Class Note2_Jellur

Class Note2_Jellur - Lecture Notes on Structure of Matter...

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Lecture Notes on Structure of Matter by Mohammad Jellur Rahman, Department of Physics, BUET, Dhaka-1000 27 Lecture 5: Crystal planes and Miller Indices Index 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 [n 1 n 2 n 3 ]. If the numbers n 1 n 2 n 3 have a common factor, this factor is removed. For example, [111] is used rather than [222], or [100], rather than [400]. 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 a 1 , a 2 and a 3 . Let’s these intercepts be x , y , and z , so that x is fractional multiple of a 1 , y is a fractional multiple of a 2 and z is a fractional multiple of a 3 . Therefore we can measure x , y , and z in units a 1 , a 2 and a 3 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.
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Lecture Notes on Structure of Matter by Mohammad Jellur Rahman, Department of Physics, BUET, Dhaka-1000 28 The equation of plane in 2-D and 3D having the intercepts a, b and a, b, c respectively are 1 1 c z b y a x and b y a x Crystal direction is the direction (line) of axes or line from the origin and denoted as [111], [100], [010] etc. How to find Miller Indices : To determine the indices for the plane p in Figure 2, -first we have to find the intercepts with the axes along the basis vector c b a , , . Let these intercepts be x, y, z. We form the fractional triplet   c z b y a x , , . -Take reciprocal to this set.
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This note was uploaded on 01/31/2012 for the course PHYS 418 taught by Professor Somalwar during the Spring '11 term at Rutgers.

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Class Note2_Jellur - Lecture Notes on Structure of Matter...

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