Answer any THREE QUESTIONS The numbers in square brackets in
the right-hand margin indicate the provisional allocation of maximum marks
per sub-section of a question.
Explain what is meant by the terms lattice and basis
as applied to the description of crystal structures. Using as an example
the structure of caesium chloride, or another appropriate substance of
your choice, indicate on a diagram the presence of both lattice points
and the basis.
A simple atomic substance is found to exist in three different forms
at different pressures, namely simple cubic, body centred cubic, and face
centred cubic. Letting the radius of the spherical atom in each case be
r, calculate the unit cell parameter in the case of each structure
in terms of r. Hence, demonstrate that the fraction of empty space
in each structure is in the ratio
Which structure would you expect to be that observed at the highest pressure,
Explain how the Miller index hkl can be used to describe planes in
a crystal. Sketch the 110 plane of each of the three cubic structures mentioned
in (b) above, showing in each case the atoms in the structures that have
their centres on the respective 110 planes. Mark clearly on each diagram
the points of contact between the spherical atoms. Sketch also the 111
plane of the face centred cubic structure, again showing the atoms with
their centres in this plane and their points of contact.
Draw the unit cell of the NaCl structure, and the corresponding
unit cell projection. What is the lattice type and why? How many nearest
neighbours does each sodium ion have, and what is the sodium ion's co-ordination
polyhedron? In the NaCl structure, show, for the case in which the
nearest neighbour larger anions and cations just touch, the cation/anion
radius ratio is 0.414.
PHYS3C25/1997 PLEASE TURN OVER
Crystalline solids are usually considered in terms of four idealised
categories. Yet in all these four kinds of crystal, it is the electrostatic
Coulomb interaction that provides the attractive forces that produce the
observed diversity of solid structures. By considering each of the four
types of idealised crystalline solids in turn, and by paying particular
attention to the outer valence electrons, discuss how the Coulomb force
is operating in each case.
Explain what is meant by the terms cohesive energy and lattice
sum. For a Lennard-Jones system, scale parameter ,
well depth ,
show that the molar cohesive energy U is given by
where NA is Avogadro's number, r0
the nearest neighbour distance, and A12 and A6
the lattice sums for the repulsive and attractive contributions to the
Show also that the equilibrium nearest neighbour separation is
For a two dimensional square lattice, show that the first four terms
in A6 are
4 + 0.500 + 0.062 + 0.064
PHYS3C25/1997 CONTINUED truecm
Quantum mechanics tells us that the energy of an electron of mass m
in a cubic box of side L is given by:
where nx, ny, nz
are quantum numbers that are allowed to take only integer values.
Consider an 8 atom metal in which each atom is confined in a cubic box
of side a, with the whole system itself being confined in a (obviously
larger) cubic box. One valence electron can be detached from its parent
atom. If each electron is confined within the box occupied by its
parent atom, show that the total energy of the 8 atom system is
E1 = 3h2/ma2
By now allowing each electron to move freely throughout the whole 8-atom
metal, show that the total energy of the system is now
E2 = 0.44E1
Suggest how your calculation is relevant to the free electron theory
For a finite sized metal, discuss how the free electron model leads
to the concept of the Fermi surface. By taking as an example one
physical property of your choice (e.g. electrical conductivity, thermoelectric
effects, Pauli paramagnetism, etc), show how the concept of the Fermi surface
is useful in explaining the observed behaviour.
Each quantum state occupies a volume
in k space, where V is the volume of the metal. Show that
the number of quantum states with k values in the range k,
Hence, knowing the relationship between E and k for an
electron, show that the density of states
Consider a piece of metal of volume 1cm3. How many
quantum states are there with energy in the range 1 to 1.001eV?
PHYS3C25/1997 PLEASE TURN OVER
Explain what is meant by electronic bands and band gaps in solids.
Under what circumstances can band gaps have a major effect on the properties
of a solid? Discuss, using appropriate examples, how the electrical and
optical properties of materials can depend on electronic band gaps.
Atoms A and B each carry a single 1s electron
with an energy
respectively when isolated. A diatomic molecule is formed by bringing atoms
A and B together. The Hamiltonian for single, mutually non-interacting
electrons in the molecule is H. Representing the single electron
a linear combination of the atomic orbitals
on atom A and
on atom B, solve Schrödinger's equation
to calculate the energies E available to electrons in the molecule.
(Consider only the following set of matrix elements: ,
In an infinite linear chain of A and B atoms ()
with equal spacings R between each atom, the energies of electrons
in the system are given by ,where
k is the wavevector of the electron state. What is the band gap
in the electronic band structure for this system? How would you expect
the electrical and optical properties of this structure to depend on
(N.B. The following question deals with material some of which is now
dropped from the syllabus).
The resultant amplitude A scattered by an assembly of n
atoms placed at vector positions
is given by
is the scattering vector and
the atomic scattering factor of the nth atom. Explain with the aid
of the scattering diagram how
relates to the scattering geometry, and show that ,
is the scattering angle and
the wavelength of the radiation.
For the case of scattering of x-rays, sketch the form of
for a typical atom.
Explain (i) its value at ,
and (ii) the physical reasons for the form of the curve.
By considering the scattering from a crystalline arrangement of atoms,
show how the scattering from a crystal can be considered as the product
of two terms, the first relating to the lattice, the second to the scattering
from the unit cell. Indicate how the lattice term behaves as the number
of unit cells in the crystal increases, and suggest how this might be used
to estimate crystallite size from the diffraction pattern.
Explain how systematic absences in the scattering from a crystal can
arise from the symmetry of the arrangement of atoms within a unit cell.
Starting from the expression for the structure factor show for a two dimensional
centred rectangular lattice that those `reflections' for which h
+ k is odd are absent.