PROBLEM SET 8

# PROBLEM SET 8 - XRD Family of planes in a 2-D lattice...

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MSE 110 - UA Family of planes in a 2-D lattice : d-spacings XRD It is also clear that each family of planes is separated by a specific d-spacing value. An example of 2-D lattice illustrates that many families of planes with different directions can be identified. •The collection of d- spacing (d 1 , d 2 , d 3 , d 4 , d 5 …) define entirely the structure.

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MSE 110 - UA Identifying families of planes (lines) in 2-D Miller Indices • Families of planes (lines in 2-D) are identified by a formalism based on where they intersect the unit cell edges. a b Intersects at 1 b, lies along a Intersects at 1 a, 1 b. Intersects t 1a, lies long b Intersects at ½ a, 1 b. Intersects at 1 a, ½ b
MSE 110 - UA X-Ray pattern: XRD BCC Fe λ = 2d·sin θ Using XRD we can measure a set of θ from the X-ray pattern and use Bragg’s equation to calculate the set of d corresponding to the crystal structure. We need to be able to identify the different atomic planes in the crystal in order to assign them the d-spacings and determine the structure. We can identify atomic planes based on their orientation in the unit cell using the Miller indices .

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MSE 110 - UA Structure determination: XRD •F o r a s e t o f d (hkl) there can only be one value of a with h, k, l being integers which satisfy the above equation for all d (hkl) . This analysis then provides the cell parameter a as well as the Miller indices for each planes. From Bragg’s law λ =2d·sin θ and the values of θ on the XRD pattern, we can extract a set of d values corresponding to planes with different orientation (hkl). 2 2 2 ) ( l k h a d hkl + + = In fact, the three are directly related by geometry and in a cubic system: For any crystalline lattice, the d-spacing depends on its orientation (hkl) and on the size of the cell parameter a .
MSE 110 - UA Miller Indices: rules To determine the Miller indices of a plane we take the following steps: MILLER INDICES ( ) 2 1 0 5. Cite the three integers in parenthesis placing bars over negative indices. 4. Reduce the reciprocals to the smallest integers having the same ratio. 3. Determine the reciprocal of these numbers.

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## This note was uploaded on 10/06/2009 for the course MSE 110 taught by Professor Lucas during the Spring '08 term at Arizona.

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PROBLEM SET 8 - XRD Family of planes in a 2-D lattice...

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