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Nuclear Induction1,2
Resonance phenomena in alternating magnetic fields of radio frequency, coinciding with the frequency of the Larmor precession of nuclear spins, were originally studied by the method of molecular beams. The molecular-beam method made it possible to measure the magnetic moments of the neutron and of a number of nuclei.3 At the present time, results have been published on the successful application of resonance methods to macroscopic samples of matter in the solid, liquid, and gaseous states.
The electromagnetic effects observed in resonance may be divided into three groups:
1) an increase in magnetic susceptibility,4
2) an increase in the absorption of electromagnetic energy,5
3) the peculiar effect, indicated by Bloch, of elliptical polarization of the magnetic-induction vector, analogous to the Faraday rotation of the plane of polarization of a light wave.
The essence of the method and the results obtained are most fully set forth in the cited papers of Bloch and collaborators. The published results refer to hydrogen nuclei. A hydrogen-containing substance (water, paraffin, aqueous solutions of paramagnetic salts), in an amount of the order of one gram, is placed in a magnetic field with components
\[ H_x = 2H_1 \cos \omega t,\qquad H_y = 0,\qquad H_z = H_0. \]
Here \(H_0\) is a constant or slowly varying field, to which there corresponds a Larmor frequency \(\omega_0/2\pi\) lying in the range of radio frequencies convenient for observation (in Bloch’s principal experiments \(H_0 = 1826\) gauss, \(\omega_0/2\pi = 7.76 \times 10^6\) cycles/sec); \(H_1\) is the amplitude of the radio-frequency magnetic field (of the order of \(10\) gauss), with \(\omega\) the cyclic frequency of the section, close to \(\omega_0\).
The resulting macroscopic magnetic moment of the hydrogen nuclei in the field \(H_0\) at room temperature can be estimated from the usual formulas for paramagnetic susceptibility.
This estimate gives a magnetization of the order of \(10^{-6}\) gauss, i.e., a quantity inaccessible to static methods of observation. Also difficult to observe directly is the reaction of the sample on the circuit producing the radio-frequency field. An electromotive force is observed arising in a second coil, the axis of which is parallel to the \(OY\) axis. In the absence of elliptical polarization of the magnetic-induction vector, the electromotive force in this coil (at least according to the theoretical scheme) is zero, which makes it possible to observe comparatively small e.m.f.’s arising as a result of the forced precession of the nuclear spins in the field of frequency \(\omega\).
Theoretical estimates, as well as discussion of the experiment, show that the relaxation times characterizing the establishment of equilibrium values of the resultant magnetic moment of the nuclei, unlike the usual phenomena of paramagnetism, may be of the order of seconds and even hours, so that, when considering processes occurring in rapidly varying fields, one may partially or completely neglect the interaction of the magnetic moments
of the nuclei with the remaining degrees of freedom characterizing the given substance. The case of long relaxation times, when the interaction of the magnetic moments of the nuclei with the external magnetic field must be taken into account, has been considered theoretically. For the magnetic moment per unit volume \(M\) one obtains the equation
\[ dM/dt=\gamma[MH], \tag{1} \]
where \(\gamma\) is the gyromagnetic ratio, i.e., the ratio of the magnetic moment of the nuclei to the mechanical moment. For frequencies \(\omega\) close to \(\omega_0\), the alternating magnetic field \(H_x=2H_1\cos \omega t\) may be replaced by its circular component corresponding to the direction of Larmor precession. After this replacement, the solution of equation (1) is obtained in elementary fashion in the form
\[ M_x=M\frac{\cos \omega t}{\sqrt{1+\delta^2}},\qquad M_y=\pm M\frac{\sin \omega t}{\sqrt{1+\delta^2}},\qquad M_z=M\frac{\delta}{\sqrt{1+\delta^2}}, \tag{2} \]
where \(\delta=(H_0-H^*)/H_1\), \(H^*=\omega/\gamma\) is the resonant value of the field for the specified frequency \(\omega\), and \(M\) is constant. The sign “minus” or “plus” in the expression for \(M_y\) corresponds to positive or negative values of the gyromagnetic ratio \(\gamma\).
