Physics 481
Spring 2006
Problem Set 3
Soeren Prell
1. A system of indistinguishable spin S = 1/2 particles can accommodate two states
in each energy level, spin-up and spin-down, i.e. with Sz = ±1/2. How many states
can a particle with S = 3/2 have? Calculate the total, average, and Fermi energy of
N such particles in a three-dimensional infinite well. Generalize the results to spin
5/2, 7/2, and so forth.
2. Consider a heavy nucleus consisting of Z = np protons and N = Nn neutrons so
that the total atomic number is A = Z +N . If each nucleon has a volume V0 and the
nucleons are bound so that they are “just touching”, the volume of such a nucleus
will go as V = V0 A so that its radius will be given by R = R0 A1/3 . With R0 ≈ 1.2
to 1.4 fm, this form works well for most nuclei.
(a) Instead of considering all the many nucleon-nucleon interactions, model this
system as A nucleons in a three-dimensional infinite square well of volume
V . Using equation [5.45] in Griffith, find the total zero point energy of a
system of A = Z + N nucleons, i.e., E(Z, N ), recalling that both protons and
neutrons are spin-1/2 particles and are distinguishable from each other. For
simplicity, assume that the neutron and proton have the same mass, roughly
mc2 ≈ 940 MeV/c2 .
(b) For fixed A, minimize this energy, and show that equal numbers of neutrons
and protons are favored.
(c) Assuming that ∆ ≡ N − Z ¿ A, show that the zero point energy can be
approximated by an expression of the form
E(Z, N ) ≈ Emin (A) + Esym
∆2
+ ...
A
and find an expression for Esym . Using the values above, give the numerical
value of Esym . Note: This contribution to the total nuclear energy (or rest
mass) is often called the symmetry contribution and is part of the so-called
semi-empirical mass formula of nuclear physics.
3. Problem 5.16 & 5.17, page 223 in Griffith - Fermi energy, electron velocity, Fermi
temperature, and bulk modulus for Copper.
Due Friday, February 3, 5 pm. Scores for late problem sets will be divided by 2.
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