# 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

${\displaystyle N_{0}=N+N_{D}\,\!}$ dimensionless dimensionless
Decay rate, activity of a radioisotope A ${\displaystyle A=\lambda N\,\!}$ Bq = Hz = s−1 [T]−1
Decay constant λ ${\displaystyle \lambda =A/N\,\!}$ 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

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

s [T]
Number of half-lives n (no standard symbol) ${\displaystyle n=t/T_{1/2}\,\!}$ dimensionless dimensionless
Radioisotope time constant, mean lifetime of an atom before decay τ (no standard symbol) ${\displaystyle \tau =1/\lambda \,\!}$ 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 ${\displaystyle H=DQ\,\!}$

Q = radiation quality factor (dimensionless)

Sv = J kg−1 (Sievert) [L]2[T]−2
Effective dose E ${\displaystyle E=\sum _{j}H_{j}W_{j}\,\!}$

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

${\displaystyle \sum _{j}W_{j}=1\,\!}$

Sv = J kg−1 (Sievert) [L]2[T]−2

## Equations

### Nuclear structure

Physical situation Nomenclature Equations
Mass number
• A = (Relative) atomic mass = Mass number = Sum of protons and neutrons
• N = Number of neutrons
• Z = Atomic number = Number of protons = Number of electrons
${\displaystyle A=Z+N\,\!}$
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
• ${\displaystyle M_{\Sigma }=Zm_{p}+Nm_{n}\,\!}$
• ${\displaystyle M_{\Sigma }>M_{N}\,\!}$
• ${\displaystyle \Delta M=M_{\Sigma }-M_{\mathrm {nuc} }\,\!}$
• ${\displaystyle \Delta E=\Delta Mc^{2}\,\!}$

r0 ≈ 1.2 fm

${\displaystyle r=r_{0}A^{1/3}\,\!}$ 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,
{\displaystyle {\begin{aligned}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{aligned}}} 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
Statistical decay of a radionuclide:

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

${\displaystyle N=N_{0}e^{-\lambda t}\,\!}$

Bateman's equations ${\displaystyle c_{i}=\prod _{j=1,i\neq j}^{D}{\frac {\lambda _{j}}{\lambda _{j}-\lambda _{i}}}}$ ${\displaystyle N_{D}={\frac {N_{1}(0)}{\lambda _{D}}}\sum _{i=1}^{D}\lambda _{i}c_{i}e^{-\lambda _{i}t}}$
• I0 = Initial intensity/Flux of radiation
• I = Number of atoms at time t
• μ = Linear absorption coefficient
• x = Thickness of substance
${\displaystyle I=I_{0}e^{-\mu x}\,\!}$

### 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:

${\displaystyle \sigma (E)={\frac {\pi g}{k^{2}}}{\frac {\Gamma _{ab}\Gamma _{c}}{(E-E_{0})^{2}+\Gamma ^{2}/4}}}$

Spin factor:

${\displaystyle g={\frac {2J+1}{(2s_{a}+1)(2s_{b}+1)}}}$

Total width:

${\displaystyle \Gamma =\Gamma _{ab}+\Gamma _{c}}$

${\displaystyle \tau =\hbar /\Gamma }$

Born scattering
• r = radial distance
• 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:

${\displaystyle {\frac {d\sigma }{d\Omega }}=\left|{\frac {2\mu }{\hbar ^{2}}}\int _{0}^{\infty }{\frac {\sin(\Delta kr)}{\Delta kr}}V(r)r^{2}dr\right|^{2}}$

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):

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

Scattering potential energy (α = constant):

${\displaystyle V=-\alpha /r}$

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

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

### Fundamental forces

These equations need to be refined such that the notation is defined as has been done for the previous sets of equations.

Name Equations
Strong force {\displaystyle {\begin{aligned}{\mathcal {L}}_{\mathrm {QCD} }&={\bar {\psi }}_{i}\left(i\gamma ^{\mu }(D_{\mu })_{ij}-m\,\delta _{ij}\right)\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\\&={\bar {\psi }}_{i}(i\gamma ^{\mu }\partial _{\mu }-m)\psi _{i}-gG_{\mu }^{a}{\bar {\psi }}_{i}\gamma ^{\mu }T_{ij}^{a}\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\,,\\\end{aligned}}\,\!}
Electroweak interaction :${\displaystyle {\mathcal {L}}_{EW}={\mathcal {L}}_{g}+{\mathcal {L}}_{f}+{\mathcal {L}}_{h}+{\mathcal {L}}_{y}.\,\!}$
${\displaystyle {\mathcal {L}}_{g}=-{\frac {1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\frac {1}{4}}B^{\mu \nu }B_{\mu \nu }\,\!}$
${\displaystyle {\mathcal {L}}_{f}={\overline {Q}}_{i}iD\!\!\!\!/\;Q_{i}+{\overline {u}}_{i}^{c}iD\!\!\!\!/\;u_{i}^{c}+{\overline {d}}_{i}^{c}iD\!\!\!\!/\;d_{i}^{c}+{\overline {L}}_{i}iD\!\!\!\!/\;L_{i}+{\overline {e}}_{i}^{c}iD\!\!\!\!/\;e_{i}^{c}\,\!}$
${\displaystyle {\mathcal {L}}_{h}=|D_{\mu }h|^{2}-\lambda \left(|h|^{2}-{\frac {v^{2}}{2}}\right)^{2}\,\!}$
${\displaystyle {\mathcal {L}}_{y}=-y_{u\,ij}\epsilon ^{ab}\,h_{b}^{\dagger }\,{\overline {Q}}_{ia}u_{j}^{c}-y_{d\,ij}\,h\,{\overline {Q}}_{i}d_{j}^{c}-y_{e\,ij}\,h\,{\overline {L}}_{i}e_{j}^{c}+h.c.\,\!}$
Quantum electrodynamics ${\displaystyle {\mathcal {L}}={\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\;,\,\!}$

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