Find the translation vectors of the primitive cell

[Anisotrop]

Homework Statement

Find the translation vectors of the primitive cell. The figures are provided.

Homework Equations

The Attempt at a Solution

I found these are more difficult than what I am learning from Kittel. I did number 1, but not sure if it is correct. Answer or hint for any of the figures is appreciated.

 

[ehild]

1. The figures are provided



2. Answer or hint for any of the figures is appreciated.
View attachment 55827


3. I did number 1 but not sure if it is correct

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Show you work. How did you find the translation vectors in case 1?

ehild

 

[Anisotrop]

Here is the one 1 I did for number 1.
I also need to find the location of basis and generate the structure.

 

[Anisotrop]

I think my vector a1= sqrt(3)*a in x-direction...

 

[ehild]

I think my vector a1= sqrt(3)*a in x-direction...

Yes, but a2 was not correct
The picture shows the primitive cells and its content, the basis, repeated in the crystal. The basis is not necessarily a single atom. It can be two as in the first figure, or it can be a complicated molecule.

ehild

 

[Anisotrop]

Thanks for response! Could you please tell me what I need to change in the a2? Or should I take a different reference point (origin).

 

[ehild]

What are the vectors making the sides of the cell shown in the picture?

If you take the vector a2 you have shown, you would get atoms even at the empty places at the centres of the hexagons.

ehild

 

[Anisotrop]

Thank you Ehild for your great help. I have changed my origin to the vertex of the Rhombus provided. I will upload it once I am done with it. Thanks again! :)

Anisotrop

 

[ehild]

Find the smallest shape you can use to "tile" the whole plane. In the first figure, you can cover the plane with the hexagons. You can put the vertex of the primitive cell at the centre of the hexagons, and the length of the primitive translation vectors are equal to the distance between the centres. The primitive translation vectors determine the primitive cell. The cell contains the basis, the group of atoms, that are repeated in the crystal.

ehild