• (David?)

# Finding the correct frequency

In order to start looking for the optimum millimeter/microwave frequency, the variables below are important to know. Note that here the frequencies are given in GHz for finding mmwave frequency, as the frequency for 5 T fields will always be around 140 GHz, and in MHz for nuclei to aid in finding NMR signals.

• Electron Gyromagnetic Ratio: $\frac{\gamma_e}{2\pi}=28.024~951~6~\mathrm{GHz/T}$
• Proton Gyromagnetic Ratio: $\frac{\gamma_p}{2\pi}=0.042~577~478~92~\mathrm{GHz/T}=42.577~478~92~\mathrm{MHz/T}$
• Deuteron Gyromagnetic Ratio: $\frac{\gamma_D}{2\pi}=0.006~535~902~311~\mathrm{GHz/T}=6.535~902~311~\mathrm{MHz/T}$

In order to find both the ideal mm-wave and NMR frequencies, some combination of the above numbers are multiplied by the magnetic field strength $B$ in T. This gives the particle's Larmor frequency, $\nu_i = \frac{\gamma_i}{2\pi}B$, that is used in the descriptions below.

## Proton Enhancement

The image below shows the energy level splitting of the proton, along with the expected NMR spectra along the top. These energy splittings are a useful guide for determining a good initial estimate of the polarization enhancement frequency for the mm-waves.

1. Find & measure TE using the Larmor Frequency
• $\nu_p = \frac{\gamma_p}{2/pi}B$
• Use the central value of the measured TE to extract the magnetic field for the calculations below
2. To calculate the best positive enhancement frequency for mm-waves:
• $f_{\uparrow}=\nu_e-\nu_p$
3. To calculate the best negative enhancement frequency for mm-waves:
• $f_{\downarrow}=\nu_e+\nu_p$

Note that the above should be considered starting points. The actual frequency will shift slightly due to the number and position of free radicals in the material (this causes $\nu_e$ to shift slightly). It's good practice to:

1. Start at the frequencies calculated above
2. Carefully go up in frequency and note if the maximum polarization increases or decreases
• Typically the above calculations tend to be slightly lower than the actual polarization numbers, though this may not always be the case