MODERN THEORY OF MAGNETISM*)

S. V. Vonsovsky

II. MAGNETISM OF MATTER—WEAKLY MAGNETIC BODIES

CONTENTS

1. Phenomenological description of the magnetic properties of matter.
2. Classification of magnetic substances according to experimental data.
3. Diamagnetism. 7. Magnetic properties of superconductors. 8. Paramagnetism.
4. Magnetic cooling.

4. PHENOMENOLOGICAL DESCRIPTION OF THE MAGNETIC PROPERTIES OF MATTER

All macroscopic bodies around us, without exception, are magnetically active to one degree or another, i.e., they are magnetics. This was to be expected by virtue of the very fact of their atomic structure, since, as was set forth in detail in Part I, atoms and their constituent parts—electrons and nuclei—are elementary carriers of magnetic moment.

Before setting forth the content of the modern microscopic explanation of the magnetic properties of macroscopic bodies from the point of view of their atomic structure, it is necessary, at least in a very condensed form, to recall the basic points of a purely external phenomenological description of these properties and to give a certain classification of magnetic substances.

In order to say anything about the magnetic properties of a given body, we must first of all study its behavior under the action of magnetic forces from another magnetic substance or from an electric current flowing through a conductor. Thus, from the very beginning, it is necessary to establish the nature of the magnetic forces created by currents and magnetic substances.

a) Magnetic actions of an electric current. An electric current flowing through a conductor creates a magnetic field in the surrounding space. The magnitude of this field is determined by the strength of the current in the conductor and also depends on the distance. If the dis—

*) Continuation. See Uspekhi Fizicheskikh Nauk, Vol. XXXV, No. 4, p. 514.

If one considers a small element of length \(dl\) of a thin conductor carrying a current \(i\), then the intensity, or field strength, of the magnetic field at a point at a distance \(r\) from this element in a direction making an angle \(\Theta\) with the direction of the element \(dl\) is customarily called the quantity

\[ dH=\frac{i\,dl\sin\Theta}{cr^{2}} \qquad \text{(Biot–Savart law).} \tag{4.1} \]

The direction of this field vector is perpendicular to the plane formed by the vectors \(dl\) and \(\mathbf r\). The magnetic field of a conductor of finite length and thickness is obtained as the result of a simple addition of the vectors \(d\mathbf H\). If, instead of the current strength \(i\), one introduces the current-density vector \(\mathbf j\), then the total field of the current in a finite conductor is, by virtue of (4.1),

\[ \mathbf H=\int d\mathbf H=\int \frac{[\mathbf{jr}]}{cr^{3}}\,d\tau, \tag{4.2} \]

where \(d\tau\) is an element of the volume of the conductor, and the integration extends over its entire volume. However, such an integral expression is not always convenient. More often one uses the differential equations, which can be obtained from (4.2) and, in the case of a constant magnetic field, have the form:

\[ \operatorname{div}\mathbf H=0,\qquad \operatorname{rot}\mathbf H=\frac{4\pi}{c}\,\mathbf j. \tag{4.3} \]

The physical meaning of the first of equations (4.3) consists in the assertion of the absence of true magnetic charges, while the second equation states that around each current line there are created “vortices” (closed lines of force) of the magnetic field.

b) Magnetic actions of magnetic substances. Historically, the phenomenon of magnetism was discovered already in antiquity, but not in the form of the magnetic field of an electric current, rather in the form of the magnetic field of so-called natural permanent magnets. For a long time it was believed that the sources of the magnetic field of magnetic substances are special magnetic charges of two signs. However, experiment showed that it is impossible to isolate these magnetic charges of different signs. Therefore the elementary magnetic particle came to be regarded as a magnetic dipole—a system of two magnetic charges equal in magnitude and opposite in sign, inseparably connected (poles). The principal quantitative characteristic of a magnetic dipole is the vector of its magnetic moment, which may conventionally be represented as the product of the “magnetic charge” \(m\) by the vector of the dipole “length” \(\mathbf l\), i.e.,

\[ \boldsymbol{\mu}=m\mathbf l. \tag{4.4} \]

The magnetic properties of a body are determined by the collective behavior of such elementary magnetic dipoles. After Oersted’s discovery (1821) of the magnetic field of an electric current, the hypothesis was advanced (Ampère) that these elementary magnetic dipoles are—

are molecular circular currents. It can be shown that an elementary circular current behaves in an external magnetic field exactly like an elementary magnetic dipole (the theorem of equivalence of magnets and currents). In this case, for complete quantitative agreement it is necessary that the following equality hold:

\[ \boldsymbol{\mu}=iS\mathbf{n}_0, \tag{4.5} \]

where \(i\) is the strength of the circular current, \(S\) is the area enclosed by the current, and \(\mathbf{n}_0\) is the unit vector normal to this area. Thus,

Fig. 20. Magnetization curves of weakly magnetic substances.

the product of the strength of an elementary circular current by the area enclosed by it is equal to the absolute magnitude of the magnetic moment of this current (see (2.3)). To characterize the magnetic state of a macroscopic specimen it is convenient to choose the magnitude of the resultant magnetic moment referred either to unit volume, or to unit mass, or to one gram. The resultant magnetic moment of unit volume is commonly called the vector of magnetization \(\mathbf{I}\):

\[ \mathbf{I}=\sum_{\text{unit vol.}}\boldsymbol{\mu}. \tag{4.6} \]

Experiment shows that, in various bodies, the magnetization is a function of the external magnetic field. For some substances, within a certain range of fields and temperature, this dependence has a simple linear character, i.e.,

\[ \mathbf{I}=\chi\mathbf{H}. \tag{4.7} \]

The coefficient of proportionality \(\chi\) is called the magnetic susceptibility. If the susceptibility is negative \((\chi < 0)\), then such substances are called diamagnetics.

The absolute value of the susceptibility of diamagnetics is, as a rule, very small \((\sim 10^{-6})\). Typical representatives of these substances are inert gases, many organic compounds, and a number of metals. Substances with positive susceptibility \((\chi > 0)\) are called paramagnetics. Their susceptibility is also small \((\sim 10^{-3}—10^{-6})\). Typical paramagnetics are gases—molecular oxygen \((\mathrm{O}_2)\), nitric oxide \((\mathrm{NO})\),—numerous salts of the rare earths and of the elements of the iron group, and the alkali metals.

Fig. 21. Magnetization curve \((OB)\) and hysteresis loop \((BCDEFGB)\) of a ferromagnetic substance.

Alongside these so-called weakly magnetic bodies there exists a number of substances (iron, nickel, cobalt, gadolinium, their compounds and alloys, alloys of chromium and manganese) for which the magnetization, as a rule, is not a linear function of the field; rather, a more complex functional nonlinear relation exists, and, moreover, it is not single-valued. It is precisely to this class of substances, called ferromagnetics, that permanent magnets (pieces of magnetite ore), discovered already in antiquity, belonged.

The relation between magnetization and field is represented graphically by the so-called magnetization curve. In the case of dia- and paramagnetics these curves have the form of straight lines (Fig. 20), whereas for ferromagnetics the curve has a much more complicated form (Fig. 21). From Fig. 21 it is seen that the magnetization curve has a steep rise and then reaches saturation. The magnitude of this saturation \(I_s\) decreases with increasing temperature, and above a certain temperature \(\Theta\) (the Curie point) for the given material all ferromagnetic properties disappear altogether, and the ferromagnetic becomes a paramagnetic with a linear magnetization curve. In the case of ferromagnetics one may retain the concept of magnetic susceptibility either as the ratio \(I/H\) (total susceptibility) or as the derivative \(dI/dH\) (differential susceptibility). However, for ferromagnetics these magni-

...cease to be constant, and depend in a complicated way on the magnitude of the magnetic field. From Fig. 22 it is seen that the curve \(\chi_{\mathrm{dif}}(H)\) begins at \(H=0\) from a certain finite value \(\chi_a\) (initial susceptibility), reaches the greatest value \(\chi_{\max}\) (maximum susceptibility), corresponding to the steepest rise of the curve \(I(H)\), and then tends to zero as the magnetization tends to saturation \(I_s\). The ambiguity of the relation between \(I\) and \(H\) in a ferromagnet appears when one attempts to demagnetize a specimen magnetized to saturation. If one begins to decrease the field which has brought the specimen to saturation \(+I_s\), then the magnetization, beginning with some value of the field, will “lag behind” in its decrease relative to the decrease of the field, and at \(H=0\) will differ from zero. A residual magnetization \(I_R\) appears in the specimen, not equal to zero, but somewhat less than saturation \((I_R \leq I_s)\).

Fig. 22. Curve of the magnetic susceptibility \(\chi_d(H)\) of a ferromagnetic substance.

In order to make the magnetization of the alloy equal to zero, it is necessary to apply an opposite magnetic field of magnitude \(-H_C\) (the so-called coercive force). A further increase of the negative field will ultimately bring the specimen to saturation in the opposite direction \((-I_s)\). Returning back (decreasing the negative field), at \(H=0\) we obtain \(I=-I_R\), at \(H=+H_C\), \(I=0\), and then again obtain saturation \(+I_s\). Thus, in a complete cycle we describe a closed curve, which is called the magnetic hysteresis loop (Fig. 21).

Along with the vectors \(\mathbf I\) and \(\mathbf H\), another vector is introduced—the magnetic induction

\[ \mathbf B=\mathbf H+4\pi\mathbf I . \tag{4.8} \]

Substituting \(\mathbf I\) from (4.7), we obtain:

\[ \mathbf B=(1+4\pi\chi)\mathbf H=\mu\mathbf H . \tag{4.8'} \]

The quantity \(\mu=1+4\pi\chi\) is called the magnetic permeability. Formula (4.8′) is also valid for ferromagnets, if \(\mu\) is regarded as a function of the field. In diamagnets \(\mu<1\), while in paramagnets \(\mu>1\).

In ferromagnets, where the magnitude of the magnetization is comparable with, and sometimes greater than, the field itself (see Fig. 21), one must take into account one more circumstance. Let us consider, for example, a ferromagnetic...

body in the form of a rectangular bar. Let us place it in an external magnetic field \(\mathbf{H}_e\), directed along an edge of the bar. Then the bar will be magnetized in this direction up to the value \(\mathbf{I}\) (for simplicity we shall assume that the magnetization is uniform throughout the volume of the bar). In this case, on the end surfaces there will arise so-called “poles.” The density of the magnetic charge on these poles will be equal to \(+I\) or \(-I\). These “charges,” together with the external field \(\mathbf{H}_e\), will create their own magnetic field \(\mathbf{H}_0\), which inside the bar will be directed opposite to the external field \(\mathbf{H}_e\) and opposite to the magnetization \(\mathbf{I}\), i.e., it will play the role of a demagnetizing field. The total true field inside the bar will be equal to

\[ \mathbf{H}=\mathbf{H}_e+\mathbf{H}_0 . \tag{4.9} \]

In many cases it may be approximately assumed that the magnitude of the demagnetizing field \(H_0\) is proportional to the magnitude of the magnetization and opposite to it in sign, i.e.,

\[ \mathbf{H}_0=-N\mathbf{I}, \tag{4.10} \]

where \(N\) is the demagnetizing factor, which is determined, approximately, only by the geometrical shape of the specimen. From (4.10) and (4.9) we find:

\[ \mathbf{H}=\mathbf{H}_e-N\mathbf{I}. \tag{4.11} \]

But, by virtue of (4.7), we have:

\[ \mathbf{H}=\frac{1}{(1+\chi N)}\mathbf{H}_e \quad \text{and} \quad \mathbf{I}=\chi\mathbf{H}=\frac{\chi}{1+\chi N}\mathbf{H}_e=\chi_0\mathbf{H}_e . \tag{4.12} \]

Here \(\chi\) is called the magnetic susceptibility of the substance, for it is determined only by the physical nature of the given material, while \(\chi_0\) is called the magnetic susceptibility of the body, since it depends not only on the nature of the material (through \(\chi\)), but also on the shape of the body (through \(N\)). Analogous definitions may be made also for permeability (\(\mu\) is the permeability of the substance and \(\mu_0\) is the permeability of the body). The most complete discussion of the question of the various methods of defining magnetic permeabilities was given by V. K. Arkad’ev^27.

Arkad’ev^27 also developed a general macroscopic theory of the electromagnetic field in ferromagnetic metals.

c) Foundations of the thermodynamics of magnets. The fundamental equation of the first and second laws of thermodynamics for infinitely small reversible processes, as applied to magnets, taking into account the change of their magnetic state, has the form:

\[ \delta U=T\delta S-p\delta V+\frac{1}{4\pi}\mathbf{H}\delta\mathbf{B}, \tag{4.13} \]

where \(\delta U\) is the change in total energy, \(\delta S\) the change in entropy, \(\delta V\) the volume of the magnet, \(p\) the external pressure, and \(T\) the temperature. The last term on the right-hand side of (4.13) is equal to the work of the magnetic forces.

In an adiabatic-isochoric process \((\delta S=\delta V=0)\) this quantity is equal to the change in the total energy of the magnet. With another choice of independent variables (for example, \(T, V, I\) or \(T, p, H\)), as the basic thermodynamic potential instead of the total energy \(U\) it is more convenient to consider the “free energy”

\[ F=U-\frac{1}{8\pi}H^2-TS \tag{4.14} \]

or the “thermodynamic potential”\(^*\)

\[ \Phi=F+pV-\mathbf{I}\mathbf{H}. \tag{4.15} \]

With such a choice, (4.13) accordingly takes the form:

\[ \delta F=-S\delta T-p\delta V+\mathbf{H}\delta\mathbf{I}, \tag{4.16} \]

\[ \delta\Phi=-S\delta T+V\delta p-\mathbf{I}\delta\mathbf{H}. \tag{4.17} \]

Using the basic differential relations of thermodynamics, one can obtain general relations between various quantities characterizing the macroscopic properties of a magnet. Knowing any one of the thermodynamic potentials as a function of its corresponding variables, for example \(\Phi(T,p,\mathbf{H})\), one can determine the relation between the magnetization and the field, i.e., the “magnetic equation of state.” In particular, from (4.17) we obtain:

\[ I_x=-\frac{\partial\Phi}{\partial H_x},\qquad I_y=-\frac{\partial\Phi}{\partial H_y},\qquad I_z=-\frac{\partial\Phi}{\partial H_z}. \tag{4.18} \]

If the magnet is isotropic, then the directions of the vectors \(\mathbf{I}\) and \(\mathbf{H}\) coincide, and (4.18) can be written in the simpler form:

\[ I=-\frac{\partial\Phi}{\partial H}. \tag{4.19} \]

The connection between the thermodynamic relations set forth above and atomic theory is established with the aid of the formulas of statistical mechanics, which gives a relation between the thermodynamic potential and the value of the phase sum \(Z\), which is computed from the known microstates of the system. This relation has the form

\[ \Phi=-kT\ln Z, \tag{4.20} \]

where

\[ Z=\sum_n g_n e^{-\frac{\varepsilon_n}{kT}}. \tag{4.21} \]

\(^*\) Strictly speaking, the free energy in the usual definition is equal to \(F=U-TS\), and the thermodynamic potential \(\Phi=U+pV\). It should, however, be emphasized that the choice of the basic thermodynamic potential is not physically essential and is determined only by the choice of independent variables.

