MOLECULAR BEAMS AND THE MAGNETIC RESONANCE METHOD.

RADIOFREQUENCY SPECTRA OF ATOMS AND MOLECULES *)

J. B. M. Kellogg and S. Millman **)

MOLECULAR BEAMS AND THE MAGNETIC RESONANCE METHOD

1. General Considerations

The new so-called “magnetic resonance method,” which made possible sufficiently precise spectroscopy in the region of low frequencies, usually called radiofrequencies, was first proposed in 1938 by Rabi, Zacharias, Millman, and Kusch ^{R6, R5}. In this method the operations of ordinary spectroscopy are, as it were, reversed, and instead of analyzing radiation emitted by atoms or molecules, the change in the energy of the atomic system occurring as a result of the action of radiation is investigated directly.

Establishing the fact of a change in energy is associated with the use of molecular beams.

The first experiments were undertaken with the aim of measuring nuclear magnetic moments by a new method. However, it immediately became clear that the applicability of the new method is not limited to measuring only this quantity.

The purpose of the present review is to consider the most important applications of the magnetic resonance method, thanks to which the successful solution of a number of problems becomes possible.

Historically, molecular beams were first applied to the investigation of various problems of atomic and nuclear physics. The first successful experiments—the classical experiments of Stern and Gerlach ^{S4, G1}—

) The present article was written in 1941 and has not been revised since. Therefore possible applications of the magnetic resonance method in the region of higher frequencies are not considered here, nor are those technical improvements that developed during the war years. Ingenious modifications of the magnetic resonance method, published not long ago, have been applied to measuring the magnetic moment of the proton and to refining the structure of the energy levels of the cesium atom in the ground state. (Authors’ note.)*

**) J. B. M. Kellogg and S. Millman, Rev. Mod. Physics, 18, 323 (1946). Translated from English by I. S. Shapiro.

were devoted to testing the rules of spatial quantization for an atom situated in a magnetic field.

An atom placed in a magnetic field, which we may take to be directed along the \(z\)-axis, undergoes a change in energy \(\Delta W=-\mu_z H\), where \(\mu_z\) is the component of the magnetic moment in the direction of the field.

The force acting on the atom in the direction of the \(z\)-axis, for constant \(\mu_z\), will be:

\[ F_z=\left(-\frac{\partial W}{\partial z}\right)=\mu_z\left(\frac{\partial H}{\partial z}\right). \]

If, furthermore, the nature of the field is such that \(\dfrac{\partial H}{\partial z}\ne 0\), while \(\dfrac{\partial H}{\partial x}=\dfrac{\partial H}{\partial y}=0\), then \(F_z\) will be the only magnetic force acting on this atom. Thus, an atom moving along the \(x\)-axis with velocity \(v\) and traversing a distance \(l\) in such a field undergoes a deflection \(d\), given by the expression:

\[ d=\frac{1}{2}at^2=\frac{1}{2}\frac{F_z}{m}t^2=\frac{\mu_z}{4E}\frac{\partial H}{\partial z}t^2, \]

where \(E\) is the kinetic energy of the translational motion of the atom with velocity \(v\).

Although the atom is accelerated only while it is in the field, the total deflection acquired by it is composed of the deflection obtained during motion in the magnetic field and the deflection accumulated after leaving the field.

The formula given above must, of course, be corrected so as to take this second component of the total deflection into account; this will lead only to replacing \(l^2\) by a certain factor \(G\), depending quite simply on the geometry of the apparatus.

In reality, atoms flying through the field have a very diffuse distribution of velocities, which must also be taken into account in experiments of this kind. If the deflection \(d\) is measured, one can determine \(\mu_z\). In the experiments of Stern and Gerlach the molecular beam consisted of silver atoms. From the standpoint of classical physics, a continuous series of values of \(\mu_z\), and consequently also of \(d\), is possible. According to quantum mechanics, only certain definite orientations of \(\mu\) are possible, and from spectroscopic data for silver atoms one should expect only two possible orientations. The Stern–Gerlach experiment showed that a beam of silver atoms, on passing through an inhomogeneous magnetic field, splits into two parts; i.e. \(\mu_z\) has only two values. Further, the magnitude of the deflection causing the splitting of the beam into two parts turned out to be such that the magnetic moment was equal to 1 Bohr magneton,

\[ \left(\frac{eh}{4\pi mc}\right). \]

This first work initiated a whole series of determinations of the magnetic moments of atoms by various investigators.

The field of application of the method of deflection of a molecular beam was subsequently extended by Stern, Estermann, and Frisch[^5], by their experiments on determining the magnetic moment of the proton, which demonstrated the possibility of measuring magnetic moments of the order of the nuclear magneton

\[ \left(\frac{eh}{4\pi Mc}\right), \]

which is approximately

\[ \frac{1}{1800} \]

of the Bohr magneton. They worked with a molecular beam consisting of hydrogen molecules passing through a strong magnetic field with a very high gradient. The hydrogen serving as the source of the beam was cooled to a very low temperature. Under these conditions the magnetic moment of the electron shell of the hydrogen molecule is zero, and the magnetic moment of the molecule is made up of the intrinsic magnetic moments of the two protons and the moment arising from the rotation of the nuclei. The rotational magnetic moment was estimated from experiments on pure parahydrogen, in which the intrinsic magnetic moments of the protons are compensated. The magnitude of the rotational magnetic moment proved to be of the order of 0.9 nuclear magneton per unit of rotational quantum number. A correction for the rotational moment was then introduced into the data of the experiments performed with orthohydrogen (natural hydrogen was used to form the beam).

The experiments of Stern, Estermann, and Frisch led to a completely unexpected result: the magnetic moment of the proton, in order of magnitude, proved to be equal not to one nuclear magneton, but to 2.5 nuclear magnetons.

Another important contribution was made by Breit and Rabi[^84], who indicated a method by which experiments with molecular beams can be used to determine nuclear spins. In experiments of the Stern–Gerlach type, in which strong fields are used, the beam is split into \(2J+1\) components, where \(J\hbar\) is the mechanical moment of the electron shell of the atom. If the nucleus possesses a mechanical moment \(I\hbar\) and a magnetic moment different from zero, each of these \(2J+1\) components consists, in turn, of \(2I+1\) separate parts that overlap one another in strong fields. Breit and Rabi showed that if the inhomogeneous field is not too large (so that the interaction between the electron shell and the nucleus is not disturbed), the beam splits into \((2J+1)(2I+1)\) components. If \(J\) is known, the nuclear spin can be determined from the number of components. These considerations were used in the experiment of Rabi and Cohen[^83], whose aim was to show that the spin of the \(Na\) nucleus is \(3/2\).

The method of Breit and Rabi was very soon replaced by a more powerful modification, proposed by Rabi and first carried out experimentally by Cohen[^61], [^64], [^82], [^62], [^46]. In modified form

this method makes it possible to measure the hyperfine structure of the energy levels of atoms in the ground state and, in general, in all those cases where the hyperfine splitting of the terms is so small that it cannot be studied spectroscopically. Further improvements of the method made it possible to determine the character of the hyperfine structure—whether it is normal or inverted \(^{R1, K3}\). Although measurements of hyperfine structure could be carried out with an accuracy of up to 1%, it does not appear possible to calculate the magnetic moment of the nucleus with similar accuracy, except in the case of hydrogen, since for this it is necessary to know the wave function of the corresponding state of the atom.

With the exception of the magnetic resonance method, all applications of molecular beams to the determination of atomic and nuclear moments, to problems of the kinetic theory of gases, to the diffraction of molecules from crystal surfaces, to the measurement of electric dipole moments, etc., are excellently presented in two books by Fraser \(^{F3, F4}\).

Some time later the resonance method was introduced. This method, in addition to the previously developed technique of working with molecular beams, uses the fact that a mechanical moment and the magnetic moment associated with it can be reoriented under the action of a rotating magnetic field, if the frequency of rotation of the field is equal to the Larmor frequency of precession of the mechanical moment. Such reorientation processes, or nonadiabatic transitions as they are called, were considered \(^{G4, M1}\) by G. Güttinger, Majorana and, specifically for the general case of a rotating field, by Rabi \(^{R2}\).

Fig. 1. Simple vector representation of a system possessing identically directed mechanical and magnetic moments, situated in a strong constant magnetic field \(H\) and a weak rotating field \(H_1\).

The principle on which this method is based is valid not only for nuclear magnetic moments, but in general for any system possessing mechanical and magnetic moments. It is very easy to form a qualitative picture of the reorientation process by starting from the classical model of a rotating gyroscope possessing mechanical moment \(I\hbar\) and magnetic moment \(\mu\). Such a system is shown in Fig. 1, where, for simplicity, the mechanical and magnetic moments are shown as having the same orientation. The vector \(I\) will pre-

precess around \(H\) with the Larmor frequency

\[ \nu=\frac{\mu H}{Jh}. \]

Let us now apply, at some initial instant of time, perpendicular to \(H\) and \(J\), the field \(H_1\) itself, rotating so that its angular velocity coincides in direction with the angular velocity of the Larmor precession. It is easy to see that the moment of rotation imparted to the vector \(J\) by applying the field \(H_1\) will cause an increase in the angle between \(H\) and \(\mu\). If \(H_1\) rotates with frequency \(\nu\), this effect will accumulate, so that the angle between \(J\) and \(H\) can be made large. If the perturbing frequency \(f\) of rotation of \(H_1\) relative to \(H\) differs appreciably from \(\nu\), the effect will be small, since the motion of the vector \(J\), caused by the additional moment of rotation, will very quickly get out of phase with the precession. The smaller the ratio

\[ \frac{H_1}{H}, \]

the more sharply the effect will depend on the closeness of the frequencies \(\nu\) and \(f\). Further, if the direction of rotation of \(H_1\) is opposite to the direction of precession, the additional moment of rotation will increase the angle between \(H\) and \(J\) during one half-period and decrease it during the other half-period, owing to which the resultant effect will be very small. This last circumstance plays a primary role in determining the sign of the magnetic moment. Thus, if \(\mu\) is oriented oppositely to \(J\) (negative magnetic moment), the precession will take place in the direction opposite to that indicated in Fig. 1; consequently, for the reorientation of \(\mu\), the direction of rotation of \(H_1\) must likewise be changed to the opposite one.

Any method \(G^2\) which makes it possible to detect the fact of a change in the orientation of the mechanical moment relative to \(H\) at a given frequency of rotation of \(H_1\) makes it possible to establish whether the relation \(f=\nu\) holds, which can be used for measuring the precession frequency and thereby the gyromagnetic ratio \(g=\dfrac{\mu}{J}\), if \(H\) is known. Exact observance of the initial conditions described above is not at all essential, and we may regard \(H_1\) as directed at the initial instant at an angle \(\varphi\) to the plane determined by \(H\) and \(J\), but perpendicular to \(H\). In fact, according to quantum mechanics one should consider, instead, initial conditions for an ensemble of systems uniformly distributed in \(\varphi\), but with a definite projection of the mechanical moment \(J\) on the direction \(H\). Since most of the systems of interest to us have small mechanical moments \((<10\hbar)\), the above classical considerations must be reconsidered from the standpoint of quantum mechanics. The process of reorientation, if it is defined more precisely, consists in the fact that a system possessing mechanical moment \(J\hbar\) and magnetic moment \(\mu\), which at the initial instant of time is in a state with magnetic quantum number \(m\), passes to another magnetic level \(m'\).

An exact calculation of the transition probability for the case of a rotating field of arbitrary strength was given by Rabi \(^{2}\). He considered a system with magnetic moment \(\mu=g\mu_{0}J\), where \(J\) is the total mechanical angular momentum and \(\mu_{0}\) is the Bohr magneton. If \(g\) is positive, the total angular momentum is negative, as in the case of the rotating electron. For negative \(g\) the angular momentum is positive. In a magnetic field \(H\) the system will precess with the Larmor frequency \(\nu=g\mu_{0}H/h\).

For the special case \(J=\dfrac{1}{2}\), Rabi’s formula has the form:

\[ P_{(1/2,-1/2)}= \frac{\sin^{2}\vartheta}{1+q^{2}-2q\cos\vartheta}\, \sin^{2}\pi tf\,(1+q^{2}-2q\cos\vartheta)^{1/2}. \tag{1} \]

Here \(P_{(1/2,-1/2)}\) is the probability of finding the system, which at the initial instant was in the state \(m=\dfrac{1}{2}\), after a time \(t\) in the state \(m=-\dfrac{1}{2}\), when the magnetic-field strength is equal to \((H^{2}+H_{1}^{2})^{1/2}\), with \(H_{1}\) rotating with frequency \(f\); \(q=\nu/f\) and is taken positive if \(H_{1}\) rotates in the direction of precession, and negative if the rotation takes place in the opposite direction; \(\operatorname{tg}\vartheta=\dfrac{H_{1}}{H}\).

If the experimental conditions are such that \(\dfrac{H_{1}}{H}\ll 1\), formula (1) is written as follows:

\[ P_{(1/2,-1/2)}= \frac{\vartheta^{2}}{(1-q)^{2}+q\vartheta^{2}}\, \sin^{2}\pi tf\,[(1-q)^{2}+q\vartheta^{2}]^{1/2}. \tag{2} \]

From the last equation and from the definition of \(q\) it follows that, when \(H_{1}\) rotates in the direction of Larmor precession and when there is a resonance relation between the frequencies \(\nu\) and \(f\), i.e. when \(q=+1\), the probability of reorientation of the magnetic moment of the system with respect to the direction of the magnetic field, equal to \(P_{(1/2,-1/2)}=\sin^{2}\pi f\vartheta\), can be made very close to unity by choosing suitable values of \(t\) and \(\vartheta\). If the direction of rotation of \(H_{1}\) is changed to the opposite one (or the direction of Larmor precession by reversing the direction of \(H\)), then, with all other conditions unchanged, we obtain for the transition probability the value

\[ P_{(1/2,-1/2)}= \frac{\vartheta^{2}}{4-\vartheta^{2}}\, \sin^{2}\pi tf\,(4-\vartheta^{2})^{1/2}, \]

which is considerably less than unity. On the basis of this sharp qualitative difference, when the direction of rotation of \(H_{1}\) is known, one can determine the sign of the magnetic moment. The preceding formula was derived for spin \(1/2\). For higher spins the general formula was obtained

by Majorana C4, M1:

\[ P_{(m,m')}\left(\cos \frac{\alpha}{2}\right)^{4J} (J+m)!(J+m')!(J-m)!(J-m')!\times \left[ \sum_{\lambda=0}^{2J} \frac{(-1)^\lambda\left(\operatorname{tg}\frac{\alpha}{2}\right)^{2\lambda-m+m'}} {(\lambda-m+m')!(J+m-\lambda)!(J-m'-\lambda)!} \right]^2 . \]

In this equation \(\alpha\) is determined by the relation \(\sin^2 \frac{1}{2}\alpha = P_{(1/2,-1/2)}\), where \(P_{(1/2,-1/2)}\) is found from equation (1) for a system with spin \(1/2\) with the same ratio \(\frac{\mu}{J}\), placed under the same conditions as the system with spin \(J\). The terms entering under the summation sign and containing factorials of negative quantities must be omitted. It should be noted that use of Majorana’s formula is necessary only in those cases when it is required to know quantitatively the distribution of molecules over the states \(m'\) into which they have passed from the state \(m\). For measuring the Larmor precession frequency of the nucleus, however, it is sufficient merely to use the fact that for \(f=\nu\) the probability of reorientation reaches a maximum.

