1000 Solved Problems in Modern Physics

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

AEF, where

Ais the mass number andEFis the Fermi energy. For a nucleus of volume

VwithN = Z= A/2.A =KVEF^3 /^2 whereKis a constant. Using the

thermodynamic relation,p=−

(

∂U

∂V

)

s

, show that the pressure is given by

p=

2

5

ρnEF, whereρnis the nucleon density.

8.25 Assuming that in a nucleusN=Z=A/2, calculate the Fermi momentum,
Fermi energyEF, 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 itsjandlvalues?

8.27 The shell model energy levels are in the following way
[1s 1 / 2 ][1p 3 / 2 , 1 p 1 / 2 ][1d 5 / 2 , 2 s 1 / 2 , 1 d 3 / 2 ][1f 7 / 2 ][2p 3 / 2 , 1 f 5 / 2 , 2 p 1 / 2 , 1 g 9 / 2 ]
[1g 7 / 2 , 2 d 5 / 2 , 2 d 3 / 2 , 3 s 1 / 2 , 1 h 11 / 2 ]...
Assuming that the shells are filled in the order written, what spins and pari-
ties 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 1p 1 / 2 and 1d 5 / 2 neutron shells for nuclei with mass
numberA≈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^126 C and the next three isotopes of increasingA.Give
the spin and parity assignment for the ground state of^136 C and compare this
with the equivalent assignments for the ground state of^156 C.
8.32 How does the shell model predict

7 −

2

for the ground state spin parity of^4120 Ca.

What does the model predict about the spin and parities of the ground states

of^3014 Si and^147 N?

[University of Cambridge, Tripos 2004]