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Estimation of Precise Lattice Parameter of Cubic Crystals
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Bragg’s Law   λ = 2dsinθ

 

Precision in ‘d’ is a function of sin θ, and cot θ.

 

Differentiating Bragg’s law with respect to θ, we obtain  

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mo»§#9651;«/mo»«mi mathvariant=¨bold-italic¨»d«/mi»«/mrow»«mi mathvariant=¨bold-italic¨»d«/mi»«/mfrac»«mo»=«/mo»«mo»-«/mo»«mi mathvariant=¨bold-italic¨»c«/mi»«mi mathvariant=¨bold-italic¨»o«/mi»«mi mathvariant=¨bold-italic¨»t«/mi»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«mo».«/mo»«mi mathvariant=¨bold-italic¨»d«/mi»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mo»§#9651;«/mo»«mi mathvariant=¨bold-italic¨»a«/mi»«/mrow»«mi mathvariant=¨bold-italic¨»a«/mi»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mo»§#9651;«/mo»«mi mathvariant=¨bold-italic¨»d«/mi»«/mrow»«mi mathvariant=¨bold-italic¨»d«/mi»«/mfrac»«mo»=«/mo»«mo»-«/mo»«mi mathvariant=¨bold-italic¨»c«/mi»«mi mathvariant=¨bold-italic¨»o«/mi»«mi mathvariant=¨bold-italic¨»t«/mi»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«mo».«/mo»«mi mathvariant=¨bold-italic¨»d«/mi»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/math»

Other relationships user:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mo»§#9651;«/mo»«mi mathvariant=¨bold-italic¨»a«/mi»«/mrow»«mi mathvariant=¨bold-italic¨»a«/mi»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mo»§#9651;«/mo»«mi mathvariant=¨bold-italic¨»d«/mi»«/mrow»«mi mathvariant=¨bold-italic¨»d«/mi»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi mathvariant=¨bold-italic¨»a«/mi»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»a«/mi»«mi mathvariant=¨bold-italic¨»o«/mi»«/msub»«/mrow»«msub»«mi mathvariant=¨bold-italic¨»a«/mi»«mi mathvariant=¨bold-italic¨»o«/mi»«/msub»«/mfrac»«mo»=«/mo»«mi mathvariant=¨bold-italic¨»K«/mi»«msup»«mi mathvariant=¨bold¨»cos«/mi»«mn»2«/mn»«/msup»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold-italic¨»a«/mi»«mo»=«/mo»«msub»«mi mathvariant=¨bold-italic¨»a«/mi»«mi mathvariant=¨bold-italic¨»o«/mi»«/msub»«mo»+«/mo»«msub»«mi mathvariant=¨bold-italic¨»a«/mi»«mn»0«/mn»«/msub»«mi mathvariant=¨bold-italic¨»K«/mi»«msup»«mi mathvariant=¨bold¨»cos«/mi»«mn»2«/mn»«/msup»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mo»§#9651;«/mo»«mi mathvariant=¨bold-italic¨»d«/mi»«/mrow»«mi mathvariant=¨bold-italic¨»d«/mi»«/mfrac»«mo»=«/mo»«mfenced»«mrow»«mo»-«/mo»«mfrac»«mi mathvariant=¨bold-italic¨»D«/mi»«mi mathvariant=¨bold-italic¨»R«/mi»«/mfrac»«/mrow»«/mfenced»«mfrac»«mrow»«msup»«mi mathvariant=¨bold¨»cos«/mi»«mn»2«/mn»«/msup»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/mrow»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/mrow»«/mfrac»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mo»§#9651;«/mo»«mi mathvariant=¨bold-italic¨»d«/mi»«/mrow»«mi mathvariant=¨bold-italic¨»d«/mi»«/mfrac»«mo»=«/mo»«mfenced»«mi mathvariant=¨bold-italic¨»K«/mi»«/mfenced»«mfenced»«mrow»«mfrac»«mrow»«msup»«mi mathvariant=¨bold¨»cos«/mi»«mn»2«/mn»«/msup»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/mrow»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«msup»«mi mathvariant=¨bold¨»cos«/mi»«mn»2«/mn»«/msup»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/mrow»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/mfrac»«/mrow»«/mfenced»«/math»

The lattice parameters obained is plotted against cos2, cos2θ/sinθ and the Nelson-Riley function

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mfrac»«mrow»«msup»«mi mathvariant=¨bold¨»cos«/mi»«mn»2«/mn»«/msup»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/mrow»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«msup»«mi mathvariant=¨bold¨»cos«/mi»«mn»2«/mn»«/msup»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/mrow»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/mfrac»«/mrow»«/mfenced»«/math»

an the extrapolation of the curves to cos2 = 0 will give true value of lattice parameter, i.e., ao 

 

By this process two different errors may arise:

(i)

Large systematic errors, small random errors 

(ii)

Small systematic errors, large random errors

 

Note:

 

(i) Systematic errors can be reduced by proper extrapolation function

 

(ii) Random errors can be reduced by Cohen's method (for further information about cohens method see Cullity text book)

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