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PHYSICAL CHEMISTRY
R. Kh. TIMEROV
THEORY OF ELECTRON PARAMAGNETIC RESONANCE IN SOLUTIONS
(Presented by Academician B. A. Arbuzov on 16 IX 1961)
In paramagnetic solutions the EPR lines are narrowed as a result of the combined averaging action of the thermal motion of the particles of the solution and the exchange interaction between paramagnetic particles. According to (1), the Hamiltonian \(\hat{\mathscr H}\) of a system of \(N\) identical paramagnetic particles in solution can be represented in the form
\[ \hat{\mathscr H}=\hat{\mathscr H}_1+\hat{\mathscr H}_2+\hat{\mathscr H}'=\hat{\mathscr H}_0+\hat{\mathscr H}'; \tag{1} \]
\[ \hat{\mathscr H}_1=\hat{\mathscr H}_z^u,\qquad \hat{\mathscr H}_2=\hat{\mathscr H}_k+\hat{\mathscr H}_e,\qquad \hat{\mathscr H}'=\hat{\mathscr H}_{SS}+\hat{\mathscr H}_T+\hat{\mathscr H}_{\mathrm{st}}+\hat{\mathscr H}_z^a, \tag{2} \]
i.e., the unperturbed Hamiltonian \(\hat{\mathscr H}_0\) includes the isotropic part of the Zeeman interaction \(\hat{\mathscr H}_z^u=g\beta\left(H_0\sum_k \hat S^k\right)\) and the modulating term \(\hat{\mathscr H}_2\), containing the kinetic energy \(\hat{\mathscr H}_k\) and the exchange interaction
\(\hat{\mathscr H}_e=\frac{1}{2}\hbar\sum_{k\ne l}J(r_{kl})(\hat S^k\hat S^l)\); the perturbation \(\hat{\mathscr H}'\), responsible for the broadening of the EPR line, includes the following interactions: magnetic dipole–dipole
\[ \hat{\mathscr H}_{SS} = g^2\beta^2\sum_{k>l} r_{kl}^{-3} \left[ (\hat S^k\cdot \hat S^l) - 3r_{kl}^{-2}(\hat S^k\cdot r_{kl})(\hat S^l\cdot r_{kl}) \right], \]
Stark
\[ \hat{\mathscr H}_T=\hbar\sum_k \hat S^k D_k\hat S^k, \]
hyperfine
\[ \hat{\mathscr H}_{\mathrm{st}}=\hbar\sum_k \hat I^k A_k\hat S^k \]
and anisotropic Zeeman
\[ \hat{\mathscr H}_z^a=\beta\sum_k H^0 G_k\hat S^k-\beta g\left(H_0\cdot \sum_k \hat S^k\right). \]
For simplicity, throughout what follows the symmetry of the local electric field at the site of the magnetic particle is assumed to be axial (2). The condition of strong exchange \(\langle|\hat{\mathscr H}_e|^2\rangle \gg \langle|\hat{\mathscr H}'|^2\rangle\) is assumed to be satisfied, so that exchange narrowing occurs (1).
According to Kubo and Tomita (1), the longitudinal relaxation time \(T_1\) and the transverse relaxation time \(T_2\) can be expressed in the form
\[ T_1^{-1} = \frac{1}{2} \sum_{\gamma\ne0}\sigma_{z\gamma}^{2} \int_{-\infty}^{\infty} \exp(-i\gamma\omega_0\tau)\,f_{z\gamma}(\tau)\,d\tau; \tag{3} \]
\[ T_2^{-1} = \frac{1}{2} \sum_{\gamma}\sigma_{+\gamma}^{2} \int_{-\infty}^{\infty} \exp(-i\gamma\omega_0\tau)\,f_{+\gamma}(\tau)\,d\tau, \tag{4} \]
where
\[ \sigma_{\lambda\gamma}^{2} f_{\lambda\gamma}(\tau) = \frac{ \langle[\hat S_\lambda,\hat{\mathscr H}'_\gamma(\tau)] [\hat{\mathscr H}'_{-\gamma}(0),\hat S_{-\lambda}]\rangle }{ \hbar^{2}\langle|\hat S_\lambda|^{2}\rangle } = \frac{ \langle|[\hat S_\lambda,\hat{\mathscr H}'_\gamma(0)]|^{2}\rangle }{ \hbar^{2}\langle|\hat S_\lambda|^{2}\rangle } f_{\lambda\gamma}(\tau); \tag{5} \]
\[ \hat{\mathscr H}'_\gamma(\tau) = \exp(it\hat{\mathscr H}_2/\hbar)\, \hat{\mathscr H}'_\gamma\, \exp(-it\hat{\mathscr H}_2/\hbar), \tag{6} \]
