Radio Spectroscopy

Gorter*)

1. Introduction

One of the most interesting fields of modern physics is spectroscopy in the range of meter and centimeter waves. Just as in the entire interval from gamma rays to infrared radiation, here we are dealing with processes of transition between energy levels in nuclear, atomic, ionic, or molecular systems, which in what follows we shall denote by the single word “atom.” These transition processes are accompanied either by spontaneous emission, leading to a decrease in the energy of the atom, or by absorption, associated with an increase in the energy of the atom by a quantum of radiation \(h\nu\), or, finally, by the stimulated emission of a quantum \(h\nu\), caused by the presence of an external electromagnetic field with frequency \(\nu\).

Einstein\(^{1}\) showed that, for Planck’s law of black-body radiation to be valid, it is necessary to admit the existence of stimulated emission. Such radiation is negligible in comparison with spontaneous radiation so long as the spectra of gamma, X-ray, ultraviolet, and visible radiation are considered; in the radio-wave region, however, the reverse situation is observed. This is because, in the emission of a quantum of visible light, the mean lifetime of the excited state is, in order of magnitude, \(10^{-8}\) sec, whereas in the emission of a quantum in the meter or centimeter range it turns out to be equal to hours, years, and even centuries.

In the radio range, spontaneous emission of quanta may be completely neglected. At ordinary temperature the processes of absorption and stimulated emission, caused by the so-called black electromagnetic radiation, likewise

) C. J. Gorter, Experientia 9*, No. 5, 161 (1953). Lecture delivered in November 1951 at the Faculté des Sciences de Paris.

are of no special significance. The radiation induced by hertzian waves becomes significant because, by applying radio-engineering methods, it proves possible to concentrate a colossal radiation energy in a narrow frequency band. If the frequency band coincides with the frequency corresponding to a spectral line of the atom, then the hertzian waves can cause millions of acts of absorption and stimulated emission per second².

Until now we have considered only the interaction of the atom with the electromagnetic field. Let us now admit the presence of an interaction of the atom with the medium. If the interaction is weak, it can be described in first approximation as a perturbation superposed on the state of the isolated atomic system. This perturbation will produce two effects. On the one hand, it will lead to a displacement or splitting of the energy levels and spectral lines (if the displacement is not selective, then a broadening of the spectral line is observed). On the other hand, the perturbation may cause new transitions between energy levels (if the number of new transitions is comparable with the number of transitions taking place in the isolated atom, then in this case also a broadening of the spectral lines will be observed). Both these effects are well known and often complicate the investigation.

Transitions between energy levels may lead to an increase or decrease in the energy of the atom. The thermodynamic temperature of the surrounding medium often determines the relative frequency with which processes of each type occur. In this case the interaction with the medium leads to the preservation or, in some cases, to the restoration of the Boltzmann statistical distribution over energy levels for an ensemble of individual atoms. Sometimes the statistical distribution over levels corresponds to an effective temperature \(T_{\mathrm{eff}}\), entering into the Boltzmann factor and not equal to the temperature of the surrounding medium. In this case the interaction leads to a decrease in the difference between \(T_{\mathrm{eff}}\) and the temperature of the medium; the exponential time dependence of the difference between these quantities is characterized by a relaxation time, which is of very great importance in the study of radiation processes.

Let us note that if the statistical distribution over energy levels can be described by means of an effective temperature, then one can also define the specific effective heat capacity of the ensemble, equal to the energy required to raise the temperature by one degree. If the energy levels of the atoms are situated considerably below the energy level \(kT_{\mathrm{eff}}\), then the specific effective heat capacity will be inversely proportional to the square of \(T_{\mathrm{eff}}^{3}\).

It should be noted that for an ensemble of atoms it is not always possible to establish a single value of \(T_{\mathrm{eff}}\). In some cases \(T_{\mathrm{eff}}\) may

may take unexpected values. Thus, for example, Purcell^4 succeeded in achieving a preferential population of the upper energy levels, which corresponds to a negative \(T_{\mathrm{eff}}\). From thermodynamic considerations it follows that these negative temperatures should be regarded not as less than absolute zero, but rather as exceeding infinitely large ones.

To summarize: radiospectroscopy deals with quanta whose energies are smaller than the thermal energy \(kT_{\mathrm{eff}}\). In contrast to optical spectroscopy, which considers regions of large energy quanta, in radiospectroscopy Einstein’s forced emission plays an important role. The consideration of processes is facilitated by introducing the concept of effective temperature and specific effective heat capacity.

Experiments show that in most cases (though not always) transitions are due to the presence of an alternating magnetic field. Usually magnetic dipole transitions occur, rather than electric dipole or quadrupole transitions, as in optical spectra^2.

One should distinguish between the methods applied in the region of meter and centimeter waves. When working in the meter-wave range, the magnetic fields imposed on the atomic systems under study are produced by high-frequency alternating currents. If the atoms possessed more highly populated levels, then the probability of these states can be reduced as a result of forced transitions. In the case of atoms in an atomic or molecular beam, the reduction of the preferential population can be observed directly by applying the method of Rabi and his collaborators^5. Kastler^6 proposed other methods, one of which has already been successfully applied. If the preferential population of certain levels is due to the Boltzmann factor, then a decrease in the population will correspond to an increase in the effective temperature, which assumes an infinitely large value when the population is reduced to zero. Such an increase in temperature may lead to a transfer of energy to the medium. If the atoms are in a condensed substance, then heat is released.

