Interaction of Nuclei with Electromagnetic Radiation & Multipole Transitions

1. Electromagnetic Interaction Hamiltonian

The interaction between an ensemble of nucleons and the quantized electromagnetic radiation field is governed by the minimal coupling Hamiltonian $\vec{p} \to \vec{p} - q\vec{A}$. In the Coulomb gauge ($\nabla\cdot\vec{A} = 0$):

$$\hat{H}_{\text{int}} = -\int \vec{j}(\vec{r})\cdot\vec{A}(\vec{r}) d^3r - \int \vec{M}(\vec{r})\cdot\vec{B}(\vec{r}) d^3r$$

where $\vec{j}(\vec{r}) = \sum_{k} \frac{e_k}{2M} (\vec{p}_k \delta(\vec{r}-\vec{r}_k) + \delta(\vec{r}-\vec{r}_k)\vec{p}_k)$ is the convection current density of protons, and $\vec{M}(\vec{r}) = \sum_k \mu_N g_s^{(k)} \vec{s}_k \delta(\vec{r}-\vec{r}_k)$ is the magnetization density due to nucleon intrinsic spins.

2. Transition Rates from Fermi's Golden Rule

For a nucleus decaying from initial excited state $|i\rangle$ of energy $E_i$ to final state $|f\rangle$ of energy $E_f$ with emission of a photon of energy $\hbar\omega = E_i - E_f$ and wavevector $k = \omega/c$, time-dependent perturbation theory yields Fermi's Golden Rule:

$$\lambda_{fi} = \frac{2\pi}{\hbar} |\langle f; 1_{\vec{k},\epsilon}| \hat{H}_{\text{int}} |i; 0\rangle|^2 \rho(E)$$

where the density of final photon states in solid angle $d\Omega$ is $\rho(E) = \frac{V \omega^2 d\Omega}{(2\pi)^3 \hbar c^3}$.

Because nuclear dimensions ($R \sim 5\text{ fm}$) are vastly smaller than the wavelength of typical gamma rays ($E_\gamma \sim 1\text{ MeV} \implies \lambdabar = \frac{\hbar c}{E_\gamma} \approx 200\text{ fm}$):

$$k R = \frac{R}{\lambdabar} = \frac{5\text{ fm}}{200\text{ fm}} \approx 0.025 \ll 1$$

This long-wavelength condition justifies expanding the vector potential $\vec{A}(\vec{r}) \propto e^{i\vec{k}\cdot\vec{r}}$ in spherical multipoles of order $(k r)^\lambda$.

1. Spherical Multipole Expansion of the Radiation Field

The radiation field can be decomposed rigorously into orthogonal vector spherical harmonics carrying definite total angular momentum $\lambda\hbar$ and parity $\pi$:

• Electric Multipole of Order $\lambda$ ($E\lambda$): Generated by the oscillating nuclear electric charge distribution $\rho(\vec{r})$ and convection currents. The transition operator is: $$\hat{\mathcal{M}}(E\lambda, \mu) = \sum_{k=1}^A e_k r_k^\lambda Y_{\lambda\mu}(\theta_k, \phi_k)$$
• Magnetic Multipole of Order $\lambda$ ($M\lambda$): Generated by the circulating orbital current loops and intrinsic nucleon spin magnetic moments. The transition operator is: $$\hat{\mathcal{M}}(M\lambda, \mu) = \mu_N \sum_{k=1}^A \left[ \frac{2}{\lambda+1} g_l^{(k)} \vec{l}_k + g_s^{(k)} \vec{s}_k \right] \cdot \nabla \left( r_k^\lambda Y_{\lambda\mu}(\hat{r}_k) \right)$$

2. Multipolarity Terminology

Order $\lambda$	Multipolarity Name	Electric Mode	Magnetic Mode
$\lambda = 1$	Dipole	$E1$	$M1$
$\lambda = 2$	Quadrupole	$E2$	$M2$
$\lambda = 3$	Octupole	$E3$	$M3$
$\lambda = 4$	Hexadecapole	$E4$	$M4$

1. Angular Momentum Conservation Selection Rule

A photon emitted in a multipole mode of order $\lambda$ carries an intrinsic angular momentum of $\lambda\hbar$ (with $\lambda \ge 1$, since photons are transverse vector bosons with helicity $\pm 1$ and cannot exist in a scalar $\lambda = 0$ state).

