How to show all Brillouin zones have same volume

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SUMMARY

The volume of the first Brillouin zone is calculated as (2π)³/V, where V represents the volume of the unit cell in the real lattice. The discussion highlights a common assertion that all Brillouin zones possess the same volume, yet lacks a definitive proof. The user initially attempted to verify this in 2D but encountered discrepancies, particularly with the third Brillouin zone, which appeared to have double the volume of the first two. Clarification on the calculation method is sought to resolve these inconsistencies.

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aleeds
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It is quite easy to calculate the volume of the first brillouin zone to be (2pi)^3/V if V is the volume of a unit cell in the real lattice. In many places one can also find the statement that all Brillouin zones have the same volume. I have not however found a proof of this anywhere. The 3D case seemed complicated so I first tried solving hte problem for the 2D case but must have done something wrong as I found that the first and second Brillouin zones have the same volume (or area in the 2D case) but then for the 3rd Brillouin zone I foudn it to be twice of this. Does anybody know a solution to this (either in 2D or even better in 3D)?
Thanks
 
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By its definition the volume of the reciprocal cell is 1/V, where V is the direct cell volume. V=(a.[bc])
 
Sorry, it comes from the definition a*=[b x c]/V, b*= [c x a]/V, ...
 
aleeds said:
It is quite easy to calculate the volume of the first brillouin zone to be (2pi)^3/V if V is the volume of a unit cell in the real lattice. In many places one can also find the statement that all Brillouin zones have the same volume. I have not however found a proof of this anywhere. The 3D case seemed complicated so I first tried solving hte problem for the 2D case but must have done something wrong as I found that the first and second Brillouin zones have the same volume (or area in the 2D case) but then for the 3rd Brillouin zone I foudn it to be twice of this. Does anybody know a solution to this (either in 2D or even better in 3D)?
Thanks
It may be that you include areas which does not belong to the third zone.
How did you do the calculation?
 

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