Problem 1 (20 points) Consider a two-dimensional lattice on the x-y plane. The lattice can be described by the primitive

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Problem 1 20 Points Consider A Two Dimensional Lattice On The X Y Plane The Lattice Can Be Described By The Primitive 1 (261.37 KiB) Viewed 32 times

Problem 1 (20 points) Consider a two-dimensional lattice on the x-y plane. The lattice can be described by the primitive vectors ā; = 2î and az = i + 2y. (a) Use a graph paper (you can use the one provided on the last page), dot out the lattice. Identify the Bravais lattice type. (b) Find the area of the primitive unit cell. (c) Using the given primitive vectors and introduce an additional a3 = ź, which is a unit vector perpendicular to the plane of the system. Construct the vectors ,,b2,bz. Use bı and by as the primitive vectors, dot out the reciprocal lattice on a separate graph paper. Identify the lattice type of the reciprocal lattice (it must belong to one of the 5 possible Bravais lattice types in 2D). (d) Find the "area" of the primitive cell of the reciprocal lattice. How is this area related to the area of the primitive cell in the direct lattice, i.e. the result in (b)? (e) Go back to the direct lattice in (a). Sketch a set of parallel crystal planes (try some less trivial ones). Identify the Miller's indices (hk) or (hko) of these planes. (f) On the graph of the reciprocal lattice in part (c), draw a reciprocal lattice vector Ĝ(hk) = h bị + k bị for the (hk) in part (e). (g) Don't need to write down anything for this part. Just do it privately. Put the graph paper with the direct lattice on top of the graph paper with the sketched reciprocal lattice (look at them through strong light if you are using hard copies of the graph paper), do you see that the reciprocal lattice vector G(hk) in part (f) is perpendicular to the set of planes you drew in part (e)? (h) Use the graph of the reciprocal lattice. Take a lattice point and construct the first Brillouin zone.