In (4.21) the summation is over all microstates of the system with energies \(\varepsilon_n\) and statistical weights \(g_n\) (\(k\) is Boltzmann’s constant). Here it should be pointed out that carrying out such a program proved to be fundamentally impossible within the framework of classical statistics. As Bohr and Lorentz showed for special cases, and Van Leeuwen \(^{28}\) and Terletsky \(^{29}\) in general form, the magnetic moment of any magnet considered as a classical ensemble of moving elementary charges placed in an external constant magnetic field is, in the stationary state, exactly equal to zero. The positive result obtained for a number of cases in the classical statistical theory of magnetism (for example, the theory of paramagnetism) is always the result of an implicit allowance for the quantum nature of the atom (the stability of stationary states, etc.), which is foreign to the laws of classical physics. Only taking into account the quantum nature of atomic laws made it possible to remove this fundamental difficulty of classical attempts to construct an atomic theory of magnetism and to build a unified quantum theory of magnetism on the basis of the general laws of the microworld.

5. CLASSIFICATION OF MAGNETICS ACCORDING TO THE BASIC EXPERIMENTAL DATA

As already indicated above, according to their magnetic properties magnetics can be divided, more or less distinctly, into three main classes: diamagnetics, paramagnetics, and ferromagnetics. However, each of these three classes can in turn be subdivided into subgroups, depending on the finer features of their magnetic properties.

Diamagnetics

a) “Classical” diamagnetics: all inert gases, a number of metals (zinc, gold, mercury, etc.), nonmetallic elements (silicon, phosphorus, sulfur, etc.), and many organic compounds. Their susceptibility has a normal small value \(\sim - (0.1—10)\cdot 10^{-6}\) and practically does not depend on temperature (in some cases jumps occur, for example at the melting temperature).

b) “Anomalous” diamagnetics: bismuth, gallium, antimony, graphite, iodine, gallium, \(\gamma\)-phases of alloys of systems of the Cu–Zn type, and a number of others. The atomic susceptibility of these diamagnetics is 10–100 times greater than in the classical ones and depends on temperature. In addition, for example in bismuth there is a further series of anomalies: the susceptibility is a periodic function of the field, depends especially strongly on temperature, etc.

c) Superconductors: a number of metals at very low temperatures (from \(1^\circ\) K to \(\sim 10^\circ\) K) possess anomalous electrical and magnetic properties. In particular, when passing through

S. V. VONSOVSKII

...electric current from an extraneous e.m.f. they behave like ideal conductors with zero resistance (hence the term superconductors). However, apparently, the primary property of superconductors is their special magnetic nature. Experiment shows that inside a superconductor (with the exception of a very thin, \(\sim 10^{-5}\) cm, surface layer) the magnetic induction is always equal to zero, i.e. \(\mathbf{B}=0\). This condition can, at first glance, on the basis of (4.7), be reduced to the condition that

\[ \mathbf{I}=-\frac{1}{4\pi}\mathbf{H}. \]

Thus, it would seem to follow that superconductors are at the same time also superdiamagnetics (or ideal diamagnetics) with a colossal susceptibility (in comparison with normal diamagnetism)

\[ \chi_{\text{s.p.}}=-\frac{1}{4\pi}. \]

It must be noted, however, that such an identification of a superconductor and a superdiamagnetic has only a formal character. The point is that a superconductor is a body over whose surface a macroscopic current circulates. This current screens the bulk of the superconductor from magnetic fields, and therefore there always \(\mathbf{B}=\mathbf{H}=0\) (apart from the surface layer), while the permeability \(\mu\) has its ordinary value, close to unity. There is no diamagnetic magnetization inside the superconductor. It can be conditionally likened only to a single atom or molecule of a diamagnetic substance, in which the role of the precession of the electron shells is played by a surface macroscopic current\({}^{30}\).

Paramagnetics

a) Normal paramagnetism of free atoms, ions, or molecules possessing a resultant moment different from zero. Typical representatives of this group are, for example, gases, \(\mathrm{O}_2\), NO, platinum, palladium, rare earths, salts of iron, cobalt, and nickel, as well as these ferromagnetic metals themselves at sufficiently high temperatures. The susceptibility of these substances depends on temperature according to Curie’s law

\[ \chi=\frac{C}{T} \tag{5.1} \]

(\(C\) is the Curie constant). However, few paramagnets obey this law exactly (\(\mathrm{O}_2\), NO, gadolinium sulfate); the majority of them obey the Curie–Weiss law

\[ \chi=\frac{C'}{T-\Delta}, \tag{5.2} \]

where the constant \(\Delta\) may be either greater or less than zero. In very strong external fields and at sufficiently low temperatures the magnetic properties of these paramagnets become strongly complicated (saturation phenomena, cryomagnetic anomalies).

b) Paramagnetism of metals, independent of temperature: typical representatives of this group are the alkali metals (Li, Na, K, Rb). They are very weakly magnetic \((\chi \sim 10^{-7} — 10^{-6})\), the dependence of their magnetization on the field is strictly linear, and the susceptibility is almost independent of temperature (at least up to the melting point).

c) Antiferromagnets: this type includes crystalline bodies made of elements of the transition groups or of their chemical compounds and alloys. The possibility of such a type of paramagnets follows from the modern theory of ferromagnetism (see § 10). All of them possess a certain critical temperature (the antiferromagnetic Curie point \(\Theta_{\mathrm{a.f.}}\)), above which they behave almost like normal paramagnets obeying the Curie–Weiss law with \(\Delta < 0\). Below this temperature their susceptibility again decreases and tends to zero as \(T \to 0^\circ\mathrm{K}\), i.e., at \(T = \Theta_{\mathrm{a.f.}}\) the susceptibility has a maximum; here too maxima of anomalies of other properties of antiferromagnets (for example, heat capacity) are observed. To this group of magnets belong compounds of manganese (MnO, MnS), chromium (NiCr, CrS — Cr\(_2\)S, Cr\(_2\)O\(_3\)), vanadium (VO\(_2\)) and iron (FeS\(_2\), FeS — FeS\(_2\)).

d) Metamagnetics. It was recently discovered that in paramagnetic salts of the type CoCl\(_2\), FeCl\(_2\), CaCl\(_2\) there is a series of anomalies at very low temperatures. Namely, at relatively small values of the magnetization (in comparison with ferromagnets) the susceptibility of these salts shows a sharp dependence on the field and on temperature, and the phenomenon of magnetic hysteresis, typical of ferromagnets, is observed.

Ferromagnets

This group of substances has, qualitatively, more homogeneous properties. For all of them the form of the magnetization curve and the hysteresis loop shown in Fig. 21 are characteristic. They may be divided according to a purely quantitative criterion: ferromagnets for which the area of the loop is not very large and the coercive force does not exceed tens of oersteds are customarily called soft ferromagnetic substances, while ferromagnets with a wide hysteresis loop and with a coercive force of hundreds and even thousands of oersteds are called hard or high-coercivity ferromagnetic materials.

6. DIAMAGNETISM

a) Theory

The cause of the diamagnetic properties of matter is the effect of electromagnetic induction of elementary molecular currents, produced in atomic electron shells by an external magnetic field. Therefore the phenomenon of diamagnetism is universal and is inherent in all bodies without exception. However, in many cases we do not observe this

phenomenon; for a weak diamagnetic effect may be masked by a stronger paramagnetic magnetization (see § 8).

Diamagnetism is observed in all those cases where atoms, ions, or molecules have no resultant magnetic moment, i.e., are in \(S_0\) or \(\Sigma_0\) states (see § 2). Substances composed of such particles, when placed in an external magnetic field, lose their magnetic neutrality. The magnetic field produces an induction effect and the electron shell acquires an additional angular velocity of Larmor precession (see (2.23)) and the corresponding magnetic moment, directed opposite to the field and equal, according to (2.3), to

\[ \Delta \mu=-\frac{e^2 S}{4\pi m c^2}\,H . \tag{6.1} \]

Replacing in (6.1) \(S\)—the area of the projection of the orbit onto the plane perpendicular to the magnetic field \(H\)—by \(\pi r_1^2\), where \(\overline{r_1^2}\) is the time average of the square of the radius of the projection of the electronic orbit, and noting that for a spherically symmetric electron shell of an atom \(\overline{r^2}=\frac{3}{2}\overline{r_1^2}\), we find

\[ \Delta \mu=-\frac{e^2 H}{6mc^2}\sum_{n=1}^{Z}\overline{r_n^2}. \tag{6.2} \]

where the summation is over all \(Z\) electrons of the given atom.

Thus, for the atomic diamagnetic susceptibility we obtain:

\[ \chi_A=-\frac{Ne^2}{6mc^2}\sum_{n=1}^{Z}\overline{r_n^2} =-2.832\cdot 10^{10}\sum_{n=1}^{Z}\overline{r_n^2}. \tag{6.3} \]

Here \(N=6.02\cdot 10^{23}\) is Avogadro’s number. From (6.3) it is clear that diamagnetism depends on the radii of the electronic orbits and therefore does not depend on temperature. Taking \(\overline{r_n^2}\sim 10^{-16}\ \mathrm{cm}\), we find that \(\chi_A\sim -10^{-6}Z\)—in qualitative agreement with experiment.

The quantum-mechanical theory of diamagnetism was developed by Van Vleck\({}^{28}\). In the case of atoms and molecules with a spherically symmetric electron shell, the same formula (6.3) is obtained, except that the calculation of \(\overline{r_n^2}\) is carried out according to the laws of quantum mechanics. If the condition of spherical symmetry is violated, then a positive term (paramagnetic!), which somewhat reduces the absolute value of \(\chi_A\), is added to the right-hand side of (6.3).

Classical theory did not make it possible to calculate \(\chi_A\) for diamagnetic molecular gases. Quantum mechanics, at least in principle, makes it possible to obtain a complete solution of the prob-

which, in the case of the simplest molecules (for example, \(H_2\)), also gives a quantitatively correct result. According to Van Vleck\(^{28}\), the molar diamagnetic susceptibility of polyatomic molecules that have no resultant magnetic moment is equal to

\[ \chi_M = -\frac{Ne^2}{6mc^2}\sum_n r_n^2 + \frac{2}{3}N\sum_{n'\ne n} \frac{\left[m^0(n',n)\right]^2}{h\nu(n',n)} , \tag{6.4} \]

where \(m^0(n',n)\) is the off-diagonal matrix element of the angular momentum of the molecule, and \(\nu(n',n)\) is the frequency (\(h\nu\) is the energy) corresponding to the transition from the state \(n'\) to the state \(n\). Both terms in (6.4)

Fig. 23. Atomic diamagnetic susceptibility of elements at room temperature.

\(\sim 10^{-6}\). Consequently, \(\chi_M\) is small and does not depend on temperature. A substance is dia- or paramagnetic depending on which of the terms in (6.4) is larger.

b) Principal experimental data

Because of the smallness of the diamagnetic effect, its measurement presents certain experimental difficulties. The most advanced method for measuring the diamagnetic susceptibility of gases and vapors was developed by the Soviet physicists Janus and Shur\(^{31}\).

1) Diamagnetism of atoms. In Fig. 23 a graph is shown of the atomic susceptibilities of diamagnetic elements according to the latest experimental data. Typical representatives of monatomic diamagnetic gases are the inert gases. Table IV gives the experimental and theoretically calculated values of the atomic susceptibilities of these substances according to the data of various investigators\(^{32}\). From Table IV it is seen that \(\chi_A\) increases regularly with the atomic number of the element, and that the calculated values are sufficiently close to

observed. Apart from the inert gases, Shur measured \(\chi_A\) for the monatomic mercury vapor. It proved to be equal to \(-78.2\cdot 10^{-6}\); the theoretical calculation gives the value \(-84.6\cdot 10^{-6}\).

Table IV

Element	\(-\chi_A\cdot 10^6\), experiment	\(-\chi_A\cdot 10^6\), theory
He	1.88—1.91	1.54—1.90
Ne	6.66—7.65	5.7—8.6
Ar	18.13—19.72	18.9—24.8
Kr	28.02—29.2	31.7—42.0
Xe	42.40—44.1	48.0—66.0

2) Diamagnetism of ions. The theory gives the same formulas for calculating the diamagnetic susceptibility of ions as for atoms. However, in this case experiment always gives either the molecular susceptibility (if a molecular gas is being investigated), or the susceptibility of a solution, or, finally, the susceptibility of a solid crystal. Therefore, in determining ionic susceptibility there is always a certain inaccuracy connected with the influence of the binding forces between the ion and the atoms surrounding it on the diamagnetic susceptibility. On the other hand, by comparing the magnetic susceptibilities obtained for ions from measurements on their various compounds, and also with their atomic susceptibility, we can obtain definite information about the binding forces between ions and about the role of valence electrons in diamagnetism. Thus, for example, Shur^31 obtained for the atomic susceptibility of mercury vapor the value \(-78.2\cdot 10^{-6}\), which is almost twice as large as the experimental^31 value of \(\chi_A\) for the ion \(\mathrm{Hg}^{++}\), which is equal to \(-40.4\cdot 10^{-6}\). This indicates the substantial contribution made by the valence electrons to the diamagnetism of mercury.

The diamagnetism of many salts, for example NaCl and the like, is explained by the fact that the ions (in this case \(\mathrm{Na}^+\) and \(\mathrm{Cl}^-\)) of which they are built have a closed electron shell, close to the shell of the atoms of the inert gases (for \(\mathrm{Na}^+\) and \(\mathrm{Cl}^-\) these will be, respectively, Ne and Ar). Measurements are usually carried out either on solid salts, or more often on solutions (aqueous). Despite the above-mentioned inaccuracy associated with the indirectness of determining the ionic susceptibility, the latter, determined by different methods, proves to be sufficiently constant.

Figure 24 shows the dependence of the ionic susceptibility for ions of similar structure on the number \(n\) of electrons in the ion shell.

A sharper increase in the interval from \(n=10\) to \(n=18\) is apparently connected with the increase of the ionic radius (see (6.3)). Ions with the same value of \(n\), but belonging to different curves in Fig. 24 (for example \(Br^-\), \(Kr\), \(Rb^+\), \(Sr^{++}\)), differ from one another by the charge of the nucleus \(+Ze\). In this case, the larger \(Z\) is, the stronger is the “attraction” of the electron cloud

Fig. 24. Dependence of the ionic diamagnetic susceptibility of ions of similar structure on the number of electrons \((n)\) in the atomic shell.

to the nucleus and the smaller is \(\sum_{n=1}^{Z} \overline{r_n^2}\) in (6.3). The diamagnetism of ionic solutions, in general, follows the additivity rule

\[ \chi_{\text{soln}} = C_s \chi_s + (1-C_s)\chi_{\mathrm{H_2O}}, \]

where \(\chi_s\) and \(\chi_{\mathrm{H_2O}}\) are, respectively, the susceptibility of the salt and of water, and \(C_s\) is the salt concentration.

However, there may also be exceptions to this rule, as is shown, for example, in Fig. 25\(^{32}\) for a solution of LiCl, whose susceptibility shows noticeable deviations from the additivity rule.

3) Diamagnetism of molecules. The calculation of the molecular susceptibility for diatomic molecules (except \(H_2\)), as well as for more complex ones, presents great difficulties. Schur\(^{31}\) measured

susceptibility of vapors of molecular bromine Br\(_2\), which proved to be equal to \(-74\cdot 10^{-6}\). The theoretical data for bromine ions Br\(^{-}\) and Br\(^{+}\) together give values within the limits \((-62.24 \div 89.81)\cdot 10^{-6}\), which agrees quite well with the value found for the molecular susceptibility of bromine.

Of special interest is the measurement of the diamagnetic susceptibility of water\(^{32}\) (H\(_2\)O), since it is very often used for purposes of calibrating apparatus and as a solvent. The value of its susceptibility at 20° C is equal to \(\chi_M=-0.7218\cdot 10^{-6}\) with an accuracy

Fig. 25. Dependence of the diamagnetic susceptibility on the concentration of an aqueous solution.

