Ewald sphere vs Brillouin Zone

In summary, the Ewald sphere and Brillouin Zone are both representations of the condition that k'=k+G in a periodic potential. However, they differ in their visual representation and the information they convey. The Brillouin zone is a volume in reciprocal space that characterizes the crystal structure, while the Ewald sphere is a surface whose size is determined by the wavelength of diffracting radiation. Both constructs illustrate the general concept of momentum conservation in a periodic potential.
  • #1
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Is it correct to say that the Ewald sphere and Brillouin Zone are both representations of k'=k+G?

I'm comfortable with the construction of the Ewald sphere, but don't quite see how a BZ represents k'=k+G.

Can anyone explain how the construction of a BZ represents k'=k+G and whether it provides any information that one can't represent with an Ewald sphere?


Thanks
 
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  • #2
I am not quite sure if I understand your question. The Ewald sphere E is a sphere in the reciprocal lattice around the point k with radius |k|, i.e. all vectors q=k'-k with |k'|=|k|. Obviously, it is possible to fold back the Ewald sphere into the first Brillouin zone. The vectors on q of E which coincide with a lattice vector G are mapped onto the origin 0 of the Brillouin zone.
 
  • #3
What I mean to ask is whether the Ewald sphere and BZ both represent the condition that k'=k+G.

If so, why do they look different?

Do they convey different information?
 
  • #4
The Brillouin zone depends only on the crystal structure, and contains the minimum number of reciprocal space points which are required to fully characterize the system's behavior in momentum space. (That may not be the most clear way to state it). The Brillouin zone represents a volume in reciprocal space.

The Ewald sphere is only a surface in reciprocal space, and the interior volume does not have much significance to the construction. Its size depends only on the wavelength of the diffracting radiation (ie x-rays).

k'=k+G is a general statement of momentum conservation in a periodic potential, and as such it is going to show up in either construct. I don't consider either construct to simply be an illustration of k'=k+G.
 

1. What is the difference between the Ewald sphere and Brillouin Zone?

The Ewald sphere and Brillouin Zone are both concepts used in the study of crystal structures and diffraction patterns. The main difference between them is that the Ewald sphere is a mathematical construct used to visualize the scattering of X-rays by a crystal, while the Brillouin Zone is a geometric representation of the possible electron energy levels in a crystal lattice.

2. How are the Ewald sphere and Brillouin Zone related?

The Ewald sphere and Brillouin Zone are related in that they both provide important information about the structure and properties of crystals. The Ewald sphere helps to determine the diffraction pattern produced by a crystal, while the Brillouin Zone helps to understand the electronic properties and behavior of the crystal.

3. What is the purpose of using the Ewald sphere and Brillouin Zone in crystallography?

The Ewald sphere and Brillouin Zone are essential tools in crystallography as they help to analyze and interpret diffraction patterns and electronic properties of crystals. They provide valuable information about the atomic arrangement and symmetry of a crystal, which is crucial in understanding its physical and chemical properties.

4. Can the Ewald sphere and Brillouin Zone be used for any type of crystal?

Yes, the Ewald sphere and Brillouin Zone can be used for any type of crystal as long as it has a well-defined crystal structure. They are fundamental concepts in crystallography and are applicable to all types of crystals, including metals, minerals, and biological crystals.

5. How does the Ewald sphere and Brillouin Zone help in the determination of crystal structure?

The Ewald sphere and Brillouin Zone provide important information about the symmetry, diffraction pattern, and electronic properties of a crystal. This information can be used to determine the crystal structure by comparing it to known crystal structures and using techniques such as X-ray diffraction and electron diffraction. The Ewald sphere and Brillouin Zone serve as powerful tools in the process of crystal structure determination.

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