Formula for Interplanar Distance in Cubic Lattice

[potatowhisperer]

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 .

 

[mathman]

It looks like you need to clarify the definitions of a and a*.

 

[potatowhisperer]

a refers to the parameter of the elementary lattice , as a, b, c , of the simple cubic lattice .

a* is the parameter of the reciprocal lattice , as in a* , b* , c* .

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 .

 

[potatowhisperer]

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 ?

 

[potatowhisperer]

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 .