ORIENTED KERNELS

G. R. Khutsishvili

I. OBTAINING ORIENTED KERNELS

1. Introduction

Kernels are called oriented if the ensemble of nuclear spins has some predominant direction. Under ordinary conditions, owing to thermal motion, the distribution of the nuclear spins of a macroscopic body over directions is completely chaotic. The production of targets with oriented kernels is of considerable interest for nuclear physics, since, by carrying out experiments with oriented kernels, one can obtain much very valuable information about the spin dependence of nuclear forces, about the spins, parities, and magnetic moments of excited states of nuclei, etc.

In the first part of the present review we shall consider methods by means of which oriented kernels can be obtained. In the second part we shall consider experiments that have been carried out, as well as experiments that can be carried out with oriented kernels. There too the question will be discussed of what information can be extracted from experiments with oriented kernels.

2. Parameters characterizing the degree of orientation of kernels¹ ²

Let us direct the \(Z\) axis along the quantization axis*) of the nuclear spin (this direction may be the direction of an external magnetic field or an axis of symmetry of the intracrystalline electric field). Denote by \(m\) the projection of the nuclear spin on the quantization axis.

*) We restrict ourselves to the case in which the quantization of the nuclear spin has axial symmetry; for the general case the parameters characterizing the degree of orientation of kernels are given in work ³.

(in units of $\hbar$), and by $I$ the maximum projection (in this case the magnitude of the spin is equal to $\sqrt{I(I+1)}$). It is well known that $m$ can take $2I+1$ different values $I, I-1, \ldots, -I$, corresponding to the $2I+1$ possible orientations of the nuclear spin.

The degree of orientation of nuclei is usually characterized by the following two quantities$^{1,2}$:

\[ f_1=\frac{\overline{m}}{I} \tag{1} \]

and

\[ f_2=\frac{3}{I(2I-1)} \left[\overline{m^2}-\frac{I(I+1)}{3}\right], \tag{2} \]

where the averaging is performed over the ensemble of nuclei of the given type in the sample. $f_1$ is called the nuclear polarization, and $f_2$ the nuclear quadrupolarization. For a chaotic distribution of nuclear spins over directions, $f_1$ and $f_2$ vanish (taking into account that in this case $\overline{m}=0$ and $\overline{m^2}=\dfrac{I(I+1)}{3}$); for oriented nuclei, however, $f_1$ and $f_2$ take nonzero values. It is easy to see that the maximum absolute values of $f_1$ and $f_2$ are equal to unity.

The introduction of the quantity $f_2$ is necessary because, when certain methods are applied, samples are obtained in which the numbers of nuclei in states differing only in the sign of $m$ are equal, and in such a case

\[ f_1=0. \]

For the consideration of some questions it is also necessary to introduce parameters of higher order $f_3, f_4, \ldots, f_{2I}$. We give the expressions for $f_3$ and $f_4$*:

\[ f_3=\frac{5}{I(I-1)(2I-1)} \left\{\overline{m^3}-\frac{1}{5}[3I(I+1)-1]\overline{m}\right\}, \tag{3} \]

\[ f_4=\frac{35}{2I(I-1)(2I-1)(2I-3)} \left\{\overline{m^4}-\frac{1}{7}[6I(I+1)-5]\overline{m^2} +\frac{3}{35}I(I-1)(I+1)(I+2)\right\}. \tag{4} \]

Nuclei are called polarized if at least one of the odd $f$’s is different from zero; if all the odd $f$’s are zero, but at least one of the even $f$’s is different from zero, the nuclei are called aligned (in this case the numbers of nuclei in the states $+m$ and $-m$ are equal).

* The parameters $f_n$ are normalized in such a way that their maximum values are equal to unity.

3. Polarization by an external field

In principle, the simplest method for obtaining oriented nuclei is a method based on applying a strong magnetic field to the sample at ultralow temperature. In the field \(H\) the energy levels of the nucleus split. The additional energy in the field is equal to \(E_m=-m\dfrac{\mu H}{I}\) (\(\mu\) is the nuclear magnetic moment), i.e. we obtain \(2I+1\) equidistant levels with an energy difference between neighboring levels of \(\dfrac{\mu H}{I}\). A simple calculation gives\({}^{1,2}\):

\[ f_1=B_I\left(\frac{\mu H}{kT}\right), \tag{5} \]

where \(B_I\) is the so-called Brillouin function:

\[ B_I(y)=\frac{I+\frac{1}{2}}{I}\operatorname{cth}\left(\frac{I+\frac{1}{2}}{I}y\right)-\frac{1}{2I}\operatorname{cth}\left(\frac{y}{2I}\right). \tag{6} \]

In particular, for \(\mu H \ll kT\), we have:

\[ f_1=\frac{I+1}{3I}\frac{\mu H}{kT}. \tag{7} \]

Figure 1 gives the dependence of \(f_1\) on \(\dfrac{\mu H}{kT}\) for \(I=\dfrac{1}{2},\ \dfrac{3}{2}\), and \(\dfrac{5}{2}\).

However, owing to the smallness of nuclear magnetic moments, to obtain a significant degree of orientation, extremely large values of \(\dfrac{H}{T}\) are required.

We shall cite numerical data pertaining to \(T=0.01^\circ\text{K}\). To obtain \(f_1=20\%\) in the case of protons (\(I=\dfrac{1}{2}\), \(\mu=2.8\) nuclear magnetons), \(H=20\) kilooersteds is required; and in the case of \(\mathrm{Li}^7\) (\(I=\dfrac{3}{2}\), \(\mu=3.25\) nuclear magnetons), \(H=30\) kilooersteds is required. To obtain \(f_1=75\%\) in the same cases, 100 and, respectively, 200 kilooersteds are required.

Fig. 1.

The advantage of the method under consideration is its universality, since this method is applicable to any nuclei*).

*) For polarization by an external field of nuclei of elements of the iron group and rare earths, it is necessary that \(H\) be greater than the internal magnetic field produced at the nucleus by an unfilled \(3d\) or \(4f\) shell (see § 5).

Its drawback, however, is that it requires excessively large \(H/T\).

It is well known that ultra-low temperatures are obtained by adiabatic demagnetization of paramagnetic salts. Therefore, when polarization is produced by an external field, the experiment will proceed as follows: the specimen in which we wish to obtain oriented nuclei is brought into good thermal contact with a cooled paramagnetic salt, and then a strong external field is applied to the specimen (but not to the salt, otherwise heating will occur).

For experiments one must take substances in which the relaxation time of the nuclear spin (i.e., the time required for the system of nuclear spins to come into equilibrium with the lattice) at ultra-low temperatures is sufficiently small. This condition is satisfied by metals (according to experimental data, for copper the relaxation time is of the order of 1 second at \(0.1^\circ\mathrm{K}\) and increases inversely proportionally to the temperature as the latter decreases), as well as by paramagnetic and ferromagnetic substances. In the case of other substances, in order to reduce the relaxation time it is necessary to add to them a little paramagnetic salt (in this case the relaxation time of the nuclear spin is reduced owing to the interaction of the nuclear spin with the spin of the electronic shell of the paramagnetic ion).

4. Quadrupole Method

Let a nucleus possessing a quadrupole moment be placed in an axially symmetric electric field. We take the axis of symmetry of the electric field as the quantization axis of the nuclear spin. The spin electric levels of the nucleus will have the form (see, for example, \(^{5}\)):

\[ E_m=\frac{3e^2 q Q}{4I(2I-1)} \left[ m^2-\frac{I(I+1)}{3} \right], \tag{8} \]

where \(Q\) is the electric quadrupole moment of the nucleus (in \(\mathrm{cm}^2\)), \(eq=\left.\dfrac{\partial^2\varphi}{\partial z^2}\right|_{\text{at the nucleus}}\), \(\varphi\) is the potential of the electric field. Thus, in an electric field the energy levels of a nucleus possessing a quadrupole moment are split.

Usually these splittings, owing to the smallness of nuclear quadrupole moments, are negligibly small; however, for halogen nuclei in molecular crystals, in molecules in which there are covalent C–halogen bonds, these splittings are anomalously large (for reasons on which we shall not dwell here) and reach values of the order of \(0.005\text{–}0.01^\circ\mathrm{K}\). Such substances are, for example, \(\mathrm{CH_3Br}\), \(\mathrm{C_2H_2Br_2}\), etc.

For the case \(I={}^{3}/_{2}\) the level scheme is shown in Fig. 2 (for \(qQ<0\)). It is clear that at a sufficiently low temperature the states with \(m=\pm {}^{3}/_{2}\) will be populated more than the states with \(m=\pm {}^{1}/_{2}\); in this case \(f_1\) will be equal to zero, while \(f_2\) will be different from zero, i.e., oriented nuclei will be obtained. This method was proposed by Pound\(^6\).

