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**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.

- 1.
- (a)
- 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. - (b)
- 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, and why? - (c)
- 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.
- (d)
- 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. - 2.
- (a)
- 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.
- (b)
- 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 - (c)
- For a two dimensional square lattice, show that the first four terms
in
*A*are_{6} - 3.
- (a)
- Quantum mechanics tells us that the energy of an electron of mass
*m*in a cubic box of side*L*is given by: - (b)
- 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. - (c)
- 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*,*k*+*dk*, is - 4.
- (a)
- 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.
- (b)
- Atoms
*A*and*B*each carry a single 1*s*electron with an energy and 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 wavefunction as 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. - (c)
- 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 and ? - 5.
- (a)
- The resultant amplitude
*A*scattered by an assembly of*n*atoms placed at vector positions is given by - (b)
- 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.
- (c)
- 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.

PHYS3C25/1997 **PLEASE TURN OVER**

where *N _{A}* is Avogadro's number,

Show also that the equilibrium nearest neighbour separation is

4 + 0.500 + 0.062 + 0.064

PHYS3C25/1997 **CONTINUED** truecm

where *n _{x}*,

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

*E _{1}* = 3

By now allowing each electron to move freely throughout the whole 8-atom metal, show that the total energy of the system is now

*E _{2}* = 0.44

Suggest how your calculation is relevant to the free electron theory of metals.

Hence, knowing the relationship between *E* and *k* for an
electron, show that the density of states

Consider a piece of metal of volume 1*cm ^{3}*. How many
quantum states are there with energy in the range 1 to 1.001

PHYS3C25/1997 **PLEASE TURN OVER**

(Consider only the following set of matrix elements: , , , , , with both and positive).

PHYS3C25/1997 **CONTINUED**

(N.B. The following question deals with material some of which is now dropped from the syllabus).

where
is the scattering vector and
the atomic scattering factor of the *n*th atom. Explain with the aid
of the scattering diagram how
relates to the scattering geometry, and show that ,
where
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.

PHYS3C25/1997 **END OF PAPER**