Unit 6 : Equations & Terms

Unit 6 : Equations & Terms#

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

The time for half the sample to decay (half-life)

\[T_{1/2} = \tau \textnormal{ln}(2) = 0.693\tau\]

Mass Deficit

(34)#\[\begin{equation} Q = M(A,Z)^{2}c^{2} - [M(A-X,Z-Y)c^{2} + M(X,Y)c^{2}] > 0 \end{equation}\]

Beta Decay Example

\[ \bf _{83}^{210}Bi \rightarrow _{84}^{210}Po + e^- + \bar{\nu}_e \]

Q Value Beta- Decay

(35)#\[\begin{equation} Q_{\beta^{-}} = [ M(A,Z) - M(A,Z-1) ]c^{2} \end{equation}\]

Q Value Beta+ Decay or EC

(36)#\[\begin{equation} Q_{\beta^{+}} = [ M(A,Z) - M(A,Z-1) - 2m_{e}]c^{2} \end{equation}\]

Lambda and Matrix Element

(37)#\[\begin{equation} \lambda = \frac{2\pi}{\hbar} ~| M_{if} |^{2} ~\frac{dn_{f}}{dE_{f}} \end{equation}\]

(38)#\[\begin{equation} |M_{fi}| = \int \psi^{*} \hat{H} \psi dV. \end{equation}\]

Beta Flux

(39)#\[\begin{equation} N(p) = \frac{C}{c^{2}} p^{2} (Q-T_{e})^{2} \end{equation}\]

Beta Superallowed Transition

\[ \Delta I = 0, \quad \Delta \pi = \textnormal{none} \]

Beta Allowed Transition

\[ \Delta I = 0 \text{ or } \pm 1, \quad \Delta \pi = \textnormal{none} \]

except for \(0 \rightarrow 0\) transitions, which are forbidden.

Beta Forbidden Transitions

\[ \Delta I = 0, \pm 1, \pm 2, \quad \Delta \pi = 1 \]

except for \(0 \rightarrow 0\) transitions, which are still completely forbidden.

In general, the “n-th forbidden” transition is characterized by:

\[ \Delta I = n, n \pm 1, n \pm 2, ..., \quad \Delta \pi = (-1)^{n} \]

Alpha Geiger Nutall

\[ \log_{10} t_{1/2} = b_{1} \frac{Z}{Q^{1/2}} + b_{2} \]

or

\[ \ln\lambda = -a_{1} \frac{Z}{Q^{1/2}} + a_2 \]

Alpha Tunnelling Transmission Probability

(40)#\[\begin{equation} X = Ae^{-αL} \end{equation}\]