DYNAMIC METHOD FOR STUDYING NUCLEAR PARAMAGNETISM

N. M. Pomerantsev

INTRODUCTION

The dynamic method for studying nuclear paramagnetism[^1], in contrast to the static method[^2], has recently become widely used. It is enough to note that, during the period from 1946 to 1953, more than 300 papers devoted to this method were published. Such success of the dynamic method is due to the fact that its application opens up a new field in the study of matter. Whereas the static method does not make it possible, at ordinary temperatures \((\sim 300^\circ \mathrm{K})\), to detect the component of magnetization due to the paramagnetism of nuclei, the dynamic method, based on the resonance properties of nuclear magnetic moments, makes it possible to observe the behavior of this component in an alternating magnetic field over a wide range of temperatures, with solid, liquid, and gaseous substances.

The foundations of the dynamic method were laid as early as 1913 by V. K. Arkad’ev[^3], who observed the dependence of the magnetic permeability of ferromagnets on frequency in an alternating electromagnetic field. The first successful experiments on observing resonant absorption and dispersion of radio waves in matter, caused by the presence of a magnetic moment in atomic nuclei, were carried out by E. Purcell[^16] and independently by F. Bloch[^7] (1946). Using the dynamic method of F. Bloch, S. D. Gvozdover and A. A. Magazanik[^4] studied the paramagnetism of atomic nuclei and further developed the theory explaining the form of the signals observed in the experiment (1950).

The essence of the dynamic method for studying nuclear paramagnetism consists in the fact that, in addition to a constant magnetic field, which produces paramagnetic polarization of the nuclei, a radio-frequency magnetic field, perpendicular to the constant field, acts on the sample of the substance (Fig. 1).

N. M. POMERANTSEV

If the frequency of the radio-frequency field is close to the Larmor precession frequency of the nuclear magnetic moments, then the radio-frequency field excites precession of the resultant vector of nuclear magnetization*) about the direction of the magnetic field (Fig. 1). The amplitude of this precession will be the greater, the better the frequencies coincide.

The precessional motion of the resultant vector of nuclear magnetization induces an alternating electromotive force in the coil of the radio-frequency circuit in which the sample is placed. This emf can be amplified with the aid of special radio-receiving devices. Thus it becomes possible to observe the resonant absorption and dispersion of radio waves in a substance, due to the frequency dependence of the nuclear component of the magnetic susceptibility. By periodically and comparatively slowly varying, within small limits, the strength of the constant magnetic field near its resonant value, one can obtain the periodic occurrence of resonance and then display the process of the appearance of resonance signals on the screen of an electronic oscilloscope. Instead of varying the strength of the magnetic field, one may also periodically vary the frequency of the radio-frequency field.

Fig. 1. Magnetic fields imposed on the sample in experiments on magnetic resonance of atomic nuclei.

The dynamic method, relying on the resonant properties of nuclear magnetic moments, makes it possible to observe the behavior of the nuclear component of magnetization, reducing the influence of the remaining components to a negligible, nonresonant action on the system of nuclear magnetic moments. The most important advantage of the method is that, from the Larmor precession frequency known from the resonance conditions, it makes it possible to determine very accurately the gyromagnetic ratios of atomic nuclei.

The principal difficulty of the experiment lies in the necessity of observing exceedingly small emfs, comparable with the noise of the circuits,

*) By the resultant vector of nuclear magnetization we shall mean the total magnetic moment per unit volume of the substance, due to the paramagnetism of atomic nuclei. The Larmor precession frequency is equal to \(\omega = \gamma H_0\), where \(\gamma\) is the gyromagnetic ratio and \(H_0\) is the strength of the constant magnetic field.

in the presence of a radio-frequency field stronger by several orders of magnitude. Another experimental difficulty lies in the need to have a very homogeneous magnetic field, since the width of the resonance lines is very small and in individual cases may be of the order of thousandths of an oersted.

In the works of various authors the dynamic method of studying nuclear paramagnetism is called by different names: “nuclear induction” (F. Bloch), “nuclear magnetic resonance absorption” (E. Purcell, N. Bloembergen), “magnetic resonance of atomic nuclei” (C. D. Gorter). Nevertheless, the essence of all such experiments reduces to the above-described excitation of the precession of the resultant vector of nuclear magnetization. We shall use the last name, which, as it seems to us, expresses the essence of the phenomenon better than the others.

At present the study of the problem is proceeding in the following directions: measurement of the magnetic moments of atomic nuclei, investigation of relaxation processes occurring during the motion of a system of nuclear magnetic moments in a variable magnetic field, investigation of the magnetic shielding of nuclei, investigation of the structure of crystals and molecules, and application of the magnetic resonance of atomic nuclei for purposes of precision measurement and stabilization of a magnetic field.

The aim of the present review is to acquaint the reader with the basic theories of the magnetic resonance of atomic nuclei and the physical ideas connected with this phenomenon, and also to give an account of the principal experiments characterizing each of the directions listed above.

I. THEORIES OF THE MAGNETIC RESONANCE OF ATOMIC NUCLEI

§ 1. Phenomenological theories

On the basis of qualitative ideas similar to those set forth in the introduction, a phenomenological theory may be constructed which describes the behavior of the resultant vector of nuclear magnetization in a variable magnetic field. In the static case, in the absence of a variable field, the component of the magnetic susceptibility due to nuclear paramagnetism can be readily calculated. A calculation based on the use of the Boltzmann statistical distribution leads to an expression analogous to Curie’s law for paramagnets (see, for example, ⁵, pp. 68 and 135):

\[ \chi_0=\frac{N\gamma^2\hbar^2 I(I+1)}{3kT}, \tag{1} \]

where \(N\) is the number of particles, \(\gamma\) is the gyromagnetic ratio, \(\hbar\) is Planck’s constant, \(I\) is the spin of the particle, \(k\) is Boltzmann’s constant, and \(T\) is the absolute temperature.

In the dynamic case, as was first shown by V. K. Arkad’ev\(^6\), it is necessary to introduce a complex magnetic susceptibility. The usual formulas (in which \(\mathbf M\) in our case is the resultant vector of nuclear magnetization)

\[ \mathbf B=\mu \mathbf H=\mathbf H+4\pi \mathbf M,\qquad \mathbf M=\chi \mathbf H,\qquad \mathbf B=(1+4\pi\chi)\mathbf H, \tag{2} \]

for a magnetic field of the form

\[ H_x+iH_y=H_1 e^{-i\omega t},\qquad H_z=H_z(t), \tag{3} \]

and under the assumption

\[ M_x+iM_y=(u-iv)e^{-i\omega t} \tag{4} \]

take the form

\[ B_x+iB_y=\mu (H_x+iH_y) =\left[1+4\pi\left(\frac{u-iv}{H_1}\right)\right]H_1 e^{-i\omega t}. \tag{5} \]

The quantity

\[ \chi=\chi'-i\chi''=\frac{u-iv}{H_1} \tag{6} \]

will then be the complex nuclear magnetic susceptibility.

As is seen from expressions (3) and (4), the quantities \(u\) and \(v\), proportional to the real and imaginary parts of the nuclear magnetic susceptibility, are components of the resultant vector of nuclear magnetization in a coordinate system rotating with frequency \(\omega\) about the \(z\)-axis. They are sometimes called, respectively, the dispersion and the absorption.

To calculate the complex susceptibility it is no longer possible to use expressions valid in the static case, since in a variable magnetic field there appears a probability of transition from one energy state to another, the greater the closer the frequency of the radio-frequency field is to the frequency of the Larmor precession of the nuclear magnetic moments. F. Bloch\(^7\) proposed a system of equations giving the dependence of the components of the resultant vector of nuclear magnetization on time. Initially this system was obtained on the basis of classical ideas, and only later\(^8\) was it shown that the laws of quantum mechanics also lead to it, if a number of approximations are used. Bloch starts from the usual law of classical mechanics for angular momentum: the time derivative of the angular momentum is equal to the sum of the torques of the acting forces. If only the action of the external magnetic field is taken into account, then the equations will have the form

\[ \frac{d\mathbf M}{dt}=\gamma[\mathbf M\mathbf H], \tag{7} \]

where \(\mathbf M\) is the resultant vector of nuclear magnetization, \(\gamma\) is the gyromagnetic ratio, and \(\mathbf H\) is the magnetic-field strength. In\(^9\) it is shown that (7), with an appropriate change of variables, can

DYNAMIC METHOD FOR INVESTIGATING NUCLEAR PARAMAGNETISM

can be transformed into Pauli equations for noninteracting particles with spin \(1/2\). The equivalence of the quantum and classical solutions of the given problem for the case of free particles is thus proved without any assumptions.

However, the system of nuclear magnetic moments, in addition to the external field, is under the influence of interaction forces, which lead (according to Bloch) to the occurrence of relaxation processes in the system. This makes it necessary to add to the right-hand side of (7) terms which, in the classical treatment, play the role of the moments of the interaction forces of the system of nuclear magnetic moments with the surrounding medium.

The equations take the form

\[ \frac{d\mathbf{M}}{dt} = \gamma[\mathbf{M}\mathbf{H}] -\mathbf{i}\frac{M_x}{T_2} -\mathbf{j}\frac{M_y}{T_2} -\mathbf{k}\frac{M_z-M_0}{T_1}, \tag{8} \]

where \(M_0=\chi_0 H_z\) is the equilibrium value of the resultant vector of nuclear magnetization. The transition to the quantities \(T_1\) and \(T_2\), to the components \(u\) and \(v\), observed experimentally, is carried out by means of substitutions (3) and (4).

In coordinate notation, the equations for the components \(u, v\), and \(M_z\) have the form

\[ \begin{cases} \displaystyle \frac{du}{dt}+\frac{u}{T_2}+\left(|\gamma|H_z-\omega\right)v=0,\\[6pt] \displaystyle \frac{dv}{dt}+\frac{v}{T_2}-\left(|\gamma|H_z-\omega\right)u+|\gamma|H_1M_z=0,\\[6pt] \displaystyle \frac{dM_z}{dt}+\frac{M_z-M_0}{T_1}-|\gamma|H_1v=0. \end{cases} \tag{9} \]

The quantities \(T_1\) and \(T_2\), appearing in equations (8) and (9), are called relaxation times.

The time \(T_1\), characterizing the process of energy exchange between the system of nuclear magnetic moments and other degrees of freedom (the magnetic moments of molecules, paramagnetic ions, etc.) and expressing the tendency of the component \(M_z\) of the resultant magnetization vector toward the equilibrium value \(M_0\), has been called the longitudinal relaxation time. The time \(T_2\) has been called the transverse relaxation time. In addition to heat-exchange processes, it is connected with the violation of the phase coherence of the precession of the system of nuclear magnetic moments in a constant magnetic field, arising from the interaction of the nuclear magnetic moments with one another. Thus \(T_2\) expresses the tendency of the transverse component of the resultant vector of nuclear magnetization to decrease in a constant field.

Theoretically, the values of the relaxation times \(T_1\) and \(T_2\) are not calculated, for which reason the theory may be called phenomenological.

For comparison of the theory with experiment it is necessary to have a solution of the system (9) for the components \(u\) and \(v\). The system (9) is linear inhomogeneous and contains terms proportional to the components \(u\) and \(v\). Owing to these terms, the solution of the homogeneous system will have exponentially decaying factors, i.e., a process of signal establishment will take place. The established process will, evidently, be described by the solution of the inhomogeneous system, if the initial instant of time is taken to be sufficiently remote \((t_0 \to -\infty)\). Experimentally, both established processes and non-established ones may be observed. Accordingly, the form of the signals will be determined by one or another solution.

In view of the presence of several parameters in the system of equations (9), reducing its solution to a numerical result for arbitrary values of the parameters is associated with great computational difficulties, and tabulating the functions representing the solutions is practically difficult to carry out. Therefore at present approximate solutions exist for a number of special cases\(^{10, 11, 12, 13, 14, 73}\). The solution of the system (9) in the most general form was given in the work of S. D. Gvozdover and A. A. Magazanik\(^{4}\). The connection of the system of equations (7) with the Riccati and Hill equations, as well as the transition from equations (7) to the quantum-mechanical Pauli equations, were found in work\(^{9}\). Solutions of equations (9) for sinusoidal modulation of the longitudinal component of the magnetic field are given in work\(^{10}\).

