Unit 4 : Equations & Terms

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Unit 4 : 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.

PARITY COMBINATION

(22)#\[\begin{equation} \pi_{total}=\pi_{1}\pi_{2}\pi_{3}\pi_{4} = \prod_{A}(-1)^{l} = + (\textnormal{even})~~\textnormal{or}~~-(\textnormal{odd}) \end{equation}\]

  • if two neutrons (or protons) occupy a level with the same \(j\) and \(l\), then their total angular momenta couple to give zero contribution.

Magnetic Moment Free Nucleon

(23)#\[\begin{equation} \mu = g_{I} \frac{e\hbar}{2m_{p}} I = g_{I} I \mu_{N} \end{equation}\]

Nuclear Magneton

(24)#\[\begin{equation} \mu_{N} = \frac{e\hbar}{2m_{p}} = 5.05\times 10^{-27}\textnormal{J/T} = 3.25 \times 10^{-8} \textnormal{eV/T} \end{equation}\]

Magnetic Moment and Projection

(25)#\[\begin{equation} \mu = g_{j}~j~\mu_{N} \end{equation}\]
(26)#\[\begin{equation} \mu_{obs} = g_j j_z \mu_N \end{equation}\]

Magneton Estimates

(27)#\[\begin{equation} g = g_{l} \frac{j(j+1)+l(l+1)-s(s+1)}{2j(j+1)} + g_s \frac{j(j+1)+s(s+1)-l(l+1)}{2j(j+1)} \end{equation}\]

We arrive at only two possible scenarios for \(g\):

(28)#\[\begin{equation} \frac{\mu}{\mu_N} = (j-\frac{1}{2})g_l + \frac{1}{2}g_s \rightarrow (stretched) \end{equation}\]
(29)#\[\begin{equation} \frac{\mu}{\mu_N} = ((j+\frac{3}{2})g_l - \frac{1}{2}g_s))\frac{j}{j+1} \rightarrow (jackknife) \end{equation}\]

G Factors

(30)#\[\begin{split}\begin{equation} g_l = 1~~\textnormal{(proton)} ~~~~~~~ g_l = 0 ~~\textnormal{(neutron)}\\ g_{s} = 5.85 ~~\textnormal{(proton)} ~~~~~~ g_{s} = -3.826 ~~\textnormal{(neutron)} \end{equation}\end{split}\]

Quadrupole Operations

(31)#\[\begin{equation} eQ_{EQM} = e\int \psi^{*} (3z^{2} - r^{2})\psi dV \end{equation}\]

Calculations to find the expectation value \(< \psi >^{2}\) end up giving as values for \(Q\) for single particle missing/addition states of:

(32)#\[\begin{equation} Q_{EQM} = - < r^{2} > \frac{2j-1}{2(j+1)} \end{equation}\]

For unfilled shells with more than one particle missing:

(33)#\[\begin{equation} < Q_{EQM} > = < Q_{sp} > \left [ 1-2\frac{n-1}{2j-1} \right ] \end{equation}\]
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