Crystal Structure-II

Dr. Saloni Sharma

Department of Physics

Crystal Structure & Translational Vector

• A solid is said to be a crystal if atoms are arranged in such a way that their positions are exactly periodic. This concept can be illustrated using a two-dimensional (2D) structure as in Figure

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• A perfect crystal maintains this periodicity in both the x and y directions from -∞ to +∞. As follows from this periodicity, the atoms A, B, C, etc. are equivalent.
• Hence for an observer located at any of these atomic sites, the crystal appears exactly the same.

Crystal Structure & Translational Vector

• Periodicity of atoms can also expressed by saying that a crystal possesses a translational symmetry, i.e., if the crystal is translated by any vector joining two atoms, say T, the crystal appears exactly the same as it did before the translation. In other words the crystal remains invariant under any such translation.
• The structure of all crystals can be described in terms of a lattice, with a group of atoms attached to every lattice point.
• On the other hand, if we replace each atom by a geometrical point located at the equilibrium position of that atom, we obtain a crystal lattice.
• The crystal lattice has the same geometrical properties as the crystal, but it is devoid of any physical contents.

• Similarly, a space lattice is defined as an infinite array of points in three dimensional space in which each point is identically located with respect to the other.
• The atoms or group of atoms occupying the points in the lattice are known as basis.
• The space lattice when combines with the basis generates a unit cell.Thus space lattice + basis = unit cell.
• A unit cell is defined as the basic structural part in the composition of materials. The figure shows the lattice and unit cell in two and three dimension.

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Primitive and Non-Primitive Unit Cells

• Primitive cells are those unit cells which contain atoms at corner lattice points only. So these cells have least number of total atoms and the least volume of atoms per unit cell.
• All unit cells namely simple cube (SC), simple tetragon (ST), simple orthorhombic (SO), simple rhombohedral (SR) etc. are primitive cells.
• All those unit cells which do not fall under this category are non-primitive cells.
• The choice of the unit cell is not unique.

Wigner-Seitzs unit cell

The primitive cell may be chosen

as shown in Fig.4.

(i) Draw lines to connect a given

lattice point to all nearby lattice points.

(ii) At the midpoint and normal to these lines, draw new lines

(planes in 3D).

The smallest volume enclosed is the

Wigner-Seitz primitive cell.

All the space of the crystal

may be filled by these primitive cells, by translating the

unit cell by the lattice vectors.

Symmetry and Symmetry Operation

• (i) Translation Symmetry:
• A lattice point under a lattice translation operation performed it gives another point which is exactly identical to initial point .
• (ii) Rotational Symmetry:
• If a crystal is rotated through a point by an angle Ɵ, it transform the lattice to another lattice which is again itself in appearance.
• For simplest example is the crystal lattice is rotated by an angle Ɵ is 360⁰ its lattice arrangement remains same. Possible value of n are 1,2,3,4,6 only. multiplicity of rotational axis. Lattice can be found as one, two, three, four and six fold rotation about an axis which carries the lattice to itself. n=5,7,9 not possible and lattice cannot transform into itself under such rotation thus 5,7,9 fold symmetry are not possible

(iii) Reflection:

If the lattice have a plane or line in two dimensional which divide the lattice into two halves which are mirror image of each other then this is called reflection symmetry such a plane is symbolically represented by letter small m.

(iv) Inversion:

Inversion is a symmetry operation which is applicable in three dimensional lattice structure only. In this symmetry if we consider a point as centre of symmetry, and locate all points by lattice vector r then -r (inversion of sign) give the same lattice. The centre of inversion of lattice is denoted by symbol and read as one bar

Crystal Direction and Planes

• It is necessary to locate the directions and planes for its analysis. In a crystal lattice the directions are given by the coordinates of first whole number point

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Miller Indices

• Crystal is made up an aggregate of a large number of parallel equidistance planes.

• The orientation of the planes are first defined by Miller, and known as Miller Indices.

