Electron Paramagnetic Resonance

S. A. Al’tshuler and B. M. Kozyrev

§ 1. Introduction

1.1. Paramagnetic resonance and the history of its discovery

Paramagnetic resonance is the aggregate of phenomena associated with quantum transitions occurring between the energy levels of macroscopic systems under the action of an alternating magnetic field of resonant frequency. Usually the energy splittings lie in the radio-frequency region \((10^7—10^{11}\ \text{Hz})\) and are produced by an external static magnetic field directed perpendicular to the weak oscillating magnetic field. Observable effects occur in substances containing a sufficient number of paramagnetic particles (atoms, molecules, atomic nuclei); in general, however, these substances may also be diamagnetic.

For paramagnetic resonance, besides the interaction of the paramagnetic particles with the radiation, their interaction with one another and with the surrounding diamagnetic particles is extremely important.

Let us first suppose that we have an ensemble of isolated particles possessing spin magnetism, and, for simplicity, let us assume that their spin \(S\) is equal to \(1/2\). In a static magnetic field \(H\), the energy level of a particle will split into two sublevels separated by an interval \(g\beta H\). Here \(\beta\) is the magneton, and \(g\) is the so-called spectroscopic splitting factor, in our case equal to 2. If, perpendicular to the field \(H\), an alternating magnetic field of frequency \(\nu\) is applied that satisfies the condition

\[ h\nu = g\beta H, \tag{1} \]

then, under the action of the radio-frequency radiation, transitions from the lower sublevel to the upper one and back will be stimulated with equal probability. As a result, irrespective of the initial distribution of the particles over the sublevels, the populations of the latter will ultimately become equal, after which the net absorption or emission of energy by the substance will cease (so-called saturation will set in).

The situation will be different if the interaction between particles is taken into account. In this case we may assume that initially the substance placed in the field \(H\) is in a state of thermodynamic equilibrium. Upon application of the alternating magnetic field, owing to the greater population of the lower sublevels, stationary absorption of the energy of this field by the substance should be observed, provided that there is a sufficiently effective mechanism ensuring restoration of the equilibrium distribution through the transfer of the energy of the excited particles into heat (the so-called spin-lattice interaction).

Resonance absorption will naturally be accompanied by anomalous paramagnetic dispersion and anomalous paramagnetic rotation of the plane of polarization of radio-frequency waves. A more detailed discussion of these phenomena will be given later; for now we turn to the history of the question.

Paramagnetic resonance was discovered by E. K. Zavoisky^1 in 1944 in Kazan; his first experiments concerned resonance absorption in salts of ions of the iron group. Zavoisky’s discovery was preceded by certain theoretical assumptions about the nature of the expected effect. After the well-known experiments of Stern and Gerlach on space quantization, Einstein and Ehrenfest^2 expressed a number of considerations concerning quantum transitions between magnetic sublevels of atoms under the influence of equilibrium radiation. Relying on these considerations, Dorfman in 1923 suggested the possibility of resonance absorption of electromagnetic waves by paramagnets, calling this phenomenon the photomagnetic effect.^3

In 1932 there appeared, carried out at Pauli’s suggestion, the fundamental work of I. Waller,^4 containing the quantum theory of paramagnetic relaxation in solids. This work served as the basis for all subsequent development of the theory of dynamic phenomena in paramagnets, in particular of paramagnetic resonance.

From the mid-1930s, Gorter and his collaborators^5 began a systematic study of the absorption and dispersion of radio-frequency electromagnetic waves by paramagnets at frequencies \(10^6—3\cdot10^7\) Hz in the presence of parallel and perpendicular static fields. However, their attempts to detect paramagnetic resonance proved fruitless^6 because of the imperfection of the method and the use of insufficiently high frequencies.

Zavoisky^1 developed new, highly sensitive methods for studying paramagnetic resonance: instead of measuring the amount of heat released by the paramagnet, as Gorter had done, he began to measure the attenuation of the energy of a high-frequency electromagnetic field as a result of absorption. To obtain fully resolved lines of paramagnetic resonance absorption, he extended the range of frequencies used up to \(3\cdot10^9\) Hz. All this predetermined the success of his experiments. He succeeded not only in discovering the phenomenon of paramagnetic resonance, but also in studying a number of its regularities, and also in considerably expanding the field of investigation of paramagnetic relaxation.

The first theoretical interpretation of Zavoisky’s experiments was proposed by Ya. I. Frenkel.^7

A natural continuation of the study of paramagnetic resonance due to the magnetic moments of electrons was the discovery of an analogous effect in atomic nuclei, made by Purcell^8 and Bloch^9 with collaborators two years after the publication of Zavoisky’s work. Finally, in 1950, paramagnetic resonance caused by transitions between quadrupole energy levels of nuclei in crystals in the absence of an external magnetic field was discovered by Dehmelt and Kruger.^10

In the postwar years, in connection with the great progress of radio and microwave technology, on the one hand, and with the very valuable applications that emerged for the method of paramagnetic resonance in solving problems of solid-state physics, nuclear physics, chemistry, and, finally, technology, an enormous number of works based on this method appeared. Therefore it is hardly expedient to continue the further exposition in a historical aspect.

There are two lines of research that use the idea of magnetic resonance. One of them is connected with the study of changes occurring

with matter as a result of magnetic resonance. The beginning of these investigations was laid by Rabi^11, who, at Gorter’s suggestion, developed a method for determining the magnetic moments of atomic nuclei in molecular beams, based on changes in spin orientation under conditions of magnetic resonance. To the same line may be assigned the experiments of Alvarez and Bloch on determining the magnetic moment of the neutron^12, Deutsch’s experiments with the positron^13, and the optical effect of Kastler^14. All the experiments listed were carried out in beams of magnetic particles, between which there are no appreciable interactions.

Another line of investigation concerns macroscopic systems, in which internal interactions play an essential role. In this case it is most convenient to study the changes that occur not with the substance, but with the electromagnetic field. This direction was initiated by Zavoisky’s experiments. A major advantage of methods of this type is the possibility of obtaining a whole series of important data concerning interatomic interactions. In our definition of paramagnetic resonance, given at the beginning of the article, we wish to emphasize this aspect of the phenomenon as well.

1.2. Paramagnetic resonance as part of the theory of magnetism

For the present stage of the theory of paramagnetism, the characteristic feature is the transition from the study of the magnetic properties of matter under static conditions to phenomena observed in alternating magnetic fields. The modern theory of dynamic paramagnetism is developing along three lines:

1) adiabatic demagnetization, 2) paramagnetic relaxation, and 3) paramagnetic resonance. Between these lines there is so deep a connection that some authors^15 consider, for example, paramagnetic resonance to be part of the theory of paramagnetic relaxation, whereas others^16, conversely, regard paramagnetic relaxation as paramagnetic resonance associated with the transition to zero frequency. This close connection makes it possible to carry out a general theoretical treatment of a number of questions relating to all three lines, and to obtain mutually complementary information about various physical constants, such as the magnetic heat capacity of substances, relaxation times, etc.

If, in studying the behavior of substances in constant magnetic fields, the principal characteristic is the static susceptibility \(\chi_0\), then in passing to dynamic phenomena it is convenient to regard the susceptibility as a complex quantity \(\chi = \chi' - i\chi''\); the part of the magnetization that varies in phase with the field is determined by the dynamic susceptibility \(\chi'\), while absorption of the energy of the alternating field by a paramagnet is determined by the coefficient \(\chi''\). The task of the theory of paramagnetic absorption and dispersion is to establish the dependence of the coefficients \(\chi''\) and \(\chi'\) on the frequency of the alternating field and on the strength of the applied static field. The general relation between the coefficients \(\chi'\) and \(\chi''\) is given by the Kramers—Kronig relations^17:

\[ \left. \begin{aligned} \chi'(\nu) &= \frac{2}{\pi}\int\limits_{0}^{\infty} \frac{\nu_1 \chi''(\nu_1)}{\nu_1^2-\nu^2}\,d\nu_1 + c,\\[6pt] \chi''(\nu) &= -\frac{2}{\pi}\int\limits_{0}^{\infty} \frac{\nu \chi'(\nu_1)}{\nu_1^2-\nu^2}\,d\nu_1. \end{aligned} \right\} \tag{2} \]

Dispersion formulas in closed form have been obtained only for gases[^18]. Naturally, for condensed systems, with their very complex internal interactions, a simple solution of this problem is hardly possible. Therefore one has to use approximate formulas. It is convenient to introduce the absorption-line shape function \(g(\nu)\), satisfying the relation

\[ \int_0^\infty g(\nu)\,d\nu=1. \tag{3} \]

Comparison with relations (2) at \(\nu=0\) shows that

\[ g(\nu)=\frac{2}{\pi}\frac{\chi_0''(\nu)}{\chi_0}. \tag{4} \]

In comparison with experiment, to describe \(g(\nu)\) one usually adopts either functions of Gaussian type, for example

\[ g(\nu)=\frac{1}{\sqrt{2\pi}\sigma} \left\{ e^{-\frac{(\nu-\nu_0)^2}{2\sigma^2}}+ e^{-\frac{(\nu+\nu_0)^2}{2\sigma^2}} \right\} =g_1(\nu)+g_2(\nu), \tag{5} \]

or of Lorentzian type:

\[ g(\nu)=\frac{\Delta\nu}{2\pi} \left\{ \frac{1}{(\nu-\nu_0)^2+\frac{1}{4}\Delta\nu^2} + \frac{1}{(\nu+\nu_0)^2+\frac{1}{4}\Delta\nu^2} \right\} =g_1(\nu)+g_2(\nu). \tag{6} \]

Here \(\nu_0=g\beta H/h\), \(\sigma=\Delta\nu/2\sqrt{2\ln 2}\) is the half-width of the absorption line at high frequencies. The second terms of the right-hand sides of formulas (5) and (6) vanish for \(\nu\gg\Delta\nu\); the necessity of introducing them is determined by the evenness of the absorption effect with respect to the field \(H\)[^19].

Experiments on electron paramagnetic resonance are carried out by studying the dependence of \(\chi'\) and \(\chi''\) on the magnitude of the field \(H\) at \(\nu=\mathrm{const}\). Therefore the Kramers–Kronig relations must be modified[^19]:

\[ \left. \begin{aligned} \chi_0-\chi'(H) &=\frac{1}{\pi}\int_0^\infty \frac{F(H+H_1)-F(H-H_1)}{H_1}\,dH_1,\\[6pt] F(H) &=\frac{1}{\pi}\int_0^\infty \frac{\chi'(H+H_1)-\chi'(H-H_1)}{H}\,dH_1, \end{aligned} \right\} \tag{7} \]

where \(F(H)=\chi''(H)-\pi\chi_0 g_2(\nu)\) is denoted. Since \(g_2(\nu)\) is a monotonically decreasing function, it is immaterial what expressions are adopted for it. Paramagnetic resonance in most cases is studied by measuring \(\chi''(H)\). Paramagnetic dispersion under resonance conditions was first observed by Zavoiskii[^20] on the salt MnSO\(_4\); further measurements of \(\chi'(H)\) are described in works[^21]. Figure 1 gives typical curves of resonant paramagnetic absorption and dispersion.

Paramagnetic resonance can be detected not only by measurements of \(\chi'\) and \(\chi''\), but also by observing the rotation of the plane

polarization of microwaves in paramagnets under the influence of a static magnetic field. A number of authors have considered the theory of this effect[^22] and carried out the corresponding measurements[^23]. A typical curve \(\varphi(H)\) is shown in Fig. 2.

Between the angle of rotation of the plane of polarization \(\varphi\) and paramagnetic absorption there is a simple integral relation[^24]:

\[ \varphi=\frac{4\pi\nu\sqrt{\varepsilon}}{c}\int_{0}^{\infty} \frac{\chi''(H+H_1)-\chi''(H-H_1)}{H_1}\,dH_1 . \tag{8} \]

The integral relations (7) and (8) given above make it possible to check the correctness of the shape of the experimental curves of paramagnetic resonance. Recently, investigations have also begun of other analogues of magneto-optical phenomena in microwaves under conditions of magnetic resonance, for example the Cotton–Mouton effect[^25].

Fig. 1. Resonant paramagnetic absorption and dispersion of susceptibility in manganese sulfate at \(\lambda \simeq 16\ \mathrm{cm}\)[^20].

Fig. 2. Resonant paramagnetic rotation of the plane of polarization in \(\mathrm{MnCO}_3\) at \(\lambda \simeq 3.2\ \mathrm{cm}\)[^23].

The totality of results obtained by means of paramagnetic resonance gives characteristics of various substances that are very important for the study of magnetism. It is sufficient to mention here: the magnetic and mechanical moments of atoms, molecules, and atomic nuclei; times of paramagnetic relaxation, etc.

1.3. Paramagnetic Resonance and Spectroscopy

Paramagnetic resonance is a component part of spectroscopy, since it makes it possible to determine the positions of the energy levels of magnetic particles. It is of interest to consider the special features of paramagnetic resonance in comparison with spectroscopy in the region of optical frequencies.

1. Let us note first of all that the range of frequencies used in magnetic-resonance experiments lies between \(10^6\) and \(10^{11}\) cycles/sec. The use of these frequencies, lying beyond the infrared part of the spectrum, makes it possible to investigate with great accuracy such small splittings of energy levels as are inaccessible, or almost inaccessible, to optical methods.

2. For the radio-frequency region the probability of spontaneous transitions is very small, since it is proportional to \(\nu^3\). Therefore, in the study

in paramagnetic resonance one has to deal only with induced absorption and emission.

1. If optical spectra in the overwhelming majority of cases are due to electric dipole transitions between energy levels, then the lines of paramagnetic-resonance absorption arise exclusively from magnetic dipole transitions. Because of this, the Einstein coefficients for induced absorption and emission in the case of paramagnetic resonance will be lower by approximately 4 orders of magnitude.

2. As a consequence of what has been said, the paramagnetic-resonance effect is very subtle; the possibility of observing it, apart from the high sensitivity of radio-engineering detection methods, is connected with the enormous number of photons involved. Thus, at an incident power of 1 mW, about \(n \sim 10^{20}\) photons are produced per second.

3. From the uncertainty relations between the number of photons and the phase of an electromagnetic wave it follows that, in our case, owing to the enormous magnitude of \(n\), the phase will be determined with very great accuracy. A consequence of this is the possibility of treating the electromagnetic field in radiospectroscopy as a classical quantity.

4. In the region of optical frequencies the line width is always very small in comparison with the fundamental frequency. In studies of paramagnetic resonance the relation between these quantities becomes quite different, since the interactions causing line broadening may be of the same order as the energy splittings that determine the resonance frequencies. Therefore, in paramagnetic-resonance lines the width is often comparable with the fundamental frequency and can be measured with great accuracy. This opens broad possibilities for investigating various types of interactions in paramagnets by analyzing the shape and width of paramagnetic-resonance lines and the nature of their dependence on various factors.

The most important factors determining the line width are magnetic dipole interactions, exchange forces, the influence of the local electric field created by the environment of the magnetic particles, and the action of thermal motion. It is obvious that the natural line width of radio-frequency spectra is utterly negligible.

1. In contrast to the conditions of optical experiments, in radiospectroscopy one usually uses radiation so monochromatic that the generated frequency band proves to be incomparably narrower than the width of the absorption line.

2. Paramagnetic-resonance spectra are studied not by changing the frequency of the incident radiation, but by changing the intrinsic frequencies of the absorbing systems. This change is produced by varying the static magnetic field.

Paramagnetic resonance is studied in a variety of substances, in which the carriers of magnetism may be ions in crystals, liquids, and gases; conduction electrons in metals and semiconductors; and, finally, molecules of free radicals. We shall return later to a detailed consideration of the spectra of separate classes of paramagnets.

1.4. Experimental Methods

The method currently used for investigating paramagnetic resonance is based on determining changes in some parameter of a radio-engineering circuit associated with changes in the high-

frequency field, occurring as a result of paramagnetic absorption in the substance under study or as a result of dispersion of the susceptibility.

For lack of space we cannot dwell here in detail on the measurement technique and the apparatus used; therefore we shall have to confine ourselves to only a few remarks, referring those interested to Gorter's book^26 and to the numerous original papers describing modern installations, for example^27.

The measurement technique is different for the frequency ranges \(10^6\)—\(10^9\) cps, on the one hand, and \(10^{10}\)—\(10^{11}\) cps, on the other; this gives some authors^26 grounds for distinguishing between radio-frequency and microwave magnetic spectroscopy. This distinction, however, seems to be poorly justified, since the nature of the phenomena studied is the same in both cases.