For experiments one needs a single crystal with the same direction of \(z\) at all halogen nuclei (\(z\) is the symmetry axis of the intracrystalline electric field, whose direction coincides with the direction of the line of the covalent C–halogen bond)*). Such a single crystal must be brought into good thermal contact with a cooled paramagnetic salt and, at a sufficiently low temperature, a considerable degree of orientation of the halogen nuclei will be obtained. However, a little paramagnetic salt must be mixed into the crystal in order to reduce the relaxation time of the nuclear spins.

        ± 1/2
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      e²qQ
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Fig. 2.

One may also proceed in another way, namely, grow a crystal which itself contains paramagnetic ions; then cooling is also easier (adiabatic demagnetization is carried out on the very sample being studied), and relaxation is fast.

5. Orientation of nuclei in paramagnetic salts

a) Paramagnetic salts and paramagnetic resonance. In obtaining oriented nuclei, the use of paramagnetic salts plays a major role because the electron shell of a paramagnetic ion produces an enormous internal magnetic field at the nucleus (since the shell of a paramagnetic ion has a magnetic moment owing to the presence of an unfilled \(3d\) or \(4f\) subshell).

Paramagnetic resonance absorption, discovered by E. K. Zavoisky\(^7\), makes it possible to study the energy levels of paramagnetic ions in condensed media. The experiments with crossed fields played the greatest role; we shall briefly discuss their principle.

A specimen (a paramagnetic salt in the solid or liquid state) is placed in a strong homogeneous magnetic field \(H\). In this case, for each paramagnetic ion, a certain system of spin levels is obtained. Perpendicular to the main field

* For this reason crystals of the type \(\mathrm{CH_3Br}\) or \(\mathrm{C_2H_2Br_2}\) are unsuitable, but suitable organic compounds can be selected.

impose a weak alternating field with a frequency in the microwave region. Then, by modulating the magnitude of the main field and measuring the amount of energy absorbed from the alternating field, Zavoisky obtained a curve for the dependence of the absorption coefficient on \(H\); in the general case one obtains a curve with several maxima, and the absorption is maximal when \(h\nu\) (\(\nu\) is the frequency of the alternating field) is equal to the difference between some two spin levels of the paramagnetic ion.

Many experiments have been devoted to paramagnetic resonance in chromium and iron alums. Alums are compounds of the following form:

\[ M^{I}M^{III}(\mathrm{SO}_{4})_{2}\cdot 12\mathrm{H}_{2}\mathrm{O}, \]

where \(M^{I}\) is an ion of a monovalent metal (K, Rb, Na, \(\mathrm{NH}_{4}\)), and \(M^{III}\) is an ion of a trivalent metal of the iron group (Cr, Fe).

A considerable number of experiments have also been devoted to paramagnetic resonance in Tutton salts (compounds of the type \(M_{2}^{I}M^{II}(\mathrm{SO}_{4})_{2}\cdot 6\mathrm{H}_{2}\mathrm{O}\) (\(M^{II}\) is an ion of a divalent metal of the iron group—Co, Cu, Mn, Ni)) and fluorosilicates \(M^{II}\mathrm{SiF}_{6}\cdot 6\mathrm{H}_{2}\mathrm{O}\). Recently, experiments have also been carried out on paramagnetic resonance in ethyl sulfates \(M^{III}(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{SO}_{4})_{3}\cdot 9\mathrm{H}_{2}\mathrm{O}\) (\(M^{III}\) is an ion of a trivalent rare-earth element, for example Ce, Nd, Pr) and double nitrates \(M_{3}^{II}M_{2}^{III}(\mathrm{NO}_{3})_{12}\cdot 24\mathrm{H}_{2}\mathrm{O}\).

As we have already indicated, the paramagnetic-resonance absorption line usually has several maxima. Having found experimentally the absorption line, one can reconstruct the picture of the levels of the paramagnetic ion. In sufficiently strong external fields the basic structure of the levels is obtained owing to the interaction of the spin of the electron shell of the paramagnetic ion with the external magnetic field. The interaction of the shell spin with the intracrystalline electric field, however, gives the fine structure of the spin levels and, correspondingly, the fine structure of the absorption line.

Let us introduce the concept of the so-called effective spin of the electron shell of a paramagnetic ion. Suppose that at \(H=0\) some levels of the shell have energies smaller than or of the order of \(kT\), while other levels have energies much larger than \(kT\); it is clear that the second group of levels will play no role, since these levels are unoccupied. Let us equate the number of levels with energies smaller than on the order of \(kT\) (counting each level as many times as its degeneracy) to the quantity \(2S+1\); the resulting quantity \(S\) is the so-called effective spin of the electron shell of the paramagnetic ion.

For example, the configuration of the electron shell of the \(\mathrm{Co}^{++}\) ion is \(3d^{7}\), and therefore the ground state of the free cobalt ion is the state \({}^{4}F\). However, in paramagnetic salts

of cobalt the intracrystalline electric field, together with the spin–orbit interaction, leads to a partial lifting of the degeneracy, and the lowest level proves (in the absence of an external field) to be doubly degenerate; the nearest other levels lie 400–500° K higher and therefore are not populated at low temperatures (all these results have been obtained from studies of the magnetic susceptibility and paramagnetic resonance of cobalt salts). Therefore in paramagnetic salts the effective spin of Co++ is equal to one half.

Paramagnetic salts are widely used for obtaining ultralow temperatures by adiabatic demagnetization. The lowest temperature achieved (obtained by demagnetizing potassium chromium alums) is equal to 0.0015° K.

b) Interaction of the nuclear spin with the spin of the electron shell of a paramagnetic ion. In those cases where the resonance absorption of salts is studied whose paramagnetic ions contain nuclei possessing spins, the absorption line obtained experimentally has a hyperfine structure.

The study of the fine and hyperfine structures of paramagnetic resonance shows that the following Hamiltonian of the paramagnetic ion describes the experimental data well², ⁸, ⁹ (we neglect quadrupole effects and the direct action of the external magnetic field on the nuclear spin):

\[ V = \beta \{ g_{\parallel} H_z S_z + g_{\perp} (H_x S_x + H_y S_y) \} + D \left( S_z^2 - \frac{1}{3} S(S+1) \right) + \]
\[ + \{ A S_z I_z + B(S_x I_x + S_y I_y) \}. \tag{9} \]

The first term describes the interaction of the shell spin of the paramagnetic ion with the external field, the second—its interaction with the intracrystalline electric field, and the last term gives the interaction of the shell spin with the nuclear spin. Formula (9) holds for axial symmetry of the intracrystalline field at the paramagnetic ion; \(z\) is the symmetry axis, which plays the role of the quantization axis for the shell and nuclear spins. We note that in paramagnetic salts the symmetry of the intracrystalline electric field at the ion is usually rhombohedral or tetragonal, but quite close to axial.

In formula (9), \(S\) is the effective spin of the electron shell of the paramagnetic ion, \(I\) is the nuclear spin, \(g_{\parallel}\) and \(g_{\perp}\) are the \(g\)-factors of the shell spin along and, respectively, perpendicular to the \(z\) axis; \(A\), \(B\), and \(D\) are constants determined from experimental data (the term with \(D\) vanishes for \(S = \frac{1}{2}\)); \(\beta\) is the Bohr magneton.

There are cases when \(A=B\) (alums, Tutton salts of manganese); there are also cases when \(A\ne B\). In particular, in the case of Tutton salts and fluosilicates of cobalt and copper, \(A\) is considerably greater than \(B\).

To obtain oriented nuclei, salts are required for which \(A\) and \(B\) are sufficiently large, and for this reason alums are unsuitable. In the most favorable cases (salts of cobalt, copper, and manganese, as well as salts of certain rare-earth elements) \(A\) and \(B\) are of the order of \(0.01\ \mathrm{cm}^{-1}\) (we note that \(1\ \mathrm{cm}^{-1}=1.45^\circ \mathrm{K}\)*).

It is easy to see that in the case \(A\gg B\), \(S=\frac{1}{2}\), the magnetic field produced by the shell of the paramagnetic ion and acting on the nucleus is equal to \(H'=\frac{AI}{2\mu}\), and if \(A\) is of the order of \(0.01\ \mathrm{cm}^{-1}\), then \(H'\) reaches several hundred kilooersteds (this internal field has such an enormous magnitude because in the electron shell of the paramagnetic ion there are unfilled subshells \(3d\) or \(4f\)). \(H'\) is the same for different isotopes of one and the same element, and hence it is clear that the values of \(A\) (and also of \(B\)) for different isotopes are proportional to their gyromagnetic ratios.

Let us give numerical data for some salts of cobalt, copper, and manganese\(^9\).

