Heisenberg Spin Exchange Between Nitroxide Probes Diffusing in a Percolation Network

Heisenberg spin exchange (HSE) between nitroxide (Tempone) spin probes has been measured as a function of concentration in the aqueous phase of the hydrated ion exchange membrane Nafion 117. The observed fast-motional electron paramagnetic resonance spectra were analyzed in terms of the perturbation expressions given by Bales and Peric based on the earlier monograph of Molin et al. , as well as by the full stochastic Liouville equation lineshape calculation of Freed and coworkers. Differences between the methods for determining HSE from the spectrum are presented and discussed. In acidic Nafion membranes, the spin probes are deactivated over time, allowing simultaneous measurement of the decay kinetics and HSE as a function of paramagnetic probe concentration. Both processes deviate from the behavior that would be expected based on classical diffusion and chemical kinetics in isotropic media. The results are discussed in terms of currently available models for diffusion and reaction in a percolation network.


Introduction
Heisenberg spin exchange (HSE) between paramagnetic particles is one of the few elementary bimolecular processes in solutions that is readily accessible to detailed experimental studies and to strict theoretical description.It arises when a colliding pair of radicals exchange electronic spin states at a rate proportional to their collision rate [1][2][3][4].Because it depends on the mutual translational diffusion of spin probes, HSE allows one to study elementary molecular collisions in detail and has been widely used to address various problems of chemistry, physics, and molecular biology.One can obtain quantitative information on collision rates of molecules in solutions, including information about the structure of the colliding pair.Particularly valuable is the possibility of using spin exchange to study collisions in such complex systems as polymer solutions, multicomponent and heterogeneous mixtures, , and especially biological systems.
In isotropic media, the rate of HSE between identical radicals is related to the radical concentration by the equation [1]  ex () =  ⋅  ⋅ 4 (1) where  is the concentration and  is the collision radius of the radical,  is the relative translational diffusion coefficient,  is a steric factor, and  is the probability of effective exchange per collision.
Most detailed studies of the concentration dependence of HSE focus on the phenomenon in isotropic phases for which the dimensionality is well-defined, such as liquid solution.The Hubbell group has used the sensitivity of HSE to measure electrostatic interactions between charged nitroxide labels and probes to measure electrostatic potentials at biological interfaces.[5][6][7][8][9][10][11][12][13][14].HSE has also been extensively used by the Hubbell group and by others to determine the solvent accessibility of spin labels attached to proteins by measuring broadening or spin relaxation due to HSE with paramagnetic species in solution [15][16][17] or in membranes.[18] Although HSE between paramagnetic probes has been studied in several complex and heterogeneous systems in which Eq. ( 1) may not apply, we not aware of any systematic study of the detailed concentration dependence of HSE in such a system.One type of system that has been intensively studied is the percolation network, [19,20] which has applications in porous geological formations, aerogels, amorphous semiconductors, and biological membranes to name only a few.Specifically, the kinetics of elementary chemical reactions of a single species A, such as unimolecular decay or trapping A * → A, bimolecular coalescence A + A → A, or annihilation A + A → 0, in such systems has been extensively characterized theoretically and verified experimentally.[19][20][21][22] In contrast to classical rate expressions for these reactions in isotropic homogeneous media, in percolation networks concentrations may appear with fractional exponents that depend upon the "fracton", or "spectral" dimension of the network.
These considerations motivated us to identify a suitable system that might be expected to exhibit an effective dimensionality different from that of an isotropic solution and carry out a systematic study of spin probe concentration.A common and accessible example of such a network is a conductive ionic polymer such as Nafion, which has seen extensive use as conductive membranes in fuel cell applications.
As revealed by cryo-electron tomography [23], the hydrated Nafion membrane consists of a network of aqueous channels lined with hydrophilic sulfonic acid groups interspersed with hydrophobic fluorocarbon backbone regions.The channels in the hydrated membrane are reported to have an average diameter 2.5 nm with a center-to-center spacing of 5.1 nm.
This system offers the additional advantage of an internal consistency check of the results.Most spin probe studies of ion exchange membranes are performed after exchange of the acidic proton on the internal sulfonate groups of the polymer with metal cations.[24][25][26] Without this pretreatment, the nitroxides are deactivated in the acidic environment, presumably via reduction of the nitroxide to the 1hydroxy compound.Thus, by initially loading the membrane with a high concentration of radical, it is possible to measure HSE over a wide range of concentrations without disturbing the sample, while at the same time measuring the kinetics of the radical disappearance.
This work presents a preliminary such study on a hydrated Nafion 117 membrane.We find that the rate of Heisenberg spin exchange as a function of probe concentration deviates from Equation (1), and further that the decay kinetics of Tempone in the membrane does not follow standard first-order kinetics.
These results are discussed in the context of available percolation theories.