In investigations in the microwave range (\(10^{10}\)—\(10^{11}\) cps) the substance is placed in a resonant cavity located between the poles of an electromagnet in such a way that the static and alternating magnetic fields acting on the substance are mutually perpendicular. A position is chosen for the specimen where the alternating electric field is minimal and the magnetic field maximal. During the measurements the frequency of the microwave generator exciting electromagnetic oscillations in the cavity is kept constant, while the strength of the static field is varied. The choice of this form of experiment is due to the fact that studying the dependence \(\chi''(\nu)\) at \(H=\mathrm{const}\) would introduce additional experimental difficulties associated with changing the power of the generator when changing the frequency of the radiation it produces.

Two main methods are used for measuring paramagnetic resonance at microwaves. In the first, the paramagnetic resonance absorption is detected from the change in the power passing through the resonant cavity (the transmitted-wave method), and in the second from the change in the power reflected from the cavity (the reflected-wave method). For very broad absorption lines the measurements are made “point by point”: at a given strength of the static field \(H\) the generator is tuned into resonance with the cavity, and the power emerging from the cavity is fed to a crystal detector operating in the square-law regime. The rectified signal produces galvanometer deflections proportional to the power incident on the detector.

For studying not too broad lines of paramagnetic absorption, it is very convenient to use modulation of the static magnetic field by a field of audio frequency, first applied by E. K. Zavoisky^1. If the modulation amplitude, at a given strength of the static field, covers the region of paramagnetic resonance, then there will be a corresponding modulation of the power passing through the cavity (or reflected from it). This modulation can be amplified and fed to an oscilloscope, whose horizontal sweep is synchronized with the modulation field. The possibilities for amplifying the signal are limited chiefly by the noise of the crystal detector at audio frequencies; to eliminate this, in some installations the crystal detector is replaced by a bolometer. An example of a block diagram of a microwave spectroscope for observing paramagnetic resonance is given in Fig. 3.

At low frequencies (\(10^6\)—\(10^9\) cps) the substance under study is usually placed not in a resonant cavity, but in a self-induction coil, comprising

...forming part of the circuit of an electronic self-oscillator, or inductively coupled to it. Measurements in this frequency region are most often made by means of the reaction method on the generator, proposed by Zavoisky1. Provided certain known conditions are observed, a change in the watt load on the generator produces a proportional change in the grid or anode current; by studying the dependence of one of these parameters on the strength of the static magnetic field, one can obtain the paramagnetic-resonance absorption curve \(\chi''(H)\); since in the microwave range modulation of the magnetic field is widely used in this frequency region as well. An example of a block diagram for studying paramagnetic resonance at low frequencies is shown in Fig. 4.

Fig. 3. Block diagram of a microwave spectroscope for observing paramagnetic resonance[^37]:
1 — stabilized-voltage source; 2 — klystron generator; 3 — wavemeter; 4 — attenuator; 5 — phase shifter; 6 — hybrid ring; 7 — piston; 8 — crystal detector; 9 — loop; 10 — resonator; 11 — specimen; 12 — low-frequency amplifier; 13 — sweep; 14 — electromagnet; 15 — oscilloscope.

The sensitivity of modern setups is very high. Detection of resonance in samples containing \(10^{13}\) magnetic particles with an absorption-line width of about 1 gauss is already no longer a record. For substances giving broader lines, the signal intensity \(\chi''_{\max}\), naturally, decreases, and the limiting detectable number of paramagnetic centers in the sample becomes larger.

The accuracy of determining the position of the resonance depends primarily on the accuracy with which the strength of the static magnetic field at resonance is measured. These measurements are made with fluxmeters constructed on the principle of proton paramagnetic resonance. Along with this, standard substances giving narrow absorption lines with a well-defined \(g\)-factor are used for field calibration. Most often, the free radical diphenylpicrylhydrazyl is used for this purpose.

Fig. 4. Block diagram of a radiospectroscope for observing paramagnetic resonance at frequencies \(10^7—10^8\) Hz1:
1 — modulation winding of the electromagnet; 2 — to the low-frequency amplifier.

The source of the static magnetic field is, for the most part, electromagnets. Since in most cases the electron paramagnetic absorption lines are rather broad, especially stringent requirements are not imposed on the homogeneity and stability of the field, except in work with free radicals.

§ 2. SPECTRA OF IONIC CRYSTALS

Among the various classes of paramagnets, the most thoroughly studied are ionic crystals. Paramagnetic properties are characteristic of ionic crystals containing elements of the transition groups, for only the atoms of these elements retain incomplete electron shells in the process of crystal formation.

In constructing a theory of the energy spectra of ionic paramagnetic crystals, one must first of all take into account the interaction of the electrons with one another and with the nucleus within each ion; then the electrostatic, magnetic, and exchange interactions between different ions; and, finally, the action of the external magnetic field. The magnetic and exchange forces create narrow quasi-continuous energy bands, since these forces are small in not too magnetically concentrated substances, while the number of possible orientations of the moments of the magnetic particles of the crystal relative to one another is enormous. As a result, magnetic and exchange interactions, as a rule, do not affect the form of the paramagnetic-resonance spectrum*), but merely cause a broadening of individual lines. We shall therefore consider these interactions in the following section, devoted to the shape of the absorption lines.

The starting point of our further theoretical discussion will be free ions; the electrostatic interaction between them will be taken into account approximately, by assuming that each ion is in a certain mean electric field created by all the surrounding particles. This field we shall briefly call the crystalline field.

The action of the crystalline field is always weaker than the Coulomb interaction between electrons within the atom. Therefore we may use the self-consistent-field method and speak of the configuration of the electrons forming the incomplete shell of the paramagnetic ion. The different transition groups correspond to the following electronic configurations of the ions: the iron group (from Ti to Cu) — \(3d^n\), the palladium group (from Zr to Ag) — \(4d^n\), the rare earths (from Ce to Yb) — \(4f^n\), the platinum group (from Hf to Au) — \(5d^n\), the actinides (from U and onward) — \(6d5f^n\). The self-consistent-field method does not fully take into account the electrostatic interaction between electrons. Therefore, for calculations, usually carried out by the perturbation method, it is necessary to know in what ratio to one another stand the unaccounted part of the electrostatic repulsion between electrons, the magnetic couplings between their spin and orbital moments, and the forces of the crystalline field. Three cases are distinguished. The crystalline field is called weak if it is not capable of disrupting the coupling between the orbital and spin moments of the entire incomplete electron shell. The field is considered intermediate if its action is stronger than the spin-orbit coupling of the electrons, but much weaker than the interaction between individual electrons. Finally, the field is called strong if its action is more significant than the coupling between the electrons of the incomplete shell that leads to the formation of terms. The first two cases are realized in hydrated salts of the rare-earth elements and of the iron group, respectively. The case of a strong field does not occur in pure form, for if the crystalline field becomes more significant than the interaction between electrons, then the covalent bond of the paramagnetic atom with its nearest environment always begins to play a noticeable role.

*) We shall discuss some exceptions below (see 2.1.).

For the entire theory of paramagnetic-resonance spectra, Kramers’ theorem \(^{28}\) is of fundamental importance; according to it, electric forces are not able completely to remove the degeneracy of the energy level of a system containing an odd number of electrons. It follows from this that the energy levels of ions containing an even number of electrons, in crystals with an internal electric field of low symmetry, will be simple, and therefore observation of paramagnetic resonance (at radio frequencies) may prove impossible. Conversely, for ions with an odd number of electrons, under suitable temperature conditions paramagnetic resonance is always accessible to observation.

The energy of an ion in the electric field of a crystal may be represented in the form

\[ \mathcal{H}_{c}=\sum_i -eV(x_i,y_i,z_i), \tag{9} \]

where \(V\) is the potential of the crystal field, and \(x_i, y_i, z_i\) are the coordinates of the \(i\)-th electron of the unfilled shell. It is convenient to expand the potential \(V\) in a series in spherical functions:

\[ V=\sum_{n,m} A_n^m r^n Y_n^m(\vartheta,\varphi). \tag{10} \]

This expression can be greatly simplified by retaining in it only a few terms of the series. In calculating the perturbation matrix \(H_c\) with the aid of the wave functions of \(d\)-electrons, those spherical functions for which \(n>4\) will give matrix elements equal to zero. In exactly the same way, in the case of \(f\)-electrons, the terms of series (10) with \(n>6\) may be omitted. Owing to the invariance of the electronic wave functions with respect to inversion transformation, one must also discard the terms of the series with odd \(n\). The term with \(n=0\) gives an inessential additive constant. Finally, from the reality of \(V\) it follows that \(A_n^m=\overline{A_n^{-m}}\). Further simplifications can be achieved if the symmetry of the crystalline field is taken into account. It is easy to verify that, to characterize a field of cubic symmetry, only one constant is required if we are dealing with \(d\)-electrons, and two constants are sufficient in the case of \(f\)-electrons. To describe a field of the lowest symmetry, 8 and 15 constants, respectively, are required.

The energy splittings under the influence of the crystalline field are calculated by the perturbation method; the necessary calculations of the matrix elements of the crystal-field potential (10) are usually carried out with the aid of equivalent operators \(^{29,30}\).

After all these preliminary remarks, we may proceed to a consecutive consideration of the various types of paramagnetic ionic crystals.

2.1. Hydrated salts of elements of the iron group

The Hamiltonian for a paramagnetic ion of the iron group, situated in crystalline and external magnetic fields, has the form:

\[ \mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{c}+\mathcal{H}_{LS}+\mathcal{H}_{SS}+\mathcal{H}_{z}. \tag{11} \]

Here \(\mathcal{H}_{0}\) includes all interactions in the free atom that do not depend

from the spin variables; \(\mathscr{H}_{LS}=\lambda \hat{\mathbf{L}}\hat{\mathbf{S}}\) and

\[ \mathscr{H}_{SS}=-\rho\left[(\hat{\mathbf{L}}\hat{\mathbf{S}})^2+\frac{1}{2}(\hat{\mathbf{L}}\hat{\mathbf{S}})-\frac{1}{3}L(L+1)S(S+1)\right] \]

are the spin-orbit and spin-spin interactions, respectively; \({}^{31}\)

\(\mathscr{H}_{z}=\beta(\hat{\mathbf{L}}+2\hat{\mathbf{S}})\mathbf{H}\) is the energy of the electrons in an external magnetic field (Zeeman energy). The problem of the energy spectrum of a paramagnetic ion is solved by successive application of perturbation theory, which is possible if one takes into account the order of the splittings caused by the various terms: \(\mathscr{H}_{c}\sim 10^{4}\ \mathrm{cm}^{-1}\), \(\mathscr{H}_{LS}\sim 10^{2}\ \mathrm{cm}^{-1}\), \(\mathscr{H}_{SS}\sim 1\ \mathrm{cm}^{-1}\), \(\mathscr{H}_{z}\sim 1\ \mathrm{cm}^{-1}\). In most of the salts studied, the crystalline field can be decomposed into two components: a strong field of cubic symmetry and a weaker field of lower symmetry, for example trigonal or tetragonal. Thus, the energy is expressed as the sum \(\mathscr{H}_{c}=K+T\). The cubic field \(K\) is usually produced by six water molecules situated at the vertices of an octahedron, whose center is occupied by the paramagnetic ion. The field \(T\) has a dual nature: first, it is produced by all the ions of the crystal and has the symmetry of the latter; second, it arises from deformation of the octahedron occurring as a result of the Jahn–Teller effect. \({}^{32}\)

For free ions of the elements of the iron group, only \(S\)-, \(D\)-, and \(F\)-terms occur. Ions in \(S\)-states will be considered separately. The pattern of splitting of the orbital energy levels of ions in \(D\)- and \(F\)-states under the influence of a cubic field is shown in Fig. 5. In this connection it must be borne in mind that, depending on the sign

Fig. 5. Scheme of the splitting of \(D\)- and \(F\)-terms in an electric field of cubic symmetry.

of the crystalline-field constant, the order of the levels may also be reversed. Usually experiments are carried out at such temperatures that the populated energy levels may be taken to be those lying no more than several hundred inverse centimeters from the ground level. Therefore, only the lowest orbital level arising in the cubic field is of interest. It is essential whether this level is singlet or degenerate.

Let us first suppose that the lower orbital level is singlet. If the electron spin is taken into account, a \(2S+1\)-fold degeneracy appears. The field \(T\), not acting on the electron spin, can produce only an insignificant shift of the level. In a singlet orbital state the mean moment \(L\) is equal to zero, and therefore in the first approximation the spin-orbit interaction is also equal to zero. Consequently, it is necessary to take into account the second approximation, which gives a splitting.

of the orbital level, equal to approximately \(\lambda^2/\Delta \sim 1\ \mathrm{cm}^{-1}\), i.e., of the same order of magnitude as the Zeeman energy and the spin–spin interaction.

In works \(^{33}\) and \(^{34}\) a method was developed for calculating the splittings of the ground energy level of a magnetic ion, a method that has found wide application in experimental studies of paramagnetic resonance and has come to be called the spin-Hamiltonian method. The method consists in the following. The usual perturbation-theory procedure is carried out in two stages. The matrix elements of the perturbation are first calculated with the aid of coordinate wave functions, which is possible because the unperturbed Hamiltonian does not depend on spin variables. As a result, the perturbation energy turns out to be a function of the spin operator \(S\); this function is called the spin Hamiltonian.

It is not difficult to show that the spin Hamiltonian has the form:

\[ \hat{\mathcal{H}} = D_{ij}\hat{S}_i\hat{S}_j + \beta g_{ij}H_i\hat{S}_i, \tag{12} \]

where \(i, j = x, y, z\), and the tensors \(D_{ij}\) and \(g_{ij}\) are certain functions of \(A_m^n\), \(\bar r^n\), \(\lambda\), and \(\rho\). If \(\mathcal{H}_c\) possesses tetragonal or trigonal symmetry, then the tensors \(D_{ij}\) and \(g_{ij}\) are characterized by two principal values, corresponding to two directions—parallel and perpendicular to the axis of symmetry. Taking the symmetry axis as the \(z\)-axis, we have

\[ \hat{\mathcal{H}} = D\left[\hat{S}_z^2 - \frac{1}{3}S(S+1)\right] + \beta g_{\parallel}H_z\hat{S}_z + \beta g_{\perp}(H_x\hat{S}_x + H_y\hat{S}_y). \tag{13} \]

Deviations from \(T\)-symmetry can be taken into account by adding the term \(E(\hat{S}_x^2-\hat{S}_y^2)\) and replacing \(g_{\perp}\) by the coefficients \(g_x\) and \(g_y\). The part of the spin Hamiltonian proportional to \(D\) and \(E\) determines the splitting of the orbital level in the absence of an external magnetic field. The terms proportional to \(H_x\), \(H_y\), \(H_z\) indicate the anisotropy of the magnetic moment of the atom in the crystal; the deviations of the \(g\)-factor from the value \(g = 2.0023\) mean that a small part of the moment associated with orbital motion is added to the spin moment.

The spin-Hamiltonian method makes it possible to characterize the paramagnetic-resonance spectrum by a small number of constants: \(D\), \(E\), \(g_{\parallel}\), \(g_{\perp}\). Determining these constants from the form of the spectrum constitutes the main task of experiments in the field of paramagnetic resonance. The task of theory is to obtain these constants on the basis of a definite model of the crystal.

The theory set forth here is primarily applicable to ions whose lowest orbital level in a cubic field is a singlet. These include the ions \(\mathrm{Cr}^{3+}\), \(\mathrm{V}^{2+}\), \(\mathrm{Ni}^{2+}\). The same group of ions should also include \(\mathrm{Cr}^{2+}\), \(\mathrm{Mn}^{3+}\), and \(\mathrm{Cu}^{2+}\), if the field \(T\) is tetragonal. In the last three ions \(L=2\), and the lower level after the action of the cubic field becomes an orbital doublet (Fig. 5), on which the spin–orbit coupling \(\mathcal{H}_{LS}\), splitting the field \(T\), has no effect. Thus, as before, under the action of the crystal field a simple lower orbital level arises.

If the lower orbital level is degenerate and, moreover, the spin–orbit coupling is nonzero in first approximation, then the perturbations \(\mathcal{H}_{LS}\) and \(T\) must be taken into account simultaneously. Owing to the Jahn–Teller effect, the paramagnetic ion after the action of these forces must possess a minimal degeneracy. If, in addition, Kramers’ theorem is taken into account,

then the following conclusion may be drawn: under the influence of \(\mathscr H_{LS}\) and \(T\), the splitting of the energy levels of paramagnetic ions possessing an even number of electrons will be complete. Now not only the orbital, but also the spin levels will be simple, separated, as a rule, by intervals of \(\sim 100\ \text{cm}^{-1}\), and, consequently, observation of paramagnetic resonance is impossible.

