Nuclear Shell Model and Schrödinger Equation

Nuclear Shell Model

• Nuclei with proton or neutron numbers 2, 8, 20, 28, 50, 82, and 126 exhibit unique behavior compared to neighboring nuclei, designated as "magic numbers".
• The shell model explains nuclear properties like spin, parity, and magnetic moment.
• It is the quantum mechanical form of the independent particle model.
• The independent particle model assumes nucleons move independently in a common potential well, which is the spherical average of nuclear potential by other nucleons.

Schrödinger Equation

• Energy Eigen values are derived by solving the Schrödinger equation.
• 1/r² d/dr(r² d/dr Ψ(r,θ, φ)) + (2μ/ħ²)[E – U(r)]Ψ(r,θ, φ) = 0.
• Solving the equation requires knowledge about the potential, specifically assuming a well shape for U(r): U(r) = 0 for 0 < r < ro, and U(r) = ∞ for r ≥ ro.
• With the central potential dependent on r only, and the total wave function dependent on (θ, φ), Ψ(r,θ, φ) = ψ(r) Y™(θ,φ)
• ψ(r) = R(r)/r, allowing transformation into: -ħ²/2m (d²/dr²) R(r) + [U(r) + ħ²l(l+1)/2mr²] R(r) = ER(r).
• R(r) represents the radial wave function, where Y™ (θ,φ) are spherical harmonics, and m symbolizes the mass of a nucleon.

Equations for r < ro

• -ħ²/2m (d²/dr²) R(r) + [ħ²l(l+1)/2mr²] R(r) = ER(r)
• Solution to the equation: R(r) = rı(K,ro).
• K = √(2mE/ħ²).
• Applies the boundary conditions R(r = 0) = 0 and R(r = ro) = 0.
• rolı(Kro) = 0 where ro ≠ 0 and lı(Kro) ≠ 0

Infinite Square Well Potential

• The infinite square well potential predicts nucleon numbers 2, 8, 20, 34, 40, 58, 92, and 138, which are not magic numbers.
• It isn't a good approximation for the nuclear potential.
• Separating a neutron or proton requires infinite energy.
• Nuclear shell model labels such as s, p, d, f, g, h... denote the orbital angular momentum quantum number (l) of nucleons in the nucleus where
• l = 0:s
• l = 1: p
• l = 2: d
• l = 3: f
• l = 4: g
• l = 5: h
• l = 6: i
• l = 7: j
• Occupancy = 2(2l + 1)

Harmonic Potential

• The radial equation for 3D harmonic oscillator potential is given by U(r) = (1/2)mw²r² with boundary conditions U(r = ∞) = 0 and U(r = 0) = 0.
• (-ħ²/2m d²/dr² + (1/2)mw²r² + ħ²l(l+1)/2mr²)R(r) = ER(r)
• The boundary condition energy of a nucleon E: E = (2n' + l + 3/2)ħω, involving l = 0, 1, 2... and n' = 0, 1, 2... representing the radial quantum number.
• E = (N + 3/2)ħω uses the number of modes n in the interval [0,∞], n = n' + 1 = 1,2,3... where N = 2n' + l = 0, 1, 2.......
• Closed shell configurations of nucleon numbers predicted are 2, 8, 20, 40, 70, 112, and 168 but aren't experimentally obtained magic numbers
• The harmonic oscillator requires infinite separation and doesn't have a sharp edge.

Woods-Saxon Potential

• A more realistic model is the Woods-Saxon Model.
• The Woods–Saxon potential is defined as U(r) = -Uo / (1 + exp((r - R) / a))
• R = 1.25 A^(1/3)
• a = 0.52 fm,
• Uo = 50 MeV
• This removes the l degeneracy of the magic numbers to the model yields 2, 8, and 20.

L-S Coupling

• The L-S Coupling was suggested by Mayer, Havel, and Suess.
• Splitting energy levels are due to spin (S) and orbital angular momentum (L).
• Spin-orbit coupling introduces L.S term into energy
• j = l + 1/2 level is below j = l – 1/2 level
• Splitting increases with l, naturally from L.S form of spin orbit
• Adjusting the Woods-Saxon Model involves adding a spin-orbit coupling term.
• Ueff = U(r) − f(r) L.S = -Uo /(1 + exp((r - R) / a)) - f(r) L.S
• Considering that (ħ²/2) L.S = j(j + 1) – l(l + 1) - s(s + 1) and assuming a nucleon with s = 1/2, when j = (l + 1/2): (ħ²/2) L.S = l

More on L-S Coupling

• When j = (l - 1/2): (ħ²/2) L.S = -(l + 1)
• With Ueff = U(r) − f(r)(ħ²/2) {l, if j = l + 1/2 spin up; -(l+1), if j = l − 1/2 spin down},
• The splitting is larger given larger splitting
• Sub-level sequences can alter without affecting closed shell configuration and magic numbers; energy levels' sequence in a shell are inferred from experimental data.
• Proton energies (Z > 20) > neutron energies.
• Filling order differences exist when Z > 50.
• Filling order for protons in the 6th shell: 1g9/2, 2d5/2, 1h11/2, 2d3/2, and 3s1/2
• Filling order for neutrons: 2d5/2, 1g7/2, 1h9/2, 3s1/2, 2d3/2
• Light nuclei see Columbic interaction as ineffective, leading to almost the same energy level schemes.
• Larger nuclei have effective Columbic interaction
• Nucleons fill from bottom up while keeping the nucleus's total energy at a minimum

Spin-Parity of Nuclei

• The shell model can predict nuclear spin and parity of nuclei in their ground state
• Nuclei with N and Z both equal to a magic number have a closed configuration.
• The Pauli Exclusion Principle couples intrinsic spin and orbital angular momentum vectors of the nucleons, giving zero total angular and magnetic momentum, in agreement w/ measured nuclear spin (j).
• Pauli Exclusion Principle predicts the nucleus' total angular momentum equals the total angular momentum of the unpaired nucleon only when any of the nucleons is one type of nucleon is equal to a magic number and another type is equal to one number above or below the magic number.

Spectroscopic Notation Examples

• Nitrogen-15 (¹⁵N):
  • p = 7, n = 8, j = 1/2, P = (-1)¹ = -1
  • Spectroscopic notation: ¹⁵N is 1/2⁻
• Oxygen-17 (¹⁷O):
  • p = 8, n = 9, j = 5/2, P = (-1)² = 1
  • Spectroscopic notation: ¹⁷O is 5/2⁺
• Carbon-13 (¹³C):
  • p = 6, n = 7, j = 1/2, P = (-1)¹ = -1
  • Spectroscopic notation: ¹³C is 1/2⁻

More on Spectroscopic Notation, Pairing Energy and Quadrupole Moment

• The "pairing energy" is the phenomenon where energy is reduced when two nucleons (neutrons/protons) pair up in the same energy level with opposing spins. The amount the total energy reduces is of a greater magnitude in larger I states.
• Nucleons paired in the same orbital with opposite spins have total spin canceled, stabilizing the nucleus.
• When calculating the formula uses:
• Q is the quadrupole moment
• j is the nuclear spin
• <r²> is the mean square radius of the nucleus
• For nucleus w/ unpaired nucleons:
• Q = (2j-1)/(2j+1) <r²> for j = l ± 1/2
• The shell model strengths: predicts magic numbers, ground-state properties, and single nucleon states for unpaired nucleons outside a closed shell.
• The shell model weakness: simplifies spherical symmetry