8.2 Problems
437
8.2.6 Fermi Gas Model
8.24 In the Fermi gas model the internal energy is given by U =
3
5
AE F , where
A is the mass number and E F is the Fermi energy. For a nucleus of volume
V with N = Z = A/2. A = KVE F
3/2 where K is a constant. Using the
thermodynamic relation, p = −
∂U
∂ V
s
, show that the pressure is given by
p =
2
5
ρ n E F , where ρ n is the nucleon density.
8.25 Assuming that in a nucleus N = Z = A/2, calculate the Fermi momentum,
Fermi energy E F , and the well depth.
8.2.7 Shell Model
8.26 A certain odd-parity shell model state can accommodate up to a maximum of
12 nucleons. What are its j and l values?
8.27 The shell model energy levels are in the following way
[1s 1/2 ][1 p 3/2 , 1 p 1/2 ][1d 5/2 , 2s 1/2 , 1d 3/2 ][1 f 7/2 ][2 p 3/2 , 1 f 5/2 , 2 p 1/2 , 1g 9/2 ]
[1g 7/2 , 2d 5/2 , 2d 3/2 , 3s 1/2 , 1h 11/2 ] . . .
Assuming that the shells are filled in the order written, what spins and parities should be expected for the ground state of the following nuclei?
7
3 Li,
16
8 O,
17
8 O,
39
19 K,
45
21 Sc.
8.28 Find the gap between the 1 p 1/2 and 1d 5/2 neutron shells for nuclei with mass
number A ≈ 16 from the total binding energy of the
15 O (111.9556 MeV),
16 O (127.6193 MeV) and
17 O (131.7627 MeV) atoms.
8.29 Compute the expected shell-model quadruple moment of
209 Bi(9/2
− )
8.30 From the shell model predictions find the ground state spin and parity of the
following nuclides:
3
2 He;
20
10 Ne;
27
13 Al;
41
21 Sc;
8.31 Making use of the shell model, write down the ground state configuration of
protons and neutrons for
12
6 C and the next three isotopes of increasing A. Give
the spin and parity assignment for the ground state of
13
6 C and compare this
with the equivalent assignments for the ground state of
15
6 C.
8.32 How does the shell model predict
7
−
2
for the ground state spin parity of
41
20 Ca.
What does the model predict about the spin and parities of the ground states
of
30
14 Si and
14
7 N ?
[University of Cambridge, Tripos 2004]
437
8.2.6 Fermi Gas Model
8.24 In the Fermi gas model the internal energy is given by U =
3
5
AE F , where
A is the mass number and E F is the Fermi energy. For a nucleus of volume
V with N = Z = A/2. A = KVE F
3/2 where K is a constant. Using the
thermodynamic relation, p = −
∂U
∂ V
s
, show that the pressure is given by
p =
2
5
ρ n E F , where ρ n is the nucleon density.
8.25 Assuming that in a nucleus N = Z = A/2, calculate the Fermi momentum,
Fermi energy E F , and the well depth.
8.2.7 Shell Model
8.26 A certain odd-parity shell model state can accommodate up to a maximum of
12 nucleons. What are its j and l values?
8.27 The shell model energy levels are in the following way
[1s 1/2 ][1 p 3/2 , 1 p 1/2 ][1d 5/2 , 2s 1/2 , 1d 3/2 ][1 f 7/2 ][2 p 3/2 , 1 f 5/2 , 2 p 1/2 , 1g 9/2 ]
[1g 7/2 , 2d 5/2 , 2d 3/2 , 3s 1/2 , 1h 11/2 ] . . .
Assuming that the shells are filled in the order written, what spins and parities should be expected for the ground state of the following nuclei?
7
3 Li,
16
8 O,
17
8 O,
39
19 K,
45
21 Sc.
8.28 Find the gap between the 1 p 1/2 and 1d 5/2 neutron shells for nuclei with mass
number A ≈ 16 from the total binding energy of the
15 O (111.9556 MeV),
16 O (127.6193 MeV) and
17 O (131.7627 MeV) atoms.
8.29 Compute the expected shell-model quadruple moment of
209 Bi(9/2
− )
8.30 From the shell model predictions find the ground state spin and parity of the
following nuclides:
3
2 He;
20
10 Ne;
27
13 Al;
41
21 Sc;
8.31 Making use of the shell model, write down the ground state configuration of
protons and neutrons for
12
6 C and the next three isotopes of increasing A. Give
the spin and parity assignment for the ground state of
13
6 C and compare this
with the equivalent assignments for the ground state of
15
6 C.
8.32 How does the shell model predict
7
−
2
for the ground state spin parity of
41
20 Ca.
What does the model predict about the spin and parities of the ground states
of
30
14 Si and
14
7 N ?
[University of Cambridge, Tripos 2004]