If a paramagnetic ion has an odd number of electrons, then the twofold Kramers degeneracy is preserved. In this case the splitting of the energy level in an external magnetic field can be calculated by introducing an effective spin equal to \(1/2\). The spin Hamiltonian will have the following simple form:

\[ \hat{\mathscr H}=\beta\{g_xH_x\hat s'_x+g_yH_y\hat s'_y+g_zH_z\hat s'_z\}, \tag{14} \]

where \(\hat s'_i\) are the Pauli matrices. We shall not dwell on the connection of the coefficients \(g_x\), \(g_y\), \(g_z\) with \(\lambda\) and the constants of the crystal field.

A special place is occupied by ions which have electronic configurations \(3d^5\) and are in the \({}^6S\) state \((\mathrm{Mn}^{2+}, \mathrm{Fe}^{3+})\). The resultant orbital angular momentum of the electrons is equal to zero, and therefore the electric field of the crystal should not split the ground levels of these ions. In reality, small splittings have been established both from experiments on adiabatic demagnetization and from observations of paramagnetic resonance. This fact may be explained by various causes. According to Van Vleck and Penney\(^{35}\), simultaneous allowance for the spin–orbit interaction and the action of the electric field of cubic symmetry leads, in the fifth-order approximation, to a splitting of the ground term of the \(d^5\) configuration. Abragam and Pryce\(^{34}\) showed that if the field has tetragonal symmetry, then another mechanism will be more important. The magnetic dipole interaction of the electron spins inside the paramagnetic atom depends not only on their relative orientation, but also on the electronic coordinates. If the electron cloud has spherical or cubic symmetry, then, upon averaging, the energy of the spin–spin interaction proves to be independent of the orientation of the spins with respect to one another; as a result the ground state of the paramagnetic ion will be completely spin-degenerate. Under the action of a field of trigonal or tetragonal symmetry the electron cloud is slightly deformed, acquiring an ellipsoidal form. In this case the energy of the spin–spin interaction averaged over the electron cloud will depend on the mutual orientation of the spins. Splittings of the ground level of the paramagnetic ion then arise already in the second approximation and will be proportional to the quantity \((\hat S_z^2-35/12)\).

The complexity of the processes leading to the splitting of the energy levels of ions in the \(S\)-state makes attempts at direct calculations hopeless. All calculations are therefore carried out with the aid of a spin Hamiltonian, to which the following form may be assigned:

\[ \hat{\mathscr H} = D\left(\hat S_z^2-\frac{35}{12}\right) + E\left(\hat S_x^2-\hat S_y^2\right) + \frac{a}{6}\left(\hat S_1^4+\hat S_2^4+\hat S_3^4\right) + g\beta HS. \tag{15} \]

Here the first term takes into account the action of a tetragonal or trigonal field with an axis of symmetry directed along \(z\); the second term is connected with small deviations toward lower symmetry; the third term specifies the action of a field of cubic symmetry; \(S_1\), \(S_2\), \(S_3\) are the components of spin,

referred to the cubic axes. For ions in an \(S\)-state the \(g\)-factor is isotropic.

In some copper salts interesting exchange effects have been observed. The sharp anomalies in the magnetic behavior of copper acetate, established both by static measurements\(^{36}\) and by the method of paramagnetic resonance\(^{37}\), found the following explanation. In the crystal cell of copper acetate there are two ions close to one another, the exchange interactions between which are stronger than the spin-orbit coupling. Therefore the two copper atoms behave like a single “molecule,” possessing spin \(1/2 + 1/2\). The exchange interactions split the lower level of the “molecule” into a spin triplet and a singlet separated from one another by approximately \(300\ \text{cm}^{-1}\). In this way it is possible to explain all the known facts concerning the static magnetic properties of copper acetate and its paramagnetic-resonance spectrum.

2.2. Hyperfine structure of paramagnetic-resonance spectra

The theory of the hyperfine structure of atomic spectra has long been developed. What is new in its application to the paramagnetic-resonance spectra of crystals is the need to take into account the influence of the crystalline field and of certain other effects that are not essential for optical investigations. The first calculations of the hyperfine structure of paramagnetic-resonance spectra\(^{38,39}\), which concerned copper salts, revealed disagreement with the experimental data. The contradictions were removed with the aid of the hypothesis of “\(s\)-configuration interaction.”\(^{40}\) It is well known\(^{41}\) that the magnetic interaction of \(s\)-electrons with the nucleus is much stronger than that of electrons with \(l>0\). It was assumed that in the ground state of the paramagnetic ion, in addition to the usually adopted configuration \(3d^n\), there is a small admixture of a configuration containing an unpaired \(s\)-electron, namely \(3sp^6d^n4s\). This makes it possible to bring the theoretically expected hyperfine splittings up to the observed values, which is especially important for salts of elements of the iron group, since in these crystals the reduction of the electron-nuclear interaction is further promoted by the “quenching” of the electronic orbits.

The general theory of hyperfine structure in paramagnetic crystals\(^{34}\) leads to the following spin Hamiltonian, which takes into account the interaction of the nuclear moments with the electron shell and with the external magnetic field:

\[ \mathscr{H}_N = A_{ij}\hat S_i \hat I_j + P_{ij}\hat I_i \hat I_j - g_N\beta_N \mathbf{H}\hat{\mathbf I}. \tag{16} \]

Here \(\hat{\mathbf I}\) is the nuclear spin, and \(g_N\beta_N\) is its magnetic moment. If the resulting crystalline field has trigonal or tetragonal symmetry, then

\[ \mathscr{H}_N = A\hat S_z\hat I_z + B\left(\hat S_x\hat I_x+\hat S_y\hat I_y\right) + P\left(\hat I_z^2-\frac{1}{3}I(I+1)\right) - g_N\beta_N\mathbf{H}\hat{\mathbf I}. \tag{17} \]

We shall not dwell on the relation of the coefficients \(A\), \(B\), \(P\) to the constants of the crystalline field and to the nuclear moments, but it is evident that the constants of the magnetic hyperfine structure \(A\) and \(B\) are proportional to \(g_N r^{-3}\), while the constant of the quadrupole interaction \(P\) is proportional to \(q r^{-3}\).

2.3. Salts of Rare-Earth Elements

The spin–orbit coupling in ions of rare-earth elements is much stronger than the influence of the crystalline field. As a result, in calculations of the energy spectra of paramagnetic rare-earth ions by the perturbation method one starts from the ground state of the free ion, in which the conserved quantities may be taken to be the total angular momentum \(J\), the orbital and spin angular momenta \(L\) and \(S\). Sometimes higher approximations, taking into account the influence of excited multiplet levels, are important.

In contrast to hydrated salts of the elements of the iron group, in which the paramagnetic ion is usually surrounded by an octahedron of water molecules producing a strong electric field of cubic symmetry, in the rare-earth salts that have been studied the environment of the paramagnetic ion creates a field of trigonal symmetry\({}^{42}\). A group-theoretical treatment\({}^{43}\) shows that in such a field a \(2J+1\)-fold energy level of a free ion with an odd number of electrons is split into \(J+\tfrac12\) doublets; if the number of electrons is even, singlets and doublets arise. Since the intervals between energy levels in the crystalline field are somewhat larger than the Zeeman splittings in practically used magnetic fields, the action of the magnetic field on each level is considered separately. Since the spin–lattice interaction in rare-earth salts is very strong at room temperature (see § 3), experiments have to be carried out at temperatures so low that, for the most part, only the lowest level is appreciably populated. It is clear that observation of paramagnetic resonance is possible if this level is a doublet. With the aid of the wave functions of such a doublet, the matrix elements of the perturbation

\[ \mathscr{H}_z=\beta H\left(\hat{L}+2\hat{S}\right). \]

are calculated. The matrix of the perturbation of rank 2 has a trace equal to zero, and may be represented in the form \(\beta H\cdot g\cdot \hat{S}'\), where \(\hat{S}'\) is a Pauli matrix vector, and \(g\) is a certain tensor with principal values \(g_{\parallel}\), \(g_{\perp}\), and \(g_{\perp}\). If the wave functions of our doublet, which we shall symbolically denote by \(|+\rangle\) and \(|-\rangle\), are chosen so that the matrix \(L_z+2S_z\) is diagonal, then

\[ g_{\parallel}=2\left|\langle +|L_z+2S_z|+\rangle\right|,\qquad g_{\perp}=2\left|\langle +|L_x+2S_x|-\rangle\right|. \tag{18} \]

Thus the paramagnetic-resonance spectrum can be interpreted with the aid of a spin Hamiltonian with effective spin \(s'=\tfrac12\):

\[ \mathscr{H}=g_{\parallel}H_zs'_z+g_{\perp}(H_xs'_x+H_ys'_y). \tag{19} \]

Ions with an even number of electrons require special consideration. In the present case it is easy to show that the nondiagonal matrix element \(\langle +|L_x+2S_x|-\rangle=0\). It follows that paramagnetic resonance should be absent. Indeed, if the field \(\mathbf{H}\) is parallel to the trigonal axis of the crystal, then magnetic dipole transitions between magnetic sublevels will be forbidden; if, however, \(\mathbf{H}\) is perpendicular to the trigonal axis, then \(g_{\perp}=0\). In experiment, however, in praseodymium salts, for example, paramagnetic resonance absorption is observed. This is explained by the Jahn–Teller effect\({}^{32}\), owing to which, in crystals containing paramagnetic ions with an even number of electrons, the symmetry of the electric field is lowered so much that the degeneracy is completely—

ness, and the doublets turn out to be split. These splittings in rare-earth ions are very small^44 and do not hinder the observation of paramagnetic resonance at ordinary magnetic-field strengths. The paramagnetic-resonance spectrum of ions with an even number of electrons can be calculated with the aid of the spin Hamiltonian:

\[ \mathcal{H}=g_{\parallel}\beta H_z S'_z+\Delta_x S'_x+\Delta_y S'_y, \tag{20} \]

where \(\Delta=\sqrt{\Delta_x^2+\Delta_y^2}\) is the splitting of the doublet in the absence of the magnetic field \(H\), due to the Jahn—Teller effect.

The crystal field in salts of rare-earth elements is characterized by a large number of constants \(A_n^m\), which makes an unambiguous interpretation of the observed paramagnetic-resonance spectra difficult. Therefore one usually also makes use of optical data, the results of investigations of the dependence of the static magnetic susceptibility on temperature, and information on the Faraday effect. A numerical determination of the coefficients \(A_n^m r^n\) led to an unexpected result: the quantities \(A_6^m r^6\) proved to be 1–2 orders of magnitude larger than \(A_2^0 r^2\), which contradicts the natural assumption of proportionality \(A_n^m r^n \sim \frac{1}{d^n}\) (\(d\) is the lattice constant). Apparently, no quantitatively reliable results can be obtained from simple model representations.

The ions \(\mathrm{Gd}^{3+}\) and \(\mathrm{Eu}^{2+}\) are in the \({}^{7}S\)-state, and therefore their spectra can be explained by means of a spin Hamiltonian similar to (15), but also containing the sixth powers of \(S_x\), \(S_y\), \(S_z\).

2.4. Covalent bond; \(3d\)-, \(4d\)-, \(5d\)-transition groups

Ionic crystals whose paramagnetism is due to elements of the \(d\)-transition groups often contain octahedral complexes \(\mathrm{MX}_6\); at the center of such a complex is an atom \(M\) with an unfilled \(d\)-shell, and at the vertices of the octahedron are water molecules, CN radicals, or atoms of chlorine, fluorine, etc. The bond within the \(\mathrm{MX}_6\) complex often has a covalent character, as was first pointed out by Pauling^45, who attempted to explain the peculiarities of the magnetic properties of potassium ferrocyanide by means of the theory of localized pairs. Van Vleck^46 showed that the properties of an octahedral complex can best be studied by the method of molecular orbitals. Later, detailed calculations were undertaken for elements with electronic configurations from \(d^1\) to \(d^4\). A further impetus to the development of the theory of the covalent bond within the \(\mathrm{MX}_6\) complex was given by the discovery of an unusual hyperfine structure in the paramagnetic-resonance spectrum of iridium^48. It turned out that the absorption line of iridium, which is part of \([\mathrm{IrCl}_6]^{--}\), has a structure caused by the magnetic moment of the chlorine nucleus; this fact clearly indicates the covalent character of the bond within the complex. A detailed examination shows that the hyperfine structure of the Ir spectrum cannot be explained by the \(\sigma\)-bond, already studied by Van Vleck, and therefore one must assume that a noticeable role is played by the \(\pi\)-bond between iridium and chlorine^49.

Soon, data on the absorption of light by hydrated salts of the elements of the iron group were compared with experimental results on paramagnetic resonance in these substances^50. Contradictions were found, which could be eliminated by assuming that in this case as well the bond in the octahedral complex is partly covalent in character.

For the construction of the theory, the relative magnitude of the interaction energy of the electrons of a free atom and of the interval \(\Delta\) between the energy levels arising in a crystal field of cubic symmetry is important. In hydrated salts of elements of the iron group the magnitude of \(\Delta\) is much smaller than the interval between the various terms of the free paramagnetic ion; in cyanides and some other salts of elements of this same group, and in compounds of elements of the \(4d\)- and \(5d\)-transition groups, the reverse relation holds.

For hydrated salts of elements of the iron group, the scheme for calculating the paramagnetic-resonance spectrum is the same as in § 21, but the matrix elements of the perturbation must be calculated taking account of the presence of covalent bonds. For this purpose the wave function of the entire unfilled electron shell must be expanded in terms of the \(d\)-functions of the individual electrons, and then, in place of the latter, one must take molecular orbitals, which are combinations of these \(d\)-functions with the \(p\)-functions of the atoms \(X_6\). Owen\({}^{50}\), taking into account only \(\sigma\)-bonds, showed that the systematic discrepancies between optical and magnetic data on the intervals \(\Delta\) can be eliminated if appropriate values of the coefficient \(\alpha\) are chosen. This coefficient shows how strongly the \(\psi\)-functions of the central atom and of the surroundings are mixed; if \(\alpha=1\), the bond is purely ionic, while if \(\alpha^2=0.5\), the electrons are distributed with equal probability between \(M\) and \(X_6\). For \(\mathrm{Ni}^{2+}\), for example, according to the purely ionic theory set forth by us earlier, \(g=2.0023-8\lambda/\Delta\), whereas if the covalent \(\sigma\)-bonds are taken into account, one obtains

\[ g=2.0023-\alpha^2\cdot 8\lambda/\Delta . \tag{21} \]

This result may be interpreted as follows. Each of the two unpaired electrons is in the nickel atom with probability \(\alpha^2\) and with probability \(1/6(1-\alpha^2)\) in each water molecule. As a result, the spin-orbit coupling will decrease, and instead of \(\lambda\) we shall have \(\lambda'=\alpha^2\lambda\). For the complex \([\mathrm{Ni}(\mathrm{H}_2\mathrm{O}_6)]^{2+}\) the experimental values of \(\Delta\) and \(\lambda\), taken from optical observations, and the values obtained by paramagnetic-resonance measurements, lead, according to \({}^{21}\), to \(\alpha=0.83\). Covalent bonds must also reduce the hyperfine splitting, which has indeed been established for copper salts\({}^{51}\).

In the case of a strong crystal field it is necessary first of all to consider its action on each electron, and only then to take into account the interactions between them. In a field of cubic symmetry, as we know (Fig. 5), a triplet \(d\varepsilon\) and a higher-lying doublet \(d\gamma\) arise\({}^{43}\). The electrons usually fill only the lower triplet level; the interaction between them leads to the appearance of terms, for the lowest of which Hund’s rule can be proved to hold. This fundamental term may be characterized by the values of the total electron spin \(S\) and of the effective orbital angular momentum \(L'\), namely: \((d\varepsilon)^1\) and \((d\varepsilon)^5\), \(L'=1,\ S=\frac{1}{2}\);

\[ (d\varepsilon)^2\ \text{and}\ (d\varepsilon)^4,\quad L'=1,\ S=1;\qquad (d\varepsilon)^3,\quad L'=0,\ S=\frac{3}{2};\qquad (d\varepsilon)^6,\quad L'=0,\ S=0 . \]

We see that the ground states of all configurations, with the exception of \((d\varepsilon)^3\), possess a threefold orbital degeneracy. Consequently, the spin-Hamiltonian for complexes with three \(d\)-electrons will have the same form as in the case of intermediate crystal fields (for example, hydrated salts of trivalent chromium). If there is one or five \(d\)-electrons, then, after allowance for the spin-orbit coupling and the action of a crystal field of low symmetry, all levels turn out to be Kramers doublets. Calculation of the splitting of the doublets in a magnetic field shows that,

that, owing to the covalent bond, the \(g\)-factor will decrease. Thus, for \([\mathrm{IrCl}_6]^{2-}\), for example,

\[ g = 2 - \frac{2}{3}(1-\beta^2), \tag{22} \]

where the coefficient \(\beta\) shows how the \(\psi\)-functions of the central atom and of the environment are mixed in the case of a \(\pi\)-bond.

