Combined Equation Sheet

Atomic Mass Unit

\[\begin{equation} u = 1.66056\times 10^{-27}~\textnormal{kg} = 931.5 \textnormal{MeV/c}^2. \end{equation}\]

Nuclear Mass Deficit

\[\begin{equation} \Delta M(A,Z) = M(A,Z) - (ZM_{p} + NM_{n}). \end{equation}\]

Binding Energy Equation

\[\begin{equation} -\Delta M(A,Z) = B \end{equation}\]

Nuclear Mass Density

\[\begin{equation} \rho_{m} = \frac{M}{V} = \frac{3m_{n}}{4\pi r^{3}_{0}} \end{equation}\]

Nuclear Charge Density

\[\begin{split}\begin{equation} \rho_{c} = \frac{Q}{V} = \frac{3Ze}{4\pi r^{3}_{0} A}\\ \end{equation}\end{split}\]

Cross-sectional Area

\[\begin{equation} \sigma = \pi (2R)^{2} \end{equation}\]

Mean Free Path

\[\begin{equation} \lambda = \frac{1}{n\sigma}. \end{equation}\]

Beam Intensity Reduction in Target

\[\begin{equation} N = N_0 e^{-x/\lambda} \end{equation}\]

N Collisions in Target

\[\begin{equation} C=N_0-N=N_0(1-e^{-x/\lambda}). \end{equation}\]

Rutherford Scattering

\[\begin{split}\begin{equation} \frac{d\sigma}{d\Omega_{R}} = \frac{z^{2}Z^{2}\alpha^{2}\hbar^{2}c^{2}}{4E^{2}} \frac{1}{sin^{4}(\theta/2)} \\ \end{equation}\end{split}\]

Mott Scattering Form

\[\begin{equation} \frac{d\sigma}{d\Omega_{M}} = \frac{d\sigma}{d\Omega_{R}} \times [1-\beta^{2} sin^{2} (\theta/2)] \end{equation}\]

Form Factor Modification

(44)\[\begin{equation} \left(\frac{d\sigma}{d\Omega}\right)_{\textnormal{distributed}} = |F({\bf q}^{2})|^{2}\times \left(\frac{d\sigma}{d\Omega}\right)_{\textnormal{point-like}}, \end{equation}\]

Relation between Form Factor and Charge Distribution

(45)\[\begin{equation} F(q) = \int_{-\infty}^{\infty} e^{i~{\bf q} \cdot {\bf r} / \hbar} ~\rho({\bf r})~d^{3} {\bf r} \end{equation}\]

Saxon Woods Distribution

(46)\[\begin{equation} \rho(r) = \frac{\rho_0}{1+{\textnormal{exp}}[\frac{r-a}{d}]} \end{equation}\]

Typical Values For the charge density distribution

(47)\[\begin{equation} \rho_{\textnormal{charge}}^{0} = \frac{\rho^0_{\textnormal{charge}}}{1+\textnormal{exp}[\frac{r-a}{d}]} \end{equation}\]

Typical Values For the mass density distribution

(48)\[\begin{equation} \rho_{\textnormal{mass}}(r) = \frac{\rho^0_{\textnormal{mass}}}{1+\textnormal{exp}[\frac{r-a}{d}]} \end{equation}\]

Coulomb Energy Estimate

(49)\[\begin{equation} E_{e} = \frac{3}{5} \frac{Z^{2}e^{2}}{4\pi \epsilon_{0}R} \end{equation}\]

Coulomb Constant assuming spherical integration

(50)\[\begin{equation} a_{c} = \frac{3}{5} \times \frac{e^{2}}{4\pi \epsilon_{0} r_{0}} \approx 0.7~\textnormal{MeV} \end{equation}\]

Nucleon Pairing Term

(51)\[\begin{split}\begin{align} \delta_{\textnormal{pair}} &= +a_{p}A^{-1/2} ~~~~\textnormal{for N odd AND Z odd}\\ \delta_{\textnormal{pair}} &= 0 ~~~~~~~~~~~~~~~~~~~\textnormal{for N even OR Z even} \\ \delta_{\textnormal{pair}} &= -a_{p}A^{-1/2} ~~~~\textnormal{for N even AND Z even}\\ \end{align}\end{split}\]

Nucleon Asymmetry Term

(52)\[\begin{align} B_{A} = -a_{a} (A-2Z)^{2}A^{-1} \end{align}\]

MAGIC NUMBERS

(53)\[\begin{equation} \bf \textnormal{Magic Nuclei -}~ Z~\textnormal{or}~N=2, 8, 20, 28, 50, 82, 126... \end{equation}\]

