Magnetic dipole and electric quadrupole radiation: multipole expansion of radiation fields

Intuition Beginner

The electric dipole radiation discussed in 10.07.02 is the strongest type of radiation from a small source. But it is not the only type. Just as a charge distribution can have higher multipole moments (dipole, quadrupole, octupole), the radiation it emits has higher multipole components.

The next two terms in the expansion are magnetic dipole (M1) and electric quadrupole (E2) radiation. Both are suppressed compared to electric dipole (E1) by a factor of roughly $(kd{)}^2$ where $d$ is the source size and $k$ is the wave number. For an atom ($d\sim 10^{-10}$ m) emitting optical light ($k\sim 10^7$ m${}^{-1}$): $(kd{)}^2\sim 10^{-6}$. The higher multipoles are a million times weaker than the electric dipole.

However, if the electric dipole moment is zero (by symmetry), the leading contribution comes from M1 and E2. This happens in many atomic and nuclear transitions. The "forbidden" transitions in spectroscopy are exactly those where E1 is forbidden but M1 or E2 are allowed.

Magnetic dipole radiation comes from oscillating current loops (equivalently, oscillating magnetic moments). Electric quadrupole radiation comes from charge distributions that have no net dipole but have a quadrupole moment — like two dipoles arranged in opposition.

Visual Beginner

Multipole	Symbol	Source	Relative power
Electric dipole	E1	Oscillating charge separation	1
Magnetic dipole	M1	Oscillating current loop	$(kd{)}^2$
Electric quadrupole	E2	Oscillating quadrupole moment	$(kd{)}^2$
Magnetic quadrupole	M2	Higher order	$(kd{)}^4$

Worked example Beginner

A hydrogen atom transition from the 2s to the 1s state has zero electric dipole matrix element (by parity). The leading contribution is E2 (electric quadrupole), suppressed by $(k{a}_0{)}^2\sim (2\pi \times 10^7\times 10^{-10}{)}^2\sim 4\times 10^{-6}$.

The transition rate is reduced by this factor compared to a typical E1 transition. Instead of a lifetime of $10^{-8}$ s (typical for E1), the 2s state has a lifetime of about 0.1 s — it is metastable. This is why the 2s state of hydrogen is called a "forbidden" state: the E1 transition is forbidden by selection rules, and the higher multipole transitions are very slow.

Check your understanding Beginner

Formal definition Intermediate+

Multipole expansion of the radiation field. The retarded vector potential for a localised source of size $d\ll \lambda$ is expanded in powers of $kd$:

$$\mathbf{A}(\mathbf{r},t)=\frac{{\mu }_0}{4\pi r}\left[\dot{\mathbf{p}}(t_r)+\frac{1}{2c}\ddot{\mathbf{Q}}(t_r)\cdot \hat{\mathbf{r}}+\dots \right]$$

where $\mathbf{p}$ is the electric dipole moment and $\mathbf{Q}$ is the electric quadrupole moment tensor. The magnetic dipole contribution is:

$${\mathbf{A}}_{M1}=\frac{{\mu }_0}{4\pi cr}\hat{\mathbf{r}}\times \dot{\mathbf{m}}(t_r)$$

where $\mathbf{m}=\frac{1}{2}\int {\mathbf{r}}^{\prime }\times \mathbf{J}({\mathbf{r}}^{\prime })d^3{r}^{\prime }$ is the magnetic dipole moment.

Magnetic dipole radiation. The radiation fields for an oscillating magnetic dipole $\mathbf{m}(t)=m_0{e}^{-i\omega t}\hat{\mathbf{z}}$:

$${\mathbf{B}}_{M1}=\frac{{\mu }_0{k}^2{m}_0}{4\pi }\frac{e^{i(kr-\omega t)}}{r}\sin \theta \hat{\mathbf{\theta }},{\mathbf{E}}_{M1}=-\frac{{\mu }_0{k}^2\omega m_0}{4\pi }\frac{e^{i(kr-\omega t)}}{r}\sin \theta \hat{\mathbf{\varphi }}$$

Note the E-field is in the $\hat{\mathbf{\varphi }}$ direction (perpendicular to the E1 case, where E is in $\hat{\mathbf{\theta }}$). The total radiated power:

$$P_{M1}=\frac{{\mu }_0{m}_0^2{\omega }^4}{12\pi c^3}$$

Electric quadrupole radiation. The electric quadrupole moment tensor is:

$$Q_{ij}=\int (3r_i^{\prime }{r}_j^{\prime }-r^{\prime 2}{\delta }_{ij})\rho ({\mathbf{r}}^{\prime })d^3{r}^{\prime }$$