We have already indicated that, as a result of covalent bonds, the lines of paramagnetic resonance exhibit a hyperfine structure due not only to the moment of the nucleus \(M\), but also to the moments of the nuclei \(X_6\). The hyperfine structure can be calculated with the aid of a spin Hamiltonian containing an additional series of terms that take into account the spins of the nuclei \(X_6\). Thus, if a strong magnetic field is applied along the \(z\) axis and only the octahedral symmetry of the crystal field is taken into account, then the spin Hamiltonian has the form

\[ \mathscr{H} = g_{\parallel}\beta_0 H \hat{S}_z + A(\hat{S}_z\hat{I}_z)_0 + A'\{(\hat{S}_z\hat{I}_z)_3+(\hat{S}_z\hat{I}_z)_6\}, \tag{23} \]

where

\[ A'=-\frac{32}{15}\beta^2(1-\beta^2)g_0\beta_0\beta_N\,\overline{r^{-3}}. \]

Here the index 0 refers to the central atom, and the indices 3 and 6 to the atoms \(X\) situated on the \(z\) axis.

In conclusion, it should be noted that the spin–orbit interaction can be described with the aid of a single constant \(\lambda\) only in the case of octahedral symmetry. Deviations of the symmetry from octahedral will cause anisotropy of the spin–orbit interaction.

2.5. Actinides

At present it has been firmly established that the transition group of elements beginning near uranium contains a partially filled \(5f\)-shell[^52]. The actinides differ from the \(4f\)-transition group of the rare earths in their tendency to form compounds containing chemically very stable complexes, similar to the uranyl ion \((\mathrm{UO}_2)^{2+}\). A systematic study of the magnetic properties of the actinides and, in particular, of the phenomenon of paramagnetic resonance has begun only recently, and so far only compounds containing \(\mathrm{UO}_2\), \(\mathrm{NpO}_2\), and \(\mathrm{PuO}_2\) have been well studied. The experimental data concerning these complexes have found a theoretical interpretation in works[^53][^54][^55].

Let us begin with the consideration of the \(\mathrm{UO}_2\) complex, although it does not possess normal paramagnetism and therefore does not give a paramagnetic-resonance effect.

The structure of this complex is linear: \(\mathrm{U}-\mathrm{O}-\mathrm{U}\). A free uranium atom has the closed radon core and 6 valence electrons, forming the configuration \(5f^3 6d\,7s^2\). In \((\mathrm{UO}_2)^{2+}\) two electrons are lost, and the four remaining ones create a strong covalent bond with the oxygen atoms. In the simplest model, used in[^53][^54], only a \(\sigma\)-bond is admitted. Linear combinations of the \(5f_\sigma\)-, \(6d_\sigma\)-, and \(7s\)-functions form orbitals, very elongated in the directions of the oxygen atoms, which overlap strongly with the \(sp_\sigma\)-orbitals of oxygen. Thus, in the ground state of \((\mathrm{UO}_2)^{2+}\) no unpaired electrons remain and, consequently, compounds containing uranyl will either be diamagnets or will possess weak, temperature-independent paramagnetism.

The ions \((\mathrm{NpO}_2)^{2+}\), \((\mathrm{PuO}_2)^{2+}\), and \((\mathrm{AmO}_2)^{2+}\) have structure and chemical properties analogous to \((\mathrm{UO}_2)^{2+}\). It is natural to suppose that the character

the bond in all these ions is the same, and that the additional electrons fill the \(5f\) shell, similarly to the \(4f\) electrons of the trivalent ions Ce, Pr, and Nd. However, transuranium compounds differ strongly from solid lanthanide salts in that the magnetic properties of the former are greatly influenced by the crystal field, whereas for the \(5f\) electrons of the actinides the axially symmetric field produced by the bonding electrons of the complex is of dominant importance. In the first approximation the magnetic properties of a compound containing a transuranium complex will be the same as those of a linear molecule; the crystal field introduces small corrections.

The complex \((\mathrm{NpO}_2)^{2+}\) contains one unpaired \(f\)-electron, which moves in a strong field of axial symmetry. Therefore, in the first approximation the conserved quantities will be the components of the total \((j_z)\), orbital \((l_z)\), and spin \((s_z)\) angular momenta along the symmetry axis, which we shall take to be the \(z\)-axis. In an axial field, different energy levels will correspond to all possible values \(|l_z|=3,2,1,0\). The lowest level, separated from the neighboring one by approximately \(10^4\ \mathrm{cm}^{-1}\), corresponds to the state with \(|l_z|=3\), since in this case the charge of the unpaired electron is located in the equatorial plane, so that its repulsion from the electrons forming the \(\sigma\)-bond will be minimal. This fourfold-degenerate level \(\left(l_z=\pm 3,\ S_z=\pm \dfrac{1}{2}\right)\), owing to the spin-orbit interaction, splits into two doublets: \(j_z=\pm {}^{5}/_{2}, \pm {}^{7}/_{2}\). The first doublet lies below the second by \(3000\)–\(4000\ \mathrm{cm}^{-1}\), and therefore it alone is responsible for the paramagnetism of neptunyl.

The paramagnetic-resonance spectrum can be calculated with the aid of the following simple spin Hamiltonian:

\[ \hat{\mathscr H} = g_{\parallel}\beta H_z \hat S'_z + g_{\perp}\beta (H_x \hat S'_x + H_y \hat S'_y) + A \hat I_z \hat S'_z + B(\hat I_x \hat S'_x + \hat I_y \hat S'_y) + \]

\[ + P\left[\hat I_z^2-\frac{1}{3}I(I+1)\right] - \gamma \beta_N \mathbf{H}\mathbf{I}. \tag{24} \]

Here \(g_{\parallel}=2\langle +|\hat l_z+2\hat S_z|+\rangle\), \(g_{\perp}=2\langle +|\hat l_x+2\hat S_x|-\rangle\); by \(+\rangle\) and \(-\rangle\) are denoted the wave functions of the lower doublet, and by \(S'\) the effective spin, equal to \(1/2\). In an approximation taking into account only the \(\sigma\)-bond, \(g_{\parallel}=4,\ g_{\perp}=0\). If, however, the possibility of a \(\pi\)-bond is also taken into account, then, as we saw in the preceding paragraph, the orbital moment is reduced; instead of \(l_z\) one must introduce \(k l_z\), where \(k<1\). Now \(g_{\perp}\ne 0,\ g_{\parallel}=6k-2\). Comparison with the experimental data shows that \(k=0.9\).

Finally, it should be noted that, because of the large gradient of the electric field created by the electrons forming the covalent bond, the superhyperfine structure due to the quadrupole moment of the Np nucleus will be large.

The complex \((\mathrm{PuO}_2)^{2+}\) contains two unpaired electrons, upon whose motion the perturbing action is exerted primarily by the axial field and by their electrostatic repulsion from each other. For the same reasons as in the case of neptunyl, it would seem that the unpaired electrons should occupy the state \(l_z=\pm 3\). In reality, however, owing to electrostatic repulsion from each other, the ground state of the “electron configuration” \(5f^2\) is determined by a modified Hund rule: the projection of the electron spin must be maximal, \(S_z=1\); the projection of the orbital moment must have the maximal value,

compatible with \(S_z=1\), namely \(|l_{1z}|=3\), \(|l_{2z}|=2\), and, consequently, \(L_z=\pm 5\). The spin-orbit interaction causes a further splitting of the energy levels, after which the lower level becomes a doublet with \(j_z=\pm(5-1)=\pm4\). An elementary calculation shows that, if one again introduces the effective spin \(s'=1/2\), then for this doublet \(g_{\parallel}=6\), \(g_{\perp}=0\). In this case the probability of a transition between the magnetic sublevels proves to be zero, independently of the direction of the external magnetic field \(H\). A detailed consideration shows that allowance for various corrections does not change \(g_{\perp}=0\). As a result, the paramagnetic-resonance effect is maximal when the external magnetic field is oriented parallel to the \(z\) axis. This is explained by the fact that the low-symmetry crystal field, which we have not taken into account earlier, mixes the wave functions with \(j_z=\pm4\). It should be borne in mind that the doublet under consideration is not a Kramers doublet, since the number of unpaired electrons is even. Thus the spin Hamiltonian will have the form

\[ \hat{\mathcal H} = g_{\parallel}\beta H_z \hat{s}'_z + A\hat{s}'_z\hat{I}_z - P\left[\hat{I}_z^2-\frac{1}{3}I(I+1)\right] + \Delta_x\hat{s}'_x + \Delta_y\hat{s}'_y . \tag{25} \]

The last two terms take into account the splitting caused by the low-symmetry crystal field.

2.6. Experimental results relating to the spectra of ionic crystals

The first experiments on paramagnetic resonance were carried out with polycrystalline samples of undiluted paramagnets at room temperature. Zavoisky’s experimental discovery^56 of the fine structure of paramagnetic-absorption lines, which received its first interpretation in the work of Weiss^57, and Penrose’s discovery^58 of the superhyperfine structure of lines in solid paramagnets led to a broad development of investigations of paramagnetic-resonance spectra, owing to their obvious importance for nuclear physics and for the theory of the solid state.

The experimental study of these spectra is naturally carried out, wherever possible, on single crystals. In doing so, one tries to choose compounds that crystallize well, have a known crystal structure, and contain in the unit cell a minimal number of magnetically nonequivalent ions. To obtain the best resolution of the absorption-line structure, “magnetically diluted” crystals are often used, in which a large fraction of the paramagnetic ions has been replaced by suitable diamagnetic ones. Usually “magnetic dilutions” of the order of \(1:100\) are used. In a number of cases, despite dilution, which reduces the magnetic dipole interactions between ions, the line width proves too large because of strong spin-lattice interactions; these are weakened by cooling the sample under study to a temperature at which the spin-lattice relaxation time becomes sufficiently large. Finally, to reduce the magnetic dipole interactions of the paramagnetic ion with neighboring atomic nuclei in hydrated salts, \(H_2O\) is sometimes replaced by \(D_2O\).

By the present time a very large number of paramagnetic ionic crystals has already been investigated. A detailed summary of the data obtained up to the middle of 1955 is contained in reviews^59 and ^60. Here we shall confine ourselves to indicating only the principal results.

2.61. Salts of ions of the iron group (3d)

Among these salts the following types have been studied most extensively: alums
$M^{\mathrm I}M^{\mathrm{III}}(SO_4)_2\cdot 12H_2O$; Tutton salts
$M_2^{\mathrm I}M^{\mathrm{II}}(SO_4)_2\cdot 6H_2O$; double nitrates
$M_3^{\mathrm{II}}M_2^{\mathrm{III}}(NO_3)_{12}\cdot 24H_2O$; fluorosilicates
$M^{\mathrm{II}}SiF_6\cdot 6H_2O$ and some others. In the formulas given, $M^{\mathrm I}$, $M^{\mathrm{II}}$, and $M^{\mathrm{III}}$ denote, respectively, mono-, di-, and trivalent cations. In all the named types of salts the paramagnetic ion is acted upon by a strong cubic field produced by an octahedron of the nearest water molecules. To this field, as was indicated above, there is added a weak field of lower symmetry, due to partial distortion of the octahedron and to the influence of more distant neighbors. In the case of alums it has trigonal symmetry for $M^{\mathrm{III}}$; in Tutton salts it has tetragonal or rhombic symmetry for $M^{\mathrm{II}}$, and in double nitrates and fluorosilicates it is trigonal for $M^{\mathrm{II}}$. Establishing these types of symmetry was in most cases the result of analysis of paramagnetic-resonance spectra. Recently the study has begun of spectra given by ions of the iron group in purely cubic fields^61; for this purpose paramagnetic impurities are introduced into the lattice of single crystals of MgO, ZnS, CaF$_2$, etc.

A number of works were devoted to complex cyanides of the type $M_3^{\mathrm I}M^{\mathrm{III}}(CN)_6$ and $M_4^{\mathrm I}M^{\mathrm{II}}(CN)_6$. In them the divalent or trivalent paramagnetic ion is surrounded by an almost regular octahedron of $(CN)^-$ anions. The bonds of the central ion with $(CN)^-$ are covalent to a high degree. The bonds in compounds of the type K$_2$MnO$_4$ and certain analogous compounds are likewise covalent; in these compounds the paramagnetic atom is not in an octahedral but in a tetragonal field of O atoms. The investigation of such complexes (so far only in polycrystalline specimens) has also begun at the present time^62.

Below are given some experimental results for individual paramagnetic ions of the iron group.

$3d^1$, $^2D_{3/2}$. For ions of this type the lower orbital triplet, obtained as a result of the action of the cubic field, is split by the trigonal field and by spin-orbit coupling into three Kramers doublets, the separation between which for weakly distorted octahedral complexes is small ($\sim 100\ \mathrm{cm}^{-1}$). At low temperatures only the lower doublet with effective spin $s=1/2$ is populated; the $g$-factor differs strongly from 2, and spin-lattice interactions are very strong (see § 3). For non-octahedral complexes the separation between orbital doublets is much greater, as a result of which the $g$-factor is close to 2 and the spin-lattice interactions are small.

a) Ti$^{+++}$. Single crystals of the alum CsTi(SO$_4$)$_2\cdot 2H_2O$ and of the oxalate KTi(C$_2$O$_4$)$_2\cdot 2H_2O$ have been investigated. The alum gives an effect only at helium temperatures; $g_{\parallel}=1.25$, $g_{\perp}=1.14$. In the oxalate the field is not octahedral; therefore the effect was observed at 90 °K; $g_{\parallel}=1.86$, $g_{\perp}=1.96$^63.

b) V$^{++++}$. Some salts of the vanadyl ion VO$^{++}$ have been investigated^60; in single crystals of the double oxalate K$_2$(VO)$_2$(C$_2$O$_4$)$_3\cdot 4H_2O$ (the magnetic complex is not octahedral) the following were determined: $g_1=1.954$, $g_2=1.985$, $g_3=1.967$.

$3d^3$, $^4F_{3/2}$. The lower orbital singlet obtained after the action of the cubic field has fourfold spin degeneracy and is split by a crystal field of lower symmetry into two Kramers doublets. The effective $g$-factor is expressed by the formula

\[ g=2-\frac{72\lambda}{5\Delta}, \]

where $\Delta$ is the total splitting of the orbital levels.

a) V$^{++}$. The following have been investigated: 1) single crystals of the Tutton salt $(NH_4)_2\cdot (V,Zn)(SO_4)_2\cdot 6H_2O$^64. The constants determined were: $D=0.158\ \mathrm{cm}^{-1}$,

\(\varepsilon = 0.049\ \mathrm{cm}^{-1}\), \(A = 0.0088\ \mathrm{cm}^{-1}\) (for \({}^{51}\mathrm{V}\)); \(g = 1.951\); 2) single crystals of cyanide \(K_4(V,Fe)(CN)_6 \cdot 3H_2O\) \((V:Fe = 1:100)\). The constants have the values: \(D = 0.0264\ \mathrm{cm}^{-1}\), \(\varepsilon = 0.0072\ \mathrm{cm}^{-1}\), \(A = 0.0056\ \mathrm{cm}^{-1}\) (for \({}^{51}\mathrm{V}\)), \(g = 1.992\). In a cyanide containing \({}^{50}\mathrm{V}\), for this isotope the nuclear spin was determined to be \(I = 6\); \(A = 0.0021\ \mathrm{cm}^{-1}\) \(^{65}\).

b) \(Cr^{+++}\). A series of alums \(^{60}\) has been studied in detail in the temperature range from 290 to \(20^\circ K\). In all of them the \(g\)-factor is close to 1.98. In \(KCr(SO_4)_2 \cdot 12H_2O\), upon cooling, a gradual change of the crystal is observed: at \(290^\circ K\), \(D = 0.060\ \mathrm{cm}^{-1}\); at \(193^\circ K\), \(0.027\ \mathrm{cm}^{-1}\); at \(160^\circ K\), \(0.017\ \mathrm{cm}^{-1}\); below \(160^\circ K\) the crystals contain two magnetically nonequivalent complexes in the unit cell, and at \(90^\circ K\), \(D_1 = 0.130\ \mathrm{cm}^{-1}\) and \(D_2 = 0.075\ \mathrm{cm}^{-1}\). In \(NH_4Cr(SO_4)_2 \cdot 12H_2O\) the phase transition occurs at \(80^\circ K\); below it two magnetic complexes have likewise been detected. In \((NH_3CH_3)Cr(SO_4)_2 \cdot 12H_2O\) the transition is observed at \(157^\circ K\); below this temperature the spectrum corresponds to rhombic symmetry of the local fields. In Rb and Cs alums no phase transition has been noted, nor in \(KCr(SeO_4)_2 \cdot 12H_2O\). In the latter salt, diluted with an Al salt and containing chromium enriched in the magnetic isotope \({}^{53}Cr\), hyperfine structure was observed with \(A = 0.00185\ \mathrm{cm}^{-1}\).