Fig. 26. Dependence of the specific diamagnetic susceptibility of antimony on temperature.

to \(\pm 0.0007\). The temperature dependence of the susceptibility of water has been investigated very carefully. The temperature coefficient

\[ \frac{1}{\chi_{\mathrm{H_2O}}}\frac{d\chi_{\mathrm{H_2O}}}{dT} \]

decreases monotonically from \(2.9\cdot 10^{-4}\) at 5° C to \(0.62\cdot 10^{-4}\) at 70° C.

It is assumed that the cause of this very weak temperature dependence of the susceptibility of water is the depolymerization of water molecules with temperature, which produces small changes in the structure of the electron shell of the water molecule. Changes of this kind should occur most sharply during melting and evaporation, which in some cases does indeed lead to a strong change in diamagnetic susceptibility, for example during the melting of antimony\(^{43}\) (see Fig. 26).

On the basis of an analysis of numerous experimental data on measurements of the diamagnetic susceptibility of compounds, especially organic ones, Pascal\(^{31}\) proposed the empirical formula

\[ \chi_M=\sum n_A\chi_A+\lambda, \]

where \(n_A\) is the number of atoms of the given kind in the molecule, \(\chi_A\) is their atomic susceptibility, and \(\lambda\) is a correction which may be positive or negative and which depends on the nature of the chemical

bonds between atoms in the molecule. Thus, for example, according to Pascal, the carbon atom enters into the composition of a molecule with \(\chi_C=-6.0\cdot10^{-6}\), the oxygen atom with \(\chi_O=-4.6\cdot10^{-6}\); the double bond \(C=O\) lowers, according to Pascal, the diamagnetism by an amount \(\lambda_2=6.35\cdot10^{-6}\), while the triple bond should give a much smaller value. On the other hand, Schur’s experiments\(^{31}\) gave:

\[ \chi_M(\mathrm{CO})=-11.8\cdot10^{-6}. \]

Thus, \(\lambda_{CO}=1.2\cdot10^{-6}\), which speaks in favor of the presence of a triple bond in the CO molecule.

It should be noted that both ionic and molecular diamagnetism in the crystalline state possess the property of magnetic anisotropy, which is caused by the forces of magnetic interaction between the electrons of the crystal.

c) Diamagnetism of metals

Metals constitute a fairly considerable group of diamagnetics. According to the latest experimental data, this group includes up to 20 metals. Table V\(^{32}\) gives the names of these metals and the magnitude of their atomic susceptibility \(\chi_A\).

It may be accepted with a certain approximation that, in the diamagnetism of solid and liquid metals, along with atomic skeletons (ions), the valence electrons also take a special part. These latter, in the crystal lattice of metals, form a kind of “gas” of conduction electrons.

Table V

Atomic susceptibility of diamagnetic metals at 20° C

Metal	\(\chi_A\cdot10^6\)	Metal	\(\chi_A\cdot10^6\)
Copper	\(-5.4\)	Indium	\(-12.36\)
Silver	\(-21.56\)	Thallium (α)	\(-49.05\)
Gold	\(-29.59\)	Germanium	\(-8.9\)
Beryllium	\(-9.02\)	Lead	\(-24.86\)
Zinc	\(-10.26\)	Arsenic	\(-5.5\)
Cadmium	\(-19.6\)	Antimony	\(-107\)
Mercury	\(-33.8\)	Bismuth	\(-285\)
Boron	\(-6.7\)	Selenium	\(-26.5\)
Gallium	\(-16.8\)	Tellurium	\(-40.8\)

If one compares the observed diamagnetic susceptibility of metals with the diamagnetic susceptibility of their ions, obtained from observations on salts containing these ions and their solutions, then always \(|\chi_A|<|\chi_{\text{ion}}|\). Thus, for example, for copper \(\chi_A=-5.5\times10^{-6}\), while the ionic susceptibility of copper \((\mathrm{Cu}^{++})\) is \(\chi_{\text{ion}}=-18.0\cdot10^{-6}\); for gold we have, respectively, \(-29.59\) and \(-21.56\), for silver \(-45.8\) and \(-31.0\).

This fact was first discovered as early as 1923 by Ya. G. Dorfman\(^{33}\), who at the same time put forward the suggestion that conduction electrons possess a paramagnetic effect.

Since the diamagnetism of metals, as a rule, does not depend on temperature, according to Dorfman the paramagnetism of the conduction electrons also should not depend on temperature. These considerations of Dorfman were fully confirmed by the further development of the quantum theory of metals. However, conduction electrons possess not only paramagnetism, but also diamagnetism. According to the general theorem of classical statistics of Van Leeuwen—Terletsky (see § 5), the diamagnetic susceptibility of an electron gas should be equal to zero. According to Bohr,^34 in the case of a metallic specimen of finite dimensions the disappearance of the diamagnetic effect is clearly explained by the fact that the diamagnetic moment induced in the external field in the electrons inside the metal (see Fig. 27) is completely compensated by the reverse moment of the “broken trajectory” of the electrons that undergo reflection from the boundary surfaces of the specimen. The same conclusion also follows from energy considerations. The magnetic field, acting on the motion of the electrons, does not change the absolute magnitude of their velocity, but only curves their trajectory. Therefore the kinetic energy of the electrons remains unchanged when the field is switched on. Landau^35 discovered the remarkable fact that the translational motion of free electrons possesses a diamagnetic effect different from zero. If we consider the action of the magnetic field on a classical gas of free electrons, then one may say that, owing to the curvature of the electron trajectories in the field, the projection of their motion onto the plane perpendicular to the direction of the magnetic field has the form of closed circles, i.e. is periodic in character. In passing to quantum mechanics, every classical periodic motion is “quantized,” and therefore, with the magnetic field switched on, the free electrons will change their energy, and as a result a diamagnetic effect different from zero will occur.

Fig. 27. Toward the “classical” proof of the disappearance of the diamagnetism of free electrons (after Bohr).

Let the field be directed along the \(z\)-axis; according to the classical theory the Larmor frequency is equal to (2.23):

\[ \nu_H=\frac{eH}{2\pi mc}. \]

The circular precessional motion about the axis can be decomposed into

the sum of two mutually perpendicular linear periodic motions along the axes \(x\) and \(y\). Quantum mechanics leads to the result\({}^{36}\) that, in the case of such motions (a linear harmonic oscillator), there is a discrete energy spectrum

\[ E_n=(n+{}^{1}/_{2})h\nu_H=\frac{e\hbar}{mc}H(n+{}^{1}/_{2})=2\mu_BH(n+{}^{1}/_{2}), \tag{6.5} \]

where \(n=0,1,2,3,\ldots\) is the quantum number of the oscillator. The motion of the electrons along the \(z\)-axis is not changed by the field, remains free, and is not quantized; its energy is equal to \(p_z^2/2m\), where \(p_z\) is the component of the electron momentum along the \(z\)-axis. The total energy is

\[ E=\frac{p_z^2}{2m}+2\mu_BH(n+{}^{1}/_{2}). \tag{6.6} \]

The character of such “partial quantization” of the energy of free electrons in a field can be visualized in the following way.

In the absence of a magnetic field, the energy spectrum corresponding to the component of the motion of a free electron in the plane \((x,y)\) [the energy of this motion is equal to \((p_x^2+p_y^2)/2m\)] is continuous, as shown by the solid band in the left-hand part of Fig. 28. In the presence of a field, however, the entire spectrum is split into separate narrow bands of width

\[ \Delta E=2\mu_BH[(n+1+{}^{1}/_{2})-(n+{}^{1}/_{2})]=2\mu_BH, \]

each of which is transformed into a single discrete level corresponding to the mean level of the band in the continuous spectrum (right-hand part of Fig. 28). Thus, each discrete level is \((2\mu_BH)\)-fold degenerate. To determine the magnetization of such a diamagnetic electron gas, we must first of all compute the phase sum \(Z\) (see (4.21)). For this purpose it is necessary to find the statistical weights of the states, i.e., in other words, to determine their number in the volume of phase space in the form of a cylindrical ring of height \(dp_z\), radius \(p=\sqrt{p_x^2+p_y^2}\), and thickness \(dp\). The volume of this ring is equal to \(2\pi p\,dp\,dp_z\).

Fig. 28. Schematic representation of the quantization of energy levels in a two-dimensional system in a magnetic field.

According to (6.6), in quantum mechanics

\[ \frac{p^2}{2m}\to E-\frac{p_z^2}{2m}=2\mu_BH(n+{}^{1}/_{2}) \]

and

\[ p\,dp\to 2m\mu_BH\,\Delta n=2m\mu_BH. \]

If we also recall that the magnitude of an elementary cell in phase space is equal to \(h^3/V\), where \(V\) is the volume of the metal, and take into account the electron spin, then as a result for the statistical weight we shall have, replacing \(\mu_B\) according to (1.2),

\[ g_n=\frac{2VeH}{ch^2}\,dp_z. \tag{6.7} \]

Substituting (6.7) and (6.6) into formula (4.21) for the phase sum, we find:

\[ Z=\sum_{n=0}^{\infty}\int_{-\infty}^{\infty} dp_z\, \frac{2eV}{c\hbar^2}\, e^{-\frac{\left[\mu_B H(2n+1)+\frac{p_z^2}{2m}\right]}{kT}} = \frac{eVH}{c\hbar^2}\, \frac{\sqrt{2\pi m kT}}{\operatorname{sh}\frac{\mu H}{kT}}. \tag{6.8} \]

According to (4.19), for the magnetization we obtain:

\[ I = NkT\,\frac{d\ln Z}{dH} = -N\mu_B\left[\operatorname{cth}\left(\frac{\mu H}{kT}\right)-\frac{kT}{\mu H}\right]. \tag{6.9} \]

In the case of weak fields and not very low temperatures, i.e. under the condition

\[ H\mu_B \ll kT, \]

(6.9) assumes a simpler form, and for the diamagnetic susceptibility of the electron gas we obtain:

\[ \chi_{\mathrm{dia}}=-\frac{1}{3}\frac{N\mu_B}{kT}. \tag{6.10} \]

Below we shall show (see § 8) that the paramagnetic effect due to the spin magnetic moments of the electron gas gives the susceptibility

\[ \chi_{\mathrm{para}}=+\frac{N\mu_B}{kT}, \tag{6.11} \]

i.e. a quantity three times larger in absolute value than \(\chi_{\mathrm{dia}}\).

It should be noted that in deriving (6.10) we allowed a substantial inaccuracy, since we assumed that the electron gas obeys classical statistics, whereas it obeys Fermi statistics. It is possible, however, to obtain from (6.10) the correct expression for \(\chi_{\mathrm{dia}}\) by an elementary method, without carrying out the statistical calculation anew. From the character of the Fermi distribution it follows that the electron gas at all temperatures up to the evaporation temperature of the metal is, as a rule, strongly degenerate. Therefore only the electrons in the narrow region of smearing near the Fermi edge behave “classically” (in the statistical sense). The number of these electrons in the first approximation is given by the formula

\[ N' = N\frac{T}{\Theta}, \tag{6.12} \]

where \(T\) is the absolute temperature, and \(\Theta\) is the degeneracy temperature, related to the number of particles of the electron gas by the known relation\({}^{34}\):

\[ \Theta=\frac{h^2}{2mk}\left(\frac{3n}{8\pi}\right)^{2/3}. \tag{6.13} \]

Substituting (6.13) into (6.12) and then into (6.10) in place of \(N\), we find:

\[ \chi_{\mathrm{dia}}=-\frac{4m\mu_0^2}{h^2}\left(\frac{\pi}{3}\right)^{3/2}N^{1/3} \tag{6.14} \]

i.e., the expression obtained by Landau\({}^{35}\). Formula (6.14) shows that the quantum gas of free electrons possesses a diamagnetic effect which likewise cannot be observed directly, since it is masked by a positive spin paramagnetism three times stronger (see below, § 8).

The calculation given above does not take into account that the conduction electrons in the metal lattice do not form a gas of “free” particles, but are under the strong influence of the periodic potential of the ions. In a first approximation this can be allowed for by substituting in (6.14), instead of the electron mass \(m\), its effective mass \(m^*\) \((\ne m)\). A more exact calculation\({}^{37}\) shows that the diamagnetic susceptibility of a metal in fact consists of the sum of the diamagnetic susceptibilities of the ions and of the conduction electrons [in the weak-binding limit passing into (6.14)] and, finally, of one further additional term (determined by the magnitude of the interaction of the electrons and ions, and also vanishing in the limiting cases of strong and weak binding).

The anomalous magnitude and temperature dependence of the diamagnetism of bismuth (see Table V above) find their explanation in the specific character of the electronic energy spectrum of this metal, which has a crystal lattice not typical of the overwhelming majority of metals. In addition, a periodic change of the diamagnetic moment with change in the magnetic-field strength in the region of low temperatures was discovered. This phenomenon can be explained quantitatively if one abandons the limitation imposed by the inequality \(H\mu_B \ll kT\), and carries out a rigorous quantum-mechanical solution. Qualitatively this can be understood by referring to Fig. 28. The degree of degeneracy of the discrete levels is determined by the magnitude of the magnetic field according to (6.7). If this number is greater than the total number of electrons \(N\), then they all “fit” on a single level with \(n=0\); as the field decreases, the number of places on the level decreases and may become less than \(N\), whereupon the electrons begin to move over to the next level with \(n=1\), etc. Therefore the magnetic properties of the electrons must vary periodically with change in the magnitude of the magnetic field; in this case the susceptibility may change not only in magnitude but also in sign, as is observed experimentally\({}^{38}\).

7. MAGNETIC PROPERTIES OF SUPERCONDUCTORS

The peculiarity and essential significance of the magnetic properties of superconductors—this special state of certain metals and alloys at very low temperatures—are already sufficiently clearly seen from the very fact of the influence of a magnetic field on the phase transition of matter

from the normal to the superconducting state. This influence consists primarily in the shift, with increasing external magnetic field, of the temperature \(\Theta_{\mathrm{sp}}\) of the phase transition superconductor—normal metal toward lower temperatures; that is, in other words, this temperature is a function of the field \(\Theta_{\mathrm{sp}}(H)\), and

\[ -\frac{d\Theta_{\mathrm{sp}}(H)}{dH}<0. \tag{7.1} \]

In Fig. 29 the phase diagram in the \(H,T\) plane is schematically shown by the curve \(\Theta_{\mathrm{sp}}(H)\). At \(H=0\) the temperature of the phase transition is maximal, \(\Theta_{\mathrm{sp}}(0)\); with increasing field it decreases monotonically, and at a certain maximum critical value of the magnetic field \(H_k^0\) and above it \(\Theta_{\mathrm{sp}}(H)=0\) \((H\geq H_k^0)\), i.e., the superconducting phase does not exist at any temperatures at all. In different superconductors the curves of the phase-transition temperature \(\Theta_{\mathrm{sp}}(H)\), or the curves of the critical field \(H_k(T)\), intersect the \(T\) and \(H\) axes at different values of \(\Theta_{\mathrm{sp}}(0)\) and \(H_k(0)\), which are characteristic constants of the superconducting state of each given substance. The numerical values of the temperature \(\Theta_{\mathrm{sp}}(0)\) in different bodies range from fractions of a degree to \(10^\circ\mathrm{K}\), and \(H_k(0)\) from hundreds to several thousand oersteds.

Fig. 29. Schematic phase diagram for a superconductor in the \(H\)–\(T\) plane.

At \(H=0\) the phase transition from the normal to the superconducting state is a typical phase transition of the 2nd kind, in which there is no discontinuous change in the specific volume and no latent heat of transformation. In the presence of a magnetic field \((H\ne0)\), however, this transformation becomes a phase transition of the 1st kind, and therefore latent heat appears. If by \(F_{\mathrm{n}}^0\) we denote the free energy of the normal state, and by \(F_{\mathrm{sp}}^0\) that of the superconducting state at \(H=0\), then at the phase-transition point the equality holds

\[ F_{\mathrm{sp}}^0(\Theta_{\mathrm{sp}}^0)=F_{\mathrm{n}}^0(\Theta_{\mathrm{sp}}^0), \tag{7.2} \]

moreover, at \(\Theta_{\mathrm{sp}}^0\) the entropies of both phases are also equal:

\[ S_{\mathrm{sp}}^0(\Theta_{\mathrm{sp}}^0)=S_{\mathrm{n}}^0(\Theta_{\mathrm{sp}}^0); \tag{7.3} \]

this means that there is no latent heat of transformation.

