Image Analysis Textual Content: 22.101 Quiz Review 11/12 Shell Model shell model diagram Spin-orbit Coupling Splits energy levels H = p^2 / 2m + V(r) + V_so(r) * S.L To find S.L: Define J = L + S => J^2 = L^2 + S^2 + 2L.S => L.S = 1/2(J^2 - L^2 - S^2) Transform |l,m_l,s,m_s> -> |j,m_j,l,s> J^2 |j,m_j,l,s> = ħ^2 j(j+1) |j,m_j,l,s>, |l-s| ≤ j ≤ l+s J_z |j,m_j,l,s> = ħm_j |j,m_j,l,s>, -j ≤ m_j ≤ j Values differ by 1 e.g., -7/2, -5/2, -3/2, -1/2, 1/2, 3/2, 5/2, 7/2 Degeneracy of 2j+1 for level j 3p3/2 3p state, l = 1, j = 3/2 => L and S parallel Binding Energy Curve binding energy curve Empirical Mass Formula B(A, Z) = avA - asA^(2/3) - ac(Z^2)/A^(1/3) - aa(N-Z)^2/A + δ Volume term: Each nucleon contributes a constant energy due to its binding with surrounding nucleons. Surface term: Nuclei on the surface contribute less because they are not completely surrounded. The number of nucleons on the surface is related to r^2 and ~ A^(2/3). Coulomb term: Protons repel others, removing binding energy. This term is related to Coulomb potential (Z^2)/A^(1/3). Asymmetry term: Owing to Pauli exclusion, neutrons and protons fill nuclear wells separately. Extra neutrons must be included if protons fill a well. Pairing term: If nucleons pair up, they reduce energy. Radioactive decay A → B dN_A/dt = -λN_A ⇒ N_A = N_A0 * e^(-λt) A → B → C dN_A/dt = -λ_1N_A ⇒ N_A = N_A0 * e^(-λ_1t) dN_B/dt = λ_1N_A - λ_2N_B ⇒ N_B = (λ_1N_A0)/(λ_2 - λ_1) * (e^(-λ_1t) - e^(-λ_2t)) Stopping Power -dE/dx = 4πe^4Z^2n/mv^2 * ln(2mv^2/I) (Non-relativistic collisions) -dE/dx = 2πe^4Z^2n/mv^2 * [ln(2mv^2/I) - β^2] (Relativistic e- collisions) What is the effect of Z of absorber? What scale? (-dE/dx) = n(T + mec^2)σ_tot (radiative loss of e-) Range R = ∫(0 to ∞) (dx/(-dE/dx)) dT I = w ∫(0 to ∞) (-dE/dx) dx graph showing stopping power vs. energy