MODULE 2 Crystals, Unit Cell, Lattices, Point groups and Space groups LEARNING OBJECTIVES What is a crystal? Definition of unit cell Lattice and Bravais lattices Symmetry Lattice directions and planes Miller Indices Point groups and Space groups 2.1. INTRODUCTION The difference between the solid and a crystal is that, in crystals, there is a three-dimensional periodicity. Based on concepts like unit cell, lattices, point groups and space groups, the basics of crystallography was formulated by many scientists. Most of these concepts are for easier understanding of the diffraction pattern and also for interpreting it. Now, more than 6 lakhs of small molecular structures and more than 1 lakh of macromolecular structures have been determined using single crystal X-ray diffraction studies with these foundations. 2.2. Crystals Crystals are substances where there is a three dimensional periodicity of molecules. A crystal selected for single crystal X-ray diffraction purpose should be a single crystal of about 0.2 mm dimension in length, breadth and thickness. Crystals should not have any cracks and there should not be any cluster formation. Once, diffraction quality crystals are obtained, the rest of the process in getting the molecular structure is more or less automatic. 2.3. Unit Cell Unit cell is the smallest repeating unit in a crystal in three dimensions and molecules composed of various atomic species lie in the unit cell. Unit cell is characterized by six parameters and three are the cell edges and three are angles. Based on the distribution among the six unit cell parameters, crystal systems can be classified into seven. 2.4. Lattices The concept of the lattice is useful as a simple way of describing the periodicity of a crystal structure. The crystal structure is built up by repetition of the repeat unit and this can be completely and very simply described by replacing each repeat unit by a lattice point placed at an exactly equivalent point in each and every repeat unit. All these lattice points have the same environment in the same orientation and are indistinguishable from one another. Unit cells are of two types, the simple or primitive cell (symbol P or R) and non-primitive cells (Body centered (I), Face centered (F), Base centered (C)). Primitive cells have only one lattice point per cell while non-primitive have more than one. 2.5. Bravais lattice Bravais, the French crystallographer in 1848 worked on the problem of lattice types and demonstrated that there are fourteen possible point lattices only. These are termed as Bravais lattice or point lattice. If a point is placed at the centre of each cell of a cubic point lattice, the new array of points also forms a point lattice. Similarly, another point lattice can be used as a cubic unit cell having lattice point at each corner and in the centre of each face. The fourteen Bravais lattices are described in the following table and figure (Table 2.1 and Fig. 2.1). Table 2.1 Figure 2.1 A lattice point in the interior of a cell 'belongs' to that cell, while one in a cell face is shared by two cells and one at a corner is shared by eight unit cells. The number of lattice points per cell is therefore given by N = Ni + Nf/2 + Nc/8 where Ni = number of interior points, Nf = number of points on faces and Nc = number of points on corners. Thus, any cell containing lattice points on the corners only is therefore primitive, while the one containing additional points in the interior or on faces is non-primitive. The symbols F and I refer to face- centered and body-centered cells, respectively, while A, B, C refer to base-centered cells, centered on one pair of opposite faces of A, B or C (‘A’ face is the face defined by the b and c axes, etc.). 2.6. Symmetry If certain operations (like translation, rotation about an axis, reflection, rotation followed by the translation, etc.) can be performed by a structure which will bring it into coincidence with itself, then these are termed 'symmetry operations'. As a simple example, we can consider a 'cube'. This has several planes of symmetry as described in the following figure (Fig. 2.2). If a rotation of 360/n brings the body or structure into self-coincidence, the body is said to have an n-fold rotational symmetry. Figure 2.2 In any point lattice, a unit cell may be chosen in an infinite number of ways and may contain one or more lattice points per cell. Unit cells do not 'exist' as such in a lattice; they are imaginary construct and can accordingly be chosen at our convenience. The following figure (Fig. 2.3) explains the position of any lattice point