If an increase of \(T_{\mathrm{eff}}\) takes place, the investigation can be conducted by measuring the reaction in the radiating or receiving circuit. Such studies require the construction of alternating-current bridges or generators whose characteristics will be very sensitive to small changes in the working circuits. When using Rabi’s methods, where each atom moves along an individual trajectory, one can record both the processes of forced emission and the processes of absorption. Other methods

make it possible to determine only changes in the statistical distribution caused by the difference between the number of acts of induced emission and absorption. Whereas the effects observed by almost all methods are inversely proportional to the temperature or, more precisely, to the effective temperature, the Rabi methods are suitable even for infinitely high temperatures.

When working in the centimeter-wave range, improved methods used in radar are employed. Work is carried out with klystron generators. One studies either the influence of a sample on the attenuation of radio waves in a cavity resonator, or the weakening of electromagnetic waves propagating along a waveguide. The latter method is applicable only to the study of gases and vapors. The attenuation and weakening of electromagnetic waves are inversely proportional to the effective temperature.

Let us now briefly consider the various lines of research and their historical development.

In 1934 Cleeton and Williams8 discovered a remarkable band in the absorption spectrum of the ammonia molecule, corresponding to a wavelength of 1.5 cm. These authors used a magnetron as the radiation generator and measured the absorption of the latter in a gas enclosed in a rubber balloon. The absorption was very slight, and the structure of the absorption band was completely smeared out as a result of collisions.

In 1946 a number of scientists simultaneously resumed these investigations, applying methods developed during the Second World War for radar. It proved possible to resolve the complex structure by using low pressures9. Encouraged by this success, researchers undertook the study of the rotational spectra of simple molecules. Particularly great attention was attracted by pronounced isotopic effects, as well as by the hyperfine structure caused by the magnetic and quadrupole moment of the nucleus10. In 1950 Dehmelt and Krüger11 discovered transitions between the quadrupole levels of the chlorine nucleus in a crystalline organic compound.

In the investigation of atomic spectra one often uses a regulated constant magnetic field. As a result of the Zeeman effect, the energy levels shift by several megacycles per oersted. (In the nuclear Zeeman effect this shift amounts to several kilocycles per oersted.) The frequency of the generator is kept stable. The constant magnetic field, however, is chosen in such a way that the frequency of some spectral line coincides with the frequency of the generator. Using this method, Rabi, Lamb, and their collaborators12, investigating individual atoms in an atomic beam, obtained fundamental results for hydrogen, deuterium, and other atoms.

In most works, transitions were observed between Zeeman sublevels obtained as a result of the magnetic splitting of a single energy level. If one denotes by \(\gamma\) the gyromagnetic ratio for an atom situated on such a level, then for the energy difference \(\Delta W\) between two neighboring Zeeman sublevels we obtain:

\[ \Delta W=\gamma \frac{Hh}{2\pi}, \]

i.e. the frequency of the line of the so-called magnetic resonance will be equal to

\[ \nu=\gamma \frac{H}{2\pi}. \]

After fruitless attempts by Dutch scientists\({}^{13}\), this line was found for Rabi nuclei\({}^{5}\)—in work with atomic beams, by Alvarez and Bloch\({}^{14}\)—in work with a beam of neutrons and, finally, by Purcell\({}^{15}\) and Bloch\({}^{16}\)—in the study of condensed matter. At the present time precise measurements of the magnitude \(\gamma\) are being carried out, which makes it possible to calculate the magnetic moment of the nucleus if its mechanical moment is known. Quadrupole moments of nuclei are also being studied\({}^{17}\). In condensed matter, a whole series of small shifts, splittings and broadenings of lines, and energy exchange between the spins of nuclei and the surrounding medium are observed. These studies arouse extraordinarily great interest.

Nuclear magnetic resonance in fields with strengths of the order of thousands of oersteds can be observed on meter waves. For the observation of splittings caused by the electronic Zeeman effect, however, in fields of the same strength, frequencies a thousand times greater must be used. After several fruitless attempts by Dutch scientists\({}^{18}\), Zavoisky\({}^{19}\) in 1946 and Griffiths\({}^{20}\) succeeded in detecting paramagnetic and ferromagnetic resonances. Finally, in 1951 in Leiden\({}^{21}\) antiferromagnetic resonance was discovered (see § 4). Since in these investigations the band widths turn out to be considerably greater, it is clear that sufficiently short waves and strong magnetic fields must be used.

Ferromagnetic resonance (as well as other properties of ferromagnets) depends strongly on the shape of the specimen placed in the high-frequency field. Since almost all ferromagnets possess good electrical conductivity, then, as Kittel showed\({}^{22}\), the presence of the skin effect significantly complicates the picture. For most substances the value of \(\gamma\) obtained from experiment differs from the value found for a free spin. This may be connected with the influence of the orbital component on the effective mechanical and magnetic moments. It should be noted that the width

the line turns out to be considerably larger than the value given by the modern theory[^23].

In the first studies of paramagnetic resonance in powdered salts containing water of crystallization, the lines obtained were rather broad. In anhydrous salts the lines were significantly narrower. This effect is explained by exchange interaction between magnetic ions[^24]. In almost all modern works, single crystals are studied in which the magnetic ions are strongly diluted by nonmagnetic ones. In this case the line width is very small, which makes it possible to study splittings caused by the crystalline field, and also by the hyperfine structure.

The first attempts to detect antiferromagnetic resonance at centimeter wavelengths in oxides and anhydrous powdered salts gave negative results. Recently a rather complex resonance was discovered in \(\mathrm{CuCl_2\cdot 2H_2O}\), but there is as yet no exhaustive interpretation of this phenomenon[^21].