Methods and Materials
Sample Preparation and Measurement Spectra were acquired until the doubly integrated intensity of the spectrum was at most 1% of its initial value, over a period of at least 24 hours.Two experimental runs were evaluated with 1831 and 2182 field scans, respectively.

Measurement of HSE
Bales and Peric [27,28] have given an expression for the lineshape of a nitroxide radical in the fasttumbling limit and in the presence of HSE with frequency  ex .The expression is an elaboration of results summarized earlier in the monograph of Molin et al. [29] In terms of the standard Lorentzian lineshape parameters  and , where  determines the width of the line and  is the position of the center of the line, this expression may be rewritten as Equation (2): Molin et al. [29] identified three effects of HSE on the fast-motional spectrum of a nitroxide: (1) it produces a homogeneous width  ex in addition to the inhomogeneous and rotationally-induced linewidths; (2) it introduces a dispersion component of the   = ±1 lines of the spectrum, and (3) it causes a shift of the outer hyperfine lines towards the center of the spectrum.Bales and Peric tested these predictions experimentally and demonstrated that  ex can be quantitatively measured from all three of these effects when  ex ≲ 0.3  N,0 .They also identified a further effect of HSE, namely a change in the relative intensities of the   = ±1 and the   = 0 lines resulting from magnetization transfer by the HSE.[28] Bales and Peric cast their expression in terms of quantities that could be directly measured from the experimental EPR spectrum.The equivalence of their expression to Equation ( 2) is demonstrated in the Appendix, which also indicates how  ex is obtained from the parameters in Equation (2) for each of the methods identified above, as follows: In addition to the parameters appearing in Equation ( 2),   is the electron gyromagnetic ratio,  ,0 is the isotropic 14 N hyperfine interaction in the absence of HSE, and  N, is the effective or apparent isotropic 14 N hyperfine interaction in the presence of HSE.The parameter  in Equation ( 2) is the ratio of peak intensities of the   = ±1 to the   = 0 line.The theoretical approach of Molin et al. [29] considered individual lines in the fast motional spectra of radicals in an encounter pair and solved a reduced stochastic Liouville density matrix equation for two transitions in the encounter pair.In principle, a spectrum calculated by the full stochastic Liouville equation (SLE) for an individual radical but including HSE with surrounding radicals should exhibit the above features as well: i.e., line-broadening, a dispersion component, and shifting of the outer hyperfine lines.Specifically, the SLE program of Freed and coworkers [30,31] and the analogous calculation in the EasySpin package by Stoll [32,33] do include HSE, and produce fast-motional spectra in the presence of high exchange rates that exhibit all these features.This affords a fourth approach to determining  ex from an experimental spectrum, which is the direct fitting of the calculated lineshape using the SLE program.
The advantage of this approach is that it is not restricted to the fast-motional regime, although it also cannot reproduce the magnetization transfer between the   = 0 and   = ±1 hyperfine lines observed by Bales and Peric.[28] Least-squares fitting for a series of test simulations and for the full 2-D time series of experimental data was automated in Matlab [34] using a modified Nelder-Mead simplex search algorithm (fminsearch).
The Bales-Peric functions were programmed in a Matlab macro, and the SLE calculations were performed using the chili function of the EasySpin Matlab package.[32,33] For the fitting of the SLE lineshape calculations, all parameters for the HSE-free spectrum were fixed at the values noted above and only the  ex was varied.All fits also included a small correction for the instrumental microwave phase.

