How are allowed and forbidden reflections calculated?

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SUMMARY

The discussion focuses on the calculation of allowed and forbidden reflections in cubic lattices (Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC)) based on Bragg's law. The rules established are that SC allows any combination of Miller indices (h, k, l), BCC requires that the sum of h, k, and l be even, and FCC requires that all indices be either odd or even. The calculations involve using the equations 2dsin(θ)=nγ and d=a/√(h²+k²+l²) to determine constructive or destructive interference for specific planes such as 001, 011, 111, 200, 210, and 211.

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  • Understanding of Bragg's law and its application in crystallography.
  • Familiarity with Miller indices and their significance in crystal planes.
  • Knowledge of cubic lattice structures: Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC).
  • Basic proficiency in mathematical calculations involving trigonometric functions.
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  • Study the derivation and application of Bragg's law in X-ray diffraction.
  • Learn how to calculate Miller indices for various crystal planes.
  • Explore the implications of constructive and destructive interference in crystallography.
  • Investigate the differences in reflection rules among various crystal structures beyond cubic lattices.
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Students and professionals in materials science, crystallography, and solid-state physics who are interested in understanding the principles of X-ray diffraction and the behavior of different cubic lattices under incident radiation.

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Question:
Explain the rules of reflection for different cubic lattices (SC, BCC, FCC)
based on the Bragg ́s law. Help: If we have incident of X-rays, which planes have constructive/ distractive reflection? Why? Please calculate it for the following crystal planes. 001, 011, 111, 200, 210, 211

Equations:
2dsin(θ)=nγ
d=a/√(h^2+k^2+l^2)
γ=lambda (cant find the symbol sorry)
∴
sin2(θ)=(γ2/4a2)*(h2+l2+k2)Attempt At Solution:
I know the rules. SC=any h,k,l BCC=h+k+l=even. FCC=h,k,l= all odd or even. And that it has to do with the satisfaction of Bragg's law that leads to constructive interference which leads to normal reflection. Although, I'm having trouble understanding exactly what to calculate. Any help would be greatly appreciated, If I could be pointed in the right direction or given an example for just one calculation that would be great.
 
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Perhaps "calculate" is a bad choice of wording. I believe the question is asking you to state whether each of the given planes for each of the lattices exhibits constructive or destructive interference and explain your reasons. The explanation part will require some calculation involving h, k and l.
 

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