SHELL MODEL POTENTIAL

(54)\[\begin{equation} V(r) = \frac{-V_{0}}{1+\textnormal{exp}(\frac{r-b}{a})} \end{equation}\]

SHELL QUANTUM NUMBERS AND PROJECTIONS

(55)\[\begin{equation} \langle l^{2} \rangle = \hbar^{2}l(l+1) \end{equation}\]

(56)\[\begin{equation} \langle l_{z} \rangle = \hbar m_{z} ~~~~\textnormal{where}~~~~ m_z=0,\pm1,\pm2,...,\pm l \end{equation}\]

(57)\[\begin{equation} \langle s^{2} \rangle = \hbar^{2} s(s+1) \end{equation}\]

(58)\[\begin{equation} \langle s_z\rangle = \hbar m_{s} ~~~~\textnormal{where}~~~~ m_{s} = \pm \frac{1}{2}. \end{equation}\]

SHELL MODEL NUMBER OF STATES

\[ n_{\textnormal{states}} = (2j+1) \]

SPIN ORBIT COUPLING POTENTIAL OPERATOR

\[ V=V(r) \hat{L} \cdot \hat{S}. \]

PARITY COMBINATION

(59)\[\begin{equation} \pi_{total}=\pi_{1}\pi_{2}\pi_{3}\pi_{4} = \prod_{A}(-1)^{l} = + (\textnormal{even})~~\textnormal{or}~~-(\textnormal{odd}) \end{equation}\]

Magnetic Moment Free Nucleon

(60)\[\begin{equation} \mu = g_{I} \frac{e\hbar}{2m_{p}} I = g_{I} I \mu_{N} \end{equation}\]

Nuclear Magneton

(61)\[\begin{equation} \mu_{N} = \frac{e\hbar}{2m_{p}} = 5.05\times 10^{-27}\textnormal{J/T} = 3.25 \times 10^{-8} \textnormal{eV/T} \end{equation}\]

Magnetic Moment and Projection

(62)\[\begin{equation} \mu = g_{j}~j~\mu_{N} \end{equation}\]

(63)\[\begin{equation} \mu_{obs} = g_j j_z \mu_N \end{equation}\]

Magneton Estimates

(64)\[\begin{equation} g = g_{l} \frac{j(j+1)+l(l+1)-s(s+1)}{2j(j+1)} + g_s \frac{j(j+1)+s(s+1)-l(l+1)}{2j(j+1)} \end{equation}\]

(65)\[\begin{equation} \frac{\mu}{\mu_N} = (j-\frac{1}{2})g_l + \frac{1}{2}g_s \rightarrow (stretched) \end{equation}\]

(66)\[\begin{equation} \frac{\mu}{\mu_N} = ((j+\frac{3}{2})g_l - \frac{1}{2}g_s)\frac{j}{j+1} \rightarrow (jackknife) \end{equation}\]

G Factors

(67)\[\begin{equation} g_{s} = 5.85 ~~\textnormal{(proton)} ~~~~~~ g_{s} = -3.826 ~~\textnormal{(neutron)} \end{equation}\]

(68)\[\begin{equation} g_l = 1~~\textnormal{(proton)} ~~~~~~~ g_l = 0 ~~\textnormal{(neutron)} \end{equation}\]

Quadrupole Operations

(69)\[\begin{equation} eQ_{EQM} = e\int \psi^{*} (3z^{2} - r^{2})\psi dV \end{equation}\]

(70)\[\begin{equation} Q_{EQM,sp} = - < r^{2} > \frac{2j-1}{2(j+1)} \end{equation}\]

(71)\[\begin{equation} < Q_{EQM, \textnormal{unfilled}} > = < Q_{EQM,sp} > \left [ 1-2\frac{n-1}{2j-1} \right ] \end{equation}\]

NUCLEAR SURFACE BASED ON BETA

\[ R(\theta, \phi) = R_{av} [Y_{00}+\beta Y_{20} (\theta, \phi)] \]

BETA BASED ON ELONGATION

\[ \beta = \frac{4}{3} \sqrt{\frac{\pi}{5}} \frac{\Delta R}{R_{av}} \]

INTRINSIC ELECTRIC QUADRUPOLE MOMENT

\[ Q_{0} = \frac{3}{\sqrt{5\pi}} R_{av}^{2} Z \beta (1+0.16\beta) \]

ENERGY LEVELS ROTATIONAL SPIN J

\[ \frac{\hat{L}^{2}}{2I} \psi = E_{J} \psi \]

\[ \hat{L}^{2} Y_{JM}(\theta,\phi) = J(J+1) \hbar^{2} Y_{JM} (\theta, \phi) \]

\[ E_{J} = \frac{\hbar^{2}}{2I}J(J+1)~~~~~J=0,2,4,\ldots \]

