Crystal Structure and Lattice types.pptx

Explains about crystal structure, lattice types

1. Crystal Structure and Lattice types Mrs.P.Kanmani M.Sc., M.Phil ., Assistant Professor of Physics, V.V.Vanniaperumal College for Women, Virudhunagar

2. Overview Lattice Basis and the Crystal Structure 2D & 3D Lattice types Index system for Crystal Planes

3. Lattice A lattice is a three-dimensional periodic array of identical building blocks. The building blocks are atoms or groups of atoms. The periodicity of crystals is well established by the experimental studies of X-ray, neutron and electron diffraction patterns. A solid is a crystal if the positions of the atoms in it are exactly periodic.

4. Bravais and Non-Bravais Lattice Types

5. Lattice Translation Vectors consider the lattice shown in the figure. The position vector of any lattice point is given as The vectors should be non-collinear and are known as the basis vectors of the lattice. All lattice points can be expressed by the above boxed equation.

6. Lattice Translation Vectors All vectors expressed by the above equation are known as lattice vectors, i.e. lattice vectors is the set of vectors. Lattice is invariant under group of translation expressed by the above equation. Thus the lattice has translational symmetry under all displacements specified by lattice vectors Choice of basis vectors is not unique.

7. Lattice Translation Vectors The parallelogram represented by the basis vectors is called a unit cell of the lattice. When the cell is translated, the whole area of the lattice can be covered — only once. unit cell is the parallelogram with the smallest area that spans the lattice. All unit cells have the same area.

8. Lattice Translation Vectors A lattice transformation — translation here, defined by the following vector equation represents a 3-dimensional lattice. The cells of the parallelepiped constructed by primitive basis vectors. has minimum volume. This volume is given by

9. Wigner Seitz Cell All primitive cells do not have all the possible symmetries of a Bravais lattice. Wigner-Seitz cell is one example of a primitive cell which possesses all possible symmetries of a Bravais lattice. This is because in constructing a W-S cell one does not refer to any particular choice of primitive or basis vectors

10. Wigner Seitz Cell A particular lattice point is chosen. It is then connected to all neighboring closest points. At the mid-point of these joining lines planes or new lines are drawn so that they bisect the joining line, perpendicularly. The resulting minimal volume or area is the W-S cell. Here are two diagrams that show the W-S cell and the way to construct it.

11. Wigner Seitz Cell

12. 2D & 3D Lattice types To understand the various types of lattices, one has to learn elements of group theory: Point group consists of symmetry operations in which at least one point remains fixed and unchanged in space. Space group consists of both translational and rotational symmetry operations of a crystal. T = group of all translational symmetry operations R = group of all symmetry operations that involve rotations.

13. 2D & 3D Lattice types The most common symmetry operations are listed below: C2 = two-fold rotation or a rotation by 180° C3 = three-fold rotation or a rotation by 120° C4 = four-fold rotation or a rotation by 90° C6 = six-fold rotation or a rotation by 180° σ = reflection about a plane through a lattice point i = inversion, I.e. rotation by 180°that is followed by a reflection in a plane normal to rotation axis

14. 2D & 3D Lattice types Two-dimensional lattices, invariant under C 3, C 4 or C 6 rotations are classified into five categories: Oblique lattice and Special lattice types (square, hexagonal, rectangular and centered rectangular). Three-dimensional lattices – The point symmetry operations in 3D lead to 14 distinct types of lattices: The general lattice type is triclinic. The rest of the lattices are grouped based on the type of cells they form. One of the lattices is a cubic lattice, which is further separated into: - simple cubic (SC) - Face-centered cubic (FCC) - Body-centered cubic (BCC)

15. 2D & 3D Lattice types

16. 2D & 3D Lattice types

17. Index System for Crystal Planes The orientation of a crystal plane is determined by three points in the plane that are not collinear to each other. It is more useful to specify the orientation of a plane by the following rules: Find the intercepts of the axes in terms of lattice constants a1, a2 and a3. Take a reciprocal of these numbers. Then reduce to three integers having the same ratio. The result ( hkl ) is called the index of a plane. Planes equivalent by summetry are denoted in curly brackets around the indices { hkl }.