6. Nuclear Properties From Shell Model Flashcards
What are special about the magic numbers?
They are nuclei with a specific number of protons and neutrons that have a larger binding energy
Describe how the magic numbers arise
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
Describe the notation for the spin and parity of the ground state
( n l_j ) ^K e.g. (1s_1/2)^2
where K is the number of nucleons
What is the value of m_j for a filled shell?
0 as the positive and negative m_j’s will cancel out
Which type of nuclei always have spin 0?
Any even Z and even N nucleus
Why do any even Z and even N nuclei have a total spin of 0?
Even though the shell is not full, the sub shells are filed in order which pair nucleons together to minimise m_j
How can we calculate the spin of an even-odd nuclei?
By looking at the total angular momentum of the shell where the unpaired nucleon is
- The spins of filled sub shells are 0
How do we calculate the spin of an odd-odd nucleus?
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 |
Define parity
The reflection of the spatial coordinates
r -> -r
State the effects of the parity operator acting on the spherical harmonics
P Y^m _l (theta, phi) = (-1)^l Y^m _l (theta, phi)
What is the intrinsic parity of a nucleon, and how can the parity be calculated for a multi-particle state?
Defined to be positive for a nucleon
- Only the parities of unpaired nucleons will determine the nucleus parity
P = product (-1)^l
What is the notation for the parity and ang. mom. of the nucleus
J^P
e.g. 5/2 ^ +
What are the two sources for a nucleus haveing a non-0 spin, and what is a consequence of this?
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
What is the equation for the nuclear magnetic moment?
μ = g_j * j * μ_N
g_j - Landé g factor
j - Nuclear spin
μ_N - Nuclear magneton
In which cases is the nuclear magnetic moment calculation most accurate?
For even-odd nuclei, where there is one unpaired nucleons
Between which two values do most nuclear magnetic moments lie?
Between j = l + 1/2 and j = l - 1/2
Why doesn’t the nuclear magnetic moment calculation work well for odd-odd nuclei?
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
Define lowest excitation
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
Which assumption leads to the limitation of the accuracy of the nuclear shell model?
That nucleons are moving freely, independent of other nucleons in a spherically symmetric potential