The solution obtained remains valid not only for constant \(\omega\), \(H_0\), and, consequently, \(\delta\), but also for slowly varying \(\omega\) and \(H_0\). On passing through resonance, the vector of the resultant magnetic moment of the nuclei does not change in modulus, but its components \(M_x\) and \(M_z\) increase sharply. It follows from the form of the solution that the sharpness of resonance is characterized by the magnitude of the ratio \(H_1/H_0\). The observed e.m.f. is proportional to the component of magnetization \(M_y\); under the experimental conditions e.m.f.’s of the order of 1 millivolt were obtained.
The phenomena accompanying passage through resonance were observed directly in the experiment. The field \(H_0\) was modulated by a small additional field of frequency 60 cycles/sec; the e.m.f. proportional to \(M_y\) was observed with the aid of a cathode-ray oscillograph, to the second pair of plates of which a voltage of the same frequency, 60 cycles/sec, was applied. The screen of the oscillograph directly displayed the resonance curve corresponding to the above expression for \(M_y\). The most interesting of these experiments is the phenomenon of “memory” of the sample. The sign and magnitude of the effect depend on the law of variation of the field with time before passage through resonance. In formulas (2) this corresponds to a change in the value of the constant \(M\).
In addition to the resonance curve of the form (2), phenomena corresponding to relaxation times comparable with or smaller than the time of passage through resonance are also observed. These cases have also been considered theoretically. It proves possible to give a qualitatively correct picture of the phenomena by characterizing the substance by two relaxation times, \(T_1\) and \(T_2\), respectively for the change of the components of magnetization parallel and perpendicular to the field \(H_0\). Instead of equations (1), one obtains the system
\[ \begin{aligned} \dot M_x-\gamma(M_yH_z-M_zH_y)+\frac{1}{T_2}M_x&=0,\\ \dot M_y-\gamma(M_zH_x-M_xH_z)+\frac{1}{T_2}M_y&=0,\\ \dot M_z-\gamma(M_xH_y-M_yH_x)+\frac{1}{T_1}M_z&=\frac{1}{T_1}M_0, \end{aligned} \tag{3} \]
where \(M_0\) is the equilibrium value of the magnetization in the field \(H^*\).
Approximate solutions of this system are given. Of greatest interest is the stationary solution, i.e., the solution for passage through resonance
over a time interval large in comparison with the relaxation times. For the observed component of the magnetization \(M_y\), in this case one obtains the expression
\[ M_y = \pm M_0 \frac{2\sin \omega t}{T_1/T_2 + \gamma^2}. \tag{4} \]
In contrast to the case considered above, the effect reaches a maximum at two points near resonance, and for the resonant value of the field it vanishes, changing sign. The observed phenomena no longer depend on the prehistory of the sample, i.e., on the value of the field in which it was before passing through resonance.
In intermediate cases of relaxation times comparable with the time during which the field changes near the electromagnet that creates the field \(H_0\), it proves possible to estimate the relaxation time from the time during which the nonequilibrium sign of the effect is replaced by the equilibrium one. The relaxation time depends substantially on the chemical composition of the sample. Thus, traces of oxygen dissolved in water reduce the relaxation time from 15 to 5 seconds. Substances significant in the chemical respect reduce the relaxation time to values of the order of \(10^{-4}\)–\(10^{-5}\) seconds. The influence of paramagnetic molecules on the relaxation time is explained by the “catalytic” action of their magnetic moments, which facilitate the exchange of energy between the magnetic moments of the nuclei and the other degrees of freedom of the substance.
K. Vladimirsky
REFERENCES
- F. Bloch, Nuclear Inductions, Phys. Rev. 70, 460 (1946).
- F. Bloch, W. W. Hansen, and M. Packard, Nuclear Induction Experiment, Phys. Rev. 70, 474 (1946).
- I. I. Rabi, Phys. Rev. 51, 652 (1937); L. W. Alvarez and F. Bloch, Phys. Rev. 57, 111 (1940); I. I. Rabi, S. Millman, P. Kusch and T. R. Zacharias, Phys. Rev. 53, 318 (1938); 55, 526 (1939).
- A. Roberts, Y. Beers and A. G. Hill, Phys. Rev. 70, 112 (1946).
- F. M. Purcell, H. C. Torrey and R. V. Pound, Phys. Rev. 69, 37 (1946).