In addition to the alums, cyanide \(K_3[(Cr, Al)(CN)_6]\) was studied at \(90\text{–}12^\circ K\). The following were found: \(g = 1.993\); \(D = 0.054\ \mathrm{cm}^{-1}\), \(\varepsilon = 0.012\ \mathrm{cm}^{-1}\), \(A = 0.0014\ \mathrm{cm}^{-1}\) \(^{60}\). Further, the ion \(Cr^{+++}\) was investigated in a single crystal of corundum \(Al_2O_3\) (artificial ruby) \(^{66,67}\). Assuming trigonal field symmetry, it was found \(^{67}\): \(g_{\parallel} = 1.989\); \(g_{\perp} = 1.987\); \(D = 0.141\ \mathrm{cm}^{-1}\); \(A = 0.0017\ \mathrm{cm}^{-1}\). In the MgO lattice \(^{61}\), one very narrow \(Cr^{+++}\) line was found with \(g = 1.980\) and with hyperfine structure resolved even in a non-enriched sample \((9.4\%\,{}^{53}Cr)\); \(A = 0.00016\ \mathrm{cm}^{-1}\); the magnetic moment of the \({}^{53}Cr\) nucleus was estimated as \(0.475\,\mu_{\mathrm{nuc}}\). Many \(Cr^{+++}\) compounds have been investigated in powder form, in particular \(CrCl_3\) \((g = 1.99)\) and the hydrated isomers \(CrCl_3 \cdot 6H_2O\) \(^{21a}\); \(CrF_3\); sulfates, phosphates, rhodanides, and salts of certain organic acids.

\(3d^4,\ ^5D_0,\ Cr^{++}\). In a cubic field the lower orbital doublet remains; a trigonal field removes the orbital degeneracy; the remaining fivefold spin degeneracy of the lower orbital level is completely lifted by a rhombic field. \(CrSO_4 \cdot 5H_2O\) has been studied at \(290^\circ K\) \((g_{\parallel} = 1.95;\ g_{\perp} = 1.99;\ D = 2.24\ \mathrm{cm}^{-1};\ E = 0.10\ \mathrm{cm}^{-1}\) \(^{68})\).

\(3d^5,\ ^6S_{5/2}\). In hydrated magnetic complexes the electric field splits the sixfold spin-degenerate lower orbital singlet into three Kramers doublets with a separation usually less than \(1\ \mathrm{cm}^{-1}\). In compounds with covalent bonding (for example, in cyanides) the ground state is only doubly spin-degenerate; there are low-lying excited levels.

a) \(Mn^{++}\). Single crystals of Tutton salt, double nitrate, fluorosilicate, heptahydrate sulfate, carbonate (natural calcite \(CaCO_3\) with an admixture of \(Mn^{++}\)), sulfide (ZnS with an admixture of \(Mn^{++}\)), and other salts \(^{60}\), diluted with isomorphous Mg or Zn salts, have been studied in detail. In \(NH_4(Mn,Zn)\cdot(SO_4)_2 \cdot 6H_2O\) \((Mn:Zn = 1:1000)\) the symmetry of the magnetic complex proved to be rhombic, with \(g = 2.00\); \(D = 0.0277\ \mathrm{cm}^{-1}\), \(E = 0.005\ \mathrm{cm}^{-1}\), the splitting constant by the cubic field \(a = 0.0008\ \mathrm{cm}^{-1}\); \(A = -0.00193\ \mathrm{cm}^{-1}\) at \(20^\circ K\) \(^{69}\).

In double nitrates the field symmetry is trigonal; in the crystal there are two magnetic complexes; at \(90^\circ K\) for the first \(D = -0.0211\ \mathrm{cm}^{-1}\), \(a = 0.0008\ \mathrm{cm}^{-1}\), \(A = 0.0090\ \mathrm{cm}^{-1}\); for the second \(D = 0.0064\ \mathrm{cm}^{-1}\), \(a = 0.0010\ \mathrm{cm}^{-1}\), \(A = 0.0089\ \mathrm{cm}^{-1}\); for both complexes \(g = 1.99\). In sulfide \((Mn:Zn\) from \(1:10^3\) to \(1:10^5)\), \(D = 0.001\ \mathrm{cm}^{-1}\), \(A = 0.0065\ \mathrm{cm}^{-1}\), \(g = 2.0024\); the anomalously small value of \(A\) indicates a noticeable fraction of covalent bonding \(^{70}\).

In the purely cubic lattice of MgO the crystalline field splits the sixfold-degenerate level of Mn++ into a quartet and a (lower) doublet. The spectrum is described by the constants: \(g = 2.0014\); the splitting constant in the cubic field \(a = 0.00186\ \mathrm{cm}^{-1}\), \(A = -0.0081\ \mathrm{cm}^{-1}\). The values of the constants led to the conclusion that the bond of manganese with oxygen is 20% covalent \(^{61}\).

Among Mn++ compounds with a strongly pronounced covalent bond, the cyanide \(K_4[(\mathrm{Mn}, \mathrm{Fe})(\mathrm{CN})_6]\cdot 3\mathrm{H}_2\mathrm{O}\) \((\mathrm{Mn}:\mathrm{Fe}=1:200)\) has been studied at \(15^\circ\mathrm{K}\). The results are described (for the effective spin \(s' = 1/2\)) by the following constants: \(g_x = 2.624\); \(g_y = 2.188\); \(g_z = 0.72\); \(A_x = 0.00845\ \mathrm{cm}^{-1}\); \(A_y = 0.00465\ \mathrm{cm}^{-1}\); \(A_z = 0.0083\ \mathrm{cm}^{-1}\) \(^{60}\). In addition to single crystals, a number of Mn++ salts have been investigated in the form of powders, in particular \(\mathrm{MnCl}_2\cdot 4\mathrm{H}_2\mathrm{O}^{21a}\); \(\mathrm{MnF}_2\); variously hydrated sulfates, phosphates, and salts of certain organic acids.

b) Fe+++. Certain ferric alums, diluted with the corresponding aluminum alums, have been studied in detail \(^{60}\). In all of them \(g = 2.003\); the constant \(a\) is negative and much larger in absolute value than in Mn++ compounds. Thus, for \(\mathrm{NH}_4(\mathrm{Fe}, \mathrm{Al})(\mathrm{SO}_4)_2\cdot 12\mathrm{H}_2\mathrm{O}\) \((\mathrm{Fe}:\mathrm{Al}=1:80)\) at \(4^\circ\mathrm{K}\), \(D = 0.016\ \mathrm{cm}^{-1}\), \(a = -0.0128\ \mathrm{cm}^{-1}\); an attempt to determine the spin of the \(^{57}\mathrm{Fe}\) nucleus in the salt \(\mathrm{K}(\mathrm{Fe}, \mathrm{Al})(\mathrm{SeO}_4)_2\cdot 12\mathrm{D}_2\mathrm{O}\) was unsuccessful \(^{71}\). We note a work devoted to the study of the Fe+++ spectrum in beryl \(\mathrm{Be}_3\mathrm{Al}_2(\mathrm{SiO}_3)_3^{126}\). For this substance it was found: \(g = 2.00\); \(a = -0.014\ \mathrm{cm}^{-1}\); \(D = 0.0165\ \mathrm{cm}^{-1}\); \(F = 0.0007\ \mathrm{cm}^{-1}\). The spectrum obtained in this crystal is shown in Fig. 6.

Fig. 6. Fine structure of the paramagnetic resonance line in beryl with the optical axis of the crystal oriented parallel to the constant magnetic field. \(\lambda \simeq 3.2\ \mathrm{cm}^{126}\). a) observed spectrum; b) calculated spectrum.

A number of Fe+++ salts have been investigated in the form of powders. Among complex compounds, a single crystal of cyanide \(\mathrm{K}_3\mathrm{Fe}(\mathrm{CN})_6\) has been studied, both pure and diluted with cobalt \((\mathrm{Fe}:\mathrm{Co}=1:100)\). In the latter, at \(20^\circ\mathrm{K}\), \(g_x = 2.35\), \(g_y = 2.10\), and \(g_z = 0.91\) (effective spin \(s' = 1/2\) \(^{60}\)). We mention here in passing interesting results obtained in the study of a number of hemin compounds of Fe+++, namely hemoglobin derivatives. In them the \({}^{6}S_{5/2}\)-state is also split into 3 doublets with a large separation between them \((>10\ \mathrm{cm}^{-1})\) and with an effective spin \(s' = 1/2\) for the lower level. The cause of this large splitting is either an admixture of a covalent \(\pi\)- or \(\sigma\)-bond, or simply the action of a strong field of low symmetry; the \(g\)-factors are sharply anisotropic \((g_{\parallel}=2.0,\ g_{\perp}=6.0)^{72}\).

\(3d^6,\ ^5D_4,\ \mathrm{Fe}^{++}\). The orbital levels split, as in \(3d'\); the cubic field gives a lower orbital triplet with fivefold spin degeneracy. The rhombic field removes all degeneracy completely. Only a single crystal of diluted fluoride \((\mathrm{Fe}, \mathrm{Zn})\cdot \mathrm{F}_3\) \((\mathrm{Fe}:\mathrm{Zn}=1:3000)\) has been studied in detail. In it the ground state is a weakly split doublet. The effect is observed only at low temperatures \(^{73}\).

\(3d^7,\ ^4F_{9/2},\ \mathrm{Co}^{++}\). The cubic field gives a lower orbital triplet with fourfold spin degeneracy; fields of lower symmetry and spin-orbit coupling split it into Kramers doublets. In hydrated salts \(s' = 1/2\); the \(g\)-factor is anisotropic, as is the hyperfine structure due to the spin \(I = 7/2\) of the \(^{59}\mathrm{Co}\) nucleus. Thus, in Tutton’s salt \(\mathrm{K}_2(\mathrm{Co}, \mathrm{Zn})\cdot\)

\((\mathrm{SO}_4)_2 \cdot 6\mathrm{H}_2\mathrm{O}\) (Co:Zn from 1:500 to \(1:10^5\)) \(g\) changes from \(g_z = 6.56\) to \(g_{\min}=2.50\); \(A_z = 0.0286\ \mathrm{cm}^{-1}\); \(A_{\min}=0.0065\ \mathrm{cm}^{-1}\) 74.

In addition to this salt, the double nitrate, fluorosilicate \(\mathrm{CoSiF}_6 \cdot 6\mathrm{H}_2\mathrm{O}\), etc. have been studied. The effect is observed only at low temperatures. In the salt \(\mathrm{K}_2(\mathrm{Co},\mathrm{Zn})(\mathrm{SO}_4)_2 \cdot 12\mathrm{H}_2\mathrm{O}\), the spin \(I=4\) and magnetic moment \(\mu = 3.848\,\mu_{\mathrm{nuc}}\) were determined for the radioactive isotope \({}^{56}\mathrm{Co}\) 75. A special case is represented by the salt \(\mathrm{Cs}_3\mathrm{CoCl}_5\) with a tetrahedral magnetic complex. Here the cubic field has the opposite sign compared with the octahedron, and the lower level is an orbital singlet with fourfold spin degeneracy. In fields of lower symmetry the spin quadruplet is split into two doublets. The effective spin \(s'\) is equal to \(3/2\) 60.

\(3d^8,\ ^3F_4,\ \mathrm{Ni}^{++}\). The cubic field gives a lower orbital singlet with threefold spin degeneracy; the spin triplet is split by a trigonal or tetragonal field into a doublet and a singlet; a rhombic field gives three spin singlets; \(s' = s = 1\); the \(g\)-factor is almost isotropic, and no hyperfine structure of \({}^{61}\mathrm{Ni}\) was observed.

Single crystals of various Tutton salts, the double nitrate, fluorosilicate, etc. 60 have been investigated. In the latter \(g=2.3\); \(D=-0.50\ \mathrm{cm}^{-1}\); \(E=0\) (at \(290^\circ\mathrm{K}\)) and \(g=2.29\); \(D=-0.12\ \mathrm{cm}^{-1}\); \(E=0\) (at \(20^\circ\mathrm{K}\)).

\(3d^9,\ ^2D_{5/2},\ \mathrm{Cu}^{++}\). The cubic field splits the ground state into an upper triplet and a lower doublet. A tetragonal or rhombic field splits the orbital doublet into two Kramers doublets; resonance is observed for the lower of them. In the case of such fields, the approximate value is \(g = 2 - (4 \pm 4)\lambda/\Delta;\ 2 - (4 \pm 2)\lambda/\Delta\) (the sign depends on which of the orbital singlets is lower). In dilute crystals an anisotropic hyperfine structure is observed, due to the spin \(I=3/2\) of the nuclei \({}^{63}\mathrm{Cu}\) and \({}^{65}\mathrm{Cu}\). Besides the nuclear spin, the spectrum in \(\mathrm{Cu}^{++}\) salts is also affected by the nuclear quadrupole moment, which causes additional splittings of the hyperfine-structure lines.

For salts with a trigonal component of the field, the \(g\)-factor and the hyperfine structure are close to isotropic.

Of the compounds with symmetry lower than trigonal, a large number of single crystals of Tutton salts 60 have been studied. In particular, for the salt \((\mathrm{NH}_4)_2(\mathrm{Cu},\mathrm{Zn})(\mathrm{SO}_4)_2 \cdot 6\mathrm{H}_2\mathrm{O}\) (Cu:Zn up to 1:1000) at \(20^\circ\mathrm{K}\) it was found: \(g_x=2.12\), \(g_y=2.05\), \(g_z=2.46\); \(A_x=A_z=0.0025\ \mathrm{cm}^{-1}\), \(A_y=0.0035\ \mathrm{cm}^{-1}\), quadrupole-splitting constant \(p=0.0011\ \mathrm{cm}^{-1}\). The hyperfine structure in the Tutton salt \(\mathrm{K}_2(\mathrm{Cu},\mathrm{Zn})(\mathrm{SO}_4)_2 \cdot 6\mathrm{H}_2\mathrm{O}\) (Cu:Zn = 1:200) was studied at \(90^\circ\mathrm{K}\) under intermediate-field conditions. The experiment agrees with the theory for \(g_{\parallel}=2.47,\ g_{\perp}=2.08;\ A_{\parallel}=-0.083\ \mathrm{cm}^{-1},\ A_{\perp}=0.045\ \mathrm{cm}^{-1};\ p=0.001\ \mathrm{cm}^{-1}\) 76.

In addition to Tutton salts, single crystals of \(\mathrm{CuSO}_4 \cdot 5\mathrm{H}_2\mathrm{O}\); \([\mathrm{Cu}(\mathrm{NH}_3)_4]\mathrm{SO}_4 \cdot \mathrm{H}_2\mathrm{O}\); \(\mathrm{Cu}(\mathrm{CH}_3\mathrm{COO})_2 \cdot 2\mathrm{H}_2\mathrm{O}\), etc., with field symmetry lower than trigonal 60, have also been studied.

Among salts with a trigonal field, dilute fluorosilicates, bromates, and double nitrates have been investigated. In \((\mathrm{Cu},\mathrm{Mg})_2\mathrm{Bi}_3(\mathrm{NO}_3)_{12} \cdot 24\mathrm{H}_2\mathrm{O}\) (Cu:Mg = 1:100) it was found at \(90^\circ\mathrm{K}\): \(g_{\parallel}=2.219,\ g_{\perp}=2.217;\ A_{\parallel}=0.0027\ \mathrm{cm}^{-1},\ A_{\perp}=0.0026\ \mathrm{cm}^{-1}\). On cooling to \(20^\circ\mathrm{K}\), the field symmetry in this salt becomes tetragonal and the anisotropy of the constants increases 77.