Distinct from the questions discussed above, which relate to the nuclear spectra of atoms and molecules, is the field of radiospectroscopy that considers the rotational frequencies of a free charged particle situated in a homogeneous magnetic field. This frequency, usually called the “cyclotron frequency,” is equal to

\[ \frac{eH}{2\pi mc}, \]

where \(e\) and \(m\) are the charge and mass of the particle, and \(c\) is the speed of light. By comparing the value of this frequency with one of the magnetic-resonance frequencies in a field of the same strength \(H\), one can obtain a very accurate value of \(\gamma\), expressed in \(e/mc\)[^25].

Let us summarize. Radiospectroscopy has already made it possible to obtain extraordinarily interesting results in various fields of physics[^26]. In particular, the physics of elementary particles and nuclei owes to radiospectroscopy a whole series of fundamental discoveries. The performance of these investigations became possible thanks to extremely accurate frequency measurements. We have in mind, for example, work on determining the magnetic moment of the electron and on the fine structure of the hydrogen spectrum. With the aid of radiospectroscopic methods it is possible to measure some atomic constants with great accuracy, to obtain data on the magnetic and quadrupole moments of nuclei, on the weak interaction of nuclei with the electron shell and with condensed matter, and also data on the structure of liquids and solids.

The application of radiospectroscopy in chemistry and crystallography is extremely fruitful. Apparently, radiospectroscopy will soon find applications in technology as well.

2. NUCLEAR MAGNETIC RESONANCE IN CONDENSED SUBSTANCES

From experiments on nuclear magnetic resonance one can obtain the value of the constant \(\gamma\)—the ratio of the spin magnetic and mechanical moments. Multiplying \(\gamma\) by the mechanical moment \(Ih/2\pi\), known from observations of hyperfine structure, we obtain the magnetic moment of the nucleus. In the expression

\[ \omega = 2\pi\nu = \gamma H \]

the frequency \(\nu\) of the line (usually a very sharp one) is determined by the usual methods of UHF technique with a relative accuracy of the order of \(10^{-5}\). The accuracy in determining \(H\) is of the order of \(10^{-3}\). Thus, the determination of the ratio of the values of \(\gamma\) for two nuclei is, as a rule, considerably more accurate than the determination of their absolute values. However, even the absolute accuracy of the measurements turns out to be considerably greater than the accuracy of determining the corresponding quantities from optical hyperfine structure (approximately 10%). This is explained by the presence of the magnetic fields of the valence electrons, whose influence is difficult to estimate.

When considering the table in which known values of \(\gamma\) or of magnetic moments are given, striking regularities become apparent, such as, for example, those occurring in the series \(D^2\), \(Li^7\), \(N^{14}\), where the successive addition of \(\alpha\)-particles is accompanied by the same decrease in the value of \(\gamma\).

On the basis of this table, as well as of the table of quadrupole-moment values obtained from other investigations carried out using radio waves, numerous hypotheses and theories of nuclear structure have been created. It should be noted that the accuracy of the experimental data is still considerably higher than the possibilities of theoretical calculation.

The inhomogeneous electric field created by the environment of the nucleus does not act directly on the spin. Nevertheless, splitting of lines is sometimes observed, caused by the quadrupole moment of the nucleus \(^{17}\).

It was established comparatively early that the magnetic field acting on the nucleus is not completely identical with the applied external field. It must be taken into account that polarization of the electron shell by the applied field leads to the appearance of an additional field in the region of the nucleus. This effect occurs in the free atom \(^{27}\); in the case of a molecule one must also take into account its rotation as a whole and the influence of neighboring atoms.

For liquids and solids the influence of the environment of the nucleus manifests itself in many ways. The shape of the specimen appears in the form of the so-called demagnetizing field. These additional fields

usually proportional to the applied field and therefore give a constant relative correction to the value of \(\gamma\). In substances which, in the normal state, possess diamagnetic properties, the correction does not exceed \(10^{-4}\) [28]. However, as Ramsey [29] has shown, one should expect line shifts caused by paramagnetism not depending on the temperature introduced by Van Vleck [30]. In individual cases, for example in complex compounds containing the trivalent cobalt ion [28], relative shifts of order \(10^{-2}\) are observed. Quite recently, Bloch and co-workers [31] discovered relative shifts of order \(10^{-6}\), caused by protons occupying different positions in the molecules of organic compounds; this makes it possible to expect interesting applications in the field of organic chemistry. On the other hand, Knight [32] found considerably larger shifts in metals, caused by the paramagnetism of conduction electrons.

In those cases where the additional field is due not to polarization but to permanent magnetic moments, one should expect a splitting of the lines; if the magnitude and direction of the shift of the split lines are different, then line broadening should be observed. The interaction between two protons in a molecule of water of crystallization leads to a line splitting which does not depend on the absolute value of the frequency, but does depend on the angle between the line joining the protons and the magnetic field. Splitting does not occur if this angle is equal to \(54^{\circ}\frac{1}{4}\), since then the field of one proton at the point where the second proton is located is perpendicular to the applied field. Pake [33] studied the splitting caused by protons in gypsum \(\mathrm{Na_2SO_4} + 10\mathrm{H_2O}\), where it corresponds to a magnetic field of order 10 oersteds. Line broadening is not observed, however, in such liquids as water, liquid paraffin, alcohol, and in solids (paraffin and ice) at not very low temperatures. Thus the width of the resonance for protons in these substances is at least a hundred times smaller than expected. Bloembergen, Purcell, and Pound [34] established that the decrease in line width is caused here by the rapid relative motion of neighboring protons. This also leads to the secular perturbations corresponding to very weak fields. A more or less analogous picture is observed in the study of nuclear magnetic resonance in a substance containing ions with a permanent magnetic moment. In solutions of paramagnets and in many paramagnetic solid salts the broadening is considerable, and in some cases so large that observation of the band becomes impossible. However, in solid salts in which the concentration of paramagnetic ions is sufficiently large for exchange interactions with high frequency to appear, the line width decreases and becomes suffici-

exactly small\(^{24}\). At low temperature there occurs a splitting of this narrow band, caused by the positive sign of the mean moment of each ion\(^{35}\). We shall return to this question later.