Before considering the most general solution, obtained by S. D. Gvozdover and A. A. Magazanik\(^{4}\), let us dwell on the solution of the inhomogeneous system (9) in the case \(H_z = H_0 = \mathrm{const}\), which well illustrates the resonance character of the action of the radio-frequency field on a system of nuclear magnetic moments. From (9) it is easy to see that a particular solution of the inhomogeneous system (which in this case becomes a system with constant coefficients) will be constant, so that

\[ \frac{du}{dt}=\frac{dv}{dt}=\frac{dM_z}{dt}=0, \tag{10} \]

Substituting these values into (9), we find:

\[ \begin{aligned} u&=\frac{M_0|\gamma|H_1T_2^2\left(|\gamma|H_0-\omega\right)} {1+T_2^2\left(|\gamma|H_0-\omega\right)^2+(\gamma H_1)^2T_1T_2},\\[6pt] v&=-\frac{M_0|\gamma|H_1T_2} {1+T_2^2\left(|\gamma|H_0-\omega\right)^2+(\gamma H_1)^2T_1T_2},\\[6pt] M_z&=\frac{M_0\left[1+T_2^2\left(|\gamma|H_0-\omega\right)^2\right]} {1+T_2^2\left(|\gamma|H_0-\omega\right)^2+(\gamma H_1)^2T_1T_2}. \end{aligned} \tag{11} \]

If the condition is satisfied

\[ (\gamma H_1)^2 T_1 T_2 \ll 1, \tag{12} \]

which is possible in the case of a weak radio-frequency field, then the solution takes the form

\[ \left. \begin{aligned} \frac{u}{M_0|\gamma|H_1T_2} &= \frac{\delta}{1+\delta^2},\\ \frac{v}{M_0|\gamma|H_1T_2} &= -\frac{1}{1+\delta^2},\\ M_z &= M_0, \end{aligned} \right\} \tag{13} \]

where \(\delta = T_2(|\gamma|H_0-\omega)\) is a quantity proportional to the difference between the Larmor precession frequency of the nuclear magnetic moments and the frequency of the radio-frequency field.

The graphs of the curves \(\dfrac{\delta}{1+\delta^2}\) and \(\dfrac{1}{1+\delta^2}\) are presented in Fig. 2.

From the graphs it is evident that the function \(u(\delta)\) has the form of a dispersion curve, while the function \(v(\delta)\) has the form of an absorption curve. The indicated form of the absorption curve is usually called Lorentzian.

Fig. 2. Simplest form of magnetic-resonance signals of atomic nuclei.

With an increase in the strength of the radio-frequency field, the phenomenon of saturation occurs, leading to a decrease in the signal.

Here, owing to an increase in the temperature of the system of nuclear magnetic moments, the absorption of energy from the radio-frequency field by the system decreases. This can be illustrated by means of the solution (11), which at resonance has the form

\[ \left. \begin{aligned} u &= 0,\\ v &= -\frac{M_0|\gamma|H_1T_2}{1+(\gamma H_1)^2T_1T_2},\\ M_z &= \frac{M_0}{1+(\gamma H_1)^2T_1T_2}. \end{aligned} \right\} \tag{14} \]

From (14) it is clear that, with increasing \(H_1\), the quantity \(v\) decreases. By analogy with (1), the expression for \(M_z\) may be written in the form

\[ M_z=\frac{N\gamma^2\hbar^2 I(I+1)H_0}{3kT^*}, \tag{15} \]

where

\[ T^*=T\left[1+(\gamma H_1)^2T_1T_2\right] \]

is the effective temperature, which increases with increasing intensity of the radio-frequency field \(H_1\).

Owing to a number of reasons determined by the specifics of the experiment, in experiments on magnetic resonance of atomic nuclei signals of such a simple form as those shown in Fig. 2 can be obtained only in exceptional cases\(^*\). Usually, in experiments the longitudinal component \(H_z\) of the magnetic field does not remain constant, which leads to a change in the shape of the signal. The answer to the question of the form of the signal in this case is given by the solution of C. D. Gvozdover and A. A. Magazanik,4 to the consideration of which we now turn.

To solve system (9) in the general case, the functions \(u\) and \(v\) are expressed in terms of \(M_z\) from the first two equations of system (9). The equations obtained in this way have the form

\[ \left. \begin{aligned} u(t)={}&\left[u_a\cos f(t,t_a)-v_a\sin f(t,t_a)\right]e^{-\frac{t-t_a}{T_2}} \\ &+|\gamma|H_1\int_{t_a}^{t} M_z(t')e^{-\frac{t-t'}{T_2}}\sin f(t,t')\,dt', \\[1.2em] v(t)={}&\left[u_a\sin f(t,t_a)+v_a\cos f(t,t_a)\right]e^{-\frac{t-t_a}{T_2}} \\ &-|\gamma|H_1\int_{t_a}^{t} M_z(t')e^{-\frac{t-t'}{T_2}}\cos f(t,t')\,dt', \\[1.2em] M_z(t)={}&M_z(t_a)e^{-\frac{t-t_a}{T_1}} +\frac{1}{T_1}\int_{t_a}^{t} M_0(t')e^{-\frac{t-t'}{T_1}}\,dt' \\ &+|\gamma|H_1\int_{t_a}^{t} e^{-\frac{t-t'}{T_1}}v(t')\,dt'. \end{aligned} \right\} \tag{16} \]

where \(u_a\) and \(v_a\) are the values of the functions at the initial instant of time \(t_a\), and the argument appearing under the signs \(\sin\) and \(\cos\) has the form

\[ f(t,t')=\int_{t'}^{t}\left(|\gamma|H_z-\omega\right)\,dt'' . \tag{17} \]

\[ \overline{\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}} \]

\(^*\) Recently, apparatus has been developed,64 making it possible to observe the form of signals described by equations (13).

Turning to the consideration of steady-state processes, it is necessary to set \(t_a \to -\infty\). Equations (16) become

\[ \left. \begin{aligned} u(t)&=|\gamma|H_1 \int_{-\infty}^{t} M_z(t') e^{-\frac{t-t'}{T_2}}\sin f(t,t')\,dt',\\[4pt] v(t)&=-|\gamma|H_1 \int_{-\infty}^{t} M_z(t') e^{-\frac{t-t'}{T_2}}\cos f(t,t')\,dt',\\[4pt] M_z(t)&=M_0+|\gamma|H_1\int_{-\infty}^{t} e^{-\frac{t-t'}{T_1}}v(t')\,dt'. \end{aligned} \right\} \tag{18} \]

From (18) one can obtain for \(M_z\) an integral equation of the Volterra type:

\[ M_z=M_0-\gamma^2H_1^2\int_{-\infty}^{t} M_z(t')K(t,t')\,dt' \tag{19} \]

with kernel

\[ K(t,t')=\int_{t'}^{t} e^{-\frac{t-t''}{T_1}}e^{-\frac{t''-t'}{T_2}}\cos f(t'',t')\,dt''. \tag{20} \]

The system can be solved by the method of successive approximations. The zero approximation has the form

\[ u_0=v_0=0,\qquad M_z=M_0 \tag{21} \]

and the first approximation

\[ \left. \begin{aligned} u_1&=|\gamma|H_1M_0\int_{-\infty}^{t} e^{-\frac{t-t'}{T_2}}\sin f(t,t')\,dt',\\[4pt] v_1&=-|\gamma|H_1M_0\int_{-\infty}^{t} e^{-\frac{t-t'}{T_2}}\cos f(t,t')\,dt',\\[4pt] M_z&=M_0. \end{aligned} \right\} \tag{22} \]

For the particular case when the modulation of the longitudinal component of the magnetic field occurs according to a linear law,

\[ |\gamma|H_z-\omega=at, \tag{23} \]

where

\[ a=|\gamma|\frac{dH_z}{dt}=\operatorname{const}, \tag{24} \]

the function (17) has the form

\[ f(t,t')=\frac{a}{2}(t^2-t'^2). \tag{25} \]

The solution of system (9) in the first approximation will be:

\[ u_1(t)=|\gamma|H_1M_0T_2U(\tau), \qquad v_1(t)=-|\gamma|H_1M_0T_2V(\tau), \tag{26} \]

where

\[ \left. \begin{aligned} U(\tau)&=A\int_{-\infty}^{\tau} e^{A(w-\tau)} \sin \frac{A}{2}(\tau^2-w^2)\,dw,\\[4pt] V(\tau)&=A\int_{-\infty}^{\tau} e^{A(w-\tau)} \cos \frac{A}{2}(\tau^2-w^2)\,dw,\\[4pt] \tau&=aT_2t,\qquad A=\frac{1}{aT_2^2}. \end{aligned} \right\} \tag{27} \]

Graphs of the functions \(U(\tau)\) and \(V(\tau)\) are given in work \(^{11}\). If the parameter \(A\) is small, then the solution (26) has the form of oscillating functions with damping. Graphs of these solutions are pre-

Fig. 3. Graph of the function \(v(t)\) for linear modulation of the longitudinal component of the magnetic field. The numbers on the curves indicate the values of the parameter

\[ \left(|\gamma|\,\frac{dH_z}{dt}\right)^{1/2}T_2. \]

sented in Figs. 3 and 4. The solutions (26) are also applicable in the case of sinusoidal modulation of the longitudinal component of the magnetic field, if the oscillatory processes have time to decay before the onset of the next modulation period.

In paper \(^{10}\) the solutions are obtained for two limiting cases under sinusoidal modulation of the longitudinal component of the magnetic field. Depending on the behavior of the longitudinal component of the resultant vector of nuclear magnetization, these cases are called the cases with absence and with presence of inversion of the longitudinal component of the resultant vector of nuclear magnetization. The absence or presence of inversion is determined by the magnitude of the parameter

\[ \lambda=\frac{\omega_m H_m}{|\gamma|H_1^2}, \]

where \(\omega_m\) and \(H_m\) are the angular frequency and the modulation amplitude.

Fig. 4. Graph of the function \(u(t)\) for linear modulation of the longitudinal component of the magnetic field. The numbers by the curves indicate the values of the parameter

\[ \left(|\gamma|\frac{dH_z}{dt}\right)^{1/2}T_2 . \]

For \(\lambda \gg 1\) the vector \(\mathbf{M}\) deviates only slightly from the equilibrium position; for \(\lambda \ll 1\) it turns over. In the first case (\(\lambda \gg 1\)) the solution is analogous to that given above in (26). However, under sinusoidal modulation of the longitudinal component of the magnetic field there is also a form of signal in which the oscillations continue throughout the entire modulation period.

Graphs for this case are presented in Figs. 5 and 6.

In the second case (in the presence of inversion) the equations have a small parameter multiplying the derivatives. Oscillatory processes are absent. The function \(u(x)\) is an order of magnitude larger than the function \(v(x)\). The longitudinal component of the magnetization vector changes sign. These solutions are shown in the graphs of Figs. 7 and 8.

Comparison of the solutions of system (9) with experimental data shows that the theory based on the system of equations (9) does not take into account a number of factors influencing the behavior of the resultant

vectors of nuclear magnetization. Thus, for example, a small difference in the resonance frequencies in different chemical compounds of one and the same element is observed experimentally. This difference is due to the fact that the role of the internal fields acting on nuclear magnetic moments is not limited merely to the excitation

![Figure 5 and Figure 6]

Fig. 5. Graph of the function \(u(x)\) (\(x=\omega_m t\)) for sinusoidal modulation of the longitudinal component of the magnetic field. Parameter values: \(\sigma=100,\ \mu=-1\).

Fig. 6. Graph of the function \(v(x)\) for sinusoidal modulation of the longitudinal component of the magnetic field. The parameter values are the same as for Fig. 5.

of relaxation processes, as is assumed in the theory, but also leads to the effect of magnetic shielding of nuclei, which will be discussed in more detail below. In addition, a fine structure of absorption lines is observed experimentally in crystals, in organic compounds of complex structure, etc.,

![Figure 7 and Figure 8]

Fig. 7. Graph of the function \(u(x)\) in the presence of inversion of the longitudinal component of the resultant vector of nuclear magnetization. Parameter values: \(\mu=1,\ \nu=0.1,\ k=10\).

Fig. 8. Inversion of the longitudinal component of the resultant vector of nuclear magnetization. The parameter values are the same as for Fig. 7.

where, owing to the arrangement of the nuclei in groups more or less separated from one another, a splitting of the energy levels into several components takes place. From equations (9) there follows only a broadening of the lines about a single frequency.

In Figs. 9, 10, and 11 are shown oscillograms, obtained by the author of the present review, of proton magnetic-resonance signals in water with different contents of paramagnetic copper ions. The frequency of the radio-frequency field was equal to \(12.9\) MHz. In this case the theory set out above proves to be fully applicable.

The presence of copper ions in water changes the relaxation time of the nuclear magnetic moments, which corresponds to a change in the parameter

Fig. 9. Oscillogram of the signal \(v'_x(t)\). Sample: \(0.7\ \mathrm{cm}^3\) of a one-molar solution of \(\mathrm{CuSO}_4\). Modulation sinusoidal, modulation amplitude \(0.6\) oersted, frequency \(50\ \mathrm{Hz}\). Pass band of the amplifying channel from \(20\ \mathrm{Hz}\) to \(150\ \mathrm{kHz}\).

