Formula sheet: Nuclei

1.1 How to use this sheet

Nuclear numericals in this chapter are about careful bookkeeping. Most marks are lost by using the wrong particle count, forgetting that the tabulated masses are for atoms and not bare nuclei, or dropping a factor of 931.5 in the final conversion. Keep the conversion line visible in every answer.

1.2 Nuclear size and density

The radius of a nucleus follows

R = R0 A to the power one-third, with R0 = 1.2 fm = 1.2 × 10 to the -15 m.

Since the volume goes as R cubed and therefore as A, the nuclear mass density is nearly the same for every nucleus:

density = mass/volume, which gives about 10 to the 17 kg per cubic metre. This near-constancy is the reason the chapter says nuclear density is independent of the mass number.

1.3 Atomic mass unit and masses

1 u = 1/12 of the mass of a carbon-12 atom = 1.660539 × 10 to the -27 kg, and 1 u of mass is equivalent to 931.5 MeV.

m proton = 1.00727 u, m neutron = 1.00866 u, m electron = 0.00055 u, mass of hydrogen atom = 1.00783 u.

1.4 Mass defect and binding energy

For a nucleus of mass M with Z protons and A - Z neutrons:

mass defect ΔM = [Z m p + (A - Z) m n] - M

binding energy E b = ΔM c squared = ΔM × 931.5 MeV when ΔM is in u

binding energy per nucleon E bn = E b/A

For the middle range 30 < A < 170 the value is nearly constant at about 8 MeV per nucleon, reaching a maximum near 8.75 MeV for A = 56; for A = 238 it is about 7.6 MeV.

1.5 Q value of a reaction

Q = (total mass before - total mass after) × c squared, which is also final kinetic energy minus initial kinetic energy. A positive Q means the reaction releases energy.

1.6 Radioactive decay

The number left after time t:

N = N0 e to the power -λt

The half-life and the decay constant are linked by

T one-half = 0.693/λ, and the mean life τ = 1/λ.

Activity, the decays per second:

R = λN, with the becquerel as its unit, and 1 curie = 3.7 × 10 to the 10 becquerel.

1.7 Useful conversions

1 MeV = 1.6 × 10 to the -13 J, 1 u = 931.5 MeV/c squared, 1 fm = 10 to the -15 m, Avogadro number 6.023 × 10 to the 23 per mole, 1 year = 3.154 × 10 to the 7 s.