We indicated above that an indiscriminate, or variable, displacement of the magnetic-resonance line leads to its broadening, provided only that the changes of the perturbations do not occur too rapidly. This is not the only cause of broadening. One must also take into account broadening caused by spontaneous transitions between energy levels. Bloch introduced the concept of a characteristic time interval \(t_2 = 1/2\pi \Delta \nu\), where \(\Delta \nu\) is the full width of the band.

Usually one considers interaction with a system possessing a large heat capacity, which promotes the establishment or restoration of the Boltzmann distribution over the energy levels under consideration. The set of transitions is most often characterized by a relaxation constant \(t_1\), which determines the time of return to the temperature \(T\) for the energy levels under consideration. In 1946 Bloch\(^{16}\) gave his famous classical calculation of the magnetic gyroscope. The equation of motion of the gyroscope has the following form:

\[ \frac{d\boldsymbol{\mu}}{dt}=\gamma[\boldsymbol{\mu}H], \]

where \(\boldsymbol{\mu}\) is the magnetic moment, and the square brackets denote the vector product.

The gyroscope precesses about the direction of the constant field; the cyclic frequency of precession is equal to

\[ \omega=\gamma H. \]

This precession proves to be substantially perturbed when an additional weak field \(H_1\), perpendicular to \(H\) and rotating about \(H\) with an angular frequency close to \(\omega\), is applied. In this case the perpendicular component of \(\boldsymbol{\mu}\) follows the field \(H_1\), whereas the component in the direction of the constant field oscillates with considerable amplitude.

Bloch introduced into the dynamical equations two terms to account for absorption: one for the component parallel to the constant field, and the other for the perpendicular component. The first term, associated with the transfer of potential energy, is characterized by the relaxation time \(t_1\) (to avoid confusion, following Purcell\(^{4}\), we write \(t_1\) and \(t_2\) instead of \(T_1\) and \(T_2\), reserving these designations for absolute temperature), and the second by the relaxation time \(t_2\). Sometimes the terms parallel and perpendicular relaxation times are used. It can be shown that the width of the absorption band (expressed in frequencies) is equal to \((1+\gamma^2 H_1^2 t_1 t_2)^{-1/2}\cdot t_2^{-1}\),

and the total intensity is proportional to \((1+\gamma^2 H_1^2 t_1 t_2)^{-1/2}\). Thus, for small amplitudes \(H_1\) the line width is equal to \(t_2^{-1}\); for large values of \(H_1\) the line broadens and, finally, disappears.

The Bloch equations are valid for an ensemble of nuclei for which the resultant vector of angular momentum is very large in comparison with Planck’s constant. They are indispensable in determining the phase of precession, and also in considering transitions and other effects. Some results derived from these equations, however, can be obtained by means of simpler arguments based on the concepts of energy levels and effective temperature. Let us consider an example. When the energy of the alternating field, proportional to \(H_1^2\), increases so much that the number of transitions induced by the field becomes comparable in order of magnitude with the number of spontaneous transitions, then the energy levels, and consequently the spectral lines, must broaden. The initial distribution changes in such a way that the effective temperature rises, i.e., the observed excess of absorption over induced Einstein emission decreases inversely proportional to the effective temperature.

Fig. 1. Resonance line in an alternating field of small (solid curve) and large (dashed curve) amplitude \(H_1\).

The observed line width directly gives the value \(1/t_2\). It is more difficult, but also more interesting, to measure the value \(t_1\), which characterizes the energy interaction between the spins of the nuclei and their surroundings. There are two methods for determining the value \(t_1^{34,35}\). In the first method, a very strong alternating magnetic field is first applied, which considerably increases the effective temperature. Then the intensity of the alternating field is sharply reduced. With the aid of an oscillograph one observes the recovery with time of the absorption or dispersion in the line itself (or near it), which almost completely

disappeared under the action of a strong field. In this way it is possible to measure values of \(t_1\) greater than 1 sec. The second method is used

Fig. 2. Time dependence of the absorption coefficient upon a sharp decrease of the initially substantial amplitude of the alternating field \(H_1\).

Fig. 3. Increase of the absorption coefficient as a function of time after a sharp decrease of the amplitude of the alternating field \(H_1\). Substance CaF\(_2\); temperature 2.1° K.

especially for determining the ratios between \(t_1\) for a number of substances, for which \(t_2\) may be regarded as being of the same order of magnitude. With the aid of this method the dependence of the absorption

Fig. 4. Decrease of the absorption coefficient in water and in mixtures of water with glycerin (60 and 98% glycerin) as a function of the square of the amplitude \(H_1\) of the alternating field. Room temperature. Logarithmic scale for \(H_1^2\).

or of the dispersion on \(H_1^2\) is studied. The resulting decrease of the corresponding quantity is determined by the product \(\gamma^2 H_1^2 t_1 t_2\). By measuring the values of \(H_1^2\) that produce this decrease, one can calculate the ratio of the products \(t_1 t_2\). In this way it was possible to measure values of \(t_1\) down to \(10^{-5}\) sec.