2.62. Compounds of elements of the palladium group (\(4d\)) and platinum group (\(5d\))

The elements of these groups usually form covalent bonds with their neighbors in magnetic complexes; only a small number of compounds have been investigated 60.

\(4d^1\), \(\mathrm{Mo}^{+++++}\). A single crystal of \(\mathrm{K}_3[\mathrm{InCl}_6]\cdot 2\mathrm{H}_2\mathrm{O}\), containing \(\mathrm{Mo}^{5+}\) as an impurity, was studied. According to the analysis of the observed paramagnetic-resonance spectrum, the magnetic complex has a field of axial symmetry and is described by a Hamiltonian with \(s' = 1/2\) (\(I = 5/2\) for the odd isotopes \({}^{95}\mathrm{Mo}\) and \({}^{97}\mathrm{Mo}\)).

\(4d^3\), \(\mathrm{Mo}^{+++}\); \(5d^3\), \(\mathrm{Re}^{++++}\). Measurements were carried out at 90 and \(20^\circ\) K in a single crystal of \(\mathrm{K}_3[(\mathrm{Mo},\mathrm{In})\mathrm{Cl}_6]\) (\(\mathrm{Mo}:\mathrm{In}=1:200\)). The complex is octahedral with a rhombic displacement; \(s' = 3/2\) (and \(I=5/2\) for the odd Mo isotopes). The unit cell contains many magnetically nonequivalent complexes. Measurements were also made in several other \(\mathrm{Mo}^{+++}\) salts and in the rhenium salt \(\mathrm{K}_2[(\mathrm{Re},\mathrm{Pt})\mathrm{Cl}_6]\) (\(\mathrm{Re}:\mathrm{Pt}=1:200\)). In the latter, six lines were observed, probably due to the hyperfine structure of the odd isotopes \({}^{185}\mathrm{Re}\) and \({}^{187}\mathrm{Re}\).

\(4d^5\), \(\mathrm{Ru}^{+++}\); \(5d^5\), \(\mathrm{Ir}^{++++}\). Of the \(\mathrm{Ru}^{+++}\) compounds, single crystals of \([(\mathrm{Ru},\mathrm{Co})(\mathrm{NH}_3)_6]\mathrm{Cl}_3\) (\(\mathrm{Ru}:\mathrm{Co}=1:200\)) were studied. It was found that the complex in them is octahedral with rhombic displacements; \(s' = 1/2\) and \(I=5/2\) (for \({}^{99}\mathrm{Ru}\) and \({}^{101}\mathrm{Ru}\)). There are three nonequivalent complexes in the cell.

Of the \(\mathrm{Ir}^{++++}\) compounds, \(\mathrm{K}_2[\mathrm{IrBr}_6]\), \(\mathrm{K}_2[\mathrm{IrCl}_6]\), \((\mathrm{NH}_4)_2[\mathrm{IrCl}_6]\), \(\mathrm{Na}_2[\mathrm{IrBr}_6]\cdot 6\mathrm{H}_2\mathrm{O}\), and \(\mathrm{Na}_2[\mathrm{IrCl}_6]\cdot 6\mathrm{H}_2\mathrm{O}\), diluted with the corresponding Pt salts (\(\mathrm{Ir}:\mathrm{Pt}=1:200\)), were investigated. The first salt contains three magnetically nonequivalent complexes in the cell; in the others all complexes are equivalent; they are octahedral, with various rhombic displacements; \(s' = 1/2\), \(I = 5/2\) (for \({}^{191}\mathrm{Ir}\) and \({}^{193}\mathrm{Ir}\)). In the last three salts, besides the hyperfine structure due to the spins of the iridium nuclei, a hyperfine structure due to the spins of the Cl or Br nuclei was observed. In \((\mathrm{NH}_4)_2[\mathrm{IrCl}_6]\) at \(20^\circ\) K, \(g = 1.775\); \(A = 0.0265\ \mathrm{cm}^{-1}\) (for Ir) and \(A' = 0.00088\ \mathrm{cm}^{-1}\) (for \({}^{35}\mathrm{Cl}\)).

\(4d^9\), \(\mathrm{Ag}^{++}\). Measurements were made only in powders of certain compounds, for example \(\mathrm{Ag}'(\mathrm{C}_5\mathrm{H}_5\mathrm{N})_4\mathrm{S}_2\mathrm{O}_8\). The effective \(g\)-factor for this compound is equal to 2.08.

2.63. Compounds of ions of the rare-earth group (\(4f\))

The spectra of paramagnetic resonance were studied in single crystals of double nitrates, ethyl sulfates \(\mathrm{M}^{III}(\mathrm{C}_2\mathrm{H}_5\mathrm{SO}_4)_3\cdot 9\mathrm{H}_2\mathrm{O}\), and sulfates \(\mathrm{M}_2^{III}(\mathrm{SO}_4)_3\cdot 8\mathrm{H}_2\mathrm{O}\). For the first two types of salts it was established that the magnetic complex has not cubic (octahedral), but trigonal symmetry. For all ions, with the exception of \(\mathrm{Gd}^{+++}\) and \(\mathrm{Eu}^{++}\), which have the ground state \({}^{8}S_{7/2}\), paramagnetic resonance, because of strong spin-lattice interactions, is observed only at very low temperatures (liquid hydrogen and below). At such temperatures only the lower doublet is populated, as a result of which the effective spin is \(s' = 1/2\).

\(4f^1\), \({}^{2}F_{5/2}\), \(\mathrm{Ce}^{+++}\). In the double nitrate \(\mathrm{Mg}_3\mathrm{Ce}_2(\mathrm{NO}_3)_{12}\cdot 24\mathrm{H}_2\mathrm{O}\) at 4.2 K it was found that \(g_{\parallel}=0.25\), \(g_{\perp}=1.84\). In ethyl sulfate diluted with lanthanum, at 4.2 K paramagnetic resonance was also observed on the second doublet, with \(g_{\parallel}=3.72\) and \(g_{\perp}=0.20\); the lower doublet gives \(g_{\parallel}=0.955\) and \(g_{\perp}=2.185^{60}\).

\(4f^2\), \({}^{3}H_4\), \(\mathrm{Pr}^{+++}\). Since the number of electrons is even, the lower spin doublet may be nondegenerate. However, it is not split by the trigonal field in the double nitrate and ethyl sulfate. The observed spectrum can be described by putting \(g_{\perp}\simeq 0\) and taking into account in the spin Hamiltonian a term \(\Delta s\), small deviations of the crystalline field (due to thermal motion) from trigonality, which cause weak splittings of the doublet. In ethyl sulfate diluted with yttrium, \(g_z=1.525\); \(A=0.075\ \mathrm{cm}^{-1}\) (\(\Delta\) has a Gaussian distribution with mean value 0), and in this case \(s'=1/2\), \(I=5/2\) (for the nucleus \({}^{141}\mathrm{Pr}\))\(^{78}\).

$4f^3,\ ^4I_{9/2},\ \mathrm{Nd}^{+++}$. Ethyl sulfate and double nitrate diluted with La salts have been studied. For the nitrate at $4.2^\circ\mathrm{K}$ it was found that $g_\parallel=0.45$, $g_\perp=2.72$, $I=7/2$, $A_\parallel=0.0052\ \mathrm{cm}^{-1}$, $A_\perp=0.0312\ \mathrm{cm}^{-1}$ (for the nucleus ${}^{143}\mathrm{Nd}$) and $I=7/2$, $A_\parallel=0.0032\ \mathrm{cm}^{-1}$, $A_\perp=0.0194\ \mathrm{cm}^{-1}$ (for ${}^{145}\mathrm{Nd}$) $^{60}$.

$4f^5,\ ^6H_{5/2},\ \mathrm{Sm}^{+++}$. Ethyl sulfate and double nitrate were investigated. For the former, at $4.2^\circ\mathrm{K}$, $g_\parallel=0.569$, $g_\perp=0.604$; $I=7/2$, $A_\parallel=0.060\ \mathrm{cm}^{-1}$, $A_\perp=0.0251\ \mathrm{cm}^{-1}$ (for ${}^{147}\mathrm{Sm}$); $I=7/2$, $A_\parallel=0.049\ \mathrm{cm}^{-1}$, $A_\perp=0.0205\ \mathrm{cm}^{-1}$ (for ${}^{149}\mathrm{Sm}$) $^{60}$.

$4f^7,\ ^8S_{7/2}$.

a) $\mathrm{Gd}^{+++}$. As in salts of $\mathrm{Gd}^{+++}$, so also in $\mathrm{Eu}^{++}$, paramagnetic resonance can be observed even at room temperatures. The fine structure of the $\mathrm{Gd}^{+++}$ spectrum has been analyzed in detail for ethyl sulfate, double nitrate, and $\mathrm{Gd}_2(\mathrm{SO}_4)_3\cdot 8\mathrm{H}_2\mathrm{O}$; the $g$-factor is isotropic and equal to 1.99. In addition to the named substances, the fine-structure spectrum was observed in $\mathrm{Gd}(\mathrm{NO}_3)_3\cdot 6\mathrm{H}_2\mathrm{O}$ $^{78}$ and in some other salts. Owing to the smallness of its constant, the hyperfine structure was detected only recently $^{79}$, and for both magnetic isotopes ${}^{155}\mathrm{Gd}$ and ${}^{157}\mathrm{Gd}$ the nuclear spin proved to be $I=3/2$.

b) $\mathrm{Eu}^{++}$. The hyperfine-structure spectrum was observed in SrS powder with an EuS impurity; $g=1.991$ was found. For the stable isotopes ${}^{151}\mathrm{Eu}$ and ${}^{153}\mathrm{Eu}$ the constant $A$ is respectively $0.00308\ \mathrm{cm}^{-1}$ and $0.00138\ \mathrm{cm}^{-1}$. Recently, in the same compound it has been possible to observe the hyperfine structure of radioactive ${}^{152}\mathrm{Eu}$, and it was found $^{82}$ that $I=3$, $\mu=2.03\,\mu_{\mathrm{nuc}}$, $A=0.00139\ \mathrm{cm}^{-1}$.

$4f^8,\ ^7F_6,\ \mathrm{Tb}^{+++}$. Terbium ethyl sulfate diluted with yttrium (1:1000) was investigated. The trigonal field slightly splits the lower spin doublet, since the number of electrons in the ion is even. The spectrum could be described by putting $g_\perp\simeq 0$ and introducing into the Hamiltonian a term $\Delta s_x$, characterizing this splitting. At $20^\circ\mathrm{K}$ it was found: $g_\parallel=17.72$, $g_\perp<0.3$; $\Delta=0.387\ \mathrm{cm}^{-1}$, $A=0.209\ \mathrm{cm}^{-1}$ (for ${}^{159}\mathrm{Tb}$) $^{60}$.

$4f^9,\ ^8H_{15/2},\ \mathrm{Dy}^{+++}$. In acetate $(\mathrm{Dy},\mathrm{Y})(\mathrm{CH}_3\mathrm{COO})_3\cdot 4\mathrm{H}_2\mathrm{O}$ the hyperfine structure of the absorption line has recently been investigated, and $I=5/2$ was found for the isotopes ${}^{161}\mathrm{Dy}$ and ${}^{163}\mathrm{Dy}$ $^{83}$.

$4f^{10},\ ^5I_8,\ \mathrm{Ho}^{+++}$. Only holmium ethyl sulfate diluted with yttrium was studied; for it, at $13^\circ\mathrm{K}$, $g_z=15.36$; $A=0.334\ \mathrm{cm}^{-1}$, $\Delta=0.065\ \mathrm{cm}^{-1}$ for $s'=1/2$ and $I=7/2$. The magnetic moment of the ${}^{165}\mathrm{Ho}$ nucleus was estimated to be $3.29\,\mu_{\mathrm{nuc}}$ $^{84}$.

$4f^{11},\ ^4I_{15/2},\ \mathrm{Er}^{+++}$. Erbium ethyl sulfate diluted with lanthanum was studied. At $4^\circ\mathrm{K}$ it was obtained: $g_\parallel=1.47$, $g_\perp=8.85$; $I=7/2$; $A_\parallel=0.0052\ \mathrm{cm}^{-1}$, $A_\perp=0.0314\ \mathrm{cm}^{-1}$, $p=0.0030\ \mathrm{cm}^{-1}$ (for ${}^{167}\mathrm{Er}$) $^{60}$.

2.64. Compounds of ions of the actinide group

Among compounds of this group, paramagnetic resonance has been studied in more or less detail only in certain salts of complex cations of the type $[\mathrm{M}^{\mathrm{VI}}\mathrm{O}_2]^{++}$, in particular plutonium $[\mathrm{PuO}_2]^{++}$ and neptunium $[\mathrm{NpO}_2]^{++}$. In addition, there are data for polycrystals of some paramagnetic uranium compounds of valence below 6 ($\mathrm{UF}_3$ and $\mathrm{UF}_4$) $^{60}$.

$5f^1$, neptunium ion ($[\mathrm{NpO}_2]^{++}$). The double nitrate $[(\mathrm{Np},\mathrm{U})\mathrm{O}_2]\mathrm{Rb}(\mathrm{NO}_3)_3$, $(\mathrm{Np}:\mathrm{U}=1:10)$ was investigated; in the temperature range from 20 to $12^\circ\mathrm{K}$ it was found: $g_\parallel=3.40$, $g_\perp=0.205$; $A_\parallel=0.1654\ \mathrm{cm}^{-1}$, $A_\perp=0.0178\ \mathrm{cm}^{-1}$, $p=-0.0301\ \mathrm{cm}^{-1}$. The spin of the ${}^{237}\mathrm{Np}$ nucleus is $5/2$ $^{60}$.

$5f^2$, plutonium ion ($\mathrm{PuO}_2^{++}$). The following were investigated: double nitrate and double acetate $[\mathrm{PuO}_2]\mathrm{Na}(\mathrm{CH}_3\mathrm{COO})_3$. The situation is analogous to that which has

site for $\mathrm{Pr}^{+++}$ in crystals with trigonal field symmetry. For the first salt, diluted with $\mathrm{UO}_2$ salt (1:200), it was found at a temperature from 20 to 12° K: $g_{\parallel}=5.32$, $g_{\perp}\leqslant 0.4$; $I=\dfrac{1}{2}$, $A=0.0862\ \mathrm{cm}^{-1}$ (for ${}^{239}\mathrm{Pu}$) and $I={}^{5}/_{2}$, $A=0.0609\ \mathrm{cm}^{-1}$ (for ${}^{241}\mathrm{Pu}$)$^{60}$.

§ 3. LINE SHAPE OF PARAMAGNETIC RESONANCE IN IONIC CRYSTALS

The width of the lines of paramagnetic-resonance absorption in ionic crystals is determined mainly by two factors: dipole and exchange interactions between magnetic particles (spin-spin interaction) and interactions of the system of magnetic moments of the ions (spin system) with lattice vibrations (spin-lattice interaction). These same interactions determine the spin-spin relaxation time $\tau_s$ and the spin-lattice relaxation time $\tau_l$ in paramagnets$^{5}$. The half-width of the absorption line is often taken to be equal to $\Delta\nu=\dfrac{1}{\tau_s}+\dfrac{1}{\tau_l}$; in reality, the relation between the width and the relaxation times is more complicated and cannot be expressed by such a simple formula.

For lack of space we cannot dwell here on the theory of spin-spin and spin-lattice interactions, and are compelled to give only the main results.

3.1. Spin-spin interaction

If two neighboring magnetic atoms are at a distance $r$ from one another, then each Zeeman energy level, owing to dipole interaction, will be broadened by an amount $h/(\beta^2 r^{-3})$. This may be visualized as follows. In addition to the external magnetic field $H$, each atom is acted upon by a local field $H_{\mathrm{loc}}$, produced by neighboring particles. Therefore the resonance condition takes the form: $h\nu=g\beta(H+H_{\mathrm{loc}})$. Since the average spread of possible values of $H_{\mathrm{loc}}$ is of the order of $\beta/r^3$, it is clear that for $\Delta\nu$ one obtains the value given above.

If all the magnetic particles are identical, then, in addition to the “magnetostatic” broadening mechanism just considered, another broadening mechanism—“dynamic”—will also operate. Let us consider two precessing dipoles with oppositely directed moments. Each of them will create, at the position of the other, an alternating field of resonant frequency, under the influence of which an exchange of orientations of the moments is possible, since the total energy is then conserved. The limitation of the lifetime of each particle on a definite Zeeman energy level will lead to a broadening having again, according to the uncertainty relation, a magnitude $\sim h/(\beta^2 r^{-3})$.