In the presence of a magnetic field \((H \ne 0)\), equality (7.2) must be replaced, in the case of massive specimens whose thickness is \(>10^{-5}\,\mathrm{cm}\), by another one\(^{30,39}\):

\[ F_{\mathrm n}^{0}(\Theta_{\mathrm{sp}})=F_{\mathrm{sp}}^{0}(\Theta_{\mathrm{sp}})+\frac{H_k^2(\Theta_{\mathrm{sp}})}{8\pi}. \tag{7.4} \]

From (7.4) it is seen that for \(H\ne 0\) the free energy of the normal phase practically does not change (weakly magnetic bodies), whereas the free energy of the superconducting state depends substantially on the magnetic field. In this case the latent heat is different from zero and instead of (7.3), by virtue of (7.4), we have:

\[ Q=T(S_{\mathrm n}^{0}-S_{\mathrm{sp}})=-\frac{T H_k}{4\pi}\frac{dH_k}{dT}. \tag{7.5} \]

According to (7.1), \(dH_k/dT<0\) and, consequently, \(Q>0\), i.e. in the transition from the normal state to the superconducting state heat is absorbed, and in the reverse transition it is liberated.

Such a substantial influence of an external magnetic field on the very process of the appearance of the superconducting state is closely connected with the magnetic properties of this phase. Numerous experiments, beginning in 1933–1934 (Meissner–Ochsenfeld, Rjabinin–Shubnikov), showed that inside massive superconductors

\[ \mathbf B=\mathbf H=0. \tag{7.6} \]

On the other hand, ever since the discovery of the phenomenon of superconductivity (1911) it had been considered that the primary property of superconductors is the absence of electrical resistance in them. Assuming the validity of Ohm’s law, it follows at once from this supposition that inside a superconductor the electric field is equal to zero,

\[ \mathbf E=0, \tag{7.7} \]

but then from Maxwell’s equation (the law of induction)

\[ \operatorname{rot}\mathbf E=-\frac{1}{c}\frac{d\mathbf B}{dt} \tag{7.8} \]

it follows that \(\mathbf B=\mathrm{const}\). From this conclusion, it would seem, one might expect that in superconductors there should be the so-called effect of “freezing in” of the magnetic field which existed in the metal at \(T>\Theta_{\mathrm{sp}}\). Experiment, however, has shown that (7.6) takes place, i.e. the magnetic field is not “frozen in,” but is “expelled” from the superconductor. (7.6) is not a consequence of (7.7) and must with equal right be regarded as a primary fundamental property of a superconductor.

If one considers a superconductor in the form of a cylinder at \(T<\Theta_{\mathrm{sp}}^{0}\), then its “magnetization curve,” for a field parallel to the axis of the cylinder, has the form (assuming that the normal phase is weakly magnetic—

na, i.e. \(\mu \sim 1\) and \(H = B\)), shown in Fig. 30. For \(H \ll H_k(T)\), \(B = 0\), while for \(H \geq H_k(T)\), \(B = H\). In real cylindrical superconductors, because of unavoidable impurities and distortions of the crystal lattice, the form of the \(B(H)\) curve becomes more complicated—magnetic hysteresis occurs. However, the purer the material, the less pronounced the hysteresis.

A more detailed consideration of the magnetic properties of superconductors shows that the magnetic field penetrates into the superconductor to an appreciable depth \(\sim 10^{-5}\) cm (i.e. to thousands of interatomic distances!), and therefore in very thin superconducting films, whose thickness does not exceed \(10^{-4}\)—\(10^{-5}\) cm, strictly speaking, one can no longer say that (7.6) holds.

Fig. 30. Magnetization curve of a superconducting cylinder (infinitely long) in a field parallel to its axis

In § 5 it was already pointed out that there is no reason to identify a superconductor with a “super”-diamagnetic. The equality to zero of the magnetic field inside a massive superconductor is the result of the fact that, when the external field is “switched on,” or when the body is cooled in the presence of a field to a temperature below the critical \(\Theta_{\mathrm{sp}}(H)\), a macroscopic electric current begins to flow in the surface layer of the superconductor. The magnetic field of this current inside the body is opposite in direction to the external field, and in the sum these fields in the bulk of the superconductor give a resultant field equal to zero. Thus one may say that in superconductors the external magnetic field is screened by a surface macrocurrent. In an “ideal” diamagnetic \(B = 0\) because throughout its entire volume there exists a diamagnetic magnetization “opposite” to the external field,

\[ I = - \frac{1}{4\pi} H, \]

while the external field \(H\) itself is then not equal to zero. The equations of electrodynamics in the case of superconductors hold with only the difference from normal media that, to Ohm’s law relating the density of the electric current to the electric-field vector, there is added a system of equations. First of all, it is assumed that the total electric current \(\mathbf{j}\) in a superconductor is, in the general case, composed of the superconducting current \(\mathbf{j}_{\mathrm{sp}}\) and the normal current \(\mathbf{j}_{\mathrm{n}}\), i.e.

\[ \mathbf{j} = \mathbf{j}_{\mathrm{sp}} + \mathbf{j}_{\mathrm{n}}, \tag{7.9} \]

where

\[ \mathbf{j}_{\mathrm{n}}=\sigma \mathbf{E}, \tag{7.10} \]

and \(\mathbf{j}_{\mathrm{sp}}\) satisfies the new equations:

\[ \frac{\partial \Lambda \mathbf{j}_{\mathrm{sp}}}{\partial t}=\mathbf{E}, \tag{7.11} \]

\[ \operatorname{rot}\Lambda \mathbf{j}_{\mathrm{sp}}=-\frac{1}{c}\mathbf{H}, \tag{7.12} \]

where \(\Lambda\) is a new characteristic constant for the superconducting state*).

If the equations of electrodynamics and (7.11), (7.12) are applied, for example, to a plane massive superconductor, then the field in it decreases along the normal to the surface into the body according to the law\({}^{30}\)

\[ H=H_0 e^{-x/\delta}, \tag{7.13} \]

where \(x\) is the distance from the surface, \(H_0\) is the field at the surface, and \(\delta\) is a constant determining the depth of penetration of the magnetic field into the superconductor, related to the constant \(\Lambda\) by the formula

\[ \delta=\sqrt{\frac{\Lambda c^2}{4\pi}}. \tag{7.14} \]

As \(T\to\Theta_{\mathrm{sp}}\), \(\delta\to\infty\), and consequently \(\Lambda\to\infty\); thus at \(T=\Theta_{\mathrm{sp}}\) the superconducting current is equal to zero, as follows from (7.11) and (7.12).

The magnetization curve of a superconductor shown in Fig. 30 has such a simple form only for bodies in the form of a sufficiently long cylinder or rod with lateral surfaces parallel to the external magnetic field. If, however, the form of the body is different—for example, a sphere, or the same cylinder but with generators situated at right angles to the direction of the external field—then such superconducting bodies, while in an external field, will distort it to one degree or another. In the case of a spherical superconductor, one may say that its distorting influence on the external field \(H_0\) outside its volume (inside \(H=0!\)) is equivalent to the field of a magnetic dipole placed at the center of the sphere and having the moment

\[ \mathbf{M}=-\frac{R_0^3\mathbf{H}_0}{2} \]

(\(R_0\) is the radius of the sphere). Thus, the total field outside the sphere is equal to

\[ \mathbf{H}_e=\mathbf{H}_0+\frac{3(\mathbf{R}\mathbf{M})\mathbf{R}}{R^5}-\frac{\mathbf{M}}{R^3}. \tag{7.15} \]

* In a very crude, so-called inertial model of a superconductor (treated there as an ideal conductor with \(\sigma=\infty\)), \(\Lambda=(m/e^2)n\), where \(n\) is the number of electrons participating in the superconducting current.

At the equator of the sphere ($b—b$ in Fig. 31) we shall have the maximum value of the external field

\[ H_{e_{\max}}=\frac{3}{2}H_0, \tag{7.16} \]

whereas at the poles ($a—a$ in Fig. 31) we have a minimum value, equal to zero.

The existence of an inhomogeneous magnetic field around superconductors can be shown very vividly by using the phenomenon of the so-called levitation of magnets. This circumstance was first pointed out by Arkad’ev\(^{42}\), who carried out an effective experiment on the levitation of permanent magnets made of an iron–nickel–aluminum alloy above a superconducting specimen of lead.

An analogous phenomenon will occur for a body of any shape with a demagnetizing factor different from zero for the direction parallel to the external field. A characteristic feature

Fig. 31. Distortion of a uniform external magnetic field $H_0$ by a superconducting sphere.

Fig. 32. Incorrect picture of the division of a sphere into the normal and superconducting phases in an external field $|H_0|>\frac{2}{3}H_k$.

of the process of “magnetization” of such superconductors is that, with a gradual increase in the intensity of the external magnetic field, it reaches the critical value not at once over the entire surface of the body. For a sphere, for example, the critical field will first be reached at the equator ($b—b$ in Fig. 31), where the external field is maximal, according to (7.16). Therefore, at first glance it is quite natural to suppose that the process of destruction of the superconducting state should begin at the equator (at $H_0=\frac{2}{3}H_k$) and will gradually penetrate into the volume of the sphere at $H_0>\frac{2}{3}H_k$ (Fig. 32). Thus the sphere would have to divide into two parts: a superconducting core and a normal shell. However, it is easy to see that this supposition is incorrect. Indeed, the position of the boundary is determined by the condition that on it the magnetic field is equal to the critical field,

whereas in the normal phase it must be greater than the critical value. The latter condition is not fulfilled in the present case, since the field in the interior (the hatched region in Fig. 32) is smaller than at its surface, and consequently smaller than the critical value; therefore there the superconducting state must again be restored. A more rigorous quantitative calculation likewise proves the impossibility of such a transition from the superconducting to the normal state.

In the case of a sphere the magnetization curve, instead of having the form of Fig. 30, assumes a different form (Fig. 33). For values of the external field \(H_0\) from 0 to \({}^{2}/_{3}H_k\), \(B=0\); in the interval from \(H_0={}^{2}/_{3}H_k\) to \(H_0=H_k\) we have a linear increase of the induction from \(B=0\) to \(B=H_k\), and then the usual straight line \(B=H\). Thus the interval of fields for the superconducting state of the sphere has decreased, in comparison with the cylinder (magnetized along its generatrix), from \((0—H_k)\) to \((0—{}^{2}/_{3}H_k)\). The normal phase still begins for fields \(H \geq H_k\). The interval \(({}^{2}/_{3}H_k—H_k)\), when the magnetization curve differs both from the normal and from the superconducting state, came to be attributed to a new intermediate state\({}^{40}\). The upper boundary of the intermediate state \(H_2\) is determined by the field \(H_k\), while the lower boundary \(H_1\) depends on the shape of the body (its demagnetizing factor) according to the formula

\[ H_1=\left(1-\frac{N}{4\pi}\right)H_k. \]

Fig. 33. Magnetization curve of a superconducting sphere.

Thus, for a sphere \((N={}^{4}/_{3}\pi)\), \(H_1={}^{2}/_{3}H_k\); for a cylinder \((N=2\pi\) under transverse magnetization), \(H_1={}^{1}/_{2}H_k\); for an infinite flat plate (under transverse magnetization, \(N=4\pi\)), \(H_1=0\), i.e. the whole region of fields belongs to the intermediate state.

Landau, in a number of fundamental papers\({}^{41}\), showed quite clearly that there is no special intermediate phase, but that the intermediate state is a mixture of superconducting and normal layers of the substance, alternating with one another (see, for example, Fig. 34). In this case, for example, for a sphere the magnitude of the induction in the range of fields \({}^{2}/_{3}H_k < H < H_k\) is determined by the relative volume of the sphere occupied by the normal phase. In the case of a plate Landau gave a very accurate quantitative calculation of the dimensions of these alternating layers of the superconducting and normal phase. In a thick plate these layers are plane-parallel with

thicknesses \(s\) (for the superconducting phase) and \(n\) (for the normal phase). The sum of these thicknesses inside the plate is approximately equal to

\[ s+n \simeq 4\pi^{1/3}\alpha^{1/3}\left(\frac{L}{H_k}\right)^{2/3}, \tag{7.17} \]

where \(L\) is the thickness of the plate (along the field), and \(\alpha\) is the coefficient of surface tension between the normal and superconducting phases. Thus, the “geometry” of the intermediate state depends on the dimensions of the specimen (\(L\)), on the magnitude of the critical field \(H_k\), and on a quantity \(\alpha\) which is as yet unknown.* Near the surface of the plate the layered structure becomes more complicated: individual normal layers begin to branch, and at the very surface the thickness of the individual layers begins to be comparable with the penetration depth \(\delta\) from (7.14). Therefore, at the surface, according to Landau’s theory, there arises a new macroscopic state, which he provisionally called “mixed.”

Fig. 34. Form of the superconducting and normal layers of a superconducting plate in the intermediate state (after Landau).

In brilliant works—works which in a number of cases are masterpieces in the subtlety of the experiment—Soviet physicists Shalnikov, Alekseevskii, Meshkovskii, and Nakhutin\(^{30,39}\) demonstrated experimentally the fundamental correctness of Landau’s ideas on the nature of the intermediate state. At the same time, these experiments showed that in real superconducting specimens, which, as a rule, are always inhomogeneous, a more complicated picture of layers takes place. Evidently, just as in the case of ferromagnets (see below, § 12), the boundaries between different regions of spontaneous magnetization are located in places with a minimum boundary energy; the boundaries between superconducting and normal regions of the intermediate state occupy places with a minimum boundary energy \(\alpha\). Under the influence of a change in the magnitude of the external magnetic field, both reversible and irreversible displacements of the boundaries between regions occur, which can lead—and in fact does lead—to hysteresis phenomena in the magnetization curves of superconductors. For more detailed data we refer the reader to the original papers, references to which are given in the cited reviews\(^{30,39}\).

* The quantity \(\alpha\) has a certain purely formal similarity to the boundary energy between regions of spontaneous magnetization in ferromagnets. It can be determined directly from investigations of the magnetic properties of thin superconducting films or from experiments on the kinetics of the onset of the superconducting state at \(H \ne 0\).

8. PARAMAGNETISM

Paramagnetism of atoms and ions

a) Theory. A characteristic and necessary feature of the paramagnetic state of a substance is the presence, in the atoms constituting this substance, of permanent magnetic moments, independently of the presence of an external magnetic field. However, in the general case, in the absence of an external field \((H_e=0)\), the disorienting action of thermal motion, as a rule, does not allow the formation of a spontaneous ordered orientation of these permanent atomic magnetic moments. Below (see § 11) it will be shown that only in the presence of a special internal interaction between the elementary carriers of the magnetic moment is such a spontaneous orientation possible at \(H_e=0\), which is realized in ferromagnets.

In paramagnets at \(H_e=0\) the resultant magnetization of the body is always equal to zero. Magnetization arises and begins to increase only when the strength of the external magnetic field \(H_e\) is switched on and increased. If the field is not very large, so that the energy of the elementary magnets \((\sim \mu H)\) is small in comparison with their mean thermal energy \((kT)\), i.e. \(\mu H/kT \ll 1\), then the magnetization increases directly proportionally to the magnitude of the external magnetic field

\[ I \sim \varkappa H_e, \]

where the paramagnetic susceptibility \(\varkappa\) does not depend on the field \(H\), but depends strongly on temperature. Experiment in a number of cases confirms this general conclusion. However, alongside this there are many cases in which the paramagnetic susceptibility is practically independent of temperature (for example, the alkali metals).

