Unit Cell & Reciprocal Lattice of a 2D Oblique Lattice

[Howlin]

Homework Statement

a 2d oblique lattice is described by the following vectors:
a = 3i
b = 0.5i + j
sketch the unit cell of this lattice

Sketch accurately the reciprocal lattice of the cell and justify the direction and relative magnitudes of each reciprocal lattice vector

The Attempt at a Solution

Isnt the unit cell and reciprocal lattice be the same?
for the unit cell you plot the points and then plot a point out from 3i up 1j and over 0.5i

Is this right?

 

[nate808]

Thank you for your post. it is my pleasure to assist you with this problem.

Firstly, let's clarify the difference between a unit cell and a reciprocal lattice. A unit cell is the smallest repeating unit of a crystal lattice, while a reciprocal lattice is a mathematical construct used to describe the periodic arrangement of points in a crystal lattice in terms of their reciprocal vectors.

In the case of a 2D oblique lattice, the unit cell and reciprocal lattice are not the same. The unit cell is defined by the vectors a and b, while the reciprocal lattice is defined by the reciprocal vectors a* and b*.

To sketch the unit cell accurately, you can follow the steps outlined in your attempt. First, plot the points of the lattice using the vectors a and b. Then, extend the vector a by 1 unit in the direction of vector b and draw a line from the endpoint of a to the endpoint of this extended vector. This will give you the unit cell of the lattice.

To sketch the reciprocal lattice, you can follow the same steps. Plot the points of the reciprocal lattice using the reciprocal vectors a* and b*. Then, extend the vector a* by 1 unit in the direction of vector b* and draw a line from the endpoint of a* to the endpoint of this extended vector. This will give you the reciprocal lattice of the original lattice.

The direction and relative magnitudes of each reciprocal lattice vector are determined by the underlying symmetry of the original lattice. In this case, the reciprocal lattice vectors will be perpendicular to the corresponding unit cell vectors and their magnitudes will be inversely proportional to the lengths of the unit cell vectors. You can refer to the concept of Bragg's law for more information on this relationship.

I hope this helps. Let me know if you have any further questions.