List of equations in nuclear and particle physics

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


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

  dimensionless dimensionless
Decay rate, activity of a radioisotope A   Bq = Hz = s−1 [T]−1
Decay constant λ   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


s [T]
Number of half-lives n (no standard symbol)   dimensionless dimensionless
Radioisotope time constant, mean lifetime of an atom before decay τ (no standard symbol)   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  

Q = radiation quality factor (dimensionless)

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

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


Sv = J kg−1 (Sievert) [L]2[T]−2


Nuclear structureEdit

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
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
Nuclear radius r0 ≈ 1.2 fm  

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,
  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 decayEdit

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:



Bateman's equations    
Radiation flux
  • I0 = Initial intensity/Flux of radiation
  • I = Number of atoms at time t
  • μ = Linear absorption coefficient
  • x = Thickness of substance

Nuclear scattering theoryEdit

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


Spin factor:


Total width:


Resonance lifetime:


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:


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):


Scattering potential energy (α = constant):


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


Fundamental forcesEdit

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  
Electroweak interaction  
Quantum electrodynamics  

See alsoEdit



  • B. R. Martin, G.Shaw. Particle Physics (3rd ed.). Manchester Physics Series, John Wiley & Sons. ISBN 978-0-470-03294-7.
  • D. McMahon (2008). Quantum Field Theory. Mc Graw Hill (USA). ISBN 978-0-07-154382-8.
  • P.M. Whelan, M.J. Hodgeson (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1.
  • G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2.
  • A. Halpern (1988). 3000 Solved Problems in Physics, Schaum Series. Mc Graw Hill. ISBN 978-0-07-025734-4.
  • R.G. Lerner, G.L. Trigg (2005). Encyclopaedia of Physics (2nd ed.). VHC Publishers, Hans Warlimont, Springer. pp. 12–13. ISBN 978-0-07-025734-4.
  • C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. ISBN 0-07-051400-3.
  • P.A. Tipler, G. Mosca (2008). Physics for Scientists and Engineers: With Modern Physics (6th ed.). W.H. Freeman and Co. ISBN 978-1-4292-0265-7.
  • J.R. Forshaw, A.G. Smith (2009). Dynamics and Relativity. Wiley. ISBN 978-0-470-01460-8.

Further readingEdit