Download Nuclear Physics Problems 2.3. “Mirror nuclei” are pairs of nuclei in

Nuclear Physics Problems
2.3. “Mirror nuclei” are pairs of nuclei in which the proton number in one equals
the neutron number in the other and vice versa. The simplest examples are provided
by odd-A nuclei with one odd nucleon. In one of the mirror nuclei the charge is
Z = (A + 1)/2 and the neutron number is N = (A − 1)/2; whilst in the other the
charge is Z = (A − 1)/2 and the neutron number N = (A + 1)/2. Examples are 13
7 N
13
31
31
and 6 C or 16 S and 15 P. The nuclei in these pairs differ from each other only in that
a proton in one is exchanged for the neutron in the other. There is strong reason to
believe that the force between nucleons does not differentiate between neutrons and
protons; consequently, the “nuclear part” of the binding energy of two mirror nuclei
must be the same. Hence, the mass difference of two mirror nuclei can be due only
to the difference between the proton mass and the neutron mass, and the different
Coulomb energies of the two. This can be used to find the radius of the two nuclei
(assumed to be the same), thus:
1. Show that the electrostatic potential energy (Coulomb energy) of a uniformly
charged sphere of radius R and charge Ze is 3Z 2 e2 /(20π0 R).
21
2
2. The nucleus 21
11 Na is more massive than its “mirror partner” 10 Ne by 4.02 MeV/c .
What is the difference between their Coulomb energies? Use this to estimate a
value for R. (The neutron-proton mass difference is 1.29 MeV/c2 .)
2.5. Define the terms atomic mass unit, mass defect, and binding energy of a nucleus.
Show that 1u (atomic mass unit) = 931.50 MeV/c2 . (The mass of a 12 C atom is
1.99267 × 10−26 kg.)
The binding energy of 181
73 Ta is 1454 MeV. What is the mass defect of this nucleus?
(Mass defects of neutron and proton are 0.008667u and 0.007279u respectively.)
2.6. The semi-empirical mass formula (SEMF) for nuclear masses is
(A − 2Z)2
Z2
+
a
+ δP
A
A1/3
A
where mp and mn are the masses of the free proton and neutron, and the coefficients
aV , as , ac , aA have the values (in MeV/c2 ) 15.56, 17.23, 0.697, 23.285 respectively.
M (Z, A) = Zmp + (A − Z)mn − aV A + as A2/3 + ac
1. What expression does this formula give for the binding energy per nucleon of
the nucleus (A, Z)?
2. Explain briefly the physical basis of the various terms in the SEMF. How does
the last term δP depend on A and Z?
3. Use the calculation you did in question 2.3(1) to predict a value for the coefficient
ac , assuming that the nuclear radius is given by R = 1.24 × A1/3 fm.
4. Use the SEMF to calculate the binding energy of the nucleus 135
56 Ba. Given
135
that the atomic mass of 56 Ba is 134.904553u, the mass of the hydrogen atom
is 1.007825u, and the mass of the neutron is 1.008667u, make a second estimate
of the binding energy of 135
56 Ba. Can you suggest reasons for any discrepancy
between your two results?
1
2.7. The Fermi Gas model of the nucleus assumes that the nucleus is a spherical
“box” of radius R, and that the neutrons and protons are gases of fermions which (in
the nuclear ground state) are “at absolute zero”—i.e., they fiill up the lowest energy
levels. Given that
√ the density of levels for spin-half fermions of mass m in a volume
V is V (2m)3/2 /(2π 2 h̄3 ) (this includes the spin degeneracy factor), show that in a
nucleus containing Z protons and N neutrons, the Fermi energy of the proton gas is
EF =
h̄2
2mr02
9πZ
4A
2/3
where R = r0 A1/3 , and m is the nucleon mass (assumed the same here for neutrons
and protons). Write down a similar expression for the Fermi energy of the neutron
gas.
Calculate the total energy of the proton gas and the neutron gas. Now consider the
case where Z and N add to a fixed total A (take A to be even). Show that the total
energy E of the nucleons has a minimum when N = Z = A/2. By expanding E about
this minimum value, and retaining only the quadratic term in (Z − A/2), derive an
expression for the coefficient aA in the SEMF. Evaluate aA , taking r0 to be 1.24 fm.
How does it compare with the value quoted in question 6?
3.6. Explain what the “shell model” of the nucleus is. What experimental evidence
is there for it?
What would the first five “magic numbers” be if nucleons were assumed to move in a
three-dimensional harmonic oscillator potential? What additional component in the
potential is required to explain the observed magic numbers? Show how this extra
term accounts for the first five observed magic numbers.
3.7. Use the shell model to predict the spin and parity of the ground states of the
195
167
87
27
20
following nuclei: 15
7 N, 10 Ne, 12 Mg, 38 Sr, 68 Er, and 80 Hg. Explain any assumptions
you make.
The observed spin-parities are
ancies.
Can the spin-parity of
38
19 K
1−
,
2
0+ ,
1+ 9+ 7+
,2 ,2 ,
2
and
1−
.
2
Comment on any discrep-
be predicted by the shell model?
2