How to Find the Shortest Distance to the Brillouin Zone Boundary?

[Robbas]

Hi, I just can't understand the basics with BZ.

How do I find the shortest distance to the BZ boundary, how do I compare the electron energy between the last electron in the 1st BZ with the first electron in the 2nd BZ?

I think I need a visual how to calculate these things, does anyone know any good site with illustrations?

Here's an example:
Q: For what minimum electron concentration Z does the free electron Fermi sphere touch the first Brillouin zone boundary of a BCC metal?
A: Calculating the primitive reciprocal lattice vectors b_i of BCC we find the shortest distance to the BZ boundary |b_i|/2 = √2(π/a).

How do I know the shortest distance is |b_i|/2? From here I know how to finish.

If someone could show me some examples how to solve these types of questions I would be grateful!

Sorry for any grammatic errors, English is not my native language.

 

[Astronuc]

Perhaps this will help.

http://www.msm.cam.ac.uk/doitpoms//tlplib/brillouin_zones/zone_construction.php

 

[Robbas]

Thanks for your reply.

Actually I know how to draw the BZ in 2D-lattice, it's just like the WZ-cell.

How do I apply this to calculate the shortest distance to the 1st BZ for a BCC or FCC?
Is it always half of the reciprocal lattice vectors b_i? Or is that specific for a BCC?

Do I understand this correct:
For a BCC is the shortest dist to the 1st BZ .5*(2pi/a)*|0,1,1|=sqrt(2)*pi/a
and for FCC .5*(2pi/a)*|1,1,1|=sqrt(3)*pi/a?

 

[nickjer]

The shortest distance to the BZ doesn't necessarily mean it will have the lowest energy. Only if you make the assumption that the Fermi surface is a sphere (for free electrons), which isn't true when you have a potential.