The calculation methods developed up to the present time make it possible to compute only the moments of the resonance-absorption curves. By the $k$-th moment of an absorption line is meant the following quantity:

\[ M_k=\int(\nu-\nu_0)^k g(\nu)\,d\nu . \tag{25} \]

For estimating the magnitude of the spin-spin interaction this method was first applied by Waller$^{4}$, and then by Broer$^{85}$. Analysis of the line shape of paramagnetic-resonance absorption by the method of moments was carried out

Van Vleck[^86]. Van Vleck’s theory is based on the following assumptions: a) the magnetism of the particles is purely spin magnetism, b) there is no ferromagnetism, c) the frequency of the oscillating field is so high that the Zeeman energy is much greater than the mean energy of the spin-spin interaction of neighboring particles, d) the exchange forces are isotropic, e) the spin system is under adiabatic conditions, there is no exchange of energy with the lattice vibrations, f) the temperature is so high that all Zeeman levels are populated equally.

Owing to dipole interactions, in addition to the main line of frequency \(g\beta H\), weak satellites will appear at the frequencies \(0\), \(2g\beta H\), and \(3g\beta H\). For the main line Van Vleck calculated the 2nd and 4th moments; Glebashев[^87] later calculated the 6th moment. The odd moments are equal to zero and, consequently, the absorption line is symmetric. The second moment was found to be

\[ M_2=\frac{3}{4}g^4\beta^4h^{-2}S(S+1)\sum_n r_n^{-6}(3\cos^2\vartheta_n-1)^2, \tag{26} \]

where \(n\) enumerates all magnetic particles of the lattice, \(r_n\) is the distance from some atom taken as the origin to the \(n\)-th atom, and \(\vartheta_n\) is the angle of \(r_n\) with the direction of \(H\). For a crystalline powder:

\[ M_2=\frac{3}{5}g^4\beta^4h^{-2}S(S+1)\sum_n r_n^{-6}. \tag{27} \]

For a simple cubic lattice whose constant is \(d\),

\[ \sum_n r_n^{-6}=8.5d^{-6}. \tag{28} \]

If the quantum-mechanical calculations are replaced by magnetostatic ones, and thus the influence of the dynamic broadening mechanism named by us is discarded, then for \(M_2\) the same expression (26) is obtained, reduced by a factor of \(9/4\). This reduced value should appear when one is dealing with broadening caused by interactions of dipoles of different kinds, for example paramagnetic atoms with nuclear spins surrounding diamagnetic particles. A calculation was also made of the 2nd moment \(\widetilde M_2\) of the absorption curve that includes not only the main line, but also the additional lines at the frequencies \(0,2g\beta H,3g\beta H\). It turned out that \(\widetilde M_2=\dfrac{10}{3}M_2\), and this relation, as Brur[^85] had already pointed out, does not depend on \(H\), since the heights of the additional absorption curves are inversely proportional to \(H\), while their frequencies are approximately linear with respect to \(H\).

Let us return to consideration of the main line of paramagnetic resonance. Isotropic exchange forces have no effect at all on the magnitude of the 2nd moment of the absorption line. Therefore, in order to judge the influence of exchange forces on the line shape, it is necessary to invoke higher moments. In the case of purely dipole interactions it turns out that the ratios of the moments are close to the values obtained for a Gaussian function, namely: \(M_6^{1/6}:M_4^{1/4}:M_2^{1/2}=1.57:1.32:1\). If the exchange interactions are much greater than the dipole interactions, then \(M_4^{1/4}:M_2^{1/2}\gg1\), and consequently the line acquires a Lorentzian shape. Since the area under the absorption curve and its 2nd moment do not change, this will mean that the line becomes sharper, becoming narrower at the center and correspondingly less steep at the edges.

It should be said that the narrowing of lines under the influence of isotropic exchange forces is a consequence of the assumptions underlying Van Vleck’s theory, which for real crystals in most cases cannot be accepted.

Price and Stevens88 developed a general method for calculating the zeroth and second moments of resonance lines, applicable also when the magnetism is not purely spin in character. In this case the isotropic exchange forces also enter into the second moment, and the question of whether they lead to narrowing or broadening of the absorption line requires special consideration. Specific calculations based on the general theory of Price and Stevens have been carried out only for fluorosilicate89 and Tutton’s nickel salt90.

Van Vleck’s theory of dipolar broadening was extended to the case of solid paramagnetic solutions91. It was found that if the concentration of paramagnetic atoms is \(f > 0.1\), the line retains a Gaussian form and its width is proportional to \(\sqrt{f}\); whereas if \(f < 0.01\), the line shape becomes Lorentzian and its width is \(\sim f\).

The method of moments has also been used to estimate the width of the line of resonant paramagnetic absorption that arises, in the absence of an external static magnetic field, from transitions between closely spaced energy sublevels92.

Finally, a number of works93 should be noted in which the form of the lines of paramagnetic resonance absorption is considered on the basis of the theory of stochastic processes.

3.2. Spin–lattice interaction

The most widely used method for calculating the magnitude of the spin–lattice interaction was given by Waller4, who at the same time carried out, for substances with purely spin magnetism and with \(S = 1/2\), detailed calculations of the probability of a change in the spin direction under the action of lattice vibrations. It was assumed that the transfer of the Zeeman energy of atoms to the lattice vibrations occurs as a result of a change, under the influence of these vibrations, in the magnetic interaction of the spins. The calculations showed that at low temperatures the principal role is played by first-order processes, consisting in the excitation of a quantum of lattice vibrations at the expense of the magnetic energy of the atoms. At high temperatures the spin–lattice interaction is determined by second-order processes, namely by combination scattering of phonons.

The values of the paramagnetic relaxation times obtained by Waller proved to be several orders of magnitude larger than those given by experiment. An especially sharp discrepancy between theory and experiment was found for titanium alums. Therefore Kronig94 proposed another mechanism of spin–lattice interaction, consisting in the modulation by lattice vibrations of the crystal’s electric field, which in turn acts on the orbital motion and, through it, on the spin of the electrons.

In this way, with the aid of rather crude estimates, he obtained the correct order of magnitude for the spin–lattice relaxation time for titanium alums. Similar but more detailed calculations were carried out by Van Vleck95 for titanium and chromium alums.

Akhiezer and Pomeranchuk120 considered the question of relaxation in paramagnetic dielectrics at low temperatures by the method of elementary excitations.

In connection with studies of paramagnetic resonance, interest has grown in the theory of spin-lattice interaction. A number of works have been devoted to compounds of rare-earth elements[^96], salts whose magnetic ions are in an \(S\)-state[^97], and salts of ions of the iron group containing an even number of electrons[^98].

Finally, a generalization and refinement of Waller’s theory was made, showing that in some cases the relaxation mechanism considered by him may play the determining role[^97].

Let us present some final results of calculations of the spin-lattice interaction. Denote by \(A_1\) the probability that, under the influence of lattice vibrations, in 1 second a magnetic particle will make a transition from one Zeeman level to another, if this involves the creation (or disappearance) of only one phonon. We shall denote by \(A_2\) the analogous transition probability due to the combination scattering of phonons (two-phonon processes). If \(\beta H \ll kT\), then for these probabilities the following expressions are obtained:

\[ A_1 = C_1 K_1 \frac{kT}{\rho v^5}, \qquad A_2 = C_2 K_2 \frac{I_n}{\rho^2 v^{10}} . \tag{29} \]

Here \(\rho\) is the density of the crystal, \(T\) its temperature, \(v\) the mean velocity of sound, \(C_1\) and \(C_2\) numerical factors, and \(K_1\) and \(K_2\) quantities depending on the structure of the energy levels of the magnetic particles and therefore also on the applied magnetic field \(H\). By \(I_n\) is denoted

\[ I_n = \int_0^{k\theta/h} \frac{\nu^n e^{h\nu/kT}}{\left(e^{h\nu/kT}-1\right)^2}\,d\nu, \tag{30} \]

where \(\theta\) is the Debye temperature. If the exchange of energy between the vibrations of the lattice and the system of spins occurs by means of Waller’s mechanism, then[^97]

\[ \left. \begin{aligned} K_2 &= z \left[\frac{g^3 \beta^2}{R^3}(2S+1)(S+1)\right]^2,\\ K_1 &= \left(\frac{g\beta H}{h^2}\right)^2 K_2; \qquad n=6, \end{aligned} \right\} \tag{31} \]

where \(R\) is the equilibrium distance between two neighboring magnetic particles, and \(z\) is the number of nearest neighbors. From formulas (31) we arrive at the entirely natural conclusion: magnetic forces will determine the spin-lattice interaction in substances with large magnetic moments of the atoms and with a high density of the latter. Such substances apparently include some crystals containing magnetic ions in an \(S\)-state.

Now suppose that the relaxation mechanism is determined by modulation of the electric field of the crystal, produced by lattice vibrations. For salts of elements of the iron group one obtains expressions of the form:

\[ \left. \begin{aligned} K_1 &= \left(\frac{\lambda}{\Delta}\right)^2 \left(\frac{r_0}{a}\right)^4 \left(\frac{ee'}{a\Delta}\right)^2 (g\beta H)^4;\\ K_2 &= h^2 \left(\frac{\lambda}{\Delta}\right)^2 \left(\frac{r_0}{a}\right)^8 \left(\frac{ee'}{a\Delta}\right)^4; \qquad n=8. \end{aligned} \right\} \tag{32} \]

Here \(\Delta\) is the interval between two lower energy sublevels arising in the electric field of the crystal, and \(r_0\) is the mean distance

the \(3d\)-electron from the nucleus, \(a\) is the equilibrium distance from the center of the magnetic particle to the nearest diamagnetic ion, and \(e'\) is the effective charge of this ion. The factor \(\left(\dfrac{\lambda}{\Delta}\right)^2\) appears in (32) because changes in the electric field of the crystal cannot directly cause a reorientation of the electron spin, but act through the orbital moment. Since the orbital motion is usually “frozen,” the matrix element of the spin-lattice interaction proves to be “composite”; it differs from zero only in higher approximations of perturbation theory. It is known from experiment that the spin-lattice interaction times may differ by several orders of magnitude for different elements. This is explained mainly by the fact that the interval \(\Delta\) may vary in going from one ion to another from \(\sim 10^2\ \mathrm{cm}^{-1}\) to \(\sim 10^4\ \mathrm{cm}^{-1}\).

In copper salts a strong anisotropy of the spin-lattice relaxation time at room temperatures has been observed \(^{100}\), which could be explained by assuming an anisotropy of the spin-orbit interaction constant \(^{101}\). If there is axial symmetry and the field \(H\) makes an angle \(\vartheta\) with the crystal axis, then in the formulas for \(A_1\) and \(A_2\), instead of \(\lambda^2\) there enters

\[ \frac{1}{3}\left[\lambda_{\parallel}^{2}+\lambda_{\perp}^{2}\left(1+\cos^{2}\vartheta\right)\right]. \]

In Section 2.1 we saw that the spectra of ions with an even number of electrons possess a number of peculiarities. In connection with this, detailed theoretical calculations of the spin-lattice interaction were undertaken for crystals with such ions \(^{99}\). Special consideration was also required \(^{97}\) for ions in an \(S\)-state. It turned out that

\[ \left. \begin{aligned} K_1 &= \frac{D^2}{h^4}(g\beta H)^2,\\ K_2 &= D^2,\\ n &= 6, \end{aligned} \right\} \tag{33} \]

where \(D\) is a parameter of the spin Hamiltonian that determines the small energy splittings caused by the crystal field (see § 2.1).

In compounds of rare-earth elements, lattice vibrations, by changing the crystal field, can directly change the direction of the moment of the paramagnetic ion, since in this case the coupling between the spin and orbital moments is stronger than the action of the electric field of the crystal. Therefore, for \(A_1\) and \(A_2\) one obtains expressions of type (32), but without the factor \(\left(\dfrac{\lambda}{\Delta}\right)^2\). Since the interval \(\Delta\) in rare-earth ions is relatively small, of the order of \(10\)–\(100\ \mathrm{cm}^{-1}\), the spin-lattice interaction proves to be very strong, despite the fact that for \(4f\)-electrons \(r_0\) is much smaller than for the valence electrons of the iron group. In ions with an even number of \(f\)-electrons, the spin-lattice interaction due to direct processes proves to be especially large if there is non-Kramers degeneracy of the ground energy level. In this case we have:

\[ K_1=\frac{1}{h^4}\left(\frac{ee'}{a}\right)^2\left(\frac{r_0}{a}\right)^4(g\beta H)^2. \tag{34} \]

Here the matrix element of the spin-lattice interaction differs from zero already in the first approximation of perturbation theory.

We have not given the values of the numerical factors \(C_1\) and \(C_2\), since the complexity of the theory of the spin-lattice interaction makes it possible to judge only the order of magnitude of this interaction.

Experimental studies concerning the shape of paramagnetic-resonance lines and the magnitude of the spin-lattice interaction are very few in number. Therefore it is hardly appropriate to dwell here on a comparison of experimental data with theory. We shall confine ourselves only to mentioning Soviet experimental work on the study of spin-spin interactions^116 and the influence of exchange forces on the width of absorption lines^135.

§ 4. ELECTROLYTE SOLUTIONS

The study of paramagnetic resonance in liquid electrolyte solutions is of interest not only for elucidating their magnetic properties, but also from the point of view of the information that can be obtained concerning certain details of their microstructure. Unfortunately, the number of works in this field is still small, although electron paramagnetic absorption in solutions of manganese salts was first observed by E. K. Zavoisky^102 as early as 1944.

Of the inorganic compounds*) in liquid solutions, chiefly salts of ions of the iron group have been studied. Water was used as the solvent, as well as ethyl alcohol, glycerin, acetone, etc. A resonance effect accessible for measurements was found in solutions containing the ions Mn++^103,104,105,108,106, VO++^104, Cu++^106,107, and Cr+++^106. In addition, there are data on the observation of the effect in solutions of salts of Gd+++^106 and \([W(CN)_8]^{----}\)^108.

The paramagnetic-resonance lines investigated in solutions are single, or else exhibit hyperfine structure. The fine-structure peaks are not resolved, although, as is known, in some polycrystals, for example in powdered chrome alum, they can be observed.

The position of the hyperfine-structure peaks under strong-field conditions is described, for solutions containing \(^{55}\mathrm{Mn}\), \(^{51}\mathrm{VO}^{++}\), and \(^{63,65}\mathrm{Cu}^{++}\), by the Hamiltonian:

\[ \mathcal{H}=g\beta \hat{H}\hat{S}+A\hat{I}\hat{S}. \tag{35} \]

The form of the spectrum for a solution of \(\mathrm{MnCl}_2\) in water is shown in Fig. 7. The values of the \(g\)-factors lie close to 2, and their exact magnitude depends essentially on the immediate environment of the ion. The hyperfine-structure constant \(A\) can also change considerably when this environment changes. Such a case occurs, in particular, in solutions of Cu++ salts. Whereas in aqueous solutions containing hydrated Cu++ ions the constant \(A\) is so small that resolution of the hyperfine structure does not occur^106, a solution of \(\mathrm{CuCl}_2\) in ethylenediamine^107 makes it possible to observe this structure easily. An analogous fact has been noted for VO++, where aqueous and water-acetone solutions give somewhat different values of \(A\)^104.

Under conditions corresponding to the Zeeman effect on the hyperfine structure in weak fields, i.e., at low frequencies of the oscillating magnetic field, in aqueous solutions of Mn++ salts a single peak with \(g=1.00\) is observed. The position of this peak is described by the formula

\[ h\nu=g_F\beta H, \tag{36} \]

*) On solutions of free radicals, see below, § 5.

where \(F\) is the quantum number of the resultant moment of the electron shell and the nucleus, and

\[ g_F=\frac{F(F+1)+J(J+1)-I(I+1)}{2F(F+1)} . \]

Indeed, for \(J=1=5/2\) for \({}^{55}\mathrm{Mn}\) we obtain \(g_F=1\). This effect, discovered in 1948[^103], was the first evidence of the influence of nuclear spin on electron paramagnetic resonance lines. The applicability of formula (36) to the description of the effect in aqueous solutions of \(\mathrm{Mn}^{++}\) salts shows that the fine splittings in this case are very small in comparison with the hyperfine ones, which is apparently due to the very high symmetry of the hydrate shell of \(\mathrm{Mn}^{++}\). Measurements in solutions of other ions in weak fields showed that there is no agreement with formula (36).

