Group of Wave Vector for k - Action of Space Group

In summary, the group of wave vector for a specific vector ##k## is defined as all the space group operations that leave ##k## invariant or turn it into ##k+K_m## where ##K_m## is a reciprocal vector. The translation operator ##\tau## acts on a wave vector by modifying its length, as the dimension of a wave vector is ##1/length##. This is different from the translation operator in physical space, as the space group symmetries can exist without having dimensions of length. To apply a translation operator in reciprocal space, a specific translation operator for reciprocal space must be used.
  • #1
hokhani
483
8
For a specific wave vector, ##k##, the group of wave vector is defined as all the space group operations that leave ##k## invariant or turn it into ##k+K_m## where ##K_m## is a reciprocal vector. How the translation parts of the space group, ##\tau##, can act on wave vector? Better to say, the dimension of a wave vector is ##1/length## while the translation operator acts on the lengths!
 
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  • #2
You are using the word space in space group as physical space. You can have space group symmetries without having dimensions of length.

Your question makes me wonder if you know what a wave vector looks like in reciprocal space and what it represents in a wave function.
 
  • #3
Dr_Nate said:
You are using the word space in space group as physical space. You can have space group symmetries without having dimensions of length.

Your question makes me wonder if you know what a wave vector looks like in reciprocal space and what it represents in a wave function.
Many thanks for your answer.
I don't know about that. Could you please help me with how a translation operator ##\tau## act on a wave vector ##k##?
 
  • #4
It sounds like you are trying to apply a spatial-translation operator to reciprocal space. There's more than one type of translation operator. For example, there is also a time-translation operator. Instead you want a translation operator specific to reciprocal space.
 
  • #5
Dr_Nate said:
It sounds like you are trying to apply a spatial-translation operator to reciprocal space. There's more than one type of translation operator. For example, there is also a time-translation operator. Instead you want a translation operator specific to reciprocal space.
My question still remains.
 
  • #6
It's real easy. You've pretty much gave it in your original post. Just write out the definition for a spatial-translation operation for a crystal and just substitute the appropriate variables that are in reciprocal space.
 

1. What is the Group of Wave Vector for k?

The Group of Wave Vector for k, also known as the wave vector space group, is a mathematical representation of the symmetry properties of a crystal lattice. It describes the allowed wave vectors for a given crystal structure and is used to analyze the diffraction patterns of X-rays or electrons from a crystal.

2. How is the Group of Wave Vector for k determined?

The Group of Wave Vector for k is determined by the symmetry elements of the crystal lattice, such as rotation, reflection, and translation. These symmetry elements are represented by mathematical operations and combined to form a group, which is then used to determine the allowed wave vectors for the crystal structure.

3. What is the significance of the Group of Wave Vector for k?

The Group of Wave Vector for k is important in crystallography as it allows scientists to predict and interpret the diffraction patterns of crystals. It also provides information about the symmetry properties of a crystal and can be used to determine the crystal structure and orientation.

4. How does the Group of Wave Vector for k relate to the Action of Space Group?

The Action of Space Group is a mathematical representation of the symmetry operations that relate different crystal structures. The Group of Wave Vector for k is a subgroup of the Action of Space Group, and together they provide a complete description of the symmetry properties of a crystal lattice.

5. Can the Group of Wave Vector for k change for different crystal structures?

Yes, the Group of Wave Vector for k can vary for different crystal structures. This is because the symmetry elements and operations of a crystal lattice can differ, leading to different allowed wave vectors. However, the Action of Space Group remains the same for related crystal structures.

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