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(23) These effects are consistent with the observations of critical orientatlonal fluctuatlons observed1lbfor PD-Tempone in MBBA. (24) We have attempted to calculate values of TR and N following the prescriptiin of Luckhurstand Yeate%= We obtained from our spectra
values of T R 5 lo-' s for all temperatures and the mean value of Nywas 0.4 for Brownian diffusion. For strong jump diffusion, we could not get any meaningful solutions for T R or~ N. (25) G. R. Luckhurst and A. Sanson, Mol. Phys., 24, 1297 (1972).
An Electron Spin-Lattice Relaxation Mechanism Involving Tunneling Modes for Trapped Radicals in Glassy Matrices. Theoretical Development and Application to Trapped Electrons in y-Irradiated Ethanol Glasses Michael K. Bowman" and Larry Kevan Department of Chemistty, Wayne State UniversiW, Detrolt, Michigan 48202 (Recelved October 15, 1976)
A new electron spin-lattice relaxation mechanism for molecular radicals in glassy matrices is developed theoretically and tested experimentally. The mechanism depends on modulation of the electron nuclear dipolar interaction between a trapped radical and nearby magnetic nuclei by the motion of tunneling nuclei or groups of nuclei in the disordered glass. In glassy systems it appears that modulation by tunneling modes is much more effective than modulation by lattice phonons for electron spin-lattice relaxation in low and intermediate temperature ranges, typically to 100 K. The quantitative mechanism predicts: (a) that the spin-lattice relaxation rate T1-l is linearly proportional to temperature, (b) that Tr' is dependent on glass preparation to the extent that this affects the number and distribution of tunneling groups, (c) that Ti-l is sensitive to the isotopic composition of the glass, (d) that T1-l for a given radical is larger by several orders of magnitude in a glassy environment than in a crystalline one, and (e) that Tl-I a u - where ~ w is the EPR frequency. Predictions (a) to (d) have been tested and supported by Tl measurements vs. temperature on trapped electrons in C2H60H, C&OH, and C2HSOD glasses. The measurements were made with a pulsed EPR spectrometer by the saturation recovery technique.
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I. Introduction The electron spin-lattice relaxation of transition metal ions and rare earth metal ions in ionic crystals has been extensively studied in the past 45 years since Waller's original paper on spin-lattice relaxation in 1932.l As a result, nearly all the spin-lattice relaxation mechanisms that have been investigated both experimentally and theoretically are constrained by the conditions found in these systems. That is, strong crystalline fields and spin-orbit coupling are assumed for the ion in the crystal and the dynamics of the lattice is described in terms of the Debye model with only few modifications for the effects of phonon lifetime and crystal defects.2 The study of the spin-lattice relaxation of molecular radicals in molecular crystals and glasses is still in its infancy. The first report of the temperature dependence of the spin-lattice relaxation of an organic radical was made in 1957.3 The spin-lattice relaxation time (TI) was measured at four temperatures between 1.2 and 300 K and no attempt was made to explain either the magnitude of Tl, which was many orders of magnitude longer than that of transition metal ions, or the spin-lattice relaxation mechanisms involved. Nine years later, three surveys of different classes of radicals (organic, organosulfur, and peroxy) were reported>+ Over the limited range of temperature in these studies, Tlwas proportional to T n where n varied from 1 to 3. More importantly, for determining the relaxation mechanism, the spin-lattice relaxation rates in each study were found to be highly correlated with the square of the g factor deviation from
* Present Address: Argonne National Laboratory, Chemistry Division, Argonne, Ill. 60439 The Journal of Physical Chemistry, VoL 8 1, No. 5, 1977