By conservation of angular momentum $\vec{I}_i = \vec{I}_f + \vec{\lambda}$, the triangle inequality requires:

$$|I_i - I_f| \le \lambda \le I_i + I_f$$

Absolute Exclusion of Single-Photon $0 \to 0$ Transitions: If $I_i = 0$ and $I_f = 0$, then $|0 - 0| \le \lambda \le 0 + 0 \implies \lambda = 0$. Because there are no longitudinal or scalar photons in free space, single-photon transitions between two spin-zero states ($0^+ \to 0^+$ or $0^- \to 0^+$) are strictly forbidden by conservation of angular momentum. Such states decay via internal conversion (IC) or internal pair creation ($e^+ e^-$).

2. Parity Selection Rules

The electromagnetic multipole operators transform under spatial inversion $\vec{r} \to -\vec{r}$ according to their parity:

$$\pi(E\lambda) = (-1)^\lambda, \qquad \pi(M\lambda) = (-1)^{\lambda+1}$$

Therefore, the parity of the nuclear states $\pi_i$ and $\pi_f$ must satisfy:

$$\text{For Electric Transitions }(E\lambda): \quad \pi_i \pi_f = (-1)^\lambda \implies \Delta\pi = \begin{cases} \text{No parity change}, & \lambda = 2, 4, 6 \dots (E2, E4) \\ \text{Parity change}, & \lambda = 1, 3, 5 \dots (E1, E3) \end{cases}$$ $$\text{For Magnetic Transitions }(M\lambda): \quad \pi_i \pi_f = (-1)^{\lambda+1} \implies \Delta\pi = \begin{cases} \text{No parity change}, & \lambda = 1, 3, 5 \dots (M1, M3) \\ \text{Parity change}, & \lambda = 2, 4, 6 \dots (M2, M4) \end{cases}$$

3. Summary Decision Matrix

Transition Character	Parity Change ($\pi_i \pi_f$)	Allowed Multipoles	Dominant Mode
$2^+ \to 0^+$	No ($\Delta\pi = +1$)	$E2, M3, E4 \dots$	$E2$
$1^- \to 0^+$	Yes ($\Delta\pi = -1$)	$E1, M2, E3 \dots$	$E1$
$2^+ \to 1^+$	No ($\Delta\pi = +1$)	$M1, E2, M3 \dots$	$M1 + E2$ (Mixed)
$3^- \to 0^+$	Yes ($\Delta\pi = -1$)	$E3, M4, E5 \dots$	$E3$

1. Derivation of the Weisskopf Units (W.u.)

Victor Weisskopf derived standard baseline transition rates by assuming a single valence proton moves from a single-particle orbital $j_i$ to $j_f$ in a spherical nucleus of radius $R = R_0 A^{1/3}$ ($R_0 \approx 1.2\text{ fm}$), with constant radial wavefunctions:

$$\langle r^\lambda \rangle \approx \frac{\int_0^R r^\lambda r^2 dr}{\int_0^R r^2 dr} = \frac{3}{\lambda + 3} R^\lambda$$

The resulting Weisskopf single-particle transition rates $\lambda_W = T_W$ in $\text{s}^{-1}$ for photon energy $E_\gamma$ in MeV are:

$$\lambda_W(E1) = 1.023 \times 10^{14} A^{2/3} E_\gamma^3\text{ s}^{-1}$$ $$\lambda_W(E2) = 7.28 \times 10^7 A^{4/3} E_\gamma^5\text{ s}^{-1}$$ $$\lambda_W(E3) = 3.39 \times 10^1 A^2 E_\gamma^7\text{ s}^{-1}$$ $$\lambda_W(M1) = 3.15 \times 10^{13} E_\gamma^3\text{ s}^{-1}$$ $$\lambda_W(M2) = 2.24 \times 10^7 A^{2/3} E_\gamma^5\text{ s}^{-1}$$ $$\lambda_W(M3) = 1.04 \times 10^1 A^{4/3} E_\gamma^7\text{ s}^{-1}$$

2. Hierarchy and Multipole Dominance

Because $k R \ll 1$, each increase in multipole order $\lambda \to \lambda + 1$ suppresses the transition probability by a colossal factor of $(k R)^2 \sim 10^{-4} \text{ to } 10^{-6}$:

$$\frac{\lambda_W(E(\lambda+1))}{\lambda_W(E\lambda)} \sim (k R)^2 \approx 10^{-4}$$ $$\frac{\lambda_W(M\lambda)}{\lambda_W(E\lambda)} \approx 10^{-2}$$

Therefore, in any electromagnetic transition, the lowest allowed multipole completely dominates the decay rate unless hindered by selection rules.

3. Physical Significance of Weisskopf Units

• If an experimental transition rate matches $\sim 1\text{ W.u.}$, the transition is confirmed to be an independent single-particle transition.
• If $B(E2) \gg 1\text{ W.u.}$ (often $10\text{ to }300\text{ W.u.}$ in deformed rare-earth and actinide nuclei), the transition is collective, involving the coherent motion of dozens of nucleons in a rotating or vibrating nuclear core.
• If $B(E1) \ll 1\text{ W.u.}$ ($10^{-3}\text{ to }10^{-6}\text{ W.u.}$), the transition is strongly hindered, typically by isospin selection rules ($\Delta I = 0$ in self-conjugate $N=Z$ nuclei) or shape changes.

