In Project 8, we adopt the dictum of Arthur Schawlow: "Never measure anything but frequency!." In this case, we take advantage of the motion of charged particles in magnetic fields to make a measurement of their energies.

In a constant magnetic field, \(B\), the corresponding cyclotron motion of the electron occurs at a frequency that depends on the kinetic energy (\(K_e\)) of the charged particle:

\begin{equation} \label{eq:cyclotron} f_\gamma \equiv \frac{\omega_c}{2\pi \gamma} = \frac{1}{2\pi}\frac{e B}{m_e+K_e/c^2} \end{equation}where \(e\ (m_e)\) is the electron charge (mass) and \(f_\gamma\) is the radiated cyclotron frequency. The baseline angular frequency \(\omega_c\) is \(1.759\times 10^{11}\) rad/s/T, which means at magnetic fields near 1 T, the emission is in the Ka microwave band. As the cyclotron frequency depends on the electron Lorentz factor, \(\gamma\), it is dependent on the kinetic energy of the electron. The corresponding radiation emitted by the electron motion is small enough not to significantly affect the electron's energy, yet still carries the information of the kinetic energy itself.

For a freely-radiating electron undergoing cyclotron motion, the total power \((P)\) radiated is given by the Larmor formula:

\begin{equation} P \propto \frac{2}{3} \frac{q^2 \omega_c^2 p_\perp^2}{m_e^2 c^3} \end{equation}where \(p_\perp\) is is the relativistic transverse momentum of the trapped electron. For a 1 T magnetic field, the amount of power radiated by 30 keV electrons is approximately 1 fW.

In the limit of a perfectly uniform magnetic field, the achievable resolution on the total predicted by this technique depends the total observation time of the electron:

\begin{equation} \Delta \omega = \Gamma \equiv \frac{1}{\tau} \end{equation}where \(\tau\) is the mean observation time of the electron.

This technique, which we call Cyclotron Radiation Emission Spectroscopy, or CRES, offers distinct advantages over other spectroscopic techniques that are used in direct neutrino mass measurements:

**Source = Detector**-- There is no longer a need to separate the electrons from the radioactive source (a potential energy loss mechanism);**Frequency Measurement**-- Frequency standards are some of the most accurate measurements we can make to date;**Full Spectrum Sampling**-- We can in principle measure the entire beta decay spectrum at once, allowing us to leverage greater stability and statistics.

The first step in our science program is to demonstrate the feasibility of CRES on single electrons emitted from a gaseous source. Future incarnations will move toward measuring the electron energy spectrum from a gaseous tritium source.