In measurements of \(t_1\) for protons in a series of oils, alcohols, and glycerin–water mixtures at various temperatures, a striking correlation was found between \(t_1\) and viscosity. The relaxation time \(t_1\) is usually inversely proportional to the viscosity; since the latter is approximately proportional to the Debye dielectric relaxation time \(\tau\), the product \(t_1\tau\) is approximately constant. This correlation supports the hypothesis according to which spontaneous transitions are due to the random motion of magnetic nuclei in a liquid. Indeed, these displacements create rapid and random changes of the local magnetic field, causing acts of induced absorption and emission, which in turn lead to the restoration of the Boltzmann distribution. As the viscosity increases, the spectrum of these random changes shifts into the region of low frequencies and, consequently, \(t_1\) decreases.

The explanation given is confirmed by the fact that \(t_1\) passes through a minimum as the viscosity increases and the quantity \(\tau\) becomes, in order of magnitude, equal to \(\omega^{-1}\).

Fig. 5. Spectra of random displacements in a liquid for three different values of the viscosity, i.e., of the relaxation constant \(\tau\). As is seen from the figure, the intensity of the spectrum reaches a maximum at \(\omega \tau \sim 1\). In this case the relaxation time \(t_1\) passes through a minimum.

By order of magnitude it becomes equal to \(\omega^{-1}\). Analogous results are also obtained for some solids (paraffin, ice).

Investigations of a number of nonconducting crystals, carried out over a very wide temperature interval, showed the enormous influence of negligible amounts of paramagnetic impurities\({}^{37}\). Replacement of \(0.004\%\) of aluminum ions in alum by chromium ions already reduces the value of \(t_1\) by a factor of two. Theoretically, one would expect a very weak interaction between the nuclear spins and the thermal vibrations of the lattice of a pure crystal: in this case the value of \(t_1\) should be of the order of \(10^3\) sec at room temperature. On the other hand, it is clear that paramagnet-

… impurities lead to disordered disturbances of the local magnetic field, thereby causing a rapid redistribution of energy, analogous to what takes place in liquids. Thus, values of \(t_1\) of the order of magnitude of \(1\) sec

Fig. 6. Temperature dependence of the relaxation time \(t_1\) for the proton in quartz containing traces of \(\mathrm{Cr}^{+++}\). Curve \(I\) refers to the parent substance, containing only traces of iron. Curves \(II\), \(III\), \(IV\), and \(V\) correspond to substances in which 0.004, 0.03, 0.26, and 3.5% of aluminum ions are replaced by chromium ions.

(which is observed at room temperature) can probably be attributed to the presence of negligible paramagnetic impurities. Working with crystals of exceptional purity, Purcell\(^4\) succeeded in obtaining values of \(t_1\) reaching \(100\) sec even at room temperature: it was precisely in such crystals that it proved possible to create and maintain negative effective temperatures for a considerable interval of time.

In metals the magnitude \(t_1\) is inversely proportional to the absolute temperature\(^ {38}\), which is in agreement with the theoretical considerations of Heitler and Teller\(^ {39}\) on the interaction of conduction electrons with nuclear spins. It is clear that this interaction is decisively influenced by the value of the wave function…

valence electrons near the nucleus. This same value determines the shift of the nuclear magnetic resonance line[^32]. Korringa[^40] showed that, under certain conditions, the relation

\[ t_1(\Delta \nu)^2=\frac{h}{2\pi^2 kT}\left(\frac{e}{2mc}\right)^2 . \]

holds.

Experimental verification confirmed the validity of this relation in order of magnitude.

Values of \(\gamma\) \((\mathrm{sec}^{-1}\ \mathrm{oersted}^{-1})\) for some light nuclei

3. PARAMAGNETIC RELAXATION AND PARAMAGNETIC RESONANCE

To describe a resonant system (magnetic, electrical, or mechanical) with damping caused by random impacts or collisions, the following expression is valid:

\[ \chi=\chi_0 \frac{1+\tau^2\omega_0^2+i\tau\omega}{\tau^2\omega_0^2+(1+i\tau\omega)^2}, \tag{1} \]

where \(\chi\) may, in particular, denote the variable part of the magnetic susceptibility \(\chi\). The real and imaginary parts give, respectively, the dispersion and damping as functions of \(\omega\). In the case of weak damping, when it is sufficient to take into account only frequencies close to resonance, this formula of Van Vleck, Weisskopf[^41], and Fröhlich[^42] becomes the classical Lorentz formula

\[ \chi=\chi_0 \frac{\omega_0^2}{\omega_0^2-\omega^2+i\omega_0^2\tau\omega}. \tag{2} \]

With considerable damping, the region of dispersion and strong absorption is shifted, as is evident from formula (1), toward high frequencies; in this case the Wagner–Debye formula[^43] is obtained

\[ \chi=\chi_0 \frac{1}{1+i\tau\omega}. \tag{3} \]

The Lorentz formula would lead to the opposite shift and to an unrealistic form of the dispersion curve.

Under the combined action of the inhomogeneous electric field caused by neighboring atoms and of an external magnetic field—

Fig. 7. Dependence of the real \((\chi')\) and imaginary \((\chi'')\) components of the magnetic susceptibility \(\chi=\chi'-i\chi''\) on \(\omega\). On the left: damping curves for an oscillator due to friction (classical Lorentz formula). On the right: damping curves due to collisions (Van Vleck, Weisskopf, and Fröhlich formula). Curves are given for several values of \(\tau\omega_0\). For small values of \(\tau\omega_0\) (strong damping), the curves on the right-hand side of the figure approach the Wagner–Debye curves.