Although in Rabi’s theory the field \(H_1\), rotating in a plane perpendicular to \(H\), is considered, in none of the experiments carried out up to the present time has a rotating field been used. In practice it is much more convenient to use an oscillating field as \(H_1\). In the case of an oscillating field the picture is not as clear as it was for rotation of \(H_1\); however, Bloch and Siegert B\(^3\) showed that the expression for the transition probability in weak oscillating fields with frequencies close to resonance undergoes almost no change. It turns out that in equation (1) for \(P_{(1/2,-1/2)}\) it is only necessary to replace \(\mathfrak{H}\) by \(\frac{\mathfrak{H}}{2}\). An oscillating field can be represented as the result of the superposition of two fields rotating with the same frequency in opposite directions. The amplitude of each of the rotating fields is equal to half the amplitude of the oscillating field. This also explains the necessity of replacing \(\mathfrak{H}\) by \(\frac{\mathfrak{H}}{2}\). Reorientation is caused by that component of the oscillating field which rotates in the direction of the Larmor precession. The second component produces no effect. The use of an oscillating field has its negative aspects; in particular, the fact that determining the sign of the magnetic moment becomes impossible, since it cannot be said which of the rotating components causes the transition. It turns out, however, M\(^5\), that in reality in the experiment a rotating field of known direction is created, which can be used to determine the sign. It owes its origin to edge effects of the oscillating field. A more detailed consideration of this effect will be given after a detailed description of the apparatus.

Everything said up to now pertains to a simple system possessing only one vector of mechanical angular momentum and the magnetic moment associated with it, interacting with an applied external field.

With a more general case in view, it is necessary to consider complex systems with two or more mutually coupled mechanical moments, interacting with one another and with an external field. For such a treatment it is simplest to make use of the circumstance that resonance reorientation will occur if the frequency of the oscillating field satisfies Bohr’s frequency relation:

\[ h\nu_{nm}=W_n-W_m, \tag{3} \]

where \(W_n\) and \(W_m\) are the energies of two states of the entire molecular system in the magnetic field.

The selection rule for the transitions of interest to us will be \(\Delta m=\pm 1\), where \(m\) is the magnetic quantum number of the system. It should perhaps be noted here that in the method under consideration one takes into account not only transitions from state \(m\) to state \(n\), but also the reverse transitions from \(n\) to \(m\). One of these transitions corresponds to absorption, the other to induced emission. As Einstein showed, the two processes are equally probable.

For the case in which there is no interaction within the system, everything stated above for the simple system is valid, since the difference in energy between successive spin states is always equal to \(\mu H/J\), whence it follows that the Larmor frequency is equal to the frequency determined by equation (3).

2. Apparatus

Experimentally, the fact of reorientation can be observed with the aid of the apparatus described below. This apparatus must provide for the generation and detection of the molecular beam, the creation of the conditions for reorientation, and, finally, the possibility of detecting the reorientation.

Figure 2 shows the arrangement of the magnetic fields and diaphragms in an apparatus of this kind. Molecules emerging from the source \(O\) enter a region of high vacuum. A very small fraction of these molecules, after passing through the collimating diaphragms \(S\), reaches the detector \(D\). In the absence of a deflecting inhomogeneous magnetic field, the molecules move along the straight line \(OSD\) and form the so-called “straight” beam.

The source is usually a small furnace made of a suitable material and provided with slit diaphragms. The temperature of the furnace is maintained so that the pressure of the vapor filling it amounts to several tenths of a millimeter of mercury. From such a source the molecules fly out in all directions—

tions, the intensity in a given direction being proportional to \(\cos \alpha\) (the angle \(\alpha\) is shown in Fig. 2).

Since it is necessary to use narrow collimating slits, not exceeding a few hundredths of a millimeter, the number of molecules going into the formation of the beam constitutes a very small fraction of the total number of molecules leaving the source. At the same time, of course, measures must be taken with respect to particles that do not form the beam. These measures usually consist either in “freezing out the molecules,” i.e., depositing the particles on the walls of the apparatus, or, in the case of gases, in removing them by means of pumps providing a high pumping speed.

Fig. 2. Schematic drawing of the apparatus and examples of trajectories of beam molecules

The two solid curves in the upper part of the figure show the paths of two molecules moving with different velocities and having different components of their moments along the \(z\)-axis, which remain unchanged along the entire path of the beam. The two dotted curves in the region of magnet \(B\) show possible changes in the trajectory of one of these molecules if the \(z\)-component of its magnetic moment increases or decreases during passage through field \(C\). The motion in the \(z\)-direction on each of the curves shown is greatly exaggerated.

The application of such procedures is necessary in order to maintain a sufficiently low pressure in the apparatus, so as to avoid scattering of the beam molecules by other molecules. Diaphragms \(F_1\) and \(F_2\) do not confine the beam, since they are usually 5–10 times wider than the exit aperture of the source and the collimating slits. They serve to isolate the greater part of the volume of the apparatus from the relatively high (\(\sim 10^{-5}\) mm Hg) pressure in the space directly adjacent to the furnace.

The “beam intensity,” which may be defined as the number of molecules crossing a unit area of the detector per unit time, depends on many factors. Under the assumption that mo-

molecules, moving in the beam, do not undergo collisions, the intensity will be proportional to the pressure in the source, and also to its useful area, and inversely proportional to the square of the distance from the source. Consequently, when large intensities are required, it is necessary to work with beams as short as possible. The intensity of the beam increases with increasing pressure in the source. However, there is a limit to increasing the pressure, determined by the pressure at which the mean free path of the molecules becomes comparable with the width of the exit slit of the source. When this limit is exceeded, a “cloud” forms in front of the slit, acting as a source of much greater width than the actual source. Further, since the intensity is proportional to the area of the source and since it is usually necessary to work with narrow beams, the slits are made as long as the other parts of the apparatus allow, chiefly the inhomogeneous magnetic field. They try to make the width of the slit as small as possible, but nevertheless such as to ensure sufficient beam intensity and ease of adjustment. Slits of width less than \(0.01\ \mathrm{mm}\) are rarely used. For such slits the diffraction effect is negligibly small, since the de Broglie wavelength of the moving molecules is of the order of one angstrom. The use of such narrow diaphragms requires exact parallelism of the source slit and the collimator. This can be easily achieved by setting both slits parallel to a plumb line by means of a telemicroscope. One might think that the slits could be set horizontally, with the aid of a level. However, the spreading of the beam caused by the action of the earth’s gravitational field rules out this possibility*). The final adjustment of the parallelism of the diaphragms always proves possible by means of the molecular beam itself. The most convenient method for this purpose is described in M6.

Let us now return to the consideration of Fig. 2. Magnets \(A\) and \(B\) produce inhomogeneous magnetic fields, whose gradients \(\left(\dfrac{\partial H}{\partial z}\right)_A\) and \(\left(\dfrac{\partial H}{\partial z}\right)_B\) are directed as shown by the arrows. A molecule with magnetic moment \(\mu\) will be deflected in the direction of the field gradient if \(\mu_z\), the projection of \(\mu\) on the direction of the field, is positive, and in the opposite direction if \(\mu_z\) is negative. Molecules for which \(\alpha=0\) will therefore, if their moment is not too small and their velocities are not very large, be deflected by the field \(A\) to one side or the other and will not pass through the collimator slit. Molecules, however, flying out of the furnace in directions different from \(OS\), will be able to pass through the slit. Generally speaking, for a molecule possessing some

*) In one of the experiments the deflection of a horizontal beam in the earth’s gravitational field was used as a useful effect. See O. Stern, Phys. Rev., 51, 852 (1937).

value \(\mu_z\) and speed \(v\), it is always possible to choose such an initial direction of the velocity upon emergence from the source that it can pass through the collimator. Such a trajectory is shown in the figure by a solid line.

The deflection \(d_A\) from the line \(OSD\) at the detector, due to the action of field \(A\), may be written in the form

\[ d_A=\mu_z\left(\frac{\partial H}{\partial z}\right)_A \frac{G_A}{4E}, \]

where \(G_A\) is a factor depending on the geometry of the apparatus. If \(\mu_z\) remains constant during the molecule’s passage through fields \(A\) and \(B\), then, owing to the mutual opposition of the directions of the gradients of fields \(A\) and \(B\), the deflection

\[ d_B=\mu_z\left(\frac{\partial H}{\partial z}\right)_B \frac{G_B}{4E}, \]

caused by field \(B\), will be opposite to the deflection \(d_A\). Thus, if \(d_A=-d_B\), the molecules will reach the detector, forming a “refocused” beam, and for the most part will be registered by it. The focusing does not depend on the distribution of the molecules by velocity and is achieved when the single requirement

\[ \left(\frac{\partial H}{\partial z}\right)_A G_A = \left(\frac{\partial H}{\partial z}\right)_B G_B \]

is fulfilled, which is easy to realize.

It has been shown experimentally that, with properly balanced fields \(A\) and \(B\), the number of molecules reaching the detector in the presence of fields \(A\) and \(B\) is almost exactly the same as when the fields are switched off. Magnet \(C\) creates a homogeneous field \(H\). In this same region of homogeneous field there is placed a pair of parallel wires through which an oscillating current flows, producing an oscillating magnetic field perpendicular to \(H\).

If, as a result of the reorientation described above, the value of \(\mu_z\) changes, the condition determining the possibility for the molecule to reach \(D\) under the action of field \(B\) will no longer be satisfied. The molecule will then fly along one or another dashed trajectory, depending on whether \(\mu_z\) becomes “more positive” or changes sign. In any case, in this situation the molecule will not enter the detector, and we shall observe a decrease in the detector reading, which will make it possible for us to establish the fact of reorientation.

As a numerical example, let us consider an apparatus of the following dimensions: the width of the exit slit of the source and of the collimator slits is \(0.01\) mm, the length of field \(A\) is \(25\) cm, its distance from the source slit is \(10\) cm, the length of field \(B\) is \(30\) cm, its distance from the source is \(52\) cm, the collimator slits are located at a distance of \(37\) cm from the source, the distance between the furnace and the de

tor—92 cm. Under these conditions, the ratio of the geometrical factors is \(G_A/G_B=1.11\), and in order to accomplish focusing it is necessary that the ratio of the corresponding gradients be equal to the reciprocal value. Suppose that the field gradient \(A\) is \(5\cdot 10^8\ \mathrm{gauss/cm}\). Then, if the beam consists of molecules with \(J=\frac{1}{2}\) and \(\mu=1\) nuclear magneton, emerging from an oven at room temperature, with velocities distributed according to Maxwell’s law, the intensity at the detector, in the case where reorientation occurs in the region between the fields \(A\) and \(B\), will amount to about 10% of the intensity that would have occurred in the absence of reorientation. For molecules with the most probable velocity the deviation from the focus is only \(0.05\) mm. The smallness of the deviation also necessitates work with narrow slits.

Inhomogeneous fields are usually produced by iron electromagnets with a copper winding well cooled by water, through which a large current can be passed from high-capacity storage batteries connected in parallel. The cross section of the pole pieces of such a magnet is shown in Fig. 3.

Fig. 3. Cross section of the poles of a magnet producing an inhomogeneous field.

The effective space of the interpole gap is cut out by two circular cylinders intersecting at points separated from one another by \(2a\). Such iron pole pieces are magnetic equipotential surfaces for the field produced by two parallel conductors through which currents flow in mutually opposite directions and whose centers coincide with the points of intersection of the cylindrical surfaces. For rough calculations we may therefore assume that the field distribution in the gap will be the same as in the case of the currents mentioned above.

The properties of the field of the currents and its gradient can easily be calculated \([^{34}], [^{11}]\) and are considered ideal for such experiments. At \(z=1,2a\) the gradient remains approximately constant in the region \(-0.7a<y<+0.7a\), and the ratio of the gradient to the field over this section is approximately equal to \(1/z\). An iron “yoke,” with an interpole gap of 1 mm, with a winding of 4 turns at a current of 200 amperes, gives a field with an intensity of about 10,000 gauss. With \(a\) approximately equal to \(1/8\) cm, a gradient of 80,000 gauss/cm is readily attainable.

Magnet \(C\), which produces a homogeneous field, may be an ordinary iron core “yoke.” This magnet must be carefully annealed in a hydrogen atmosphere. The pole pieces are grounded. The size of the gap is about \(0.6\ \mathrm{cm}\).

Although the gradients of the two inhomogeneous fields, in order to achieve focusing, must be directed oppositely to one another, it is very important that the direction of all three magnetic fields be the same and that the magnets be placed as close as possible to one another, in order to avoid reorientation of the molecules in the region of the weak, rapidly changing field between the magnets. To reduce this effect, iron plates are attached to the ends of the magnets producing the inhomogeneous field.

Fig. 4. Perspective drawing of the “hairpin” made of copper tubes through which radio-frequency current is passed.

They increase the field in the interval between the magnets and make its change more gradual. Such an arrangement provides a sufficiently strong field on that segment of the beam path where a change in the total moment of the molecule leads to a disruption of the focusing. Thanks to this, transitions from one quantum state to another can occur only in the region of the oscillating field, where it is possible to control them and observe them.

The oscillating field \(R\) can be produced by a high-frequency current flowing through two copper tubes bent in the form of a “hairpin,” as shown in Fig. 4. These tubes are inserted into the interpolar gap of magnet \(C\) in such a way that the high-frequency field is approximately perpendicular to the homogeneous field. The beam passes between the tubes, as shown in the figure by the dashed line. The resulting field in this region is therefore oscillatory in character, oscillating with a frequency equal to the frequency of the current flowing through the tubes. A copper pipe is connected to the tubes, making it possible to supply water (for cooling) and electric power from outside the apparatus. The current supply is carried out

through a coupling coil connected to the oscillatory circuit of the generator. The frequencies used in the various experiments lie in the interval from 0.3 to 1100 megacycles.

Since the value of the magnetic moment of any nucleus is calculated from the known magnetic field and the observed frequency, it is very important to know these quantities with a high degree of accuracy. The frequency of the oscillating field can easily be determined, with an error less than 0.01% of the measured frequency, by means of a technical crystal-heterodyne frequency meter. Calibration of the homogeneous magnetic field of magnet C as a function of the strength of the current flowing through the winding can be carried out in the usual manner with the aid of a ballistic galvanometer, giving a throw when a small coil connected to it is removed from the field. In one of the examples of an atomic spectrum given below, another, more accurate method of measuring the magnetic field, more suitable for these experiments, will be described.

At present, three types of molecular-beam detectors are used for measuring low intensities: the Pirani manometer \(F^{5, E3, K4, E1}\), the ionization manometer \(D^{1, J2, H5}\), and the surface-ionization detector \(T^1\). Each of these detectors is limited in its application, and only the surface-ionization detector can be used for measuring extremely low intensities.

The Pirani manometer and the ionization manometer are slow detectors. They are used for recording beams consisting of molecules of light gases, whereas the surface-ionization detector has been used, at least up to the present time, only for heavy elements. The surface-ionization detector is used for detecting atoms of alkali metals, indium, barium, and gallium, as well as molecules containing some alkali metal.

The readings of all three detectors depend linearly on the intensity of the beam. The Pirani manometer and the ionization manometer measure the change in pressure caused by the entry of beam molecules through a narrow slit into a chamber that is otherwise sealed. In the Pirani manometer the effect consists in an increase in the heat transfer from a heated nickel ribbon to the walls of the chamber in which it is located. If the nickel ribbon forms one of the arms of a balanced Wheatstone bridge, the change in its resistance due to the change in temperature will cause a deflection of the galvanometer needle proportional to the intensity of the beam. In the ionization manometer, electrons emitted by an incandescent filament are accelerated by the electric field of a positively charged grid. As a result of collisions of the electrons with atoms or molecules, positive ions are formed, which are collected by a negatively charged electrode. The change in positive ion current caused by molecules entering the manometer

the beam molecules can be measured directly; it is proportional to the change in pressure in the volume of the manometer.

