Adiabatic Invariant

Source lecture(s): pc368_lec08_adiabatic_invariant

Intuition

If you slowly squeeze a magnet, particles trapped in its field lines find themselves in tighter orbits. The magnetic moment $\mu$—the ratio of the perpendicular kinetic energy to the field strength—stays nearly constant. This invarianc governs the mirror trapping of particles in the Earth’s radiation belts and the heating of fusion plasmas.

Formal Definition

An adiabatic invariant is a quantity that remains constant when external parameters vary slowly compared to the particle’s internal dynamical timescale. For magnetized particles, the first adiabatic invariant is the magnetic moment:

\[\mu = \frac{m v_\perp^2}{2 B}\]

Mathematical Formulation

For a particle in a slowly varying B-field with $\Omega_c \gg \omega_{\text{drift}}$:

\[\mu = \frac{p_\perp^2}{2 m B} = \text{const along orbit}\]

The moment arises from averaging the perpendicular energy over one cyclotron period. The condition for validity is:

\[\frac{1}{B} \frac{dB}{dt} \ll \Omega_c\]

Derivation

Write the equation of motion perpendicular to B:

$$m \dot{\mathbf{v}}_\perp = q \mathbf{v}_\perp \times \mathbf{B}$$

The perpendicular kinetic energy is $W_\perp = \frac{1}{2}m v_\perp^2$.
Differentiate $W_\perp/B$ using the slow-field condition:

$$\frac{d}{dt}\Bigl(\frac{W_\perp}{B}\Bigr) \approx \frac{1}{B}\frac{dW_\perp}{dt} - \frac{W_\perp}{B^2}\frac{dB}{dt}$$

Because the gyration is periodic and the field changes adiabatically, $\langle dW_\perp/dt \rangle = 0$ over one gyroperiod.
Therefore $d\mu/dt = 0$ to leading order.

Worked Example

Magnetic mirror trap: A particle has $v_\parallel = 0$ at the mirror midplane where $B = B_0$. Its equatorial magnetic moment is $\mu_0 = m v_{\perp 0}^2/(2B_0)$. At the mirror throat $B_m$, the parallel velocity is:

\[v_\parallel^2 = \frac{2}{m}\Bigl(\mu_0 B_m - \frac{1}{2}m v_\perp^2\Bigr)\]

Reflection occurs when $v_\parallel \to 0$, i.e. when $\sin^2\theta_0 = B_0/B_m$. Particles with small enough pitch angle escape.

Common Mistakes

$\mu$ is always constant. It fails if $B$ changes on the timescale of the gyroperiod ($\omega_B/\Omega_c \not\ll 1$).
Confusing invariants. The first invariant is $\mu$; the second involves

the bounce action; the third is magnetic flux enclosed.
Mirror trapping is perfect. In real scenarios, collisions and non-adiabatic scattering destroy trapping.

Quiz Questions

Conceptual: Why does $\mu$ conservation imply that particles move to regions of weaker $B$ when they gain perpendicular energy?
Computational: A proton with $W_\perp = 5$ keV enters a region where $B$ doubles adiabatically. What is its new $W_\perp$?
MCQ: The adiabatic invariant applies when:
A) $|dB/dt| \gg \Omega_c B$
B) $|dB/dt| \ll \Omega_c B$
C) $B$ is zero
D) The particle is unmagnetized