The first theory of paramagnetism was developed in 1905 by Langevin \(^{43}\), within the framework of the classical statistical theory of matter. In this first version the theory was given under the assumption that there is no interaction between atoms and, of course, the phenomenon of spatial quantization, unknown at that time, was not taken into account; i.e. an ideal classical gas of magnetic arrows was considered. The magnetization was calculated by the general methods of statistical physics [see (4.21)].

Let us denote by \(\Theta\) the angle between the field and the atomic magnetic moment, and by \(\varphi\) the azimuth about the direction of the field. Then the energy of an atom with respect to the field will be equal to \(-\mu_0 H=-\mu_0 H\cos\Theta\), and the phase sum (4.21) will take the form:

\[ Z(H)=\left[\int_{0}^{2\pi} d\varphi \int_{0}^{\pi} e^{\mu_0 H\cos\theta/kT}\sin\theta\,d\theta\right]^N = \left(\frac{4\pi kT}{\mu_0 H}\operatorname{sh}\frac{\mu_0 H}{kT}\right)^N . \tag{8.1} \]

The part of the phase sum that depends on the kinetic energy may be disregarded, since it does not depend on the field. The share of thermo-

of the dynamic potential, depending on the field, according to (4.20), is equal to

\[ \Phi(H)=-NkT\ln\left(\frac{4\pi kT}{\mu_0H}\operatorname{sh}\frac{\mu_0H}{kT}\right), \]

and from (4.19) we shall have:

\[ I=N\mu_0\left(\operatorname{cth}\frac{\mu_0H}{kT}-\frac{kT}{\mu_0H}\right). \tag{8.2} \]

In the case of weak fields, i.e. for \(\mu_0H\ll kT\), from (8.2) it follows that

\[ \chi=\frac{I}{H}=\frac{N\mu_0^2}{3kT}. \tag{8.3} \]

Relation (8.3) gives a theoretical explanation of the experimental Curie law (5.1). At very low temperatures or strong fields, when \(\mu_0H\gg kT\), the linear relation between \(I\) and \(H\) is violated and the magnetization, according to (8.2), approaches with increasing \(H\) its maximum value \(N\mu_0\), i.e. saturation.

The refinement of this elementary classical theory proceeds in two ways: 1) by taking account of the quantum effect of spatial quantization, and 2) by taking account of the interaction between the elementary magnetic moments of paramagnetic substances.

The allowance for spatial quantization amounts to the fact that in (8.1) \(\cos\theta\), according to (2.13), assumes not arbitrary values, but only a discrete series of possible values. Assuming, for simplicity, that the atoms of the paramagnet have one valence electron actively participating in paramagnetism, formula (8.1) should be transformed to the form:

\[ I=N\mu_B \frac{ g\displaystyle\sum_{m_j=-j}^{m_j=+j} m_j e^{\frac{\mu_BH}{kT}m_jg} }{ \displaystyle\sum_{m_j=-j}^{m_j=+j} e^{\frac{\mu_BH}{kT}m_jg} } = N\mu_B g L_j\left(\frac{g\mu_BH}{kT}\right), \tag{8.4} \]

where \(L_j(x)\) is the so-called generalized Langevin function. For \(j\to\infty\), (8.4) passes into the classical formula (8.2). If the elementary carrier of magnetism in paramagnetic atoms is only the electron spin \(\left(l=0,\ j=s=\frac{1}{2}\right)\), then the number of possible orientations is reduced to two and (8.4) assumes the simpler form

\[ I=N\mu_B g L_{1/2}\left(\frac{\mu_0Hg}{kT}\right) = N\mu_B g\,\operatorname{th}\left(\frac{g\mu_BH}{kT}\right). \tag{8.5} \]

In the case of weak fields \((\mu_0H\ll kT)\) from (8.4) we obtain:

\[ \chi=\frac{N\mu_B^2g^2}{3kT}(j+1)j, \tag{8.6} \]

i.e., again Curie’s law (5.1) or (8.3), with \(\mu_0^2\) replaced by \(g^2\mu_B^2(j+1)j\). Using (2.20), one can determine the constant of Curie’s law (5.1), namely

\[ C=\frac{N}{3k}\left[\mu_B^2 g^2 j(j+1)\right]. \tag{8.7} \]

Thus, knowing from experiment \(C\) and the universal constants \(k, N\) and \(\mu_B\), one can determine the number of magnetons per one atom of the paramagnet,

\[ p_p=g\sqrt{j(j+1)}. \tag{8.8} \]

The rigorous quantum-mechanical theory of paramagnetism, developed by Van Vleck\(^{41}\), leads externally to the same expression for the paramagnetic susceptibility (8.6) as the classical theory and the elementary quantum vector model of the atom,

\[ \chi=\frac{N\overline{\mu^2}}{3kT}+N\overline{\alpha}, \tag{8.9} \]

where \(\overline{\mu^2}\) is the square of the low-frequency part of the vector of the magnetic moment, averaged over time and over the various states of the atoms (for in deriving expression (8.6) in the vector model it is assumed that all atoms are in the same state with given values of the quantum numbers \(L, S\) and \(J\)); \(N\overline{\alpha}\) is the temperature-independent part of the susceptibility and is equal to the sum of the high-frequency elements of the paramagnetic moment and the diamagnetic part. Instead of \(\mu\) it is convenient to introduce the quantity \(p_p\mu_B\); then (8.9) takes the form:

\[ \chi=N\left(\frac{\mu_B^2 p_p^2}{3kT}+\overline{\alpha}\right). \tag{8.9'} \]

As was already indicated above, the magnetic moment of the atom consists of two parts—orbital and spin (see § 2). In the various stationary states of atoms the share of these two types of moments is different. Therefore, although it is customary to speak of “constant” magnetic moments of atoms, it must be remembered that in a number of cases they may change with temperature.

When comparing the general formula (8.9) with experimental data, it must be written in a more concrete form, so that it contains quantities directly measurable in experiment. For this purpose it is convenient to consider three special cases, when the energy differences \(h\nu(JJ')\) of the multiplet structure (see § 2) of the electronic levels of atoms are small in comparison with \(kT\), large in comparison with \(kT\), and comparable with \(kT\).

1) \(h\nu(JJ') \ll kT\). The high-frequency part of the paramagnetic moment, as Van Vleck\(^{28}\) showed, is absent. Therefore, if one also neglects the diamagnetic susceptibility, which everywhere, as a rule, is small in comparison with the paramagnetic susceptibility, then only the low-frequency ...

... part. Owing to the smallness of the energy differences of the multiplet structure, atoms can, with high probability, be in states with different \(J\) (because the statistical factors \(e^{-E/kT}\) for these energetically close states differ little from one another). On the other hand, the smallness of the energies \(h\nu(JJ')\) makes it possible to assume that the coupling of the orbital and spin moments of the atom is small in comparison with their coupling to the magnetic field. This is equivalent to the assumption that the atom is as if in a strong field (the Paschen–Back effect, see above, § 2), and both these moments are quantized in the field independently of one another. Therefore the energy takes the form (2.29), and the expression for the susceptibility becomes equal to:

\[ \chi=\frac{\partial}{\partial H} \left\{ N\mu_B \left[ \left( \frac{\displaystyle\sum_{m_L=-L}^{+L} m_L e^{m_L\mu_B H/kT}} {\displaystyle\sum_{m_L=-L}^{+L} e^{m_L\mu_B H/kT}} \right) + \left( \frac{\displaystyle\sum_{m_S=-S}^{+S} 2m_S e^{2m_S\mu_B H/kT}} {\displaystyle\sum_{m_S=-S}^{+S} e^{2m_S\mu_B H/kT}} \right) \right] \right\}; \tag{8.10} \]

by virtue of the condition \(m_L\mu_B H \ll kT\) we find:

\[ \chi=\frac{N\mu_B^2}{3kT}\{4S(S+1)+L(L+1)\} \tag{8.11} \]

and

\[ p_p=\sqrt{4S(S+1)+L(L+1)}. \tag{8.12} \]

Thus, Curie’s law \(\left(\sim \frac{1}{T}\right)\) is again obtained, but the effective number of magnetons \(p_p\) differs from (8.8). This case is realized for ions of the elements of the iron group.

2) \(h\nu(JJ') \gg kT\). In this case almost all particles must be in the energetically lowest state, since the statistical factor already for the first excited normal state (i.e., a state with the same principal quantum number as the normal state, but with \(J'\ne J_0\)) is, owing to the condition \(h\nu(JJ')\gg kT\), negligibly small. Thus the low-frequency part of the susceptibility has the classical form (8.3). However, in this case the high-frequency part is also different from zero; it has the form

\[ N\alpha= \frac{N\mu_B^2}{6(2J+1)} \left[ \frac{F(J+1)}{h\nu(J+1;J)} - \frac{F(J)}{h\nu(J+1;J)} \right], \tag{8.13} \]

where

\[ F(J)=\frac{1}{J}\left[(S+L+1)^2-J^2\right]\left[J^2-(S-L)^2\right]. \tag{8.14} \]

This case is realized for the majority of ions of the rare-earth elements.

3) \(h\nu(JJ') \sim kT\). In this intermediate case it may be assumed that the total number of atoms \(N\) is divided into groups with a given value of the quantum number \(J\), i.e. \(N=N_{J_1}+N_{J_2}+\cdots\). Further, one may take

\[ N_J = N(2J+1)e^{-W_J^0/kT}, \]

where \((2J+1)\) gives the number of components of the multiplet, and \(W_J^0\) is the energy at \(H=0\) for the state with the given \(J\). As a result we obtain the following formula for the susceptibility:

\[ \chi = \frac{ N\sum_{J=(L-S)}^{L+S} \left\{\left[g_J^2\mu_B^2 J(J+1)/3k\right]+\alpha_J\right\}(2J+1)e^{-W_J^0/kT} }{ \sum (2J+1)e^{-W_J^0/kT} }, \tag{8.15} \]

where the index \(J\) on \(g\) and \(\alpha\) indicates that the Landé factor and the high-frequency part \(\alpha\) of the susceptibility depend on \(J\).

In the limiting case of narrow intervals of the multiplet structure, (8.15) goes over into (8.12). In general, however, (8.15) gives a temperature dependence \(\chi(T)\) quite different from that which follows from the classical law \(\sim 1/T\).

b) Basic experimental data. 1. Monatomic paramagnetic gases. The formulas given above apply, strictly speaking, to the case of monatomic gases in which the interatomic interaction is minimal. Unfortunately, however, the ordinary monatomic gases (the inert gases) which exist in the natural state (at normal temperatures and pressures—densities) are all diamagnetic (with a closed electron shell); vapors of the alkali metals, although paramagnetic, have a negligibly small vapor pressure at normal temperatures. Measurements at very high temperatures (\(>500^\circ\text{C}\)) are technically very difficult and inaccurate, since the paramagnetic effect itself is very small \((\sim 1/T!)\). Nevertheless, measurements have been made for potassium and thallium vapors (see \(^{28}\)), which showed that, for example, for potassium vapor in the temperature interval from \(600^\circ\) to \(800^\circ\text{C}\) and, correspondingly, in the pressure interval from 0.5 to 30 mm Hg, the atomic paramagnetic susceptibility obeys Curie’s law and is equal to \(0.38/T\). It is natural to assume that the normal state of the potassium atom is \({}^2S\), i.e. that there is only one valence electron with an uncompensated spin moment; then, according to (8.11),

\[ \chi = \frac{N\mu^2}{kT} = 0.372/T. \]

Thus the agreement between theory and experiment is quite good. Still, it must be stated that experiments on measurements of the susceptibility of paramagnetic monatomic gases are entirely insufficient, and this question remains to this day one of the urgent tasks for experimental verification of the theory set forth above.

2. Ions of the rare-earth elements. At present the most thoroughly studied are the paramagnetic properties of ions of the rare-earth elements from cerium \((\mathrm{Ce}, Z=58)\) to ytterbium \((\mathrm{Yb}, Z=70)\), whose compounds (salts and salt solutions) are strongly paramagnetic substances obeying the simple laws of “atomic” paramagnetism. The latter is connected with the fact that, practically always, in these substances the ions of the rare-earth elements are responsible for the paramagnetism—namely, the resultant magnetic moment of the incomplete \(4f\) shell of the electron shell of these rare-earth atoms. The electronic configuration in the shell of the rare-earth atoms has the form

\[ \ldots 4f^{\,c-14}5s^2 5p^6 5d^2 6s . \]

Fig. 35. Atomic effective magnetic moments of rare-earth ions.

It is natural to suppose that the closed outer electron shells \((5s^2 — 5p^6)\) screen the incomplete and magnetically active \(4f\) shell sufficiently well from the influence of neighboring atoms, and therefore these atoms (or ions) can behave magnetically like the atoms of rarefied (ideal) gases. Such screening of the \(4f\) shell from the external influences of neighboring atoms is indicated by the similarity of the chemical properties and by the fact that the absorption bands and the magnetic susceptibility depend only very slightly on the change of the “environment” of a rare-earth ion in passing from one compound to another or from a solution to a solid salt. In most cases the ions of the rare earths are trivalent (they give up 3 valence electrons, \(5d^2\) and \(6s\)).

The first, most extensive and profound work on the theoretical calculation of the paramagnetic susceptibility of rare-earth ions belongs to J. H. Van Vleck\(^ {45}\). This work was then continued by Hund\(^ {46}\) and Van Vleck\(^ {28}\). In Fig. 35 the experimental values for the atomic effective moment of rare-earth ions are given (circles on the graph with vertical segments showing the scatter of the points according to data from various investigations), and also the theoretical curve calculated by Van Vleck’s formulas (8.9). For most of the elements it proved possible, in good agreement with experiment, to use the simplified formulas (8.3) and (8.13), which assume that the distances between neighboring energy levels of the multiplet structure are large in comparison with \(kT\).

The values of the quantum numbers \(J, S\), and \(L\) for the ground state are determined from spectroscopic considerations according to Hund. For two elements—samarium (\(\mathrm{Sm}, Z=62\)) and europium (\(\mathrm{Eu}, Z=63\))—this assumption is not fulfilled; the energy spectrum of the electronic shell of these atoms is such that \(h\nu (JJ')\sim kT\)*. Therefore, in these two cases one must use the more complicated formula (8.15). The temperature dependence of \(\chi\) for these ions also differs from Curie’s law and, at not very high temperatures, can be well approximated by formula (8.15). It would be very interesting to investigate \(\chi(T)\) in the region of very low temperatures (liquid helium) for a more detailed test of the theory. Drozdhina and Janus\(^{47}\) investigated the paramagnetic properties of the purest (in the sense of ferromagnetic impurities) samples of cerium and praseodymium in the solid state and found that these rare-earth metals obey the Curie–Weiss law. The effective atomic magnetic moments turn out to be somewhat smaller than is observed for trivalent ions in measurements with salts and than is obtained from Van Vleck’s theory. This can quite possibly be explained qualitatively by the presence of unavoidable ferromagnetic impurities, and also by the fact that the valence electrons in the metal have different magnetic properties (they form an electron-conduction gas) than in salts and solutions; moreover, the influence of the electric field of the crystal on the orbital moments of the \(4f\)-shell in the case of the metal and the salt may be entirely different. The technical difficulties in obtaining pure and large samples of rare-earth metals do not make it possible to carry out their magnetic investigation on a broad scale, although the study of the magnetic properties of these metals, especially at ultra-low temperatures, is of very great interest.

