3. Volume of Brillouin zone. Show that the volume of the first Brillouin zone is (27)³/V where V is the volume of a crystal primitive cell. Hint: The volume of a Brillouin zone is equal to the volume of the primitive parallelepiped in Fourier space. Recall the vector identity (cxä)×(axb) = č. (axb)ā.

Elements Of Electromagnetics
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Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
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3. Volume of Brillouin zone. Show that the volume of the first Brillouin zone is
(27)³/V where Ve is the volume of a crystal primitive cell. Hint: The volume of a
Brillouin zone is equal to the volume of the primitive parallelepiped in Fourier space.
Recall the vector identity (×ā) × (axb) = c.(axb)ā.
Transcribed Image Text:3. Volume of Brillouin zone. Show that the volume of the first Brillouin zone is (27)³/V where Ve is the volume of a crystal primitive cell. Hint: The volume of a Brillouin zone is equal to the volume of the primitive parallelepiped in Fourier space. Recall the vector identity (×ā) × (axb) = c.(axb)ā.
Expert Solution
Step 1

let the volume of the parallelopiped primitive cell :

VC=a.(b×c)=b.(c×a)=c.(a×b)

The reciprocal lattice primitive  translation vectors are :

A=2π.(b×c)a.(b×c)A=2πVc(b×c)B=2πVc(c×a)C=2πVc(a×b)

 

 

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