where \(\hat S_z=\sum_k \hat S_z^k,\ \hat S_\pm=\sum_k(\hat S_x^k \pm i\hat S_y^k)\). \(f_{\lambda\gamma}(\tau)\) has the meaning of a correlation function and describes the rate of change in time of the perturbation under the action of the modulating term \(\hat{\mathscr H}_2\). At low concentrations of magnetic particles (exchange may be neglected), \(\hat{\mathscr H}_2=\hat{\mathscr H}_k\), and one usually takes \(f_{\lambda\gamma}(\tau)=\exp(-|\tau|/\tau_c)\) \({}^{(3,1)}\). Conversely, in the absence of motion (solids, supercooled liquids) \(\hat{\mathscr H}_2=\hat{\mathscr H}_e\), and one may take \(f_{\lambda\gamma}(\tau)=\exp(-\omega_e^2\tau^2/2)\) \({}^{(1)}\). In the general case \(\hat{\mathscr H}_2=\hat{\mathscr H}_k+\hat{\mathscr H}_e\); direct calculation of \(f_{\lambda\gamma}(\tau)\) from (5) is difficult because, in a liquid, the exchange-interaction energy \(\hat{\mathscr H}_e\) is itself modulated by the thermal motion \(\hat{\mathscr H}_k\) with a certain characteristic time \(\tau_1\), so that, approximately, one may put \({}^{(4)}\)
\[ \left\langle \exp(i\tau \hat{\mathscr H}_k/\hbar)\hat{\mathscr H}_e \exp(-i\tau \hat{\mathscr H}_k/\hbar)\hat{\mathscr H}_e^* \right\rangle = \left\langle \hat{\mathscr H}(\tau)\hat{\mathscr H}^* \right\rangle = \left\langle |\hat{\mathscr H}_e|^2 \right\rangle \exp(-|\tau|/\tau_1). \tag{7} \]
One may expect that \(\tau_1 \lesssim \tau_c\), since the exchange interaction is very sensitive to changes in the mutual positions of the particles. Taking into account that in solutions the exchange energy is much smaller than the kinetic energy, i.e. \(\langle|\hat{\mathscr H}_e|^2\rangle \ll \langle|\hat{\mathscr H}_k|^2\rangle\), (6) can be expanded in a series in powers of \(\hat{\mathscr H}_e\). Retaining only the first three terms in this series and substituting them into (5), we obtain for \(f_{\lambda\gamma}(\tau)\) the approximate expression
\[ f_{\lambda\gamma}(\tau)=\exp(-|\tau|/\tau_c)\exp(-\omega_e^2 F(\tau)); \tag{8} \]
where
\[ F(\tau)=\int_0^\tau (\tau-\tau')\exp(-|\tau'|/\tau_1)\,d\tau'; \tag{9} \]
\[ \omega_e^2=\left\langle |[\hat{\mathscr H}_e,[\hat S_\lambda,\hat{\mathscr H}'_\gamma]]|^2 \right\rangle \big/ \hbar \left\langle |[\hat S_\lambda,\hat{\mathscr H}'_\gamma]|^2 \right\rangle . \tag{10} \]
It is easy to verify that
\[ F(\tau) \begin{cases} |\tau|\tau_1, & \text{for } \tau_1^{-1},\tau_c^{-1}\gg \sigma;\\ \tau^2/2, & \text{for } \tau_1^{-1},\tau_c^{-1}\ll \sigma. \end{cases} \tag{8^1} \tag{8^2} \]
Thus, for rapid thermal motion in the system, \(\tau_c^{-1}\gg\sigma\), one should use a correlation function of the form
\[ f_{\lambda\gamma}(\tau)=\exp[-|\tau|(\tau_c^{-1}+\tau_1\omega_e^2)] =\exp(-|\tau|/K), \]
\[ K=\tau_c/(1+\tau_1\tau_c\omega_e^2), \tag{11} \]
whose characteristic time \(K\) takes into account the simultaneous action of exchange and motion. For slow thermal motion, \(\tau_c^{-1}\ll\sigma\) (very viscous liquids), one must use the function
\[ f_{\lambda\gamma}(\tau)=\exp(-\tau^2\omega_e^2/2). \tag{12} \]
It follows from (12) that in viscous liquids only exchange narrowing is effective (the quantity \(\omega_e\) coincides with the definition of the exchange frequency \({}^{(1)}\)). We note that the study of the temperature dependence of \(T_1\) and \(T_2\) makes it possible to choose between (11) and (12). In particular, a weak or even complete independence of \(T_1\) and \(T_2\) from temperature in solutions often indicates the predominant role of exchange narrowing (provided, of course, that the concentration of magnetic particles is sufficient for its occurrence), the theory and applications of which have been developed in \({}^{(1,4,5)}\).