1. The Thermal Neutron Radiative Capture Process

When thermal neutrons ($E_n \approx 0.025\text{ eV}$) interact with hydrogen, capture occurs via gamma emission:

$$n + p \to {}^2\text{H} + \gamma \quad (E_\gamma = B = 2.2246\text{ MeV})$$

The incident thermal neutron has $l = 0$ ($s$-wave), so the initial continuum state has positive parity $\pi_i = +1$. The bound deuteron has $J^\pi = 1^+$.

Because there is no change in parity ($\pi_i = \pi_f = +1$), electric dipole radiation ($E1$, which requires $\Delta\pi = -1$) is strictly forbidden. The dominant decay mode is magnetic dipole radiation ($M1$).

2. The Spin-Flip Capture Mechanism

The transition proceeds from the unbound ${}^1S_0$ continuum state ($J=0, S=0$) to the ${}^3S_1$ component of the deuteron bound state ($J=1, S=1$). The magnetic dipole transition operator is:

$$\vec{\mathcal{M}}(M1) = \mu_N (g_p \vec{s}_p + g_n \vec{s}_n) = \frac{\mu_N}{2} \left[ (\mu_p + \mu_n)(\vec{\sigma}_p + \vec{\sigma}_n) + (\mu_p - \mu_n)(\vec{\sigma}_p - \vec{\sigma}_n) \right]$$

The spin-flip transition between singlet ($S=0$) and triplet ($S=1$) is driven by the term proportional to $(\mu_p - \mu_n)$:

$$\langle {}^3S_1 | (\vec{\sigma}_p - \vec{\sigma}_n) | {}^1S_0 \rangle \ne 0$$

Because $(\mu_p - \mu_n) = 2.793 - (-1.913) = +4.706\text{ }\mu_N$ is exceptionally large, the transition amplitude is greatly enhanced.

3. Cross-Section Discrepancy & Meson Exchange Currents

The cross section calculated from the simple single-particle wavefunctions without meson exchange currents is:

$$\sigma_{\text{calc}}^{(0)} = 302 \pm 4\text{ mb}$$

However, high-precision thermal neutron capture experiments yield:

$$\sigma_{\text{exp}} = 334.2 \pm 0.5\text{ mb}$$

The famous $10\%$ discrepancy ($\Delta\sigma \approx 32\text{ mb}$) is resolved by including Meson Exchange Currents (MEC): during the collision, virtual charged pions ($\pi^\pm$) in flight between the proton and neutron directly couple to the photon field ($\gamma \pi \pi$ and $\gamma N N \pi$ contact vertices), providing conclusive proof of sub-nucleonic meson degrees of freedom in nuclei.

1. The Microscopic Mechanism of Internal Conversion

In an excited nucleus, gamma decay is not the only electromagnetic de-excitation mechanism. The oscillating nuclear electromagnetic multipole field extends into the atomic electron cloud. A bound atomic electron (most commonly from the innermost $K$ shell, $n=1$) can undergo direct electromagnetic interaction with the nucleus, being ejected into the continuum:

$${}^A_Z X^* \to {}^A_Z X^+ + e_{\text{IC}}^-$$

This process is Internal Conversion (IC).

The kinetic energy of the ejected conversion electron is:

$$T_e = E_\gamma - B_e$$

where $E_\gamma = E_i - E_f$ is the nuclear transition energy and $B_e$ is the electron atomic binding energy ($B_K, B_L, \dots$). Consequently, conversion electron spectra consist of discrete, sharp monoenergetic lines, in stark contrast to continuous beta spectra.

2. The Internal Conversion Coefficient (ICC)

The internal conversion coefficient $\alpha$ is defined as the ratio of the conversion electron decay rate ($\lambda_e$) to the gamma-ray emission rate ($\lambda_\gamma$):

$$\alpha \equiv \frac{\lambda_e}{\lambda_\gamma} = \alpha_K + \alpha_{L_I} + \alpha_{L_{II}} + \alpha_{L_{III}} + \alpha_M + \dots$$

The total transition probability is $\lambda_{\text{total}} = \lambda_\gamma + \lambda_e = \lambda_\gamma (1 + \alpha)$.