the free electron value of 2.0023 and hence with spin-orbit coupling. This suggested that the Kronig-VanVleck spin-lattice relaxation mechanism was important. Other studies have supported thk7p8 The internal motions and hindered rotations of a molecular radical have also been suggested as an important relaxation rnechanism.+l2 At this same time, careful studies of the spin-lattice relaxation mechanisms of F centers (trapped electrons) in alkali halide crystals have shown that the electron nuclear dipolar (END) and isotropic hyperfine interactions between the unpaired electron and ita surrounding magnetic nuclei can make important contributions to the electron relaxation.13J4 An extensive study of the spin-lattice relaxation of a number of hydro-, deuterio-, and fluorocarbon radicals produced by ionizing radiation in single crystals of the parent compound has been made by Dalton, Kwiram, and Cowen.l5~l6They report that the Kronig-VanVleck relaxation mechanism is responsible for the spin-lattice relaxation of all but the fluorinated radicals. The spinlattice relaxation of the fluorinated radicals is dominated by modulation of the END interaction. In addition, they report observing an Orbach-like process in some of the samples in which the intermediate state is suggested to be an excited vibrational state of the radical instead of an excited electronic d a t e as in the usual Orbach process. Although these studies have demonstrated several important relaxation mechanisms for molecular radicals in single crystals, they by no means include all the dominant mechanisms in molecular glasses. DPPH in polystyrene has been the subject of two spin-lattice relaxation studies.17J8Between 1and 300 K, the relaxation rate is proportional to the temperature. This is a surprising
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Electron Spin-Lattice Relaxation in Glassy Matrices
result. Whatever the relaxation mechanism, if the low temperature relaxation represents a direct process (absorption or emission of lattice phonons resonant with the EPR transition), then a much stronger temperature dependence is expected above at least 50 K due to a Raman, Orbach, or other higher order process involving multiple phonon scattering. What is almost more surprising is that the T1of the triphenylmethyl radical3 in a single crystal host is about four orders of magnitude longer at 1.2 K than the T I of DPPHI7 in glassy polystyrene at the same temperature. Although it has been noticed that nuclear spin-lattice relaxation is usually faster in glassy samples than in crystalline samples of the same compound,lS2' the paucity of careful relaxation studies of the same radical in both glassy and crystalline states has prevented such L t i comparisons in EPR. Reinecke and NgaP have proposed a model for nuclear relaxation in which nuclear spin energy Flgure 1. Symbolic diagram of the double potential well of the tunneling is removed by a "Raman" type process involving two particle. The distance /separates the two minima which differ in energy by 2 8 . The barrier between minima is Voand the energy difference tunneling modes and Rubinstein and ReisingZ2have between the two tunneling states is 26. considered the effect of tunneling modes on the scattering of low energy phonons in connection with nuclear spincoupled tunneling states of the multiple potential well. lattice relaxation. For our purposes, we will describe these tunneling modes We have recently investigated the electron spin-lattice in the same manner as Phillips43 did. The tunneling relaxation of trapped electrons and trapped hydrogen particle is in a potential field which can be represented by atoms in a number of molecular glasses.23 It is found that the one-dimensional potential function in Figure 1. It is the relaxation rate at low temperature is proportional to formed from two harmonic oscillator potentials with the temperature and is several orders of magnitude faster than reported for those radicals in single ~ r y s t a l s . ~ ~ ? minima ~ ~ , ~ ~displaced by distance 1 and energy 2A. Between the two minima is an energy barrier of height Vo.The two Here, we wish to present a new relaxation mechanism, lowest energy levels (the only ones populated at the unique in some respect to the glassy state, which may be temperatures considered here) have energies of ft = (&02 responsible for the low temperature spin-lattice relaxation A2)ll2where A,, = hs2(a/?r)1/2e-u and s2 is the frequency of many molecular radicals in molecular glasses. of oscillation in one of the isolated harmonic potentials 11. Theory comprising the total potential, u = (rnVoh-z)l/zl, and m is Electron spin-lattice relaxation occurs due to a timethe mass of the tunneling