Nuclear magnetic resonance
We have said that atomic paramagnetism is very small and that nuclear magnetism is even a thousand times smaller. Yet it is relatively easy to observe the nuclear magnetism by the phenomenon of “nuclear magnetic resonance.” Suppose we take a substance like water, in which all of the electron spins are exactly balanced so that their net magnetic moment is zero. The molecules will still have a very, very tiny magnetic moment due to the nuclear magnetic moment of the hydrogen nuclei. Suppose we put a small sample of water in a magnetic field $\FLPB$. Since the protons (of the hydrogen) have a spin of $1/2$, they will have two possible energy states. If the water is in thermal equilibrium, there will be slightly more protons in the lower energy states—with their moments directed parallel to the field. There is a small net magnetic moment per unit volume. Since the proton moment is only about one-thousandth of an atomic moment, the magnetization which goes as $\mu^2$—using Eq. (35.22)—is only about one-millionth as strong as typical atomic paramagnetism. (That’s why we have to pick a material with no atomic magnetism.) If you work it out, the difference between the number of protons with spin up and with spin down is only one part in $10^8$, so the effect is indeed very small! It can still be observed, however, in the following way.  Suppose we surround the water sample with a small coil that produces a small horizontal oscillating magnetic field. If this field oscillates at the frequency $\omega_p$, it will induce transitions between the two energy states—just as we described for the Rabi experiment in Section 35-3. When a proton flips from an upper energy state to a lower one, it will give up the energy $2\mu_zB$ which, as we have seen, is equal to $\hbar\omega_p$. If it flips from the lower energy state to the upper one, it will absorb the energy $\hbar\omega_p$ from the coil. Since there are slightly more protons in the lower state than in the upper one, there will be a net absorption of energy from the coil. Although the effect is very small, the slight energy absorption can be seen with a sensitive electronic amplifier.  Just as in the Rabi molecular-beam experiment, the energy absorption will be seen only when the oscillating field is in resonance, that is, when \begin{equation*} \omega=\omega_p=g\biggl(\frac{q_e}{2m_p}\biggr)B. \end{equation*} It is often more convenient to search for the resonance by varying $B$ while keeping $\omega$ fixed. The energy absorption will evidently appear when \begin{equation*} B=\frac{2m_p}{gq_e}\,\omega. \end{equation*}  A typical nuclear magnetic resonance apparatus is shown in Fig. 35-8. A high-frequency oscillator drives a small coil placed between the poles of a large electromagnet. Two small auxiliary coils around the pole tips are driven with a $60$-cycle current so that the magnetic field is “wobbled” about its average value by a very small amount. As an example, say that the main current of the magnet is set to give a field of $5000$ gauss, and the auxiliary coils produce a variation of $\pm1$ gauss about this value. If the oscillator is set at $21.2$ megacycles per second, it will then be at the proton resonance each time the field sweeps through $5000$ gauss [using Eq. (34.13) with $g=5.58$ for the proton].  The circuit of the oscillator is arranged to give an additional output signal proportional to any change in the power being absorbed from the oscillator. This signal is fed to the vertical deflection amplifier of an oscilloscope. The horizontal sweep of the oscilloscope is triggered once during each cycle of the field-wobbling frequency. (More usually, the horizontal deflection is made to follow in proportion to the wobbling field.)  Before the water sample is placed inside the high-frequency coil, the power drawn from the oscillator is some value. (It doesn’t change with the magnetic field.) When a small bottle of water is placed in the coil, however, a signal appears on the oscilloscope, as shown in the figure. We see a picture of the power being absorbed by the flipping over of the protons!  In practice, it is difficult to know how to set the main magnet to exactly $5000$ gauss. What one does is to adjust the main magnet current until the resonance signal appears on the oscilloscope. It turns out that this is now the most convenient way to make an accurate measurement of the strength of a magnetic field. Of course, at some time someone had to measure accurately the magnetic field and frequency to determine the $g$-value of the proton. But now that this has been done, a proton resonance apparatus like that of the figure can be used as a “proton resonance magnetometer.”  We should say a word about the shape of the signal. If we were to wobble the magnetic field very slowly, we would expect to see a normal resonance curve.  The energy absorption would read a maximum when $\omega_p$ arrived exactly at the oscillator frequency. There would be some absorption at nearby frequencies because all the protons are not in exactly the same field—and different fields mean slightly different resonant frequencies.  One might wonder, incidentally, whether at the resonance frequency we should see any signal at all. Shouldn’t we expect the high-frequency field to equalize the populations of the two states—so that there should be no signal except when the water is first put in? Not exactly, because although we are trying to equalize the two populations, the thermal motions on their part are trying to keep the proper ratios for the temperature $T$. If we sit at the resonance, the power being absorbed by the nuclei is just what is being lost to the thermal motions. There is, however, relatively little “thermal contact” between the proton magnetic moments and the atomic motions. The protons are relatively isolated down in the center of the electron distributions. So in pure water, the resonance signal is, in fact, usually too small to be seen. To increase the absorption, it is necessary to increase the “thermal contact.” This is usually done by adding a little iron oxide to the water. The iron atoms are like small magnets; as they jiggle around in their thermal dance, they make tiny jiggling magnetic fields at the protons. These varying fields “couple” the proton magnets to the atomic vibrations and tend to establish thermal equilibrium. It is through this “coupling” that protons in the higher energy states can lose their energy so that they are again capable of absorbing energy from the oscillator.  In practice the output signal of a nuclear resonance apparatus does not look like a normal resonance curve. It is usually a more complicated signal with oscillations—like the one drawn in the figure. Such signal shapes appear because of the changing fields. The explanation should be given in terms of quantum mechanics, but it can be shown that in such experiments the classical ideas of precessing moments always give the correct answer. Classically, we would say that when we arrive at resonance we start driving a lot of the precessing nuclear magnets synchronously. In so doing, we make them precess together. These nuclear magnets, all rotating together, will set up an induced emf in the oscillator coil at the frequency $\omega_p$. But because the magnetic field is increasing with time, the precession frequency is increasing also, and the induced voltage is soon at a frequency a little higher than the oscillator frequency. As the induced emf goes alternately in phase and out of phase with the oscillator, the “absorbed” power goes alternately positive and negative. So on the oscilloscope we see the beat note between the proton frequency and the oscillator frequency. Because the proton frequencies are not all identical (different protons are in slightly different fields) and also possibly because of the disturbance from the iron oxide in the water, the freely precessing moments soon get out of phase, and the beat signal disappears.  These phenomena of magnetic resonance have been put to use in many ways as tools for finding out new things about matter—especially in chemistry and nuclear physics. It goes without saying that the numerical values of the magnetic moments of nuclei tell us something about their structure. In chemistry, much has been learned from the structure (or shape) of the resonances. Because of magnetic fields produced by nearby nuclei, the exact position of a nuclear resonance is shifted somewhat, depending on the environment in which any particular nucleus finds itself. Measuring these shifts helps determine which atoms are near which other ones and helps to elucidate the details of the structure of molecules. Equally important is the electron spin resonance of free radicals. Although not present to any very large extent in equilibrium, such radicals are often intermediate states of chemical reactions. A measurement of an electron spin resonance is a delicate test for the presence of free radicals and is often the key to understanding the mechanism of certain chemical reactions. 