Parametric dependence of $\alpha$:

$$\alpha(E\lambda) \propto Z^3 \left( \frac{m_e c^2}{E_\gamma} \right)^{\lambda + 7/2}, \qquad \alpha(M\lambda) \propto Z^3 \left( \frac{m_e c^2}{E_\gamma} \right)^{\lambda + 5/2}$$

Key physical properties:

• Heavy Nuclei ($Z^3$ Scaling): IC dominates in heavy elements ($Z \ge 50$) because $K$-shell electrons have wavefunctions strongly concentrated at the nucleus ($\psi_e(0) \propto Z^{3/2}$).
• High Multipolarity ($\lambda \ge 3$): Because $\lambda_\gamma$ is heavily suppressed for high multipolarities while electron conversion near the origin is less hindered, high-$\lambda$ transitions have massive conversion coefficients ($\alpha \gg 1$).
• Low Transition Energy ($E_\gamma \le 200\text{ keV}$): Low energy transitions proceed almost entirely via conversion electrons.

3. Electric Monopole ($E0$) Transitions

When both the initial and final states have spin-parity $0^+$ (such as in ${}^{16}\text{O}^*(6.05\text{ MeV}) \to {}^{16}\text{O}(\text{g.s.}, 0^+)$ or ${}^{72}\text{Ge}^*(691\text{ keV}) \to {}^{72}\text{Ge}(\text{g.s.}, 0^+)$), single-photon emission is strictly forbidden ($\lambda = 0$).

Because the spherically symmetric Coulomb monopole operator $\hat{\mathcal{M}}(E0) = e \sum r_p^2$ can overlap with atomic $s_{1/2}$ electrons entering the nuclear interior, these states de-excite via pure $E0$ internal conversion (or electron-positron pair creation if $E_\gamma > 2 m_e c^2 = 1.022\text{ MeV}$).

1. The Phenomenon of the Giant Dipole Resonance

When atomic nuclei are bombarded with high-energy photons ($E_\gamma \approx 10\text{ to }25\text{ MeV}$), the total photoabsorption cross section $\sigma_{\text{abs}}(E_\gamma)$ does not exhibit isolated narrow Breit-Wigner peaks. Instead, it displays a colossal, universal, broad peak known as the Giant Dipole Resonance (GDR):

• Resonance Energy: Systematically decreases with mass number $A$: $$E_{\text{GDR}} \approx 78 A^{-1/3}\text{ MeV} \quad (\text{or } 31.2 A^{-1/3} + 20.6 A^{-1/6}\text{ MeV})$$ In light nuclei ($A \sim 16$), $E_{\text{GDR}} \approx 22\text{ to }25\text{ MeV}$; in heavy nuclei ($A \sim 208$), $E_{\text{GDR}} \approx 13.5\text{ MeV}$.
• Resonance Width ($\Gamma$): Typically $\Gamma \approx 4\text{ to }6\text{ MeV}$ in spherical magic nuclei, broadening in deformed nuclei.
• Exhaustion of the TRK Sum Rule: The integrated photoabsorption cross section exhausts the classical Thomas-Reiche-Kuhn (TRK) dipole sum rule: $$\int_0^\infty \sigma_{\text{abs}}(E_\gamma) dE_\gamma = \frac{2\pi^2 e^2 \hbar}{M c} \frac{N Z}{A} (1 + \kappa) \approx 60 \frac{N Z}{A}\text{ MeV}\cdot\text{mb}$$ where $\kappa \approx 0.2\text{ to }0.4$ represents meson exchange current enhancements.

2. Macroscopic Hydrodynamic Models: Goldhaber-Teller vs Steinwedel-Jensen

The GDR is a collective macroscopic vibration of all protons against all neutrons:

1. Goldhaber-Teller (GT) Model: The protons are treated as a rigid sphere oscillating out of phase with a rigid neutron sphere against the restoring force of the nuclear symmetry energy. It predicts $E_{\text{GDR}} \propto A^{-1/6}$.
2. Steinwedel-Jensen (SJ) Model: The protons and neutrons form compressible fluids within a fixed spherical nuclear boundary. Acoustic compressional density waves oscillate against each other ($n_p - n_n \ne 0$). It predicts $E_{\text{GDR}} \propto A^{-1/3}$.

3. Resonance Splitting in Deformed Nuclei

In deformed, prolate nuclei (such as the rare-earths ${}^{160}\text{Gd}$ or actinides ${}^{238}\text{U}$), the GDR splits into two distinct peaks:

• Low-Energy Peak ($E_a$): Collective oscillation along the longer major axis ($a$). Since the wavelength is longer, the frequency is lower.
• High-Energy Peak ($E_b$): Collective oscillation along the shorter minor axes ($b$).

The ratio of peak energies directly yields the nuclear deformation axis ratio: $\frac{E_b}{E_a} \approx \frac{a}{b} = 1 + 0.95 \beta_2$.