Fig. 10. Oscillogram of the signal \(v(t)\). Sample: \(0.7\ \mathrm{cm}^3\) of a \(0.2\)-molar solution of \(\mathrm{CuSO}_4\). The remaining data are the same as for Fig. 9.

Fig. 11. Oscillogram of the signal \(v(t)\). Sample: \(0.7\ \mathrm{cm}^3\) of distilled water. The remaining data are the same as for Fig. 9.

of the theoretical curves. Comparison of Figs. 9 and 10 with Fig. 3, and also Fig. 11 with Fig. 6, shows a sufficiently good qualitative agreement between theory and experiment.

A very strong influence on the shape of the signals of magnetic resonance of atomic nuclei is exerted by the inhomogeneity of the magnetic field. In the presence of inhomogeneity, different parts of the sample enter into resonance at different moments of time. This is due to the fact that each point of the sample will correspond to its own Larmor precession frequency.

Owing to the fact that the signals from different parts of the sample will have different phases, the resultant signal is deformed in such a way that its principal maximum is broadened, while the oscillatory processes are damped. According to the theory[^11], the resultant signal will have the form

\[ v+iu=-|\gamma|H_1\int_{-\infty}^{t}dt'\exp\left[-\frac{t-t'}{T_2} -i\int_{t'}^{t}\Delta\omega(t'')\,dt''\right] \int_{H'} M_z(t',H')e^{-i|\gamma|(t-t')H'}\,dH' . \]

This expression is the result of summing the signals from individual parts of the sample having the Larmor precession frequency \(|\gamma|(H_0+H')\), where \(H'\) is the deviation of the magnetic field from the resonance value \(H_0=\dfrac{\omega}{|\gamma|}\) at the given point of the sample.

In the case of a weak field (see condition (12)) one may put

\[ M_z(t',H')=M_0N(H'), \]

where \(N(H')\) is the distribution function of the magnetic-field inhomogeneity.

In the theory[^11] the Lorentzian form of the distribution function is considered,

\[ N(H')=\frac{1}{\pi}\frac{\Gamma}{(H')^2+\Gamma^2}, \]

where \(\Gamma\) is the effective value of the field inhomogeneity in the volume of the sample. In this case the shape of the signals retains the form (27), but \(T_2\) is replaced by the quantity \(T_2^*\), determined from the relation

\[ \frac{1}{T_2^*}=\frac{1}{T_2}+|\gamma|\Gamma . \]

The signals presented in the oscillograms of Figs. 9 and 10 were used to estimate the magnetic-field inhomogeneity. As follows from the theory set forth below[^16], the quantity \(\dfrac{1}{T_2}\) is proportional to the concentration of paramagnetic ions. Therefore, by constructing...

plot \(\dfrac{1}{T_2^*}=f(N_{\mathrm{ion}})\) and extrapolating to the value \(N_{\mathrm{ion}}=0\), we obtain the value of the quantity \(\gamma \Gamma\). For the oscillograms in Figs. 9 and 10 the quantity \(\Gamma\) is equal to 0.02 gauss.

§ 2. Statistical theories

The theories presented in this section, in contrast to phenomenological theories, do not contain constants determined only from experiments on the magnetic resonance of atomic nuclei. The constants appearing in these theories are atomic or molecular constants. The main difficulty of such theories is that a rigorous investigation, based on the laws of quantum mechanics, of the question of the shape of magnetic-resonance signals of atomic nuclei requires solving the many-body problem and is connected with insurmountable mathematical difficulties. Therefore, in the theories set out below, quantities indirectly characterizing the shape of the signals are calculated, and the calculations are simplified on the basis of physical considerations.

The following concepts underlie these theories\(^{15,16,17}\). The exchange of energy between a system of nuclear spins situated in a constant magnetic field and a “heat reservoir,” consisting of other degrees of freedom (“lattice”), brings the system into a state of thermal equilibrium at a finite temperature. In this state the system can absorb energy from an applied radio-frequency field of the “resonance” frequency. However, as a result of energy absorption the temperature of the spin system rises, and the absorption of energy decreases. “Saturation” sets in, determined by the relaxation time \(T_1\), which characterizes the transfer of energy from the system of nuclear spins to the “heat reservoir,” i.e., the tendency of the system toward thermal equilibrium with the surrounding medium. The presence of a finite relaxation time \(T_1\) is one of the causes of broadening of the absorption line. On the other hand, the interaction of the magnetic moments of nuclei with one another, with which the relaxation time \(T_2\) is associated, makes its own contribution to the width of the line.

The theory proceeds from a definite, prescribed in advance, line shape, which does not always agree well with the experimentally observed line shape. In addition, the theory does not consider the processes by which signals are established. As has already been said, it is assumed that the line width is due to two causes: thermal motion and magnetic dipole interaction. The line shape due to thermal motion is taken in the form\(^{18}\)

\[ g(\omega)=\frac{2}{p}\,\frac{1}{1+\left(\dfrac{\Delta\omega}{p}\right)^2}, \tag{28} \]

where \(p\) is the probability of transition from one energy state to another, associated with the relaxation time \(T_1\). This relation is established as follows. The change in time of the number of particles with spin directed along the field (for particles with spin \(1/2\)) is determined by the equation

\[ \frac{dN^{+}}{dt} = N^{-}p e^{\frac{\gamma\hbar H_0}{2kT}} - N^{+}p e^{-\frac{\gamma\hbar H_0}{2kT}}, \tag{29} \]

where \(N^{+}\) and \(N^{-}\) are the numbers of particles with spin directed respectively along the field and against the field, and \(p\) is the same probability as in (28). In deriving the equation it is assumed that the distribution of the number of particles over states is close to equilibrium. Expanding the exponential in a series (which can be done in view of the smallness of the exponent) and taking into account that \(N^{+}+N^{-}=N\), we obtain the solution

\[ N^{+} = Ce^{-2pt} + \frac{N}{2} \left( 1-\frac{\gamma\hbar H_0}{2kT} \right), \tag{30} \]

indicating the presence of a relaxation process with relaxation time

\[ T_1=\frac{1}{2p}. \tag{31} \]

In an analogous way it can be shown (see \(^{16}\)) that relation (31) also holds for particles with higher spin.

The probability \(p\) is calculated in the usual way from the theory of quantum transitions, with the use of the perturbation method (see \(^{19}\), pp. 327–331), and has the form

\[ p_{mm'} = \frac{1}{\hbar^2} \left|W_{mm'}(\nu_{mm'})\right|^2, \tag{32} \]

where

\[ W_{mm'}(\nu_{mm'}) = \int_{-\infty}^{\infty} W_{mm'}(\tau)\, e^{2\pi i\nu_{mm'}\tau}\, d\tau \tag{33} \]

is the spectral density of the matrix element of the perturbation energy.

It follows from (28) and (31) that the half-width of the line (the value \(\Delta\omega\) at \(g(\omega)=\frac{1}{2}[g(\omega)]_{\max}\)) for broadening due to thermal motion is equal to \(\frac{1}{2T_1}\). The calculation of \(T_1\) for protons in water on the basis of formula (32) according to \(^{16}\) will be considered below. For solids \(T_1\) is sufficiently large and may be neglected.

Another cause of line broadening is the interaction of the magnetic moments of nuclei with one another. The calculation of the line shape in this case is associated with insurmountable mathematical difficulties. However, applying the method of diagonal sums \(^{17}\), it is possible

compute the parameter characterizing the line width—the mean-square frequency deviation:

\[ \overline{\Delta\omega^{2}}=\int_{-\infty}^{\infty}(\Delta\omega)^{2}f(\Delta\omega)\,d(\Delta\omega). \tag{34} \]

If one assumes that the line shape is Gaussian, i.e., if the distribution function \(f(\Delta\omega)\) in (34) has the form

\[ f(\Delta\omega)=(2\pi)^{-3/2}\left(\overline{\Delta\omega^{2}}\right)^{1/2} e^{-\frac{(\Delta\omega)^{2}}{2\overline{\Delta\omega^{2}}}}, \tag{35} \]

then the mean-square deviation is equal to the distance between the points of maximum slope. For a Lorentzian line shape, the mean-square frequency deviation does not exist, since in this case the integral (34) diverges. The width of the Lorentzian form is characterized by the parameter \(T_{2}\) (see Fig. 2). (On the relation of \(\overline{\Delta\omega^{2}}\) to \(T_{2}\), see below (40).)

The calculation of the mean-square frequency

\[ \overline{\omega^{2}}= \frac{\displaystyle\int_{0}^{\infty}\omega^{2}f(\omega)\,d\omega} {\displaystyle\int_{0}^{\infty}f(\omega)\,d\omega} \tag{36} \]

is based on the relations \(^{20,21,22}\)

\[ \left. \begin{aligned} \int_{0}^{\infty} f(\omega)\,d\omega &= \operatorname{Sp}(\mathbf{M})^{2},\\ \int_{0}^{\infty} \omega^{2}f(\omega)\,d\omega &= \operatorname{Sp}\left(\frac{d\mathbf{M}}{dt}\right)^{2}, \end{aligned} \right\} \tag{37} \]

which, together with the quantum-mechanical equation of motion for the operator \(\mathbf{M}\), give:

\[ \overline{\omega^{2}}= \frac{\operatorname{Sp}[\mathbf{H}\mathbf{M}-\mathbf{M}\mathbf{H}]^{2}} {\hbar^{2}\operatorname{Sp}(\mathbf{M})^{2}}. \tag{38} \]

The mean-square frequency deviation is calculated by the formula

\[ \overline{\Delta\omega^{2}}= \overline{(\omega-\gamma H_{0})^{2}} = \overline{\omega^{2}}-(\gamma H_{0})^{2}. \tag{39} \]

The full half-width of the absorption line, according to the theory being presented, is equal to

\[ \frac{1}{T_2}=\frac{1}{2T_1}+\sqrt{\overline{\Delta\omega^2}} . \tag{40} \]

It should be noted that there is a certain inconsistency in the theory, which, taking the form of line broadening due to thermal motion to be Lorentzian, and the form of broadening due to the interaction of nuclei to be Gaussian, considers the full half-width of the line as the sum of the half-width of the Lorentzian line and half the distance between the points of maximum slope of the Gaussian line. However, both the Lorentzian and Gaussian line forms are idealizations of lines observed in reality. Therefore the distinction between the half-width and half the distance between the points of maximum slope is not very significant, in view of the approximate character of the whole theory.

The calculation of the relaxation times, as is seen from expressions (32) and (38), reduces to a transformation of the Hamiltonian of the system of interacting particles.

The Hamiltonian is taken in the following form:

\[ \hat{\mathscr H} = \sum_{1\leq i\leq N}\gamma_i\hbar \mathbf I_i\mathbf H_0 + \sum\sum_{1\leq i<j\leq N} \frac{\gamma_i\gamma_j\hbar^2}{r_{ij}^3} \left\{ (\mathbf I_i\mathbf I_j)-3(\mathbf I_i\mathbf r_{ij}^{0})(\mathbf r_{ij}^{0}\mathbf I_j) \right\}, \tag{41} \]

where \(\gamma_i,\gamma_j\) are the gyromagnetic ratios of the interacting particles, \(\mathbf H_0\) is the intensity vector of the constant magnetic field, \(\mathbf I\) is the vector operator of the particle spins, \(r_{ij}\) is the distance between the \(i\)-th and \(j\)-th particles, and \(\mathbf r_{ij}^{0}\) is the unit vector of the direction from the \(i\)-th particle to the \(j\)-th. The first term of the Hamiltonian represents the energy of the system of particles in the external magnetic field. The second term characterizes the dipole interaction of the particles and is the quantum analogue of the classical expression for the energy of a dipole in the field of another dipole (see \(^{23}\), § 56, p. 4). The unit vector \(\mathbf r_{ij}^{0}\) has as coordinates the direction cosines \(\alpha_{ij}\), \(\beta_{ij}\), and \(\gamma_{ij}\) with the axes \(x\), \(y\), and \(z\):

\[ \mathbf r_{ij}^{0}=\mathbf i\alpha_{ij}+\mathbf j\beta_{ij}+\mathbf k\gamma_{ij}. \tag{42} \]

If, instead of the direction cosines, one introduces the polar and azimuthal angles \(\vartheta_{ij}\) and \(\varphi_{ij}\), using the relations

\[ \left. \begin{aligned} \alpha_{ij}&=\sin\vartheta_{ij}\cos\varphi_{ij},\\ \beta_{ij}&=\sin\vartheta_{ij}\sin\varphi_{ij},\\ \gamma_{ij}&=\cos\vartheta_{ij}, \end{aligned} \right\} \tag{43} \]

then the interaction energy \(W_{ij}\) in the Hamiltonian (41), represented in the form

\[ \hat{\mathcal H}=\sum_{1\leq i\leq N}\gamma_i\hbar I_i H_0+ \sum_{1\leq i<j\leq N} W_{ij}, \tag{44} \]

can be written as follows:

\[ W_{ij}=\frac{\gamma^2\hbar^2}{r_{ij}^3}(A+B+C+D+E+F), \tag{45} \]

where

\[ \begin{aligned} A+B&=-\frac{1}{2}(I_i I_j-3I_{z_i}I_{z_j})(1-3\cos^2\vartheta_{ij}),\\ C&=-\frac{3}{2}\{(I_{x_i}+iI_{y_i})I_{z_j}+I_{z_i}(I_{x_j}+iI_{y_j})\}\times\\ &\qquad\qquad\times \sin\vartheta_{ij}\cos\vartheta_{ij}e^{-i\varphi_{ij}},\\ E&=\frac{3}{4}(I_{x_i}+iI_{y_i})(I_{x_j}+iI_{y_j})\sin^2\vartheta_{ij}e^{-2i\varphi_{ij}},\\ D&=C^*,\qquad F=E^* . \end{aligned} \tag{46} \]

The first term of the Hamiltonian \((A+B)\) in (45) is used to calculate the mean-square frequency deviation by the method of diagonal sums. The calculation carried out in work \(^{17}\) leads to the expression

\[ \overline{\Delta\omega^2}= \frac{3}{4}\gamma^4\hbar^2 I(I+1) \sum_j(1-3\cos^2\vartheta_{ij})^2 r_{ij}^{-6}. \tag{47} \]

For polycrystalline substances, the following expression was obtained there:

\[ \overline{\Delta\omega^2}= \frac{3}{5}\gamma^4\hbar^2 I(I+1)\sum_j r_{jk}^{-6}; \tag{48} \]

formulas (47) and (48) are used to calculate the line width in crystals \(^{24}\).

