Atomic and Nuclear Physics - Formula Sheet

Atomic Structure Formulas

\(r_n = a_0 \frac{n^2}{Z}\) (Radius of nth electron orbit)
where \(a_0 = 0.529 \AA\) is the Bohr radius, \(n\) is the principal quantum number, and \(Z\) is the atomic number.

\(v_n = v_0 \frac{Z}{n}\) (Velocity of electron in nth orbit)
where \(v_0 \approx 2.19 \times 10^6 m/s\) is the velocity in the hydrogen ground state.

\(E_n = -13.6 \frac{Z^2}{n^2} eV\) (Total energy in nth orbit)
The negative sign indicates the electron is bound to the nucleus.

\(\frac{1}{\lambda} = R \left( \frac{1}{n^2} - \frac{1}{m^2} \right)\) (Rydberg formula for wave number)
where \(R \approx 1.097 \times 10^7 m^{-1}\) is the Rydberg constant and \(m > n\).

Nuclear Physics Formulas

\(R = R_0 A^{1/3}\) (Nuclear radius empirical formula)
where \(R_0 \approx 1.2 F\) and \(A\) is the mass number.

\(\Delta m = [Z m_p + N m_n] - M\) (Nuclear mass defect)
where \(m_p\) is proton mass, \(m_n\) is neutron mass, and \(M\) is the actual nuclear mass.

\(BE = \Delta m c^2\) (Total binding energy)
Commonly calculated using \(1 u = 931.5 MeV\).

Radioactivity Formulas

\(N = N_0 e^{-\lambda t}\) (Radioactive decay law)
where \(N_0\) is the initial count, \(\lambda\) is the decay constant, and \(t\) is time.

\(T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.6931}{\lambda}\) (Half-life period)

\(\tau = \frac{1}{\lambda}\) (Mean life of a radioactive nucleus)

\(R = \lambda N\) (Activity or decay rate)
Measured in Becquerels (Bq) or Curies (Ci).