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Nuclear Reactions

Field: Nuclear Physics

Processes in which two nuclei or a nucleus and a subatomic particle collide to produce one or more nuclides.

Sequence of Expressions

Definition

Resonance

The energy dependence of the cross-section σ(E)\sigma(E) near a resonance energy E0E_0 is described by the Breit-Wigner formula, assuming a narrow width Γ\Gamma and a constant background cross-section σ0\sigma_0:\nσ(E)=πℏ2k2gΓ2(E−E0)2+(Γ/2)2+σ0\sigma(E) = \frac{\pi \hbar^2}{k^2} g \frac{\Gamma^2}{(E - E_0)^2 + (\Gamma/2)^2} + \sigma_0
Define the total energy EE and the rest mass m0m_0 of a particle. The relationship between them is given by the fundamental equation:\nE=m0c2E = m_0 c^2
Consider a potential barrier V(x)V(x) defined over x∈[a,b]x \in [a, b], and a particle with energy E<V(x)E < V(x). The transmission probability TT through the barrier, using the WKB approximation, is given by:\nT≈exp⁡(−2∫ab2m(V(x)−E)ℏdx)T \approx \exp\left( -2 \int_{a}^{b} \frac{\sqrt{2m(V(x) - E)}}{\hbar} dx \right)\nwhere mm is the particle mass and ℏ\hbar is the reduced Planck constant.
Define the differential cross-section dσdΩ\frac{d\sigma}{d\Omega} as the probability per unit solid angle for a scattering event. The total cross-section σ\sigma for a reaction A+B→C+DA + B \to C + D is obtained by integrating the differential cross-section over all solid angles Ω\Omega and summing over all possible final states ff:\nσ=∫dσdΩdΩ=∑f∫dσdΩdΩ\sigma = \int \frac{d\sigma}{d\Omega} d\Omega = \sum_{f} \int \frac{d\sigma}{d\Omega} d\Omega
The binding energy B(A,Z)B(A, Z) of a nucleus with AA nucleons and ZZ protons, according to the Semi-Empirical Mass Formula (SEMF), is given by:\nB(A,Z)=avA−asA2/3−acZ(Z−1)A1/3−asym(A−2Z)2A+Correction TermsB(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} + \text{Correction Terms}
For a transition from an initial state ∣i⟩|i\rangle to a final state ∣f⟩|f\rangle induced by a perturbation H′\mathcal{H}', the transition rate Γi→f\Gamma_{i \to f} is given by Fermi's Golden Rule:\nΓi→f=2Im(⟨f∣H′∣i⟩))ℏρ(Ef)\Gamma_{i \to f} = \frac{2\text{Im}(\langle f | \mathcal{H}' | i \rangle))}{\hbar} \rho(E_f) \nwhere ⟨f∣H′∣i⟩\langle f | \mathcal{H}' | i \rangle is the matrix element of the perturbation, and ρ(Ef)\rho(E_f) is the density of final states at the energy EfE_f.
Let the system before and after the reaction be denoted by the sets of particles SinitialS_{initial} and SfinalS_{final}, respectively. The total energy EE is defined as the sum of the rest mass energy and the kinetic energy: E=12∑imivi2+∑imic2E = \frac{1}{2} \sum_{i} m_i v_i^2 + \sum_{i} m_i c^2. Conservation requires that the total energy remains invariant: \n∑i→initialEi=∑j→finalEj\sum_{i \to initial} E_i = \sum_{j \to final} E_j
Let the system before and after the reaction be denoted by the sets of particles SinitialS_{initial} and SfinalS_{final}. The total momentum P\mathbf{P} is defined as the vector sum of the individual momenta: P=∑ipi=∑imivi\mathbf{P} = \sum_{i} \mathbf{p}_i = \sum_{i} m_i \mathbf{v}_i. Conservation requires that the total momentum remains invariant: \n∑i→initialpi=∑j→finalpj\sum_{i \to initial} \mathbf{p}_i = \sum_{j \to final} \mathbf{p}_j
The strong interaction potential Vstrong(r)V_{strong}(r) between two nucleons (N) is modeled by a potential that includes a short-range attractive component and a long-range repulsive component (due to the Pauli exclusion principle and tensor forces). The Hamiltonian density H\mathcal{H} governing the interaction is:\nHstrong=Vstrong(r)=Vresidual(r)+Vtensor(r)\mathcal{H}_{strong} = V_{strong}(r) = V_{residual}(r) + V_{tensor}(r) \nWhere Vresidual(r)V_{residual}(r) is typically approximated by a Yukawa potential or a more complex form incorporating saturation effects, and rr is the separation distance.
The weak interaction is described by the effective Lagrangian Lweak\mathcal{L}_{weak} involving the exchange of W±W^{\pm} and Z0Z^0 bosons. For a process like beta decay (n→p+e−+νˉen \to p + e^- + \bar{\nu}_e), the interaction Hamiltonian density Hint\mathcal{H}_{int} is derived from the coupling of the leptonic current JLJ_L and the hadronic current JHJ_H: \nHint=GF2JLμJHμ+h.c.\mathcal{H}_{int} = \frac{G_F}{\sqrt{2}} J_L^{\mu} J_{H}^{\mu} + \text{h.c.} \nWhere GFG_F is the Fermi constant, and JLJ_L and JHJ_H are the respective weak currents.