Unit 5 : Equations & Terms

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Unit 5 : 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.

NUCLEAR SURFACE BASED ON BETA

\[ R(\theta, \phi) = R_{av} [Y_{00}+\beta Y_{20} (\theta, \phi)] \]

BETA BASED ON ELONGATION

\[ \beta = \frac{4}{3} \sqrt{\frac{\pi}{5}} \frac{\Delta R}{R_{av}} \]

INTRINSIC ELECTRIC QUADRUPOLE MOMENT

\[ Q_{0} = \frac{3}{\sqrt{5\pi}} R_{av}^{2} Z \beta (1+0.16\beta) \]

ENERGY LEVELS ROTATIONAL SPIN J

\[ \frac{\hat{L}^{2}}{2I} \psi = E_{J} \psi \]
\[ \hat{L}^{2} Y_{JM}(\theta,\phi) = J(J+1) \hbar^{2} Y_{JM} (\theta, \phi) \]
\[ E_{J} = \frac{\hbar^{2}}{2I}J(J+1)~~~~~J=0,2,4,\ldots \]

Note that \(I\) here is the moment of inertia of the perturbed nucleus.

ENERGY PREDICTION USING E2 State

\[ E_{J} = \frac{1}{6} J(J+1)E_{2}~~~~~~~~~J=0,2,4,\ldots \]

VIBRATIONAL VARIATIONS

\[ R(t, \theta,\phi) = R_{av} + \sum_{\lambda>1}^{\lambda=\infty} \sum_{\mu=-\lambda}^{\mu=+\lambda} \alpha_{\lambda \mu}(t) Y_{\lambda \mu} (\theta, \phi) \]

VIBRATIONAL ENERGY LEVELS

\[ E_{N} = \hbar \omega_{l} \left(\frac{2l+1}{2} +N \right) \]

Here \(N\) is the number of oscillator quanta, and \(l\) is the quantized oscillator state (0-monopole, 1-dipole, etc).

GAMMA RULES

  • If \(\vec{J_i}\) (initial spin vector) is equal to \(\vec{J_f}\) (final spin vector) then the transition can’t take place. Here the \(J\) terms correspond to the vector components of the nuclear spin. When we say that the an equal \(\vec{J}_{i}\) and \(\vec{J}_{f}\) state mean that the transition can not take place we mean it in terms of the vector components since \(\vec{J}_{i} = \vec{J}_{f} + \vec{L}\). We can however have cases where the overall nuclear spin \(I\) is identical before and afterwards, in this case we expect the allow magnitude of \(\vec{L}\) to be

\[ | I_{f} - I_{i} | \leq L \leq|I_{f} + I_{i}| \]

  • Parity is conserved. Electric multipole radiation has parity \((-1)^l\), while magnetic multipole radiation has parity \((-1)^{(L+1)}\).

GAMMA TRANSITIONS

  • Electric Dipole (E1) Transitions: These transitions involve \(ΔL = 1\) and result in a parity change of π = \((-1)^1\) = -1 (parity-changing). Electric dipole transitions are more common than other multipole transitions and have relatively higher probabilities.

  • Magnetic Dipole (M1) Transitions: These transitions involve \(ΔL = 1\) and result in a parity change of π = \((-1)^{1+1}\) = 1 (parity-preserving). Magnetic dipole transitions are less common than E1 transitions but still occur in certain nuclear decays.

  • Quadrupole (E2) Transitions: Quadrupole transitions involve \(ΔL = 2\) and result in a parity change of π = \((-1)^2\) = 1 (parity-preserving). Quadrupole transitions are less probable than E1 and M1 transitions and are associated with higher-order nuclear excitations.

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