List of equations in nuclear and particle physics

This article summarizes equations in the theory of nuclear physics and particle physics.

Definitions

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Number of atoms N = Number of atoms remaining at time t

N0 = Initial number of atoms at time t = 0
ND = Number of atoms decayed at time t

$N_0 = N + N_D \,\!$ dimensionless dimensionless
Decay rate, activity of a radioisotope A $A = \mathrm{d} N /\mathrm{d} t \,\!$ Bq = Hz = s−1 [T]−1
Decay constant λ $\lambda = A/N \,\!$ Bq = Hz = s−1 [T]−1
Half-life of a radioisotope t1/2, T1/2 Time taken for half the number of atoms present to decay

$t \rightarrow t + T_{1/2} \,\!$
$N \rightarrow N / 2 \,\!$

s [T]
Number of half-lives n (no standard symbol) $n = t / T_{1/2} \,\!$ dimensionless dimensionless
Radioisotope time constant, mean lifetime of an atom before decay τ (no standard symbol) $\tau = 1 / \lambda \,\!$ s [T]
Absorbed dose, total ionizing dose (total energy of radiation transferred to unit mass) D can only be found experimentally N/A Gy = 1 J/kg (Gray) [L]2[T]−2
Equivalent dose H $H = DQ \,\!$

Q = radiation quality factor (dimensionless)

Sv = J kg−1 (Sievert) [L]2[T]−2
Effective dose E $E = \sum_j H_jW_j \,\!$

Wj = weighting factors corresponding to radiosensitivities of matter (dimensionless)

$\sum_j W_j = 1 \,\!$

Sv = J kg−1 (Sievert) [L]2[T]−2

Equations

Nuclear structure

Physical situation Nomenclature Equations
Mass number
• A = (Relative) atomic mass = Mass number = Number of protons and neutrons
• N = Number of neutrons
• Z = Atomic number = Number of protons = Number of electrons
$A = Z+N\,\!$
Mass in nuclei
• M'nuc = Mass of nucleus, bound nucleons
• MΣ = Sum of masses for isolated nucleons
• mp = proton rest mass
• mn = neutron rest mass
• $M_\Sigma = Zm_p + Nm_n \,\!$
• $M_\Sigma > M_N \,\!$
• $\Delta M = M_\Sigma - M_\mathrm{nuc} \,\!$
• $\Delta E = \Delta M c^2\,\!$

r0 ≈ 1.2 fm

$r=r_0A^{1/3} \,\!$ hence (approximately)
• nuclear volume ∝ A
• nuclear surface ∝ A2/3
Nuclear binding energy, empirical curve Dimensionless parameters to fit experiment:
• EB = binding energy,
• av = nuclear volume coefficient,
• as = nuclear surface coefficient,
• ac = electrostatic interaction coefficient,
• aa = symmetry/asymmetry extent coefficient for the numbers of neutrons/protons,
\begin{align} E_B = & a_v A - a_s A^{2/3} - a_c Z(Z-1)A^{-1/3} \\ & -a_a (N-Z)^2 A^{-1} + 12\delta(N,Z)A^{-1/2} \\ \end{align} where (due to pairing of nuclei)
• δ(N, Z) = +1 even N, even Z,
• δ(N, Z) = −1 odd N, odd Z,
• δ(N, Z) = 0 odd A

Nuclear decay

Physical situation Nomenclature Equations
• N0 = Initial number of atoms
• N = Number of atoms at time t
• λ = Decay constant
• t = Time

$\frac{\mathrm{d} N}{\mathrm{d} t} = - \lambda N$

$N = N_0e^{-\lambda t}\,\!$

Bateman's equations $c_i = \prod_{j=1, i\neq j}^D \frac{\lambda_j}{\lambda_j - \lambda_i}$ $N_D = \frac{N_1(0)}{\lambda_D} \sum_{i=1}^D \lambda_i c_i e^{-\lambda_i t}$
• I0 = Initial intensity/Flux of radiation
• I = Number of atoms at time t
• μ = Linear absorption coefficient
• x = Thickness of substance
$I = I_0e^{-\mu x}\,\!$

Nuclear scattering theory

The following apply for the nuclear reaction:

a + bRc

in the centre of mass frame, where a and b are the initial species about to collide, c is the final species, and R is the resonant state.

