Calculating the Shortest Distance to the BZ Boundary for Electron Energy Comparison

[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.

 

[paranormal]

Hello,

To calculate the shortest distance to the BZ boundary, you will need to first understand the concept of reciprocal lattice vectors. These vectors represent the periodicity of the crystal structure in reciprocal space, which is the Fourier transform of the real space. In simple terms, they represent the spacing between planes of atoms in the crystal.

To find the shortest distance to the BZ boundary, you will need to calculate the magnitude of the reciprocal lattice vectors (|b_i|) and divide it by 2. This will give you the distance from the origin to the BZ boundary. This distance is important because it determines the energy of the electrons at the boundary.

To compare the electron energy between the last electron in the 1st BZ and the first electron in the 2nd BZ, you will need to calculate the energy at the BZ boundary using the shortest distance you calculated earlier. This energy can be compared to the energy of the last electron in the 1st BZ to determine the minimum electron concentration Z.

There are many resources available online with illustrations and examples of how to calculate these quantities. Some good sites to check are those of universities or research institutions that specialize in solid state physics or materials science.

I hope this helps and good luck with your calculations!