Lecture Notes on Structure of Matter by Mohammad Jellur Rahman, Department of Physics, BUET, Dhaka-100027 Lecture 5: Crystal planes and Miller IndicesIndex system for crystal directions and planes Crystal directions:Any lattice vector can be written as that given by Eq.(1.2). The direction is then specified by the three integers [n1n2n3]. If the numbers n1n2n3 have a common factor, this factor is removed. For example,  is used rather than , or , rather than . When we speak about directions, we mean a whole set of parallel lines, which are equivalent due to transnational symmetry. Opposite orientation is denoted by the negative sign over a number. For example: Crystal planes: The orientation of a plane in a lattice is specified by Miller indices. They are defined as follows. We find intercept of the plane with the axes along the primitive translation vectors a1, a2 and a3. Let’s these intercepts be x, y, and z, so that x is fractional multiple of a1, y is a fractional multiple of a2 and z is a fractional multiple of a3. Therefore we can measure x, y, and z in units a1, a2 and a3 respectively. We have then a triplet of integers (x y z). Then we invert it (1/x 1/y 1/z) and reduce this set to a similar one having the smallest integers by multiplying by a common factor. This set is called Miller indices of the plane (hkl). For example, if the plane intercepts x, y, and z in points 1, 3, and 1, the index of this plane will be (313). The orientation of a crystal plane is determined by three points in the plane, provided they are not collinear. If each point lay on a different crystal axis, the plane could be specified by giving the co-ordinates of the points in terms of the lattice constants a, b, c. A notation conventionally used to describe lattice points (sites), directions and planes is known as Miller Indices. A crystal lattice may be considered as an assembly of equidistant parallel planes passing through the lattice points and are called lattice planes. In order to specify the orientation one employs the so called Miller indices. For simplicity, let us start with a two dimensional lattice and then generalized to three dimensional case.
has intentionally blurred sections.
Sign up to view the full version.