The Rabi molecular-beam method
We would now like to describe an improved apparatus for the measurement of magnetic moments which was developed by I. I. Rabi and his collaborators. In the Stern-Gerlach experiment the deflection of atoms is very small, and the measurement of the magnetic moment is not very precise. Rabi’s technique permits a fantastic precision in the measurement of the magnetic moments. The method is based on the fact that the original energy of the atoms in a magnetic field is split up into a finite number of energy levels. That the energy of an atom in the magnetic field can have only certain discrete energies is really not more surprising than the fact that atoms in general have only certain discrete energy levels—something we mentioned often in Volume I. Why should the same thing not hold for atoms in a magnetic field?  It does. But it is the attempt to correlate this with the idea of an oriented magnetic moment that brings out some of the strange implications of quantum mechanics.  When an atom has two levels which differ in energy by the amount $\Delta U$, it can make a transition from the upper level to the lower level by emitting a light quantum of frequency $\omega$, where \begin{equation} \label{Eq:II:35:7} \hbar\omega=\Delta U. \end{equation} The same thing can happen with atoms in a magnetic field. Only then, the energy differences are so small that the frequency does not correspond to light, but to microwaves or to radiofrequencies. The transitions from the lower energy level to an upper energy level of an atom can also take place with the absorption of light or, in the case of atoms in a magnetic field, by the absorption of microwave energy. Thus if we have an atom in a magnetic field, we can cause transitions from one state to another by applying an additional electromagnetic field of the proper frequency. In other words, if we have an atom in a strong magnetic field and we “tickle” the atom with a weak varying electromagnetic field, there will be a certain probability of knocking it to another level if the frequency is near to the $\omega$ in Eq. (35.7). For an atom in a magnetic field, this frequency is just what we have earlier called $\omega_p$ and it is given in terms of the magnetic field by Eq. (35.4). If the atom is tickled with the wrong frequency, the chance of causing a transition is very small. Thus there is a sharp resonance at $\omega_p$ in the probability of causing a transition. By measuring the frequency of this resonance in a known magnetic field $B$, we can measure the quantity $g(q/2m)$—and hence the $g$-factor—with great precision.  It is interesting that one comes to the same conclusion from a classical point of view. According to the classical picture, when we place a small gyroscope with a magnetic moment $\mu$ and an angular momentum $J$ in an external magnetic field, the gyroscope will precess about an axis parallel to the magnetic field. (See Fig. 35-3.) Suppose we ask: How can we change the angle of the classical gyroscope with respect to the field—namely, with respect to the $z$-axis? The magnetic field produces a torque around a horizontal axis. Such a torque you would think is trying to line up the magnet with the field, but it only causes the precession. If we want to change the angle of the gyroscope with respect to the $z$-axis, we must exert a torque on it about the $z$-axis. If we apply a torque which goes in the same direction as the precession, the angle of the gyroscope will change to give a smaller component of $\FLPJ$ in the $z$-direction. In Fig. 35-3, the angle between $\FLPJ$ and the $z$-axis would increase. If we try to hinder the precession, $\FLPJ$ moves toward the vertical.  For our precessing atom in a uniform magnetic field, how can we apply the kind of torque we want? The answer is: with a weak magnetic field from the side. You might at first think that the direction of this magnetic field would have to rotate with the precession of the magnetic moment, so that it was always at right angles to the moment, as indicated by the field $B'$ in Fig. 35-4(a). Such a field works very well, but an alternating horizontal field is almost as good. If we have a small horizontal field $B'$, which is always in the $x$-direction (plus or minus) and which oscillates with the frequency $\omega_p$, then on each one-half cycle the torque on the magnetic moment reverses, so that it has a cumulative effect which is almost as effective as a rotating magnetic field. Classically, then, we would expect the component of the magnetic moment along the $z$-direction to change if we have a very weak oscillating magnetic field at a frequency which is exactly $\omega_p$. Classically, of course, $\mu_z$ would change continuously, but in quantum mechanics the $z$-component of the magnetic moment cannot adjust continuously. It must jump suddenly from one value to another. We have made the comparison between the consequences of classical mechanics and quantum mechanics to give you some clue as to what might happen classically and how it is related to what actually happens in quantum mechanics. You will notice, incidentally, that the expected resonant frequency is the same in both cases.  