g-Factor of electron centered in symmetric cavity

The cyclotron frequency of an electron localized in the center of a cylindrical microwave cavity differs slightly from that in free space. To see this, the system is modeled by a series resonant lc circuit for the cyclotron motion connected across the center of a section of two-wire transmission line (the cavity). When the transmission line is infinitely long, its input impedance at terminals in its center is strictly ohmic and equal to half its characteristic impedance. This ohmic impedance completes the lc circuit, essentially without changing its resonant frequency but damping it. This is the analog of the cyclotron motion in free space that is damped by spontaneous emission. Free space may be mimicked by a short section of transmission line that is terminated at both ends by resistors equal in value to its characteristic impedance. A slightly lossy cavity is modeled by a section of line terminated by resistors much smaller than the characteristic impedance. When the frequency of the lc circuit/cyclotron motion is chosen thus that the line/cavity contains exactly an even number of half-waves, its input impedance will be ohmic and much smaller than its characteristic impedance. Therefore, the frequency of the lc circuit/cyclotron motion will be the same as in vacuum but spontaneous emission and associated line broadening will be much reduced. The g-factor of the electron measured under these conditions practically equals the free space value. This follows because g is determined by the ratio of spin and cyclotron frequencies and the spin practically does not couple to the radiation field.