in a cell in terms of its coordinates. Figure 2.3 If point A is taken as the origin having coordinates 000, then lattice points B, C and D will have coordinates 0 ½ ½, ½ 0 ½, and ½ ½ 0, respectively. Points E and D are equivalent and have the coordinates as ½ ½ 1. The coordinates of equivalent points in different unit cells can always be made identical by the addition or subtraction of a set of integral coordinates; subtraction of 001 from the coordinates ½ ½ 1 of E gives ½ ½ 0, the coordinates of D and this is the reason that points D and E are equivalent points. The centering translations – Body Centered: 000, ½ ½ ½ . Face Centered: 0 0 0, 0 ½ ½, ½ 0 ½, ½ ½ 0. Base Centered: 000, ½ ½ 0. An 'n-fold rotation axis of symmetry' is defined as a line, rotation above which produced congruent positions, namely, positions indistinguishable for the initial position, after rotation through 2/n. One fold, two fold, three fold, four fold and six fold rotation axes are denoted by the symbols 1, 2, 3, 4, 6. Rotation followed by a translation is called 'screw axis'. 21 screw means that there is a 2-fold rotation about the conventional 'b' axis (rotation of 180 = 360/2) followed by a translation of 1/2 along the 'b axis'. Mirror reflection is another symmetry element. 'Inversion axis of symmetry' is a distinct type of symmetry axis that combines rotation about a line through 2/n with inversion through a point. These axes are also called 'inverse n-fold axes', where n = 1, 2, 3, 4, 6. During inversion through the origin of coordinates, every point with coordinates x, y, z becomes a point with coordinates -x, -y, -z. If in a structure, every atom with coordinates x, y, z is duplicated by an atom of the same element with coordinates -x, -y, -z (i.e., -x, -y, -z) the structure is said to possess ' centre of symmetry' at the origin. The symbol is 1̅. In space group diagrams, the position of centre of symmetry is denoted by a ' small open circle'. The symmetry operator as a mirror plane or plane of symmetry is assigned the symbol 'm' and the operation of the mirror plane amounts to rotation through 2/2 = 180 followed by the inversion through the centre. Operation of the inverse triad (3̅) is equivalent to the operation of a triad (3) combined with a centre of symmetry (i.e., 3̅ = 3 + 1̅); the operation of the inverse hexad is equivalent to that of a triad combined with a perpendicular mirror plane (i.e., 6̅ = 3 + m). A glide plane is a translation symmetry element representing simultaneous reflection, as in a mirror plane, and a translation through half a lattice repeat in a direction parallel to the plane. 2.7. Lattice directions and planes Let a point has coordinates u v w in the lattice. The direction of any line in a lattice may be described by first drawing a line through the origin parallel to the given line and then giving the coordinate of the point on the line through the origin. The [uvw], written in square brackets are the 'indices' of the direction of the line. They are also the indices of any line parallel to the given line. u,v,w are always converted to a set of smallest integers by multiplication or division throughout. for eg., [½ 1 ½], [1 2 1] and [2 4 2] all represent the same direction but [1 2 1] is the preferred. Following figure (Fig. 2.4) represents the various indices of directions. Figure 2.4 2.8. Miller indices These are the fractional intercepts which the plane makes with the crystallographic axes converted to their nearest integers. Following figure (Fig. 2.5) shows the plane designation by Miller indices for the axial length 4Å, 8Å and 3Å. The four steps involved are listed below to arrive at the Miller indices. Figure 2.5 Axial length 4Å 8Å 3Å Intercept length 1Å 4Å 3Å Fractional intercepts ¼ 4/8 = ½ 1 (intercepts divided by the respective axis length) Miller indices (the 1 2 4 above fractions converted into the nearest integer) 2.9. Crystallographic point groups A combination of symmetry operators (eg., rotation, translation, reflection etc.) can operate on a lattice. The operators or the symmetry elements of a finite body or a structure must pass through a point which is taken as the centre of the body; such a combination or a group of symmetry elements is known as 'point group'. A point group operating on a crystalline solid must be such that every symmetry element of the group can operate on a lattice; such a point group is known as 'crystallographic point group'. Crystallographic point group can be defined as a group of symmetry elements that can operate on an infinite three-dimensional lattice so as to leave one point unmoved.