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List of equations in nuclear and particle physics

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Nuclear physics
Models of the nucleus
Nuclides' classification
Nuclear stability
Radioactive decay
Nuclear fission
Capturing processes
High-energy processes
Nucleosynthesis and
nuclear astrophysics
High-energy nuclear physics
Scientists

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 {\displaystyle N_{0}=N+N_{D}\,\!} dimensionless dimensionless
Decay rate, activity of a radioisotope A A = λ N {\displaystyle A=\lambda N\,\!} Bq = Hz = s
Decay constant λ λ = A / N {\displaystyle \lambda =A/N\,\!} Bq = Hz = s
Half-life of a radioisotope t1/2, T1/2 Time taken for half the number of atoms present to decay

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

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

Q = radiation quality factor (dimensionless)

Sv = J kg (Sievert)
Effective dose E E = ∑ j H j W j {\displaystyle E=\sum _{j}H_{j}W_{j}\,\!}

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

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

Sv = J kg (Sievert)

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
 A = Z + N {\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
  • M Σ = Z m p + N m n {\displaystyle M_{\Sigma }=Zm_{p}+Nm_{n}\,\!}
  • M Σ > M N {\displaystyle M_{\Sigma }>M_{N}\,\!}
  • Δ M = M Σ − M n u c {\displaystyle \Delta M=M_{\Sigma }-M_{\mathrm {nuc} }\,\!}
  • Δ E = Δ M c 2 {\displaystyle \Delta E=\Delta Mc^{2}\,\!}
Nuclear radius r0 ≈ 1.2 fm r = r 0 A 1 / 3 {\displaystyle r=r_{0}A^{1/3}\,\!}

hence (approximately)

  • nuclear volume ∝ A
  • nuclear surface ∝ A
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,
 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 δ ( N , Z ) A − 1 / 2 {\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
Radioactive decay
  • N0 = Initial number of atoms
  • N = Number of atoms at time t
  • λ = Decay constant
  • t = Time
 Statistical decay of a radionuclide:

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

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

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

Nuclear scattering theory

The following apply for the nuclear reaction:

a + b ↔ R → c

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:

σ ( E ) = π g k 2 Γ a b Γ c ( E − E 0 ) 2 + Γ 2 / 4 {\displaystyle \sigma (E)={\frac {\pi g}{k^{2}}}{\frac {\Gamma _{ab}\Gamma _{c}}{(E-E_{0})^{2}+\Gamma ^{2}/4}}}

Spin factor:

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

Total width:

Γ = Γ a b + Γ c {\displaystyle \Gamma =\Gamma _{ab}+\Gamma _{c}}

Resonance lifetime:

τ = ℏ / Γ {\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:

d σ d Ω = | 2 μ ℏ 2 ∫ 0 ∞ sin ⁡ ( Δ k r ) Δ k r V ( r ) r 2 d r | 2 {\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):

d σ d Ω = ( α 4 E ) [ csc 4 ⁡ χ 2 + sec 4 ⁡ χ 2 + A cos ⁡ ( α ℏ ν ln ⁡ tan 2 ⁡ χ 2 ) sin 2 ⁡ χ 2 cos ⁡ χ 2 ] 2 {\displaystyle {\frac {d\sigma }{d\Omega }}=\left({\frac {\alpha }{4E}}\right)\left^{2}}

Scattering potential energy (α = constant):

V = − α / r {\displaystyle V=-\alpha /r}

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

d σ d Ω = ( 1 n ) d N d Ω = ( α 4 E ) 2 csc 4 ⁡ χ 2 {\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 L Q C D = ψ ¯ i ( i γ μ ( D μ ) i j − m δ i j ) ψ j − 1 4 G μ ν a G a μ ν = ψ ¯ i ( i γ μ ∂ μ − m ) ψ i − g G μ a ψ ¯ i γ μ T i j a ψ j − 1 4 G μ ν a G a μ ν , {\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 L E W = L g + L f + L h + L y . {\displaystyle {\mathcal {L}}_{\mathrm {EW} }={\mathcal {L}}_{g}+{\mathcal {L}}_{f}+{\mathcal {L}}_{h}+{\mathcal {L}}_{y}.\,\!}
L g = − 1 4 W a μ ν W μ ν a − 1 4 B μ ν B μ ν {\displaystyle {\mathcal {L}}_{g}=-{\frac {1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\frac {1}{4}}B^{\mu \nu }B_{\mu \nu }\,\!}
L f = Q ¯ i i D / Q i + u ¯ i c i D / u i c + d ¯ i c i D / d i c + L ¯ i i D / L i + e ¯ i c i D / e i c {\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}\,\!}
L h = | D μ h | 2 − λ ( | h | 2 − v 2 2 ) 2 {\displaystyle {\mathcal {L}}_{h}=|D_{\mu }h|^{2}-\lambda \left(|h|^{2}-{\frac {v^{2}}{2}}\right)^{2}\,\!}
L y = − y u i j ϵ a b h b † Q ¯ i a u j c − y d i j h Q ¯ i d j c − y e i j h L ¯ i e j c + h . c . {\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 L Q E D = ψ ¯ ( i γ μ D μ − m ) ψ − 1 4 F μ ν F μ ν , {\displaystyle {\mathcal {L}}_{\mathrm {QED} }={\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\;,\,\!}

See also

Footnotes

Sources

Further reading

SI units
Base units
Derived units
with special names
Other accepted units
See also
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