Physics 141A Midterm March 19th 2009
¯h= 1.054 ×10−34 Js kB= 1.38 ×10−23 J/K NAvogadro = 6.022 ×1023 mol−1
1. (25 points) The crystal structure of barium titanate is shown below with the Ti, O
and Ba atoms occupying the body-center, face-center and corner positions of a cubic
cell, respectively. This so-called perovskite structure is very important in the field of
ferroelectricity. Let us denote the nearest neighbor Ba-Ba distance by d.
(a) Write down the explicit expressions (in terms of d) for a set of primitive translation
vectors a1,a2, and a3and for the corresponding set of primitive reciprocal lattice vectors
b1,b2, and b3.
(b) What is the shape and volume of the first Brillouin zone? If there are 1022 BaTiO3
units in the sample, how many allowed ~
k-states are there in the first Brillouin zone and
why?
(c) How many optical phonon branches are there for this material?
(d) Determine the structure factors for the first two sets of non-zero reciprocal lattice
vectors assuming that the atomic form factors are in the ratios 3:2:1 for Ba:Ti:O.
2. (25 points) In this problem no calculations are required. Consider a chain of atoms, all
with the same mass, but with two different spring constants (C1and C2) that alternate
along the chain. In answering each of the questions below illustrate your answer with
a sketch that includes at least four atoms (use X’s and O’s to represent the two atoms
in a unit cell). Sketch, using arrows near each of the four atoms, the normal mode
displacements for
(a) The zone-center acoustic mode;
(b) The zone-center optical mode;
(c) The zone-boundary acoustic mode;
(d) The zone-boundary optical mode.
