3. Nuclear Shells Flashcards
In an atom, electronic energy levels are __________ filled by electrons.
Successively
Why are there discontinuities in the graph of atom ionisation energy?
The discontinuities are at the noble gasses which have a higher ionization energy because they’re stable.
Does a nucleus have energy levels?
Yes
What are nuclear energy levels?
The energy levels of the nucleus, determined by solving the Schrodinger equation for the nuclear potential. They are DIFFERENT to electronic energy levels.
State the equation for neutron separation
How can the neutron separation energy be calculated?
The required energy can be calculated by finding the mass difference between initial and final states.
The separation energy is smaller for final nuclei with _, __, and __ neutrons.
8, 20, 28
What are the ‘magic numbers’ for two neutron and two proton separation energies?
N or Z = 2, 8, 20, 28, 50, 82, 126
What do the ‘magic numbers’ for separation energies represent?
The effects of filled nuclear shells
Give the classical equation for the angular momentum of a particle
l = angular momentum
r = position
p = momentum
(cross product)
What is the quantum mechanical operator for momentum in the x-direction ?
What is the quantum mechanical operator for angular momentum in the x-direction?
Give the equation for the square of angular momentum
l² = square of angular momentum
l_x, l_y, l_z = angular momentum operators
State the unsolved equation for the expectation value of angular momentum squared
l = angular momentum
Ψ = wavefunction
Ψ* = wavefunction complex conjugate
r = position
t = time
State the solved equation for the expectation value of angular momentum squared
l² = square of angular momentum
l = orbital angular momentum quantum number
What is the notation for the l = 0 angular momentum state?
s
What is the notation for the l = 1 angular momentum state?
p
What is the notation for the l = 2 angular momentum state?
d
What is the notation for the l = 3 angular momentum state?
f
In quantum mechanics, the ________ of the angular momentum and ___ component can be measured.
Magnitude
One
Describe how the angular momentum is visualised when the z-component is measured
It is thought of as precessing around the z-axis as x and y are uncertain.
Give the equation for the expectation value of the z-component of angular momentum
l_z = z-component of angular momentum
m_l = magnetic quantum number
What values can the magnetic quantum number take?
m_l = magnetic quantum number
l = orbital angular momentum quantum number
How many possible z-components of angular momentum are there?
The number of possible z-components of angular momentum determines the _________.
Degeneracy
Give the equation for the expectation value of spin squared
s² = square of spin
s = spin angular momentum quantum number
Give the equation for the expectation value of the z-component of spin
s_z = z-component of spin
m_s = spin quantum number
What values can the spin quantum number take
m_s = spin quantum number
s = spin angular momentum quantum number
How many possible z-components of spin are there?
How many total states are there per ℓ level in the shell model? Why?
Because nucleons have spin s = 1/2 so there are 2 possible spin states.
What does the degeneracy of each ℓ level in the shell model represent?
The total number of possible states per level.
The total angular momentum is the sum of ________ ________ and _____.
Angular momentum
Spin
Give the equation for the total angular momentum
j = total angular momentum
l = orbital angular momentum
s = spin angular momentum
Give the equation for the expectation value of total angular momentum squared
j² = total angular momentum squared
j = total angular momentum quantum number
Give the equation for the expectation value of the z-component of total angular momentum
j_z = z-component of total angular momentum
m_j = total angular momentum quantum number
State the two rules for the allowed values of j
- The allowed values are in integer steps from |ℓ - s| to |ℓ + s|
- The quantum numbers can’t be negative
What is the allowed total angular momentum quantum number when s = 1/2
j = ℓ ± 1/2
But j must be positive
How can the nuclear energy levels be determined?
By numerically solving the Schrödinger equation for the nuclear potential.
Nucleons orbit as if they are ___________ because they are in occupied discrete energy levels so energy isn’t transferred.
Transparent
What is spin-orbit coupling?
A coupling between spin and angular momentum that contributes to the shell model of nuclear potential. It shows that the nuclear potential is also spin-dependent and that there is a change to potential depending if the spin is parallel (widens the potential) or anti-parallel (narrows the potential) to angular momentum.
State the spin-orbit interaction term
What does the spin-orbit interaction term depend on?
It has radial dependence as well as depending on l-s coupling
Write l.s in terms of j, l, and s
j = total angular momentum
l = orbital angular momentum
s = spin angular momentum
State the expectation value of l.s
l = orbital angular momentum
s = spin angular momentum
j = total angular momentum quantum number
l = orbital angular momentum quantum number
s = spin angular momentum quantum number
What does spin-orbit coupling result in?
A change in potential energy from the spin-orbit term, depending on the orientation of the spin. This splits each ℓ level into ℓ ± 1/2 levels.
State the formula that equates degeneracy before and after splitting due to spin-orbit coupling
Describe the energy level diagram of the shell model
All the magic numbers are successfully reproduced in this version of the model. LHS is before splitting and RHS is after splitting.
How are shells of protons and neutrons filled?
- Work out the number of protons and neutrons
- Remember that protons and neutrons both fill in their own set of energy levels (as they aren’t identical)
- Start with the lowest available energy and fill sequentially
- Degeneracy gives the allowed number of nucleons