Salt	Isotope	\(S\)	\(I\)	\(g_{\parallel}\)	\(g_{\perp}\)	\(D\ (\mathrm{cm}^{-1})\)	\(A\ (\mathrm{cm}^{-1})\)	\(B\ (\mathrm{cm}^{-1})\)
\(\mathrm{Co(NH_4)_2(SO_4)_2\times 6H_2O}\) . . .	\(\mathrm{Co}^{59}\)	\(1/2\)	\(7/2\)	6.45	3.0	—	0.025	0.002
\(\mathrm{CoRb_2(SO_4)_2\cdot 6H_2O}\)	\(\mathrm{Co}^{59}\)	\(1/2\)	\(7/2\)	6.6	2.7	—	0.029	0.005
\(\mathrm{Cu(NH_4)_2(SO_4)_2\times 6H_2O}\) . . .	\(\mathrm{Cu}^{63,65}\)	\(1/2\)	\(3/2\)	2.5	2.0	—	0.012	0.003
\(\mathrm{Mn(NH_4)_2(SO_4)_2\times 6H_2O}\) . . .	\(\mathrm{Mn}^{55}\)	\(5/2\)	\(5/2\)	2.0	2.0	\(-0.028\)	0.009	0.009

c) Blin’s method.\(^8\) Let us consider a paramagnetic salt (the paramagnetic ions of which contain nuclei possessing spins) in the absence of an external magnetic field. In the case when \(A\) is not equal to \(B\), different projections of the nuclear spin on the \(z\)-axis will corres-

* Differences of level energies are often expressed in reciprocal centimeters. If \(\nu\) and \(T\) are one and the same energy, measured in reciprocal centimeters and, respectively, in degrees Kelvin, then it is clear that \(hc\nu=kT\), whence \(T=1.45\nu\).

different energies (although the energies corresponding to the states \(+m\) and \(-m\) coincide), and therefore at a sufficiently low temperature we obtain aligned nuclei (i.e., \(f_1=0\), \(f_2\ne 0\)).

Let us consider the simplest case: \(S=\dfrac{1}{2}\), \(A\gg B\). Neglecting in (9) the small term containing \(B\), we have \(V=AS_zI_z\) and obtain the energy levels

\[ E_m=\mp \frac{A}{2}\,m, \tag{10} \]

i.e. we obtain \(2I+1\) doubly degenerate levels; the difference of the energies of neighboring levels is \(\dfrac{A}{2}\). Figure 3 shows the pattern of levels for \(I=\dfrac{3}{2}\).

The population of states with projections of the nuclear spin \(\pm m\) is proportional to \(\operatorname{ch}\dfrac{mA}{2kT}\). In the case of \(\mathrm{Co}^{59}\) \(\left(I=\dfrac{7}{2}\right)\) we obtain 8 equidistant levels.

At \(kT=0.7A\) (which corresponds to \(T=0.025^\circ\mathrm{K}\) in the case of ammonium cobalt Tutton salt) we obtain the following relative populations of the different spin states:

\[ \pm \frac{7}{2}:\pm \frac{5}{2}:\pm \frac{3}{2}:\pm \frac{1}{2}=1:0.53:0.32:0.21, \]

i.e. a noticeable predominance of nuclear spins directed along and against the \(z\)-axis.

The calculations give:

\[ f_2=\frac{2(I+1)}{2I-1}-\frac{3\,\operatorname{cth}\left(\dfrac{A}{4kT}\right)}{2I-1}\,B_I\left(\frac{AI}{2kT}\right). \tag{11} \]

If \(A\) is not much greater than \(B\), then, using the available formulas for the energy levels\({}^{9}\), one can find \(f_2\) by numerical methods; in the same way one can find \(f_4\), \(f_6\), etc.

For experiments one needs a monocrystal in which the directions of the \(z\)-axes are the same for all ions containing the nuclei that are to be aligned. Fluorosilicates are of this kind in particular; as for Tutton salts, they contain two nonequivalent groups of paramagnetic ions with different \(z\)-axes, as a result of which two different quantization axes arise for the two groups of nuclei; however, in studying the angular distribution of \(\beta\)- and \(\gamma\)-radiation this interfering circumstance (as we shall see below) can be overcome.

When the Blin method is used, the experiment is as follows: a specimen is taken—a single crystal of a diluted paramagnetic salt whose paramagnetic ions contain nuclei possessing spins. This specimen is brought into thermal contact with liquid helium, and a strong magnetic field (of the order of 30 kilooersteds) is applied to it. Then the thermal contact of the salt with the helium is broken, and adiabatic demagnetization of the salt is carried out down to zero field. In this way a very low final temperature is obtained and, correspondingly, a significant alignment of the nuclei (the salt must be sufficiently diluted so that the final temperature obtained in adiabatic demagnetization will be sufficiently low). The specimen under study may also be cooled with another paramagnetic salt.

Let us note that in the case of the Blin method the relaxation will be rapid, since the establishment of equilibrium occurs owing to the interaction of the spin of the shell of the paramagnetic ion with the lattice.

g) The Gorter–Rose method. The principle of this method is as follows: on a paramagnetic salt whose paramagnetic ions contain nuclei possessing spins, at an ultra-low temperature, a field of the order of several hundred oersteds is applied; this causes polarization of the spins of the shells of the paramagnetic ions, owing to which the fields they create at the nuclei are oriented, and, in view of the large magnitude

\[ \frac{H_{\text{int}}}{T} \]

a considerable polarization of the nuclei is obtained.

The experimental scheme is the same as when applying the Blin method, with the difference that the adiabatic demagnetization must be carried out not down to zero field, but down to a field equal to several hundred oersteds. It is clear that the final temperature will be higher than in the case of the Blin method, and correspondingly \(f_2\) will be smaller; however, in contrast to the Blin method, we obtain \(f_1 \ne 0\), i.e. nuclei that are not aligned, but polarized.

The most advantageous case for the Gorter–Rose method is \(A=B,\ D=0\), since in that case the shell spin is free with respect to directions and is easy to orient; this is the situation, for example, for two thirds of the cobalt ions in double nitrates (see below).

Let us note that the relaxation of the nuclear spins will be rapid for the same reasons as in the case of the Blin method.

The chief shortcoming of the indirect methods considered above is their applicability only to a limited number of elements. For example, the Pound method is suitable only for halogen nuclei, while the Blin and Gorter–Rose methods are suitable for nuclei of cobalt, copper, manganese, and some rare-earth elements. A shortcoming of these three methods is also that the specimens in which one obtains

oriented nuclei are not very suitable as targets for nuclear reactions (owing to the presence in them of a large number of different elements). In addition, the application of these three methods requires experiments to be carried out at temperatures of the order of \(0.01^\circ\) K, which greatly complicates the experiments.

Let us note that in work\({}^{10}\) expressions for \(f_1\) and \(f_2\) are given for these three methods in the case of weak orientation.

6. Polarization of nuclei in metals

Recently Overhauser proposed\({}^{11}\) a very interesting method for obtaining polarized nuclei in metals. In order to understand the principle of this method, it is first necessary to analyze two phenomena, namely, the magnetic relaxation of conduction electrons and saturation of their paramagnetic resonance.

a) Magnetic relaxation of conduction electrons.\({}^{12}\) Suppose that a sufficiently strong external magnetic field is applied to a metal. Then in the metal we shall have two groups of conduction electrons: electrons with spins parallel to the field (we shall denote them by the sign \(+\)), and electrons with spins directed opposite to the field (we denote them by the sign \(-\)). Each of these groups of electrons will have its own Fermi distribution. We shall denote the numbers of these electrons per unit volume, respectively, by \(N_+\) and \(N_-\), the corresponding Fermi boundaries by \(\varepsilon_+\) and \(\varepsilon_-\), and the Fermi boundary in the absence of an external field by \(\varepsilon_0\). We have:

Fig. 4.

\[ \left. \begin{aligned} N_- + N_+ &= N,\\ N_- - N_+ &= D, \end{aligned} \right\} \tag{12} \]

where \(N\) is the total number of conduction electrons per unit volume of the metal, and \(\beta D\) is the magnetization caused by the conduction electrons.

At equilibrium the limiting energies \(\varepsilon_+\) and \(\varepsilon_-\) are equal to one another (and equal to \(\varepsilon_0\)), and the distributions have the form shown in Fig. 4 (the functions \(n_+\) and \(n_-\) denote the distribution functions for two

groups of electrons, $\varepsilon$ is the total energy of the electron, including the energy in the magnetic field; scales are not observed; in reality $\beta H \ll \varepsilon_0$).

For the equilibrium excess $D_0$ it is easy to obtain:

\[ D_0=\frac{3}{2}N\frac{\beta H}{\varepsilon_0}. \tag{13} \]

If there is no equilibrium (i.e., $D\ne D_0$), then, owing to the interaction of the spins of the conduction electrons with the lattice, equilibrium is established, and the approach to equilibrium will proceed according to the simple relaxation law

\[ \dot D=\frac{D_0-D}{T_1}. \tag{14} \]

$T_1$ is called the time of magnetic relaxation of the conduction electrons (the dot over $D$ denotes differentiation with respect to time).

