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Indexing of Powder Patterns
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For cubic system, expected lines of a diffraction pattern can be generated from the following realtions:

 

 

(i) The relation between miller indices of a particular peak and the inter planner spacing (d) can be written as:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨»d«/mi»«mrow»«mi mathvariant=¨bold-italic¨»h«/mi»«mi mathvariant=¨bold-italic¨»k«/mi»«mi mathvariant=¨bold-italic¨»l«/mi»«/mrow»«/msub»«mo»=«/mo»«mfrac»«msub»«mi mathvariant=¨bold-italic¨»a«/mi»«mi mathvariant=¨bold-italic¨»o«/mi»«/msub»«msqrt»«mrow»«msup»«mi mathvariant=¨bold-italic¨»h«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi mathvariant=¨bold-italic¨»k«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi mathvariant=¨bold-italic¨»l«/mi»«mn»2«/mn»«/msup»«/mrow»«/msqrt»«/mfrac»«/math»

 

(ii) Bragg’s Law

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold-italic¨»§#955;«/mi»«mo»=«/mo»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»d«/mi»«mi mathvariant=¨bold¨»sin«/mi»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/math»

 

 

For indexing the powder diffraction patterns above two equations are used, i.e., 

 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msup»«mi»§#955;«/mi»«mn»2«/mn»«/msup»«mrow»«mn»4«/mn»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi»S«/mi»«mi»i«/mi»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«mi»§#952;«/mi»«/mrow»«msqrt»«mrow»«msup»«mi mathvariant=¨bold-italic¨»h«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi mathvariant=¨bold-italic¨»k«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi mathvariant=¨bold-italic¨»l«/mi»«mn»2«/mn»«/msup»«/mrow»«/msqrt»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi»S«/mi»«mi»i«/mi»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«mi»§#952;«/mi»«/mrow»«mi»s«/mi»«/mfrac»«/math»

 

In above equation:

 

(i) «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msup»«mi mathvariant=¨bold-italic¨»§#955;«/mi»«mn»2«/mn»«/msup»«mrow»«mn»4«/mn»«msup»«mi mathvariant=¨bold-italic¨»a«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»§nbsp;«/mo»«/math»is always constant for a given crystal.

(ii) «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold-italic¨»s«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfenced»«mrow»«msup»«mi mathvariant=¨bold-italic¨»h«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi mathvariant=¨bold-italic¨»k«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi mathvariant=¨bold-italic¨»l«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§nbsp;«/mo»«/math»will always be an integer regardless of the sign of each of the h, k, l values.

 

Each of the four common cubic lattice types has characteristic sequence of X-ray diffraction lines described by their successive " s " values:

 

(i) Simple cubic: 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13,..................

 

(ii) Body centered cubic: 2, 4, 6, 8, 10 ,12 ,14, 16..............

 

(iii) Face centered cubic: 3, 4, 8, 11, 12, 16............

 

(iv) Diamond cubic: 3, 8, 11, 16............

 

 

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