Results and Discussion
The different methods of measuring HSE noted above were first compared.Fig. 1(a) shows nitroxide spectra calculated for a series of  ex values using the SLE (chili) function in EasySpin.[33] The calculations used rotational and magnetic parameters obtained from a fit to the spectrum of Tempone in the Nafion 117 membrane at low concentration (i.e. with  ex ≈ 0), as follows: isotropic rotational correlation time  = 5.0 × 10 −9 s; principal g-values of 2.00794, 2.00594, and 2.00342, principal 14 N Fig. 1 A comparison of nitroxide spectra in the fast motion limit calculated for a range of HSE values by two methods (a) Least-squares fits of the Bales-Peric perturbation expressions (Equation 2) to a series of test spectra calculated using the stochastic Liouville equation for the exchange frequencies shown.The fits are essentially indistinguishable from the test spectra over the entire range.(b) SHE frequency  ex derived from the excess linewidth (triangles), the effective hyperfine splitting  N, (circles) and the ratio according to Equations 3. Solid line shows the  ex values used to calculate the test spectra.hyperfine values of 15.8, 15.8, and 103 MHz, and a peak-to-peak Gaussian inhomogeneous broadening of  = 0.55 mT.Equation (2) was fitted to the HSE-free spectrum to determine the linewidths    and isotropic hyperfine in the absence of HSE,  N,0 .Each of these test spectra was then fitted using Equation (2) and varying the parameters  ex (which was added width to    of each line), , the peak positions a b , and the intensity ratio .As Fig. 1a shows, the fits are essentially indistinguishable from the spectra calculated using the SLE.
Fig. 1b shows the  ex determined by each of the methods given in Equation ( 3) from the fits shown in Fig. 1a.As noted by Bales and Peric [28], the perturbation expressions are only expected to be accurate up to a value of  ex ≈ 0.3  N,0 , or about 15 MHz, even though the calculated lineshapes closely fit the SLE simulations over the full range tested.The  ex determined from , the magnitude of the outer line dispersion component, diverges from the "true"  ex quickly above this value (compare Fig. 1b to Fig. 8a of reference [28]).In contrast, the  ex determined from the effective hyperfine splitting  N, remains within about 5% of the "true" value well above   ≈ 0.3  N,0 , only diverging significantly at the high end of the range tested.This deviation most likely reflects a strong correlation between the  N, parameter and the  parameter as the peaks coalesce and become difficult to resolve.Finally, the  ex determined directly from the linewidth of the SLE simulation is consistently smaller than the "true"  ex by a factor of 1.59 ± 0.07.
Fig. 2 shows least-squares fits of the Bales-Peric expressions (Fig. 2a) and SLE lineshape calculation (Fig. 2b) to spectra at selected Tempone concentrations spanning the range of concentrations studied.Up to a nominal concentration of about 100 mM, both methods fit the experimental spectra very well.As was the case for the SLE-simulated spectra, the fits calculated using the Bales-Peric expressions closely reproduce the experimental spectra even above this concentration; small deviations are only apparent at the very highest concentrations.In contrast, the SLE lineshape does not accurately reproduce the relative intensities of the central vs. the outer lines of the experiment spectra above this concentration.
The  ex values obtained from the experimental data for each of the methods described are shown in  b a below this concentration.The behavior of the  ex determined from the Bales-Peric linewidth and dispersion () parameters is qualitatively similar to that shown in the fits to the SLE simulations in Fig. 2a.Specifically, the value determined from  N,0 diverges in the positive direction above  ex ≈ 15 MHz, while the  ex determined from the experimental linewidth is directly proportional to that obtained from the SLE fit, and is a factor of 1.64 ± 0.07 smaller.The most striking qualitative difference between Fig. 3 and Fig. 1b is the  ex determined from  N, , which closely tracks the  ex used in the SLE simulations (Fig. 1b) but is considerably larger than that obtained by the other methods from the experimental spectra.
The large deviation of the  N, method for the experimental result is most likely the result of reencounters between Tempone radicals which affect only this parameter and are not accounted for in the formulation of the SLE lineshape theory.The effect of re-encounters on the line shifts depends at least in part on the time between re-encounters and has been extensively investigated by the Bales group [35][36][37][38][39][40] and characterized theoretically by Salikhov et al. [41].It is reasonable to conclude that re-encounters between spin probes within the constricted space of the aqueous phase in the Nafion membrane account for the observed deviation.
The comparisons shown in Figs. 1 and 3 indicate that the most accurate and reliable determination of  ex from the experimental spectra over the full concentration range studied is from the fits of the SLE lineshape in the concentration range from 5-100 mM.These values of  ex are plotted vs. the concentration of the spin probe in Fig. 4a.The dashed line in Fig. 4a shows a linear fit to the values of  ex obtained at the lowest concentrations (longest times) in the experiment, making it immediately apparent that  ex does not obey the linear relationship Equation (1) over the concentration range studied.
The experimental design of monitoring the EPR spectrum as the spin probe decays in situ essentially eliminates the possibility that the observed deviation from Equation (1) arises from nonlinearity in the probe concentration.This design follows a strategy employed in the initial HSE study of Bales and Peric with the spin probe PADS, which is deactivated at higher temperatures.[27] Because successive spectra Fig. 4 (a) Plot of HSE frequency   vs. relative Tempone concentration (nominally 118 mM at   0 ⁄ = 1.Dotted line shows least-squares line fitted to data taken at the lowest concentrations (longest times) to emphasize the nonlinearity of the data over the entire range.Lines though the data points represent least-squares fits of a power law function and a double-exponential function to the data.(b) Deviations of the power law fit (lower plot) and double exponential fit (upper plot) from the experimental data with RMSD values of 0.30 and 0.16 MHz, respectively.are obtained under identical conditions, the intensity of the doubly integrated signal affords a very accurate measurement of the relative concentration of active spin probes; any nonideality in the partitioning of Tempone between the membrane and equilibration solution is immaterial.Finally, since the total number of probe molecules (both EPR-active and inactive) remains constant, any nonlinear effects that the probe molecule concentration may have on the local viscosity, as reported by Eastman et al. [42,43] is controlled for.
The kinetics of Tempone deactivation were investigated as an internal check of the fractal properties of the Nafion environment.Fig. 5a shows a plot of the doubly integrated intensity of the EPR signal of Tempone as a function of time.The dash-dot line in Fig. 5a shows the least-squares fit of the classical integrated rate expression for a first-order decay reaction, namely a single exponential function, to these data.The least-squares parameters from this fit are summarized in Table 1 together with the parameters of two other fitted functions that will be discussed shortly.The significant deviation of this function is evident both from Fig. 5a and the plot of the differences at the bottom of Fig. 5b, clearly indicating that the experimental decay is also not well-described by classical kinetics.