In liquids, molecules together with the nuclei included in them are in a state of chaotic Brownian motion. Therefore \(r_{ij}\) and the angles \(\vartheta_{ij}\) and \(\varphi_{ij}\) in (46) will be random functions of time. The theory of relaxation times in liquids, set forth in \(^{16}\), is based on this assumption.

It is assumed that the random functions entering into (46) can be expanded in a Fourier integral:

\[ F(t)=\int_{-\infty}^{\infty} A(\nu)e^{2\pi i\nu t}\,d\nu, \tag{49} \]

where

\[ A(\nu)=\int_{-\infty}^{\infty} F^*(t)e^{-2\pi i\nu t}\,dt, \tag{50} \]

and Parseval’s equality holds:

\[ \int_{-\infty}^{\infty} F(t)F^*(t)\,dt = \int_{-\infty}^{\infty} A(\nu)A^*(\nu)\,d\nu . \tag{51} \]

For the random functions entering the Hamiltonian (46), the relations can be written as

\[ \left. \begin{aligned} \overline{\sum_j (1-3\cos^2\vartheta_{ij})^2 r_{ij}^{-6}} &= \int_{-\infty}^{\infty} J_0(\nu)\,d\nu,\\ \overline{\sum_j |\sin\vartheta_{ij}\cos\vartheta_{ij}e^{i\varphi_{ij}}|^2 r_{ij}^{-6}} &= \int_{-\infty}^{\infty} J_1(\nu)\,d\nu,\\ \overline{\sum_j |\sin^2\vartheta_{ij}e^{2i\varphi_{ij}}|^2 r_{ij}^{-6}} &= \int_{-\infty}^{\infty} J_2(\nu)\,d\nu, \end{aligned} \right\} \tag{52} \]

where \(J(\nu)\) is the square of the spectral density of the random function[^26]. This quantity enters formula (32) for the transition probability. The matrix elements of the spin operators have the form[^25]

\[ \left. \begin{aligned} (m|I_x+iI_y|m')&=\sqrt{(I+m)(I-m+1)}\,\delta_{m'm-1},\\ (m|I_x-iI_y|m')&=\sqrt{(I-m)(I+m+1)}\,\delta_{m'm+1},\\ (m|I_z|m')&=m\delta_{m'm}. \end{aligned} \right\} \tag{53} \]

Using formulas (32), (46), (52), and (53), one can obtain the following expression for the transition probability:

\[ P_{m_i+1\leftarrow m_i} = \frac{3}{8}\gamma^4\hbar^2 I(I+1)(I-m_i)(I+m_i+1) \times \]

\[ \times\,[J_2(-2\nu_0)+2J_1(-\nu_0)], \tag{54} \]

which represents an average over all \(m_j\).

In carrying out the averaging, the following relations were used:

\[ \frac{1}{2I+1}\sum_{m_j=-I}^{I} m_j^2 = \frac{1}{3}I(I+1), \qquad \sum_{m_i=-I}^{I} m_j=0. \tag{55} \]

The relaxation time \(T_1\) is related to (54) by relation (31). Since (31) is equally valid both for particles with spin \(1/2\) and for particles of higher spin, one may put \(m_i = 1/2\). The expression for the relaxation time takes the form

\[ \frac{1}{T_1} = \frac{3}{4}\gamma^4\hbar^2 I(I+1)\,[J_2(2\nu_0)+2J_1(\nu_0)]. \tag{56} \]

For calculating the transverse relaxation time, from (47) and (52) one may obtain:

\[ \overline{\Delta\omega^2} = \frac{3}{4}\gamma^4\hbar^2 I(I+1) \int_{-\frac{1}{\pi T_2'}}^{\frac{1}{\pi T_2'}} J_0(\nu)\,d\nu, \tag{57} \]

where the integration is carried out over the width of the line, and it is assumed that

\[ \frac{1}{T_2'} = \sqrt{\overline{\Delta\omega^2}}. \tag{58} \]

Thus, the problem of calculating relaxation times for liquids is reduced to determining the functions \(J_0(\nu)\), \(J_1(\nu)\), and \(J_2(\nu)\). These functions, which characterize random processes in liquids caused by the Brownian motion of molecules, can be expressed through the correlation function considered in the theory of random processes\({}^{26}\),

\[ k(\tau) = \lim_{T\to\infty} \frac{1}{T} \int_0^T F(t)F^*(t+\tau)\,dt. \tag{59} \]

The relation of the correlation function \(k(\tau)\) to the square of the spectral density of the random function \(J(\nu)\) is given in the theory of random processes (see, for example, \({}^{27}\), p. 189) by the relation

\[ J(\nu) = \int_{-\infty}^{\infty} k(\tau)e^{2\pi i\nu\tau}\,d\tau, \tag{60} \]

i.e. \(J(\nu)\) is the spectral density of the correlation function.

To find the correlation function of the Brownian rotation of the protons of a water molecule, in \({}^{16}\) a physical picture was used analogous to that adopted in the Debye theory of absorption and dispersion of electromagnetic waves in dielectrics\({}^{28}\).

A water molecule undergoing Brownian rotation in a liquid with viscosity \(\eta\) and temperature \(T\) is considered. From the Fokker—Planck equation

\[ \frac{\partial f}{\partial t}+D\Delta f(\vartheta,\varphi)=0, \tag{61} \]

where \(D\) is the diffusion coefficient, the probability \(f(t,\vartheta,\varphi)\) is determined for finding, at time \(t\), the axis connecting the two protons of the molecule inside the solid angle \(\sin\vartheta\,d\vartheta\,d\varphi\). The distance between the protons bound in the molecule is assumed to be unchanged. The correlation function is found from (59), with time averaging replaced by averaging over initial states (according to the ergodic hypothesis):

\[ k(\tau)=\frac{1}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi} F(\vartheta_0,\varphi_0)\sin\vartheta_0\,d\vartheta_0\,d\varphi_0 \times \int_{0}^{\pi}\int_{0}^{2\pi} F^{*}(\vartheta,\varphi)\, f(\vartheta,\varphi,\tau,\vartheta_0,\varphi_0)\sin\vartheta\,d\vartheta\,d\varphi. \tag{62} \]

The solution of equation (61) has the form

\[ f(t,\vartheta,\varphi,\vartheta_0,\varphi_0) = \sum_{l,m} C_{lm}(\vartheta_0,\varphi_0)\, Y_{lm}(\vartheta,\varphi)\, e^{-\frac{tDl(l+1)}{a^2}}, \tag{63} \]

where \(C_{lm}\) and \(Y_{lm}\) are spherical functions, and \(a\) is the radius of the sphere of molecular action. As already stated above, the calculations performed in \(^{16}\) refer to the case of Brownian rotation of a water molecule.

Substitution of (63) into (62) gives the following expressions for the correlation function:

\[ \left. \begin{aligned} \overline{F_1(t)F_1^{*}(t+\tau)} &= \frac{2}{15}\,b^{-6}e^{-\frac{|t|}{\tau_c}}, \\[4pt] \overline{F_2(t)F_2^{*}(t+\tau)} &= \frac{8}{15}\,b^{-6}e^{-\frac{|t|}{\tau_c}}. \end{aligned} \right\} \tag{64} \]

From (56) and (57) the expressions for the relaxation times are obtained:

\[ \left. \begin{aligned} \frac{1}{T_1} &= K_1\left[ \frac{\tau_c}{1+4\pi^2\nu^2\tau_c^2} + \frac{2\tau_c}{1+16\pi^2\nu^2\tau_c^2} \right], \\[6pt] \frac{1}{T_2'} &= \sqrt{ \frac{1}{\pi}K_0\,\operatorname{arctg}\frac{2\tau_c}{T_2'} }. \end{aligned} \right\} \tag{65} \]

\[ \frac{1}{T_2}=\frac{1}{2T_1}+\frac{1}{T_2'}. \tag{66} \]

In these formulas \(b\) is the distance between the protons in the water molecule,

\[ K_0=3K_1,\qquad K_1=\frac{2}{5}\gamma^4\hbar^2 I(I+1)b^{-6}, \]

\[ \tau_c=\frac{4\pi r^3\eta}{3T} \]

is the correlation time characterizing the Brownian rotation of molecules. During this time the molecule appreciably changes its orientation in the external field.

Thus, the correlation time is a quantity characterizing the relaxation processes in a system of nuclear magnetic moments. For illustration, Fig. 12 gives a plot of the dependence of the relaxation times on the correlation time according to (65) and (66).

Fig. 12. Theoretical dependence of the relaxation time on the correlation time.

It is important to note that for sufficiently small values of \(\tau_c\) both relaxation times coincide. This is in agreement with the theory presented in the preceding paragraph \(^{8}\).

The calculation of the relaxation time \(T_1\) given above refers to the case of Brownian rotation of molecules in a liquid. In addition, in \(^{16}\) the relaxation time caused by Brownian displacement of particles is considered. The resultant relaxation time is found from the expression

\[ \left(\frac{1}{T_1}\right)_{\text{rot}}+ \left(\frac{1}{T_1}\right)_{\text{trans}} = \frac{1}{T_1}. \tag{67} \]

The relaxation time \((T_1)_{\text{trans}}\) is found from (56) by integrating this expression over \(r\) within the limits from \(2a\) to \(\infty\) and multiplying by the number of particles \(N\) per unit volume (\(r\) is the distance between molecules and \(a\) is the radius of the sphere of molecular action). The correlation time \((\tau_c)_{\text{trans}}\) in this case is a function of the distance between molecules:

\[ (\tau_c)_{\text{trans}}=\frac{r^2\pi\eta a}{2kT}, \tag{68} \]

where \(\eta\) is the viscosity of the liquid. The final result has the form

\[ \left(\frac{1}{T_1}\right)_{\text{trans}} = 0.9\pi^2\gamma^4\hbar^2\frac{N\eta}{kT}. \tag{69} \]

For protons in water the value obtained is

\[ (T_1)_{\text{trans}}=10\ \text{sec.}, \]

whereas

\[ (T_1)_{\text{rot}}=5.2\ \text{sec.} \]

The total relaxation time turns out to be

\[ T_1=3.4\ \text{sec.}, \]

which agrees with experiment.

The presence in the specimen of paramagnetic ions possessing a magnetic moment large in comparison with the nuclear one strongly changes the value of the relaxation time \((T_1)_{\text{trans}}\). If in the case of pure water one has

\[ (T_1)_{\text{rot}} < (T_1)_{\text{trans}}, \]

then in the presence of paramagnetic ions this inequality is reversed and assumes the opposite significance, so that the quantity \((T_1)_{\text{rot}}\) may be neglected, since \(\gamma_{\text{ion}}^2\) is approximately \(10^6\) times larger than \(\gamma_p^2\). For \(T_1\) in this case the expression will be

\[ \frac{1}{T_1} = 12\pi^2\gamma_p^2\mu_{\text{eff}}^2 N_{\text{ion}} \frac{\eta}{5kT}, \tag{70} \]

where the quantity \(\mu_{\text{eff}}^2\) has been introduced in place of the expression

\[ \gamma_{\text{ion}}^2\hbar^2 S_{\text{ion}}(S_{\text{ion}}+1). \]

It should be noted that the calculations of \((T_1)_{\text{trans}}\) in the theory presented are of a very approximate character. Recently, attempts have been made\(^{29}\) to approach this question in a more substantiated way.