—the magnetic field, the principal energy level of the magnetic ion splits into a number of levels. The radio-frequency spectrum is caused by transitions between these levels. When the magnetic field predominates, approximately equidistant levels and frequencies are obtained

of the allowed transitions are equal (small shifts may be caused by an inhomogeneous electric field). These transitions—the so-called paramagnetic resonance—are magnetic dipole transitions and are polarized perpendicular to the applied field. One must also take into account zero-frequency transitions, which leave each level unchanged; they are polarized parallel to the field. If there is a magnetic or exchange interaction between the ions,

Fig. 8. Splitting of an energy level into six components under the action of a magnetic field and an electric field of cubic symmetry; the direction \((0,1)\) coincides with the direction of the magnetic field. Five values of the magnetic-field strength are indicated, giving rise to the appearance of a magnetic-resonance line of a definite frequency.

then the levels and lines broaden. Zero-frequency transitions are detected only in the presence also of other types of interaction of the ions with their environment. For example, interaction with thermal waves causes “spontaneous” transitions that establish or restore Boltzmann equilibrium corresponding to a constant temperature. On the one hand, this interaction causes broadening of the zero-frequency line, which leads to the susceptibility described by expression (3); on the other hand, the resonance lines undergo additional broadening described by formula (1). The quantity \(\tau\) in these formulas is completely analogous to the relaxation time \(t_1\) in the theory of nuclear magnetic resonance. We shall first consider the region of comparatively low frequencies, in which the paramagnetic relaxation band is located. Later we shall return to the question of paramagnetic resonance.

According to Casimir and Du Pré\(^3\), paramagnetic relaxation may also be considered from another point of view. To a system of magnetic moments one may assign an effective temperature \(T_{\mathrm{eff}}\)

Fig. 9. Absorption spectrum: \(A\)—the paramagnetic-relaxation band, \(B\)—the lines of paramagnetic resonance.

and a specific heat \(dU/dT_{\mathrm{eff}}\); these quantities are usually called the spin temperature and the spin specific heat. If an alternating magnetic field, parallel to the constant field \(H\), is now applied, the spin temperature will begin to oscillate with the frequency of the alternating field. The amplitude of these oscillations will depend on the spin specific heat, on the amplitude of the field, and, finally, on the frequency of the oscillations. If the oscillation frequency \(\omega\) is much greater than \(\tau^{-1}\), we are dealing with an “adiabatic susceptibility”; if, however, \(\omega\) is much smaller than \(\tau^{-1}\), then the spin temperature remains constant and the susceptibility becomes static. Calculations carried out by Casimir and Du Pré\(^3\) give, for substances obeying Curie’s law, the following relation:

\[ \chi=\frac{\chi_0 F}{1+i\tau\omega}+\chi_0(1-F), \tag{4} \]

where

\[ F=\frac{CH^2}{b+CH^2}, \tag{5} \]

Fig. 10. Dependence of the real (\(\chi'\)) and imaginary (\(\chi''\)) components of the magnetic susceptibility \(\chi=\chi'-i\chi''\) on \(\tau\omega\) according to the formula of Casimir and Du Pré.

\(C\) is the Curie constant \((C=\chi T)\), \((b+CH^2)/T^2\) is the spin specific heat. These formulas agree well with numerous experimental data on paramagnetic relaxation, obtained over a wide temperature range for fifty different paramagnetic salts\(^3\). It should, however, make two remar—

tions. In studies of absorption it is found that the imaginary component \(\chi\) slightly exceeds the value obtained from formula (4). Often this excess is proportional to \(\omega\);

Fig. 11. Anomaly of the susceptibility of chrome alum, discovered by de Vrier. The real \((\chi')\) and imaginary \((\chi'')\) components of the susceptibility in a constant field \(H=320\) oersted. Temperature \(20.4^\circ\mathrm{K}\).

in this case it may be considered responsible for the formation of satellites of the resonance lines, broadened as a consequence of magnetic and exchange interaction between the ions. Further, relaxation in the spin system for chrome alum, recently discovered by de Vrier \(^{44}\), cannot yet be explained in all details.

The application of formulas (4) and (5) to the experimental data made it possible to determine the relaxation time \(\tau\) as a function of temperature and the specific heat of the spin systems associated with the position of the energy levels and their broadening.

Fig. 12. Relaxation times for manganese ammonium alum at a constant field of 800 oersted and at a very large constant field. Vertical scale \(\rho=2\pi\tau\).

Let us now turn to the consideration of paramagnetic resonance. This phenomenon, discovered by Zavoisky \(^{19}\), manifests itself in a violation of the form of the relaxation curves observed at radio frequencies.

All subsequent work was carried out in the region of centimeter waves, in which the relative width of the bands is smaller. For the center of the band

\[ \gamma=\omega/H \]

or

\[ ge/2mc, \]

where \(e\) and \(m\) are the charge and mass of the electron, \(c\) is the speed of light, and \(g\) is the Landé factor. In substances, the moment

for which it is produced only by spins (salts of divalent manganese ions or trivalent iron, chromium, or gadolinium ions), \(g\) is approximately equal to 2. In the presence of orbital moments (copper or nickel salts) \(g\) may take different values, often

Fig. 13. Relaxation times for \(\mathrm{Cd_2(SO_4)_3\cdot 8H_2O}\) in a constant field of 1600 oersteds. The vertical scale is \(\beta = 2\pi\).

somewhat exceeding 2, as, for example, in the case of ferromagnetic resonance. In this case a considerable anisotropy of the crystal is often observed.

For experimental convenience one usually works at a constant frequency, varying the applied magnetic field. According to Van Vleck’s theory\(^{24}\), the width of the band (determined from the half-width) is approximately equal to \(1.3H_i\), where \(H_i\) is the root-mean-square value of the field due to the surrounding magnetic ions in the region where the given ion is located. This is valid only when

Fig. 14. Dependence of the absorption coefficient of \(\mathrm{MnSO_4\cdot 4H_2O}\) on the magnitude of the perpendicular field at a frequency of 9375 MHz.

neglect of other causes of broadening. Experience shows that the resonances for anhydrous salts are significantly sharper.