• Miller indices represent the set of parallel planes. Miller Indices of a plane is obtained by following 3 steps:

• (1) Find out the Intercepts of plane on the three crystal axis x, y, z.

• (2) Take reciprocals of these intercepts

• (3) Find out simplest ratio in integer number

(1) Intercepts of plane on the three crystal axis x, y, z are 3, 2, 2.

• (2) Reciprocals of these intercepts are 1/3 , 1/2, 1/2

• (3) Simplest ratio of reciprocals in integer number are 2, 3, 3

• Thus Miller indices (2 3 3)

• Denoted by (h k l)

• A family of planes of a particular type of is presented by enclosing Miller Indices into a {} brass.

Some planes with miller indices

Inter planer Spacing

• Crystal can be considered as arrangement of equidistance planes on which the lattice points lie.
• Interspacing distance (separation) between two planes is a significant parameter in study of crystal diffraction and can be calculated as follows:
• Consider a set of parallel planes with indices (h k l) and among these planes, one plane is passing through origin O. The next plane lies just parallel to first plane and gives intercepts a/h, b/k, c/l on x, y, z axis. a, b, c are lattice vectors of a crystal.

• ABC is plane which intercepts x, y, z axis at points A, B, C respectively.
• If we draw a perpendicular ON from O to plane ABC then ON=d.
• Let us consider perpendicular ON makes angle with x, y, z axis respectively then we can consider a OAN as shown in figure

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Cubic crystal system

• The simplest and easiest structure.

• Three types of possible crystal structure under this family named as

• simple cubic,
• body centered cubic and
• face centered cubic

Simple cubic crystal (sc)

• Lattice points are arranged at each 8 corner of cube.

• At each corner of cube, an atom is shared by 8 nearby unit cells.

• one unit cell contains 1/8×8=1 atoms.

• Each atom is surrounded by 6 nearest neighbors atoms. The number of nearest neighbors of a lattice point (or atom) in a crystal lattice is called coordinate number.

• Example Cu, Ag, Au are this types of structure.

Coordination Number

• Each atom in a crystal is surrounded by a number of atoms.
• The surrounding atoms are located at different distances.
• “The coordination number is defined as the number of nearest and equidistant atoms with respect to any other atom in a unit cell.”
• Effective number of atoms per unit cell Ne is different from total number of atoms per unit cell. The atom at the corner of a cubical unit cell has only 1/8 of it inside the boundary of that unit cell.
• The remaining 7/8 of it lies in the surrounding unit cells of the crystal.
• Similarly the atom at the face in FCC is shared 1/2 by that atom and 1/2 by the neighbouring atom.
• In BCC, the atom at the centroid is wholly occupied by that unit cell in which it lies.
• Thus the effective number of atoms are 1, 2 and 4 in SC, BCC and FCC respectively.

Effective number of atoms for different unit cells

Parameters of SC

Body centered cubic (bcc)

• In this case of cubic crystal, one atom is arranged inside the cube additional to eight atoms at eight corners this structure

• one atom is inside the unit cell entirely, and eight corners of lattice cube share1/8 part of each atoms. Therefore the number of atoms in a bcc unit cell =1+1/8=2.

• Many metals a like Li, Na, K, Cr exhibit bcc

BCC Structure

Parameters of bcc crystal

Face centered cubic crystal

• 8 atoms are arranged at eight corners of the cubic lattice and 6 atoms are arranged at the centre of eight faces of cube

• Each atom of 8 corners is shared by 8 neighbor unit cells therefore one corner of cube share 1/8 atom; each atom at the faces of cube is shared by 2 unit cell and each face shared 1/2 atom and total 6 faces share 6×1/2=3 atoms. Therefore net atoms inside a unit cell is equal to 1/8×8+½×6=4. • Co-ordinate number of fcc crystal is 12.

• Example of fcc Crystal are Al, Cu, Au, Ag etc.

THANKS