These observed deviations from classical expressions will be considered in the context of percolation theory after a brief review of the relevant aspects of that theory.The reader is referred to the monographs references [19,20] for greater detail.A percolation system is modeled as a network of nodes or links that may be open with probability .At some critical threshold   , which depends on the network geometry, enough sites are open to form an "infinite cluster" that provides an open path through the network.Two important characteristics of the network are  ∞ , the probability that a node belongs to the infinite cluster, and , the characteristic length of finite clusters in the network.These are governed respectively by the critical exponents  and  as follows: Note that  ∞ → 1 and  → ∞ as  →   .The fractal dimension of the network is then defined as where  is the Euclidean dimension in which the network is embedded.In Nafion, these properties are generally expressed in terms of water concentration [44] instead of occupation probability, but the principles are the same.The deactivation of the Tempone spin probe is an example of the "trapping" reaction that has been extensively characterized in percolation networks, and for which theoretical rate expressions are available [21,45,46].It is represented by the reaction A + B → B, where A is assumed to be diffusing within the network with concentration  A , and B is a stationary trap with concentration  B distributed evenly throughout the network.In the present case, A represents the ESR-active nitroxide, and B represents the stationary sulfonate groups in the polymer.The following expression, a "stretched exponential", has been derived for the time dependence of the concentration of  A at the percolation threshold  ~   : [21,45,46] In this expression   is the so-called "spectral" or "fracton" dimension of the percolation network.It measures the interplay between the fractal geometry of the conductive phase of the membrane and the dynamics of the diffusion, and it plays a role equivalent to the dimensionality of standard Euclidean systems.Explicitly, where   is the dimension of a random walk within the network.
We are aware of no theoretical treatment that fully addresses the physics of our experimental SHE observation, i.e. an expression for the rate of collisions between identical species A in a percolation network that includes stationary traps B that destroy A. The most analogous case for which a theoretical rate expression is available is the diffusion-limited bimolecular coalescence reaction A + A → A in a percolation network.This reaction has a time-dependent rate constant for which the expression near the percolation threshold is [47] where  0 is proportional to the collision frequency at  = 0.
If the rate of disappearance of spin probe A due to trapping is not too different from the A-A collision rate, Equation ( 8) might approximate the HSE case.By analogy, Equation (1) becomes where  0 is the HSE rate at zero time and includes all the additional factors appearing in Equation (1).
We now apply these results from percolation theory to the experimental data, starting with the Tempone decay curve for which there is an exactly applicable rate expression.The dashed line in Fig. 5a shows the least squares fit of the stretched exponential function Equation ( 6) to the Tempone decay curve.
The functional form (assuming constant   in Equation ( 5)) and least-squares parameters are given in Table 1.As is evident both from the dashed line in Fig. 5a and in the differences shown in the middle plot of Fig. 5b, this function also deviates significantly from the experimentally observed decay curve.Fits to two datasets also produced considerable uncertainty in the value of the fracton dimension   (Table 1); moreover, the average value obtained for   is physically unrealistic since   ≤ 2 for fractal systems.[47] The percolation theory result for the experimentally measured   () is somewhat more satisfactory.
Fig. 4a shows a fit of Equation ( 9) to the experimental  ex () curve.The fit overlaps with the experimental data closely enough that it is not clearly resolvable in the plot.The differences between data and fit are made clearer in the lower plot of Fig. 4b.The value obtained for the fracton dimension is consistent between the datasets studied, which give a physically reasonable value of 1.980.08.
Nevertheless, the systematic deviations of the fractal kinetics expressions from the observed decay of Tempone (middle plot of Fig. 5b) and somewhat smaller deviations from the observed   () (lower plot of Fig. 4b) motivate us to consider alternative models.In particular, the residuals from both the single exponential and stretched exponential fits (Fig. 5b) suggest the presence of a second decaying component.
It has been noted [48,49] that macroscopic systems can exhibit a "crossover time"  × at which the observed decay switches from a stretched exponential (fractal kinetics) to a simple exponential decay (classical kinetics).This behavior can be qualitatively understood in the following way.The self-diffusion coefficient of particles diffusing in a percolation network varies with time, [50] approaching a constant value when the distance scale of the diffusion is larger than the characteristic length of the finite clusters in the network, which is the length scale at which the network can be considered a homogeneous medium.
Thus, at long times, separate ensembles of diffusing particles may emerge [20], one consisting of particles in clusters of finite size, for which there is an upper limit on the particle's displacement, and the other diffusing freely in the infinite cluster over long distance scales and exhibiting classical kinetics.
To test this so-called "scaling" model, [49] a double exponential function was fitted to the experimental decay curve (solid line in Fig. 5a).The fit is quite good, and the plot of residuals (top, Fig. 5b) shows essentially no systematic variations.The least-squares parameters are given in The calculated fit of Equation ( 10) is shown in Fig. 4a, and the least-squares values of  1 and ω 2 are given in Table 2.Although the   (()) is calculated from Equation ( 10) is nearly indistinguishable from the fit of Equation ( 9) in Fig. 5a, differences are evident from the residuals that are compared in Fig. 5b.The RMSD for the fit of Equation ( 10) is also somewhat lower than that for Equation (9).I.The relative collision rates found for the two components are consistent with the scaling model: at early times (corresponding to high concentrations), radicals diffusing locally in finite clusters undergo fractal diffusion so that the collision rate per unit concentration as reflected by  1 is significantly lower than the  2 observed at longer times.However, there are two additional effects that are not accounted for by existing scaling models.First, a higher decay rate is expected for the ensemble that dominates at early times since radicals diffusing in finite clusters would be unable to escape traps.Second, the confined spaces of the clusters lead to a high rate of re-encounters in this ensemble, which is reflected in the higher apparent  ex that is derived from the shift of the outer 14 N hyperfine lines in the presence of SHE (cf.Fig. 3).It is not possible to determine the individual contributions of these effects to the observed early time dependence of  ex (()) from the available data.