In work$^{12}$ the author theoretically considered six possible mechanisms of magnetic relaxation of conduction electrons. Among these mechanisms, the strongest proves to be the interaction of the spin of a conduction electron with the orbital moments of other conduction electrons, i.e., with the currents caused by the translational motion of other conduction electrons. The calculation shows that, for this mechanism, the relaxation time is inversely proportional to the absolute temperature. Theoretically, for lithium at room temperature one obtains $T_1 \simeq 10^{-6}$ sec, whereas according to experiment $T_1 \simeq 10^{-8}$ sec. Thus, one must conclude that, in addition to the relaxation mechanisms considered in work$^{12}$, there exists some still stronger mechanism. Thus, in work$^{12}$ the most important mechanism of magnetic relaxation of conduction electrons was not identified, and therefore for the time being nothing can be said about the temperature dependence of the relaxation time either (there have as yet been no measurements of the temperature dependence of $T_1$).

Among the relaxation mechanisms considered in work$^{12}$, of particular interest is the interaction of the spin of a conduction electron with the spin of the nucleus. The author assumes that this interaction has the form:

\[ V_{\mathrm{int}}=-\frac{8\pi}{3}\mu\mu_e\delta(\mathbf r), \tag{15} \]

where $\mu_e$ is the magnetic moment of the conduction electron, $\mathbf r$ is its radius vector relative to the nucleus, and $\delta$ is the delta function. (At first glance this expression raises some doubts. The point is that the interaction energy of the spin of an $s$-electron of an isolated atom with the spin of the nucleus has this form. However, detailed consideration shows that in the case of a metal as well formula (15) is valid to a high degree of accuracy.)

The expression (15) is proportional to \(\mathbf{SI}\) (\(\mathbf{S}\) is the spin of the conduction electron). On the other hand, it is easy to see that the operator \(\mathbf{S}+\mathbf{I}\) commutes with \(\mathbf{SI}\). It follows that \(\mathbf{S}+\mathbf{I}\), in the case of the interaction given by (15), is an integral of motion. In particular, we obtain that, in transitions caused by this interaction, the total projection of the spins of the conduction electron and the nucleus on the external field is conserved, i.e., when the electron spin is flipped from the direction parallel to the field into the antiparallel direction (the transition \(+\to-\)), the spin projection of one of the nuclei increases by one unit.

According to the calculation carried out in \({}^{12}\), the relaxation time of this process does not depend on temperature and for lithium is approximately \(10^{-4}\) sec. Since this time is much greater than the relaxation time of the electron spin, this process is insignificant for the magnetic relaxation of conduction electrons.

Such reorientations, as emphasized in work \({}^{11}\), are possible only in the case of a metal, since it is necessary that, when the spin is reoriented, the electron be able to change its kinetic energy by the small amount \(2\beta H\), in order to compensate the change in magnetic energy.

b) Saturation of the paramagnetic resonance of conduction electrons. Let, in addition to the constant field \(H\), an alternating field also be applied to the metal (perpendicular to the constant field), whose frequency \(\nu\) is equal to the Larmor frequency of the electron spin in the field \(H\). In other words,

\[ h\nu = 2\beta H . \tag{16} \]

For \(H=10\) kilooersteds we obtain \(\nu=2.8\cdot 10^{10}\) hertz, i.e., a wavelength of about 1 centimeter. We shall denote the amplitude of the alternating field by \(2H_1\). We assume that \(H_1 \ll H\).

It is easy to see that the probability that the alternating field transfers a conduction electron from the state with spin parallel to the field into the antiparallel state is equal to the probability of the reverse process. However, because initially \(N_- > N_+\), the number of transitions \(-\to+\) will be greater and the excess \(D=N_- - N_+\) will decrease.

Usually, in the experimental investigation of magnetic resonance, one measures the amount of energy of the source of the alternating field that is absorbed (owing to magnetic resonance) per unit time. It is easy to obtain an expression for this quantity.

The alternating field tends to reduce \(D\) to zero, whereas the interaction of the spins of the conduction electrons with the lattice tends to make \(D\) equal to \(D_0\). It is easy to obtain:

\[ \dot D=\frac{D_0-D}{T_1}-2WD, \tag{17} \]

where \(W\) is the probability of reorientation of the electron spin under the influence of the alternating field per unit time. It can be derived that

\[ W=\frac{1}{2}(\gamma_e H_1)^2T_2, \tag{18} \]

where \(\gamma_e\) is the gyromagnetic ratio of the electron \(\left(\gamma_e=\frac{2\beta}{\hbar}\right)\), and \(\frac{1}{2T_2}\) is the width of the resonance absorption line in oersteds.

In the stationary case (17) and (18) give:

\[ D=\frac{D_0}{1+2WT_1}=\frac{D_0}{1+(\gamma_e H_1)^2T_1T_2}, \tag{19} \]

where \(D_0\) is determined by formula (13). We see that, for a sufficiently large amplitude of the alternating field, \(D\) is considerably smaller than \(D_0\), i.e., the so-called saturation of the paramagnetic resonance of conduction electrons takes place.

For the power absorbed per unit volume we obtain (in the stationary case):

\[ P=Dh\nu W, \]

or

\[ P=D_0h\nu\frac{W}{1+2WT_1} =D_0\frac{h\nu}{2}\frac{(\gamma_e H_1)^2T_2}{1+(\gamma_e H_1)^2T_1T_2}. \tag{20} \]

In particular, for

\[ H_1\gg\frac{1}{\gamma_e\sqrt{T_1T_2}} \]

\[ P_\infty=D_0\frac{h\nu}{2T_1} =\frac{3}{2}\,N\frac{(\beta H)^2}{\varepsilon_0T_1}. \tag{21} \]

Thus, for small \(H_1\), \(P\) is proportional to \(H_1^2\), while for sufficiently large \(H_1\), \(P\) does not depend on \(H_1\). One may introduce a critical value of the amplitude of the alternating field \(H_1^c\):

\[ H_1^c=\frac{1}{\gamma_e\sqrt{T_1T_2}}. \tag{22} \]

For \(H_1\ll H_1^c\) the role of the alternating field is small, while for \(H_1\gtrsim H_1^c\) saturation effects are substantial. According to work \(^{11}\), for lithium at room temperature \(H_1^c=2\text{--}3\) oersteds. As the temperature decreases, \(T_1\) and \(T_2\) increase, and therefore at low temperatures \(H_1^c\) will be smaller.

For a quantitative characterization of the saturation of the resonance of conduction electrons, work \(^{11}\) introduces the quantity

\[ s=1-\frac{D}{D_0} =\frac{(\gamma_e H_1)^2T_1T_2}{1+(\gamma_e H_1)^2T_1T_2}. \tag{23} \]

For \(H_1\ll H_1^c\), \(s\simeq0\), while for \(H_1\gg H_1^c\), \(s\simeq1\) (complete saturation).

According to the theory of the skin effect (see, for example, \(^{13}\)), the thickness of the skin layer \(\delta\) is equal to (\(\sigma\) is the conductivity)

\[ \delta=\frac{c}{2\pi\sqrt{\nu\sigma}} . \tag{24} \]

Since in paramagnetic-resonance experiments \(\nu\) is large, \(\delta\) turns out to be very small. For example, at \(\nu=2.8\cdot 10^{10}\) hertz, for lithium at room temperature one obtains \(\delta=1.7\) microns. As the temperature is lowered, the conductivity increases and \(\delta\) decreases still more. Therefore, in carrying out paramagnetic-resonance experiments on metals it is necessary to use specimens consisting of very small particles.

c) Obtaining nuclear polarization in metals. As we saw above, when an alternating field of the Larmor frequency and of sufficiently large amplitude \((H_1>H_c^*)\) is applied to a metal, the excess of electrons with spins opposite to the field (i.e., \(D\)) decreases. In other words, the alternating field induces transitions of type \(A\) (see Fig. 4). (More precisely, there will be more transitions of type \(A\) than reverse transitions.) In the case of complete saturation of the resonance of the conduction electrons, \(D\) becomes zero, and we obtain two Fermi distributions shifted relative to one another by \(2\beta H\) (Fig. 5).

Fig. 5.

Let us consider the effects caused by the above-considered interaction of the spin of a conduction electron with the spin of a nucleus under saturation of the electron resonance. The corresponding transitions are denoted in Fig. 5 by \(B(+\to -)\) and \(D(-\to +)\). However, if \(\beta H\) is greater than, or of the order of, \(kT\), the \(B\) transitions will be considerably more probable than the \(D\) transitions, since the latter are hindered by the Pauli principle. After a transition of type \(B\), the electron very rapidly (in a time of order \(10^{-13}\) sec.) undergoes a transition of type \(C\), and then the alternating field again induces a transition \(A\).

But, on the other hand, when a conduction electron undergoes a transition of type \(B\), the projection of one of the nuclear spins increases by unity. Hence it is clear that if \(\beta H\gg kT\), a one-sided transfer of nuclear spins into the state with

with spins parallel to the external field (for \(\beta H \gg kT\), transitions of type \(D\) are almost completely forbidden, since the width of the smearing of the Fermi distribution is of order \(kT\)), and we obtain complete polarization of the nuclei. If, however, \(\beta H \sim kT\), then we obtain partial polarization of the nuclei.

