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Homework Help: Brillouin zone

  1. May 27, 2010 #1
    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.
  2. jcsd
  3. May 27, 2010 #2


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    Staff Emeritus
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    Perhaps this will help.

    http://www.msm.cam.ac.uk/doitpoms//tlplib/brillouin_zones/zone_construction.php [Broken]
    Last edited by a moderator: May 4, 2017
  4. May 27, 2010 #3
    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?
  5. May 28, 2010 #4
    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.
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