PHYSICS 715 PROBLEM SET 2
DUE 3/7/05
1. The chemical potential. Let us take two chunks of material, each con-
taining NSatoms. These atoms may be neutral (have equal numbers of protons and
electrons) or singly negatively ionized (have one more electron). Initially the chunks
are spatially separated and Chunk 1 is made of element 1 with electron affinity ε1,
has N1ions, while Chunk 2 is made of element 2 with electron affinity ε2and has
N2ions. The electron affinity is the energy it takes to add an electron to a neutral
atom. That is, there are N1extra electrons on Chunk 1 and N2extra electrons on
Chunk 2.The chunks are now placed in contact so that energy and electrons can
flow between them, and equilibrium is established. Assume that the extra electrons
do not interact with each other.
(a) What is the resulting distribution of electrons between the two chunks?
(b) What is the resulting chemical potential?
(c) Qualitatively, what would be the effect of repulsions among the extra elec-
trons?
2. Liquid 3He. The atom 3H e has nuclear spin 1/2 and is therefore a fermion.
Collections of these atoms remain liquid to absolute zero. The density is 0.081
gm/cm3.
(a) Calculate the Fermi velocity vF,the Fermi energy εF, and the Fermi temper-
ature TF.
(b) Calculate the specific heat at low temperatures and compare to the experi-
mental value CV= 2.89NkT . Comment on any discrepancy.
3. Zero-point pressure. Do Prob. 11.15 in Huang.
4. White dwarfs and neutron stars (application of zero-point pressure).
(Simplified version of Sec. 11.2). In a white dwarf star α-particles form a stationary
lattice and the electrons are a degenerate gas. Assume that the temperature is zero.
For a white dwarf with mass Mand radius R
(a) Calculate the gravitational self-energy, assuming uniform density.
(b) Calculate the kinetic energy of the electrons.
(c) The virial theorem states that the gravitational energy and kinetic energy are
about equal in magnitude. Deduce a relation between the mass and the radius of
the form MxRy=C. Determine the exponents xand yand the constant C.
Neutron stars consist predominantly of a liquid of neutrons at temperature zero.
(d) Repeat part (c) for neutron stars.
5. Neutrino gas.
Assume that neutrinos have zero mass and spin 1/2. Consider a universe of
volume Vand temperature Tfilled with primordial neutrinos.
What is the energy density of the neutrinos?
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