What Is the Shortest Reciprocal Vector for a BCC Lattice?

[malawi_glenn]

Homework Statement

Find the shortest reciprocal vector G, given below, v_1,...,v_3 are integers.

$$\vec{G} = \frac{2 \pi}{a}\left( (v_2 + v_3 )\vec{x} + (v_1 + v_3 )\vec{y} + (v_1 + v_2 )\vec{z} \right)$$

Homework Equations

x,y,z are ortonogal, length 1

$$l = l(v_1, v_2, v_3) = \vert \vec{G} \vert = \sqrt{\vec{G}\cdot \vec{G}}$$

$$l = \sqrt{ (v_2 + v_3 )^{2} + (v_1 + v_3 )^2 +(v_1 + v_2 )^2 }$$

The Attempt at a Solution

I want to minimize l(v_1, v_2, v_3)

$$\dfrac{\partial l}{\partial v_1} = \dfrac{2 \pi \left( (v_1 + v_3 ) + (v_1 + v_2 ) \right) }{\sqrt{ (v_2 + v_3 )^{2} + (v_1 + v_3 )^2 +(v_1 + v_2 )^2 }} = 0$$

etc. Gives me following linear equation system, it has only trivial solutions

$$\left( \begin{array}{ccc|c} 2 & 1 & 1 & 0 \\ 1 & 2 & 1 & 0 \\ 1 & 1 &2 & 0 \end{array}\right)$$

v_1 = v_2 = v_3 = 0

And that is not true, they should be something like

$$\frac{2 \pi}{a} \left( \pm \vec{x} \pm \vec{y} \right)$$

etc.


Now what have I do wrong

by the way, this is the general reciprocal lattice vetctor for bcc lattice. I want to construct the first Brillouion zone.

 

[anathema]

Thank you for your question. It seems like you are on the right track with your attempt at a solution. However, there are a few things that could be adjusted in your approach.

Firstly, it is important to note that when taking the partial derivative with respect to one of the variables (in this case, v1, v2, or v3), the other variables should be treated as constants. So your partial derivative for v1 should be:

\frac{\partial l}{\partial v_1} = \frac{2 \pi (v_1 + v_3 + v_1 + v_2)}{\sqrt{(v_2 + v_3)^2 + (v_1 + v_3)^2 + (v_1 + v_2)^2}} = 0

This simplifies to:

2\pi(v_1 + v_2 + v_3) = 0

Similarly, you can find the partial derivatives for v2 and v3 and set them equal to 0. This will give you a system of linear equations that you can solve to find the values of v1, v2, and v3.

Secondly, when solving for the values of v1, v2, and v3, it is important to remember that they are integers. This means that the solutions to the linear equations will not be exact values, but rather whole numbers. For example, if you get a solution of v1 = 2.5, this should be rounded to v1 = 3.

Lastly, it is important to keep in mind the given constraints for x, y, and z being orthogonal and of length 1. This means that the values of v1, v2, and v3 should not be too large, as this would result in vectors with lengths greater than 1. You may need to try different combinations of solutions to find the shortest reciprocal vector that satisfies the given constraints.

I hope this helps. Good luck with your calculations and constructing the first Brillouin zone for the bcc lattice!

 

[m2003]

Thank you for sharing your attempt at finding the shortest reciprocal vector for the given equation. It seems like you have correctly set up the equation and attempted to minimize it by taking the partial derivative with respect to each variable. However, the resulting linear equation system does not have any non-trivial solutions, which means that the only solution is when v1 = v2 = v3 = 0. This is not the expected result, as you have correctly pointed out.

One possible explanation for this discrepancy could be that you have made a mistake in your calculations. I would recommend double-checking your steps and equations to make sure they are correct. Another possibility is that the given equation may not be the most appropriate one for finding the shortest reciprocal vector for a bcc lattice. It is always important to carefully consider the assumptions and limitations of any mathematical model or equation before using it to solve a problem.

In order to construct the first Brillouin zone, you may need to consider other equations or methods that are specifically designed for this purpose. It may also be helpful to consult with other experts or references in the field of crystallography to ensure that you are using the most appropriate approach. Overall, it is important to carefully and critically analyze any results or solutions in order to ensure their accuracy and validity.