Experimental quantification of electron spectral-diffusion under static DNP conditions

Abstract

Dynamic Nuclear Polarization (DNP) is an efficient technique for enhancing NMR signals by utilizing the large polarization of electron spins to polarize nuclei. The mechanistic details of the polarization transfer process involve the depolarization of the electrons resulting from microwave (MW) irradiation (saturation), as well as electron–electron cross-relaxation occurring during the DNP experiment. Recently, electron–electron double resonance (ELDOR) experiments have been performed under DNP conditions to map the depolarization profile along the EPR spectrum as a consequence of spectral diffusion. A phenomenological model referred to as the eSD model was developed earlier to describe the spectral diffusion process and thus reproduce the experimental results of electron depolarization. This model has recently been supported by quantum mechanical calculations on a small dipolar coupled electron spin system, experiencing dipolar interaction based cross-relaxation. In the present study, we performed a series of ELDOR measurements on a solid glassy solution of TEMPOL radicals in an effort to substantiate the eSD model and test its predictability in terms of electron depolarization profiles, in the steady-state and under non-equilibrium conditions. The crucial empirical parameter in this model is Λ^eSD, which reflects the polarization exchange rate among the electron spins. Here, we explore further the physical basis of this parameter by analyzing the ELDOR spectra measured in the temperature range of 3–20 K and radical concentrations of 20–40 mM. Simulations using the eSD model were carried out to determine the dependence of Λ^eSD on temperature and concentration. We found that for the samples studied, Λ^eSD is temperature independent. It, however, increases with a power of ∼2.6 of the concentration of TEMPOL, which is proportional to the average electron–electron dipolar interaction strength in the sample.

---

1. Introduction

Over the years, Nuclear Magnetic Resonance (NMR) has become one of the predominant spectroscopic tools for structure determination and imaging. Despite being a very useful technique, NMR is limited by its inherently low sensitivity due to the relatively small nuclear Zeeman energies, compared to the ambient thermal energy. Dynamic Nuclear Polarization (DNP) is a process where NMR polarizations are enhanced far beyond their ambient Boltzmann values. This is accomplished by a transfer of polarization from unpaired electron spins to the nuclei in the sample during microwave (MW) irradiation. During the last two decades, it has become one of the predominant spectroscopic tools for enhancing the sensitivity of NMR and MRI,^1,2 in particular for studies in biology³ and materials science.^4,5

In solid solutions of free radicals, the DNP mechanism strongly depends on the EPR spectral width and concentration of the radicals used as polarizers. The commonly recognized DNP mechanisms, the solid effect^6,7 (SE), the cross effect^6,8–13 (CE) and thermal mixing^14–18 (TM), all play a role in nuclear enhancement. The SE is responsible for the nuclear enhancement that is induced by MW excitation of electron-nuclear zero-quantum and double-quantum transitions. The CE also transfers electron polarization to nuclear polarization, but this time it is mediated by a dipolar interaction between neighboring electrons. In the TM description, the NMR enhancement is rationalized by a spin thermodynamic process¹⁴ between the spin heat baths¹⁹ of the multi-electron system and of the nuclei.^20–28 When the polarizers are nitroxyl radicals with an in-homogeneously broadened EPR spectrum, the dominant DNP mechanisms are the SE and CE. Furthermore, the deviations from the TM predicted EPR line shape^29,30 under DNP conditions at 95 GHz indicate that the contribution of the TM mechanism is negligible.^31,32

Recently we made a distinction between the direct and the indirect CE mechanism. During a direct CE experiment one directly irradiates a pair of electron spins with a resonance frequency difference equal to the nuclear Larmor frequency (the CE conditions). During an indirect CE (iCE) enhancement process, the electron polarizations of a pair are affected by a redistribution mechanism of the electron depolarization throughout the full EPR spectrum, due to spectral diffusion.³¹ In solutions with radical concentrations above 10 mM, the MW irradiation initially burns a hole that then spreads throughout the EPR spectrum. Its lineshape reaches a steady state that is determined mainly by (i) the strength of the electron–electron dipolar interactions,^33,34 (ii) the associated electron–electron cross relaxation, (iii) the electron spin–lattice relaxation and (iv) the strength of the applied MW amplitude.^35–38 The polarization of the core and local nuclei³⁹ coupled to the electrons is then enhanced by the iCE process that transfers the polarization difference of coupled electron spins fulfilling the CE conditions,^40,41 to their nearby nuclei. In the next DNP stage, the enhanced polarizations of the local nuclei are transferred to the bulk nuclei by spin diffusion,⁴² governed by the nuclear dipolar interactions and longitudinal relaxation. Accordingly, once the depolarization profile within the EPR line shape is known and analyzed in terms of polarization gradients, it can be used to directly calculate the DNP lineshape, namely the DNP enhancement as a function of the MW irradiation frequency.