The radiated power is:

$$P_{E2}=\frac{{\mu }_0{\omega }^6}{2880\pi c^5}\sum _{i,j}|{\ddot{Q}}_{ij}{|}^2$$

Suppression factors. The ratio of M1 or E2 to E1:

$$\frac{P_{M1}}{P_{E1}}\sim (kd{)}^2,\frac{P_{E2}}{P_{E1}}\sim (kd{)}^2$$

where $d$ is the source size. For atoms ($d\sim 10^{-10}$ m) at optical frequencies ($k\sim 10^7$ m${}^{-1}$): $(kd{)}^2\sim 10^{-6}$.

Key derivation Intermediate+

Derivation (Magnetic dipole radiation from the vector potential).

Theorem. An oscillating magnetic dipole $m(t)=m_0{e}^{-i\omega t}$ radiates power $P={\mu }_0{m}_0^2{\omega }^4/(12\pi c^3)$ with the same ${\sin }^2\theta$ angular pattern as electric dipole radiation.

Proof. The vector potential for the magnetic dipole is obtained from the second term in the multipole expansion. Using the identity $\int \mathbf{J}{d}^3{r}^{\prime }=-i\omega \mathbf{p}$ and $\frac{1}{2}\int {\mathbf{r}}^{\prime }\times \mathbf{J}{d}^3{r}^{\prime }=-i\omega \mathbf{m}$:

$${\mathbf{A}}_{M1}=\frac{{\mu }_0}{4\pi r}(-i\omega )\hat{\mathbf{r}}\times \mathbf{m}(t_r)\frac{ik}{\omega }=\frac{i{\mu }_{0}k}{4\pi r}\hat{\mathbf{r}}\times {\mathbf{m}}_0{e}^{i(kr-\omega t)}$$

Computing $\mathbf{B}=\nabla \times \mathbf{A}$ in the far zone ($kr\gg 1$):

$${\mathbf{B}}_{M1}=ik\hat{\mathbf{r}}\times {\mathbf{A}}_{M1}=-\frac{{\mu }_0{k}^2}{4\pi r}(\hat{\mathbf{r}}\times (\hat{\mathbf{r}}\times {\mathbf{m}}_0))e^{i(kr-\omega t)}$$

Using $\hat{\mathbf{r}}\times (\hat{\mathbf{r}}\times {\mathbf{m}}_0)=\hat{\mathbf{r}}(\hat{\mathbf{r}}\cdot {\mathbf{m}}_0)-{\mathbf{m}}_0$: the transverse component is $-{\mathbf{m}}_{0,\perp }=-m_0\sin \theta \hat{\mathbf{\theta }}$. So ${\mathbf{B}}_{M1}=({\mu }_0{k}^2{m}_0/4\pi r)\sin \theta e^{i(kr-\omega t)}\hat{\mathbf{\theta }}$.

The Poynting vector: $⟨S⟩=({\mu }_0{k}^4{m}_0^2/(32{\pi }^{2}c{r}^2)){\sin }^2\theta$. Integrating: $P={\mu }_0{m}_0^2{\omega }^4/(12\pi c^3)$. $\square$

Bridge. The multipole expansion builds toward a complete description of radiation from arbitrary charge-current distributions. The foundational insight is that each multipole order is suppressed by $(kd{)}^2$ relative to the previous one, creating a hierarchy of radiation channels. The central message is that when the dominant E1 channel is forbidden by symmetry, the higher multipoles M1 and E2 become the leading radiation mechanism. This is exactly the situation in forbidden atomic transitions and nuclear gamma decay. Putting these together, the multipole expansion generalises the electric dipole radiation 10.07.02 to all orders, provides the selection rules used in spectroscopy and nuclear physics, and connects to the scattering amplitudes 10.07.06 where the induced multipole moments determine the cross-section.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib has spherical harmonics and tensor algebra but does not contain the multipole expansion of radiation fields, the M1 or E2 radiation formulae, the selection rules, or the quadrupole moment tensor. lean_status: none.

Advanced results Master

The full multipole expansion. The radiation field can be decomposed into electric and magnetic multipoles of order $\ell =1,2,3,\dots$:

• Electric multipole E$\ell$: radiated power $\propto k^{2\ell +2}|Q_{E\ell }{|}^2$
• Magnetic multipole M$\ell$: radiated power $\propto k^{2\ell +2}|Q_{M\ell }{|}^2$

The angular distribution of the $\ell$-th multipole involves the spherical harmonics $Y_{\ell }^m(\theta ,\varphi )$. E1 ($\ell =1$) gives ${\sin }^2\theta$. E2 ($\ell =2$) gives a pattern with four lobes involving ${\sin }^2\theta {\cos }^2\theta$.