Conclusion
This detailed study of SHE over a wide range of spin probe concentrations in in a well-characterized percolation network demonstrates the possibility of using this method to obtain quantitative information about the nanoscale properties in media with fractal properties.In principle, the ability to characterize the separate effects noted above will require a more rigorous interpretation of the concentration dependence of SHE in this type of percolation network will require development of a theory for the steady-state collision rate of probes in the presence of a trapping reaction with a different rate.Such a treatment should also include an analysis of re-encounter rates according to the methods proposed in the extensive studies of Salikhov, Bales, and coworkers.[40,[51][52][53][54] The influence of the different network parameters could then be studied by varying the degree of hydration, including values below the percolation threshold, and the concentration of traps (degree of sulfonation) in the polymer.The fracton dimension of the polymer could also be varied by using polymers with different constraints on their backbone conformations such as sulfonated polyaryl ketones.

Appendix
This Appendix demonstrates the equivalence of the fitting parameters in Equation (2) to those used by Bales and Peric [27,28], which represent empirical quantities that are directly measurable from the experimental spectra as illustrated in Fig. A1.
Their parameters include    , the integrated intensity of each 14 N hyperfine line;  pp (  ), the derivative peak-to-peak amplitude of each line;   ex (  ), the field position of each line; Δ pp , ex (  ), the derivative peak-to-peak width of each line; and  disp (  ), the amplitude of the exchange-induced admixture of the dispersion line shape for each line.Although these parameters were all varied independently in references [27] and [28], the results in that work confirmed the validity of some simplifying assumptions that allow the number of fitting parameters in Equation [2] to be reduced.
Specifically, it was shown that  disp is the same for the   = ±1 lines and zero for the   = 0 line, and that the ratio of intensities  ±1  0 ⁄ depends on   because of exchange-induced magnetization transfer between the central and outer lines, and that this ratio is the same for the   = ±1 lines.

