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 aBrillouin zone is equal to the volume of the primitive parallelepiped in Fourier space.Recall the vector identity (×ā) × (axb) = c.(axb)ā.

let the volume of the parallelopiped primitive cell : VC=a⇀.(b⇀×c)⇀=b⇀.(c⇀×a⇀)=c⇀.(a⇀×b⇀) The…

Answered: 3. Volume of Brillouin zone. Show that…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

See similar textbooks

Related questions

Concept explainers

Question

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 :

$V_{C} = \overset{⇀}{a} . (\overset{⇀}{b} \times \overset{⇀}{c)} = \overset{⇀}{b} . (\overset{⇀}{c} \times \overset{⇀}{a}) = \overset{⇀}{c} . (\overset{⇀}{a} \times \overset{⇀}{b})$

The reciprocal lattice primitive translation vectors are :

$\overset{⇀}{A} = 2 π . \frac{(\overset{⇀}{b} \times \overset{⇀}{c)}}{a . (\overset{⇀}{b} \times \overset{⇀}{c)}} \overset{⇀}{A} = \frac{2 π}{V_{c}} (\overset{⇀}{b} \times \overset{⇀}{c)} \overset{⇀}{B} = \frac{2 π}{V_{c}} (\overset{⇀}{c} \times \overset{⇀}{a)} \overset{⇀}{C} = \frac{2 π}{V_{c}} (\overset{⇀}{a} \times \overset{⇀}{b)}$

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

Similar questions