ENERGY PREDICTION USING E2 State

\[ E_{J} = \frac{1}{6} J(J+1)E_{2}~~~~~~~~~J=0,2,4,\ldots \]

VIBRATIONAL VARIATIONS

\[ R(t, \theta,\phi) = R_{av} + \sum_{\lambda>1}^{\lambda=\infty} \sum_{\mu=-\lambda}^{\mu=+\lambda} \alpha_{\lambda \mu}(t) Y_{\lambda \mu} (\theta, \phi) \]

VIBRATIONAL ENERGY LEVELS

\[ E_{N} = \hbar \omega_{l} \left(\frac{2l+1}{2} +N \right) \]

GAMMA RULES

\[ | I_{f} - I_{i} | \leq L \leq|I_{f} + I_{i}| \]

• Electric multipole radiation \(\pi=(-1)^l\).

• Magnetic multipole radiation \(\pi(-1)^{(l+1)}\).

Half Life

\[T_{1/2} = \tau \textnormal{ln}(2) = 0.693\tau\]

Mass Deficit

(72)\[\begin{equation} Q = M(A,Z)^{2}c^{2} - [M(A-X,Z-Y)c^{2} + M(X,Y)c^{2}] > 0 \end{equation}\]

Q Value Beta- Decay in AMU

(73)\[\begin{equation} Q_{\beta^{-}} = [ M(A,Z) - M(A,Z-1) ]c^{2} \end{equation}\]

Q Value Beta+ Decay or EC in AMU

(74)\[\begin{equation} Q_{\beta^{+}} = [ M(A,Z) - M(A,Z-1) - 2m_{e}]c^{2} \end{equation}\]

Lambda and Matrix Element Relation

(75)\[\begin{equation} \lambda = \frac{2\pi}{\hbar} ~| M_{if} |^{2} ~\frac{dn_{f}}{dE_{f}} \end{equation}\]

(76)\[\begin{equation} |M_{fi}| = \int \psi^{*} \hat{H} \psi dV. \end{equation}\]

Beta Flux Relation

(77)\[\begin{equation} N(p) = \frac{C}{c^{2}} p^{2} (Q-T_{e})^{2} \end{equation}\]

Alpha Decay Geiger Nutall

\[ \log_{10} t_{1/2} = b_{1} \frac{Z}{Q^{1/2}} + b_{2} \]

or

\[ \ln\lambda = -a_{1} \frac{Z}{Q^{1/2}} + a_2 \]

Alpha Tunnelling Transmission Probability

(78)\[\begin{equation} X = Ae^{-αL} \end{equation}\]

Reaction Kinetic Energy Calculation

(79)\[\begin{equation} K_b^{1/2} = \frac{\cos \theta \sqrt{m_am_bK_a} \pm [m_am_bK_a \cos^2 \theta + (m_b + m_B)((m_B - m_a)K_a + m_BQ)]^{1/2}}{m_b + m_B} \end{equation} \]

Breit Wigner Form

(80)\[\begin{equation} \sigma_{BW}(E) = \frac{\pi}{(k)^2} \frac{(2J + 1)}{(2I_a + 1)(2I_A + 1)} \frac{\Gamma_a(E)\Gamma_b(E)}{(E - E_R)^2 + \Gamma(E)^2/4}, \end{equation}\]

Compound Branching Ratio

(81)\[\begin{equation} BR(C* \rightarrow b + B) = \frac{\Gamma_b}{\Gamma}. \end{equation}\]

Fission Requirement

\[ 2a_s A^{2/3}\leq a_{C} \frac{Z^2}{A^{1/3}} \quad \Rightarrow \quad \left(\frac{Z^2}{A}\right)_{\text{crit}} \geq \frac{2a_s}{a_c} \approx 49 \]

Fission Four Factor Formula

\[ k=ηϵpf \]

Fusion Coulomb Barrier Limit

\[ \frac{Z_1Z_2e^2}{4\pi\epsilon_0(R_1 + R_2)} = 1.44 \frac{Z_1Z_2}{R_1 + R_2} \text{ MeV}, \]

Fusion Reaction Rate

\[ R_{12} = n_1 n_2 ⟨σv⟩. \]

Fusion Maxwell Boltzmann

\[ P(v)dv = \sqrt{\frac{2}{\pi}} \left( \frac{m}{k_B T} \right)^{3/2} \exp\left( -\frac{mv^2}{2k_B T} \right) v^2 dv, \]

Fusion Lawson Criteria

\[ L = \textnormal{energy output} / \textnormal{energy input} \]

Fusion Energy Out

\[ E_{out} = n_d^2 ⟨σ_{dt}v⟩ Q t_c \]