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I am reading about NMR, and from what I'm understanding it should give information on the transition energies in the spectrum of the nuclear spin in a magnetic field.

What I don't understand is how this information is accessed during the experiment.

The NMR measurement is usually described as: a magnetic field $H_0$ polarizes the sample along a certain direction; then a short pulse of an auxiliary magnetic field alters the direction of the magnetization, which then starts to precess around $H_0$. The changing magnetization induces a current the coils of the NMR machine which is measured and gives the precession frequency. The magnetization gradually aligns with $H_0$, leading to a diminishing current intensity. The decrease in intensity gives the nuclear relaxation rate.

A first question is: is the above description of the measurement correct?

A second question is: How do we extract the information on the energy levels $E_m$, or their separation, from the precession frequency? If not from the precession, how is the energy spectrum obtained?

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The description is simplified, but actually fairly accurate. You can find all the details in a specialised NMR book (other general texts usually do not go into much detail).

Quantum mechanically, you can show that the precession frequency (Larmor frequency), denoted $\omega_0$, is related to the energy difference between the up and down spin states of a spin-1/2 nucleus. The energy levels are

$$\begin{align} E_\alpha &= +\frac{1}{2}\hbar\omega_0 \\ E_\beta &= -\frac{1}{2}\hbar\omega_0 \\ \end{align}$$

and so the transition occurs at the frequency $|E_\beta - E_\alpha|/\hbar = \omega_0$, so if we can identify how much precession is happening at the frequency $\omega_0$, that tells us how strong the corresponding signal should be in the spectrum.

The initial data that you get is in the form of a free induction decay (FID), which plots the amount of detected magnetisation against time. The process of extracting the frequencies from the raw time-domain data is accomplished by a Fourier transform.

Feel free to ask if you would like any part of this to be elaborated. I intentionally did not go into great detail, partly because that is the role of a textbook, but also partly because it can get pretty complicated very quickly.

(There is also a complicated sign convention to do with the definition of $\omega_0$, and different books define it differently. Occasionally the definition even depends on the sign of the gyromagnetic ratio $\gamma$. I would advise to not worry too much about plus/minus signs.)

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  • $\begingroup$ Many thanks for your answer! Before accepting I would like your confirmation on the following. A problem I had earlier was how to extract the frequencies from the macroscopic magnetization, $M=\sum_i Tr{\left[ I_i\hat\rho \right]}$, but your mention of the FID and its fourier transform made me realize that, in the time evolution of that average, one should get factors of $exp(-\frac{it E_m}{\hbar})$ coming from the energy eigenstates, and it is from those factors we should get the oscillations in the FID (and by Fourier tr. , the spectrum). Is this how it works? $\endgroup$ – tbt Jan 13 at 18:15
  • $\begingroup$ @tbt, the spectrometer measures the $x$- and $y$-magnetisations (like in your equations, $M_x = \operatorname{Tr}(I_x\rho)$ and $M_y = \operatorname{Tr}(I_y\rho)$). We then fuse these two quantities together into a complex magnetisation $M = M_x + \mathrm i M_y$, which evolves in time $\sim \exp(i\omega t)$, or $\exp(iEt/\hbar)$ (in NMR usually everything is expressed in frequencies, so we prefer the former ;) ). So you are absolutely right! $\endgroup$ – orthocresol Jan 13 at 22:47
  • $\begingroup$ Thank you again, orthocresol. The $i$ in my formula was meant to indicate the sum over the $N$ atoms in the sample, not the spatial direction (which i left implicit) but thanks for the clarification anyways. I have accepted your answer:) $\endgroup$ – tbt Jan 14 at 14:54

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