Converting Between K-Space Sum and Integral: A Guide for Macroscopic Solids"

In summary, the conversation discussed the conversion between a k-space sum and an integral in the context of periodic boundary conditions. The number of states with a wavevector less than k0 can be calculated by multiplying the volume of a cube with radius k0, which can be translated into an integral. Alternatively, the sum can be written as an integral over a sum of delta functions, which can be replaced by their density for slowly varying test functions.
  • #1
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How is it exactly i convert between a k-space sum an integral?
Assume that we have some macroscopic solid. Periodic boundary conditions leads to kx,ky,kz = 2π/L, so each k-space state fills a volume (2π/L)3 or has a density of V/(2π)3. To then count for instance the number of state with wavevector k<k0, what do you then do?
Intuitively I would multiply the volume of a cube of radius k0, but how does this translate into an integral exactly?
 
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  • #2
##∑_k=\frac{V}{(2\pi)^3}∫d^3k##
 
  • #3
You can also write the sum as an integral over a sum of delta functions.
For slowly varying test functions, the delta functions may then be replaced by their density ##V/(2\pi)^3##.
 

1. What is K-space sum and integral?

K-space sum and integral are mathematical techniques used to describe the behavior of macroscopic solids, such as metals or crystals. They involve summing or integrating over all the possible energy states of the solid to determine its overall properties.

2. Why is converting between K-space sum and integral important?

Converting between K-space sum and integral allows scientists to better understand the behavior of macroscopic solids and predict their properties. It also helps to bridge the gap between microscopic and macroscopic descriptions of solids.

3. What are the key differences between K-space sum and integral?

K-space sum involves summing over all the possible energy states of a solid, while integral involves integrating over all the possible energy values. Additionally, K-space sum is often used to describe the behavior of crystalline solids, while integral is used for amorphous solids.

4. How can I convert between K-space sum and integral?

The conversion between K-space sum and integral can be done using mathematical equations and techniques specific to each method. It is important to understand the underlying principles and assumptions of each method in order to accurately convert between them.

5. What are the practical applications of converting between K-space sum and integral?

The conversion between K-space sum and integral has many practical applications in materials science and engineering. It can be used to predict the properties of new materials, optimize manufacturing processes, and understand the behavior of existing materials in different environments.

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