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Nuclear Structure

Field: Nuclear Physics

The study of the properties of atomic nuclei.

Sequence of Expressions

Consider the Hamiltonian H^\hat{H} for a system of AA nucleons confined within a volume VV. The allowed energy levels EkE_{k} are determined by solving the time-independent Schrödinger equation for the many-body system: \nH^Ψk(r1,…,rA)=EkΨk(r1,…,rA)\hat{H} \Psi_{k}(\mathbf{r}_1, \dots, \mathbf{r}_A) = E_{k} \Psi_{k}(\mathbf{r}_1, \dots, \mathbf{r}_A) \nIn the independent particle model, the single-particle Hamiltonian h^\hat{h} is used, leading to the eigenvalue problem: \n(−ℏ22m∇2+V(r))ψnlj(r,θ,ϕ)=Enljψnlj(r,θ,ϕ)\left( \frac{-\hbar^2}{2m} \nabla^2 + V(r) \right) \psi_{n l j}(r, \theta, \phi) = E_{n l j} \psi_{n l j}(r, \theta, \phi) \nwhere EnljE_{n l j} are the discrete energy levels.
Consider a positively charged projectile particle with charge q1q_1 and momentum p⃗1\vec{p}_1, scattering off a target nucleus with charge QQ and mass MM. The interaction potential is the Coulomb potential: V(r)=kQq1rV(r) = \frac{k Q q_1}{r}. The scattering amplitude f(θ)f(\theta) is derived from the differential cross-section dσdΩ\frac{d\sigma}{d\Omega}. For small-angle scattering, the classical trajectory analysis yields the scattering angle θ\theta and the momentum transfer q⃗\vec{q}. The differential cross-section is given by: dσdΩ=(kQq14E)21sin⁡4(θ/2) \frac{d\sigma}{d\Omega} = \left( \frac{k Q q_1}{4 E} \right)^2 \frac{1}{\sin^4(\theta/2)} where EE is the initial kinetic energy and θ\theta is the scattering angle.
The binding energy B(A,Z)B(A, Z) of a nucleus with AA nucleons and ZZ protons is modeled using the Semi-Empirical Mass Formula (SEMF), which treats the nucleus as a charged liquid drop. The binding energy is given by the sum of five terms: B(A,Z)=avA−asA2/3−acZ(Z−1)A1/3−asym(A−2Z)2A×(1+apairA1/3) B(A, Z) = a_v A - a_s A^{2/3} - a_c \frac{Z(Z-1)}{A^{1/3}} - a_{sym} \frac{(A-2Z)^2}{A} \times (1 + \frac{a_{pair}}{A^{1/3}}) where ava_v, asa_s, aca_c, and asyma_{sym} are volume, surface, Coulomb, and symmetry coefficients, respectively. The pairing term is incorporated by the factor (1+apairA1/3)(1 + \frac{a_{pair}}{A^{1/3}}), which accounts for the energetic preference for even-even nuclei.
The enhanced stability of nuclei with specific proton (ZZ) or neutron (NN) numbers (magic numbers) is explained by the closure of major energy shells in the single-particle spectrum. The single-particle energy levels En,l,jE_{n, l, j} are calculated using a mean-field potential. The magic numbers correspond to the cumulative number of nucleons required to fill the first KK energy shells, Nmagic=⨁k=1K(2jk+1)N_{magic} = \bigoplus_{k=1}^{K} (2j_k + 1), where jkj_k are the angular momentum quantum numbers of the kk-th shell. The stability enhancement is quantified by the binding energy per nucleon, B/AB/A, which exhibits local maxima when ZZ or NN equals a magic number.
Define the potential energy operator V^\hat{V} acting on the relative coordinates rij=ri−rj\mathbf{r}_{ij} = \mathbf{r}_i - \mathbf{r}_j between nucleons ii and jj. The total potential is modeled as: \nV^=∑i<jVij=∑i<j(VCoulomb(rij)+VStrong(rij))\hat{V} = \sum_{i<j} V_{ij} = \sum_{i<j} \left( V_{Coulomb}(r_{ij}) + V_{Strong}(r_{ij}) \right) \nwhere VCoulomb(rij)=e24πϵ0rijV_{Coulomb}(r_{ij}) = \frac{e^2}{4\pi\epsilon_0 r_{ij}} and VStrong(rij)V_{Strong}(r_{ij}) is the strong interaction potential, often approximated by a Yukawa form: \nVStrong(rij)=−g2rije−rij/λV_{Strong}(r_{ij}) = -\frac{g^2}{r_{ij}} e^{-r_{ij}/\lambda}
