Question

A1 The primitive lattice vectors of a 2-dimensional lattice where and are unit vectors along the horizontal and vertical, respectively. Sketch this lattice, including on your sketch the lattice points, primitive vectors, and primitive unit cell ax and p2 (a/2)( are p Write down an expression for the conventional lattice vectors, and state the number of lattice points in a conventional unit cell Draw conventional lattice vectors on your sketch 6 Marks A2 m) atoms at each site, the dispersion relation includes an acoustic and a photonic branch. Sketch this dispersion relation, marking clearly which branch is which, the edge of the 1st Brillouin [4 Marks 1-dimensional lattice with alternating light (mass = m) and heavy (mass = M > n In a Zone, and indicating any forbidden frequency ranges.

Answer 1

A unit cell C2 C1 CONVENTIONAL UNIT EELL Pi= ax a x (h +x] x 4 4 Number of latico points indicates la thce poinb Each = 2 /cal Cconvenhanal

fore constant di'stanu between A2) dis persio Ye laton is The nearest idemh'cal atom SIOn kaC ( Long wavelength imit) M 7 m (4 Cacoustic branch Cophical bramch) S 2C 2 M L c k a Ka-Th bYanch w Coptical 2 C 6yamch + 2C .coushc OL M

0 1 1 2c m m Optical branch 2c 2c m m 2c M 2c M Acoustic branch: k a