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In our EPR experiment, the signal looks like the "first derivative" part of the above picture. Why is this? What does the "first derivative" mean and why is it the quantity our instruments detect in such an experiment?

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See (Bruker know a thing or two about making spectrometers :-). I quote:

The magnetic field strength which the sample sees is modulated sinusoidally at the modulation frequency. If there is an EPR signal, the field modulation quickly sweeps through part of the signal and the microwaves reflected from the cavity are amplitude modulated at the same frequency. For an EPR signal which is approximately linear over an interval as wide as the modulation amplitude, the EPR signal is transformed into a sine wave with an amplitude proportional to the slope of the signal

So the spectrometer measures the first derivative. To get the signal you'd need to integrate the output from the spectrometer, but normally you wouldn't do that as you can get the line position and width directly from the first derivative.

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