Formula for Interplanar Distance in Cubic Lattice

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Discussion Overview

The discussion revolves around the formula for interplanar distance in cubic lattices, specifically addressing the relationship between the lattice parameter \( a \) and its reciprocal \( a^* \). Participants explore definitions, calculations, and the implications of using primitive versus non-primitive lattice vectors.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant presents the formula for interplanar distance \( d(h,k,l) = \frac{a}{\sqrt{h^2 + k^2 + l^2}} \) and expresses confusion about the transition to \( a^* \) and the relationship \( a^* = \frac{2\pi}{a} \) for cubic lattices.
  • Another participant suggests clarifying the definitions of \( a \) and \( a^* \), indicating that \( a \) refers to the lattice parameter of the elementary lattice while \( a^* \) pertains to the reciprocal lattice.
  • Further clarification is provided regarding the distinction between the lattice constant and the modulus of the lattice vectors, with one participant noting that the lattice constant is the physical dimension of unit cells.
  • A participant reflects on the importance of distinguishing between primitive and non-primitive lattice vectors, concluding that calculating reciprocal vectors directly from the body-centered lattice yields \( \frac{2\pi}{a} \) for all cubic lattices.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the definitions and relationships between \( a \) and \( a^* \). There is no consensus on the implications of using primitive versus non-primitive vectors, and the discussion remains unresolved regarding the clarity of these concepts.

Contextual Notes

Participants highlight potential confusion stemming from the terminology used, particularly the term "primitive," which affects the calculations of reciprocal vectors. The discussion reflects a need for clearer definitions and distinctions in the context of lattice theory.

potatowhisperer
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i am trying to find the formula for the inter-planar distance for the cubic .
i do know that it's :d (h,k,l)= a /√ (h² +k²+l²), i am only able to get to : 2π/(√a*²(h²+k²+l²)) , with a* being the parameter of the reciprocal lattice , the explanation given to how to go from a* to a , is that for all cubic lattices : a* = 2π/a , and this is what i don t understand , a = a* , only in the case of the simple cube , for body centered cube for example : we find a* = (2π/a)( j+k )with a*, j,k vectors ,a : parameter of the elementary lattice ; so calculating the modulus we find a*= √2 2π/a ;
and i am feeling frustrated , i know i am missing something but i don t know what .
 
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It looks like you need to clarify the definitions of a and a*.
 
a refers to the parameter of the elementary lattice , as a, b, c , of the simple cubic lattice .
lattice_parameters.gif

a* is the parameter of the reciprocal lattice , as in a* , b* , c* .
crystal-structure-analysis-46-638.jpg

a* , b* , c* are deduced from the parameters of the primitive lattice, a1 , a2 and a3 .
in the second pic a* is b1 , b*is b2 , c* is b3 , .
you can see that the modulus of a* = b 1 , is not 2π/a .
 
after a lot of searching , i noticed something , they do not actually mention the modulus of the vectors themselves but the lattice constant , i don t exactly understand what the difference is .
lattice constant is defined as the physical dimension of unit cells in a crystal lattice. so how is that different from the modulus of the lattice vector ?
 
i think i finally understood what is going on : you see i have always assumed that they were talking about the primitive vectors , for example to calculate the reciprocal vectors in the case of body centered we had to look for the primitive lattice ( which is a simple cube ) , but doing that means we re calculating the primitive reciprocal vectors . not just the reciprocal vectors .
and so if we actually try to calculate the reciprocal vectors of the actual body centered lattice,without going through the primitive lattice , we find 2π/a , and that is true for all cubic lattices .
uhhhh finally . all because of one word : primitive .
 

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