The calculation presented in paper \(^{11}\) shows that, owing to the processes considered above, a nuclear polarization is obtained in the metal corresponding to the following effective gyromagnetic ratio:

\[ \gamma_{\mathrm{eff}}=\gamma+\frac{s|\gamma_e|}{I} \tag{25} \]

(where \(\gamma\) is the true gyromagnetic ratio of the nucleus). Thus, for \(s\sim 1\), \(\gamma_{\mathrm{eff}}\) is of the order of the electronic gyromagnetic ratio, and therefore even at helium temperatures one can obtain a considerable degree of polarization of the nuclei.

For example, for lithium at \(H=10\) kilooersteds, \(T=2^\circ\mathrm{K}\), and \(s=0.8\), one obtains \(f_I=0.42\).

c) Advantages and disadvantages of the Overhauser method. An important advantage of the Overhauser method is that this method should already be effective at helium temperatures, whereas the other methods considered above require the use of temperatures of order \(0.01^\circ\mathrm{K}\).

On the other hand, the Overhauser method also has its disadvantages. Chief among them is the necessity of working with samples containing very small metal particles, for example with colloidal solutions. For this reason these samples will not be good targets for nuclear reactions. In paper \(^{11}\) it is noted that it will be advantageous to add impurities to the metal (in order to increase the thickness of the skin layer).

The method also has another disadvantage: in order for saturation of the magnetic resonance of the conduction electrons to take place, the absorbed power must be comparatively large.

We note that in paper \(^{11}\) it is shown that, upon polarization of nuclei in a metal, there should occur a shift of the resonance frequency of the conduction electrons. The author proposes using this shift for the experimental determination of the degree of polarization of the nuclei. The degree of polarization of the nuclei can also be determined by means of nuclear magnetic resonance.

7. Polarization of atoms in beams

The methods considered by us up to now make it possible to obtain oriented nuclei in condensed media. In contrast to them, the method proposed by Kastler \(^{14}\) is suitable for orienting nuclei (more precisely, atoms) in atomic beams.

For concreteness let us consider the case of a beam of sodium atoms. The ground and first excited states of the electron shell of sodium are the states \(3^2S_{1/2}\) and \(3^2P_{1/2}\). Taking the hyperfine structure into account, each of these levels will consist of two close sublevels with total atomic angular momentum \(F\) equal to one and, respectively, two (since the spin of the \(\mathrm{Na}^{23}\) nucleus is equal to \(\dfrac{3}{2}\)).

The scheme of the lower levels of the sodium atom in the absence of an external field is shown in Fig. 6.

Let a constant magnetic field be imposed on the beam, parallel to the direction of propagation of the beam, and let this field be so weak that it does not destroy the coupling of the nuclear spin with the spin of the shell. In this case we obtain sublevels, each of which is characterized by the quantum number \(m_F\) (the projection of \(F\) on the direction of the external field).

Fig. 6.

Fig. 7.

We shall irradiate the beam with circularly polarized light with the direction of propagation along the lines of force of the external magnetic field and with a frequency equal to the frequency of the transition \(3^2S_{1/2}(F=1)\to 3^2P_{1/2}(F=2)\). Let, for definiteness, the light have clockwise polarization \((\sigma^+)\). The selection rule for excitation by \(\sigma^+\) light is: \(\Delta m_F=+1\), as a result of which we obtain the transitions indicated in Fig. 7. During emission there will occur both \(\sigma^+\), and \(\sigma^-\), and \(\pi\)-transitions, owing to which we obtain an increase in the number of atoms in the state with \(m_F=+1\), i.e. we obtain a beam of polarized atoms*) (in reality the picture is somewhat more complicated because of the possibility of transitions to the level \(3^2S_{1/2}(F=2)\), but even taking this fact into account we obtain polarization of the atoms).

The situation will be completely analogous when the beam is irradiated with electromagnetic waves causing transitions between

*) In paper 14 it is shown that upon irradiation with linearly polarized or natural light, aligned atoms are obtained.

levels with the \(F=1\) and \(F=2\) states of \(3^2S_{1/2}\). In doing so, however, microwaves must be used. Let us also note that in this case the transitions will be magnetic-dipole transitions (an electric-dipole transition will be forbidden, since \(\Delta l=0\)).

In paper \(^{14}\) it is indicated that the experiments can also be carried out in the absence of a constant external magnetic field. In that case the direction of propagation of the circularly polarized wave will play the role of the axis of quantization.

If the experiments are carried out not with an atomic beam, but with a condensed body, then one hindering circumstance will be that collisions with the lattice will cause depolarization of the atoms. Similarly, in the case of a gas, depolarization will also occur owing to reorientations during collisions between molecules. In the case of an atomic beam, however, the effect of collisions is small and depolarization will hardly occur.

To determine the degree of polarization of atoms in a beam, Kastler proposes the following method: the beam passes through a region in which \(\pi\)-excitation takes place (i.e., the beam is illuminated with linearly polarized light with the electric vector parallel to the external magnetic field). \(\pi\)-excitation causes transitions with \(\Delta m_F=0\).

Kastler then proposes to measure the ratio of the intensities of the \(\sigma^+\) and \(\sigma^-\) lines upon fluorescence. It is easy to see that for an unpolarized beam of atoms this ratio will be equal to unity; in the case of a polarized beam this ratio will differ from unity and will depend on the degree of polarization of the atoms.

Kastler proposes an experiment analogous to the well-known magnetic-resonance experiments of Rabi (see, for example, \(^{15}\)), in which the inhomogeneous magnets are replaced by an optical exciter (in which \(\sigma^+\)-excitation takes place) and by an optical detector of the degree of polarization of the beam. The principle of this proposed experiment is that magnetic resonance (which takes place between the optical exciter and the optical detector) reduces the excess created by the optical exciter.

As indicated in paper \(^{14}\), the advantage of this method over Rabi’s method is its applicability to metastable states, as well as the possibility of performing experiments in weak fields.

Recently, experiments have been carried out by Kastler’s method \(^{16–18}\); however, we shall not dwell on an analysis of these experiments.

In conclusion, let us note that a beam of polarized atoms can be used as a source for obtaining polarized nuclei.

II. EXPERIMENTS WITH ORIENTED NUCLEI

8. β- and γ-Radiation of Oriented Nuclei

a) Theoretical results.

The theoretical consideration of the β- and γ-radiation of oriented nuclei was carried out in works \(^{19-26}\).

The angular distribution of the β- and γ-radiation of oriented nuclei is, generally speaking, anisotropic.

Let \(S_m^I(\vartheta)\) be the probability of emission of a γ-quantum (summed over the polarization directions of the γ-radiation) by a nucleus with spin \(I\) and spin projection \(m\), \(\vartheta\) the angle of emission of the γ-quantum relative to the quantization axis of the nuclear spins, and \(W(\vartheta)\) the total angular distribution of γ-quanta from the ensemble of nuclei observed in the experiment. It is clear that

\[ W(\vartheta)=\sum_m a_m^I S_m^I(\vartheta), \tag{26} \]

where \(a_m^I\) is the relative population of the state with nuclear-spin projection \(I\) equal to \(m\).

According to the theory \(^{20,21}\), for radiation of a multipole of order \(2^l\) (the angular distribution does not depend on whether the transition is an electric or magnetic multipole transition) one obtains:

\[ W(\vartheta)=1+c_2 f_2 P_2(\cos \vartheta)+\ldots+c_{2n} f_{2n} P_{2n}(\cos \vartheta), \tag{27} \]

where \(n\) is an integer, with \(n\le I\) and \(n\le l\), \(P_k(\cos \vartheta)\) is the \(k\)-th Legendre polynomial, and the coefficients \(c\) depend on \(I\) and on the multipolarity of the transition; their values for different cases are given in works \(^{20,21}\).

We see that only even \(f\)’s enter into the angular distribution of γ-radiation. In particular, it follows from (27) that the angular distribution of γ-radiation always possesses spherical symmetry not only for \(I=0\), but also for \(I=\frac{1}{2}\).

Usually the γ-radiation follows a β-transition, in which the degree of orientation of the nuclear spins, generally speaking, decreases. Consider the chain of transitions:

\[ (I_0,m_0)\xrightarrow{\beta}(I,m)\xrightarrow{\gamma}(I_f,m_f). \]

We obtain:

\[ a_m^I=\sum_{m_0} a_{m_0}^{I_0} P(m_0,m), \tag{28} \]

where \(a_{m_0}^{I_0}\) is the relative population of the state with projection

of the nuclear spin \(I_0\), equal to \(m_0\), and \(P(m_0,m)\) is the relative probability of the transition \(I_0,m_0 \to I,m\). The functions \(P(m_0,m)\) for various cases are given in Ref. \(^{22}\), and \(P(m_0,m)\) depends on the type of \(\beta\)-transition. The same formula (28) is also applicable in the case of two successive \(\gamma\)-transitions.