Selection rules in quantum mechanics. The quantum mechanical transition rate for multipole radiation follows from the matrix elements of the corresponding operators:

• E1: $\Delta \ell =\pm 1$, parity changes
• M1: $\Delta \ell =0$, parity preserved (spin-flip transitions)
• E2: $\Delta \ell =0,\pm 2$ (not $\Delta \ell =\pm 1$), parity preserved

These selection rules determine which atomic and nuclear transitions are "allowed" (E1) or "forbidden" (higher multipoles). Forbidden transitions are critical for lasers (the metastable upper state provides population inversion) and for astrophysical diagnostics (forbidden lines identify low-density gas in nebulae).

Synthesis. The multipole expansion of radiation fields extends the electric dipole treatment to the complete description of radiation from any source. The foundational insight is that the radiation channels form a hierarchy suppressed by powers of $(kd{)}^2$, with the dominant channel determined by the symmetry of the source. The central message is that when the dominant channel is forbidden by selection rules, the subdominant channels become physically important, producing the "forbidden" transitions that are central to atomic spectroscopy, nuclear physics, and astrophysics. Putting these together, the multipole expansion generalises the dipole radiation 10.07.02, the selection rules connect to the quantum mechanical treatment of transitions, and the scattering cross-sections 10.07.06 decompose into multipole channels that match the radiation multipoles developed here.

Full proof set Master

Proposition (M1 power formula). The total radiated power from an oscillating magnetic dipole $\mathbf{m}(t)=m_0\cos (\omega t)\hat{\mathbf{z}}$ is $P={\mu }_0{m}_0^2{\omega }^4/(12\pi c^3)$.

Proof. The time-averaged Poynting vector is $⟨S⟩=({\mu }_0{k}^4{m}_0^2)/(32{\pi }^{2}c{r}^2){\sin }^2\theta$. Integrating over solid angle: $\int {\sin }^2\theta d\Omega =8\pi /3$. So $P=({\mu }_0{k}^4{m}_0^2)/(32{\pi }^{2}c)\cdot (8\pi /3)={\mu }_0{k}^4{m}_0^2/(12\pi c)$. Using $k=\omega /c$: $P={\mu }_0{m}_0^2{\omega }^4/(12\pi c^3)$. Note the extra factor of $1/c^2$ compared to the E1 formula $P_{E1}={\mu }_0{p}_0^2{\omega }^4/(12\pi c)$: M1 is weaker by $\sim 1/c^2$ for comparable moments, reflecting the $(kd{)}^2$ suppression. $\square$

Connections Master

• Electric dipole radiation 10.07.02 is the leading term of the multipole expansion developed here.
• Larmor formula 10.07.01 is recovered from E1 for a single charge.
• Radiation reaction 10.07.04 involves the self-field at the source, which includes contributions from all multipoles.
• Thomson scattering 10.07.06 is E1 scattering; the multipole expansion provides the higher-order corrections.
• Vector potential 10.02.02 is the quantity expanded in the multipole series.

Historical & philosophical context Master

The multipole expansion of radiation fields was developed by Mie (1908) and Debye (1909) in the context of scattering from spherical particles. The application to atomic and nuclear transition rates was made by Weisskopf (1951) and Blatt and Weisskopf (1952), who derived the selection rules and the semi-classical transition rates used in nuclear physics.

The discovery of "forbidden" spectral lines in astrophysical nebulae (Bowen, 1928) was a landmark: these lines were identified as M1 and E2 transitions from metastable states of O${}^{2+}$ and N${}^{+}$ ions. The fact that these transitions are observed in nebulae but not in laboratory sources is because laboratory gas densities are too high — collisions de-excite the metastable states before they can radiate. The low density of nebulae ($n\sim 10^3$ cm${}^{-3}$) allows the forbidden transitions to proceed.

Bibliography Master

• Mie, G., "Beitraege zur Optik trueber Medien," Ann. Phys. 330, 377-445 (1908).
• Blatt, J. M. and Weisskopf, V. F., Theoretical Nuclear Physics (Wiley, 1952).
• Bowen, I. S., "The Origin of the Nebular Lines," Nature 122, 471 (1928).
• Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
• Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields, 4th ed. (Butterworth-Heinemann, 1975).