§ 3. Theory of the fine structure of absorption lines

Experimental study of the shape of absorption lines in crystals has shown the presence of fine structure. The theory\(^{17}\), set forth in the preceding section, proceeds from the fact that line broadening occurs through the formation of a very large number of levels in a narrow interval. Owing to the enormous number of interacting particles and the small interaction energy, the number of levels is so great and they are distributed so densely that, in practice, a picture of broadening of the resonance line is obtained.

If, however, the number of interacting particles is small, then the number of levels decreases, and the fine structure will be observed on an instrument of appropriate resolving power.

In crystals, when the nuclei are arranged in groups sufficiently far removed from one another, the interaction between the groups may be regarded as small. The interaction of nuclei within a group gives a splitting of the energy levels.

Fig. 13. Diagram of the shift of the energy levels caused by the interaction of two protons in the crystal lattice of gypsum.

The calculation of energy levels, set forth in ³⁰, refers to the interaction of two protons in the crystal lattice of gypsum. The paired arrangement of the protons and their remoteness from other pairs make it possible to confine oneself to consideration of the energy of two particles. The Hamiltonian in this case has the form

\[ \hat{\mathcal H} = \mu(\sigma_1+\sigma_2)\mathbf H + \mu^2 r^{-3} \{\sigma_1\sigma_2-3(\sigma_1 r_{12}^{0})(r_{12}^{0}\sigma_2)\}. \tag{71} \]

It may be represented in the form (46), quite analogously to how this was done above. The matrix elements of the perturbation energy responsible for the splitting of the levels, calculated by the perturbation method, have the form

\[ \begin{aligned} (-1|V|-1)&=-\mu^2 r^{-3}(3\cos^2\theta-1),\\ (0|V|0)&=2\mu^2 r^{-3}(3\cos^2\theta-1),\\ (1|V|1)&=-\mu^2 r^{-3}(3\cos^2\theta-1), \end{aligned} \left\} \tag{72} \right. \]

where \(\mu\) is the magnetic moment of the proton, \(r\) is the distance between the protons in the molecule, and \(\theta\) is the angle between the axis of the molecule and the direction of the magnetic field. The splitting of the levels is illustrated in Fig. 13. As is evident from the diagram, the following transitions prove possible:

\[ \begin{aligned} m=+1 &\to m=0\\ h\nu &= 2\mu H_0+3\mu^2 r^{-3}(3\cos^2\theta-1),\\ m=0 &\to m=-1\\ h\nu &= 2\mu H_0-3\mu^2 r^{-3}(3\cos^2\theta-1). \end{aligned} \left\} \tag{73} \right. \]

The fine structure, thus, depends on the orientation of the molecular axis with respect to the external field. In view of the presence in the lattice

of gypsum, two groups of differently oriented molecules form four components of the line, which change their position when the crystal is rotated with respect to the external field. The experimental results will be presented below.

In the form indicated, the theory of fine structure is applicable to single crystals. For polycrystalline substances the result of the calculations will be somewhat different. Figure 14, b presents the calculated shape of absorption lines in polycrystalline gypsum. It is assumed that the number of transitions from one energy state to another per unit time is determined by the relation

\[ N^+ = \frac{N}{2}\int_0^1 d(\cos\theta) = \]

\[ = \frac{N}{2}\int P(\Delta H)d(\Delta H). \tag{74} \]

The function \(P(\Delta H)\) determines the line shape. For calculations by formula (74) it is necessary to express \(\theta\) through \(\Delta H\). This is done with the aid of (73). The form of the function \(P(\Delta H)\), calculated in the indicated manner, is shown in Fig. 14, b. Interaction with remote molecules leads to broadening of the line. In Fig. 14, a the splitting of the absorption line caused by the pair interaction of protons in a single crystal of gypsum is presented. Fig. 14, c depicts the broadening of these lines due to interaction with other molecules of the single crystal. Fig. 14, b depicts the shape of the lines in polycrystalline gypsum, caused by the random arrangement of individual crystals, and, finally, Fig. 14, d depicts the total effect of the factors listed above.

Fig. 14. Shape of absorption lines in crystals:
a) splitting of the absorption line into two components, caused by the pair interaction of protons in a single crystal of gypsum; b) line shape for a polycrystal, obtained with allowance for the random arrangement of crystallites; c) broadening of the absorption lines in a single crystal, caused by interaction with the moments of the nuclei of remote molecules; d) line shape obtained by summing the factors taken into account for cases b) and c).

§ 4. Magnetic shielding of nuclei

Magnetic shielding arises as a consequence of the difference between the field acting on the nuclei and the external field, which is due to the superposition of the field caused by the motion of electrons in the molecule, and also the field arising as a result of the presence of paramagnetic ions in the sample. In metals shielding may be caused by the action

electrons. In paper 31 the field acting on nuclei in molecules that are in an external field and have no orbital or spin moment outside the field is calculated. For molecules having excited states of low energy, the field caused by the secondary induced paramagnetism of the molecule is of greatest importance. The calculation is associated with difficulties, since it requires knowledge of the wave functions of the excited state of the molecule. The results are applied to the case of molecular hydrogen. The total value of the magnetic shielding constant is equal to \(2.68 \cdot 10^{-5}\). In paper 32 it is suggested that the magnetic shielding for a given nucleus, chemically bonded to atoms of other elements in various simple covalent compounds, depends on the position of these elements in the periodic system. Experiments 33 confirm this hypothesis. The atomic radii for covalent bonds vary in an analogous manner. This makes it possible to relate the magnitude of the magnetic shielding directly to the magnitudes of the atomic radii. In other works 34, 35, in order to refine the gyromagnetic ratio of protons, measurements were made of the magnetic shielding in gaseous \(H_2\), water, and mineral oil. On the basis of these measurements, corrections were obtained for refining the gyromagnetic ratio of protons. Taking these corrections into account, the value of \(\gamma_p\) is

\[ \gamma_p = (2.67530 \pm 0.00006) \cdot 10^4\ \text{sec}^{-1}\ \text{oerst}^{-1}. \tag{75} \]

In metals the resonance frequencies are appreciably higher than the resonance frequencies of the same nuclei in nonmetallic compounds 36. These shifts are sufficiently large that they could be attributed to differences in magnetic susceptibility or to differences in the corrections for diamagnetism. In paper 37 it is assumed that the shift arises from the orientation of the spins of the conduction electrons in the magnetic field and the interaction of these electrons with the nuclei. The experimental results give satisfactory agreement with the results of calculations; however, the data obtained for lithium in a later paper 38 agree less well with experiment, which the authors attribute to the imperfection of the contemporary theory of the metallic state. In paper 39 a theory of relaxation time in metals is developed. Formulas are obtained that relate the relaxation time to the magnitude of the shift of nuclear magnetic resonance. The interaction of nuclei with conduction electrons, which usually have a very large probability density near the nuclei, leads in metals to a change in the effective magnetic susceptibility 40 (additional terms appear in the equation for the component \(M_z\), due to the electronic component of the susceptibility). Under ordinary conditions this change is negligibly small. However, upon saturation of the electronic component (which can be effected by the action of a resonant radio-frequency field)

this effect is very considerable and may lead to an increase of the signal by a factor of hundreds. Such an increase of the signals in metallic lithium was obtained experimentally[^41] in a field of 30.3 oersted at the nuclear-resonance frequency of 50 kHz. A saturating radio-frequency field of frequency 84 MHz was supplied from a powerful generator and had an intensity of 5 oersted. The increase of the signal obtained in these experiments was of the order of 100.

II. APPARATUS AND METHODS USED IN THE EXPERIMENT

§ 1. Distinctive features of experiments on magnetic resonance of atomic nuclei

In contrast to optical spectroscopy, where the source of electromagnetic radiation gives an entire spectrum of frequencies, in experiments on magnetic resonance of atomic nuclei practically monochromatic sources of electromagnetic oscillations are used—ordinary radio-frequency generators. To obtain a spectral picture in this case, periodic time sweep of the signals on the screen of an oscilloscope is employed, with periodic variation of the intensity of the longitudinal component of the magnetic field or of the frequency of the emitted oscillations.

Another essential distinction of the indicated experiments is the possibility of the occurrence, in the system of nuclear magnetic moments, of nonstationary processes lasting for a time sufficient for their observation. These features make possible two methods of observing signals: the method of continuous action of a high-frequency field with periodic modulation of the longitudinal component of the magnetic field or of the radio frequency, and the pulse method, in which the resonant radio-frequency field is applied in pulses at a constant magnetic field \((\omega = |\gamma|H_z = \mathrm{const})\), and nonstationary processes are observed.

§ 2. Method of continuous action

When this method is used, the radio-frequency magnetic field is continuously excited in the coil of an oscillatory circuit containing the sample and placed in a constant magnetic field. There are two ways of exciting the field in the coil. In one of them the radio-frequency circuit is fed by a high-frequency voltage from an external generator; in the other, the circuit forms part of the generator. For the periodic occurrence of resonance, a small alternating field (usually sinusoidal) is superposed on the constant magnetic field. Sometimes, instead of this, the frequency of the generator is modulated. The sweep of the oscilloscope beam is effected

synchronously with the modulating field. An approximate block diagram of an apparatus with an external generator of high-frequency oscillations is shown in Fig. 15. In these circuits, to protect the amplifier from overload by the generator’s high-frequency voltage and to reduce noise, various compensating devices are used. There exist circuits^42 in which there is a separate coil producing the high-frequency magnetic field and a separate coil for receiving signals, arranged perpendicular to one another. Compensation is obtained by the perpendicular arrangement of the coils and with the aid of special metal vanes, which induce an additional compensating emf in the receiving coil. The amplitude and phase of this emf are adjusted by rotating the vane. In other circuits^16 a radio-frequency bridge is placed between the generator and the amplifier, in one arm of which there is a circuit containing the sample. Compensation is achieved by adjusting the reactive and active resistances in the bridge arms; the appearance of a signal changes the balance. Combinations of the methods described above are possible,^43,44 as are circuits without compensation.^45,46,47.

Fig. 15. Block diagram of an apparatus with an external generator.
1 — generator, 2 — compensating device, 3 — high-frequency amplifier, 4 — detector, 5 — low-frequency amplifier, 6 — oscilloscope, 7 — source of the modulating voltage.

In all cases the compensation must not be complete. A small residual voltage, whose phase determines the nature of the signal, must be induced in the receiving coil. This voltage plays the role of the carrier frequency. Thus, at the input of the amplifier, together with the signal voltage (which is proportional to the derivative component of the resultant vector of the nuclear magnetization), there arrives a small part of the high-frequency voltage of amplitude \(E\) with phase shift \(\varphi\). To accuracy of quantities of order \(\frac{1}{\omega}\), where \(\omega\) is the angular frequency of the radio-frequency field, the signal at the receiver input has the form

\[ V=E\cos(\omega t+\varphi)-A\omega(u\sin\omega t-v\cos\omega t)= \]

\[ =\sqrt{(E\cos\varphi+A\omega v)^2+(E\sin\varphi+A\omega u)^2}\cos(\omega t+\theta), \tag{76} \]

where \(A\) is a proportionality coefficient depending on the coil of the circuit,

\[ \theta=\operatorname{arctg}\frac{E\sin\varphi+u}{E\cos\varphi+v}. \]

Under the conditions \(E \gg A\omega u,\ E \gg A\omega v\), which hold in practice, we obtain:

\[ V \simeq E\left[1+\frac{A\omega}{E}(v\cos\varphi+u\sin\varphi)\right]\cos(\omega t+\theta). \tag{77} \]

This expression shows that at the receiver input there arrives a signal amplitude-modulated with modulation coefficient \(\dfrac{A\omega}{E}\). Depending on the phase \(\varphi\), at the output of the amplitude detector the signal \(u\) or \(v\) will be separated (see (9)). After amplification the signal is fed to the input of the oscilloscope. The compensation methods indicated above make it possible to attain a certain imbalance with a definite phase shift and thus, at the experimenter’s discretion, to separate the signal \(u\) or \(v\).

The schematic diagram of an apparatus in which the emf of nuclear magnetic resonance acts directly on the generator circuit\({}^{48}\) is shown in Fig. 16. The coil of the generator circuit is placed in a magnetic field. Fig. 16 shows two circuits that make it possible to observe the signal. One circuit consists of a high-frequency amplifier, to whose input the signal comes directly from the generator circuit, a detector, a low-frequency amplifier, and an electronic oscilloscope. The second circuit consists of a low-frequency amplifier, to whose input the signal from the anode of the generator tube is fed. Detection is carried out in the grid circuit of this tube. A signal-receiving circuit usually contains one of these circuits.