Let us note that formula (28) is valid only in the case where the lifetime of the intermediate state is so short that during this time the external and internal fields do not have time appreciably to change the direction of the spin of the intermediate nucleus. In such a case the population of the spin levels in the intermediate state is completely determined by the distribution over the spin levels of the initial nucleus and by the type of transition under consideration.

In Refs. \(^{20-22}\) expressions for \(W(\vartheta)\) are given for all cases of experimental interest. In particular, expressions are given for the total angular distribution of cascade \(\gamma\)-quanta, as well as for the angular distribution of the \(\gamma\)-radiation following \(\beta\)-decay.

In Refs. \(^{20,21}\) expressions are also given for the angular correlation of two cascade \(\gamma\)-quanta emitted by oriented nuclei, and for the polarization of \(\gamma\)-radiation. It turns out that only even \(f\) enter the formula for the angular correlation of two \(\gamma\)-quanta, as well as the formula for the degree of plane polarization of \(\gamma\)-radiation, whereas only odd \(f\) enter the formula for the circular polarization of \(\gamma\)-radiation. It also follows from the theory that the polarization of \(\gamma\)-radiation (in contrast to the angular distribution and angular correlation) depends on whether the transition is an electric or magnetic multipole transition (i.e., the polarization of \(\gamma\)-radiation depends not only on the change of spin, but also on the change of parity in the \(\gamma\)-transition).

The plane polarization of \(\gamma\)-radiation can be measured by observing Compton scattering of \(\gamma\)-quanta at different angles. The study of circular polarization of \(\gamma\)-radiation is more difficult. For this it is necessary to measure the scattering of \(\gamma\)-quanta by a substance containing polarized electrons (for example, iron magnetized to saturation).

Let us now consider \(\beta\)-radiation from oriented nuclei. The theory shows \(^{23}\) that the angular distribution of \(\beta\)-radiation from oriented nuclei (summed over the polarization directions of the \(\beta\)-radiation) possesses spherical symmetry in the case of allowed transitions, as well as in the case of such forbidden transitions whose spectrum has an allowed form. For other forbidden \(\beta\)-transitions the \(\beta\)-radiation must be anisotropic; only even \(f\) enter the formulas for the angular distribution \(^{24,25}\). Let us note that the angular distribution of \(\beta\)-radiation is expressed by a formula of the type (27), with the role of \(l\) played by the total angular momentum of the emitted electron (or positron) and neutrino.

β-radiation of polarized nuclei must, generally speaking, be polarized \(^{23}\); the formulas for the degree of polarization contain only odd \(f\). We note that polarization of β-radiation occurs both for allowed and for forbidden β-transitions.

The experimental study of β-radiation from oriented nuclei is much more complicated than the study of γ-radiation, since β-radiation is strongly absorbed by the apparatus that maintains the low temperature. The study of the polarization of β-radiation from polarized nuclei presents particularly great difficulties.

Let us note that in works \(^{24,26}\) the general case is considered, when the quantization of nuclear spins does not possess axial symmetry; an expression is derived for the angular distribution of β- or γ-radiation, and also an expression for the angular correlation of successive radiations emitted by oriented nuclei.

Let us now see what information can be obtained by experimental investigation of the β- and γ-radiation of oriented nuclei.

By studying the angular distribution of γ-radiation from oriented nuclei one can obtain information on the multipolarity of γ-transitions, as well as on the spins of excited states of nuclei. In the case when there are successive γ-transitions, these same data are more easily obtained by studying the \((\gamma-\gamma)\)-angular correlation of chaotic nuclei. If the γ-transition is preceded by a β-transition with a forbidden spectrum, then these same data can be obtained by studying the \((\beta-\gamma)\)-angular correlation of chaotic nuclei \(^{27}\). In the remaining cases, however, the multipolarity of a γ-transition is usually determined by measuring the internal conversion coefficient. However, first, the data obtained in this way are not especially reliable, and, second, by measuring the internal conversion coefficient one cannot determine the spins of the excited states of nuclei. Thus, from the point of view of determining the multipolarity of a γ-transition and the spins of excited nuclear states, experimental study of the angular distribution of γ-radiation from oriented nuclei is of considerable interest in the case of simple γ-radiation not following β-decay with a forbidden spectrum.

Moreover, and this is especially important, from experimental data on the angular distribution of γ-radiation one can determine the magnetic moments of β-active nuclei (see below).

Measurement of the angular distribution of β-radiation also makes it possible to determine the spins and magnetic moments of β-active nuclei.

Let us also note that experimental study of γ-radiation from oriented nuclei makes it possible to draw certain conclusions about the form of the Hamiltonian of the β-interaction. In particular, in work \(^{22}\) it was shown that the quantities \(P(m_0, m)\) entering into (28) depend on the relative fraction of the Fermi interaction in the β-Hamiltonian (relative to the Fermi and Gamow–Teller ...).

interactions; see, for example, \(^{27}\)). A numerical calculation of the angular distribution of the \(\gamma\)-radiation of \(\mathrm{Co}^{58}\) and \(\mathrm{Co}^{60}\) as a function of the relative fraction of the Fermi interaction was carried out in work \(^{28}\).

If one measures the polarization of the \(\gamma\)-radiation of oriented nuclei, it is possible to determine whether the \(\gamma\)-transition is an electric or magnetic multipole transition; i.e., in other words, it is possible to draw a conclusion about the parities of the excited states of nuclei (if there are successive \(\gamma\)-transitions, or if the \(\gamma\)-transition is preceded by a \(\beta\)-transition with a forbidden \(\beta\)-spectrum, then the same information can be obtained by carrying out polarization-correlation experiments \(^{27}\) with randomly oriented nuclei).

Measurement of the polarization of the \(\beta\)-radiation of polarized nuclei is of enormous interest \(^{23}\), since in this way it would be possible to determine unambiguously the form of the Hamiltonian of the \(\beta\)-interaction. However, as was indicated above, experiments of this type are extremely difficult.

The \(\alpha\)-radiation of oriented nuclei must also be anisotropic \(^{29}\) if the spin of the nucleus is greater than one-half. Indeed, for \(I > 1\) the nucleus has a quadrupole moment, which means that the charge distribution of the nucleus possesses axial symmetry with an axis of rotation coinciding with the spin. In view of this, the height of the barrier will depend on the angle made by the direction of motion of the emitted \(\alpha\)-particle with the nuclear spin. Therefore, if the nuclear spins are oriented, the angular distribution of \(\alpha\)-particles will be anisotropic. Experiments of this kind would make it possible to determine the quadrupole moments of \(\alpha\)-active nuclei.

b) Experiments with \(\gamma\)-radiation of oriented nuclei. During the last three years a fairly large number of experiments have been carried out devoted to the study of the anisotropy of the \(\gamma\)-radiation of oriented cobalt nuclei. In most experiments, in order to obtain oriented nuclei, the authors used the Bleaney method.

In the experiments of the Oxford group \(^{30—32}\) the following Tutton salt was used as the specimen:

\[ (1\% \ \mathrm{Co},\ 12\% \ \mathrm{Cu},\ 87\% \ \mathrm{Zn}) \ \mathrm{Rb}_{2}(\mathrm{SO}_{4})_{2}\cdot 6\mathrm{H}_{2}\mathrm{O}. \]

Zinc is a nonmagnetic diluent, the copper ions are cooling agents, and the cobalt nuclei are oriented (for details we refer to work \(^{32}\)).

Tutton salts belong to the monoclinic system. Measurements of the susceptibility, as well as experiments on paramagnetic resonance, show that the unit cell of Tutton salts contains two nonequivalent divalent ions, each of which is located in an intracrystalline electric-

... field, possessing tetragonal (close to axial) symmetry. Let us denote the directions of the two tetragonal axes by \(T_1\) and \(T_2\). A single crystal of Tutton’s salt has three principal directions of the magnetic-susceptibility tensor: \(K_1\) and \(K_3\) are the bisectors of the two angles between the tetragonal axes \(T_1\) and \(T_2\) (\(K_3\) is parallel to the monoclinic axis \(b\)), and \(K_2\) is the direction perpendicular to the plane containing the two tetragonal axes (Fig. 8; the axis \(K_2\) is perpendicular to the plane of the drawing).

Thus, the salt under consideration contains two groups of cobalt ions with different axes of quantization of the nuclear spin (the axis of quantization of the cobalt nuclear spin is the symmetry axis of the intracrystalline electric field at the ion). For the cobalt ions in the case of the salt considered, the angle \(\alpha\) is approximately equal to \(37^\circ\).

Fig. 8.