particle. dependent perturbation of the electron spin by its enviP h i l l i p ~ ~has " ~ both ~ calculated the temperature deronment. In a calculation of the spin-lattice relaxation pendence of the relaxation time of tunneling modes and rate as a function of temperature in solids, it is necessary measured this same quantity in low temperature dielectric to have some model for the dynamics of the solid lattice. relaxation experiments on polyethylene. The relaxation The lattice dynamics is almost always described using the time of tunneling modes T is given by 1 / =~w12 + wZl Debye of a monoatomic crystal26although some where wI2and w21are transition rates between the two consideration has been given to the effects of lattice anenergy levels in each direction. This is closely related to defect^,^^-^^ and optical phoh a r m o n i ~ i t y ,crystalline ~~ the characteristic correlation time T~ for the tunneling n o n ~ . ~ ~ , ~ ~ particle given by 1/rC= nlwI2+ n2w21 where nl and n2 are Lattice Dynamics. The assumptions of the Debye model the populaticms of the two levels. If the tunneling modes seem quite reasonable for most crystals and this model has are assumed to be in thermal equilibrium at a temperature been quite successful in predicting not only the temperT then ature dependence of the spin-lattice relaxation rate of a 1 b2Ao2E number of paramagnetic metal ions in ionic crystals, but -1-aCSCll also many of the physical properties of crystals at low T~ 27 cosh2 ( e / k T ) u5p temperatures. The record of the Debye model has not ="DEcsch been as good in low temperature glasses, however. For example, although it predicts that the heat capacity at low temperatures should be proportional to the cube of the where b is the strength of coupling between the tunneling temperature for both glasses and crystals, the heat capacity mode and the lattice phonons, u is the velocity of sound of glasses is linearly proportional to the temperature and in the solid, and p is the density of the solid. is many times greater than that of a crystal of the same Tunneling Mode-Electron Spin Interaction. The substance.40 Several other physical properties of glasses motion of the tunneling particle can affect the electron spin at low temperature are equally anomalous, suggesting the of a radical in several ways. Since we are concerned here presence of add-itional low energy modes in glasses that with radicals having little spin-orbit coupling, the main are lacking in crystal^.^&^^ interaction between the electron spin and the motion of Tunneling Modes. A recent model for the low temthe tunneling particle is the END interaction. perature lattice dynamics of a glass is that of Phillips43and By expanding the dipolar interaction around the coof Anderson, Halperin, and Varma." This model envisions ordinate j along which tunneling occurs we have that, in a glass, there are a number of atoms or molecules for which the local potential has more than one minimum. If the potential for these atoms or molecules increases rapidly enough away from the minima, the only thermally populated states below room temperature will be the where ( ) denote the time average. Since j takes on only
+
(g)
(E)
The Journal of Physical Chemlstv, Vol. 81, No. 5, 1977
458
M. K.
one of two values, the spin-lattice relaxation rate between spin states lal) and laz) is given by
Bowman and L. Kevan
interesting feature to note about eq 5 is that for kT
> 2e'
1 kT - = (CD) Tl 2E
which is a different high temperature limit from the usual T1-' 0: F limit for most relaxation mechanisins. 1 027,2 Inverse E Distribution. For a distribution given by for e < A, where w is the EPR frequency. Measurements by P h i l l i ~ s ~ n (e) a e fore 2 A, on tunneling modes in polyethylene indicates that the characteristic correlation time T~ is much greater than the (we realize that there must also be a limit at large e since inverse of the EPR frequency w at x-band so that eq 2 can Jo"n(c)de must converge) eq 4 becomes be written as =w2c
+
7-C
{ 9,
This has the same temperature dependence as does spin-lattice relaxation via a symmetric double potential tunneling mode31 and via the tunneling rotation of a methyl group.'O Spin-Lattice Relaxation Rate. The calculation of the spin--latticerelaxation rate for molecular radicals in a glass can now be attempted. Since, for most molecular radicals in glasses, Tz