Fig. 16. Block diagram of an apparatus with direct action of the magnetic-resonance emf on the generator circuit.
\(1\) — generator, \(2\) — high-frequency amplifier and detector, \(3\) — low-frequency amplifier, \(4\) — oscilloscope, \(5\) — low-frequency amplifier, \(6\) — source of modulating voltage.

The generator may operate both in the regime of continuous generation and in the regime of superregeneration, in which oscillations are quenched. The latter regime is very sensitive; however, the presence of an entire spectrum of side frequencies and the related ambiguity of the signals make it applicable only for purposes of determining the gyromagnetic ratios of atomic nuclei. When the generator operates in the continuous-generation regime, it is necessary to maintain a low level of oscillations in order to avoid saturation.

To increase the signal-to-noise ratio, in many works on magnetic resonance of atomic nuclei a synchronous detector is used. The idea of this device is based on the fact that the mean value of the noise, when integrated over a sufficiently long time interval, is zero. In order to separate the useful signal from the noise, the signal is multiplied by its first harmonic. In experiments on magnetic resonance of atomic nuclei, the modulation voltage is used for this purpose. Thus, the device consists of: 1) an apparatus that multiplies the signal by its first harmonic, and 2) an integrator. As a result of these operations, a mathematical action is performed that is equivalent to calculating the first harmonic of the signal. An account of the principles of operation of such circuits is given in ^27. If the experimental conditions are such that the form of the signal approaches that shown in Fig. 2, then, as shown in ^16, the dependence of the amplitude of the signal taken from the synchronous detector on the detuning \((\Delta \omega = |\gamma|H_z - \omega)\) has the form of the derivative of the imaginary part of the nuclear magnetic susceptibility. For recording the signals of the synchronous detector, a recorder is used. The motion of the tape on which the recording is made is synchronized with the variation of the magnetic field or radio frequency.

§ 3. Pulse Methods

When pulse methods are used, the frequency of the radio-frequency field and the strength of the magnetic field remain constant, satisfying the resonance conditions \((\omega = |\gamma|H_z)\), but the radio-frequency field is applied in pulses. An approximate diagram of the apparatus for experiments of this kind is shown in Fig. 17.

Fig. 17. Block diagram of the apparatus for the pulse method of observing signals.
1 — high-frequency generator, 2 — quenching generator, 3 — high-frequency circuit, 4 — wide-band amplifier, 5 — oscilloscope, 6 — control-pulse sensor, 7 — triggered-sweep generator.

The coil with the sample 3 is placed in the magnetic field; depending on the method of observing the signals, it is either provided with com-

compensating device, or not. The coil is supplied with high-frequency voltage from generator 1. Device 2 serves for obtaining pulses. The signals are amplified by the wide-band amplifier 4 and, after detection, are observed on the screen of the oscillograph 5. The sweep of the oscillograph beam and the quenching of the high-frequency voltage in the intervals between pulses are controlled by a special generator 6.

The signals can be observed both during the application of the pulse (non-steady nutation)\(^{49}\) and in the interval between pulses (spin echoes)\(^{50}\). In the first case the use of a compensating device is necessary.

Let us consider the occurrence of non-steady nutation. In the absence of a high-frequency field, the magnetization vector arrives at its equilibrium position and is directed along the field. When the radio-frequency field

\[ H_x'' + iH_y'' = H_1 e^{i\omega t} \]

is switched on, an additional torque arises

\[ \mathbf{I} = [\mathbf{M}\mathbf{H}''], \tag{78} \]

which carries the vector \(\mathbf{M}\) out of its equilibrium position. The behavior of the resultant magnetization vector in this case can be considered with the aid of equations (7), which do not take relaxation processes into account, since, when the field is switched on rapidly, the latter do not have time to exert a noticeable influence on the behavior of the vector \(\mathbf{M}\). In a coordinate system rotating about the \(z\)-axis with the angular frequency of the radio-frequency field, equations (7) have the form\(^*\)

\[ \frac{d\mathbf{M}'}{dt} = \gamma [\mathbf{M}'\mathbf{H}'], \tag{79} \]

where \(\mathbf{M}'\) is the vector with components \(u\), \(v\), and \(M_z\) (see (4)). The components of the field-strength vector have the form

\[ H_x' = H_1,\qquad H_y' = 0,\qquad H_z' = H_z - \frac{\omega}{\gamma}. \tag{80} \]

All coefficients in (79) are constant. Equations (79) have two integrals of motion:

\[ \left. \begin{aligned} |\mathbf{M}'| &= \mathrm{const} \\ |\mathbf{M}'|\,|\mathbf{H}'|\cos\alpha &= \mathrm{const}, \end{aligned} \right\} \tag{81} \]

where \(\alpha\) is the angle between the vector \(\mathbf{M}'\) and the field \(\mathbf{H}'\), which is usually called the effective field. From (4) and (81) it follows that the vector \(\mathbf{M}\)

\(^*\) The transition to the rotating coordinate system is conveniently carried out by making use of the theorem on the local derivative known from theoretical mechanics.

participates in two motions, undergoing rapid precession (together with the rotating coordinate system) and slow nutation with \(\alpha=\mathrm{const}\) about the effective field. The angular frequency of the nutation, as follows from (79), will be:

\[ \omega_1=\gamma \sqrt{\left(H_z-\frac{\omega}{\gamma}\right)^2+H_1^2}. \tag{82} \]

When \(H_z=\frac{\omega}{\gamma}\), the vector rotates in the \(zOy\) plane, and the nutation has maximum amplitude. The behavior of the vector \(\mathbf M\) corresponding to the case under consideration is shown in Fig. 18. The emf induced by the precession of the magnetization vector in the coil of the oscillatory circuit thus proves to be modulated by the nutation. Bridge circuits are usually used as compensating devices when the pulsed method is employed.

The input of the amplifier receives the sum of the signals: the signal that has passed through the compensating device owing to the unbalance of the bridge, and the nuclear magnetic resonance signal. Everything said above concerning the influence of the phase shift between the unbalance voltage and the emf of nuclear magnetic resonance on the character of the signal also applies in the present case. As follows from the solution of equations (79), for

\[ \left(H_z=\frac{\omega}{\gamma}\right) \]

the signal \(u\) is equal to zero. Owing to the presence of relaxation processes, the nutation decays, and the signals visible on the oscilloscope screen have the form of a “damped sinusoid.” Periodic application of pulses ensures the production of a stable pattern on the oscilloscope screen.

Fig. 18. Nutation of the resultant vector of nuclear magnetization upon sudden switching-on of the radio-frequency field.

Let us consider the origin of spin echoes. This phenomenon is due to the inhomogeneity of the constant magnetic field. For simplicity of consideration, let us divide the specimen into separate elementary volumes, assuming the field within each of them to be homogeneous. For each elementary volume there will exist its own Larmor precession frequency of the nuclear magnetic moments, determined by the value of the magnetic-field intensity at the given point of the specimen. To observe spin echoes, the radio-frequency field is applied to the specimen in two successive short pulses, sepa-

with a definite time interval \(\tau\) (Fig. 19). At the moment the radio-frequency field is switched on, whose frequency is chosen on the basis of

Fig. 19. A simple vector model explaining the phenomenon of spin echoes.

the relation \(\omega = |\gamma|H_z\), the magnetization vector leaves the equilibrium position (Fig. 19, A) and begins to rotate in the plane \(zOy\). In Fig. 19 a rotating coordinate system is presented...

DYNAMIC METHOD OF INVESTIGATING NUCLEAR PARAMAGNETISM

coordinates. In the “stationary” coordinate system the vector \(\mathbf{M}\) executes a complicated motion, with its endpoint describing a spiral on the surface of a sphere of radius \(r=|\mathbf{M}|\).

The duration of the pulse is chosen so that by the moment it ends the vector \(\mathbf{M}\) is found to be oriented along the \(y\)-axis. For this it is necessary that the condition

\[ \omega_1 t_w = |\gamma| H_1 t_w = \frac{\pi}{2} \]

be satisfied (see (82) and Fig. 19, \(B\)). After the radio-frequency field is switched off, the magnetization vector will undergo Larmor precession about the direction of the constant field \(H_z\). However, owing to the inhomogeneity of this latter field, for different elementary volumes this precession will be different. In the rotating coordinate system there is formed, as it were, a fan of magnetization vectors belonging to individual elementary volumes, since some of these vectors will lead the coordinate system rotating with frequency \(\omega\), while others will lag behind it (see Fig. 19, \(C\)). With time this fan will spread more and more to both sides of the \(y\)-axis. If the specimen is now again acted upon by a pulse of the same duration, the whole fan will rotate through an angle \(\pi/2\) (Fig. 19, \(D\)), and its central part will be directed opposite to the \(z\)-axis.

After the field is switched off, the fan will begin to “fold.” One half of the fan will lag behind the coordinate system, and the other will lead it. The vectors in the fan prove to be “sorted” according to Larmor frequencies, so that during rotation the fan is deformed: its upper part moves faster than the lower. The signal \(v(t)\), observed on the screen of an oscillograph, is proportional to the projection of the vector \(\mathbf{M}\) on the \(y\)-axis of the rotating coordinate system. At the moment when the second pulse ends (Fig. 19, \(E\)) this projection is equal to zero. With time, owing to the combined action of the individual vectors of the fan, this projection increases and reaches its maximum value after a time interval \(2\tau\) from the beginning of the first pulse (Fig. 19, \(F\)). A signal appears on the screen of the oscillograph, as shown in the upper part of Fig. 19.

Recently\(^{74}\) an interesting modification of the spin-echo technique has been proposed. This modification consists in the fact that the second pulse rotates the fan of vectors belonging to the elementary volumes not by \(90^\circ\), but by \(180^\circ\). For this, the duration and amplitude of the second pulse are chosen on the basis of the relation

\[ |\gamma| H_1 t_w = \pi . \]

In this case, after being inverted, the fan will again be situated in the plane ...

velocities \(xOy\), and after the second pulse is switched off it will begin to fold up. After a time \(2\tau\) has elapsed from the instant the first pulse was switched on, all the vectors of the fan will again merge into a single vector. At this instant an oscilloscope screen will show a maximum of the spin-echo signal. When this method is used, all the vectors of the fan, in forming the signal, regress in the plane \(xOy\), which considerably increases the signal.

The spin-echo method can be successfully applied to the measurement of relaxation times, as will be discussed below.

III. RESULTS OF THE PRINCIPAL EXPERIMENTS

§ 1. Measurement of nuclear magnetic moments

The phenomenon of magnetic resonance of atomic nuclei can be used for a very accurate determination of the gyromagnetic ratios of nuclei. The application of this method has made it possible to measure a large number of gyromagnetic ratios. The accuracy of the method is determined by three factors: 1) the accuracy of measuring the magnetic field, 2) the accuracy of measuring the frequency, 3) the accuracy with which the resonance moment is fixed. The accuracy with which resonance is fixed and the accuracy of frequency measurement can be made sufficiently high with the expenditure of comparatively modest experimental means. Accurate measurement of the magnetic field, on the other hand, is very difficult. However, this difficulty can be overcome if the gyromagnetic ratio for the nuclei of one element has been measured. The gyromagnetic ratios of the nuclei of other elements are then measured with high accuracy from the ratio of the resonance frequencies in one and the same magnetic field.

A gyromagnetic ratio that has been measured very accurately is the gyromagnetic ratio for protons. The value given above (75) shows that the accuracy attained in these experiments is \(0.002\%\). In measuring the gyromagnetic ratios of nuclei of two elements, still greater accuracy can be achieved. Thus, the ratio of the moments of \(\mathrm{H}^1\) and \(\mathrm{H}^3\) was measured \(^{51}\) with an accuracy of \(0.001\%\), the ratio of the moments of \(\mathrm{H}^1\) and \(\mathrm{H}^2\) was measured \(^{52}\) with an accuracy of \(0.002\%\), and \(^{53}\) with an accuracy of \(0.0005\%\). For searches for unknown gyromagnetic ratios, special radio-frequency spectrometers have been constructed, in which a continuous recording is made of the signals arising under a comparatively slow change of the radio frequency or of the constant component of the longitudinal component of the magnetic field \(^{43,54,55,56}\).

With the aid of these installations a very large number of gyromagnetic ratios of various nuclei have been measured. Very many gyromagnetic ratios have also been measured with the aid of a super-regenerator \(^{57,58,59}\). In addition to those enumerated, there is a very large

number of papers on determining the gyromagnetic ratios of various isotopes. A summary table of nuclear moments is available in \(^{60}\), but at present it is already insufficiently complete. Among recent works one should mention \(^{75,76}\), in which very sophisticated apparatus was used, making it possible to determine the gyromagnetic ratios of the nuclei \(\mathrm{Si}^{29}\), \(\mathrm{S}^{33}\), \(\mathrm{Zn}^{67}\), \(\mathrm{As}^{75}\), \(\mathrm{Se}^{77}\), \(\mathrm{Te}^{123}\), \(\mathrm{Te}^{125}\), \(\mathrm{K}^{41}\), \(\mathrm{J}^{87}\), \(\mathrm{Ag}^{107}\), and \(\mathrm{Ag}^{109}\).