It is clear that if one measures the intensity of \(\gamma\)-radiation along directions parallel to the plane \(K_1K_2\), then the \(\gamma\)-radiation of both groups of cobalt nuclei will be measured under identical conditions (since a direction lying in the plane \(K_1K_2\) makes the same angles with the tetragonal axes \(T_1\) and \(T_2\)). In particular, the intensity of radiation along \(K_2\) corresponds to \(W\!\left(\frac{\pi}{2}\right)\), while along \(K_1\) it gives the distribution \(W(\alpha)\).

Fig. 9.

The decay scheme of \(\mathrm{Co}^{60}\), as well as the spin and parity values of the initial, final, and intermediate states, are sufficiently well known\({}^{33}\) from \((\gamma-\gamma)\)-correlation and \((\gamma-\gamma)\)-polarization-correlation experiments. In Fig. 9 we give a simplified scheme. Both \(\gamma\)-quanta are electric quadrupole quanta; the \(\beta\)-transition is allowed.

Let us describe the experiments. The authors took six single crystals of the named salt with a total weight of \(4\ \mathrm{g}\), which contained \(70\) microcuries of \(\mathrm{Co}^{60}\). With these crystals the authors carried out adiabatic

demagnetization from the initial field \(\dfrac{H}{T}=30\ \dfrac{\text{kilooersted}}{\text{K}}\) to zero field. The final temperature was about \(0.01^\circ\text{K}\).

The \(\gamma\)-radiation measurements were carried out with Geiger–Müller counters arranged along the directions \(K_1\) and \(K_2\). In parallel, the authors also measured the magnetic susceptibility (from the susceptibility the authors determined the temperature). The temperature of the crystals gradually increased; the heating was caused mainly by the absorption of \(\beta\)-particles (about \(100\ \text{erg}/\text{min}\)). Each series of measurements lasted 5 minutes.

As a result of the measurements, the authors obtained the dependence

\[ \frac{W\!\left(\frac{\pi}{2}\right)}{W(a)} \]

on temperature. Introduce the quantity \(\varepsilon\), characterizing the degree of deviation of the angular distribution from isotropy,

\[ \varepsilon = \frac{W\!\left(\frac{\pi}{2}\right)-W(a)} {W\!\left(\frac{\pi}{2}\right)} . \]

Experimental data are given in Fig. 10. Theoretically, the dependence of \(\varepsilon\) on

\[ \frac{A}{2kT} \]

was determined (assuming that the \(\beta\)-transition is an allowed \(5 \to 4\) transition and taking into account the decrease in the degree of orientation in \(\beta\)-decay). By matching the theoretical curve with the experimental one, the authors found the value of \(A\) for \(\mathrm{Co}^{60}\); taking into account that the values of \(A\) for different isotopes are proportional to their gyromagnetic ratios (see Section 5б) and using the known data for \(\mathrm{Co}^{59}\), the authors determined that the magnetic moment of \(\mathrm{Co}^{60}\) is equal to \(3.5 \pm 0.5\) nuclear magnetons.

Fig. 10.

Let us note that the obtained value of the magnetic moment of \(\mathrm{Co}^{60}\) agrees well with the predictions of the shell model. Indeed, according to the latter, the odd proton and neutron of the \(\mathrm{Co}^{60}\) nucleus are in the states \(f_{7/2}\) and, respectively, \(p_{3/2}\). In the case of parallel spins, according to the shell model we obtain the state \(5+\) with \(\mu=3.8\) nuclear magnetons.

Similar experiments were also carried out by the Leiden group \(^{34-36}\), the authors performing measurements along different directions in the \(K_1K_2\) plane.

Thus, in the experiments with \(\mathrm{Co}^{60}\) the magnetic moment of this nucleus was determined and, moreover, the decay scheme and the value were confirmed …

of spins and parities obtained earlier by correlation experiments. Similar experiments were carried out by the Oxford group\(^{37}\) also in the case of \(Co^{58}\), but in this case the situation is different. As a result of \(K\)-capture (and also, in part, of \(\beta^+\)-decay), \(Co^{58}\) passes into an excited state of \(Fe^{58}\), which then de-excites by emission of one \(\gamma\)-quantum. Since there is only one \(\gamma\)-quantum, and since the \(\beta^+\)-transition is allowed, correlation experiments can yield nothing. By measuring the angular distribution of the \(\gamma\)-radiation and comparing the results with theoretical data, the authors proved that the \(\gamma\)-transition is quadrupole, and determined that the magnetic moment of \(Co^{58}\) is equal to \(3.5 \pm 0.3\) nuclear magnetons.

In Section 8a it was noted that the \(\gamma\)-radiation of oriented nuclei must be plane-polarized. It was also indicated there that the degree of polarization of the \(\gamma\)-radiation depends on whether the transition is an electric or magnetic multipole transition. The corresponding experiments were carried out by the Oxford group\(^{38}\). In these experiments use was made of the circumstance that the differential cross section of Compton scattering depends on the polarization of the \(\gamma\)-quantum. Without dwelling on the experiments themselves, let us note that the authors confirmed that both \(\gamma\)-quanta emitted following the \(\beta\)-decay of \(Co^{60}\) are electric quadrupole quanta. Analogous experiments carried out with \(Co^{58}\) showed that the \(\gamma\)-quantum emitted in this case is likewise an electric quadrupole quantum.

Let us also consider the experiments of Halban and others\(^{39}\) with double nitrates. The authors used the salt

\[ (0.5\%Co;\ 99.5\%Mg)_3Ce_2(NO_3)_{12}\cdot 24H_2O. \]

The cerium ions play the role of cooling agents, while the cobalt nuclei are oriented.

Investigations by paramagnetic resonance show that in double nitrates, for two thirds of the cobalt ions \(A = B\), while for the remaining third \(A \gg B\). The authors’ aim was, using the Gorter–Rose method, to obtain polarized \(Co^{60}\) nuclei and to study their \(\gamma\)-radiation.

The cerium ions in double nitrates possess an interesting property: the \(g\)-factor of the cerium ion along the hexagonal axis (the single crystals of double nitrates have hexagonal symmetry) is much larger than in the perpendicular directions; therefore applying a magnetic field to the cooled salt in a direction perpendicular to the hexagonal axis almost does not raise the temperature of the salt.

The authors took 12 single crystals (of total weight 4 g) with parallel axes; the crystals contained 50 microcuries of \(Co^{60}\). With these crystals the authors carried out adiabatic demagneti-

... cooling (with a field parallel to the hexagonal axis) from the initial

\[ \frac{H}{T}=25\ \frac{\text{kiloersteds}}{^\circ\mathrm{K}} \]

down to zero field; the final temperature was about \(T=0.004^\circ\mathrm{K}\). The authors then applied a magnetic field perpendicular to the hexagonal axis and used counters to measure the intensity of \(\gamma\)-radiation along the polarizing field \((W(0))\)

Fig. 11.

and in the perpendicular direction \(\left(W\left(\frac{\pi}{2}\right)\right)\). The results of the experiments are shown in Fig. 11

\[ \left( \varepsilon= \frac{ W\left(\frac{\pi}{2}\right)-W(0) }{ W\left(\frac{\pi}{2}\right) } \right). \]

As was to be expected, the anisotropy increases with increasing polarizing field.

9. Magnetic resonance of oriented nuclei

In paper \(^{40}\) it is reported that the authors propose to carry out the following experiment: a crystal with oriented nuclei is taken; some of the oriented nuclei are radioactive and their \(\gamma\)-radiation is anisotropic. An alternating magnetic field is applied to the crystal. It is clear that if its frequency is equal to the energy difference of any two spin levels of the radioactive nucleus divided by \(h\), then the alternating field will cause a decrease in the excess of nuclei in the lower spin state and, correspondingly, the degree of anisotropy of the \(\gamma\)-radiation will decrease; thus, by varying the frequency of the alternating field and determining the frequency at which the effect decreas-

...the anisotropy of the $\gamma$-radiation takes place, we find the resonance frequency of the radioactive nuclei $\nu_0$. In the case where the orientation of the nuclei is caused by an external field, having found $\nu_0$, we immediately find the gyromagnetic ratio of the radioactive nucleus; in the case, for example, of orientation by the Blin method, comparing $\nu_0$ of the radioactive nucleus with $\nu_0$ of a stable isotope and using the value of the gyromagnetic ratio of the latter, we find the gyromagnetic ratio of the radioactive nucleus.

These experiments are intended to be carried out for the purpose of determining the magnetic moments of radioactive isotopes.

10. Experiments with Metals

Recently experiments$^{41}$ have been carried out with the aim of testing the Overhauser theory.

As a sample the authors took five cubic centimeters of lithium dispersed in oil. The lithium particles had sizes of about one micron. A constant field $H = 30.3$ oersted was applied to the sample and, perpendicular to it, an alternating field with an amplitude of 4 oersted and frequency 84 Mc (the Larmor frequency of the electron spin in a field of 30.3 oersted). The experiments were carried out at room temperature.