Fig. 15. Dependence of the absorption coefficient of CrK(SO$_4$)$_2\cdot$12H$_2$O on the magnitude of the perpendicular field.

By analogy with similar effects in nuclear magnetic resonance, this may be attributed to exchange interaction$^{24}$.

Fig. 16. Dependence of the absorption coefficient of a crystal of Fe(NH$_4$)(SO$_4$)$_2\cdot$12H$_2$O on the magnitude of the perpendicular field. Field direction (1, 0, 0). Temperature 20°K. Frequency 9200 Mc/s.

Almost all the most recent investigations have been carried out on single crystals in which $H_i$ is reduced by dilution of the magnetic ions with nonmagnetic ones$^{45,46,47}$. In this way the structure of the line produced by the inhomogeneous electric field was studied. It proved possible to compare these results with data obtained in the study of the spin specific heat$^{48,49}$. Transitions corresponding to frequencies equal to $1/2$, $1/3$, etc. of the frequency of paramagnetic resonance were also found; these transitions are forbidden by the usual selection rules and are caused by the inhomogeneous electric field$^{46,49}$. In such a diluted crystal Pake$^{47}$ discovered a hyperfine structure of the lines, which was subsequently successfully investigated by Bleaney$^{50}$.

Abragam, Pryce, and Bleaney$^{51}$ were able to explain many results obtained in the investigation of the splitting and hyperfine structure. In doing so they used a comparatively simple Hamiltonian for the electronic moment and the magnetic and quadru—

of the nuclear moments of the nucleus, located in an inhomogeneous electric field and in an applied magnetic field.

Hence a remarkable consequence is obtained: in order to explain the fine structure of the rather broad band of the copper ion spectrum,

Fig. 17. Dependence of the absorption coefficient of iron alum, diluted with aluminum \((1:80)\), on the magnitude of the perpendicular field. Field direction \((1,0,0)\). Temperature \(20^\circ\) K, frequency \(9200\) MHz.

one must assume the presence of a small admixture to the functions of the electrons of the unfilled \(3d\)-shell, due to the proper functions \(4s\).

Let us note, finally, that it is possible to construct very sensitive bridges\(^ {52}\), making it possible to study paramagnetic resonance in negligible quantities of magnetic ions (phosphorescence centers or color centers in halide compounds of alkali metals) at concentrations of the order of \(10^{-6}\).

It may be supposed that these investigations will find wide application in chemistry and technology.

Fig. 18. Dependence of the absorption coefficient of iron alum diluted with aluminum on the field magnitude. Field direction \((1,1,0)\). Temperature \(4^\circ\text{K}\), frequency \(9200\ \text{MHz}\).

Fig. 19. Dependence of the splitting of an atomic level for \(J = 1/2,\ I = 3/2\) on the magnitude of the magnetic field. Hyperfine structure of the paramagnetic resonance line.

Fig. 20. Change in the splitting shown in Fig. 19, caused by the internal electric field.

4. MAGNETIC RESONANCES IN ANTIFERROMAGNETIC CRYSTALS

For a long time it was assumed that, upon cooling, a substance that is paramagnetic under normal conditions passes below the Curie point into a ferromagnetic state. However, after the work of Woltjer[^53], J. Becquerel[^54], Shubnikov[^55] and de Haas[^56] on the cryomagnetic anomalies of anhydrous salts, and the work of Bizette, Squire, Bitter, and others[^57], carried out on oxides and sulfide compounds, it became known that, although a substance does pass through a transition temperature, below this temperature its magnetic properties very often differ greatly from the properties of a ferromagnet.

Fig. 21. Dependence of the magnetization \(M\) for \(\mathrm{CuCl_2\cdot 2H_2O}\) on the field strength \(H\) in the directions \(a(\parallel)\) and \(b(\perp)\). For a field in the direction \(b(\perp)\), the magnetization depends weakly on temperature. At the very lowest temperatures a threshold value is observed in the direction \(a(\parallel)\).

In Neél’s work[^58] a phenomenological theory of antiferromagnetism was proposed, describing the state of such substances at temperatures below the transition point, which, unlike the Curie point, may quite properly be called

temperature of Néel. According to this theory, the magnetic ions of the crystal can be divided into two systems. In addition to the applied field, the ions of each system must be acted upon by a Weiss field, proportional to the magnetization of the other system, but oppositely directed. The quantum theories of Kramers, Hulthén, and others\(^{59}\) predict a tendency of both systems toward antiparallel orientation with a negative sign of the exchange integral (or superexchange), which agrees poorly with experiment. The phenomenological theory of Néel, developed in his works, as well as in the works of Van Vleck,\(^{60}\) agrees well with the experimental data.

Fig. 22. Dependence of the magnetization for \(\mathrm{CuCl_2 \cdot 2H_2O}\) on the direction of the field in the \(ab\) plane for several values of the current in the electromagnet. A current of 21 amperes corresponds approximately to the threshold value of the field (6500 oersteds). Temperature \(2.1^\circ\mathrm{K}\).

Quite recently in Leiden, extensive experimental material\(^{61}\) was obtained for single crystals of \(\mathrm{CuCl_2 \cdot 2H_2O}\), for which the Néel temperature is \(4.3^\circ\) (above this temperature this crystal of the orthorhombic system is paramagnetic).

Fig. 23. Dependence of the frequencies \(\nu\) of the components of the proton magnetic-resonance lines for \(\mathrm{CuCl_2\cdot 2H_2O}\) on the direction of a magnetic field of strength 1700 oersteds in the \(ab\) plane. Temperature \(3.5^\circ\mathrm{K}\).

