A two-dimensional rectangular lattice has mutually perpendicular primitive vectors of…

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction · Crystal structure, bonding and diffraction

Condensed Matter Physics, Ch The reciprocal lattice and the Brillouin zone

Problem

A two-dimensional rectangular lattice has mutually perpendicular primitive vectors of length

a_1=3.00\,\mathrm{\mathring{A}}

along

x

and

a_2=4.50\,\mathrm{\mathring{A}}

along

y

. ((a)) Find the first Brillouin zone: the centre-to-face distances along

k_x

and

k_y

, and the centre-to-corner distance. ((b)) The zone's four nearest reciprocal lattice points sit on the axes, at

(\pm2\pi/a_1,0)

and

(0,\pm2\pi/a_2)

; the next-nearest sit on the diagonals, at

(\pm2\pi/a_1,\pm2\pi/a_2)

. Show, without assuming any particular ratio of

a_1

a_2

, that the perpendicular bisector of a diagonal point never cuts into the rectangle found in part (a): it always passes exactly through a corner.

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