(unit5:terms)=
## 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.