Unit 1 : Equations & Terms

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Unit 1 : Equations & Terms#

Equations#

To aid with exam revision below is a list of all equations that you should learn for this unit in the course.

Atomic Mass Unit

\[\begin{equation} u = 1.66056\times 10^{-27}~\textnormal{kg} = 931.5 \textnormal{MeV/c}^2. \end{equation}\]

Nuclear Mass Deficit

\[\begin{equation} \Delta M(A,Z) = M(A,Z) - (ZM_{p} + NM_{n} + Zm_{e}). \end{equation}\]

Binding Energy Equation

\[\begin{equation} -\Delta M(A,Z) = B \end{equation}\]

Nuclear Mass Density

\[\begin{equation} \rho_{m} = \frac{M}{V} = \frac{3m_{n}}{4\pi r^{3}_{0}} \end{equation}\]

Nuclear Charge Density

\[\begin{split}\begin{equation} \rho_{c} = \frac{Q}{V} = \frac{3Ze}{4\pi r^{3}_{0} A}\\ \end{equation}\end{split}\]

Cross-sectional Area

\[\begin{equation} \sigma = \pi (2R)^{2} \end{equation}\]

Mean Free Path

\[\begin{equation} \lambda = \frac{1}{n\sigma}. \end{equation}\]

Beam Intensity Reduction in Target

\[\begin{equation} N = N_0 e^{-x/\lambda} \end{equation}\]

N Collisions in Target

\[\begin{equation} C=N_0-N=N_0(1-e^{-x/\lambda}). \end{equation}\]

Rutherford Scattering

\[\begin{split}\begin{equation} \frac{d\sigma}{d\Omega_{R}} = \frac{z^{2}Z^{2}\alpha^{2}\hbar^{2}c^{2}}{4E^{2}} \frac{1}{sin^{4}(\theta/2)} \\ \end{equation}\end{split}\]

  • \(z\) is the charge of the projectile.

  • \(Z\) is the charge of the scatterer.

  • \(E\) is kinetic energy of the particle

  • \(\theta\) is the scattering angle

  • \(\alpha\) is the fine structure constant

Mott Scattering Form

\[\begin{equation} \frac{d\sigma}{d\Omega_{M}} = \frac{d\sigma}{d\Omega_{R}} \times [1-\beta^{2} sin^{2} (\theta/2)] \end{equation}\]
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