Fig. A1
Depiction of quantities related to the determination of  ex from experimental fast-motional spectra of a 14 N nitroxide using the perturbation equations of Bales and Peric.
The expression used by Bales and Peric for fitting their first derivative experimental spectra is Equation ( 17) from reference [27].In terms of the parameters defined above,

Fig. 3 .Fig. 2
Fig. 3. Below concentrations of about 5 mM in the membrane,  ex was comparable to the experimental linewidths in the absence of HSE and could not be determined accurately by any of the methods.Because

Fig. 3
Fig. 3 HSE values obtained from least-squares parameters of fits to experimental Tempone EPR lineshapes over a range of concentrations in Nafion 117 for each of the methods indicated in Equations (3).See text for details.

Fig. 5
Fig. 5 Time course of relative Tempone concentration in water-equilibrated Nafion 117 membranes.(a) Least-squares fits of three different functions to the data: (•) relative integrated intensity of the Tempone EPR signal; every tenth point is shown for clarity; (dashed line) single exponential, (dash dot line) stretched exponential, and (solid line) double exponential.(b) Residuals from fitted line for each of the functions in (a) with RMSD values of 0.84 (single), 0.39 (stretched), and 0.23 (double); (c) decomposition of double exponential fit showing the fast and slow components.

( 4 .
00.1)10 −5 0.0130.009(a) Reported values and uncertainties are the average and range of fits to two datasets.(b) Single exponential function  1 exp(− 1 ) +  (c) Stretched exponential function  1 exp(− 1    (  +2) ⁄ ) +  (d) Two exponential function  1 exp(− 1 ) +  2 exp (− 2 ) + Nafion 117 membranes (Ion Power, Inc., New Castle, DE) were first purified by heating to 75°C for 1 hour in 3% hydrogen peroxide followed by 1 hour in deionized water, 1 hour in 0.5M sulfuric acid, and 1 hour in deionized water.They were then cut to a size of 10×3 mm and equilibrated with a 118 mM aqueous solution of the Tempone probe (Sigma-Aldrich) that had been purged of oxygen by gently bubbling with saturated nitrogen gas.External liquid was removed from the membrane with a tissue and sealed in a EPR tube under a nitrogen atmosphere.The time evolution of the EPR spectrum was monitored in a Bruker EMX spectrometer at a microwave frequency of ~9.848 GHz.Spectra were acquired in a 2D field vs. time experiment at intervals of 90 seconds.The experimental parameters were

Table 1
Least-squares parameters(a)for different functions fitted to experimental Tempone decay Table1and the two components are individually plotted in Fig.5c.Approximately 70% of the spin probe population decays relatively quickly with a rate of 1.93 × 10 −4 s −1 , with the remainder decaying at approximately one fifth of that rate.This two-ensemble model was then compared to the  ex () data by applying Equation (1) to each population, i.e. by assuming different collision rates in the two populations that lead to different proportionality constants  1 and ω 2 between C and  ex as follows: ex (()) =  1  1 exp(− 1 ) +  2  2 exp (− 2 )

Table 2
Least-squares parameters a for different functions fitted to experimental  ex () Reported values and uncertainties are the average and range of fits to two datasets.b Time power law:  0  (1−  2 ⁄ ) () c Two exponential function:  1  1 exp(− 1 ) +  2  2 exp (− 2 ), where  1 ,  1 ,  2 , and  2 are fixed at the values given in Table (,  ex ) = ∑[  (  )   ′ () +  disp (  )   ′ ()]We wish to fit the standard Lorentzian line shape parameters    and    to the width and field position of each experimentally observed hyperfine line.The quantity    includes the effects of both Heisenberg exchange and spin-spin relaxation due to rotational diffusion.The Bales-Peric parameters may be written in terms of this quantity as follows:Finally, letting    =   ex (  ), the remaining quantity may be written