List of equations in nuclear and particle physics explained

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

N0=N+ND

dimensionless dimensionless
Decay rate, activity of a radioisotopeA

A=λN

Bq = Hz = s-1 [T]-1
Decay constantλ

λ=A/N

Bq = Hz = s-1 [T]-1
Half-life of a radioisotopet1/2, T1/2Time taken for half the number of atoms present to decay

tt+T1/2


NN/2\

s [T]
Number of half-livesn (no standard symbol)

n=t/T1/2

dimensionless dimensionless
Radioisotope time constant, mean lifetime of an atom before decayτ (no standard symbol)

\tau=1/λ

s [T]
Absorbed dose, total ionizing dose (total energy of radiation transferred to unit mass)D can only be found experimentallyN/AGy = 1 J/kg (Gray)[L]2[T]−2
Equivalent doseH

H=DQ

Q = radiation quality factor (dimensionless)

Sv = J kg−1 (Sievert)[L]2[T]−2
Effective doseE

E=\sumjHjWj

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

\sumjWj=1\

Sv = J kg−1 (Sievert)[L]2[T]−2

Equations

Nuclear structure

Physical situationNomenclatureEquations
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

    Mass in nuclei
      • Mnuc = Mass of nucleus, bound nucleons
      • MΣ = Sum of masses for isolated nucleons
      • mp = proton rest mass
      • mn = neutron rest mass

        M\Sigma=Zmp+Nmn

        M\Sigma>MN\

        \DeltaM=M\Sigma-Mnuc

        \DeltaE=\DeltaMc2\

        Nuclear radiusr0 ≈ 1.2 fm
        1/3
        r=r
        0A

        hence (approximately)
          • nuclear volume ∝ A
          • nuclear surface ∝ A2/3
          Nuclear binding energy, empirical curveDimensionless 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}EB=&avA-asA2/3-acZ(Z-1)A-1/3\\ &-aa(N-Z)2A-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 situationNomenclatureEquations
              Radioactive decay
                • N0 = Initial number of atoms
                • N = Number of atoms at time t
                • λ = Decay constant
                • t = Time
                Statistical decay of a radionuclide:
                dN
                dt

                =-λN

                N=

                t
                N
                0e

                Bateman's equations

                ci=

                D
                \prod
                j=1,ij
                λj
                λji

                ND=

                N1(0)
                λD
                D
                \sum
                i=1

                λici

                it
                e

                Radiation flux
                  • I0 = Initial intensity/Flux of radiation
                  • I = Number of atoms at time t
                  • μ = Linear absorption coefficient
                  • x = Thickness of substance

                  I=

                  -\mux
                  I
                  0e

                  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 situationNomenclatureEquations
                  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)=

                    \pig
                    k2
                    \Gammaab\Gammac
                    2+\Gamma
                    (E-E2/4
                    0)

                    Spin factor:

                    g=

                    2J+1
                    (2sa+1)(2sb+1)

                    Total width:

                    \Gamma=\Gammaab+\Gammac

                    Resonance lifetime:

                    \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\sigma
                      d\Omega

                      =\left

                      \frac\int_0^\infty\fracV(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):
                        d\sigma=\left(
                        d\Omega
                        \alpha
                        4E

                        \right)\left[\csc4

                        \chi
                        2

                        +\sec4

                        \chi+
                        2
                        A\cos\left(\alphaln\tan2
                        \chi
                        2
                        \right)
                        \hbar\nu
                        2
                        \sin
                        \chi
                        2
                        \cos\chi
                        2

                        \right]2

                        Scattering potential energy (α = constant):

                        V=-\alpha/r

                        Rutherford scatteringDifferential cross-section (non-identical particles in a coulomb potential):
                        d\sigma=\left(
                        d\Omega
                        1\right)
                        n
                        dN
                        d\Omega

                        =\left(

                        \alpha
                        4E

                        \right)2

                        4\chi
                        2
                        \csc

                        Fundamental forces

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

                        NameEquations
                        Strong force

                        \begin{align} l{L}QCD &=\bar{\psi}i\left(i\gamma\mu(D\mu)ij-m\deltaij\right)\psij-

                        1
                        4
                        a
                        G
                        \mu\nu
                        \mu\nu
                        G
                        a

                        \\ &=\bar{\psi}i(i\gamma\mu\partial\mu-m)\psii-g

                        a
                        G
                        \mu

                        \bar{\psi}i\gamma\mu

                        a
                        T
                        ij

                        \psij-

                        1
                        4
                        a
                        G
                        \mu\nu
                        \mu\nu
                        G
                        a

                        ,\\ \end{align}

                        Electroweak interaction

                        l{L}EW=l{L}g+l{L}f+l{L}h+l{L}y.

                        l{L}g=-

                        1
                        4
                        \mu\nu
                        W
                        a
                        a
                        W
                        \mu\nu

                        -

                        1
                        4

                        B\mu\nuB\mu\nu\

                        l{L}f=\overline{Q}iiD\

                        \!\/\; Q_i+ \overline_i^c iD\!\\!\/\; u^c_i+ \overline_i^c iD\!\\!\/\; d^c_i+ \overline_i iD\!\\!\/\; L_i+ \overline^c_i iD\!\\!\/\; e^c_i \,\!

                        l{L}h=

                        D_\mu h^2 - \lambda \left(h^2 - \frac\right)^2\,\!

                        l{L}y=-yu\epsilonab

                        \dagger
                        h
                        b

                        \overline{Q}ia

                        c
                        u
                        j

                        -ydh\overline{Q}i

                        c
                        d
                        j

                        -yeijh\overline{L}i

                        c
                        e
                        j

                        +h.c.\

                        Quantum electrodynamics
                        \mu
                        l{L}
                        QED=\bar\psi(i\gamma

                        D\mu-m)\psi-

                        1
                        4

                        F\mu\nuF\mu\nu,

                        See also

                        Sources

                        Further reading