The difficulty in working with detectors of this type is that one has to measure changes in pressure against a background of comparatively high pressure. When beam molecules enter such a manometer, in its volume there is ultimately established a certain equilibrium pressure, i.e., such a pressure at which the number of molecules leaving the manometer in the course of one second is equal to the number entering it. If the beam molecules, in moving from the source to the detector, undergo no scattering, and if the detector is at the same temperature as the source, the equilibrium pressure is determined by the relation:

\[ p_e=\frac{pa}{\pi r^2}, \]

where \(r\) is the distance from the source to the detector, \(p\) is the pressure in the source, and \(a\) is the area of the source slit.

Thus, for a beam \(92\ \mathrm{cm}\) long, with the pressure in the source equal to \(1\ \mathrm{mm}\ \mathrm{Hg}\), the width of the source slit \(0.01\ \mathrm{mm}\) and its length \(2\ \mathrm{mm}\), the change of pressure in the manometer when the whole beam enters it proves to be of the order of \(10^{-8}\ \mathrm{mm}\ \mathrm{Hg}\). If it is required to measure this pressure change with an accuracy of one percent, then we are faced with the problem of measuring an increment of pressure of the order of \(10^{-10}\ \mathrm{mm}\ \mathrm{Hg}\) against a background pressure of the order of \(10^{-7}\ \mathrm{mm}\ \mathrm{Hg}\).

The change of pressure in the manometer caused by the beam can be increased \(K\)-fold if the inlet aperture of the manometer is made channel-like. If this channel is placed in the direction of the beam, it will offer no resistance to the molecules flying into the manometer, but at the same time it will hinder the escape from the manometer of molecules that have entered it and move there in all directions. Pirani manometers have been successfully used with \(K=50\). However, even here there are difficulties. The point is that an appreciable time will now be required for filling the manometer and for the molecules to leave its volume. This time lag will also increase \(K\)-fold. Since the zero of manometers always “wanders” somewhat, in the Columbia Laboratory, for example, it became customary to make six readings of one and the same intensity and then take the mean. With a time lag of \(30\ \mathrm{sec}\), this means that about \(5\ \mathrm{min}\) will be spent on taking one point—a time which is, of course, much too long.

Since the “background” pressure, as a rule, fluctuates within \(10\%\), a single manometer, without a device compensating for such fluctuations, is unsuitable for detecting weak beams. Therefore two manometers are usually used, as nearly identical as possible; in one of them the inlet aperture is turned toward the beam, and in the other—in the opposite direction. Compensation of fluctuations of the “background” pressure is achieved by connecting both manometers into the corresponding electrical circuit.

The most satisfactory of all detectors currently in use is the detector with surface ionization. However, with its aid one can register molecules and atoms of only certain elements. In this method of detection, a thin tungsten filament is placed in the path of the beam. Neutral atoms and molecules striking the filament are re-evaporated as positive ions, having left one of their electrons on the surface of the filament. The probability of such a process will be very close to unity if the surface of the filament is maintained at a sufficiently high temperature and if the difference between the work function of the tungsten surface and the ionization potential of the atom or molecule is appreciably greater than \(kT\). The ions are collected by an electrode surrounding the filament. A potential of 10 volts relative to the filament is applied to this electrode.

For the reasons set forth above, namely because of the extent of the apparatus and the use of narrow slits, the resulting ion current must be amplified. This is accomplished by connecting the collecting electrode to the control grid of an FP-54 tube. The inclusion in the grid circuit of a large resistance (of the order of \(4 \cdot 10^{10}\Omega\)) leads to the appearance on the grid of a certain voltage, which is amplified by the tube. From the change in the anode current of the tube, the number of atoms or molecules colliding with the tungsten surface is determined.

The work function of a clean tungsten surface is sufficiently large (4.5 eV) for detecting beams of atoms Cs, Rb, K, since their ionization potentials are less than 4.5 eV, but not sufficiently high for registering atoms Na or Li, whose ionization potentials are respectively 5.1 eV and 5.4 eV. However, if tungsten is heated in an oxygen atmosphere at a pressure of about 1 mm Hg, its surface becomes covered with a film of stable oxide, which also serves to increase the work function of the surface. After such treatment, detection of Na, Li, and Ga atoms becomes possible.

An attempt to apply a similar operation for the purpose of registering Ba atoms, which have an ionization potential of only 5.2 eV, ended in failure. This can be explained by the fact that for the evaporation of Ba atoms a comparatively high surface temperature is required (about \(1800^\circ\)K or higher), whereas the oxide film is destroyed at high temperatures. The suitable filament temperature for Cs, Rb, and K is \(1200^\circ\)K; for Na, \(1300^\circ\)K; for Ga and In, \(1350^\circ\)K; and for Li, \(1400^\circ\)K.

It turns out, however, that it is possible to register barium beams with a clean tungsten filament heated to a temperature of \(1800^\circ\)K or higher. The mechanism of such detection is not entirely clear [N3, G3].

Registration by means of surface ionization has a number of advantages over the two previously described methods of detection. The surface detector is extremely sensitive. It can

to register an ion current of the order of \(10^{-15}\) ampere, or about 6000 atoms per second. With an area useful for detection of \(5\cdot 10^{-4}\ \text{cm}^2\), this makes it possible to register a beam of intensity \(12\cdot 10^6\) atoms per \(1\ \text{cm}^2\) per second; such a beam would produce in the chamber of a manometer a pressure of the order of \(10^{-14}\ \text{mm Hg}\)—a value considerably smaller than the detection limit of an ionization manometer and a Pirani manometer. Moreover, pressure fluctuations play no role here, since the ionization potential of the gas remaining in the apparatus is considerably higher than the limit at which registration is still possible.

The fact that there is no absolute vacuum in the apparatus leads to scattering of the beam molecules and, consequently, to a weakening of its intensity. Thus there will always be certain uncontrolled fluctuations in the number of particles reaching the detector. Therefore, in evaluating a detector one must not lose sight of such factors as the ease and speed of measurements. When working with a surface detector the resonance curve can be taken completely in ten minutes, and even faster. This makes the experimental data obtained more reliable, since the probability that the experimental conditions will change during the measurement is smaller the shorter the measurement time.

The main limitation in the application of the surface detector is the fact that with its aid one can register only the atoms of a few elements having a sufficiently low ionization potential. However, in work on the determination of nuclear moments, which will be described below, one may encounter resonance curves for such nuclei as H, Be, F, and Al, obtained on molecules which included one of the alkali metals listed by us. The alkali metal here serves to impart to the molecule a low ionization potential. It has been established experimentally that molecules containing alkali metals can be registered in exactly the same way as atoms of the corresponding alkali metals. Thus, molecules containing K, Rb, or Cs can be detected by a clean tungsten surface, while molecules containing Na or Li require, for registration, oxidation of the tungsten surface. At the same time, however, it has been found that the filament temperature necessary for registering molecules containing alkali metals is, in general, somewhat higher than the temperature sufficient for detecting atoms of the corresponding alkali metals.

As an example of the power of the surface-detection method, one may cite the fact that, in working with \(\mathrm{Li}^6\) beams, it proved possible to obtain resonance curves for the \(\mathrm{Li}_2\) molecule. In this experiment the beam was obtained by heating Li in a furnace and consisted predominantly of atomic lithium, all atoms being removed from the beam by a strongly deflecting field. Only about \(0.6\%\) of the beam consisted of

Li\(_2\), and of these \(0.6\%\), after reorientation of the Li\(^6\) nuclei in the oscillating field, only \(4\%\) in all could reach the detector.

Thus, the points lying at the minimum of the resonance curve for Li\(^6\) were obtained from

\[ \frac{1}{4000} \]

of the entire beam.

Using apparatus specially adapted for large intensities, Zacharias \(^{21}\) was able to record the radio-frequency spectrum of K\(^{40}\), using a natural mixture of isotopes (the K\(^{40}\) atoms in the natural mixture constitute only

\[ \frac{1}{800} \]

of the total amount). Such a result could not have been obtained without the use of a detector with surface ionization.

TYPICAL RADIO-FREQUENCY SPECTRA

Below we shall give examples intended to explain the essence and illustrate the possibilities of the method, by means of which information can be obtained on the magnetic moments and spins of nuclei, rotational magnetic moments, electric quadrupole moments, molecular energy levels, and hyperfine structure.

1. Allowed molecular spectra (nuclear)

The radio-frequency spectrum of the deuterium molecule, a portion of which is shown in Fig. 5, is an ideal example of a completely allowed molecular spectrum.

The experiment \(^{K1}\) was carried out with deuterium molecules emitted by a source kept at the temperature of liquid nitrogen. The deuterium nucleus has spin 1 and obeys Bose statistics. The lowest rotational state of the D\(_2\) molecule \((J=0)\) is therefore a state in which the nuclear spin wave function is symmetric, and the total spin angular momentum of the molecule \(I\) is equal to 0 or 2.

In the next, higher rotational state \((J=1)\) the spin wave function of the nuclei is antisymmetric, and the total spin angular momentum of the molecule is equal to 1. For the state \(J=2\) the wave function is again symmetric and \(I\) is equal either to 0 or to 2. Thus, the even rotational states can occur only in ortho-D\(_2\) molecules, and the odd rotational states—only in para-D\(_2\) molecules*). At the temperature of liquid nitrogen almost all D\(_2\) molecules are in these first three states. Only these, consequently, are to be considered here.

*) This will be correct if the electronic wave function of the molecule is symmetric with respect to interchange of the nuclei, which in the present case is so, since molecules in the \(^{1}\Sigma\) state are being considered. (Translator’s note.)

Relative concentrations of molecules in states with the given \(J\) and \(I\) are presented in Table I.

Table I

Relative concentration of para-\(\mathrm{D}_2\) and ortho-\(\mathrm{D}_2\) molecules

\(J\)	Total spin	Statistical weight	Relative concentration
0	0,2	6	0,559
1	1	9	0,528
2	0,2	30	0,705
3	1	21	0,001

The concentrations were calculated on the assumption that the ratio of the number of molecules in ortho states to the number of them in para states is \(2:1\),

In the figure: vertical axis — “Intensity of the beam”; horizontal axis — “Magnetic field in gauss.” Resonance-minimum labels shown: \(C_L\), \(B_L\), \(A_L\), \(H_A\), \(B_R\), \(C_R\). Inside the plot: “\(\mathrm{D}\) in \(\mathrm{D}_2\), frequency \(1{,}300\) Mc, \(l = 2{,}6\) cm.”

Fig. 5. A portion of the radio-frequency spectrum of the deuterium molecule. The central deep minimum corresponds to the reorientation of molecules with \(J = 0\), \(I = 2\). The remaining six minima are due to the reorientation of molecules with \(J = 1\), \(I = 1\). The symbols \(A_R\), \(A_L\), etc., serve to identify the resonance minima with their corresponding transitions, listed in Table III.

i.e., the normal equilibrium ratio for \(\mathrm{D}_2\) at high temperatures was adopted. Then the relative concentration of molecules occupying a specified rotational state with a specified value of \(J\),

is determined by the ratio:

\[ \frac{(2I+1)}{\sum_j (2I+1)} . \]

Thus, \(9.3\%\) of the molecules in the beam will have \(J=0,\ I=0\). It is obvious that the magnetic moment of such molecules is equal to zero, as a result of which they will not manifest themselves in the experiment.

Molecules with \(J=0,\ I=2\) will constitute \(46.7\%\) of the beam.

But for a molecule with zero rotational moment, in the \({}^{1}\Sigma\)-state, in which the resultant electronic moment is equal to zero, the interaction of any one of the nuclei with the rest of the molecule does not depend on the orientation of the nuclei in the external magnetic field.

Further, the energy of interaction of two nuclear magnetic moments, averaged over all possible orientations relative to the line joining the nuclei, is also equal to zero. Therefore, in an external magnetic field each nucleus of the molecule possesses a definite energy, depending on the orientation of its spin and not depending on the orientation of the spin of the other nucleus. But this is precisely the case for which the results cited above of the rigorous theory of transitions in an oscillating field are applicable, when reorientation occurs at equality of the frequency of the oscillating field to the Larmor precession frequency:

\[ \nu=\frac{\mu H}{Ih}. \tag{4} \]

Reorientation of molecules with \(J=0,\ I=2\) gives a deep central minimum in the resonance curve of Fig. 5. The position of this minimum, its depth, half-width, and the asymmetry of the resonance curve are of physical interest.

In these experiments the Larmor precession frequency was varied by changing the field \(H\), while the frequency of the oscillating field remained constant. The minimum of the curve corresponds to the condition \(\nu=f\). Then the quantity \(g=\frac{|\mu|}{I}\) can immediately be calculated from equation (4). From this equation the quantity \(\mu\) is obtained in units of \(\text{erg}\cdot\text{sec}/\text{gauss}\). If \(\mu\) is expressed in nuclear magnetons \(\frac{eh}{4\pi Mc}\), where \(M\) is the proton mass, and if \(g\) is defined as the ratio of the magnetic moment, expressed in nuclear magnetons, to the spin \(I\), then equation (4), under the condition \(\nu=f\), is rewritten in the form:

\[ g=\frac{4\pi}{e/Mc}\cdot\frac{f}{H}=1.3120\cdot10^{-3}\frac{f}{H}; \tag{5} \]

The specific charge of the proton in electromagnetic units, \(e/Mc\), is easily obtained directly from the Faraday number, equal to \(9650.6\) electromagnetic units per gram-equivalent of a physical (nuclear) uni-

mass unit. In these same units the atomic weight of \(\mathrm{H}^1\) is 1.00813. After introducing the correction for the electron mass, we obtain for the quantity \(e/Mc\) the value 9578.0 electromagnetic units per gram.

It should be noted that the ratio \(f/H\) determines only the quantity \(g\). To calculate the magnetic moment it is also necessary to know the spin. Substituting the values \(f=1300\) megacycles and \(H=1992\) gauss, found from the curve of Fig. 5, we obtain \(g=0.855\), and, since the spin is equal to unity, \(\mu=0.855\) nuclear magneton. The depth of the central resonance minimum agrees well with theory.

Up to now we have tacitly assumed that if the transition probability \(P_{(1/2,-1/2)}\) is close to unity for some molecule of the beam, then it will be the same for all other molecules. This, however, cannot be true, as is easily seen from the following considerations. The velocity distribution of the molecules of the beam is determined by the Maxwellian distribution which existed in the source. But the formula for the probability of a resonance transition contains the factor \(\sin^2 \pi f t \theta\), where \(t\) is the time during which the molecule is in the oscillating field. Hence, if the experimental conditions are such that this factor is equal to 1 for molecules with the most probable velocity, it will differ from unity for molecules having other velocities. Faster molecules will leave the region of the oscillating field too soon, while slower molecules will remain there too long. This effect of the velocity distribution changes the transition probability in such a way that only about 75% of all molecules are reoriented.

Since the central minimum in Fig. 5 is due to the reorientation of molecules only with \(J=0,\ I=2\), which constitute only 47% of the beam, the intensity at the minimum must be less than the full intensity by an amount equal to 75% of 47%, or by 35%, which agrees excellently with the experimental results. The half-width of the resonance minimum observed experimentally (the width of the peak at half depth) also agrees well with theory.

If in the expression for \(P_{(1/2,-1/2)}\) (for small \(\theta\)) \(\theta\) is nevertheless so large that the argument of the factor \(\sin^2 \pi f t \theta\) exceeds \(\pi\) several times, the mean value of this factor over velocities will be approximately equal to 0.5, as a result of which the values of \(H\) and of the frequency \(\nu\) at which the mean value of \(\sin^2 \pi f t \theta\) reaches a maximum will be different from the resonant ones.

The shape of the curve in this case is determined by the factor

\[ \frac{\theta^2}{(1-q)^2+q\theta^2}; \]

the half-width will be approximately equal to \(\Delta f=2\theta f\) in frequency units and \(\Delta H=H_1\) in gauss, if \(H_1\) is the intensity of the oscillating field. In this region, therefore, the half-width is proportional to \(H_1\).

