## FANDOM

153 Pages

Let us consider the energy of an electron in a magnetic field. If the momentum of the electron is $p$, then the gyration radius is $r_l = \frac{p c}{q B}$, where $c$ is the speed of light, $q$ is the electron charge and $B$ is the magnetic field. Since the angular momentum is quantised, then momentum is also quantised

$p r = \frac{c p^2}{q B} = n \hbar \Rightarrow p = \sqrt{n \hbar q B/c}$

So the energy is

$\varepsilon_n \approx \sqrt{m^2 c^4 +n \hbar q B c}$

where $m$ is the electron mass. Minimum energy is attained when $n = 0$, i.e. when the electron spin is opposite to the orbital angular momentum.

The expression above does not take into account self interaction (a QED effect). This effect slightly increases the magnetic moment of the electron by a factor of $1 + \alpha / 2 \pi$, where $\alpha$ is the finestructure constant. This is called the anomalous magnetic moment. This increase in the magnetic moment reduces the electron's energy

$\varepsilon_0 \approx \sqrt{m^2 c^4 -\frac{\alpha}{2 \pi} \hbar q B c}$

Hence there is a magnetic field above which the energy needed to generate an electron positron pair vanishes

$B_c = \frac{2 \pi}{\alpha} \frac{m^2 c^3}{\hbar q}$

It is impossible to attain a higher steady state magnetic field than $B_c$ because it gives rise to spontaneous generating of pairs that reduce the magnetic field. This limit is larger $2 \pi / \alpha$ than the Schwinger limit for an electric field.