It can be shown that the contributions to \(T_1\) and \(T_2\) from the various interactions assigned by us to the perturbation \(\hat{\mathscr H}'\) are additive, and they can be calculated by-
different. In calculating the contribution to \(T_1\) and \(T_2\) from \(\hat{\mathcal H}_z^a\), the correlation function (11) simplifies to \(f_{\lambda\gamma}(\tau)=\exp(-|\tau|/\tau_c)\), since exchange does not affect the Zeeman energy, \([\hat{\mathcal H}_e,\hat{\mathcal H}_z^a]\). On the other hand, for the isotropic part of the hyperfine interaction \(\hat{\mathcal H}_{\mathrm{ct}}^u=\frac{1}{3}(A+2B)\sum_k(\hat I^k\cdot \hat S^k)\), which does not interact with thermal motion, \([\hat{\mathcal H}_k,\hat{\mathcal H}_{\mathrm{ct}}^u]=0\), the correlation function (11) becomes \(f_{\lambda\gamma}(\tau)=\exp(-|\tau|\tau_1\omega_e^2)\), which agrees with the result of Ref. \((^4)\). Taking these simplifications into account for \(\hat{\mathcal H}_z^a\) and \(\hat{\mathcal H}_{\mathrm{ct}}^u\), and using the correlation function (11) for the remaining terms in the perturbation \(\hat{\mathcal H}'\) (2), after straightforward but rather lengthy calculations we obtain
\[ \begin{aligned} T_1^{-1}&=\sigma^2K\left(\frac{2}{5}\frac{1}{1+\omega_0^2K^2}+\frac{8}{5}\frac{1}{1+4\omega_0^2K^2}\right)\\ T_2^{-1}&=\sigma^2K\left(\frac{3}{5}+\frac{1}{1+\omega_0^2K^2}+\frac{2}{5}\frac{1}{1+4\omega_0^2K^2}\right) \end{aligned} \left\{ \begin{array}{l} \text{for } \hat{\mathcal H}_{SS}\text{ and }\hat{\mathcal H}_{T}; \end{array} \right. \tag{12^1} \]
\[ \tag{12^2} \]
\[ \begin{aligned} T_1^{-1}&=\sigma^2K\,\frac{2}{1+\omega_0^2K^2}\\ T_2^{-1}&=\sigma^2K\left(1+\frac{1}{1+\omega_0^2K^2}\right) \end{aligned} \left\{ \begin{array}{l} \text{for } \hat{\mathcal H}_{\mathrm{ct}}^u \text{ and } \hat{\mathcal H}_{\mathrm{ct}}^a=\\ \qquad =\hat{\mathcal H}_{\mathrm{ct}}-\hat{\mathcal H}_{\mathrm{ct}}^u; \end{array} \right. \tag{13^1} \]
\[ \tag{13^2} \]
\[ \begin{aligned} T_1^{-1}&=\sigma^2K\,\frac{2}{1+\omega_0^2K^2}\\ T_2^{-1}&=\sigma^2K\left(\frac{4}{3}+\frac{1}{1+\omega_0^2K^2}\right) \end{aligned} \left\{ \begin{array}{l} \text{for } \hat{\mathcal H}_{z}^{a}, \end{array} \right. \tag{14^1} \]
\[ \tag{14^2} \]
where
\[ \text{for } \hat{\mathcal H}_{SS}\qquad K=\tau_c/(1+\tau_c\tau_1\omega_e^2),\qquad \sigma^2=g^4\beta^4\hbar^{-2}S(S+1)\sum_l\langle r_{kl}^{-6}\rangle; \tag{15} \]
\[
\text{for } \hat{\mathcal H}_{T}\qquad
K=\tau_c/(1+\tau_c\tau_1\omega_e^2),\qquad
\omega_e^2=2J^2S(S+1),
\]
\[
\sigma^2=\frac{4}{15}D^2\left[S(S+1)-\frac{3}{4}\right];
\tag{16}
\]
\[
\text{for } \hat{\mathcal H}_{\mathrm{ct}}^u\qquad
K=1/\tau_1\omega_e^2,\qquad
\omega_e^2=\frac{2}{3}J^2S(S+1),
\]
\[
\sigma^2=\frac{1}{3}\left[\frac{1}{3}(A+2B)\right]^2 I(I+1);
\tag{17}
\]
\[
\text{for } \hat{\mathcal H}_{\mathrm{ct}}^a\qquad
K=\tau_c/(1+\tau_c\tau_1\omega_e^2),\qquad
\omega_e^2=\frac{2}{3}J^2S(S+1),
\]
\[
\sigma^2=\frac{1}{6}\left[\frac{2}{3}(A-B)\right]^2 I(I+1);
\tag{18}
\]
\[ \text{for } \hat{\mathcal H}_{z}^{a}\qquad K=\tau_c,\qquad \sigma^2=\frac{1}{15}\left[(g_{\parallel}-g_{\perp})\beta H_0\hbar^{-1}\right]^2. \tag{19} \]
Here \(\omega_0=g\beta H_0\hbar^{-1}\); \(g=\frac{1}{3}(g_{\parallel}+2g_{\perp})\); \(D\) and \(A,B\) are the constants of the fine and hyperfine interactions; \(g_{\parallel}\) and \(g_{\perp}\) are components of the \(G\)-tensor; \(J^2=N^{-1}\sum_{k\ne l}J_{kl}^2\) is the mean square of the exchange integral. The contribution to \(T_1\) and \(T_2\) from the anisotropy of the \(g\)-factor \((14^1)\), \((14^2)\), (19) has been studied in sufficient detail in \((^{4,6})\). Here we shall consider in greater detail expressions \((12^1)\)—\((13^2)\), from which it is evident that the dependence of \(T_1\) and \(T_2\) on the temperature and viscosity of the solution is completely determined by the characteristic time \(K\). In particular, for sufficiently strong exchange \(\tau_c\tau_1\omega_e^2\gg1\), one may put \(K=1/\tau_1\omega_e^2\). Therefore, for