§ 2. Relaxation times and structural studies

An experimental verification of the theory set forth in § 2 of Chapter I was carried out by its authors \(^{15,16}\). The method for measuring the transverse relaxation time consisted in measuring the line width. For this purpose a synchronous detector was used. As stated above, it was assumed that the form of the signals obtained with the aid of the synchronous detector is the derivative of the absorption curve. The distance between the extrema of such a signal \((\Delta \omega)\) is equal to the distance between the points of maximum slope of the absorption curve. It was assumed that \(\Delta \omega = \frac{1}{T_2}\) (see Fig. 2). This method can be applied in those cases in which the inhomogeneity of the magnetic field is less than the line width. In addition, for a large modulation amplitude the line shape will differ from that shown in Fig. 2. In the case of strong signals the line width was measured directly on the oscilloscope screen.

Two methods were used to measure \(T_1\). The first method consisted in observing the process of signal recovery after saturation caused by the action of a strong radio-frequency field. First, a strong radio-frequency field was applied to the sample, as a result of which the signal was noticeably weakened. Then the field strength was rapidly reduced, and the signal gradually increased. The process of signal recovery was photographed with a cine camera. The dependence of the signal amplitude on time was plotted. It was assumed that the exponential increase obtained in the signal amplitude is determined by the relaxation time \(T_1\). This method is applicable only to such substances in which \(T_1\) is sufficiently large. For other samples a method was used for determining \(T_1\) from the dependence of the signal amplitude on the amplitude of the radio-frequency field. In \(^{16}\) the following approximate expression was obtained for this dependence (when a synchronous detector is used):

\[ \left(\frac{\partial \chi''}{\partial H}\right)_{\max} = \left(1+(\gamma H_1)^2 T_1 T_2\right)^{-3/2}. \tag{83} \]

This expression was obtained under the assumption that the shape of the curve

corresponds to a Lorentzian form of the absorption line (Fig. 2). Since, however, the quantity \(H_1\) is not known exactly, this method was used for comparing relaxation times in two samples, in one of which \(T_1\) is sufficiently large. Applying both of the indicated methods to this sample, it was possible to measure relaxation times over a rather large range of values, from \(10^2\) to \(10^{-4}\) sec.

The results of experiments with aqueous solutions of glycerin of various concentrations and temperatures are shown in Fig. 20.

Fig. 20. Experimental dependence of the relaxation times on the ratio of viscosity to temperature (this ratio is proportional to the correlation time).

The correlation time is taken to be proportional to the ratio of viscosity to temperature. This quantity is plotted along the abscissa axis. The experimental points in Fig. 20 lie sufficiently close to the theoretical curves (see Fig. 12). The presence of a minimum on the curves of the dependence of \(T_1\) on \(\tau_c\) makes it possible to determine experimentally the value of \(\tau_c\) (at the minimum point \(\tau_c = \dfrac{1}{\omega_0\sqrt{2}}\)). The course of the curves in Fig. 20 indicates good agreement between the theoretically found value of \(\tau_c\) and the value determined experimentally.

Experiments carried out with aqueous solutions containing paramagnetic ions Fe+++, Cu++, Er+++, Cr+++, Ni++, Co++, etc., showed a linear dependence of the quantity \((T_1)^{-1}\) on the concentration of paramagnetic ions. This also agrees well with the concepts of the theory. However, experiments by other authors\({}^{77}\) on measuring \(\mu_{\mathrm{eff}}\) of paramagnetic ions, carried out with high accuracy, showed that in the case of ions

complex configuration the predictions of the theory\(^ {16}\) are not always fulfilled. The authors\(^ {77}\) suggest that in this case Stokes’ law, on which the theory\(^ {16}\) is based, is no longer applicable.

Studies of the fine structure of absorption lines have made it possible to obtain new data on the structure of crystals. In work\(^ {30}\) the fine structure of proton absorption lines in gypsum crystals was investigated.

The experimental results are in agreement with the predictions of the theory set forth above (Chap. I, § 3). In these experiments apparatus similar to that used in \(^ {16}\) was employed. A single crystal of gypsum could be rotated about an axis perpendicular to the magnetic field.

As was said above, the magnitude of the splitting depends on the angle between the axis joining the two protons of the molecule and the direction of the magnetic field. Since in the crystal lattice of gypsum there are two groups of water molecules oriented in different ways, four components are obtained. The change in the magnitude of the splitting of each of the doublets as a function of the orientation of the crystal in the magnetic field makes it possible to determine the directions of the axes joining the protons in the molecule. The theory of the chemical bond can give here only qualitative ideas. The results of the experiments described do not contradict the data of the theory of the chemical bond.

The maximum magnitude of the splitting is determined by the distance between the protons in the molecule. The experimental data make it possible to determine this distance. For CaSO\(_4\) it was found to be 1.58 Å.

Thus, the method of nuclear magnetic resonance has opened new possibilities in the investigation of the structure of crystals, in many cases inaccessible to other methods.

In subsequent works a large number of polycrystalline substances, as well as complex organic compounds, were studied. The principal attention was directed to studying the temperature dependence of the line width and the relaxation time \(T_1\). For some substances (for example, for CuSO\(_4\cdot 5\)H\(_2\)O) it was found that the fine structure of the lines arises only at low temperatures\(^ {61}\). Investigations of the temperature dependence of the line width for some substances showed the presence of jumps on the curve representing this dependence. The jumps appear at points at which the substance undergoes phase transitions\(^ {24}\). Such jumps occur if the phase transition is accompanied by a change in the structure of the substance.

The local field acting on a molecule of a given substance may be divided into an intramolecular one, caused by

by the interaction of atoms within a molecule, and the intermolecular one, caused by the interaction of molecules with one another. Such a separation was carried out in \(^{62}\). In that work the temperature dependence of the line width and of the relaxation time \(T_1\) was studied in the compounds \(C_6H_6\), \(C_6H_5D\), \(1\cdot3\cdot5\text{-}C_6H D_3\). A change in the line width upon replacing hydrogen by deuterium was observed. The magnitude of this change was also calculated theoretically. The existence of experimental data for two substances, \(C_6H_6\) and \(C_6H_5D\), made it possible to eliminate the unknowns from the expressions obtained under the assumption that the line width is the total effect of the action of intramolecular and intermolecular fields. This method made it possible to determine the distance between the protons in the indicated compounds.

The study of organic compounds of complex structure has shown in some cases the presence of a fine structure of absorption lines \(^{64,65}\). This indicates the existence of separate groups of molecules, between which rigid bonds act, restricting molecular rotation.

As an example, one may cite the splitting of the proton absorption line in ethyl alcohol. The formula of this substance may be written in the form \(CH_3\cdot CH_2\cdot OH\). In the experiments \(^{64}\), a splitting of the line into three components was observed, separated from one another by distances of about 16 millioersteds. The widths of the lines themselves were 2–3 millioersteds. The lines had different intensities, which made it possible to establish the assignment of a given line to a definite group of atoms.

In \(^{65}\) splitting of the absorption lines of \(P^{31}\) and \(F^{19}\) was observed in several liquid phosphorus-halide compounds. The number of components of the split \(F^{19}\) lines is equal to the number of quantum states of the \(P^{31}\) nuclei present in the molecule. The same is also observed for the \(P^{31}\) lines.

The variation of the relaxation times for solid \(CH_4\) was carried out in \(^{67}\). The values found are in agreement with theory \(^{16}\).

The possibility of applying nuclear magnetic resonance to structural investigations opens up a new field in the study of matter by means of radiophysical methods. Here it is necessary to note the rather high accuracy of the measurements in studying the fine structure of absorption lines. The study of relaxation processes in liquids will apparently make it possible to obtain new additional information necessary for investigating the nature of the liquid state.

The study of the fine structure of absorption lines in complex organic compounds may render substantial assistance in the investigation of various kinds of chemical bond.

§ 3. Unsteady Processes

The principal methods for observing unsteady processes are pulse methods. Above, the behavior of the resultant magnetization vector during the application of a pulse and in the intervals between pulses was described, as was the nature of the signals obtained. Unsteady nutations were observed in work \(^{49}\). The damping of nutations is determined by the relaxation times \(T_1\) and \(T_2\). It proceeds exponentially with an exponent equal to \(\frac{1}{2}\left(\frac{1}{T_1}+\frac{1}{T_2}\right)t\).

The inhomogeneity of the magnetic field increases the rate of damping, and this is taken into account in measurements of relaxation times. If the distribution function for the inhomogeneity of the magnet field has the form of a Lorentzian curve, then the influence of the inhomogeneity, as was already stated above, will be characterized by the addition to the exponent of the exponential of a term proportional to time. With another distribution of inhomogeneities their influence proves to be more complex. In practice one usually uses a Lorentzian distribution, idealizing the distribution function of the inhomogeneity of the magnetic field.

Measurement of the damping of nutation makes it possible to determine the combined action of the relaxation times. The field inhomogeneity can be estimated from measurements of large relaxation times, when the term containing the latter can be neglected, assuming that the damping is determined entirely by the field inhomogeneity.

The relaxation time \(T_1\) can be measured from independent experiments. The initial amplitude of the nutation, as follows from the solution of the equations, is determined by the value of the longitudinal component of the resultant magnetization vector at the moment the pulse is switched on. With periodic application of pulses, the initial amplitude of the nutation in subsequent pulses approaches, with time, a certain value smaller than the initial amplitude of the nutation in the first pulse, owing to the fact that the component \(M_z\) does not manage, in the interval of time between pulses, to return to its initial value. This decrease depends on the ratio between the relaxation time \(T_1\) and the time interval between pulses. By measuring this time interval and the change in the initial amplitude of the nutation, one can determine the magnitude of \(T_1\). The results of testing the method described, given in \(^{49}\), are in good agreement with the theory based on the solution of the Bloch equations.

The spin-echo method also makes it possible to measure relaxation times and, in addition, provides the possibility of observing beats between the Larmor frequencies of two or more groups of nuclei located in different local fields, caused by the complex structure of the substance under study.

The relaxation time \(T_2\) is determined from the decay of the signals following two radio-frequency pulses. As in the preceding case, inhomogeneity of the magnetic field accelerates the process of signal decay. The amplitude of the spin-echo signals decreases, as the time interval \(\tau\) between the pulses is increased, according to an exponential law with an exponent determined by the value of \(T_2\). If the specimen contains several groups of nuclei situated in different local fields and therefore possessing different Larmor frequencies, then, as a result of interference, the dependence indicated above assumes a complicated character. At certain values of the time interval \(\tau\) the signal may be equal to zero, but with further increase it appears again.

Experimentally, the dependence of the amplitude of spin-echo signals on \(\tau\) is obtained on a single oscillogram. For this purpose, signals at different values of \(\tau\) are recorded on one and the same frame. Owing to the use of a triggered sweep, the first pulse always arrives at one and the same place on the oscillogram. Since the signal always arises after a time \(2\tau\) has elapsed after the application of the first pulse, the oscillogram gives a scale of abscissas linear with respect to \(\tau\). From these oscillograms one can determine the relaxation time \(T_2\), and from the positions of the minima of the envelope of the signal amplitude—the difference of the Larmor frequencies for different groups of nuclei in the specimen. To determine \(T_1\), the method of applying three pulses is used, the third pulse being applied after a time interval \(T\) has elapsed from the beginning of the first pulse \((T > \tau)\). As theory shows, repeated spin-echo signals then arise, the largest of which occurs at the instant \(T+\tau\). The amplitude of these signals, as \(T\) is varied, decreases according to the exponential law \(e^{-T/T_1}\). Therefore, by recording, in the same way as is done in determining \(T_2\), the dependence of the signal amplitude on \(T\), one can determine the exponent of the exponential and calculate \(T_1\). Measurements of the relaxation times in water\(^{50}\), carried out by this method, gave results in agreement with measurements made by another method\(^{16}\).

The spin-echo method can be used for the direct measurement of relaxation times on the screen of an oscilloscope\(^{78}\). For this purpose, with the aid of the second beam, a scale of time marks is put on the screen at a height corresponding to a decrease of the signal envelope by a factor of \(e\). The intersection of the envelope with the scale of time marks indicates the desired value of the relaxation time. Since the relaxation time is uniquely related to the viscosity of a liquid, the method described above can be used for rapid determination of viscosity. Automatically operating installations\(^{78}\) have been developed for determining the viscosity of lubricating oils in industrial production.

