Deriving the Relationship of Cubic Crystal Structure

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SUMMARY

The relationship for a cubic crystal structure is derived using the formula d = a / √(h² + k² + l²), where 'd' is the interplanar spacing, 'a' is the lattice parameter, and (h, k, l) are the Miller indices. The discussion highlights the challenge of understanding the non-distributive nature of reciprocals in relation to Miller indices. Additionally, the query regarding the trigonometric identity that states the sum of the squares of the cosines equals one is addressed, emphasizing the need for a clearer understanding of trigonometric principles in solid-state physics.

PREREQUISITES
  • Understanding of cubic crystal structures
  • Familiarity with Miller indices
  • Basic knowledge of trigonometric identities
  • Solid-state physics concepts
NEXT STEPS
  • Study the derivation of Miller indices in crystallography
  • Learn about trigonometric identities and their applications in physics
  • Explore solid-state physics textbooks for deeper insights into crystal structures
  • Investigate the mathematical properties of reciprocals in crystallography
USEFUL FOR

Students and professionals in materials science, solid-state physicists, and anyone interested in crystallography and the mathematical foundations of crystal structures.

alexgmcm
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How can one derive the relationship for a cubic crystal:
d= \frac{a}{\sqrt{h^2 + k^2 + l^2}}

It is shown here.

This is probably trivial but I am having trouble proving it as the Miller indices are reciprocals but the reciprocal is not distributive. This isn't for homework by the way - I am just going over solid state stuff from last year trying to get a more solid understanding than I got when I just learned it by focusing on the then impending exams.

Any help would be greatly appreciated. :)
 
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