The strong nuclear force strength is characterized by the coupling constant gg (or GAG_{A} for axial coupling). In the context of the effective Hamiltonian H^eff\hat{H}_{eff} derived from Quantum Chromodynamics (QCD), the interaction term is proportional to gg: \nH^eff=H^0+∑i<jgOij\hat{H}_{eff} = \hat{H}_0 + \sum_{i<j} g \mathcal{O}_{ij} \nwhere Oij\mathcal{O}_{ij} is an operator describing the interaction between nucleons ii and jj. The binding energy per nucleon, B/AB/A, is directly related to the magnitude of gg via the semi-empirical mass formula (SEMF) approximation: \nBA≈av−asA−1/3−acZ(Z−1)/A1/3…\frac{B}{A} \approx a_{v} - a_{s} A^{-1/3} - a_{c} Z(Z-1)/A^{1/3} \dots \nwhere ava_v is proportional to gg.
Let ψi\psi_i be the wave function describing the ii-th fermion (e.g., proton or neutron) in a quantum state defined by quantum numbers (n,l,j,mj)(n, l, j, m_j). The Pauli Exclusion Principle dictates that for a system of NN identical fermions, the total wave function Ψ\Psi must be antisymmetric under the exchange of any two particles. Mathematically, this implies that the set of occupied single-particle states {ψ1,ψ2,…,ψN}\left\{ \psi_1, \psi_2, \dots, \psi_N \right\} must be distinct, such that for any two particles ii and jj (i≠ji \neq j), the state ψi\psi_i cannot be identical to ψj\psi_j. Formally, the occupation number operator n^k\hat{n}_k for any given state kk must satisfy n^k∈{0,1}\hat{n}_k \in \{0, 1\}. The Hamiltonian for the system is then constructed using second quantization: H^=∑kEkn^k\hat{H} = \sum_k E_k \hat{n}_k.
The nuclear shell model describes the energy levels En,l,jE_{n, l, j} of nucleons (protons and neutrons) within the mean field potential V(r)V(r) generated by all other nucleons. The single-particle Hamiltonian is h^=p⃗22m+V(r)\hat{h} = \frac{\vec{p}^2}{2m} + V(r). The solutions to the Schrödinger equation, h^ψ=Eψ\hat{h} \psi = E \psi, yield quantized energy levels. The effective potential V(r)V(r) is often approximated by a Woods-Saxon potential. The total energy of the nucleus is determined by filling these single-particle states according to the Pauli principle, leading to the total energy Etotal=∑i=1Z+NEiE_{total} = \sum_{i=1}^{Z+N} E_i, where EiE_i are the occupied single-particle energy eigenvalues.
Let n^i\hat{n}_{i} be the occupation number operator for a single-particle state ii. The Pauli Exclusion Principle dictates that the expectation value of the total number of fermions NN must satisfy the constraint: \n⟨N^⟩=∑i⟨n^i⟩≤g\langle \hat{N} \rangle = \sum_{i} \langle \hat{n}_{i} \rangle \le g \nwhere gg is the degeneracy of the state. The average occupation number ⟨n^i⟩\langle \hat{n}_{i} \rangle is given by the Fermi-Dirac distribution function: \n⟨n^i⟩=1e(Ei−μkBT)+1\langle \hat{n}_{i} \rangle = \frac{1}{e^{\left( \frac{E_{i} - \mu}{k_{B}T} \right)} + 1} \nwith μ\mu being the chemical potential and TT the temperature.
The collective model describes the nuclear Hamiltonian H^coll\hat{H}_{coll} using collective coordinates, typically the quadrupole deformation parameters β\beta and γ\gamma. The Hamiltonian is generally written as: \nH^coll=12∑k=13(L^k22Ik+B^k22Bk)+V(β,γ)\hat{H}_{coll} = \frac{1}{2} \sum_{k=1}^{3} \left( \frac{\hat{L}_k^2}{2I_k} + \frac{\hat{B}_k^2}{2B_k} \right) + V(\beta, \gamma) \nwhere L^k\hat{L}_k and B^k\hat{B}_k are the angular momentum and vibrational operators, respectively. The potential V(β,γ)V(\beta, \gamma) is the potential energy surface governing the equilibrium deformation, often modeled by a polynomial expansion in β\beta and γ\gamma.