If, however, \(\theta\) is so small that the factor \(\sin^2 \pi f t \theta\) can no longer be regarded as equal to 0.5, we must, in order to calculate the half-

of the line width, from the general formula for \(P_{\left(\frac12,\,-\frac12\right)}\). Taking into account the velocity distribution, this was done by Torrey\({}^{12}\). He showed that the greatest transition probability and the narrowest resonance peak occur at \(\vartheta t = 0.6\), where \(t\) is the time spent in the oscillating field by a molecule moving with the most probable velocity.

A further decrease of \(H_1\) leads not only to a decrease in the transition probability, but also to a slight broadening of the resonance line. At the optimum value of \(H_1\), the half-width of the peak is determined by the relation \(t\,\delta \nu = 1.07\), where \(\delta \nu\) is the half-width expressed in frequencies. If the minimum proves to be considerably broader, this either indicates the presence of fine structure, or shows that the intensity of the oscillating field is too large. It should be noted that the central resonance peak in Fig. 5 is not symmetric with respect to the value of the field \(H\) at which the minimum of intensity is reached. Namely, on the side of higher field values there is a certain broadening of the curve. This asymmetry is due to edge effects of the oscillating field and can serve to determine the sign of the magnetic moment \(\mathbf{M}\).

Let us consider the oscillating magnetic field \(H_1\), produced by an alternating current flowing through the wires shown in Fig. 4. For the direction of the current in the wires coinciding with the arrows drawn in the figure, the field directions along the beam are shown in the figure by arrows with small circles. In the region \(D\) the oscillating field is created mainly by the sections of the wires \(A\) and \(A'\), and its direction is approximately parallel to the beam axis, whereas over the greater part of the beam path in the oscillating field—from \(F\) to \(G\)—the field is vertical. Therefore, from the point of view of a coordinate system associated with the moving molecule, the field will rotate in the vertical plane in a direction which, together with the uniform magnetic field \(H\), forms a right-handed screw system, as is shown in the figure by the arc arrow.

For the reverse direction of the current, the field directions are shown in the figure by arrows with triangles. In this case the direction of rotation of the field remains unchanged. An analogous consideration shows that the direction of rotation of the field at the end of the wires facing the detector will be the same as at the end facing the beam source.

Thus, from the point of view of the coordinate system associated with the moving molecule, the perturbing field is strictly oscillatory only in the section \(F—G\), whereas at the edges it is a superposition of oscillating and rotating fields, the direction of rotation being known.

If, for an order-of-magnitude calculation, it is assumed that rotation of the field through \(90^\circ\) takes place over a path length equal to twice the рас-

distance between the axes of the wires \(B\) and \(B'\), then for molecules with a thermal velocity of the order of \(10^5\ \mathrm{cm/sec}\) this will be equivalent to a rotation frequency \(\Delta f\) of the order of \(3\cdot 10^4\) cycles per second. But the oscillating field may be resolved into two components, one of which rotates in the same direction as the direction of the just-described “edge rotation,” and the other in the opposite direction. Therefore the effective perturbing field in the edge regions will have rotational frequency \(f+\Delta f\), if the direction of the Larmor precession coincides with the “edge rotation,” and frequency \(f-\Delta f\), if the precession takes place in the opposite direction.

In the resonance curve expressing the dependence of the intensity on the strength of the homogeneous magnetic field \(H\), we shall obtain the principal minimum at the value \(H_0\), determined by the relation \(\nu=\dfrac{\mu H_0}{Ih}\), and an additional minimum (which may or may not be resolved from the principal minimum) either at the field strength

\[ H_0+\Delta H=\frac{(f+\Delta f)Ih}{\mu}, \]

or at the field strength

\[ H_0-\Delta H=\frac{(f-\Delta f)Ih}{\mu}. \]

Since the direction of the Larmor precession depends only on the direction of \(H\) and on the sign of the magnetic moment, we can determine the latter from the form of the resonance curve obtained for a known orientation of \(H\), or, still better, by comparing two resonance curves taken for mutually opposite directions of the homogeneous field. No quantitative data on this asymmetry of the resonance curve are required in order to determine the sign of the magnetic moment. Detailed calculations of the asymmetry considered were made by Stevenson\(^{36}\).

The asymmetry observed in our case (Fig. 5) leads to the conclusion that the magnetic moment of the deuteron has a positive sign, which agrees with the results obtained by other methods\(^{31,33}\).

The magnetic resonance method was applied by Alvarez and Bloch\(^{41}\) to the determination of the magnetic moment of the neutron. Here we again have to do with a system interacting only with the external field, owing to which we should expect the appearance of only one resonance minimum. Although in principle it is entirely possible, by means of the configuration of magnetic fields described above, to carry out a resonance experiment with a neutron beam, the production of a sufficiently narrow beam of neutrons is a practically infeasible task. Naturally, with the exception of the use of an oscillating magnetic field, an entirely different technique was used in these experiments. For a detailed acquaintance with the apparatus and with the method of detecting reorientation in these experiments, one should refer—

refer the reader to the original work of Alvarez and Bloch. After counting about 200 million neutrons they obtained \(|\mu_n|=1.935\pm0.02\) nuclear magnetons. The sign of the neutron magnetic moment was determined by Powers1.

We shall now turn to the consideration of the remaining six minima in Fig. 5. These minima are due to the reorientation of the deuteron moment in molecules with \(J=1\), \(I=1\), which constitute 33% of the beam (see Table I). If the rotational quantum number is not zero, the energy of the nucleus in an external magnetic field depends not only on its own orientation, but also on the orientation of the other nuclei and of the rotational moment of the molecule. Since in a strong magnetic field \(2J+1\) orientations of \(J\) are possible and, for each of them, \(2I+1\) orientations of \(I\), there will be nine energy states of the molecule in all. Transitions from states with \(J=1\) to states for which \(J\ne1\) will not occur in these experiments, for even if one neglects other reasons, such transitions are associated with a change in energy for which the frequency, determined by equation (3), is considerably higher than that used in the present method. In all, 12 transitions between the nine energy levels are possible. These transitions give six lines of the so-called “nuclear” spectrum, characterized by reorientation only of the nuclear moments, so that \(\Delta m_J=0\), \(\Delta m_I=\pm1\), and six lines of the “rotational” spectrum, for which \(\Delta m_J=\pm1\), \(\Delta m_I=0\). The six lines of the nuclear spectrum, shown in Fig. 5, are what we shall now discuss. The rotational spectra will be considered later.

Each of the nine levels accounts for 3.7% of the beam intensity. Since, as was already noted earlier, transitions from level \(a\) to level \(b\) and from level \(b\) to level \(a\) are recorded in exactly the same way and cannot be distinguished from one another, each of the six minima of the nuclear spectrum will be due to 7.4% of the full beam intensity if the effect of velocity distribution is not taken into account, and 5% if it is taken into account. In the region of these six lines of the \(D_2\) spectrum one may notice one very interesting circumstance, namely the very large spacing between the lines. In order to understand this question, it is necessary first to consider the results obtained from the analysis of the analogous region of the spectrum of ortho-\(\mathrm{H}_2\) molecules with \(J=1\), \(I=1\), caused by reorientation of the nuclear moments.

The situation for para-\(D_2\) molecules with \(J=1\), \(I=1\), apart from the intensities of the lines, will be exactly the same as for the ortho-\(\mathrm{H}_2\) molecules just mentioned. The six lines of the nuclear spectrum for ortho-\(\mathrm{H}_2\) molecules are shown in Fig. 6.

It has been shown that one can calculate very accurately the spacing between these lines, the asymmetry in the distribution of the lines, and also the change in the spacing between the lines as a function of the change

fields, if it is assumed that the following interactions take place: 1) interaction with an external magnetic field, 2) interaction of the magnetic moments of the nuclei (the so-called “spin” magnetic interaction), and 3) magnetic interaction between the nuclear spins and the rotation of the molecule (spin–orbital interaction). The experimental data make it possible to estimate the interaction constants.

Fig. 6. A portion of the radio-frequency spectrum of ortho-H₂ molecules with \(J=1,\ I=1\). The resonance minima correspond to reorientation of the total nuclear spin.

For the constant of the spin–orbital interaction \(H'(\mathrm{H}_2)\), which is, in essence, the strength of the magnetic field of the molecule due to rotation at the point occupied by the nucleus, a value of 27.2 gauss was found. The spin-interaction constant \(H''(\mathrm{H}_2)\) proved to be equal to 34.1 gauss. Since it is determined by the relation \(H''(\mathrm{H}_2)=\left(\dfrac{\mu_p}{r^3}\right)_{\text{av.}}\), where \(\mu_p\) is the magnetic moment of the proton and \((r^{-3})_{\text{av.}}\) is the average value of the inverse cube of the distance between the nuclei in \(\mathrm{H}_2\), we can also calculate this constant using the results of direct measurements of \(\mu_p\) and the values for \(r^{-3}\) obtained from analysis of band spectra. The values of \(H''(\mathrm{H}_2)\) obtained in the two ways are in excellent agreement with one another. If it is assumed that, in order to explain the spectrum of \(\mathrm{D}_2\), it is necessary to take into account only interactions of these two types, then, to determine the energy levels of \(\mathrm{D}_2\) in states with \(J=1,\ I=1\), one should use exactly the same formulas as for \(\mathrm{H}_2\) in states with \(J=1,\ I=1\), with, of course, different values of the constants \(H'\) and \(H''\). It is evident that \(H'(\mathrm{D}_2)\) must be equal to half of \(H'(\mathrm{H}_2)\), since the angular velocity of rotation of the \(\mathrm{D}_2\) molecule is approximately one half of this value for \(\mathrm{H}_2\), while the distance between the nuclei, to the same degree of accuracy, is the same in both molecules. The latter may be asserted with confidence, for it has been shown experimentally

accurate to within 0.2%, that the rotational magnetic moments of the molecules \(D_2\) and \(H_2\) are in the ratio \(1:2\). In exactly the same way one can find the value of the spin-interaction constant for \(D_2\),

\[ H''=\left(\frac{\mu_D}{r^3}\right)_{\mathrm{av}}, \]

since the quantities \(\frac{\mu_D}{\mu_p}\) and \(\frac{\mu_p}{r^3}\) are known. Proceeding from these assumptions, one can completely calculate the nuclear spectrum of \(D_2\). The positions of the minima calculated in this way, however, are in complete disagreement with the experimental results.

It is nevertheless possible to put forward a hypothesis explaining the position of the lines in the nuclear spectrum of deuterium that does not contradict the results obtained for the spectrum of \(H_2\), and, what is much more important, that agrees with the experimental data on the spectrum caused by the reorientation of the deuteron in the HD molecule.

Such a hypothesis is the assumption that the deuteron possesses an electric quadrupole moment\(^ {K1}\). A nucleus may have a quadrupole moment if the distribution of charge in the nucleus does not possess spherical symmetry. The quadrupole moment is taken to be positive if the charge is elongated along an axis coinciding with the spin, and negative if the opposite is the case.

The quadrupole moment of the nucleus manifests itself in the fact that the energy of the nucleus in an external electric field depends not only on the position of the nucleus, but also on its orientation with respect to the field gradient. In an atom this field is due to the distribution of electrons. In a molecule the electric field is produced by the electrons and by the other nucleus.

Proceeding from the assumption that it is necessary to take into account three kinds of interactions, i.e. spin-orbit with constant \(H'\), spin with constant \(H''\), and electric quadrupole with constant \(H'''\), expressions were obtained for the nine energy levels of para-\(D_2\) molecules in a strong magnetic field. These expressions have been calculated up to perturbations of the third order and are given in Table II. \(H'\) and \(S_D\) are first-order terms, \(c_2\) and \(c'_2\) second-order terms, and \(c_3\) third-order terms.

We give the definitions of the corresponding quantities: \(\alpha=\frac{\mu_p}{\mu_D}\) is the ratio of the rotational magnetic moment of the molecule \(D_2\) in the first rotational state to the magnetic moment of the deuteron,

\[ \begin{aligned} S_D&=(H''+H''')/5,\\ c_2&=\bigl[(H'+3S_D)^2+18S_D^2\bigr]/(1-\alpha)H,\\ c'_2&=(H'-3S_D)^2/(1-\alpha)H,\\ c_3&=(H'-3S_D)^2(H'-9S_D)/(1-\alpha)^2H^2. \end{aligned} \]

A detailed derivation of these expressions for the energy levels, as well as a detailed discussion of the experimental data, is contained in the original paper \(K\).

Table II

Expressions for the energies of para-\(D_2\) molecules in a strong magnetic field

\(m_J\)	\(m_I\)	Energy
1	1	\(-\mu_D\left(H+aH+H' - S_D\right)\)
1	0	\(-\mu_D\left(\phantom{H}+aH \phantom{+H'} +2S_D - c'_2\right)\)
1	1	\(-\mu_D\left(-H+aH-H' - S_D - c_2 + c_3\right)\)
0	1	\(-\mu_D\left(H \phantom{+aH+H'} +2S_D + c'_2\right)\)
0	0	\(-\mu_D\left(\phantom{H+aH+H'} +4S_D \phantom{+c'_2} +2c_3\right)\)
0	\(-1\)	\(-\mu_D\left(H \phantom{+aH+H'} +2S_D - c'_2\right)\)
\(-1\)	1	\(-\mu_D\left(H-aH-H' - S_D + c_2 + c_3\right)\)
\(-1\)	0	\(-\mu_D\left(\phantom{H}-aH \phantom{+H'} +2S_D + c'_2\right)\)
\(-1\)	\(-1\)	\(-\mu_D\left(-H-aH+H' - S_D\right)\)

Taking into account the selection rules \(\Delta m_J=0\), \(\Delta m_I=\pm 1\), differences of these energy expressions were formed, which were then set equal to \(hf_0\), where \(f_0\) is the frequency of the oscillating field used in the experiments. The solution of the resulting equations with respect to \(H\) gives the values of the magnetic field at which resonance minima should occur.

These values are given in Table III, where the quantity \(\dfrac{hf_0}{\mu_D}\) is denoted by \(H_0\).

Table III

Expressions for the strengths of the magnetic fields at which resonance occurs for para-\(D_2\) molecules

\(m_J\)	\(m_I\)	Magnetic field in gauss	Line designation
1	\(0\leftrightarrow 1\)	\(H_0-H' + 3S_D - c'_2 + O + O\)	\(A_R\)
1	\(-1\leftrightarrow 0\)	\(H_0-H' - 3S_D + c'_2 - c_2 + c_3\)	\(B_L\)
0	\(0\leftrightarrow 1\)	\(H_0 + O - 6S_D - c'_2 + O - 2c_3\)	\(C_L\)
0	\(-1\leftrightarrow 0\)	\(H_0 + O + 6S_D - c'_2 + O + 2c_3\)	\(C_R\)
\(-1\)	\(0\leftrightarrow 1\)	\(H_0+H' + 3S_D + c'_2 - c_2 - c_3\)	\(B_R\)
\(-1\)	\(-1\leftrightarrow 0\)	\(H_0+H' - 3S_D - c'_2 - O + O\)	\(A_L\)

The asymmetry in the positions of the six minima relative to the central minimum can be used for the identification

experimental lines with the field values given in Table III. The corresponding lines in the table and on the curve are denoted by the symbols \(A_R, A_L\), etc. (It is best to carry out this identification using a spectrum taken in a stronger field than that used to obtain the curve of Fig. 5.)

Denoting the experimentally found resonance field values by the symbols \(A_R, A_L\), etc., and neglecting the third order of perturbations, one may write the following equations:

\[ \begin{gathered} 2(3S_D-H')=A_R-A_L=89\ \text{gauss},\\ 2(3S_D+H')=B_R-B_L=146\ \text{gauss},\\ 2(6S_D)=C_R-C_L=232\ \text{gauss}. \end{gathered} \]

From these equations one can immediately find the values of \(S_D\) and \(H'\). More accurate calculations, carried out for higher frequencies with allowance for small third-order corrections, lead to the result: \(H'=14.00\) gauss and \(S_D=19.62\) gauss.