low frequencies, when
\[ T_1^{-1}=T_2^{-1}=2\sigma^2 K \quad \text{for } \omega_0^2 K^2 \ll 1, \tag{20} \]
we obtain
\[ T_1^{-1}=T_2^{-1}=2\sigma^2/\tau_1\omega_e^2. \tag{21} \]
An increase in temperature (or a decrease in viscosity) under such conditions will lead to a broadening of the absorption line and a shortening of \(T_1\), since the effectiveness of exchange decreases because of the shortening of the time \(\tau_1\) during which the exchange interaction between two magnetic particles in the solution is realized.
In strong fields \(\omega_0^2 K^2 \gg 1\)
\[ T_1^{-1}\sim \sigma^2/\omega_0^2 K,\qquad T_2^{-1}\sim \sigma^2K, \tag{22} \]
whence it follows that: 1) \(T_1 \gg T_2\) and 2) the type of dependence of \(T_1\) on temperature changes to the opposite as compared with (21).
The dependence of \(T_1\) and \(T_2\) on concentration is contained in \(K\) (through \(\omega_e^2,\tau_c,\tau_1\)), and for the contribution from \(\hat{\mathcal H}_{SS}\) also in \(\sigma\) (15). For the very lowest concentrations of the paramagnetic substance, when the exchange term in \(K\) may be neglected \((K=\tau_c^*)\), expressions \((12^1)\) and \((12^2)\) for the contribution to \(T_1\) and \(T_2\) from \(\hat{\mathcal H}_{SS}\) and \(\hat{\mathcal H}_{T}\) remain valid; while for calculating the contributions from \(\hat{\mathcal H}_{z}^{a}\) and \(\hat{\mathcal H}_{\mathrm{st}}^{a}\) one must use the formulas of Ref. \((^4)\). The latter is connected with the fact that, in the absence of exchange, \(\hat{\mathcal H}_{\mathrm{st}}^{\omega}\) should be assigned to the unperturbed Hamiltonian \(\hat{\mathcal H}_0\). In experiment this circumstance appears in the form of hyperfine structure of the EPR spectra.
Let us consider aqueous solutions of \(\mathrm{Cr}^{3+}\) ions. According to \((^7)\), the width of the \(\mathrm{Cr}^{3+}\) absorption line decreases upon dilution (at room temperature) to \(\Delta H=200\) Oe and is independent of further dilution as well as of the frequency \(\omega_0\). This indicates that the contribution of dipole–dipole interactions and of the anisotropy of the \(g\)-factor may be neglected. Therefore, using (20) and (16) and taking \(S=3/2\), \(\tau_c \simeq 2\cdot 10^{-11}\) sec, we obtain \(D=0.12\ \mathrm{cm}^{-1}\).
It should be noted that detailed measurements of \(T_1\) and \(T_2\) in paramagnetic solutions can provide information on the structure of paramagnetic centers (fine- and hyperfine-structure constants), on the magnitude of exchange interactions \((\omega_e)\), and on the character of motions in the liquid \((\tau_c,\tau_1)\).
The author expresses his gratitude to S. A. Al’tshuler for guidance of the work and to B. M. Kozyrev for discussion of the results.
Physical-Technical Institute
of the Kazan Branch of the Academy of Sciences of the USSR
Received
14 IX 1961
CITED LITERATURE
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- D. Kivelson, J. Chem. Phys., 33, 1094 (1960).
- R. Kh. Timerov, ZhETF, 40, 1101 (1961).
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- B. M. Kozyrev, Doctoral dissertation, P. N. Lebedev Physical Institute, Academy of Sciences of the USSR, Moscow, 1957.