Physical situation Nomenclature Equations
Breit-Wigner formula
• E0 = Resonant energy
• Γ, Γab, Γc are widths of R, a + b, c respectively
• k = incoming wavenumber
• s = spin angular momenta of a and b
• J = total angular momentum of R
Cross-section:

$\sigma(E) = \frac{\pi g}{k^2}\frac{\Gamma_{ab}\Gamma_c}{(E-E_0)^2+\Gamma^2/4}$

Spin factor:

$g = \frac{2J+1}{(2s_a+1)(2s_b+1)}$

Total width:

$\Gamma = \Gamma_{ab} + \Gamma_c$

$\tau = \hbar/\Gamma$

Born scattering
• μ = Scattering angle
• A = 2 (spin-0), −1 (spin-half particles)
• Δk = change in wavevector due to scattering
• V = total interaction potential
• V = total interaction potential
Differential cross-section:

$\frac{d\sigma}{d\Omega} = \left|\frac{2\mu}{\hbar^2}\int_0^\infty\frac{\sin(\Delta kr)}{\Delta kr}V(r)r^2dr\right|^2$

Mott scattering
• χ = reduced mass of a and b
• v = incoming velocity
Differential cross-section (for identical particles in a coulomb potential, in centre of mass frame):

$\frac{d\sigma}{d\Omega}=\left(\frac{\alpha}{4E}\right)\left[\csc^{4}\frac{\chi}{2}+\sec^{4}\frac{\chi}{2}+\frac{A\cos\left(\frac{\alpha}{\hbar\nu}\ln\tan^{2}\frac{\chi}{2}\right)}{\sin^{2}\frac{\chi}{2}\cos\frac{\chi}{2}}\right]^{2}$

Scattering potential energy (α = constant):

$V = -\alpha/r$

Rutherford scattering Differential cross-section (non-identical particles in a coloumb potential):

$\frac{d\sigma}{d\Omega}=\left(\frac{1}{n}\right)\frac{dN}{d\Omega} = \left(\frac{\alpha}{4E}\right)^2 \csc^4\frac{\chi}{2}$

Fundamental forces

Name Equations
Strong force \begin{align} \mathcal{L}_\mathrm{QCD} & = \bar{\psi}_i\left(i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \\ & = \bar{\psi}_i (i \gamma^\mu \partial_\mu - m )\psi_i - g G^a_\mu \bar{\psi}_i \gamma^\mu T^a_{ij} \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \,,\\ \end{align} \,\!
Electroweak interaction :$\mathcal{L}_{EW} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y.\,\!$
$\mathcal{L}_g = -\frac{1}{4}W_a^{\mu\nu}W_{\mu\nu}^a - \frac{1}{4}B^{\mu\nu}B_{\mu\nu}\,\!$
$\mathcal{L}_f = \overline{Q}_i iD\!\!\!\!/\; Q_i+ \overline{u}_i^c iD\!\!\!\!/\; u^c_i+ \overline{d}_i^c iD\!\!\!\!/\; d^c_i+ \overline{L}_i iD\!\!\!\!/\; L_i+ \overline{e}^c_i iD\!\!\!\!/\; e^c_i \,\!$
$\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2\,\!$
$\mathcal{L}_y = - y_{u\, ij} \epsilon^{ab} \,h_b^\dagger\, \overline{Q}_{ia} u_j^c - y_{d\, ij}\, h\, \overline{Q}_i d^c_j - y_{e\,ij} \,h\, \overline{L}_i e^c_j + h.c.\,\!$
Quantum electrodynamics $\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\;,\,\!$

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