In work \(^{66}\) a number of organic substances of complex structure were investigated. Owing to the presence of several groups of nuclei situated in different local fields, the envelope of the spin-echo signals has a complicated form. To explain the form of the signal envelopes observed in the experiments, work \(^{66}\) proposed a theory based on the solution of the Pauli equations for two particles with the Hamiltonian

\[ \hat{\mathcal H}=-\gamma\hbar\left[\mathbf I_1(\mathbf H_0+\mathbf h_1)+\mathbf I_2(\mathbf H_0+\mathbf h_2)\right]+\hbar J\mathbf I_1\mathbf I_2, \tag{84} \]

where \(\mathbf I_1,\mathbf I_2\) are spin operators, \(\mathbf H_0\) is the operator of the external magnetic field, \(\mathbf h_1\) and \(\mathbf h_2\) are the operators of the internal fields that produce the shift of the resonance frequency, and \(J\) is the coupling constant.

Solving the equations makes it possible to determine the energy levels and the shape of the signals. The number of energy levels is four when \(\mathbf h_1\ne\mathbf h_2\), and three when \(\mathbf h_1=\mathbf h_2\). The form of the signals is determined by the components \(u\) and \(v\), which are calculated as mean values of the spin operator. Comparison of the theoretical and experimental curves made it possible to determine the quantities \(J\) and \(\delta=\gamma(h_1-h_2)\), where \(h_1\) and \(h_2\) are the eigenvalues of the operators \(\mathbf h_1\) and \(\mathbf h_2\). The quantity \(\delta\) is the angular frequency corresponding to the chemical shift: proton 1 is in the local field \(h_1\), and proton 2 in the local field \(h_2\).

For 2-bromo-5-chlorothiophene the following values were obtained: at a frequency of 24 Mc,

\[ \frac{J}{2\pi}=3.9\ \text{rad/sec},\qquad \frac{\delta}{2\pi}=4.2\ \text{rad/sec}. \]

At a frequency of 32 Mc,

\[ \frac{\delta}{2\pi}=5.6\ \text{rad/sec}. \]

\(J\) does not depend on the external magnetic field \(H_0\), while \(\delta\) depends linearly on \(H_0\).

In work \(^{66}\) a number of other organic compounds were also investigated, and the case of three interacting spins was considered.

§ 4. Application of the magnetic resonance of atomic nuclei to the measurement and stabilization of a magnetic field

The resonance relation \(|\gamma|H_z=\omega\) makes it possible, when \(\gamma\) is known, to measure the strength of a magnetic field by determining the resonance frequency. Such measurements can be carried out with the same accuracy with which the frequency is measured. An accuracy of frequency measurement of \(0.01\%\) is usual, whereas measurements of magnetic-field strength by means of a fluxmeter or by the ballistic method cannot be made with an accuracy exceeding \(1\%\). Thus, the method of measuring magnetic-field strength using proton magnetic resonance is a precision method. The literature describes both comparatively simple \(^{68}\) and more complex \(^{55,69}\) instruments of this kind.

The disadvantages of the method are the impossibility, connected with its precision, of making measurements in fields with great inhomogeneity, and also the impossibility of making measurements in weak fields. The instrument described in \(^{69}\) gives an accuracy of \(0.005\%\) for a gradient not exceeding \(4\ \text{oersted}\cdot\text{cm}^{-1}\). At larger gradients of the magnetic field, different parts of the specimen give resonance signals at different moments of time, and the total signal becomes broadened. Its amplitude then falls sharply. The impossibility of measurement in weak fields is due to the fact that the signal is proportional to the longitudinal component of the magnetization, which in weak fields becomes too small. To obtain a signal of sufficient amplitude in this case, one has to take a specimen of large volume, which leads to an increase in the field inhomogeneity within the volume of the specimen.

It should be noted that recently an original method has been proposed for improving the homogeneity of a magnetic field \(^{79}\). It consists in rotating the specimen inside the radio-frequency coil. In this case every point of the specimen in a cross-section perpendicular to the axis of rotation will be in a variable magnetic field, which is averaged over the period of rotation. For the experiments \(^{80}\) a magnet was used having a field inhomogeneity of the order of \(10^{-3}\) gauss. Water was used as the specimen. A spherical specimen \(5\ \text{mm}\) in diameter was rotated about the \(y\)-axis. At a rotation speed of \(10\ \text{rev/sec}\) and higher, a decrease of the line width by a factor of 17 and an increase of its intensity by a factor of 7 were obtained.

In addition to the possibility of measuring a magnetic field, the phenomenon of magnetic resonance of atomic nuclei can also be used for stabilizing the field of electromagnets. For this purpose the signals of magnetic resonance of atomic nuclei are fed to a special device which regulates the current of the electromagnet so that, when the intensity of the magnetic field departs from the resonance value, the change in current is directed toward restoring the previous state.

As the signals fed to the regulating device, signals obtained from a synchronous detector are used. With a sufficiently small amplitude of modulation, these signals have the form of a discriminator curve. When the resonance conditions are satisfied, the signal is equal to zero. When the magnetic field deviates from the resonance value, an “error signal” appears, the sign of which is determined by the direction in which the magnetic field changes from the resonance value. The regulating device usually consists of a series of powerful electronic tubes, by whose anode current a special winding of the electromagnet is supplied. An “error signal” is applied to the control grids of the tubes, producing the regulating action. There are circuits in which modulation of the longitudinal-

component of the magnetic field^70,71. In other schemes^72 the radio frequency is modulated. The latter schemes are used for spin-echo experiments, where modulation of the longitudinal component of the magnetic field is absent.

REFERENCES CITED

1. Ya. G. Dorfman, Magnetic Properties of the Atomic Nucleus, Gostekhizdat, 1948.
2. B. G. Lazarev and L. V. Shubnikov, Sov. Phys. 11, 445 (1937).
3. V. K. Arkad’ev, ZhRFKhO, phys. ser. 45, 312 (1913).
4. S. D. Gvozdover and A. A. Magazanik, ZhETF 20, 705 (1950).
5. S. V. Vonsovskii, The Modern Theory of Magnetism, Gostekhizdat, 1952.
6. V. K. Arkad’ev, Electromagnetic Processes in Metals, GINTL, 1934.
7. F. Bloch, Phys. Rev. 70, 460 (1946).
8. R. K. Wangsnes and F. Bloch, Phys. Rev. 89, 728 (1953).
9. S. D. Gvozdover and N. M. Pomerantsev, Vestn. Mosk. Univ. No. 6, 85 (1953).
10. S. D. Gvozdover and N. M. Pomerantsev, Vestn. Mosk. Univ. No. 9, 79 (1953).
11. B. A. Jacobson and R. K. Wangsnes, Phys. Rev. 73, 942 (1948).
12. E. E. Salpeter, Proc. Phys. Soc. 63A, 337 (1950).
13. R. Gabillard, Comptes Rendus 232, 324 (1951).
14. Y. Ayant, Comptes Rendus 233, 39 (1951).
15. N. Bloembergen, E. M. Purcell and R. V. Pound, Phys. Rev. 73, 679 (1948).
16. N. Bloembergen, Nuclear Magnetic Relaxation, Leiden, 1948.
17. J. H. Van Vleck, Phys. Rev. 74, 1168 (1948).
18. V. Weiskopf and E. Wigner, Zeits. f. Phys. 63, 54 (1930); 65, 18 (1930).
19. D. I. Blokhintsev, Foundations of Quantum Mechanics, Gostekhizdat, 1949.
20. I. Walles, Zeits. f. Phys. 79, 370 (1932).
21. J. H. Van Vleck, Phys. Rev. 55, 924 (1939).
22. L. J. F. Broer, Physica 10, 801 (1943).
23. I. E. Tamm, Foundations of the Theory of Electricity, Gostekhizdat, 1948.
24. E. R. Andrew and R. C. Eades, Proc. Roy. Soc. 216, 398 (1953).
25. E. Condon and G. Shortley, Theory of Atomic Spectra, IL, 1949.
26. L. Landau and E. Lifshitz, Statistical Physics, Gostekhizdat, 1951.
27. A. A. Kharkevich, Spectra and Analysis, Gostekhizdat, 1952.
28. P. Debye, Polar Molecules, New York, 1945.
29. H. C. Torrey, Phys. Rev. 92, 962 (1953).
30. G. E. Pake, Journ. Chem. Phys. 16, 327 (1948).
31. N. F. Ramsey, Phys. Rev. 78, 699 (1950).
32. H. S. Gutowsky, C. J. Hoffman and McClure, Phys. Rev. 81, 305 (1951).
33. H. S. Gutowsky and C. J. Hoffman, Phys. Rev. 80, 110 (1950).
34. H. S. Gutowsky and R. E. McClure, Phys. Rev. 81, 276 (1951).
35. H. A. Thomas, Phys. Rev. 80, 901 (1950); 81, 643 (1951).
36. W. D. Knight, Phys. Rev. 76, 1259 (1949).
37. C. H. Townes, C. Herring and W. D. Knight, Phys. Rev. 77, 852 (1950).
38. W. Kohn and N. Bloembergen, Phys. Rev. 80, 913 (1950).
39. J. Korringa, Physica 16, 601 (1951).

1. Overhauser, Phys. Rev. 92, 411 (1953).
2. Carver and Slichter, Phys. Rev. 92, 212 (1953).
3. F. Bloch, W. W. Hansen and M. Packard, Phys. Rev. 70, 474 (1946).
4. W. G. Proctor, Phys. Rev. 79, 35 (1950).
5. S. D. Gvozdover and N. M. Nevskaya, JETP 25, 435 (1953).
6. K. V. Vladimirskii, DAN 58, 1625 (1947).
7. R. V. Rollin, Nature 158, 669 (1946).
8. R. Gabillard, Comptes Rendus 232, 324 (1951).
9. A. Roberts, Rev. Sci. Instr. 18, 845 (1947).
10. H. C. Torrey, Phys. Rev. 76, 1059 (1949);
11. E. L. Hahn, Phys. Rev. 80, 580 (1950).
12. F. Bloch, E. C. Graves, M. Packard and Spence, Phys. Rev. 71, 551 (1947).
13. F. Bloch, E. C. Levintal and M. Packard, Phys. Rev. 72, 1125 (1947).
14. F. Bitter, N. L. Alpert, D. E. Nagle and H. L. Poss, Phys. Rev. 72, 1271 (1947).
15. G. D. Watkins and R. V. Pound, Phys. Rev. 82, 343 (1951).
16. R. V. Pound and W. D. Knight, Rev. Sci. Instr. 21, 219 (1950).
17. Siegbahn and G. Lindstrom, Ark. f. Fysik 1, 193 (1949).
18. I. R. Zimmerman and D. Williams, Phys. Rev. 74, 1885 (1948).
19. W. H. Chambers and D. Williams, Phys. Rev. 76, 638 (1949).
20. R. E. Sheriff and D. Williams, Phys. Rev. 82, 651 (1951).
21. J. E. Mack, Rev. Mod. Phys. 22, 64 (1950).
22. N. I. Poulins, Physica 17, 392 (1951).
23. E. R. Andrew and R. C. Eades, Proc. Roy. Soc. 218, 541 (1953).
24. A. M. Sach, E. H. Turner and E. M. Purcell, Phys. Rev. 76, 466 (1949).
25. J. T. Arnold, S. S. Dharmatti and Packard, Journ. Chem. Phys. 19, 507 (1951).
26. H. S. Gutowsky and D. W. McCall, Phys. Rev. 82, 748 (1950).
27. E. L. Hahn and D. E. Maxwell, Phys. Rev. 88, 1070 (1952).
28. K. Tomita, Phys. Rev. 89, 429 (1953).
29. N. J. Hopkins, Rev. Sci. Instr. 20, 401 (1949).
30. Rev. Sci. Instr. 21, 942 (1950).
31. M. Packard, Rev. Sci. Instr. 19, 435 (1948).
32. H. A. Thomas, R. L. Driscoll and I. A. Hipple, Journ. Nat. Bur. Stand. 44, 569 (1950).
33. H. W. Knobel and E. L. Hahn, Rev. Sci. Instr. No. 12 (1951).
34. K. Halbach, Helv. Phys. Acta 27, No. 3, 259 (1954).
35. H. J. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954).
36. H. E. Weaver, Phys. Rev. 89, 923 (1953).
37. E. Brun, J. Oeser, H. H. Staub and C. G. Telechow, Phys. Rev. 93, 172 (1954).
38. B. M. Kozyrev and A. I. Rivkind, JETP 27, issue 1 (7), 69 (1954).
39. Electronics, 134, June (1954).
40. F. Bloch, Phys. Rev. 94, 496 (1954).
41. W. A. Anderson and J. T. Arnold, Phys. Rev. 94, 497 (1954).