Chemistry - Why are excess nuclei required to produce an NMR signal?
Solution 1:
"A signal is produced when a radio frequency causes one state (could be either α or β) to invert"
This isn't correct. It really isn't, even though it's often incorrectly described that way. The reality is more complicated, and it involves understanding QM as well as the fact that single spins do not necessarily exist in $\alpha$ or $\beta$ states (any superposition of them is allowed). See my answer here for more details.
Without going into too much detail, population differences* can be used to generate coherences, which are responsible for generating the detected signal. If we construct a generalised wavefunction, with two different coefficients of the two states, viz.
$$|\psi\rangle = c_\alpha|\alpha\rangle + c_\beta |\beta\rangle$$
then coherences can be (loosely) understood as an oscillation of the coefficients $c_\alpha$ and $c_\beta$. For example we could have
$$\begin{align} c_\alpha &= \cos(\omega t + \phi) \\ c_\beta &= \sin(\omega t + \phi) \end{align}$$
which mean that the coefficients, and hence the wavefunction $|\psi\rangle$, is oscillating with an associated frequency $\omega$ and a phase $\phi$. This generates a signal in the final spectrum at the frequency $\omega$. This sort of oscillation has nothing to do with individual spins flipping from $|\alpha\rangle$ to $|\beta\rangle$. Indeed, in NMR, individual spins have no meaning at all. Only the bulk behaviour of the $\sim10^{20}$ spins in your sample matters.
How does this relate to population differences? For a coherence to be generated, there is an additional requirement that across the entire ensemble of spins, there is a correlation between spins in the phase $\phi$.
A decent analogy would be having a collection of ($\sim10^{20}$) analog clocks on a wall. Each of these is oscillating at a rate of one revolution per 12 hours, since each clock returns to the same time every 12 hours. If you started all the clocks at the same time, then they would (largely) remain in sync with each other, and you could have a pretty good idea of what time it is right now. On the other hand, if they all started out of sync with each other, then you would have zero hope of telling what time it was.
This is the real key to understanding why there is a need for population differences. Without a population difference, you can generate oscillating spins, but each of them has a completely random phase and so there is no correlation between them. Consequently, the net signal sums to zero.
I don't want to go deeper because it will get very, very long, and also because it has been covered in a way that is better than I can ever do. See: James Keeler's book at Resources for learning Chemistry.
* As I've said before, populations should not be interpreted as the actual number of particles in a specific eigenstate, but rather the ensemble average of the coefficient corresponding to the specific eigenstate in the wavefunction. In other words, if each spin has a state $|\psi\rangle = c_\alpha|\alpha\rangle + c_\beta|\beta\rangle$, then we have $n_\alpha = \overline{c_\alpha^* c_\alpha}$ where the bar indicates an average over all spins in the ensemble.
Solution 2:
The NMR signal is generated by magnetization oscillating in the transverse reference plane of the experiment (orthogonal to the principal field). The oscillation induces an alternating current in a pickup coil.
From a classical perspective, an oscillating transverse magnetization is generated from an initial longitudinal magnetization to which a torque is applied by coupling the magnetization to an orthogonal RF field. As the magnetization is tipped onto the transverse plane by the RF field it begins to nutate about the principal B-field at the Larmor frequency. If the applied RF frequency matches the Larmor frequency the magnetization and RF field become resonant. This resonant coupling allows the magnetization to be rotated onto the transverse plane, generating an observable in-plane magnetization. A classical treatment of the physics is possible using the Bloch equations without need to invoke QM but many important details of the NMR phenomenon are then missing.
If there is no difference in the Zeeman populations (those of $\alpha$ and $\beta$ states for spin-1/2), then there is no net longitudinal magnetization and therefore nothing to tip over or observe.