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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
$(sc:bcc:fcc)\, 1.85:1.23:1$. 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.

PHYS3C25/1997 PLEASE TURN OVER

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 $\sigma$, well depth $\epsilon$, show that the molar cohesive energy U is given by

\begin{displaymath}U = 2\epsilon N_{A} \left[ A_{12} \left( \frac{\sigma}{r_{0}... ...^{12} - A_{6} \left( \frac{\sigma}{r_{0}} \right) ^{6} \right] \end{displaymath}

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 energy.

Show also that the equilibrium nearest neighbour separation is

\begin{displaymath}r_{0} = \sigma \left[ \frac{2A_{12}}{A_{6}} \right] ^{1/6} \end{displaymath}

(c)
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

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:

\begin{displaymath}E(n_{x}, n_{y}, n_{z}) = \frac {h^{2}}{8mL^{2}}[ n_{x}^{2} + n_{y}^{2} + n_{z}^{2}] \end{displaymath}

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 of metals.

(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 $\Delta \Omega = 4\pi ^{3}/V$ 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

\begin{displaymath}dN = \frac{Vk^{2}}{\pi ^{2}} dk \end{displaymath}

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

\begin{displaymath}g(E) = dN/dE = \frac{V \sqrt{2m^{3}E}}{\pi^{2}\hbar^{3}} \end{displaymath}

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

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 1s electron with an energy $-\epsilon$ and $+\epsilon$ 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 $\psi$as a linear combination of the atomic orbitals $\vert A\rangle$ on atom A and $\vert B\rangle$ on atom B, solve Schrödinger's equation $H\psi=E\psi$ to calculate the energies E available to electrons in the molecule.

(Consider only the following set of matrix elements: $\langle A\vert A\rangle = \langle B\vert B\rangle =1$, $\langle A\vert B\rangle = \langle B\vert A\rangle =0$, $\langle A\vert H\vert A\rangle =-\epsilon$, $\langle B\vert H\vert B\rangle =+\epsilon$, $\langle A\vert H\vert B\rangle = \langle B\vert H\vert A\rangle=-\beta$, with both $\epsilon$ and $\beta$ positive).

(c)
In an infinite linear chain of A and B atoms ($\dots ABABAB\dots$) with equal spacings R between each atom, the energies of electrons in the system are given by $E_k=\pm (\epsilon^2+4\beta^2\cos^2(kR))^{1/2}$,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 $\epsilon$ and $\beta$?

PHYS3C25/1997 CONTINUED

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

5.
(a)
The resultant amplitude A scattered by an assembly of n atoms placed at vector positions $\underline{r}_{n}$ is given by

\begin{displaymath}A(\underline{K}) = \sum_{n} f_{n}(\underline{K})e^{-i \underline{K}.\underline{r}_{n}} \end{displaymath}

where $\underline{K}$ is the scattering vector and $f_{n}(\underline{K})$ the atomic scattering factor of the nth atom. Explain with the aid of the scattering diagram how $\underline{K}$ relates to the scattering geometry, and show that $\vert\underline{K}\vert = 4\pi\sin \theta / \lambda$, where $2\theta$ is the scattering angle and $\lambda$ the wavelength of the radiation.

For the case of scattering of x-rays, sketch the form of $f_{n}(\underline{K})$ for a typical atom.

Explain (i) its value at $\vert\underline{K}\vert = 0$, and (ii) the physical reasons for the form of the curve.

(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 END OF PAPER



 


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Ian Ford
11/11/1997