 lattices exist many different crystals structures exist, depending upon the exact atomic arrangement of atoms at lattice points. However for common metals and some other '^elements of the periodic table single atoms occupy a lattice point. Crystal Systems To specify a given point in a lattice or atom in a structure, a series of crystal axes or system of axes is used to provide coordinates. The most common metallic crystals are cubic and have a 3 axis system each at right angles. To describe the 14 Bravais lattices, only 7 systems of axes are used. Each has specific equalities and inequalities in length and angles of the vectors a,b,c. These are shown in table 1, where a, b, c are vectors and α, β, γ are angles. The angle opposite a is α etc. by convection. The axes a, b, c are given, values in units of A and angles α, β, γ are specified for crystal structures, see table 1. Unit Cells Crystal axes form a parallelepiped called a Unit Cell. Repetition of a unit cell builds a crystal. Thus unit cells are a building block, and the properties of a crystal are often reflected by the properties of a unit cell. A unit cell always has lattice points at its corners. Additional ones may be at centers of faces and at the center of the unit cell. When lattice points are only at unit cell corners it is called a 'primitive cell'. Each of the 7 crystal systems has a primitive cell; numbers 1,2, 4, 8, 9, 10, 12 of Fig.2. It should be noted at this point that the hexagonal unit cell, number 8, is not hexagonal but a parallelepiped with axes a1, a2 and c. This unit cell is not immediately obvious as hexagonal, but when three are placed together a hexagonal structure is found. By convention, primitive cells have to notation P, face-centered have the notation F, body-centered the notation I. Center-faced have the notation C. Coordinates of Positions in a Unit Cell A position in a unit cell is specified from its coordinates. Let the point x, y, z in a unit cell have a vector from the origin of r xvz = xa + yb + zc. Its coordinates are therefore x, y, z. Coordinates are expressed in terms of the length of cell edges, not units of distances such as centimeters. Thus, the 2,2, 1, position is reached by moving a distance twice the unit cell vector a along the a axis, twice the vector b along the b axis and the distance of unit cell vector c along the c axis. The point will be at a cell corner. For distances part way along cell edges or in cell faces the coordinates will consist of fractions eg.

2 1/2, 1/2,1/2, is the center of a cube. Induces of lattice directions and crystal planes (a) Directions in a crystal Directions in a crystal are specified using square parentheses, ie [uvw]. This indicates the direction of a line from the origin to a point with coordinates u, v, w. Convention states that square brackets are used for a specific direction, and only integers are used inside parentheses. These integers must be the smallest which describe the direction. For example to the point 3, 3, 3, the direction could be [333]. However [111] is conventionly used, as it is parallel to [333]. To prove this draw out three unit cells and mark [333] and [111]. To find a direction in a crystal, draw a line through the origin parallel to the direction required, fig.3. Give the coordinates of a point on it, measured in cell edge lengths and convert to the smallest of a point on it, measured in cell edge lengths and convert to the smallest integers, having the same ratio, written as [uvw]. For example in fig. 3a, the coordinates are l/2a, Ib, l/2c. The direction is written [121]. \ Negative indices are written with a bar over them. For example, the coordinates of the position marked in fig. 3b are la, -Ib, Ic. its direction is therefore [1TT]. Symmetry in a crystal makes directions with the same values of [uvw] identical. For example [100] is a direction along the a axis, [010] along b, and [001] along c. Such a family is shown by arrowed parentheses: (100) represents all possibilites of [100], [010], [001], [TOO], [OTO], [OOT], There are six of these and they correspond to the edges of a cube for a unit cell in the cubic system. (100) is said to have 6 crystal equivalents. (b) Planes in a Crystal For crystal planes a similar system exists, called Miller indices. They represent the orientation of a crystal plane in a unit cell without respect to the origin, but with respect to the crystal axes. Miller indices only apply to three axes systems, and are based upon their intercepts with crystal axes. The unit of measurement is again the unit cell dimension, either a, b, or c. The intercepts of the plane on the three axes from the origin are counted. Reciprocals are then taken. These are then reduced to integers having the same ratios. Finally, these are enclosed in round parentheses ie (hkl). For example in fig.4, the shaded plane makes intercepts of a, l/2b, l/3c. Reciprocals produce a, 2b, 3c. Integers having the same ratio are 1/1, 2/1, 3/1, therefore the plane is a (123) plane. The four commonest planes for cubic crystals are shown in fig.5. Curly parentheses, {hkl} signify planes of similar form, for example: {100} = (100) + (010) + (001) + (TOO) + (OTO) + (OOT) For a single crystal type, planes of a form all have the same atomic configuration. Thus for {100} there are six crystal equivalents. For cubic crystals, the {100} family are the cube faces. The three axis system for the hexagonal system does not produce equivalent indices for equivalent atomic planes. Thus a four system notation is used, called Miller-Bravais indices. Four axes a1, a2 or a3 and c are used, fig.6. Planes are expressed as (hkil) and by convention h + k +l = 0. Directions are [uvwl] and the same rule applies of u + v + w = 0. Directions are found using the geometrical approach shown in fig.6. Thus directions, for example along the aj axis, are given by the translation along the dotted path, for [2TTO]. For planes the intercepts along a1, a2, a3 and c are used, again using cell edges as distance units reciprocals taken and reduced to the lowest integer and placed in round

3 parentheses. For example the plane cutting the c axis at a distance c from the origin, but not intercepting a1, a2, or a3, has intercepts inf, inf, inf, l, reciprocals 0, 0, 0, 1. The lowest integer with same ratio is 0, 0, 0, 1, and the plane (0001) is called 'basal' plane. The hexagonal faces are of the family {10TO}. These are termed 'prism planes'. The common hexagonal planes are shown in fig.7. Manipulation of Indices It should be noted that in both the three and four coordinate systems, planes are at 90 to directions of the same integers. For example a (111) is at 90 to [111] and (0001) is 'at 90 to [0001]. This leads to several useful properties, particularly in fields such as electron microscopy. Symmetry of Crystals Crystals posse a definite symmetry in the arrangement of its faces. This is termed 'Rotational Symmetry 1 and is represented by an axis around which the crystal can be rotated in such a way that it appears identical in several positions. For example, an axis through the center of the top and bottom faces of a cube is a four fold axis of symmetry. When a cube is viewed along this axis four positions exist in which it looks identical, ie every 90 of rotation. This is referred to as a 'tetrad 1 axis. Fig.8, shows the symmetry axes and types for cubic and hexagonal crystals. A cubic crystal has three tetrad axes along the three (100) directions, four triad axes along (111) and six diad axes along (110). Hexagonal structures also have one hexed axis along [0001]. Crystal classes are also generally divided into seven systems based upon a certain minimum rotational symmetry. These are shown below: Crystal Type Triclinic Monoclinic Orthorhomlic Tetragonal Rhombohedral Hexagonal Cubic Symmetry axes None 1 twofold 3 perpendicular two-fold axes 1 four-fold axis 1 three-fold axis 1 six-fold axis 4 three-fold axis

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