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From atoms to solids
Answer ALL parts of question 1 and TWO other questions
1
1. Using information from the periodic table provided;
a) Plot the ionization energy of atoms with Z from 11 to 19. Graph paper is not
required. Comment on the result.
[6 Marks]
b) Describe the electronic structure of copper (Cu) by drawing a diagram to show
how the 29 electrons fill the atomic shells.
[4 Marks]
c) The spectroscopic state of Neon is 1 S 0 . Explain what this means and relate it to
the ground state configuration.
[2 Marks]
d) Explain the operation of a Stern-Gerlach magnet. Neon atoms are passed
through a Stern-Gerlach magnet. Describe what would be observed and explain
your answer.
[4 Marks]
e) Explain why copper is a metal.
[4 Marks]
f) The table below gives the electrical and thermal conductivity for zinc and copper
at 300K, but has one entry missing. Estimate its value and explain your reasoning.
Element Electrical Conductivity ( −1 cm −1 ) Thermal Conductivity (Wcm −1 K −1 )
Zinc
0. 166  10 6
−
Copper
0. 6  10
4. 01
6
Describe and explain how the electrical conductivity of copper and germanium
would change with temperature.
[5 Marks]
2
2. a) For first order perturbation theory, state the formula that relates the shift in an
energy level to the perturbing potential.
State the eigenvalues of the operator L z .
B Z L , then derive
Consider an atom with spin zero. If the perturbing potential is, e2m
z
e
the energy shifts for states with orbital angular momentum, L  1. You should state
the meaning for the various terms used in this expression.
[10 Marks]
b) The energy levels of the hydrogen atom are described by the formula,
En 
−13.6eV
n2
, where n is the principal quantum number.
Calculate the wavelength of the photon emitted due to a transition between the
n  2, l  1and n  3, l  2 levels. Ignore fine-structure effects.
Draw an energy diagram to illustrate this transition. Explain why the orbital angular
momentum l cannot be the same for these two levels.
For the case of the Normal Zeeman Effect, draw an energy diagram in eV to show
how this transition is altered in a magnetic field of one Tesla. How would this
diagram change if electron spin was taken into account ?
[15 Marks]
3
3.
a) Describe the Van der Waals interaction and relate it to the bonding of solid
Krypton below 116K.
[6 Marks]
b) Define the Fermi Energy.
Calculate the Fermi velocity for Copper given that its Fermi energy is 7.0 eV.
Explain the meaning of the expression, "mean time between collisions", and relate
this to the mobility and mean free path.
Calculate the mean free path for electrons in copper at 300K. The electrical
conductivity for copper is given in the table in question 1 f). The density of copper is
8.93 gcm −3 . Comment on your result.
[14 Marks]
c) Describe the bonding mechanisms for sodium chloride and silicon.
[5 Marks]
4
4. a) Draw a suitable diagram to show how band theory explains the electrical
conduction of metals, semi-metals, semiconductors and insulators.
[8 Marks]
b) To a good approximation, in silicon one can write that
EF  Ei 
kBT
2
ln np 
Explain the meaning of all the terms in this expression.
Estimate the built-in potential at 300K in a silicon diode in which the n-type and
p-type materials are doped with 10 16 cm −3 phosphorus atoms and 10 19 cm −3 boron
atoms respectively. State any assumptions you make.
Note: The intrinsic carrier concentration of silicon at 300K is 8.72 10 9 cm −3 .
[9 Marks]
c) Draw an energy band diagram for a np junction with doping levels as given in
part b). The band gap of silicon at 300K is 1.12 eV. Use this to explain the rectifying
property of such a junction.
[8 Marks]
5