Q 15 state blochs theorem and show that the wave

q .
1.5 State Bloch’s theorem and show that the wave function
ψ
k
p r q “
ÿ
R
e
i k ¨ R
φ p r ´ R q
(where R are the points of the lattice and φ p r ´ R q is the ground-state
wave function of an electron bound to the nucleus on site R ) is of the
form required by Bloch’s theorem.
[7 marks]
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1.6 Consider barium titanate (BaTiO
3
), which is a ferroelectric material. At
high temperature it has a simple cubic Bravais lattice, where Ba atoms
lie at the cube corners, Ti atoms at the cube centres and O atoms at the
centres of the cube faces. Sketch the unit cell and indicate the number
of atoms in the basis. How many acoustic and optical branches in the
phonon dispersion curves for barium titanate would you expect?
[6 marks]
1.7 The figure below shows the measured phonon dispersion data for the
fcc metal nickel.
(a) The left hand panel shows dispersion from the centre of the zone to
the point (001), commonly designated by the X point. Why are there
two sets of data in this panel? What is indicated by the abbreviated
designations “L” and “T”? Are either of these curves degenerate?
(b) Why does the curve labelled “T” fall below the “L” curve?
(c) Make an estimate of the ratio of the sound speeds associated with
the “T” and “L” modes.
(d) In the second panel from the right, connecting (000) to (011), there
are three curves. Why is this?
(e) Phonon frequencies at (001) and (011) appear to be the same. Are
(001) and (011) equivalent points in the Brillouin zone?
7

[10 marks]