The authors measured the nuclear magnetic resonance of lithium in the absence and in the presence of saturation of the resonance of the conduction electrons. In the first case the nuclear magnetic resonance of lithium was barely noticeable—the absorption line was barely visible against the background. Upon saturation of the paramagnetic resonance of the conduction electrons, the peak of the nuclear magnetic resonance conductivity increased by about 100 times. Thus, there is qualitative agreement with Overhauser’s theory, but no quantitative agreement: according to Overhauser’s theory the nuclear resonance peak of lithium should have increased more strongly. As the authors indicate, the reason for the quantitative discrepancy is unclear to them.

11. Nuclear Reactions

Experiments on nuclear reactions involving polarized nuclei and polarized incident particles are of the greatest interest. We shall analyze this question below. For the moment we shall briefly consider methods for obtaining polarized nucleons, as well as reactions involving polarized nucleons and randomly oriented nuclei.

a) Methods for Obtaining Polarized Nucleons

We shall confine ourselves only to a brief consideration of methods for obtaining beams of polarized nucleons. For details we refer to the second part of the review by Blin-Stoyle et al.$^{2}$ (there is also a detailed bibliography there).

One of the methods for obtaining polarized beams of nucleons may be experiments of the Stern–Gerlach type. However, in this way it is difficult to obtain beams with sufficiently high intensity.

A sufficiently intense beam of polarized thermal neutrons can be obtained by passing a beam of thermal neutrons through a ferromagnet magnetized to saturation. Passing through the ferromagnet, thermal neutrons, in addition to nuclear scattering, also undergo scattering by the magnetic moments of the unfilled \(3d\) shells of the ferromagnetic atoms[^15]. The effect of polarization of the neutron beam upon passage through a ferromagnet is due to the interference of nuclear and magnetic scattering. Owing to this interference, the total scattering cross section proves to be different for neutrons with spins parallel and, respectively, antiparallel to the direction of magnetization of the ferromagnet. Therefore, when thermal neutrons pass through a magnetized ferromagnet, we obtain a polarized beam of thermal neutrons.

Let us note that this same effect can serve to determine the degree of polarization of a beam of thermal neutrons. For this it is sufficient to determine the intensity of the beam after passage through a ferromagnet in two cases: a) the direction of magnetization of the ferromagnet is parallel to the polarization of the beam, and b) these directions are opposite.

Experiments on the polarization of thermal neutrons upon passage through magnetized iron were carried out[^42], and the results were quite successful: a beam of thermal neutrons was obtained with a degree of polarization equal to \(40\%\).

A beam of polarized thermal neutrons can also be obtained as a result of reflection of a beam of thermal neutrons from a magnetized ferromagnet (reflection from a magnetized “mirror”). Such experiments were also carried out[^42].

In work[^43] a theoretical calculation is given showing that, in \((np)\)-reactions caused by polarized thermal neutrons, one can obtain a beam of protons with a sufficiently high degree of polarization (in work[^43] the author proposes carrying out such experiments with \(\mathrm{He}^3\) and \(\mathrm{N}^{13}\)). Such experiments have not yet been carried out.

In works[^43–^45] the polarization of particles emitted in nuclear reactions is calculated. In particular, it is shown that even in the case when unpolarized particles fall on chaotic nuclei, the emitted particles will be polarized if the incident partial waves with nonzero orbital angular momenta participate in the reaction. This effect is called polarization by means of spin-orbit interaction (let us note that polarization of the emitted particles also occurs in the case where ...).

spin of the target nucleus is equal to zero). The polarization of the emitted particles is perpendicular to the plane containing the directions of motion of the incident and emitted particles. Let n denote the unit vector in this direction, and P\((\vartheta)\) the polarization vector of the emitted particles \((\mathbf{P}(\vartheta)=\mathbf{n} f(\vartheta)\), where \(f(\vartheta)\) is the degree of polarization of the beam of particles emitted at an angle \(\vartheta\) to the beam of incident particles). Calculation gives\(^{44}\):

\[ P(\vartheta)=\frac{n}{I(\vartheta)}\sum_{0}^{2L-1} a_n \cos^n \vartheta \sin \vartheta, \tag{29} \]

where \(I(\vartheta)\) is the angular distribution of the emitted particles in the center-of-mass system, \(L\) is the maximum orbital angular momentum of the incident wave that plays a role in the given nuclear reaction. The coefficients \(a_n\) depend on the particular type of reaction.

Theory\(^{44,45}\) shows that if polarized nucleons are incident on chaotic nuclei, then, in the case when incident partial waves with nonzero orbital angular momenta participate in the reaction, the angular distribution of the emitted particles will possess azimuthal asymmetry. Measurement of the degree of azimuthal asymmetry makes it possible to determine the degree of polarization of the nucleon beam.

During the last few years several experiments have been carried out to obtain polarized nucleons by means of the spin-orbit interaction. In particular, work\(^{46}\) reports experiments on the double scattering of protons by helium nuclei. The scheme of these experiments is as follows. A beam of protons is scattered by helium nuclei; in this way a polarized scattered beam is obtained, which is again scattered by helium nuclei. Studying the azimuthal asymmetry in the second scattering, the authors determined the degree of polarization of the beam obtained in the first scattering. From the experimental data they drew conclusions about the arrangement of excited levels in the unstable nucleus \(\mathrm{Li}^5\).

Work\(^{47}\) reports measurement of the polarization of neutrons produced in the reaction \(\mathrm{D}(d,n)\mathrm{He}^3\). The neutrons obtained in this way were scattered by carbon, and, by measuring the degree of azimuthal asymmetry in the scattering, the authors determined the polarization of the neutron beam. In particular, it was found that, at a deuteron energy equal to \(600\ \mathrm{keV}\), neutrons emitted at an angle of \(45^\circ\) have a degree of polarization equal to \(18 \pm 7\%\).

Let us also note work\(^{48}\), in which the authors measured the polarization of protons obtained in the reaction \(\mathrm{D}(d,p)\mathrm{H}^3\), and work\(^{49}\), in which the depolarization of a beam of thermal neutrons was measured in scattering by carbon, phosphorus, and paraffin.

b) Nuclear reactions involving polarized nuclei and polarized nucleons. The performance of nuclear

reactions with polarized nuclei and polarized incident particles is of great interest.

Let us consider, in particular, a nuclear reaction caused by slow neutrons, and suppose that at a neutron energy equal to \(E_0\) the reaction cross section has a resonant maximum. It is known that, in the case of slow neutrons, only partial waves with small orbital angular momenta are essential for the reaction. Suppose, in particular, that the reaction is caused by \(s\)-neutrons. In this case the spin of the compound nucleus in the state corresponding to the neutron energy equal to \(E_0\) is equal to \(j=I+\frac{1}{2}\) or \(j=I-\frac{1}{2}\) (\(I\) is the spin of the target nucleus).

This ambiguity can be removed if experiments are carried out with polarized neutrons and polarized nuclei. Indeed, suppose that the cross section of the process at a neutron energy equal to \(E_0\) is measured in two cases: a) the neutron spins are parallel to the nuclear spins, and b) the neutron spins are antiparallel to the nuclear spins (for simplicity we consider the case of a beam of completely polarized neutrons and a target of completely polarized nuclei). It is clear that if, for example, it turns out that \(\sigma_{\uparrow\uparrow}\gg\sigma_{\uparrow\downarrow}\), then it follows from this that the compound nucleus in the state corresponding to the neutron energy equal to \(E_0\) has spin \(j=I+\frac{1}{2}\). Thus, by carrying out nuclear reactions with polarized nuclei and slow polarized neutrons, it will be possible to determine the spins of compound nuclei.

We shall describe preliminary experiments of this type, carried out by Bernstein et al.\(^{2}\).

A beam of polarized thermal neutrons (obtained by passing a beam of thermal neutrons from a uranium pile through magnetized iron) was incident on a polarized target, for which manganous Tutton salt \(\mathrm{Mn(ND_4)_2(SO_4)_2\cdot 6D_2O}\) was used. The polarization of the nuclei was obtained by the Gorter–Rose method, the nuclear polarization being approximately \(16\%\). The authors measured the cross section of the \((n,\gamma)\) reaction in two cases (when the neutron polarization was parallel and, respectively, antiparallel to the nuclear polarization). The measurements showed that \(\sigma_{\uparrow\uparrow}\) is \(3.5\%\) smaller than \(\sigma_{\uparrow\downarrow}\).

In general, by studying the dependence of the cross sections of various nuclear reactions on the relative directions of the spins of the colliding particles, and also by studying the angular distribution and polarization of particles emitted in nuclear reactions, one can obtain much very valuable information about the spin dependence of nuclear forces (we note that some data on the spin dependence of nuclear forces at low energies are known from experiments on the diffraction of thermal neutrons by crystals).

Especially interesting would be experiments on the scattering of polarized neutrons and high-energy protons by polarized protons. Such experiments would be of great help in clarifying the nature of the interaction between nucleons, and this problem is one of the central problems of nuclear physics.

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