Since the quantity \(S_D\) is defined as \((H''+H''')/5\), it is impossible from experiments with \(\mathrm{D}_2\) to find \(H''\) and \(H'''\) separately. However, \(H''\) can be obtained from experiments with \(\mathrm{H}_2\), as was indicated above. For it a value of 10.5 gauss was obtained. This gives for \(H'''\) the value 87.6 gauss. A detailed consideration shows that \(H'''\) is positive.

The same interaction constant \(H'''\) can be determined from the rotational spectra\({}^{K7}\) of \(\mathrm{D}_2\) and HD, which arise as a result of the transitions \(\Delta m_j=\pm1,\ \Delta m_I=0\), and from the nuclear spectrum of HD. Moreover, in the case of HD the interaction constants \(H''\) and \(H'''\) can be determined from the experimental data separately and, consequently, the value of \(H'''\) will not depend on the measurement of \(H''\), as occurs in the experiments with \(\mathrm{D}_2\), where \(H'''\) is determined by subtraction. From all these experiments, for \(H'''\) one obtains the value 87.2 gauss.

In theory \(H'''\) is determined as \(\dfrac{e^2 q' Q}{2\mu_D}\), where \(Q\) is the electric quadrupole moment of the nucleus and \(2eq'\) is the value of the electric-field gradient created by the other charges of the molecule at the position of the nucleus. Nordsieck\({}^{N1}\) calculated the quantity \(q'\) and obtained for it the value \(1.193\cdot10^{24}\ \mathrm{cm}^{-3}\). Since the constant \(H'''\) is given by experiment, the quantity \(Q\) can be found. It proved to be positive and equal to \(2.73\cdot10^{-27}\ \mathrm{cm}^2\).

2. Allowed molecular spectra (rotational)

In addition to the transitions considered above, there also occur transitions \(\Delta m_J=\pm1,\ \Delta m_I=0\), giving rise to radio-frequency spectra from which the very same constants \(H'\), \(H''\), and \(H'''\) and the rotational magnetic moments of \(\mathrm{H}_2\), HD, and \(\mathrm{D}_2\) can be determined. Ramsey\({}^{R7}\)

measured these rotational moments and found: \(\mu_R(\mathrm{H}_2)=0.879\), \(\mu_R(\mathrm{HD})=0.660\), \(\mu_R(\mathrm{D}_2)=0.441\) nuclear magnetons.

The analysis of the spectra is entirely analogous to that which we carried out for the case of \(\mathrm{D}_2\). Only the expressions for the energies of the levels will be different.

Examples of these levels are given in Table II. If one calculates the energy differences corresponding to rotational transitions, it immediately becomes clear that for states with \(J=1\) there is not a single simple minimum that could serve for the direct determination of \(\mu_R\). However, if by \(H_0\) one denotes the magnitude of the external field determined by the relation \(\dfrac{hf_0}{\mu_R}\), then, taking into account only first-order perturbations, we obtain that all minima are situated symmetrically with respect to \(H_0\). Thus, the value of the field corresponding to the axis of symmetry of the spectrum can be used for an approximate determination of \(\mu_R\).

The above values of the rotational magnetic moments were obtained with allowance for perturbations of higher order and for a small diamagnetic correction.

In reality the minima are not situated symmetrically with respect to \(H_0\). The small displacement present can be calculated by taking into account second-order terms and used to determine the sign of the moment, which proved to be positive.

The rotational magnetic moment is due to charges circulating in the rotating molecule. Frisch and Stern \(^{\mathrm{F5},\mathrm{E3}}\), who were the first to determine, by the deflection method in a magnetic field, the rotational magnetic moment of the \(\mathrm{H}_2\) molecule in the state \(J=1\), came to the conclusion that the result they obtained—\(\mu_R \simeq 0.9\) nuclear magneton—contradicts the assumption that the molecule rotates as a whole. Indeed, if the molecule rotated as a whole, the rotational moment would be determined by a much larger electronic component and would have a negative sign. The observed magnetic moment corresponds to the picture of a stationary electron cloud with nuclei rotating inside it. Theoretical calculations of this effect were carried out by Wick \(^{\mathrm{W1}}\), who also showed that the rotational magnetic moment must be proportional to \(J\). These calculations were then continued by Ramsey, who showed that the moment must be inversely proportional to the reduced masses. The results of these calculations agree excellently with the experimental data given above.

The proportionality of the moment to \(J\) is confirmed by experiments with \(\mathrm{H}_2\). If the temperature of the source is raised from the temperature of liquid nitrogen to that of dry ice, molecules with \(J=2\) will appear in appreciable quantity in the beam.

Since the state with \(J=2\) for \(\mathrm{H}_2\) is a parastate, the total nuclear spin quantum number of the molecule \(I\) is equal to zero and, consequently, the spin-rotational interaction is absent. Thus, ...

simultaneously, in the spectrum of H₂ at the temperature of dry ice, in addition to the six minima due to transitions between the nine levels with \(J = 1\), one more minimum appears. On the upper curve of Fig. 7, obtained for a frequency of 2.4198 megacycles with the source at the temperature of liquid oxygen, the six minima theoretically expected for molecules with \(J = 1\) are shown.

Fig. 7. Radio-frequency spectrum of H₂, corresponding to transitions in which the rotational mechanical moment of the molecule \(J\) is reoriented.
The frequency of the oscillating field used in this experiment was 2.4198 megacycles per second. The amplitude of the perturbing field \(H_1\) was about 10 gauss.

When the source was “heated” to the temperature of dry ice, the lower curve was obtained, on which an additional minimum is visible between the lines \(A_L\) and \(A_R\) (the lines \(B\) and \(C\) on the lower curve are not shown). The value \(g\) corresponding to this minimum proved to be 0.879, and since this line owes its origin to a transition with a change of \(m_j\) at \(J = 2\), the rotational magnetic moment in this state must be equal to 1.757, i.e. twice the moment of the molecule with \(J = 1\).

3. Determination of the value of \(g\) for nuclei from unresolved molecular spectra

For determining the value of \(g\) for a nucleus, complete resolution of the radio-frequency molecular spectrum is not required. The only restriction imposed on the molecules used for this purpose is that their ground (lower) state be a \({}^1\Sigma\) state, i.e. the electronic mechanical moment must be zero. If the molecules are not in a \({}^1\Sigma\) state, the interaction of the nuclear spins with the electronic mechanical

moment will prove to be of the same order as the interaction with the applied field. Moreover, the electronic moment is so much larger than the nuclear one that the deflection in the fields of magnets \(A\) and \(B\) will be determined entirely by the electronic moment, so that the apparatus will be completely insensitive to reorientation of the nuclei.

This, however, does not mean that the method cannot be applied to systems possessing nonzero electronic moments. Indeed, resonance experiments with alkali-metal atoms, for which the electronic mechanical moment is equal to \(\hbar/2\), were crowned with complete success. Such systems, however, are not the most desirable for the investigation of nuclear magnetic moments.

Fig. 8. Resonance curve for the proton in KOH.

For most nuclei the value of \(g\) was determined from spectra, each line of which represented a superposition of many lines. The constituent part of the molecule determining one or another portion of the spectrum can be found by comparing the spectra of two molecules that contain the same nucleus. For example, if there are three spectra obtained on the molecules LiCl, LiF, and NaF, then the value \(g\) common to LiCl and LiF belongs to the lithium isotope, while the value \(g\) common to LiF and NaF belongs to fluorine.

Fig. 8 shows the resonance curve for the proton obtained with KOH molecules.\(^9\) This curve was obtained by one of the methods considered above; namely, the magnetic field was kept constant, while the frequency of the oscillating field was varied. The magnetic moment of the molecule is composed of the magnetic moments of the nuclei and the rotational magnetic moment. At the furnace temperatures necessary for producing a vapor pressure of several tenths of a millimeter of mercury (600° C), rotational quantum levels with large \(J\) are represented in fairly considerable numbers. Experiment shows that, despite the large \(J\), the principal contribution to the moment is made by the hydrogen nucleus. When this molecule passes through a strong homogeneous field, the couplings between the nuclei (and the rotational moment) are disrupted, and each component precesses about the field with its own Larmor frequency. Owing to the use of a method of measuring the magnetic field which

will be described below; in experiments with beams of KOH molecules it was possible to achieve the most accurate determination of the magnetic moment of the proton.

The effect of the other moments of the molecule consists simply in a broadening of the resonance minimum. The half-width of the minimum shown in Fig. 8 is only three times greater than that which would be expected if the interaction between the nuclear and rotational moments were completely absent. From the classical point of view, this broadening is due to the presence of an additional magnetic field from the other components of the magnetic moment of the molecule. It therefore depends on the relative orientation of the magnetic moments of the component parts of the molecule and on the magnitude of the rotational magnetic dipole moment of the molecule. A simple consideration shows that such broadening does not violate the symmetry of the resonance curve.

We arrive at the same result if we consider the broadening effect from the point of view of energy levels and Bohr’s frequency rule. The moment of the proton (spin equal to \(1/2\)) can assume two orientations in a magnetic field. If there were no other interactions in the molecule, we would have two discrete energy levels. The presence of the other component parts of the molecule will lead to a splitting of each level and will thus create a very narrow multiplet. Transitions between the two multiplets will no longer correspond to a single frequency only. In the case under consideration the interaction energies are very small, and therefore the transition frequencies will not differ greatly from the fundamental frequency.

It should be noted that, in calculating the magnetic moment from the observed resonance values of the field strength and frequency, it is necessary to introduce a correction for the diamagnetism of the atoms. This correction, although negligible for hydrogen, increases with atomic number and may reach \(3/4\%\) for nuclei with large \(Z\). The Larmor rotation of the electrons in the molecule produces a magnetic field opposite to the external field. The true field acting on the nucleus is smaller than the applied one, so that the value of \(f/H\) which we obtain from the resonance curve is underestimated.

The main part of such an electronic field acting on a given nucleus is due to the electrons of the atom containing this nucleus. This field is given by the integral

\[ \int 2\pi \rho \omega r \sin^3 \vartheta \, dr\, d\vartheta = \overline{H}, \]

where \(\rho\) is the charge density, \(\omega\) is the Larmor frequency, equal to \(\dfrac{eH}{2mc}\). For one electron the expression for \(\overline{H}\) may be written in the form:

\[ \overline{H} = \omega \frac{e}{c} \left(\overline{\frac{1}{r}}\right) \frac{\int \sin^3 \vartheta\, d\vartheta}{\int \sin \vartheta\, d\vartheta} = \frac{2}{3}\,\omega \frac{e}{c} \left(\overline{\frac{1}{r}}\right) = \frac{e^2 H}{3mc^2} \left(\overline{\frac{1}{r}}\right). \]

For an electron with quantum number \(n\), if the screening effect is neglected, the quantity \(\left(\dfrac{\overline{l}}{r}\right)\) is equal to \(\dfrac{Z}{a_0 n^2}\). Here \(Z\) is the atomic number, \(a_0\) is the Bohr radius. Since for each \(n\) there are \(2n^2\) electrons, the additional electron field at the position of the nucleus will approximately be:

\[ \frac{2Ze^3H}{3mc^2a_0}\times(\text{number of electron shells}). \]

Calculations by W. E. Lamb, based on the application of the Thomas–Fermi model of the atom, led to the following expression for the correction to the field:

\[ \overline{H}=-0.320\cdot 10^{-4}Z^{1/3}H. \]

This correction must be subtracted from the measured value of the field strength \(H\). Its introduction, therefore, increases the value of the nuclear moment.

4. Atomic radio-frequency spectraK7, M8

Because the nucleus possesses spin, the ground states of many atoms constitute a system of closely spaced energy levels. Each level of the hyperfine-structure multiplet corresponds to a certain value of the total angular momentum of the atom. Hyperfine splitting is due chiefly to the weak interaction of the electric and magnetic fields of the electron shell with the magnetic and electric quadrupole moments of the nucleus. Since the magnitude of this interaction depends on the angle between the angular momentum of the nucleus and the angular momentum of the electron shell, states with different total angular momenta \(F\) will differ in energy.

An atom left to itself will make transitions between these states, emitting energy in the form of electromagnetic waves, whose frequency is determined by Bohr’s formula [equation (3)], and will thus pass into its lower energy state. This radiation has the character of magnetic dipole radiation. The frequency range extends from \(1.5\cdot 10^8\) to \(1.2\cdot 10^{10}\ \mathrm{sec}^{-1}\).

Since the radiation is low-frequency, the lifetime of the atom in any hyperfine-structure state will be very large, and the probability of a spontaneous transition very small. Therefore direct observation of such radiation is very difficult. It is possible, however, to “illuminate” the atom with electromagnetic radiation of the appropriate frequency and intensity so that the atom absorbs this radiation and, according to the Einstein process of stimulated emission, emits a quantum of the same frequency in a sufficiently short time—of the order of \(10^{-4}\ \mathrm{sec}\). Detection of such a process gives us a direct method for measuring hyperfine structure.

A method of this kind has many substantial advantages over the ordinary optical one. First, the results are much easier to interpret, since we are dealing with only one energy level. Second, a very high accuracy can be attained, since the measurements consist in determining the frequency of a radio wave. Third, it is possible to measure and study separately levels that lie very close to one another.

It is quite obvious that analogous considerations apply to any metastable states having a sufficiently long lifetime. Under certain conditions, the emission and absorption of a quantum of energy is connected with a change in the effective magnetic moment of the atom, which can be detected by means of the technique we have described.

The discussion that we shall carry out here is limited to the case in which the electronic mechanical moment \(J\) is equal to \(1/2\). If \(J\) is greater than \(1/2\), the situation becomes much more complicated. Moreover, for \(J=\frac{1}{2}\) the energies of quadrupole interaction are equal to zero, owing to which it may be assumed that the entire hyperfine structure is due to the magnetic interaction of the nucleus and the outer electrons. This condition is satisfied by the atoms of all the alkali metals and of indium, with which the resonance experiments were carried out.

The levels forming a hyperfine-structure doublet, according to quantum mechanics, are characterized by the following values of the total mechanical moment:

\[ F = I + \frac{1}{2}, \quad \text{and} \quad F = I - \frac{1}{2}, \]

where \(I\) is the mechanical moment of the nucleus. Transitions between these two levels obey the selection rules for magnetic dipole radiation: \(\Delta F = 0, \pm 1\), \(\Delta m = 0, \pm 1\), where \(m\) is the magnetic quantum number. The energy of such levels in a magnetic field, as a function of the field strength, is given by the following formula\({}^{84}\):

\[ W_{I+1/2,\,m} = -\frac{\Delta W}{2(2I+1)} + g_I\mu_0 H_m + \frac{\Delta W}{2} \left( 1+\frac{4m}{2I+1}x+x^2 \right)^{1/2} \tag{6} \]

for

\[ F=I+\frac{1}{2}, \quad m=I-\frac{1}{2},\, I-\frac{3}{2}, \ldots, -\left(I-\frac{1}{2}\right); \]

\[ W_{I+1/2,\,\pm(I+1/2)} = -\frac{\Delta W}{2I+1} \pm g_I\mu_0 H\left(I+\frac{1}{2}\right) + \frac{\Delta W}{2}(1-\alpha) \tag{7} \]

for

\[ F=I+\frac{1}{2}, \quad m=\pm\left(I+\frac{1}{2}\right); \]

\[ W_{I-1/2,\,m} = -\frac{\Delta W}{2(2I+1)} + g_I\mu_0 H_m - \frac{\Delta W}{2} \left( 1+\frac{4m}{2I+1}x+x^2 \right)^{1/2} \tag{8} \]

for

\[ F=I-\frac{1}{2}, \quad m=I-\frac{1}{2},\, I-\frac{3}{2}, \ldots, -\left(I-\frac{1}{2}\right). \]

where the parameter \(x\) is determined by the relation:

\[ x=(g_J-g_I)\mu_0\,\frac{H}{\Delta W}, \]

and \(\Delta W=h\Delta\nu\) is the energy difference of the two states with \(F=I+\frac{1}{2}\) and \(F=I-\frac{1}{2}\) in the absence of a magnetic field. In these equations, as usual, the factor \(g\) is the ratio, taken with the opposite sign, of the magnetic moment, expressed in Bohr magnetons, to the mechanical moment, expressed in units of \(\frac{h}{2\pi}\). Thus, for example,

Fig. 9. Change of the energy levels as a function of the magnitude of the magnetic field for an atom in the state \({}^{3}S_{1/2}\), with nuclear spin \(3/2\). The solid curves refer to states with \(F=2\), the dashed curves to states with \(F=1\). The parameter \(K\) is proportional to

\[ \frac{H}{\Delta W}, \]

where \(\Delta W\) is the energy difference of the states with \(F=2\) and \(F=1\) in the absence of a magnetic field.

