Nuclei formulas

Nucleus

Nuclear radius

$\displaystyle R = R_0A^{1/3}$

R₀ ≈ 1.2 fm. Nuclear density is about the same for all nuclei.

Mass-energy relation

$\displaystyle E = mc^2$

1 u ≈ 931.5 MeV.

Mass defect

$\displaystyle \Delta m = [Zm_p + (A - Z)m_n] - M$

Binding energy

$\displaystyle E_b = \Delta m\,c^2$

Radioactive decay

Decay law

$\displaystyle N = N_0e^{-\lambda t}$

Activity

$\displaystyle R = \lambda N = R_0e^{-\lambda t}$

Half-life

$\displaystyle T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$

Mean life

$\displaystyle \tau = \frac{1}{\lambda}$

Left after n half-lives

$\displaystyle N = \frac{N_0}{2^n}$

Decays

• Alpha decay: A falls by 4 and Z by 2.
• Beta-minus decay: Z rises by 1, and A stays the same.
• Gamma decay: A and Z both stay the same, and only energy is released.
• Fission splits heavy nuclei and fusion joins light ones. Both release energy because binding energy per nucleon rises.