The dependence of the line width in solutions on such factors as the concentration of magnetic ions, temperature, and viscosity is very different for different ions[^109]. Thus, for not very concentrated aqueous solutions of \(\mathrm{Mn}^{++}\), \(\mathrm{Cr}^{+++}\), and \(\mathrm{VO}^{+++}\) salts, the width initially decreases upon dilution; after a certain concentration is reached, further dilution practically does not change it; the limiting width remains considerable (of the order of 30 gauss for \(\mathrm{Mn}^{++}\) and 200 gauss for \(\mathrm{Cr}^{+++}\) at room temperature). At very high concentrations (\(N>6\) moles per liter) exchange narrowing of the line is observed for \(\mathrm{Mn}^{++}\) and \(\mathrm{VO}^{++}\)[^104][^110]. In aqueous solutions of \(\mathrm{Cu}^{++}\) salts the line width does not depend on concentration and has a value of 120–140 gauss. Raising the temperature[^140] narrows the line in solutions of \(\mathrm{Cr}^{+++}\) salts; in \(\mathrm{Mn}^{++}\) solutions narrowing is observed only up to \(70^\circ\mathrm{C}\), while on further heating the line begins to broaden; in aqueous solutions of \(\mathrm{Cu}^{++}\) salts heating only broadens the line. A change in the macroscopic viscosity of the solutions affects the width more weakly than a change in temperature. On the other hand, the line width depends strongly on the nearest neighbors of the paramagnetic ion: replacement of the solvent, or complex formation, always changes it sharply. In particular, for \(\mathrm{Mn}^{++}\) salts it is sufficiently small only in aqueous solutions; in organic solvents it is so large that the effect is not observed, except at very high concentrations, where exchange narrowing of the line apparently takes place.

Fig. 7. Hyperfine structure of the paramagnetic-resonance line in a solution of \(0.5\ \mathrm{mole/l}\) \(\mathrm{MnCl}_2\) in water at \(\lambda \cong 3.2\ \mathrm{cm}\)[^14].

The totality of experimental results shows[^140] that the principal factor determining the effect in electrolyte solutions is the electrical interaction of the ion with its environment; these interactions are only weakly averaged by the thermal motion of the liquid, which permits the conclusion that the solvation shells of ions are relatively stable. Their lifetime is in any case greater than \(10^{-7}\) sec. Therefore the mechanism of spin-lattice relaxation in electrolyte solutions must, to a certain degree, be similar to the mechanism in solids.

Measurements of paramagnetic resonance in solutions make it possible to judge the symmetry of the electric fields acting on the ion from the solvation shell[^140]. Thus, in aqueous solutions of \(\mathrm{Cu}^{++}\) salts this symmetry is cubic with an admixture of a trigonal component; in solutions of \(\mathrm{VO}^{++}\) salts the symmetry is low (the magnetic complex is not octahedral); in aqueous

in solutions of Mn++ salts the symmetry is very high, approaching spherical; in nonaqueous solutions of the same ion there is a strong symmetry field lower than cubic, etc.

In addition, measurements of the line width and the \(g\)-factor make it possible to study processes of formation of chemical complexes in solutions^140. Finally, from the magnitude of the \(g\)-factors for certain ions (for example, Cu++) one can estimate the degree of covalency of the chemical bonds between the ion and the molecules forming a magnetic complex with it.

An exhaustive theory of relaxation phenomena in electron resonance in electrolyte solutions has not yet been given. The existing attempts^111 cannot explain all aspects of the phenomenon.

The behavior of the inorganic free radical ion \(\mathrm{ON(SO_3)_2^{--}}\) in solutions^112 proved sharply different from the effects considered.

In this case the limiting line width upon dilution is only \(\sim 0.3\) gauss, i.e., 2–3 orders of magnitude smaller than in solutions of ions of the iron group. This shows that electric interactions are here entirely insignificant for the effect, which is determined mainly by magnetic dipole interactions. The theory of these interactions in liquids, developed by Bloembergen, Purcell, and Pound^113 for proton resonance, therefore proved applicable to the present case as well.

In conclusion, let us note that paramagnetic resonance has also been investigated in some supercooled solutions^114,115,99 (glasses). In this connection, for the ions \({}^{51}\mathrm{VO}^{++}\), \({}^{53}\mathrm{Cr}^{+++}\), \({}^{55}\mathrm{Mn}^{++}\), and \({}^{63,65}\mathrm{Cu}^{++}\) a hyperfine structure of the absorption lines was observed, anisotropic for \(\mathrm{VO}^{++}\) and \(\mathrm{Cu}^{++}\); in the case of \({}^{55}\mathrm{Mn}^{++}\) it is isotropic, and at low frequencies a peak with \(g = 1\) is found, as in aqueous solutions of this ion, but broader. One of the results of work in this field was the establishment^139 of the spin value \(I = \tfrac{1}{2}\) for the \({}^{57}\mathrm{Fe}\) nucleus. It is interesting that an attempt to measure this spin in a single crystal of selenium alums containing \({}^{57}\mathrm{Fe}\) was unsuccessful^71.

§ 5. FREE RADICALS

One of the most interesting applications of paramagnetic resonance is the study, with its aid, of free organic radicals, i.e., molecules in which at least one electron has an uncompensated spin. Here we are compelled to confine ourselves only to a brief indication of the principal results obtained in this direction. A review of work up to 1955 is contained in the article^117.

Paramagnetic resonance was first observed in 1947 in the case of pentaphenylcyclopentadienyl \((\mathrm{C}_{35}\mathrm{H}_{25})\)^118, and it was established that this radical possesses purely spin magnetism. Beginning in 1949, systematic investigation of this class of substances began. Perhaps the best studied of them is diphenylpicrylhydrazyl

which, owing to its high chemical stability, has found wide application as a standard substance in experiments on paramagnetic resonance. By the present time a great variety of types of free radicals have been investigated, for example various

semiquinones

\[ \left( \text{of the type } \left[ \begin{array}{c} \mathrm{O}^{-} \\ \begin{array}{cc} \mathrm{R} & \mathrm{R} \\ & \\ \mathrm{R} & \mathrm{R} \end{array} \\ \mathrm{O} \end{array} \right] \right), \]

negative hydrocarbon ions (of the type, for example, \([\mathrm{C}_{10}\mathrm{H}_8]\)), biradicals and molecules in an excited triplet state, etc.

An essential feature of free radicals is the great closeness of the \(g\)-factor to its value for a free electron, i.e., practically purely spin magnetism. The absence of orbital magnetism may be due either to the fact that the free-radical molecule has low symmetry, as a result of which the orbital degeneracy is completely removed, or, if the symmetry of the molecule is high, to the Jahn–Teller effect.

A second feature, characteristic of almost all free radicals, is the extreme narrowness of the paramagnetic-resonance lines: their width is usually of the order of 1 gauss. This value is about 100 times smaller than that calculated from magnetic dipole interactions without taking exchange into account. Thus, in free radicals we have an example of systems with enormous exchange forces. In accordance with this, their absorption line has a Lorentzian form, and the width is determined by spin-lattice interactions \(^{119}\). The narrowness of the lines makes the intensity of the resonance effect in free radicals very large, which facilitates their detection. With modern apparatus it is possible to observe this line with as few as \(10^{12}\) free-radical molecules. Therefore paramagnetic resonance is the best of the existing methods for detecting free radicals. At present, attempts are being widely made to use this method to prove the existence of unstable free radicals formed as intermediate products in the course of chemical reactions. In some cases these attempts have led to successful results. However, many difficulties still stand in the way of such investigations and, apparently, for a number of reactions the sensitivity of modern apparatus is still insufficient. The lifetime of unstable radicals is probably very short, which should lead, on the one hand, to an excessively low concentration of them and, on the other, to a broadening of the absorption line, as a result of which the intensity of the effect \((\chi''_{\max})\) should become still smaller.

Of special interest for chemistry is the study of paramagnetic resonance in solutions of free radicals. It has turned out that, at sufficient dilution, when exchange interactions are practically completely eliminated, the paramagnetic-resonance line exhibits a very complex hyperfine structure, sometimes consisting of many tens of peaks. The occurrence of such a structure is explained by partial (and in some cases complete) delocalization of the unpaired electron, interacting with the total spin \(I\) of several atomic nuclei that are part of the molecule. Thus, in a dilute solution of diphenylpicrylhydrazyl, 5 hyperfine-structure lines are observed with a relative intensity of \(1:2:3:2:1\). Since the spin of the \({}^{14}\mathrm{N}\) nucleus is equal to 1, the number of peaks and the ratio of their intensities will agree with experiment if we assume that the density of the unpaired

the electron is chiefly distributed uniformly between both N atoms.

In a solution containing the semiquinone ion,

[structural diagram of the semiquinone ion: a six-membered ring with O⁻ at the top, O at the bottom, and four ring hydrogens]

five peaks are also observed, but with an intensity ratio of \(1:4:6:4:1\); this shows that the electron spin interacts with the spins of four protons, i.e., that the electron density is distributed over the entire aromatic ring (let us recall that the nucleus \({}^{12}\mathrm{C}\) has spin equal to zero and therefore does not affect the hyperfine structure).

If the electron does not interact equally with all nuclear spins, the spectral pattern becomes more complicated. In particular, if the hyperfine-interaction constant with one group of nuclei proves to be much larger than the interaction constant with another group, then each line of the hyperfine structure arising as a result of the stronger interactions splits into \((2I+1)\) closely spaced components due to the weak interactions.

Measurement of paramagnetic resonance in solutions of free radicals is one of the most direct methods for investigating electron delocalization, although the theory of the hyperfine structure of the lines for this case cannot be regarded as complete. In particular, the very nature of the hyperfine interactions in aromatic free radicals is not entirely clear. Direct interaction between the unpaired \(\pi\)-electron and the ring protons is impossible, since the latter lie in the plane of the ring, where the density of the \(\pi\)-electron cloud is zero. An attempt to explain the effect by considering vibrations of the protons normal to the plane of the ring was unsuccessful. It was therefore assumed\({}^{121}\) that in reality the unpaired electron has an admixture of the \(\sigma\)-state. Another explanation is connected with the assumption of a Dirac exchange interaction between the unpaired electron and the electrons forming the C—H bond in the ring.

In some cases, from paramagnetic-resonance data it has been possible to calculate the density of the electron cloud on hydrogen atoms. Thus, in the radical of the naphthalene ion

[structural diagram of the naphthalene-ion radical, with positions labeled \(\alpha\), \(\alpha'\), and \(\beta\)]

for \(\alpha\) atoms it is \(0.010\), for \(\beta\) it is \(0.0035\); the sum of the densities on all atoms is \(0.054\), if the density on the nucleus of an H atom in the unexcited state is taken as unity\({}^{122}\).

Studies of free radicals in biological objects by the method of paramagnetic resonance appear very promising. In particular, paramagnetic resonance has been detected in green leav-

of plants; after they had been kept for a long time in the dark the intensity of the effect fell fivefold, while upon subsequent illumination, which restored the normal green color to the leaves, the effect increased to its initial value ^123. There is no doubt that further investigations of this kind will make it possible to obtain important information about various biochemical processes.

In conclusion to this paragraph, let us mention that paramagnetic resonance has been discovered ^124 and is being intensively investigated in various carbon-containing substances (coal, charcoal and animal charcoal, resins, etc.). The linewidth in these substances is often quite small; in anthracite, for example, it has a value from 0.3 to 0.7 gauss ^125. The paramagnetic centers in all these substances are free radicals or “broken bonds” between carbon atoms. Oxygen and other paramagnetic gases have a strong influence on the effect in coals. This influence apparently arises because the molecules of these gases absorbed by the coal cause a shortening of the relaxation time.

§ 6. CONCLUSION

As we indicated at the beginning of the article, paramagnetic resonance, under suitable conditions, can be observed in any substances containing unpaired electrons. We have considered only the most theoretically and experimentally studied types of such substances: ionic crystals, electrolyte solutions, and free radicals. In our review there remained entirely unexamined a comparatively small number of works devoted to effects observed in metals (see, for example, ^127, ^128), semiconductors ^129, and gases ^130.

In conclusion we would like to dwell briefly on what the study of paramagnetic resonance provides and what the further prospects are for the development of this new field of science.

In ionic crystals, the magnetic and mechanical moments of the electron shells of atoms are determined from the number and positions of absorption lines. From the fine splittings of a line, the structure of the lower energy levels of magnetic atoms is established; knowledge of it, in turn, makes it possible to determine such constants, of interest for low-temperature physics, as magnetic susceptibility and heat capacity. From the values of the \(g\)-factors and from the anomalous hyperfine structure of the absorption lines one can judge the role of covalent bonds in magnetic complexes. From data on the shape and width of a line one can estimate the magnitude of magnetic dipole and exchange interactions; the saturation phenomenon makes it possible to measure the magnitude of the spin-lattice interaction. Finally, with the aid of paramagnetic resonance it is possible to study defects of crystal lattices (formation of \(F\)- and \(V\)-centers).

In liquid ionic solutions one can establish the degree of stability and the character of the symmetry of the solvation shells of ions, and also study complex-formation processes.

Both in ionic crystals and in solutions, from the hyperfine structure of the lines the spins can be determined and the magnetic and quadrupole moments of nuclei estimated.

In solutions of free radicals, analysis of the hyperfine structure makes it possible to establish the character of the delocalization of unpaired electrons. The possibility of detecting, by means of the paramagnetic-resonance effect, very small quantities of free radicals often permits one to study their formation in the course of chemical reactions. Of great interest is the investigation, by means of paramagnetic resonance, of the influ-

irradiation by X-rays, $\gamma$-quanta, neutrons, etc., on various substances, in particular organic ones.

In metals, from the deviation of the $g$-factor from the value for a free electron and from the shape of the lines, one can draw interesting conclusions about the nature of the interactions of conduction electrons with one another and with the framework of the crystal lattice. Analogous data can also be obtained for semiconductors. Finally, it is necessary to note an important consequence following from the study of the interactions of conduction electrons with the magnetic moments of atomic nuclei. The Overhauser effect,¹³¹ based on these interactions, serves as one of the methods of nuclear polarization. It was examined in detail by Khaikin.¹³²

In addition to the various aspects of paramagnetic resonance that have been studied experimentally to one degree or another, there are phenomena whose existence has so far been predicted only theoretically. These include the phenomenon of resonant absorption of ultrasound by paramagnets, which is, in a certain sense, an inverted paramagnetic resonance. The theory of this phenomenon,¹³³ developed for many types of paramagnets under the assumption of different mechanisms of coupling between the spin system and lattice vibrations, has shown that both electron and nuclear acoustic resonances are in some cases quite accessible to observation. Indeed, this effect has been established experimentally for some nuclei by means of the indirect method of saturation by ultrasound.¹³⁴

Recently a theory has begun to be developed for magnetic resonance due to transitions between neighboring hyperfine sublevels of energy of paramagnetic atoms. This effect may be regarded as intermediate between purely electronic and purely nuclear paramagnetic resonance. In substances containing paramagnetic ions with an even number of electrons, an effect associated with transitions between hyperfine sublevels of singlet electronic levels is possible.¹³⁶ The study of these theoretically expected phenomena will substantially supplement the data obtained by means of ordinary paramagnetic resonance.

Besides purely scientific results, the investigation of paramagnetic resonance promises to yield certain valuable technical applications. Thus, the very narrow absorption lines found in certain paramagnets, for example in irradiated sugar, can be used to measure magnetic-field strengths.¹³⁷ Further, a very promising proposal¹³⁸ is to use paramagnetic resonance in nickel fluorosilicate and certain other salts under saturation conditions at helium temperatures for constructing microwave-oscillation generators, amplifiers, and frequency converters with an exceptionally low noise level. Apparatus based on this principle will undoubtedly be very valuable for radio astronomy, as well as for solving many other radio-engineering problems.

Finally, we point out that the high sensitivity of the paramagnetic-resonance method makes it possible to apply it, alongside other spectral methods, to the chemical analysis of certain substances.

All that has been set forth shows that the discovery and investigation of paramagnetic resonance has not only greatly broadened the boundaries of the science of magnetism, but has also provided a new and very valuable method for solving the most diverse problems of solid-state physics, the theory of liquids, nuclear physics, chemistry, and biology. There is also no doubt that in the very near future this phenomenon will be used for purely technical purposes as well. The discovery of paramagnetic resonance is an outstanding achievement of Soviet physics, which will undoubtedly attract the attention of ever wider circles of scientists and engineers.

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