Fig. 10. Dependence of the magnetic moment on the magnetic field of an atom in the state \({}^{2}S_{1/2}\), with nuclear spin \(3/2\). As in Fig. 9, the solid lines refer to the \(m\)-levels of the state \(F=2\), the dashed lines to the levels with \(F=1\).

the values of \(g_J\) for the ground state of atoms of all alkali metals are positive, whereas \(g_I\) for the same atoms is negative, since the moments are positive. For alkali metals \(g_J=2\).

Figure 9 graphically shows the behavior of the ratio \(\frac{W_{F,m}}{\Delta W}\) as a function of the parameter \(x\) for a nucleus with spin \(3/2\). The solid lines refer to levels with \(F=2\), the dashed lines to states with \(F=1\),

Dividing the energy difference of the levels between which the transitions occur by \(h\), we obtain an expression for the frequency of the line as a function of the field strength. Since the transitions are induced in the region of the oscillating field, in order to detect them it is necessary that the magnetic moment of the atom in field \(B\) differ from the magnetic moment in field \(A\). The dependence of the magnetic moment of the atom on the field strength can be found, since \(\mu_{F,m}=-\dfrac{dW_{F,m}}{dH}\). Curves showing the dependence of the magnetic moment of the atom on \(x\) for the case \(I=\dfrac{3}{2}\) are shown in Fig. 10. The solid and dashed lines have here the same meaning as in Fig. 9.

Although sometimes a strong field, characterized, for example, by \(x=3\), is preferable for magnet \(C\), in general it is undesirable in deflecting magnets, since the change of magnetic moment occurring during the transition in a strong magnetic field will be insignificant, except in the case of a change in the sign of the moment in the field of magnet \(B\). Equations (7)—(8) show that in the absence of an applied magnetic field all magnetic levels belonging to a given value of \(F\) merge, and therefore transitions \(\Delta F=1\) will give a single line

\[ \Delta \nu=\frac{\Delta W}{h}. \]

In a magnetic field the Zeeman effect will be observed. At a field strength of only a few hundredths of a gauss, the splitting of each of the levels is so large that a completely resolved Zeeman spectrum is obtained instead of a single line \(\Delta\nu\).

Fig. 11. Zeeman splitting of the line \(|\Delta F|=1\) in a magnetic field of strength about 0.05 gauss for the atom \(K^{39}\). The resonance lines correspond to transitions \(|\Delta m|=1\).

In Fig. 11 a sample Zeeman spectrum is shown, obtained in a field of 0.05 gauss for transitions \(|\Delta F|=1\) of the atom \(K^{39}\), whose nucleus has spin equal to \(3/2\). The lines correspond to transitions \(\Delta m=\pm1\), caused by the component of the oscillating field perpendicular to the constant field \(H\). Lines arising from transitions \(\Delta m=0\) are induced by the component of the oscillating field,

parallel to \(H\). These lines, however, are absent in Fig. 11, since the oscillating field was very weak and was directed almost at a right angle to the constant field, as a result of which the component parallel to \(H\) was insufficient to induce transitions. These transitions will occur at higher values of the oscillating-field strength. Therefore the spectra obtained under such conditions will be characterized by poorer resolution. At still higher oscillating-field strengths the entire spectrum will consist of one broad resonance minimum. With appropriate field directions one can, of course, arrange that the transitions \(\Delta m=0\), and not \(|\Delta m|=1\), dominate.

In weak fields, where \(x \ll 1\), in equations (6)—(8) one may neglect the terms containing \(g_i\) and \(x^2\). Then it is seen from the equations that the frequency \(\Delta \nu\) corresponds to the “center of gravity” of the lines in the pattern shown in Fig. 11. Each of the two central lines shown in Fig. 11 is a doublet consisting of two lines, the difference between whose frequencies \(2g_i\mu_0 H/h\) is too small to be observed. Thus, instead of the six lines that might have been expected from the diagram of energy levels, only four have been observed experimentally.

It should be noted that, with this method of determining \(\Delta \nu\), knowledge of the absolute magnitude of the magnetic-field strength \(H\) is not required. It is only important to know that it is small. The method we have considered for determining \(\Delta \nu\) is general, but not always easily applicable, which is connected with experimental difficulties arising in work with very high frequencies.

In the case of \(K^{39}\), the distance between the hyperfine-structure lines is about 460 megacycles. Such a frequency can readily be attained with ordinary electron tubes (for example, Western Electric 316 A). But for \(Na^{23}\), for example, in zero field the distances between the hyperfine-structure lines are 1770 megacycles. The generation of such frequencies, as well as their sufficiently accurate measurement, presented considerable difficulties at the time when these experiments were carried out. We shall now consider two methods for the precision measurement of large values of \(\Delta \nu\), not requiring the use of very high frequencies.

It is seen from Fig. 9 that for large values of \(x\) the energy levels split into two groups, in each of which the curves shown in Fig. 9 run approximately parallel to one another. This means that the frequencies of the lines corresponding to transitions between neighboring levels of one and the same group should be almost independent of \(x\), if \(x\) is large (i.e., for large \(H\)). In fact, the energy difference of such levels depends mainly on \(\Delta \nu\) and \(I\), and only to a very small degree on \(H\) and \(g_i\). Since \(H\) can readily be measured with an accuracy of up to \(1\%\), the error in its determination will have a negligibly small effect on the value of \(\Delta \nu\). Just as

the same error in the measurement of the quantity \(g_I\) (determined, for example, from resonance curves), not exceeding \(1\%\), will not affect the value of \(\Delta\nu\).

If the frequencies of the doublet lines, due by their origin to transitions of the type \((F,m)\to(F,m')\), are measured at one and the same \(H\) and, consequently, \(x\), and for both \(F\) the values of \(m\) and \(m'\) are the same, one can determine \(\Delta\nu\) without knowing \(g_I\).

The independence of the mean frequency of the doublet from \(g_I\) can be seen from Table IV, in which expressions are given for the frequencies of the lines in the case when \(I=\dfrac{3}{2}\).

Table IV

Expressions for the frequencies of atomic transitions in a magnetic field

\[ J=-\frac{1}{2},\quad I=\frac{3}{2}. \]

Transitions	Expressions for the frequencies
\((2,2)\longleftrightarrow(2,1)\)	\(-\dfrac{1}{2}\Delta\nu\bigl[(1+x)-(1+x+x^2)^{1/2}\bigr]+g_I\mu_0H/h\)
\((2,1)\longleftrightarrow(2,0)\)	\(-\dfrac{1}{2}\Delta\nu\bigl[(1+x+x^2)^{1/2}-(1+x^2)^{1/2}\bigr]+g_I\mu_0H/h\)
\((1,1)\longleftrightarrow(1,0)\)	\(\dfrac{1}{2}\Delta\nu\bigl[(1+x+x^2)^{1/2}-(1+x^2)^{1/2}\bigr]-g_I\mu_0H/h\)
\((2,0)\longleftrightarrow(2,-1)\)	\(-\dfrac{1}{2}\Delta\nu\bigl[(1+x^2)^{1/2}-(1-x+x^2)^{1/2}\bigr]+g_I\mu_0H/h\)
\((1,0)\longleftrightarrow(1,-1)\)	\(-\dfrac{1}{2}\Delta\nu\bigl[(1+x^2)^{1/2}-(1-x+x^2)^{1/2}\bigr]-g_I\mu_0H/h\)
\((2,-2)\longleftrightarrow(1,-1)\)	\(\dfrac{1}{2}\Delta\nu\bigl[(1-x+x^2)^{1/2}+(1-x)\bigr]-g_I\mu_0H/h\)

A very simple extension of these expressions to the region \(x^2\gg1\) shows that the frequencies, as \(x\) increases, approach \(\dfrac{\Delta\nu}{4}\). In the general case the frequencies tend to \(\dfrac{\Delta\nu}{2I+1}\). But this means that one can measure \(\Delta\nu\) by using frequencies lying in the region \(\dfrac{\Delta\nu}{2I+1}\).

If all transitions are observed, then in order to find \(\Delta\nu\) there is no need to know \(H\) and \(g_I\). In fact, from Table IV it can be seen that the sum of the mean frequencies of the doublets and the frequencies of the singlet lines at constant \(x\) is exactly equal to \(\Delta\nu\). This is true not only for the case \(I=\dfrac{3}{2}\), but in general for all atomic systems described by equations (6)—(8).

Thus:

\[ \begin{aligned} \Delta \nu ={}& \nu \left( I+\frac{1}{2},\; I+\frac{1}{2} \longleftrightarrow I+\frac{1}{2},\; I-\frac{1}{2} \right) + \\ &+ \nu \left( I+\frac{1}{2},\; -I-\frac{1}{2} \longleftrightarrow I-\frac{1}{2},\; -I+\frac{1}{2} \right) + \\ &+ \frac{1}{2} \sum_{m=I-\frac{1}{2}}^{-I+\frac{3}{2}} \left[ \nu \left( I+\frac{1}{2},\; m \longleftrightarrow I+\frac{1}{2},\; m-1 \right) + \right. \\ &\hspace{3.2cm}\left. + \nu \left( I-\frac{1}{2},\; m \longleftrightarrow I-\frac{1}{2},\; m-1 \right) \right], \end{aligned} \]

where, for example, the term \(\nu\left( I+\frac{1}{2},\; I+\frac{1}{2} \longleftrightarrow I+\frac{1}{2},\; I-\frac{1}{2}\right)\) denotes the frequency of the line corresponding to the transition

\[ F=I+\frac{1}{2},\quad m=I+\frac{1}{2} \longleftrightarrow F=I+\frac{1}{2},\quad m=I-\frac{1}{2}. \]

This expression for \(\Delta \nu\) is valid for a nuclear spin different from \(\frac{1}{2}\). For \(I=\frac{1}{2}\) only the first two terms must be used. All three of the methods considered have been used to investigate the hyperfine structure of the ground state of \(K^{39}\) atoms and led to identical results. If the field is known, the value \(g_I\) can be found from the separation between the lines of the doublet. Such a determination cannot be very accurate, since one has to measure the small difference between the frequencies, \(2g_I \mu_0 H/\hbar\), whereas the frequencies themselves are large in comparison with this difference.

With the aid of the magnetic-resonance method, which uses molecular beams to determine the values of the nuclear \(g\)’s, the quantity \(g_I \mu_0 H/\hbar\) is measured directly, as a first-order effect, and therefore more precisely. If the factor \(2I+1\) does not reduce the frequencies sufficiently, \(\Delta \nu\) can be determined by measuring the frequencies of two lines at one and the same field. Then we obtain two equations with two unknowns, \(x\) and \(\Delta \nu\). In order for such a method to be more sensitive, one should choose lines whose frequencies change differently with a change in the magnetic field. For example, the lines \((2,-2)\longleftrightarrow(2,-1)\) and \((2,1)\longleftrightarrow(2,2)\) at \(x=0.5\).

In this way the hyperfine structure of the ground states of \(Rb^{87}\) and \(Cs^{133}\) was investigated with an accuracy comparable with measurements made by the method of observing transitions between two levels with different \(F\) in the absence of a magnetic field.

MEASUREMENT OF THE MAGNETIC FIELD

Once \(\Delta \nu\) is known for the ground states of the atoms of the alkali metals and the spins of their nuclei, measurement of the frequencies of certain lines can be used for precise calibration of the magnetic field, since the above equations for the frequencies can be solved with respect to—

relative to \(x\) and, consequently, \(H\). Here too the term \(g_I \mu_0 H_0\) is only a small correction term.

However, not all lines are suitable for this purpose. It is necessary to choose a line whose frequency changes appreciably with a change in \(x\); for example, the line arising as a result of the transition \((2,-1)\longleftrightarrow(2,-2)\) (Fig. 9). This line can be used over the entire range of values of \(x\). The transition \((2,-2)\longleftrightarrow(1,-1)\) can be used only for values of \(x\) not exceeding 1.5 or 2; the transition \((F,0)\longleftrightarrow(F,1)\) can be used only for small values of \(x\).

Since \(\Delta\nu\) for atoms of different alkali metals covers a rather wide frequency interval, we have a method for precision calibration of the magnetic field up to field strengths of the order of 5000 gauss, even in the case where the range of frequencies supplied by the generator is limited.

For precision calibration of the field, in which the resonance curve is recorded, it is essential that the atomic transitions occur at the same field setting at which observations of the nuclear resonance spectrum were made. Since the field of magnets \(A\) and \(B\) affects the field produced by magnet \(C\), it is necessary that observation of the nuclear resonance curve take place at the same field strengths \(A\) and \(B\) as the registration of the atomic transitions.

At first sight this requirement seems impossible to fulfill, because fields with large gradients are needed to detect the reorientation of nuclei; these deflect almost all alkali-metal atoms out of the beam. Thus, most nuclear resonance curves have been obtained with a gradient of the order of \(10^5\) gauss/cm, whereas the radio-frequency spectra of atoms have been studied with gradients of about 500 gauss/cm. However, if resonance curves are studied for nuclei with a large value of \(g\), then by choosing suitable atomic states one can avoid these difficulties, i.e., observe nuclear and atomic transitions with identical values of the deflecting fields.

The study of resonance curves for the proton does not require very large gradients, since the value of \(g\) is sufficiently large, and reorientation of the nuclei leads to a comparatively large change in the component of the moment in the direction of the field. Reducing the gradient to a value at which the depth of the resonance minima begins to decrease, owing to the decrease in deflection in the fields \(A\) and \(B\), does not affect the width of the resonance peaks. Since atoms of some alkali metals possess states for which, at definite values of the magnetic-field strength, the magnetic moment is zero, it is quite possible to observe nuclear reorientations and atomic transitions without any changes in the apparatus, except for variation of the generator frequency.

In the scheme shown in Fig. 10, atoms in the states \((2,-1)\) and \((1,-1)\) have no magnetic moment. Thus, if the values of the deflecting fields are chosen so that the magnetic moment

of the atom is zero, the atom will pass through fields \(A\) and \(B\) without being deflected. If, however, one takes the transition to the state \(m=-1\) or to the state \(m=0\), the moment of the atom in the second deflecting field will be appreciably different from zero, and the atom will not reach the detector.

This method of “zero moments” was used in early experiments with atomic beams, aimed at determining the spins of the nuclei of alkali-metal atoms, \(\mathrm{Cl}\), \(\mathrm{M}\), \(\mathrm{F}_2\), \(\mathrm{M}_2\), \(\mathrm{M}_3\).

In the case of \(\mathrm{Na}^{23}\) a zero moment occurs at a field strength of 316 gauss; for \(\mathrm{Rb}^{85}\) the moment is zero for two values of the field strength: 361 gauss and 722 gauss. The moment of the \(\mathrm{Cs}^{133}\) atom can be zero at field strengths of 821 gauss, 1642 gauss, and 2463 gauss. The resonance curve for the proton, shown in Fig. 8, was taken with deflecting fields of 1642 gauss (the “second” field for Cs*). The line used in this case for calibration corresponded to the transition \((4,-2)\longleftrightarrow(4,-1)\).