Up to now it has been assumed that the magnetic-field strength is small, so that the magnetization grows linearly with the field. For the limiting case \(h\nu (JJ') \gg kT\), it is not very difficult, without expanding the exponent in a series in powers of the exponent \(g\mu_B H/kT\), to obtain an exact expression for \(I(H)\), which has the form:

\[ I = NJg\mu_B B_J\left(\frac{Jg\mu_B H}{kT}\right), \tag{8.16} \]

where

\[ B_J(y)=\frac{2J+1}{2J}\operatorname{cth}\left(\frac{2Jy+y}{2J}\right)-\frac{1}{2J}\operatorname{cth}\frac{y}{2J} \tag{8.17} \]

is the so-called generalized Langevin function (or Brillouin–Debye function). For the case \(h\nu (JJ') \ll kT\), instead of (8.16) we obtain:

\[ I=2NS\mu_B B_S(2S\mu_B H/kT)+NL\mu_B B_L(L\mu_B H/kT). \]

* To some extent this also applies to neodymium (\(\mathrm{Nd}, Z=60\)) and, apparently, to the as-yet undiscovered rare-earth element with \(Z=61\).

This formula, however, is not used, for at low temperatures, when the phenomenon of saturation can be observed experimentally, the assumption that \(h\nu(JJ')\) is small in comparison with \(kT\) is not fulfilled. For large \(J\), (8.16) passes into the classical Langevin function

\[ L(x)=\operatorname{cth}x-\frac{1}{x}. \]

The saturation value according to (8.16), for \(H\to\infty\), is equal to

\[ I_{\infty}=NJg\mu_B . \]

This value is \(\sqrt{J/J+1}\) times smaller than that given by the classical formula, which is a consequence of the uncertainty relation for the components of the angular momentum of the electron\({}^{28}\).

Fig. 36. Magnetization curve of gadolinium sulfate \([28,46]\).

The best agreement between theory and experiment was obtained for the solid salt—gadolinium sulfate

\[ \mathrm{Gd}_2[\mathrm{SO}_4]_3 + 8\mathrm{H}_2\mathrm{O}. \]

In this case \(J=7/2\). In Fig. 36 the theoretical curve

\[ \frac{I}{I_{\infty}}=B_{7/2}(7\mu_BH/kT) \]

and the experimental points according to the measurement data\({}^{48}\) are shown. The magnetic moment of saturation in this case is equal to \(0.88N\mu_{\mathrm{eff}}\).

3. Elements of the transition groups and their ions. In addition to the rare-earth elements there are four other groups of elements with an internally incomplete layer of the atomic electron shell: 1) the iron group from scandium \((\mathrm{Sc}, Z=21)\) to nickel \((\mathrm{Ni}, Z=28)\) with an incomplete \(3d\) shell; 2) the palladium group from yttrium \((\mathrm{Y}, Z=39)\) to rhodium \((\mathrm{Rh}, Z=45)\) with an incomplete \(4d\) shell; 3) the platinum group from lutetium \((\mathrm{Lu}, Z=71)\) to platinum \((\mathrm{Pt}, Z=78)\) with an incomplete \(5d\) shell; 4) the actinide group from radium \((\mathrm{Ra}, Z=88)\) to uranium \((\mathrm{U}, Z=92)\) with an incomplete \(6d\) shell\(*\). However, in contrast to

\[ \text{*} \]

\(*\) For ions these groups have a somewhat different form. Thus, for example, the iron group extends from the scandium ion \(\mathrm{Sc}^{+++}\) to the zinc ion \(\mathrm{Zn}^{++}\). The number of electrons in the \(3d\) shell then changes from 0 to 10.

of the rare-earth elements the unfilled layer (the \(d\)-layer) of the electron shell in the atoms of the elements of these groups lies closer to the periphery of the shell and therefore is not so well screened from external influences. The values of the effective magnetic moments of these ions are also smaller than those of the rare-earth ions, if only because in the present case the \(d\)-layer, which has 10 electron “places,” is responsible for the paramagnetism, whereas in the electron shell of the rare-earth atoms the magnetic moment is produced by the \(4f\)-layer with 14 “places.”

Thus, in the case of ions of the iron group, one has to take into account the influence of the “surroundings” on the magnetic behavior of the ions. It turns out that the spin and orbital parts of the magnetic moment behave differently. The best agreement between theory and experiment is obtained if it is assumed that orbital magnetism is entirely absent (the so-called phenomenon of “quenching” of the electronic orbits under the influence of the electric field inside the crystal) and that all magnetism is due to the electron spins alone. In this case, according to (8.11), the susceptibility is equal to

\[ \chi=\frac{N\mu_B^2}{3kT}[4s(s+1)], \tag{8.18} \]

and the effective magnetic moment per atom is equal to

\[ p=2\sqrt{s(s+1)}. \]

Thus, if the multiplicity \((2s+1)\), or the number of unpaired electrons \(n\), is known, then \(p\) can be calculated from the formula

\[ p=\sqrt{n(n+2)}. \tag{8.19} \]

The data for ions of the iron group are given in Table VI.

The agreement between theory and experiment here is somewhat worse than in the case of the rare earths.

Owing to the phenomenon of electron exchange (see below, Part III) and the action of the crystalline field, Curie’s law (8.3) is practically never obeyed, and in the present case the Curie–Weiss law holds:

\[ \chi \sim \frac{C}{T+\Delta}. \tag{8.20} \]

For the other groups of transition metals there is as yet too small a number of measurements to permit a detailed comparison of theory with experiment.

Paramagnetism of molecules

Most molecules containing more than one atom are diamagnetic, for they usually contain an even number of electrons, which form magnetically neutral closed shells. There are, however,

Table VI

Theoretical and experimental effective values of the number of magnetons for ions of the first group of transition metals

Ion	Number of \(3d\) electrons	Ground term	\(\mu_{\mathrm{eff}}=\sqrt{n(n+2)}\) (theory)	\(\mu_{\mathrm{eff}}\) (experiment)
\(\mathrm{Sc}^{+3}\) \(\mathrm{Ti}^{+4}\) \(\mathrm{V}^{+5}\)	0	\({}^{1}S_{0}\)	0.00	0.0 0.0 0.0
\(\mathrm{Ti}^{+3}\) \(\mathrm{V}^{+4}\)	1	\({}^{2}D_{3/2}\)	1.73	1.77–1.79
\(\mathrm{Ti}^{+2}\) \(\mathrm{V}^{+3}\)	2	\({}^{3}F_{2}\)	2.83	2.76–2.85
\(\mathrm{V}^{+2}\) \(\mathrm{Cr}^{+2}\) \(\mathrm{Mn}^{+4}\)	3	\({}^{4}F_{3/2}\)	3.87	3.81–3.86 3.68–3.86 4.00
\(\mathrm{Cr}^{+2}\) \(\mathrm{Mn}^{+3}\)	4	\({}^{5}D_{0}\)	4.90	4.80 5.0
\(\mathrm{Mn}^{+2}\) \(\mathrm{Fe}^{+3}\)	5	\({}^{6}S_{5/2}\)	5.92	5.2–5.96 5.4–6.0
\(\mathrm{Fe}^{+2}\) \(\mathrm{Co}^{+3}\)	6	\({}^{5}D_{4}\)	4.90	5.0–5.5 (2.5)
\(\mathrm{Co}^{+2}\)	7	\({}^{4}F_{9/2}\)	3.87	4.4–5.2
\(\mathrm{Ni}^{+2}\)	8	\({}^{3}F_{4}\)	2.83	2.9–3.4
\(\mathrm{Cu}^{+2}\)	9	\({}^{2}D_{5/2}\)	2.73	1.8–2.2
\(\mathrm{Cu}^{+1}\) \(\mathrm{Zn}^{+2}\)	10	\({}^{1}S_{0}\)	0.00	0.0 0.0

An exception to this rule is the paramagnetism of oxygen \((\mathrm{O}_{2})\), whose molecules have an even number of electrons and nevertheless do not form, in the normal state, a closed magnetically neutral shell. Molecules with an odd number of electrons, of which there are comparatively far fewer than “even” molecules, are, on the contrary, paramagnetic. A typical representative of this class of paramagnetism is nitric oxide \((\mathrm{NO})\).

In the simplest case of diatomic molecules, the vector scheme of the electron shell changes in comparison with the corresponding scheme

for an atom. One of the principal differences is that in the present case the resultant orbital moment is no longer an integral of the motion; the latter is only the projection of this moment on the axis of the molecule. Therefore, instead of the orbital quantum number \(L\), the quantum number \(\Lambda\) is introduced for the projection of this moment on the molecular axis (i.e., the line joining the centers of the atomic nuclei). Terms corresponding to the values \(\Lambda = 0, 1, 2, 3, \ldots\) are denoted by \(\Sigma, \Pi, \Delta, \Phi\) (instead of \(S, P, D, F\)). The total moment of the molecule with quantum number \(J\) is the result of the addition of the “parallel” projection of the orbital moment \((\Lambda)\), the spin moment of the electrons (with quantum number \(S\)), and the moment of rotation of the atoms of the molecule about an axis perpendicular to the molecular axis (with quantum number \(N\)). In this, there may be two cases of coupling: a) the coupling between the orbital moment and the spin moment is greater than the coupling between \(S\) and \(N\). Therefore the total moment \(\sqrt{J(J+1)}\,\hbar\) is obtained as the vector sum of the moment \(N\) and the resultant projection of the orbital and spin moments \(\Omega = |\Lambda+\Sigma|\) on the molecular axis (see Fig. 37, a). b) The coupling of \(S\) with \(\Lambda\) is smaller than with \(N\), and therefore the component \(\Sigma\) disappears (Fig. 37, b). \(\Lambda\) and \(N\) give a resultant vector \(K\), which is added to the spin vector \(S\).

Fig. 37. Vector model of a molecule.

Here, just as in the case of atoms, one may consider the problem for two limiting cases—small and large multiplet intervals. In the first case, when \(h\nu(JJ') \ll kT\), for states with coupling both of type \(a\) and of type \(b\), the susceptibility is equal to:

\[ \chi=\frac{N\mu_B^2}{3kT}\,[4S(S+1)+\Lambda^2]+N\alpha. \tag{8.21} \]

If, however, \(h\nu(JJ') \gg kT\), then only coupling of type \(a\) is realized, for coupling of type \(b\) already presupposes that \(h\nu(JJ')\) is much smaller than the distances between neighboring rotational levels, which are usually smaller than \(kT\). In this case the susceptibility has the form:

\[ \chi=\frac{N\mu_B^2}{3kT}(\Lambda+2\Sigma)^2. \tag{8.22} \]

In the case of oxygen \((\mathrm{O}_2)\), the ground state of the molecule is \({}^3\Sigma\) \((\Lambda = 0\) and \(S = 1)\), i.e., despite the even number of electrons, two of them remain “unpaired.” Only certain forms of sulfur vapor and a very small number of molecules of organic compounds (“biradicals”) possess the same special magnetic properties. The fact that such an anomaly is observed in molecular oxygen is proved not only by measurements of magnetic susceptibility, but also by experiments with magnetic deflection of a molecular beam and by detailed study of the structure of the molecular spectrum. The multiplet intervals in the case of oxygen are small in comparison with \(kT\); therefore, according to (8.21), for \(\Lambda = 0\) and \(S = 1\) we have:

\[ \chi = \frac{8N\mu_B^2}{3kT} = \frac{0.993}{T}. \tag{8.23} \]

Thus, for example, at \(20^\circ\mathrm{C}\), \(\chi = 339\cdot 10^{-5}\); experiment (the average over measurements by various authors) gives \(\chi_{\mathrm{exp}} = 342\cdot 10^{-5}\). At low pressures the dependence (8.23) is satisfied over a wide temperature interval \((143^\circ — 720^\circ\mathrm{K})\). However, as the pressure is increased, deviations appear, and instead of (8.23) the Curie–Weiss law \(\chi(T+\Delta)=C\) holds, with a constant \(\Delta\) depending on the density. The same takes place for the liquid and solid states. In addition, the whole phenomenon is complicated by the fact that, in the gas and in liquid solution, alongside \(\mathrm{O}_2\) molecules there are also \(\mathrm{O}_4\) molecules.\(^{49}\)

Fig. 38. Magnetization curve of nitric oxide \((\mathrm{NO})\).

The ground term of the NO molecule is a \({}^2\Pi\) term, i.e., a doublet with levels \({}^2\Pi_{3/2}\) and \({}^2\Pi_{1/2}\). It turns out that the energy difference of these two levels is of the order of \(kT\) already at room temperatures. Van Vleck, for this case, derived on the basis of the general quantum-mechanical theory a more exact formula for the susceptibility:

\[ \chi = \frac{N\mu_{\mathrm{eff}}^2}{3kT}, \tag{8.24} \]

where

\[ \mu_{\mathrm{eff}}^2 = 4\mu_B^2 \frac{1-e^x+xe^{-x}}{x+xe^{-x}}, \qquad x=\frac{\Delta E}{kT}\sim \frac{173}{T}; \tag{8.25} \]

\(\Delta E\) is the energy difference between the levels of the doublet \({}^{2}\Pi\). In Fig. 38 a comparison is given between the theoretical formula (8.24) and the experimental data. Such good agreement between theory and experiment in this particular case is customarily regarded as one of the most weighty quantitative experimental confirmations of the correctness of the quantum theory of molecular paramagnetism.

Paramagnetism of Metals

A characteristic distinction of metals from other solid bodies is the presence in them of conduction electrons, whose properties determine, in the main, the entire metallic state of matter. In § 6 the question of the diamagnetism of metals has already been considered. From Table V it is seen that the diamagnetic metallic elements belong to the series following the triads of the eighth group of Mendeleev’s table. Conversely, all metals of the short series preceding the triads, and the triads themselves, are paramagnets. A summary of the principal characteristics of these paramagnetic metals is given in Table VII.^32

As follows from Table VII, for a large number of paramagnetic metals a characteristic feature is the practical absence of a temperature dependence of the susceptibility (for example, the alkali metals). This circumstance gave J. G. Dorfman^33 the right, as early as 1923, to predict quite correctly that in these paramagnetic metals with \(d\chi/dT \sim 0\) the principal role in magnetism is played by the conduction electrons. However, a theoretical explanation of this important experimental fact became possible only after the appearance of quantum mechanics. If the conduction electrons in metals obeyed the laws of classical mechanics, then the paramagnetic properties of metals would be essentially analogous to those of gases, i.e. the susceptibility would have had to depend sharply on temperature, which is not in fact the case (if one excludes the rare-earth metals and the metals of the iron, platinum, and palladium triads). Within the framework of classical mechanics the problem of temperature-independent paramagnetism remained insoluble. Moreover, this phenomenon is one of the most convincing proofs of the inapplicability of the laws of classical physics to electrons in a metal. The works explaining the temperature independence of the magnetic properties of paramagnetic metals (Pauli^50, Frenkel^51, Dorfman^45), together with Landau’s work on diamagnetism,^55 laid the foundation not only for the quantum theory of the magnetic properties of metals, but also, in general, for the entire modern electron theory of the solid state.