One additional remark: From what we have said about quantum mechanics, there is no apparent reason why there couldn’t also be transitions at the frequency $2\omega_p$. It happens that there isn’t any analog of this in the classical case, and also it doesn’t happen in the quantum theory either—at least not for the particular method of inducing the transitions that we have described. With an oscillating horizontal magnetic field, the probability that a frequency $2\omega_p$ would cause a jump of two steps at once is zero. It is only at the frequency $\omega_p$ that transitions, either upward or downward, are likely to occur.  Now we are ready to describe Rabi’s method for measuring magnetic moments. We will consider here only the operation for atoms with a spin of $1/2$. A diagram of the apparatus is shown in Fig. 35-5. There is an oven which gives out a stream of neutral atoms which passes down a line of three magnets. Magnet $1$ is just like the one in Fig. 35-2, and has a field with a strong field gradient—say, with $\ddpl{B_z}{z}$ positive. If the atoms have a magnetic moment, they will be deflected downward if $J_z=+\hbar/2$, or upward if $J_z=-\hbar/2$ (since for electrons $\FLPmu$ is directed opposite to $\FLPJ$). If we consider only those atoms which can get through the slit $S_1$, there are two possible trajectories, as shown. Atoms with $J_z=+\hbar/2$ must go along curve $a$ to get through the slit, and those with $J_z=-\hbar/2$ must go along curve $b$. Atoms which start out from the oven along other paths will not get through the slit.  Magnet $2$ has a uniform field. There are no forces on the atoms in this region, so they go straight through and enter magnet $3$. Magnet $3$ is just like magnet $1$ but with the field inverted, so that $\ddpl{B_z}{z}$ has the opposite sign. The atoms with $J_z=+\hbar/2$ (we say “with spin up”), that felt a downward push in magnet $1$, get an upward push in magnet $3$; they continue on the path $a$ and go through slit $S_2$ to a detector. The atoms with $J_z=-\hbar/2$ (“with spin down”) also have opposite forces in magnets $1$ and $3$ and go along the path $b$, which also takes them through slit $S_2$ to the detector.  The detector may be made in various ways, depending on the atom being measured. For example, for atoms of an alkali metal like sodium, the detector can be a thin, hot tungsten wire connected to a sensitive current meter. When sodium atoms land on the wire, they are evaporated off as Na$^+$ ions, leaving an electron behind. There is a current from the wire proportional to the number of sodium atoms arriving per second.  In the gap of magnet $2$ there is a set of coils that produces a small horizontal magnetic field $\FLPB'$. The coils are driven with a current which oscillates at a variable frequency $\omega$. So between the poles of magnet $2$ there is a strong, constant, vertical field $\FLPB_0$ and a weak, oscillating, horizontal field $\FLPB'$.  Suppose now that the frequency $\omega$ of the oscillating field is set at $\omega_p$—the “precession” frequency of the atoms in the field $\FLPB$. The alternating field will cause some of the atoms passing by to make transitions from one $J_z$ to the other. An atom whose spin was initially “up” ($J_z=+\hbar/2$) may be flipped “down” ($J_z=-\hbar/2$). Now this atom has the direction of its magnetic moment reversed, so it will feel a downward force in magnet $3$ and will move along the path $a'$, shown in Fig. 35-5. It will no longer get through the slit $S_2$ to the detector. Similarly, some of the atoms whose spins were initially down ($J_z=-\hbar/2$) will have their spins flipped up ($J_z=+\hbar/2$) as they pass through magnet $2$. They will then go along the path $b'$ and will not get to the detector.  If the oscillating field $\FLPB'$ has a frequency appreciably different from $\omega_p$, it will not cause any spin flips, and the atoms will follow their undisturbed paths to the detector. So you can see that the “precession” frequency $\omega_p$ of the atoms in the field $\FLPB_0$ can be found by varying the frequency $\omega$ of the field $\FLPB'$ until a decrease is observed in the current of atoms arriving at the detector. A decrease in the current will occur when $\omega$ is “in resonance” with $\omega_p$. A plot of the detector current as a function of $\omega$ might look like the one shown in Fig. 35-6. Knowing $\omega_p$, we can obtain the $g$-value of the atom.  Such atomic-beam or, as they are usually called, “molecular” beam resonance experiments are a beautiful and delicate way of measuring the magnetic properties of atomic objects. The resonance frequency $\omega_p$ can be determined with great precision—in fact, with a greater precision than we can measure the magnetic field $\FLPB_0$, which we must know to find $g$. 