The method described by us for accurate calibration of the field requires that the atomic beam, by means of which this calibration is carried out, be observed simultaneously with the molecular beam for which the resonance curve is recorded. In some cases this is accomplished automatically. Thus, for example, if NaOH is placed in a silver furnace and heated to a temperature of \(650^\circ\), about \(2/3\) of the molecules dissociate, forming an atomic Na beam, which can be used for calibration of the field. The remaining \(1/3\) of the NaOH molecules forms a molecular beam, which can be used to obtain the resonance curve for the proton. In other cases the furnace is made in the form of two separate chambers, merging only at the exit aperture.

In doing this, one must, of course, choose substances for which the required vapor pressure is attained at approximately the same temperatures. At \(600^\circ\mathrm{C}\), KOH has approximately the same vapor pressure as Cs or Rb, provided that the latter are placed in the furnace not in pure form but in a mixture with calcium turnings and alkali-metal chlorides.

ACCURACY

The gyromagnetic deflections of nuclei are determined from the results of measurements of the magnetic field and the oscillator frequency. In the absence of molecular interactions, the resonance minimum corresponds to equality of the frequency of the oscillating field and the Larmor precession frequency. The oscillator frequency can easily be measured with an accuracy of \(0.01\%\). Therefore, the accuracy in determining the value of \(g\) is limited exclusively by the accuracy of measurement of the magnetic field.

If atomic transitions are used for calibration of the magnetic field, then the field strength, determined from the known

* The deflecting field should not be confused with the homogeneous field.

to the gyromagnetic ratio of the electron and the frequency of the observed atomic line, can be measured with an accuracy of up to 0.03%. The accuracy with which the hyperfine structure of the ground states of alkali-metal atoms can be measured is determined by the accuracy of the heterodyne frequency meter, which is somewhat better than 0.01%. For almost all atomic transitions the half-width of the lines is so small that it does not affect the accuracy of the measurements.

If the value of \(g\) for some nucleus is known with sufficient accuracy, it is possible to measure the value of \(g\) for another nucleus with an even higher degree of accuracy, if two corresponding resonance curves are observed with the same magnetic-field setting.

The accuracy of such measurements is limited by the fact that in most cases molecular interactions appreciably broaden the resonance lines. In this case it proves difficult not only to determine accurately the position of the minimum, but also to establish the resonance relation between the frequency of the oscillating field and the precession frequency, since the nature of the intramolecular interactions is not yet fully understood.

RESULTS

Tables V and VI give the results of measurements, by the resonance method, of magnetic moments and separations between levels of hyperfine structure. The values \(g\) in the second column of Table V are given without the diamagnetic correction. These corrections are given in the fifth column. They increase the values of the magnetic moments. In the third column of Table VI are given the separations between the levels of hyperfine structure of the ground states of atoms (in wave numbers), with the value \(2.99776 \cdot 10^{10}\ \mathrm{cm/sec}\) adopted for \(c\).

Of course, these tables are not a complete summary of the results of work on the determination of nuclear magnetic moments and the study of hyperfine structure. Here only results obtained by means of the magnetic resonance method are given. Readers wishing to obtain more complete data are referred to the article by Bette and Bacher\(^{B2}\) in The Reviews of Modern Physics. These authors also give tables of nuclear spins and a résumé of works published up to that time.

Hydrogen\(^{K1, K2, R7, M9}\)

The gyromagnetic ratio for the nucleus \(H^1\) was determined by measuring the frequency of its Larmor precession in the molecules HD, KOH, NaOH and from the magnetic dipole interaction of the two protons constituting the hydrogen molecule.

The values obtained by these two methods agree well with one another within the limits of experimental errors. The most accurate determination of the magnetic moment of the proton was made in experiments with the hydroxides of alkali metals. For the value of \(g\), the value \(5.5791 \pm 0.0016\) was obtained. Since the proton spin

Table V

Nuclear moments measured by the magnetic resonance method

Nucleus	Observed value \(g\)	Spin \(I\)	Magnetic moment	Diamagnetic correction (%)
\(\mathrm{H}^1\)	\(5,5791 \pm 0,0016\)	\(1/2\)	\(+2,7896\)	0
\(\mathrm{H}^2\)	\(0,8565 \pm 0,0014\)	1	\(+0,8565\)	0
\(\mathrm{Li}^6\)	\(0,8213 \pm 0,0005\)	1	\(-0,8213\)	0,01
\(\mathrm{Li}^7\)	\(2,1683 \pm 0,0010\)	\(3/2\)	\(+3,2532\)	0,01
\(\mathrm{Be}^9\)	\(0,781 \pm 0,003\)	\(3/2\)	\(-1,176\)	0,02
\(\mathrm{B}^{10}\)	\(0,598 \pm 0,003\)	\(1^*\)	\(+0,598\)	0,03
\(\mathrm{B}^{11}\)	\(1,791 \pm 0,005\)	\(3/2\)	\(+2,686\)	0,03
\(\mathrm{C}^{13}\)	\(1,402 \pm 0,004\)	\(1/2^*\)	\(+0,701\)	0,03
\(\mathrm{N}^{14}\)	\(0,403 \pm 0,02\)	1	\(+0,403\)	0,04
\(\mathrm{N}^{15}\)	\(0,560 \pm 0,016\)	\(1/2\)	\(0,280\)	0,04
\(\mathrm{F}^{19}\)	\(5,250 \pm 0,005\)	\(1/2\)	\(+2,625\)	0,06
\(\mathrm{Na}^{23}\)	\(1,4765 \pm 0,0015\)	\(3/2\)	\(+2,215\)	0,08
\(\mathrm{Al}^{27}\)	\(1,452 \pm 0,004\)	\(5/2\)	\(+3,630\)	0,10
\(\mathrm{Cl}^{35}\)	\(0,547 \pm 0,002\)	\(5/2^*\)	\(+1,368\)	0,14
\(\mathrm{Cl}^{37}\)	\(0,454 \pm 0,002\)	\(5/2^*\)	\(+1,136\)	0,14
\(\mathrm{K}^{39}\)	\(0,260 \pm 0,001\)	\(3/2\)	\(+0,391\)	0,16
\(\mathrm{K}^{40}\)		4	\(-1,290\)	0,16
\(\mathrm{K}^{41}\)	\(0,143 \pm 0,001\)	\(3/2\)	\(+0,215\)	0,16
\(\mathrm{Kr}^{83}\)	\(0,2148\)	\(9/2^*\)	\(-0,96^{**}\)	0,38
\(\mathrm{Rb}^{85}\)	\(0,536 \pm 0,003\)	\(5/2\)	\(+1,340\)	0,39
\(\mathrm{Rb}^{87}\)	\(1,822 \pm 0,006\)	\(3/2\)	\(+2,733\)	0,39
\(\mathrm{In}^{113}\)	\(1,22 \pm 0,01\)	\(9/2\)	\(+5,49\)	0,58
\(\mathrm{In}^{115}\)	\(1,22 \pm 0,01\)	\(9/2\)	\(+5,50\)	0,58
\(\mathrm{Cs}^{133}\)	\(0,731 \pm 0,002\)	\(7/2\)	\(+2,558\)	0,67
\(\mathrm{Ba}^{135}\)	\(0,554 \pm 0,002\)	\(3/2\)	\(+0,831\)	0,68
\(\mathrm{Ba}^{137}\)	\(0,619 \pm 0,0.2\)	\(3/2\)	\(+0,929\)	0,68

equals \(1/2\), and the magnetic moment equals \(2,7896\) nuclear magnetons. The proton moment has been determined with an accuracy surpassing all measurements of the magnetic moments of other nuclei made up to now. The value of \(g\) for the deuteron was determined in experiments with HD and \(\mathrm{D}_2\) molecules. The ratio

\[ \frac{\mu_p}{\mu_D}, \]

which proved to be \(3,2570 \pm 0,001\), was measured very accurately. Knowing \(\mu_p\), from this ratio we obtain for the magnetic moment of the deuteron the value \(0,8565\) nuclear magneton.

Studying the radio-frequency rotational spectra of the molecules \(\mathrm{H}_2\), HD, and \(\mathrm{D}_2\), Ramsey found that the rotational magnetic moments of these molecules in the first rotational state are respectively \(0,879 \pm 0,007\), \(0,660 \pm 0,005\), and \(0,441 \pm 0,003\) nuclear magneton. He also determined the diamagnetic correction for \(\mathrm{H}_2\) and its dependence on the orientation of the moment of the hydrogen molecule in the first

* The spin of the nucleus is not reliably known.
** Unpublished preliminary result.

Table VI

Intervals between the levels of the hyperfine structure of the ground states of atoms of the alkali metals and indium, measured by means of the magnetic resonance method

Atom	\(\Delta\nu\), \(\mathrm{sec}^{-1}\cdot 10^{-6}\)	\(\mathrm{cm}^{-1}\)
\(\mathrm{Li}^{6}\)	228.22	0.007613
\(\mathrm{Li}^{7}\)	\(803.54\pm 0.04\)	0.026805
\(\mathrm{Na}^{23}\)	\(1771.75\pm 0.07\)	0.059102
\(\mathrm{K}^{39}\)	\(461.75\pm 0.02\)	0.015403
\(\mathrm{K}^{40}\)	\(1285.7\pm 0.1\)	0.042887
\(\mathrm{K}^{41}\)	\(254.02\pm 0.02\)	0.008474
\(\mathrm{Rb}^{85}\)	\(3035.7\pm 0.2\)	0.10127
\(\mathrm{Rb}^{87}\)	\(6834.1\pm 1.0\)	0.22797
\(\mathrm{In}^{113}\)	\(11387\pm 4\)	0.3799
\(\mathrm{In}^{115}\)	\(11413\pm 3\)	0.3807
\(\mathrm{Cs}^{133}\)	\(9192.6\pm 0.5\)	0.30665

rotational state. From an analysis of the radio-frequency spectra of the HD and \(\mathrm{D}_2\) molecules, certain constants of intramolecular interactions were determined. The most interesting of these is the interaction of the deuteron quadrupole moment with the inhomogeneous electric field of the molecule. For the quadrupole moment of the deuteron the value \(2.73\cdot 10^{-27}\ \mathrm{cm}^2\) was obtained.

The presence of a quadrupole moment in the deuteron leads to the conclusion that the ground state of the deuteron is not an \(S\)-state, but a mixture of the states \({}^{3}S_1\) and \({}^{3}D_1\). On this basis, Rarita and Schwinger\({}^{88}\) derived an expression making it possible to calculate the magnetic moment of the neutron if the ratio \(\frac{\mu_p}{\mu_D}\) is known. The \(\mu_n\) calculated in this way proved to be \(-1.911\pm 0.001\). The accuracy here depends exclusively on the experimental errors in the determination of \(\mu_p\) and \(\mu_D\). With simple subtraction, \(\mu_D-\mu_p\), one obtains for \(\mu_n\) the value \(-1.933\).

These results can be compared with the experimentally found value, due to Alvarez and Bloch\({}^{41}\), for the magnetic moment of the free neutron, equal to \(-1.935\pm 0.02\). However, the accuracy achieved by Alvarez and Bloch is still insufficient for such a comparison to serve as a test of the predictions of the theory.*)

) Recently Arnold and Roberts [Phys. Rev. 70, 766 (1946)], combining the Alvarez and Bloch technique with the method of nuclear induction, measured with greater accuracy the ratios \(\frac{\mu_D}{\mu_p}\) and \(\frac{\mu_n}{\mu_p}\). The data obtained by them agree excellently with the results of Rarita and Schwinger. (Translator’s note.*)

J. B. M. Kellogg and S. Millman

Lithium \(^{6}\), \(^{7}\)

Nuclear resonance curves for the lithium isotopes were obtained in experiments with the molecules LiF, LiCl, LiBr, LiJ, and Li\(_2\). The resonance minima proved to be deep and narrow. The gyromagnetic ratio for Li\(^6\) has the value \(0.8213 \pm 0.0005\), and for Li\(^7\), \(2.1688 \pm 0.0010\). The spin of Li\(^6\) is equal to 1, the spin of Li\(^7\) is equal to \(3/2\). Thus the magnetic moments are, respectively, \(0.8213\) and \(3.2532\) nuclear magnetons.

The ratio of the magnetic moments of the two isotopes was determined very accurately by observing the corresponding resonance curves at one and the same field setting. For the ratio \(\dfrac{\mu_{\mathrm{Li}^{7}}}{\mu_{\mathrm{Li}^{6}}}\) the value \(3.9601 \pm 0.0015\) was obtained. \(\Delta \nu\) for the ground state of Li\(^6\) was determined by studying Zeeman lines in fields from 0.25 to 1.5 gauss. \(\Delta \nu\) for Li\(^7\) was determined from the Zeeman effect in a field of strength 0.15 gauss and by investigating lines characterized by the transition \(\Delta F = 0,\ \Delta m = \pm 1\) at a magnetic-field strength of 3800 gauss. For Li\(^6\), \(\Delta \nu = (228.22 \pm 0.01)\cdot 10^6\ \mathrm{sec}^{-1}\); for Li\(^7\), \(\Delta \nu = (803.54 \pm 0.04)\cdot 10^6\ \mathrm{sec}^{-1}\).

For the value of the ratio \(\dfrac{\mu_7}{\mu_6}\), calculated by the formula

\[ \frac{\mu_7}{\mu_6} = \frac{\left[\,^{7}J/(2I+1)\,\right]_7} {\left[\,2J/(2I+1)\,\right]_6} \frac{\Delta \nu_7}{\Delta \nu_6}, \]

one obtains the value \(3.9610 \pm 0.0004\). The excellent agreement of this result with the direct-measurement data given above indicates that the hyperfine structure in this case is due exclusively to the magnetic interaction of the nuclear moment with the outer electrons.

Beryllium \(^{9}\)

The nuclear resonance curve for Be\(^9\) was obtained on the molecules NaF·BeF\(_2\) and \(\Lambda\)F·BeF\(_2\). For \(g\) the value \(0.784 \pm 0.003\) was obtained. The magnetic moment of the nucleus proved to be negative. There are as yet no experimental data permitting an unambiguous determination of the nuclear spin. A theoretical consideration \({}^{9}\), taking into account the negative sign of the magnetic moment, leads with a high degree of probability to a spin equal to \(3/2\). If this value is adopted, we obtain for the magnetic moment the value \(-1.76\).

Boron \(^{10}\)

The value of \(g\) for the boron isotopes was measured from resonance curves obtained on tetraborates and metaborates of the alkali metals. For B\(^ {10}\), \(g = 0.598 \pm 0.003\); for B\(^ {11}\), \(g = 1.791 \pm 0.005\). The spins of these nuclei have not been measured.

Theoretical considerations R9, F1, S1 give spin 1 for B10 and spin 3/2 for B11. If these values are accepted for the spins, for the magnetic moment of B10 we obtain 0.598, and for B11 the value 2.686.

Carbon H3

The results of experiments with the least abundant isotope of carbon have recently been published by Hahn. The resonance curves were obtained with KCN and NaCN molecules, the beam being enriched with the isotope C13. For \(g\) the value \(1.402 \pm 0.004\) was obtained. The spin of C13 is not known with certainty. Although analysis of the intensities of the lines of the band spectrum T3 shows that spin \(3/2\) is more probable than \(1/2\), simple theoretical considerations R9 in combination with Hahn’s results make spin \(1/2\), rather than \(3/2\), preferable. If the spin is taken equal to \(1/2\), then for the magnetic moment we obtain the value 0.701.

Nitrogen K9; Z22

Resonance curves for N14 were observed with LiCN, NaCN, and KCN molecules. Resonance curves for N15 were observed with a beam, enriched with this isotope, of N2 molecules. The sign of the magnetic moment of N15 cannot be regarded as established with certainty.

