6. Nuclear Properties From Shell Model

Question 1

What are special about the magic numbers?

Answer 1

They are nuclei with a specific number of protons and neutrons that have a larger binding energy

Question 2

Describe how the magic numbers arise

Answer 2

When there is a large energy gap between the grouped SOC states
- Levels are fully filled, so the next energy level has a large gap

Question 3

Describe the notation for the spin and parity of the ground state

Answer 3

( n l_j ) ^K e.g. (1s_1/2)^2

where K is the number of nucleons

Question 4

What is the value of m_j for a filled shell?

Answer 4

0 as the positive and negative m_j’s will cancel out

Question 5

Which type of nuclei always have spin 0?

Answer 5

Any even Z and even N nucleus

Question 6

Why do any even Z and even N nuclei have a total spin of 0?

Answer 6

Even though the shell is not full, the sub shells are filed in order which pair nucleons together to minimise m_j

Question 7

How can we calculate the spin of an even-odd nuclei?

Answer 7

By looking at the total angular momentum of the shell where the unpaired nucleon is
- The spins of filled sub shells are 0

Question 8

How do we calculate the spin of an odd-odd nucleus?

Answer 8

Given by the QM sum of the contribution of the unpaired protons and neutrons
- In the range of | j_p - j_n | to |j_p + j_n |

Question 9

Define parity

Answer 9

The reflection of the spatial coordinates

r -> -r

Question 10

State the effects of the parity operator acting on the spherical harmonics

Answer 10

P Y^m _l (theta, phi) = (-1)^l Y^m _l (theta, phi)

Question 11

What is the intrinsic parity of a nucleon, and how can the parity be calculated for a multi-particle state?

Answer 11

Defined to be positive for a nucleon
- Only the parities of unpaired nucleons will determine the nucleus parity
P = product (-1)^l

Question 12

What is the notation for the parity and ang. mom. of the nucleus

Answer 12

J^P

e.g. 5/2 ^ +

Question 13

What are the two sources for a nucleus haveing a non-0 spin, and what is a consequence of this?

Answer 13

non-0 spin = non-0 magnetic dipole moment
Can be caused by:
1. Intrinsic magnetic moment of nucleons
2. Proton charge moving producing a magnetic moment

Question 14

What is the equation for the nuclear magnetic moment?

Answer 14

μ = g_j * j * μ_N
g_j - Landé g factor
j - Nuclear spin
μ_N - Nuclear magneton

Question 15

In which cases is the nuclear magnetic moment calculation most accurate?

Answer 15

For even-odd nuclei, where there is one unpaired nucleons

Question 16

Between which two values do most nuclear magnetic moments lie?

Answer 16

Between j = l + 1/2 and j = l - 1/2

Question 17

Why doesn’t the nuclear magnetic moment calculation work well for odd-odd nuclei?

Answer 17

To calculate μ = g_j * j * μ_N, we need to calculate g_j which involves combining the spin component with the orbital component
- Would be harder when considering more the magnetic moments of more than one nucleon

Question 18

Define lowest excitation

Answer 18

The lowest excitation is the movement of the last unpaired nucleon to the next subshell
- Or moving a nucleon from the last filled subshell to the subshell with the remaining unapaired nucleon

Question 19

Which assumption leads to the limitation of the accuracy of the nuclear shell model?

Answer 19

That nucleons are moving freely, independent of other nucleons in a spherically symmetric potential