Fig. 24. Dependence of the frequencies \(\nu\) of the components of the proton magnetic-resonance lines for \(\mathrm{CuCl_2\cdot 2H_2O}\) on the direction of a magnetic field of strength 8800 oersteds in the \(ab\) plane. Temperature \(3.5^\circ\mathrm{K}\).

At liquid-helium temperatures the susceptibility of the crystal in the directions \(b\) and \(c\) is completely independent of temperature, whereas the susceptibility in the direction \(a\) decreases sharply as the temperature is lowered\(^{62}\). This is the case for weak applied fields. With an increase of the field, the magnetization in the direction \(a\) grows faster than according to a linear law.

Fig. 25. Same as in Figs. 23 and 24, for a field of the order of the threshold value (7400 oersteds).

law. The slope of the curve in the region of 6500 oersteds becomes so steep that in fields exceeding 8000 oersteds the anisotropy again turns out to be, in order of magnitude, equal to the anisotropy in the paramagnetic state. At temperatures below \(2^\circ\)K the magnetic moment in the direction \(a\) is negligibly small in weak fields; it abruptly assumes the normal value when the threshold value of the field, equal to 6500 oersteds, is reached.

When the external field is rotated in the plane \(ab\), the jump becomes less sharp, but the threshold value of the field does not change. When the field is rotated in the plane \(ac\), however, the threshold value of the field increases and the slope of the curve decreases. At an angle between the direction of the field and the direction \(a\) exceeding \(55^\circ\), the threshold disappears completely.

All these results are readily explained, at least qualitatively, within the framework of phenomenological theory. Néel predicted the existence of a threshold value of the field in the plane \(ab\), while Van

Fleck calculated the temperature dependence of the anisotropy in weak fields. Together with J. Haantjes, the author\(^{63}\) performed exact calculations for a temperature of \(0^\circ\)K for crystals of rhombic symmetry.

It should be assumed that the Weiss field has the following anisotropy: the direction \(a\) is favored in comparison with the direction \(c\). In weak fields the direction of the magnetization vectors of both systems differs little from the directions \(a\) and \(-a\), which explains the sharply expressed anisotropy. At a considerable intensity of the applied field the magnetization is approximately perpendicular to the applied field, since in this case the decrease in magnetic energy will be maximal. For an arbitrary direction of the applied field, the acute angle between the field direction and one of the magnetization directions increases up to \(90^\circ\). If, however, the applied field is oriented in the plane \(ac\), then an abrupt transition to the direction \(b(-b)\) becomes possible.

Fig. 26. Dependence of the absorption coefficient in a CuCl\(_2\cdot 2\)H\(_2\)O crystal on the magnitude of the field making an angle with the direction \(a\) of only \(1^\circ\). Temperature \(3.27^\circ\)K.

Fig. 27. Dependence of the absorption coefficient for CuCl\(_2\cdot 2\)H\(_2\)O on the magnitude of the field lying in a plane with \(ab\) and making an angle of \(10^\circ\) with the direction \(a\). The two lines in Fig. 26 are brought closer together. Frequency 9200 MHz. Temperature \(3.27^\circ\)K.

Poulis\(^{64}\) discovered the phenomenon of antiferromagnetism in crystals by noting that the splitting of the magnetic resonance lines of protons is always symmetric at the temperature of liquid helium, whereas for paramagnetic crystals such symmetry is not observed. The position of a spectral component depends on the mean local field produced by the magnetic moments of the copper ions.

near some proton. The observed symmetry of the splitting directly proves the presence of two systems of ions; to each proton on which one of the systems acts there corresponds another proton, located under the oppositely directed action of the other system. On the other hand,

Fig. 28. Dependence of the absorption coefficient for CuCl₂·2H₂O on the magnitude of the field lying in the plane \(ab\) and making an angle of 16 and 18° with the direction \(a\). Both lines have merged: the intensity falls off rapidly. Frequency 9200 Mc/s. Temperature 3.27°K.

Fig. 29. Dependence of the absorption coefficient for CuCl₂·2H₂O on the magnitude of the field lying in the plane \(ac\) and making an angle of 40° with the direction \(a\). Both lines of Fig. 26 have shifted toward higher fields. Frequency 9200 Mc/s. Temperature 2.52°K.

the small width of the components indicates an extremely slow exchange of systems between places. When the field is rotated relative to the axes of the crystal, the component of the local field in the direction of the applied field assumes its initial value after a rotation through 360°; in this case the copper moments retain their orientation in the direction \(a\) and \(-a\). If, however, during rotation the moments remain perpendicular to the direction of the increased field, then the period proves to be 180°. In the region of threshold values of the field the resonance curves turn out to be extremely complex. One can investigate the magnetic saturation of the system as a function of temperature and calculate precisely the position of the proton in the crystal.

Until very recently it had not yet been possible to detect antiferromagnetic resonance in the microwave region. Paramagnetic resonance disappears at the Néel temperature even in a single crystal of MnF₂.^65 However, in CuCl₂·2H₂O Ubbink^66 discovered a sharp and rather complex resonance at wavelengths of 3 and 7 cm.

The field strength responsible for resonance at these frequencies varies noticeably depending on the orientation of the field. It differs, however, very little from the threshold field strength. This may serve to explain why, for other substances in which the threshold field strength is very large, magnetic resonance has not been observed. The resonance lines prove to be very distinctly expressed when the field is oriented in the direction \(a\). In this case two closely spaced lines with oppositely directed dispersion are observed.

At present, ways of explaining these phenomena are being outlined.

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