Question

All parts

(1) Lattice terminology. The structure on the next page displays a portion of a lattice. (As you know, a lattice is infinite in extent.) The blue points represent the lattice points. These points can all be written as Rw = u a + vb, with 1 1 a= etzez and e, and e, are the unit vectors in the x and y directions, respectively. The vectors a and bare known as primitive lattice vectors, because from them, you can generate every lattice point in the structure. In the plot, I show the unit vectors drawn to scale. • [5 points) Sketch the primitive lattice vectors on the plot. • [5 points) The primitive lattice vectors are not unique. To prove this, identify another set of primitive lattice vectors that also describe the same lattice. • [ 10 points) Construct the Wigner-Seitz primitive cell for this lattice. You can do this by hand if you like - sketching the procedure on the figure. •[ 5 points) Identify a lattice point as 00. Then identify the 23 direction for each set of primitive lattice vectors. • [5 points) Indicate the 31 point for both sets of primitive lattice vectors.

Answer 1

A) the primitive lattice vectors a and b are drawn in the figure.

1/3 e2 b=0.707e1 -1/2 el

B) The vectors a'=1/4 e₁ +1/3 e₂ and b'= 1/sqrt(2) e₁ can also be used as a primitive vector. as shown in the figure

C) The Wigner - Seitz cell has been constructed in figure below.

It is the area enclosed by the red lines.

D) Taking the point O as the origin, OA is the direction [2 $\overline 3$] for first primitive vector and OA' is for second primitive vector