If we confine ourselves to the approximation of an “ideal” gas, neglecting the energy of electron interaction, then the principal difference from the classical theory consists in the fact that the electron gas

S. V. Vonsovskii

Table VII

Atomic susceptibility of paramagnetic metals at room temperature

Metal	$\chi_A \cdot 10^6$	Temperature dependence
Lithium	25.2	practically absent
Sodium	15.6	practically absent
Potassium	21.5	practically absent
Rubidium	19.2	practically absent
Cesium	29.9	practically absent
Magnesium	6	weak
Calcium	44	weak
Strontium	92	weak
Barium	20	weak
Aluminum	16.7	not known exactly
Scandium	315	not known exactly
Yttrium	191	not known exactly
Lanthanum	140	not known exactly
Cerium	2300	Curie–Weiss law
Praseodymium	3520—5150	Curie–Weiss law
Neodymium	5600	Curie–Weiss law
Samarium	1820	Curie–Weiss law
Europium	30400	Curie–Weiss law
Gadolinium	75500	Curie–Weiss law
Terbium	115000	Curie–Weiss law
Dysprosium	1020	Curie–Weiss law
Holmium	68200	Curie–Weiss law
Erbium	44500	Curie–Weiss law
Thulium	25600	Curie–Weiss law
Ytterbium	250	Curie–Weiss law
Titanium	150	anomalous
Zirconium	120	anomalous
Thorium	130	anomalous
Gold (white)	4.4	anomalous
Vanadium	23	weak
Tantalum	145	weak
Chromium	160	weak
Molybdenum	54	unknown
Tungsten	40	unknown
Uranium	60	anomalous
Manganese	527	weak
Rhenium	68.7	weak
Ruthenium	44	Curie–Weiss law
Rhodium	113	Curie–Weiss law
Palladium	580	Curie–Weiss law
Osmium	7.6	Curie–Weiss law
Iridium	25	Curie–Weiss law
Platinum	20	Curie–Weiss law

is subject to Fermi quantum statistics, and not to classical statistics, as was assumed in Langevin’s theory. A characteristic feature of Fermi statistics is the Pauli principle, to which an ensemble of electrons must conform even when the electrostatic and magnetic interaction between the electrons is neglected. In application to a gas of “free” electrons, the Pauli principle asserts that in a “phase” cell there can be no more than two electrons with oppositely directed spins. At a temperature of \(0^\circ\mathrm{K}\) the gas must have the least energy. According to classical statistics this would mean that all particles had gathered in the phase cell with zero energy. For electrons, however, this is forbidden by the Pauli principle. Therefore at \(0^\circ\mathrm{K}\) the electrons merely fill, as densely as possible, the set of cells with the smallest possible values of kinetic energy (which for a “free” electron in the absence of external fields coincides with the total energy). The volume thus filled in phase space has the form of a sphere, the surface of which is the isoenergetic surface corresponding to the limiting maximum energy \(\varepsilon_{\max}=\zeta\), whose magnitude, at ordinary electron densities in a metal, expressed in degrees, is equal to \(20\,000^\circ\mathrm{K}\). Between \(\zeta\) and the number of electrons per unit volume \(n\) there is a simple relation. The volume of the sphere occupied by \(n\) electrons in phase space at \(0^\circ\mathrm{K}\) is equal to

\[ \frac{4\pi p_{\max}^{3}}{3} \]

(\(p_{\max}\) is the greatest momentum of the electrons at \(0^\circ\mathrm{K}\)). The magnitude of the phase cell, according to the uncertainty principle, is equal to \(\sim h^3\). Therefore

\[ n=2\,\frac{4\pi}{3h^3}\,p_{\max}^{3} \]

(the factor 2 takes into account the two spin directions). But \(p_{\max}=\sqrt{2m\zeta}\), consequently:

\[ n=\frac{8\pi}{3h^3}(2m)^{3/2}\zeta^{3/2}. \tag{8.26} \]

The number of electrons having energy in the interval from \(\varepsilon\) to \(\varepsilon+d\varepsilon\) is equal to

\[ dn=\frac{4\pi}{3}(2m)^{3/2}\varepsilon^{1/2}d\varepsilon. \]

The factor before \(d\varepsilon\) determines the density of distribution of the electrons over energies

\[ N(\varepsilon)=\frac{4\pi}{h^3}(2m)^{3/2}\varepsilon^{1/2}. \tag{8.27} \]

In the absence of an external magnetic field \(\mathbf{H}\), the total magnetic moment of the electron gas at \(0^\circ\mathrm{K}\) is equal to 0. If the field \(\mathbf{H}\) is switched on, then in order that at least one of the spins antiparallel to the field \(\mathbf{H}\) turn along the field, the latter must perform work, which goes into “pulling” the electron out beyond the boundary of the sphere with maximum

energy \(\zeta\) into the free cells of phase space. Before the magnetic field is switched on, the distribution densities \(N_{+}(\varepsilon)\) of spins parallel to the field and \(N_{-}(\varepsilon)\) of spins antiparallel to the field have the same parabolic form (8.27) (see Fig. 39). After the field is switched on, the “band” of parallel spins is displaced along the \(\varepsilon\)-axis downward by the amount \(-\mu_B H\), while the “band” of antiparallel spins is displaced upward by the amount \(+\mu_B H\). Thus, the two bands are displaced relative to one another by \(2\mu_B H\) (Fig. 39).

Fig. 39. Paramagnetism of “free” electrons.

This displacement disturbs the energetic equivalence of the two spin orientations. However, equal filling of the bands no longer corresponds to the minimum of the system’s energy. The requirement of thermodynamic equilibrium will compel part of the electrons to pass from the band with antiparallel-to-the-field spins into the band with parallel spins. The energy minimum in the new state will already correspond to a magnetized state. If we neglect the change of the function \(N(\varepsilon)\) in the interval \(\mu_B H\) (which is legitimate even for fields \(\sim 10^4\) oersteds, since \(\mu_B H\) is then \(10^5\) times smaller than \(\zeta\) in metals), then the magnetization created by the field \(H\) will be equal to

\[ I=\bigl[N_{+}(\zeta)-N_{-}(\zeta)\bigr]\mu_B H\cdot \mu_B \simeq N(\zeta)\mu_B^{2}H . \]

Using (8.27) and (8.26), we obtain for the paramagnetic susceptibility of a “free” electron gas:

\[ \chi_{p(\mathrm{el})}=\frac{12m\mu_B^{2}}{h^2}\,n^{1/3}\left(\frac{\pi}{3}\right)^{2/3}. \tag{8.28} \]

Since the degeneracy temperature of the electron gas

\[ \Theta=\frac{\zeta}{k} \tag{8.29} \]

is large (\(\Theta \gg T\), where \(T\) is the ordinary temperature of the upper limit of existence of solid metals), thermal motion has only a weak effect on the distribution of electrons in phase space. Therefore the paramagnetic susceptibility of the electron gas (8.28) must depend only very weakly on temperature, which is also confirmed by experiment. This result can be obtained from the following simple considerations. At temperatures low in comparison with the degeneracy temperature, the number of electrons “affected” by thermal motion and transferred into the upper free phase cells amounts—

comprises a small fraction of the total number, proportional to the ratio \(\dfrac{T}{\Theta}\). Thus, the density of these electrons is

\[ n' = n \frac{T}{\Theta}. \]

These electrons behave in the same way as a classical gas. Therefore, in the presence of an external magnetic field the specific magnetic susceptibility per such electron, according to (8.3), is equal to \(\dfrac{\mu_B^2}{3kT}\), while the total susceptibility of a unit volume of the metal will not depend on temperature\(^*\), and in magnitude will be equal to

\[ \chi_{p(\mathrm{el})} = n' \frac{\mu_B^2}{3kT} \simeq \frac{n\mu_B^2}{3k\Theta}, \tag{8.30} \]

which, by virtue of (8.29) and (8.26), coincides with (8.28), to within a numerical factor of order 1.

From (8.28) or (8.30) it is seen that the paramagnetic susceptibility of the electron gas is exactly 3 times greater than its diamagnetic susceptibility obtained by Landau. Therefore the latter cannot be observed directly; experiment always gives the difference of these susceptibilities, equal to \(2/3\) of (8.28). Taking into account the interaction of the valence electrons with the potential field of the lattice, under the condition \(T \ll \Theta\), practically does not change the form of relation (8.28). At very high temperatures (\(T \gg \Theta\)) the gas of free electrons should behave quite like a classical gas of magnetic needles. However, since the degeneracy temperature \(\Theta\), as a rule, is higher than the melting and even vaporization temperature of metals, it is practically impossible to observe such “classical” behavior of electrons in a metal.

In a more exact calculation of the paramagnetic susceptibility of a metal it is necessary to take into account: 1) the electrostatic, i.e. exchange and Coulomb, interactions of the valence electrons, 2) their magnetic interaction, 3) the Landau diamagnetism of the electrons, 4) the diamagnetism of the ionic cores. The theoretical values of the susceptibility thus obtained for a number of metals agree quite satisfactorily with experimental data. For example, in the case of sodium, for spin paramagnetism the theory\(^{52}\) gives \(1.11 \cdot 10^{-6}\), for Landau diamagnetism \(-0.23 \cdot 10^{-6}\), for the diamagnetism of the ionic cores \(-0.18 \cdot 10^{-6}\); thus the total susceptibility is equal to \(0.70 \cdot 10^{-6}\), while experiment\(^{53}\) gives the value \(0.63 \cdot 10^{-6}\).

It should be noted that, owing to the smallness of the susceptibility of many paramagnetic metals, the presence in them of even a negligible amount of ferromagnetic impurities may completely distort the true

\(^*\) Of course, in a more exact calculation one obtains a weak dependence on temperature \(\sim \left(\dfrac{T}{\Theta}\right)^2\).

picture of the magnetic properties. One of the ways of eliminating the influence of ferromagnetic impurities is to measure \(\chi_p\) at various values of the external magnetic field.

In particular, if measurements are made in very strong fields (when the ferromagnetic impurity is in the state of saturation), then one may approximately use the formula \(\chi=\chi_\infty+\dfrac{a}{H}\), where \(\chi\) is the measured susceptibility, \(a\) is a constant depending on the magnitude of the magnetic saturation of the impurity, and \(\chi_\infty\) is the susceptibility of the pure paramagnetic metal. However, ferromagnetic impurities may be present in the bulk of the paramagnetic metal not as isolated particles, but may enter into close interaction with the surrounding layers of the main crystal. This interaction may depend strongly on the thermal and mechanical treatment of the metal. The latter circumstance makes it possible to use such behavior of impurities for the structural study of a paramagnetic metal (for more detail see \(^{34,32}\)), and also for the study of the magnetic properties of small ferromagnetic inclusions (see §§ 11–12).

As has already been mentioned more than once, the paramagnetic susceptibility of alkali metals is practically independent of temperature. This is due to the fact that their paramagnetism is produced by valence electrons, which in these metals form a “free gas” of conduction electrons possessing the temperature-independent Dorfman–Pauli paramagnetism.

Alloys of alkali metals, in the case of ordering, give a minimum of susceptibility (for example, \(\mathrm{Na}_2\mathrm{K}\) at \(-183^\circ\mathrm{C}\) \(^{53}\)). Sometimes the susceptibility changes linearly with the concentration of the alloy, but there are cases, such as, for example, the alloys \(\mathrm{Na}—\mathrm{Cs}\) or \(\mathrm{K}—\mathrm{Cs}\), which give a sharp deviation from linearity (Fig. 40).

Alkaline-earth* paramagnetic metals (magnesium, calcium, strontium, barium) have been studied comparatively little, especially in the pure state, and therefore there are no sufficiently detailed data on the temperature dependence of their magnetic properties. There is only scattered information \(^{32}\), for example, that the susceptibility of barium at \(20^\circ\mathrm{C}\) is \(\chi_{A20^\circ}=0.147\cdot10^{-6}\), while at \(400^\circ\mathrm{C}\) it increases to the value \(\chi_{A400^\circ}=0.415\cdot10^{-6}\), and at \(350^\circ\mathrm{C}\) has a small jump; the susceptibility of strontium at \(0^\circ\mathrm{C}\) is \(1.02\cdot10^{-6}\), at \(65^\circ\mathrm{C}\) \(1.09\cdot10^{-6}\), and then begins to fall and at \(260^\circ\mathrm{C}\) is \(0.73\cdot10^{-6}\).

The magnetic properties of the paramagnetic metals of the third group of the periodic table—aluminum, scandium, yttrium, and lanthanum—have also been studied very little.

The magnetic properties of the rare-earth metals have already been described above.

The susceptibilities of titanium and zirconium have a very curious temperature dependence, which has not yet found a theoretical explanation (see Fig. 41),

The magnetic properties of vanadium, chromium, manganese, uranium, rhenium, and the platinum metals in the pure state have been studied very unsystematically. Likewise, the magnetic properties of paramagnetic alloys have been studied very poorly. At the same time, the study of the magnetic properties of paramagnetic metals and their alloys is of unquestionable practical interest, since it holds great possibilities for determining subtle

Fig. 40. Deviation from additivity for the paramagnetic susceptibility in alloys of sodium and potassium with cesium.

Fig. 41. Temperature dependence of the paramagnetic susceptibility of titanium and zirconium.

structural changes in pure metals and alloys (especially during processes of decomposition, ordering, etc.), which are inaccessible to ordinary metallographic and X-ray methods of investigation. However, for the present one must state that this method of paramagnetic analysis has not yet attained proper development.

Dispersion and absorption in paramagnets

When a paramagnet is introduced into an alternating magnetic field, there is observed the phenomenon of dispersion of magnetic permeability and absorption of energy, analogous to the phenomena in dielectrics placed in an alternating electric field. Investigation of these effects was begun in 1936 by Gorter \(^{54}\); subsequently detailed experimental studies

were carried out by Zavoisky and his collaborators^55. It should also be pointed out that the question of dispersion and absorption in magnetics in a general form had been developed by Arkad’ev still earlier^27. The most complete theory of paramagnetic dispersion and absorption was developed by Ya. I. Frenkel^56. Its further development was given by Al’tshuler, Zavoisky, and Kozyrev^57. In addition, Shaposhnikov^61 gave a thermodynamic theory of paramagnetic absorption in weak magnetic fields.

Usually experiments on paramagnetic dispersion are set up as follows: a paramagnetic substance (a solid paramagnetic salt or a solution) is placed in a constant magnetic field \(H_0\), at a certain angle to which an alternating magnetic field \(H_1\) of frequency \(\nu\) is applied; then the complex magnetic susceptibility is measured (according to Arkad’ev^27) in the direction perpendicular to the field \(H_0\),

\[ \chi_{\perp}=\chi'_{\perp}+i\chi''_{\perp} \]

or in the direction parallel to \(H_0\):

\[ \chi_{\parallel}=\chi'_{\parallel}+i\chi''_{\parallel}. \]

The coefficients of the imaginary unit (\(\chi''\)) give the magnitude of the paramagnetic absorption. Zavoisky showed^62 that the observed paramagnetic losses reveal a characteristic resonant dependence on the magnitude of the constant field \(H_0\), at a fixed frequency of oscillations of the radio-frequency field. This dependence is qualitatively well described by the theoretical formulas obtained by Frenkel^56. The idea of the theory of relaxation paramagnetic losses, given by Frenkel, consists in generalizing the theory of magnetic resonance (Dorfman, Rabi) to the case where “frictional forces” exist. The latter are caused by the interaction of the electron spins participating in the paramagnetic moment of the substance with the thermal vibrations of the lattice (phonons) and with the internal electric and magnetic fields, more precisely with their alternating component caused by vibrations of the crystal lattice. The theoretical formulas for \(\chi'_{\perp}\) and \(\chi''_{\perp}\), in the case when there is one electron spin per atom of the substance, have the characteristic resonance form

\[ \left. \begin{aligned} \chi'_{\perp} &= \frac{N\mu_B^2}{kT}\cdot \frac{\nu_0^2(\nu_H^2-\nu^2)} {(\nu_H^2-\nu^2)^2+4\nu^2\nu'^2},\\ \chi''_{\perp} &= \frac{N\mu_B^2}{kT}\cdot \frac{2\nu_H^2\nu'\nu} {(\nu_H^2-\nu^2)^2+4\nu^2\nu'^2}. \end{aligned} \right\} \tag{8.31} \]

where \(\nu_H\) is the frequency of the Larmor precession performed by the spin in the constant field \(H_0\) [see (2.23)], \(\nu_H=g\mu_B H_0/h\). The quantity \(\nu'\), entering

In these formulas, \(\tau\) is defined as the reciprocal of the relaxation time of the electronic spins,

\[ \tau=\frac{1}{\gamma'}, \]

i.e., the time required to establish their statically equilibrium distribution (along or against the constant field \(H_0\)) by the oscillatory motion of atoms in the substance under investigation. From an analysis of experimental data, Zavoisky\({}^{55}\) established the empirical relation \(\gamma'=bH_0^2\), which is also confirmed by certain theoretical considerations\({}^{60}\). Further experiments by Zavoisky\({}^{58}\) and Salikhov\({}^{59}\) showed the existence of certain deviations of experiment from the theoretical formulas (8.31). Al’tshuler, Zavoisky, and Kozyrev showed that these discrepancies can be eliminated if one takes into account the interaction between the magnetic moments of the paramagnetic ions, which affects the character of the resonance curve.

