7. Non Spherical Nuclei

Question 1

How can you describe classical electric charge distribution

Answer 1

In terms of multipole moments e.g. quadrupole moment

Question 2

What is the electric quadrupole, Q?

Answer 2

The first non zero moment

- It is 0 for spherical nuclei, and non 0 for non-spherical nuclei

Question 3

What does a non-spherical nuclei imply?

Answer 3

A non-spherical charge distribution

Question 4

How do rotations influence spherically symmetric nuclei?

Answer 4

They don’t

Question 5

Describe how rotations affect an ellipsoid nuclei

Answer 5

Axis of symmetry is the z axis

• Rotations around the z axis don’t change anything
• Rotations around a perpendicular axis introduce additional angular momentum

Question 6

What is the contribution from the total angular momentum towards to electric quadrupole moment?

Answer 6

Q_j ~ R^2 2j-1 / 2(j+1)

Question 7

What is the expected electric quadrupole moment for an even-Z even-A nucleus and why?

Answer 7

0 due to the pairing effect and that neutrons have 0 charge

Question 8

Describe how Rainwater explained the problem with the data of spherical symmetry for odd Z odd A nuclei

Answer 8

He deformed a spherical nucleus to an ellipsoid by a deformation parameter ε
a = R(1+ε)
b = R / sqrt(1+ε)
a is the semi major axis, b is the minor axis and R is the radius of a nucleus, R_0 A^1/3

Question 9

Describe how Rainwater’s assumption altered the SEMF formula

Answer 9

He altered the coefficients for a_2 and a_3

- This is because of the change in SA of the nucleus and the distances between the protons and neutrons is deformed

Question 10

What is the electric quadrupole moment equal to for an ellipsoid?

Answer 10

Q = Q_intrinsic * J(2J-1) / [ (J+1) (2J + 3) ]

Question 11

How do the spectrum of energy levels change for deformed spin-0 nuclei?

Answer 11

They change in steps of 2
E_J = -h bar^2 J(J+1) / 2I
I - Moment of Inertia

Question 12

What is the Nilsson model?

Answer 12

A single particle model with a different potential complemented by SOC
- Deformed nucleus core induces deformations

Question 13

How does the deformed nucleus core influence the energies of the different levels

Answer 13

It shifts them, proportionally to the size of the deformation

Question 14

Describe how the Nilsson model also predicts how the positions of the magic numbers will change

Answer 14

Magic numbers was based on the fact that filled levels were grouped close together, followed by a gap to the next group of levels, so adding a nucleon costs a lot of energy
- Nilsson model shifts these places of significant energy gaps

Question 15

What is an intrinsic excitation?

Answer 15

When nucleons move to higher energy levels

Question 16

What are the three types of excitations

Answer 16

Intrinsic excitation, rotations and vibrations

Question 17

What are shape oscillations?

Answer 17

A form of vibration excitation

- The density of a nucleus remains constant, but the shape changes around equilibrium

Question 18

Describe the excitations observed when l = 1 and l = 2

Answer 18

When l = 1, the centre of mass is displaced and we get a translation, not a shape change
When l = 2, we get quadrupole deformation

Question 19

What is the ground state energy?

Answer 19

E_1 = 5/2 h bar w

Question 20

Describe how the shape oscillations look on a graph

Answer 20

Equilibrium - Circle centered about the origin

Oscillations - Ellipsoids along the x and y axis

Question 21

What is the energy of the vibrational states?

Answer 21

E_N = (N + 5/2) h bar w

- N determines which excitation we have

Question 22

What is J^P equal to for one and two phonon excitation?

Answer 22

One phonon: J^P = 2^+
Two phonons: J^P = 0^+, 2^+, 4^+
For two phonons, need to add the spins

Question 23

How does the energy compare for the one and two phonon 2^+ state?

Answer 23

The one phonon 2^+ state has a smaller energy