PHYSICS

N. V. KRAVTSOV, V. N. LAZUKIN, V. A. SHANDYTSEV

“MULTIQUANTUM TRANSITIONS” IN ELECTRON PARAMAGNETIC RESONANCE

(Presented by Academician L. A. Artsimovich, 29 I 1963)

Electron paramagnetic resonance in its usual form is observed in homogeneous magnetic fields: a constant field \(\mathbf H_0\), which splits the energy levels of the system under study into Zeeman sublevels, and a microwave field \(\mathbf H_1 \cos \omega_1 t\), perpendicular to \(\mathbf H_0\), which induces transitions between the latter. In the case where the shell spins of the paramagnetic ion of the sample under study are \(S = 1/2\), two Zeeman sublevels are produced, differing in the directions of the projections of the magnetic moment on \(\mathbf H_0\). Absorption of microwave power will occur at one frequency when the condition is fulfilled

\[ \hbar \omega_1 = g \beta H_{01}, \tag{1} \]

where \(H_{01}\) is the resonance value of the field \(H_0\) for the frequency \(\omega_1\) with which the field \(H_1\) oscillates.

Fig. 1. Scheme of two-quantum transitions

The picture of resonance absorption changes substantially if, in addition to the fields \(\mathbf H_0\) and \(\mathbf H_1 \cos \omega_1 t\), another radio-frequency field \(\mathbf H_2 \cos \omega_2 t\), parallel to \(\mathbf H_0\), acts on the spin system. With the modern technique of observing EPR, such a situation necessarily occurs: a weak modulating field \(H_2\) with a frequency from the industrial frequency to several kilohertz (video spectroscopes and heterodyne-type spectroscopes) or to several megahertz (spectroscopes with high-frequency modulation) is superposed on the constant field. The influence of the modulating field on the shape of the EPR spectrum has been discussed theoretically and noted experimentally by many authors, especially for the second case, where it is manifested particularly clearly.

A formal treatment of the influence of modulation on the form of the spectrum can be carried out on the basis of general considerations of radio physics \((^1)\), from which it follows that, for a small modulation index (i.e., when \(\Delta \omega / \omega_2 \ll 1\)), the spectrum consists of the carrier frequency \(\omega_1\) and two sidebands \(\omega_1 \pm \omega_2\). In the general case the spectrum represents a set of lines with frequencies \(\omega_1 \pm k \omega_2\) of rapidly decreasing intensity. As applied to EPR, an analogous result was obtained \((^2)\): in the presence of a modulating field \(H_2\), the absorption signal is a series of lines with spacings \(\omega_2\) between them and amplitudes proportional to \([J_k(\Delta \omega/\omega_2)]^2\), where \(J_k\) is a Bessel function of the first kind.

In recent years several works \((^{3-6})\) have been carried out in which certain features of EPR were observed experimentally with account taken of the influence of the field \(H_2 \cos \omega_2 t\); absorption and emission at the frequency \(\omega_2\) were discovered. These results were interpreted as the possibility of two-quantum transitions in the spin system. From the viewpoint developed here, the essence of the phenomenon consists in the simultaneous interaction of two photons of different energy and polarization with the system of electron spins, as a result of which absorption and emission arise simultaneously at the two frequencies \(\omega_1\) and \(\omega_2\) in the fields \(H_{02}\) and \(H_{03}\), satisfying the conditions:

\[ \hbar \omega_1 + \hbar \omega_2 = g \beta H_{02}, \qquad \hbar \omega_1 - \hbar \omega_2 = g \beta H_{03}. \tag{2} \]

In other words, in the field \(H_{02} > H_{01}\) both the quantum \(\hbar\omega_1\) and the quantum \(\hbar\omega_2\) are absorbed simultaneously (Fig. 1); in the field \(H_{03} > H_{01}\) absorption of the quantum \(\hbar\omega_1\) and emission of \(\hbar\omega_2\) occur simultaneously.

The solution \((^{4,6})\) of the phenomenological Bloch equations for the case under discussion of three fields gives the value of the susceptibility at the frequency \(\omega_2\): for \(H_0 > H_{01}\) the susceptibility of the paramagnet is positive, which corresponds to absorption of energy at the frequency \(\omega_2\), while for \(H_0 < H_{01}\) the susceptibility is negative, which corresponds to emission of energy. Despite the undoubted nature of the result set forth, the concept of two-quantum transitions does not seem to us fully substantiated, since the important question of the nature of the intermediate levels of the spin system, without the introduction of which the essence of the phenomenon cannot be understood, remains unexplained. Therefore, the corresponding terminology can for the time being be used only conditionally.

Fig. 2. Block diagram of the apparatus

At the same time, this concept may prove convenient for describing splittings of EPR lines as a result of the modulation effect. We undertook experiments to observe the indicated phenomena in a free radical (diphenylpicrylhydrazyl), using recording both at the frequency \(\omega_1\) and at the frequency \(\omega_2\). To observe EPR absorption at the frequency \(\omega_1\) at room temperature, an ordinary video spectroscope of the 3-centimeter range was used (Fig. 2), with a resonator operating at the wave \(H_{011}\) and with a natural frequency of 9400 MHz. To obtain the field at the frequency \(\omega_2\), a coil connected to a GSS-6, GS-23, or IMI-2 generator was installed inside the resonator. The IMI-2 autodyne circuit made it possible to observe absorption and emission signals at the frequency \(\omega_2\). The axis of the coil was parallel to the field \(H_0\).

Fig. 3. \(A\) — action only of the field \(H_1 \cos \omega_1 t\); \(B\) — action of two fields \(H_1 \cos \omega_1 t\) and \(H_2 \cos \omega_2 t\)

Figure 3 shows: the absorption signal at the frequency \(\omega_1\) and the absorption and emission signals at the frequency \(\omega_2\), which arise when the field \(H_2 \cos \omega_2 t\) is switched on. At the same time the shape of the signal at the frequency \(\omega_1\) changes, which is due to absorption and emission of energy at the frequency \(\omega_2\). As already mentioned, with an increase in the modulation index (i.e., with an increase in the amplitude of the field \(H_2\) and with \(\omega_2 = \text{const}\)) side satellites of the EPR line at the frequencies \(\omega_1 \pm k\omega_2\) should arise in increasing number. An attempt was made to observe this effect.