For N14 \(g = 0.403 \pm 0.002\), and for N15 \(g = 0.560 \pm 0.006\). The spin of N14 is 1, the spin of N15 is \(1/2\) W2; K5. The magnetic moments are equal to 0.403 and 0.280 respectively.

Fluorine R5, R5

Resonance curves for F were obtained with LiF, NaF, and KF molecules. With a beam of NaF, deep and narrow minima are obtained. \(g = 5.250 \pm 0.005\). Since the nuclear spin is \(1/2\), the magnetic moment is 2.625.

Sodium M8, K9

Resonance curves for Na were obtained with beams of NaF, NaCN, and Na2 molecules. The difference between the resonance curves obtained in these three cases is very remarkable. In the case of NaF the resonance minima are broad and small: the drop in intensity at the minimum is only 4%. In the case of NaCN the resonance minima are considerably narrower, and the drop in intensity at the minimum is 9%. In the case of Na2 the minima are very narrow: their half-widths are comparable with those for hydrogen and lithium. The drop in intensity at the minimum reaches 55%. This difference is due to the difference in the intramolecular interactions present in the molecules listed. In all three cases, however, the same value was obtained for \(g\): \(1.4765 \pm 0.0015\). Since the nuclear spin is \(3/2\), the magnetic moment proves to be 2.215.

The distance between the hyperfine-structure levels for the ground state of the sodium atom was determined from lines characterized by transitions \(\Delta F = 0\), \(\Delta m = \pm 1\) at a field strength of 5000 gauss. It proved to be \((1771.75 \pm 0.07)\cdot 10^6\ \mathrm{sec}^{-1}\).

J. B. M. KELLOGG AND S. MILLMAN

Aluminum \(^{\mathrm{M7}}\)

Nuclear resonance curves for Al were obtained with the molecules \(\mathrm{NaCl}\cdot\mathrm{AlCl}_3\) and \(\mathrm{KCl}\cdot\mathrm{AlCl}_3\); \(g=1.452\pm0.004\).

Comparison of this result with values calculated from Goudsmit’s formula under the assumption of different spins and on the basis of experimental data on hyperfine structure, taken from the works of Jackson and Kuhn \(^{\mathrm{J1}}\) and of Heyden and Ritschl \(^{\mathrm{H4}}\), leads to the spin value \(5/2\). Then the magnetic moment is equal to \(3.630\).

Chlorine \(^{\mathrm{K6, S3}}\)

Resonance curves for chlorine isotopes were obtained with molecular beams of LiCl and RbCl. For \(\mathrm{Cl}^{35}\), \(g=0.547\); for \(\mathrm{Cl}^{37}\), \(g=0.454\). From band spectra, the value \(5/2\) \(^{\mathrm{E2, S2}}\) was obtained for the spins of both isotopes. Proceeding from these spin values, we obtain for the magnetic moments the quantities: \(1.368\) for \(\mathrm{Cl}^{35}\) and \(1.136\) for \(\mathrm{Cl}^{37}\). It should, however, be borne in mind that the determination of such large spins as \(5/2\) from band spectra cannot be regarded as reliable.

Potassium \(^{\mathrm{Z1, K9}}\)

Nuclear resonance curves for \(\mathrm{K}^{39}\) were observed with beams of \(\mathrm{K}_2\) molecules. For \(g\) the value \(0.260\pm0.001\) was obtained. Since the spin is equal to \(3/2\), the magnetic moment is \(0.391\). For the other, less abundant, potassium isotopes, analogous nuclear resonance curves were not observed. Atomic radio-frequency spectra have been studied for each of the three isotopes. The case of \(\mathrm{K}^{39}\) was considered in detail above.

For the ground state of the \(\mathrm{K}^{39}\) atom, \(\Delta\nu=(461.75\pm0.02)\times10^6\ \mathrm{sec}^{-1}\). The separation between the hyperfine-structure lines for \(\mathrm{K}^{41}\) was measured from the Zeeman splitting in fields from 0.15 to 0.5 gauss. It proved to be equal to \((254.02\pm0.02)\cdot10^6\ \mathrm{sec}^{-1}\). Since the spins of \(\mathrm{K}^{41}\) and \(\mathrm{K}^{39}\) are the same, we can use the ratio of the \(\Delta\nu\) values for these two isotopes and the magnetic moment of \(\mathrm{K}^{39}\) known to us to determine the magnetic moment of \(\mathrm{K}^{41}\). For it the value \(0.215\) is obtained.

The radio-frequency spectrum of \(\mathrm{K}^{40}\) was studied by Zacharias \(^{\mathrm{Z1}}\) in different magnetic fields. He determined the nuclear spin, which turned out to be equal to 4, measured \(\Delta\nu\) for the ground state with an accuracy comparable with the accuracy of the measurements of this quantity for the isotopes \(\mathrm{K}^{39}\) and \(\mathrm{K}^{41}\), and reliably showed that the magnetic moment of \(\mathrm{K}^{40}\) is negative. All his work was carried out with the natural mixture of isotopes, in which \(\mathrm{K}^{40}\) constitutes only \(1/8000\) part. For \(\Delta\nu\) the value \((1235.65\pm0.1)\cdot10^6\ \mathrm{sec}^{-1}\) was obtained. The magnetic moment of \(\mathrm{K}^{40}\) can be determined from the ratio of the \(\Delta\nu\) values for \(\mathrm{K}^{40}\) and \(\mathrm{K}^{39}\) and the known magnetic moment of \(\mathrm{K}^{39}\). As a result, for the magnetic moment of \(\mathrm{K}^{40}\) the value \(-1.290\) was obtained.

Krypton

The resonance curve for krypton was obtained by Rederford and Kellogg (unpublished). The value of \(g\) for \(\mathrm{Kr}^{83}\) is \(0.2148\). The spin is not known with certainty, but in all probability is equal to \(9/2\). The magnetic moment is negative.

Rubidium \(^{\mathrm{M}8,\mathrm{K}6}\)

The nuclear resonance curves for rubidium were studied on \(\mathrm{Rb}_2\) molecules. The resonance lines proved to be rather broad, as a result of which high accuracy in the determination of \(g\) could not be attained.

For \(\mathrm{Rb}^{85}\), \(g = 0.536 \pm 0.003\); for \(\mathrm{Rb}^{87}\), \(g = 1.822 \pm 0.006\). The magnetic moments are \(1.340\) and \(2.733\), respectively. Hence the ratio

\[ \frac{\mu_{87}}{\mu_{85}} \]

is equal to \(2.038 \pm 0.010\). The quantities \(\Delta \nu\) for the ground states of these atoms were determined from the frequencies of the lines characterized by the transitions \(\Delta F = 0,\ \Delta m = +1\) in a strong magnetic field. For \(\mathrm{Rb}^{85}\),

\[ \Delta \nu = (3035.7 \pm 0.2)\cdot 10^6\ \mathrm{sec}^{-1}, \]

for \(\mathrm{Rb}^{87}\),

\[ (6837.1 \pm 1.0)\cdot 10^6\ \mathrm{sec}^{-1}. \]

Starting from the values of \(\Delta \nu\) and the spins, one can calculate the ratio of the magnetic moments of the two isotopes. It turns out to be \(2.0261 \pm 0.0033\). Thus, within the limits of experimental errors, this value of the ratio

\[ \frac{\mu_{87}}{\mu_{85}} \]

agrees with the value obtained from measurements of the magnetic moments of the nuclei.

Indium \(^{\mathrm{H}1,\mathrm{H}2}\)

Radio-frequency spectra of the ground states of indium atoms for lines characterized by the transitions \(\Delta F = 0,\ \Delta m = \pm 1\) were observed for \(\mathrm{In}^{113}\) and \(\mathrm{In}^{115}\). The results show that the spins of both isotopes are equal to \(9/2\). The ratio of the quantities \(\Delta \nu\) for the ground states of the atoms \(\mathrm{In}^{115}\) and \(\mathrm{In}^{113}\) is equal to \(0.00224 \pm 0.00010\). This same quantity is also equal to the ratio of the magnetic moments of the two isotopes. For \(\mathrm{In}^{115}\), \(\Delta \nu\) is equal to

\[ (11413 \pm 3)\cdot 10^6\ \mathrm{sec}^{-1}, \]

and the magnetic moment is equal to \(5.49 \pm 0.04\) nuclear magnetons. The magnetic moments of both isotopes are positive.

Cesium \(^{\mathrm{K}9}\)

Resonance curves for cesium were obtained with the molecules CsF, CsCl, and \(\mathrm{Cs}_2\). \(g = 0.731 \pm 0.002\). Since the spin is equal to \(7/2\), the value \(2.558\) is obtained for the magnetic moment. The hyperfine structure of the ground state was measured by studying lines corresponding to the transitions \(\Delta F = 0,\ \Delta m = \pm 1\) in fields of strength about 1300 gauss. \(\Delta \nu\) proved to be equal to

\[ (9192.6 \pm 0.5)\cdot 10^6\ \mathrm{sec}^{-1}. \]

Barium^H3

Resonance curves for the odd isotopes of barium were observed by Hsü with beams of atomic barium. Since the electronic moment in this case is equal to zero, there is no interaction between the electrons and the nucleus. The resonance minima, as was to be expected, proved to be narrow. For Ba^185 \(g=-0.554 \pm 0.002\), for Ba^187 \(g=0.619 \pm 0.002\).

From spectroscopic data it is reliably known that the spins of both isotopes are equal to \(^{3}/_{2}\), which gives for the magnetic moments the values 0.831 and 0.929.

CITED LITERATURE

A1. L. W. Alvarez and F. Bloch, Phys. Rev. 57, 111 (1940).
B1. A. W. Benson and R. A. Sawyer, Phys. Rev. 52, 1127 (1937).
B2. H. A. Bethe and R. F. Bacher, Rev. Mod. Phys. 8, 82 (1936).
B3. F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940).
B4. G. Brett and I. I. Rabi, Phys. Rev. 38, 2182 (1931).
C1. V. W. Cohen, Phys. Rev. 66, 413 (1934).
D1. S. Dushman and C. G. Forind, Phys. Rev. 17, 7 (1921).
E1. A. Ellet and R. M. Zabel, Phys. Rev. 37, 1102 (1931).
E2. A. Elliot Proc. Roy. Soc. A 127, 638 (1930).
E3. I. Estermann und O. Stern, Zeits. f. Phys. 85, 17 (1933).
F1. E. Flenberg and E. Wigner, Phys. Rev. 51, 95 (1937).
F2. M. Fox and I. I. Rabi, Phys. Rev. 48, 746 (1935).
F3. R. G. J. Fraser, “Molecular rays” (The Macmillan Company, New-York, 1931).
F4. R. G. J. Fraser, “Molecular beams” (Chemical Publishing Company, New-York, 1938).
F5. R. Frisch and O. Stern, Zeits. f. Physik, 85, 4 (1933).
G1. W. Gerlach and O. Stern, Ann. d. Physik, 74, 673 (1924).
G2. G. J. Gorter, Physica, 3, 995 (1936).
G3. A. N. Guthrie, Phys. Rev. 49, 868 (1936).
G4. P. Guttinger, Zeits. f. Physik, 73, 169 (1931).
H1. T. C. Hardy, Phys. Rev. 59, 666 (1941).
H2. T. C. Hardy and S. Millman, Phys. Rev. 61, 459 (1942).
H3. R. H. Hay, Phys. Rev. 60, 75 (1941).
H4. M. Heyden and R. Ritschl, Zeits. f. Physik, 108, 739 (1938).
H5. R. D. Huntoon and A. Ellet, Phys. Rev. 49, 381 (1936).
J1. D. A. Jackson and H. Kuhn, Proc. Roy. Soc. A 164, 48 (1938).
J2. T. H. Johnson, Phys. Rev. 31, 103 (1928).
K1. J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey, Jr. and J. R. Zacharias, Phys. Rev. 56, 728 (1939).
K2. J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey, Jr. and J. R. Zacharias, Phys. Rev. 57, 677 (1940).
K3. J. M. B. Kellogg, I. I. Rabi and J. R. Zacharias, Phys. Rev. 50, 472 (1936).
K4. F. Knauer and O. Stern, Zeits. f. Physik 53, 766 (1929).
K5. H. Kruger, Zeits. f. Physik 111, 467 (1939).
K6. P. Kusch and S. Millman, Phys. Rev. 56, 527 (1939).
K7. P. Kusch, S. Millman and I. I. Rabi, Phys. Rev. 57, 765 (1940).
K8. P. Kusch, S. Millman and I. I. Rabi, Phys. Rev. 55, 666 (1939).
K9. P. Kusch, S. Millman and I. I. Rabi, Phys. Rev. 55, 1176 (1939).

M1. E. Majorana, Nuovo Cimento 9, 43 (1932).
M2. J. H. Manley, Phys. Rev. 49, 921 (1936).
M3. J. H. Manley and S. Millman, Phys. Rev. 51, 19 (1937).
M4. S. Millman, Phys. Rev. 47, 739 (1933).
M5. S. Millman, Phys. Rev. 55, 628 (1939).
M6. S. Millman and M. Fox, Phys. Rev. 50, 220 (1936).
M7. S. Millman and P. Kusch, Phys. Rev. 56, 303 (1939).
M8. S. Millman and P. Kusch, Phys. Rev. 58, 438 (1940).
M9. Millman and P. Kusch, Phys. Rev. 60, 91 (1941).
M10. S. Millman, P. Kusch and I. I. Rabi, Phys. Rev. 56, 165 (1939).
M11. S. Millman, I. I. Rabi and J. R. Zacharias, Phys. Rev. 53, 384 (1938).

N1. A. Nordsieck, Phys. Rev. 58, 310 (1940).
P1. P. N. Powers, Phys. Rev. 54, 827 (1938).
R1. I. I. Rabi, Phys. Rev. 49, 324 (1936).
R2. I. I. Rabi, Phys. Rev. 51, 652 (1937).
R3. I. I. Rabi and V. W. Cohen, Phys. Rev. 46, 707 (1934).
R4. I. I. Rabi, J. M. B. Kellogg and J. R. Zacharias, Phys. Rev. 46, 157 (1934).
R5. I. I. Rabi, S. Millman, P. Kusch and J. R. Zacharias, Phys. Rev. 55, 526 (1939).
R6. I. I. Rabi, J. R. Zacharias, S. Millman and P. Kusch, Phys. Rev. 53, 318 (1938).
R7. N. F. Ramsey, Jr., Phys. Rev. 58, 226 (1940).
R8. W. Rarita and J. Schwinger, Phys Rev. 59, 436 (1941).
R9. M. E. Rose and H. A. Bethe, Phys. Rev. 51, 205 (1937).
S1. R. G. Sachs, Phys. Rev. 55, 825 (1939).
S2. E. F. Shrader, Phys. Rev. 58, 475 (1940).
S3. E. F. Shrader, S. Millman and P. Kusch, Phys. Rev. 58, 925 (1940).
S4. O. Stern, Zeits. f. Physik, 7, 249 (1921).
S5. A. F. Stevenson, Phys. Rev. 58, 1661 (1940).
T1. I. B. Taylor, Zeits. f. Physik, 57, 242 (1929).
T2. H. C. Torrey, Phys. Rev. 59, 293 (1941).
T3. C. H. Townes and W. R. Smythe, Phys. Rev. 56, 1210 (1939).
W1. G. C. Wick, Zeits f. Physik, 85, 25 (1933).
W2. R. W. Wood and G. H. Dicke, J. Chem. Phys. 8, 351 (1940).
Z1. J. R. Zacharias, Phys. Rev. 61, 270 (1942).
Z2. J. R. Zacharias and J. M. B. Kellogg, Phys. Rev. 57, 570 (1940).

1. Powers. ↩