Is precessional frequency of hydrogen nucleus in H-NMR equal to angular frequency of magnetic component EM radiation?

Question

I asked the same question on chemistry stack exchange but got no answer over there. So, I'm posting it here in the hopes that I'll find some sort of answer:)

Suppose we have a hydrogen nucleus. Now, let's apply an external magnetic field B. As the magnetic field is applied, the hydrogen nucleus aligns itself in the direction of the magnetic field and undergoes precessional motion about its own axis with an angular frequency of, say, p.

Now suppose, we supply an electromagnetic radiation of frequency $\nu$ to the nucleus.

Then,according to Spectroscopy (third edition) by Pavia, Lampman and Kriz, resonance occurs when frequency of applied electromagnetic radiation, $\nu$ becomes equal to precessional frequency, that is: ν=p .

But in my textbook, it is written that $p=2\pi \nu$ . This is possible if precessional frequency equals the angular frequency ($\omega$) of magnetic field component of external electromagnetic radiation (as $\omega=2\pi \nu$ ) and not to the frequency ($\nu$) of EM radiation.

So, here is my doubt: At resonance, is it correct to say that precessional frequency ($p$) is equal to angular frequency of magnetic field component of applied electromagnetic radiation (ω )?

That is: $p=2\pi \nu$

Is my understanding correct?

Answer 1

Your confusion probably arises because people tend to use the term "frequency" to talk about both $\omega=2\pi\nu$ and $\nu$. Which one is used may differ between fields, and more often than not it may even differ among literature in the same field. The best practice is thus to be careful with what "frequency" means in the given literature that you read. In your case, "precession" frequency may seem vague as it may refer to either the angular frequency or the good, old frequency. Your task is, then, to carefully tell which one it is.

This applies to many things related to $\omega$ and $\nu$. You may hear someone say, "The energy separation is $2\ \mathrm{GHz}$", in which case you may wonder if it means that $E=h(2\ \mathrm{GHz})$ or $E=\hbar(2\ \mathrm{GHz})$.

If $\nu$ is the frequency of the electromagnetic radiation such that $E=h\nu$, then you saying that $\nu=p$ is incorrect since you also said that $p$ is the angular frequency for which $E=\hbar p$. It is supposed to be $p=2\pi \nu$ here, as given by your textbook.