9. MAGNETIC COOLING

In recent times the paramagnetic properties of matter have proved to be a very important factor in the important physical problem of obtaining ultralow temperatures \((<1^\circ\mathrm{K})\). The importance of the magnetic method of cooling is due to the fact that cooling methods based on the process of liquefying gases, i.e., using entropy changes during evaporation, become less and less effective when the gas pressure becomes very small, so that technically it is no longer possible to obtain an appreciable change of entropy in a finite time. The lowest temperature obtained by this method with the aid of liquid helium is \(0.71^\circ\mathrm{K}\).

The magnetic method of cooling is based on the physical phenomenon known as the magnetocaloric effect. This phenomenon consists in the fact that, upon adiabatic magnetization of a paramagnetic substance (deprived of heat exchange with the surrounding medium), first, the energy of the elementary carriers of magnetic moment is decreased owing to the appearance of an additional negative potential energy in the external magnetic field \((\mu H)\), and, second, the degeneracy of the system’s microstates is removed (Zeeman splitting), i.e., the statistical weight of the microstates is decreased \([g_n\text{—in equation }(4.21)]\). These causes, at low temperatures, lead to a decrease of the additive term in the expression for the “magnetic part” of the entropy of the system \((\sim R\ln g_0)\), where \(g_0\) is the statistical weight of the ground state of the system. By virtue of the adiabatic condition, i.e., the constancy or nondecrease of entropy, this decrease must be compensated by other sources within the system itself. The latter can be achieved by increasing the intensity of thermal motion, i.e., by heating the system from the initial temperature \(T_0\) to the final temperature \(T\). The entropy in this case increases by the amount

\[ \int_{T_0}^{T}\frac{C}{T}\,dT, \]

where \(C\) is the heat—

capacity of the substance. Conversely, during adiabatic demagnetization of a paramagnet, the energy of the elementary carriers of magnetic moment increases; therefore the magnetic part of the entropy of the body also increases, the intensity of thermal motion falling together with its entropy, which leads to a lowering of the temperature of the body.

It should be noted that such a transparent interpretation of the process of magnetic cooling is possible only in the case when the elementary magnets of the substance form an ideal gas. Otherwise the interaction between the elementary magnetic moments complicates the whole phenomenon. In particular, for example, the interaction makes it impossible, by using magnetic cooling, to reach exactly \(0^\circ\mathrm{K}\) (Nernst heat theorem).

A quantitative expression for the magnetocaloric effect, i.e. the change in temperature \(dT\) caused by an adiabatic change in the magnetic-field intensity \(dH\), can be obtained from the fundamental equation of the first and second laws of thermodynamics, using the condition of adiabaticity \((dS=0)\). Introducing the thermodynamic potential (see (4.13) and (4.14))

\[ U' = U - \frac{1}{8\pi}H^2, \tag{9.1} \]

we then obtain

\[ dU' - H\,dI = 0. \tag{9.2} \]

Choosing the temperature and the magnetic field as independent variables and neglecting the phenomenon of magnetostriction \((\partial V/\partial H=0)\), we obtain from (9.2):

\[ \left(\frac{\partial U'}{\partial T}-H\frac{\partial I}{\partial T}\right)dT+ \left(\frac{\partial U'}{\partial H}-H\frac{\partial I}{\partial H}\right)dH=0 \]

and, consequently,

\[ dT= -\frac{\left(\dfrac{\partial U'}{\partial H}-H\dfrac{\partial I}{\partial H}\right)} {\dfrac{\partial U'}{\partial T}-H\dfrac{\partial I}{\partial T}}\,dH. \tag{9.3} \]

Using the known thermodynamic relations

\[ \frac{\partial U'}{\partial H}-H\frac{\partial I}{\partial H} = T\frac{\partial I}{\partial T} \quad\text{and}\quad \frac{\partial U'}{\partial T}-H\frac{\partial I}{\partial T} = C_H, \]

where \(C_H\) is the heat capacity at constant field, we find, instead of (9.3),

\[ dT=-\frac{T\left(\dfrac{\partial I}{\partial T}\right)_H}{C_H}\,dH. \tag{9.4} \]

Let us estimate the order of magnitude of this effect\({}^{62}\). As we have seen, at high temperatures \((kT \gg \mu H)\) in paramagnets approximately

the Curie law (8.1) holds; therefore \(\dfrac{\partial I}{\partial T}=-\dfrac{CH}{T^{2}}\), and (9.4) takes the form

\[ T dT=\frac{CH}{C_H}\,dH. \tag{9.5} \]

Integrating (9.5) over the limits from \(T_1\) to \(T_2\) and from \(H=0\) to \(H=H_1\), we obtain (taking \(\Delta T=T_2-T_1 \ll T_1+T_2=2T\))

\[ \Delta T=\frac{CH_1^2}{2C_H T}. \tag{9.6} \]

For example, in the case of oxygen \(C_H=0.156\cdot 4.18\cdot 10^7\ \dfrac{\mathrm{erg}}{\mathrm{gram}\cdot \mathrm{deg}},\ C=0.0316,\ T=293^\circ\mathrm{K}\),

\[ \Delta T=0.83\cdot 10^{-11}H_1^2. \]

Even for \(H_1=30000\) gauss, \(\Delta T\sim 0.007^\circ\), i.e. the effect is clearly negligible. However, at very low temperatures the heat capacity of a body drops sharply; one may assume that at \(T\sim 1^\circ\mathrm{K}\) practically only the electronic part of the heat capacity plays a role, which is almost 100 times smaller than the heat capacity of the crystal at ordinary temperatures. Then (9.6) for \(T\sim 1^\circ\mathrm{K}\) gives

\[ \Delta T=2.43\cdot 10^{-9}H_1^2 \]

and for \(H_1=30000\) gauss, \(\Delta T\sim 2.05^\circ\).

This result is quantitatively clearly incorrect, but it shows that at very low temperatures the magnetocaloric effect may have relatively much greater significance than at room temperatures. The quantitative error made in the last calculation consists in the fact that, without any justification, we assumed the validity of the Curie law down to the very lowest temperatures. As already indicated above, at very low temperatures the magnetization of paramagnets no longer obeys the simple law (8.1), which in the present case must be replaced by the more general Curie–Weiss law:

\[ I=\frac{C}{T+\Theta}. \]

In this case, instead of (9.6), we obtain:

\[ \Delta T=\frac{CH_1^2}{2C_H(T+\Theta)}. \tag{9.7} \]

From formula (9.7) it is evident that even when \(T\) is exactly equal to zero, \(\Delta T\ne 0\) because the constant \(\Theta\) is different from zero. This means that, owing to the presence of interaction between the elementary carriers of the magnetic moment of the paramagnet, which causes the appearance of the constant \(\Theta\), absolute zero is not exactly attainable. At the same time, it follows from (9.7) that, for the purposes of magnetic cooling, the most suitable substances are those for which the constant \(\Theta\) has a minimum value.

The first experiments on magnetic cooling were carried out by Giauque^63, who, using gadolinium sulfate as the working substance, obtained a temperature only slightly below 1° K. De Haas and Wiersma^64 obtained a lower temperature—0.13° K—by demagnetizing the same preparation from a field of 27,600 oersteds at an initial temperature of 1.35° K to a field of 850 oersteds.

Even greater success was achieved when, instead of the above-mentioned substance, salts of the alum type began to be used, containing ions of elements of the iron group. The best result was obtained^65 with chromium-potassium sulfate \((K_2SO_4Cr_2(SO_4)_3 \cdot 24H_2O)\), dissolved in nonmagnetic alum \((K_2SO_4Al_2(SO_4)_3 \cdot 24H_2O)\), with which a temperature of 0.0044° K was reached. The volume of the sample in this case was 56 \(cm^3\), and the process of demagnetization was carried out from a field of 24,075 oersteds at 1.174° K to a field of 1 oersted.

The scheme of an experimental apparatus for obtaining low temperatures by magnetic cooling is shown in Fig. 42.

Fig. 42. Scheme of apparatus for magnetic cooling.

Labels in the figure: liquid helium; liquid hydrogen; tube for pumping out helium; outer Dewar; liquid helium; vessel with the working substance; induction coil for measuring the magnetization (regeneration).

In connection with the phenomenon of magnetic cooling there arises an important question concerning the kinetics of this process, i.e., the time required for thermal equilibrium to be established between the elementary carriers of magnetic moment (for example, electron spins in unfilled shells of atomic envelopes) and the crystal lattice. It should be noted that, in general, the process of magnetization of paramagnets (see also Part I, § 3) depends essentially on the interaction between atomic magnets. Theory shows that the direct action of a magnetic field on an atomic magnetic moment reduces to the precession of this moment about the direction of the magnetic field, i.e., to a diamagnetic effect. However, if one takes into account, for example, for real gases, that the atoms of a substance undergo collisions, then it is necessary to bear in mind that in each act of impact the direction of the magnetic moments of the atoms, generally speaking, must change. The methods of statistical

mechanics, as was indicated in § 8, automatically and with sufficient accuracy take this effect of thermal motion into account for the processes of magnetization of macroscopic bodies. However, in doing so we learn only the final result, which corresponds to the thermodynamically equilibrium state.

The question of the time required for the establishment of this thermodynamically equilibrium state must be decided by the methods of physical kinetics. For the case of mono- and diatomic paramagnetic gases this question was investigated in detail by Gurevich[^66], who showed that for monatomic gases whose atoms are in an S-state (vapors of alkali metals), the relaxation time is of the order of \(10^{-5}\) sec. In the case of diatomic molecules, owing to the nonspherical symmetry of their electron shell also in a \(\Sigma\)-state, “magnetization” (a change of orientation) of even an isolated molecule is possible. Gurevich found that the magnetic susceptibility corresponding to this process amounts to \(2/3\) of the full susceptibility of the gas. The remaining magnetization occurs as a result of collisions; its relaxation time, as in the case of a monatomic gas, turned out to be equal to \(10^{-5}\) sec. Such a separation of the “process” of magnetization of diatomic paramagnetic gases should lead to a peculiar dispersion of the magnetic susceptibility.

Thus, in order that, under adiabatic demagnetization, the magnetic energy \(\sim 2\mu_B H\) could pass from the crystal lattice to the elementary magnetic moments, there must exist an interaction between the spins and the vibrations of the lattice. The question of the kinetics of the establishment of thermal equilibrium between the spins and the lattice was investigated most fully and consistently by Akhiezer and Pomeranchuk[^67], who, instead of the greatly overestimated value previously obtained by Heitler and Teller[^68], found that the time after which the temperatures of the spins and of the crystal lattice will differ by less than \(1\%\), with an initial spin temperature of \(0.0001^\circ\) K and an initial lattice temperature \(\ll 0.0001^\circ\) K, does not exceed 1 second.

Thus, the applicability of the method of magnetic cooling is not limited by the practical results already attained, and one may hope that the near future will bring new significant successes in this important field of the physics of ultralow temperatures.

(For the conclusion, see Vol. XXXVI, No. 4).

CITED LITERATURE FOR PART II

§ 4

1. V. K. Arkad’ev, Electromagnetic Processes in Metals, Vols. I and II, 1936.

2. J. H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford (1932).

3. Ya. P. Terletskii, ZhETF, 9, 796 (1939).

§ 5

1. V. L. Ginzburg, Superconductivity, Publishing House of the Academy of Sciences of the USSR (1946).

§ 6

1. Ya. S. Shur, UFN, 20, 410 (1938).
2. P. W. Selwood, Magnetochemistry, New York (1943).
3. Ya. G. Dorfman, New Ideas in Physics, collection no. 11, Leningrad (1924).
4. Ya. G. Dorfman and I. K. Kikoin, Physics of Metals, GTTI (1933).
5. L. D. Landau, Zeits. f. Physik, 64, 629 (1930).
6. See, for example, D. I. Blokhintsev, Introduction to Quantum Mechanics, Gostekhizdat (1945).
7. A. Wilson, Quantum Theory of Metals, GTTI (1940).
8. W. J. de Haas and P. M. van Alphen, Proc. Acad. Sci. Amsterd., 34, 1249 (1931).

§ 7

1. E. L. Andronikashvili and K. A. Tumanov, UFN, 33, 469 (1947).
2. D. Shoenberg, Proc. Roy. Soc., 152, 10 (1935); R. Peierls, Proc. Roy. Soc., 155, 613 (1936).
3. L. D. Landau, ZhETF, 7, 371 (1937); 13, 377 (1943).
4. V. K. Arkad’ev, DAN, 47, 18 (1945).

§ 8

1. See, for example, B. A. Vvedenskii and G. S. Landsberg, Modern Theory of Magnetism, ONIZ (1929).
2. See, for example, F. Bloch, Molecular Theory of Magnetism, ONTI, 1936, ch. III, § 3.
3. Ya. G. Dorfman, Zeits. f. Physik, 23, 286 (1924).
4. P. Hund, Zeits. f. Physik, 33, 855 (1925).
5. R. I. Yanus and V. I. Drozhzhina, Sov. Phys. (1935).
6. H. R. Woltjer, Leiden Comm. 167b.
7. See L. Pauling, The Nature of the Chemical Bond, Goskhimizdat (1948), p. 253.
8. W. Pauli, Zeits. f. Phys., 41, 81 (1927).
9. Ya. I. Frenkel, Zeits. f. Physik, 49, 31 (1928).
10. I. B. Sampson, Zeits. Phys. Rev., 58, 633 (1940).
11. B. Böhm and W. Klemm, Zeits. anorg. allg. Chem., 243, 69 (1939).
12. C. J. Gorter, Physica, 3, 503, 1006 (1936).
13. E. K. Zavoiskii, ZhETF, 15, 253, 344 (1945).
14. Ya. I. Frenkel, ZhETF, 15, 409 (1945).
15. S. A. Al’tshuler, E. K. Zavoiskii and B. M. Kozyrev, ZhETF, 17, 1122 (1947).
16. E. K. Zavoiskii, ZhETF, 16, 603 (1946).
17. S. G. Salikhov, ZhETF, 17, 1070 (1947).
18. J. Brouer, Physica, 10, 801 (1943).
19. I. G. Shaposhnikov, ZhETF, 17, 824 (1947).
20. E. K. Zavoiskii, ZhETF, 17, 155 (1947).

§ 9

1. W. F. Giauque and D. P. MacDougall, Phys. Rev., 43, 768 (1933).
2. W. J. de Haas, E. C. Wiersma and Kramers, Nature, 131, 719 (1933).
3. W. J. de Haas and E. C. Wiersma, Report at the 7th International Conference on Refrigeration (1936).
4. L. E. Gurevich, ZhETF, 7, 544 (1937); 14, 348 (1944).
5. A. I. Akhiezer and I. Ya. Pomeranchuk, ZhETF, 14, 342 (1944).
6. W. Heitler and E. Teller, Proc. Roy. Soc., 155, 629 (1936).
7. Ruhemann, Low Temperature Physics (1937).