The EPR lineshape, under conditions of MW irradiation typically used in DNP experiments, can be determined by a set of electron–electron double resonance (ELDOR) experiments.^43,44 To analyze and extract the polarization gradients within the EPR spectrum, we introduced a mathematical model referred to as the electron spectral diffusion (eSD) model. This model is based on a set of coupled rate equations for the electron polarizations of the frequency bins composing the EPR spectrum. The rate constants, representing the electron spectral diffusion responsible for the electron depolarization during MW irradiation, are defined by a single eSD parameter Λ^eSD. This single parameter, together with the spin relaxation rates, has been sufficient for reconstructing ELDOR frequency profiles^31,45 and from them EPR and DNP spectra.⁴⁶ In our recent theoretical study,⁴⁷ we discussed the dependence of Λ^eSD on the electron–electron dipolar interactions in the system, and in the present study, we experimentally address the question of how it depends in practice on the temperature and the radical concentration, which is directly related to the average dipolar interaction between the electron spins.

We report on ELDOR experiments carried out on samples of TEMPOL at different concentrations, where the CE is highly efficient, namely in the range of 20–40 mM in a 50 : 50 (v/v%) DMSO/H₂O amorphous glassy solvent at different temperatures. The eSD model is then used to analyze the ELDOR spectra and to extract the concentration and temperature dependence of Λ^eSD. Additionally, we compare experimental temporal evolutions of the polarizations after MW irradiation with simulations relying on the values of Λ^eSD. This comparison also reveals that to successfully reproduce the experimental data, it is necessary to introduce time-dependent SE parameters to the eSD model.

2. Experimental and simulation procedures

2.1 Sample preparation

TEMPOL and DMSO were purchased from Sigma-Aldrich and solutions with various concentrations of TEMPOL, 20, 25, 30, 35 and 40 mM, were prepared with a 50/50 (v/v%) mixture of water and dimethyl sulfoxide (DMSO).

Measurements were carried out on three sets of samples referred to as I, II and III. Sample sets I and II were measured on our pulse EPR spectrometer (95 GHz).⁴⁸ They were placed in a 0.9 mm outer diameter (OD) quartz capillary, containing about 2–4 μl, then degassed, and immediately frozen in liquid nitrogen. These samples were then introduced cold into the precooled cryostat of the spectrometer. In this spectrometer, cooling is done with a liquid helium flow cryostat. On sample set I, a series of ELDOR measurements was performed with different detection frequencies covering the full EPR line. From these ELDOR spectra, 2D-ELDOR maps were constructed. For these measurements, the superior SNR of the pulse EPR spectrometer was of importance and enabled obtaining reasonable ELDOR data, even when detecting at the edges of the EPR spectrum. Sample set II was used to test the reproducibility and estimate the uncertainties of the measurements. To test a broad range of temperatures and concentrations, we added the third set of samples, set III, which were measured using our hybrid pulsed-EPR-NMR spectrometer⁴⁹ (henceforth called the DNP spectrometer). These measurements allowed experiments at 3 K, not possible for the EPR spectrometer. Synchronization between the EPR and NMR parts of this spectrometer enables the application of independent and simultaneous RF and MW irradiation pulse schemes. Similar to the EPR spectrometer, it operates at a magnetic field of 3.4 T, corresponding to a ¹H-Larmor frequency of 144 MHz and an electron-Larmor frequency of approximately 95 GHz. Both spectrometers have a very similar design in terms of their microwave bridge, with two channels for applying two different MW frequencies during single measurements. For measurements on the DNP spectrometer, a series of samples at different concentrations were placed in a glass sample holder containing about 20 μl of the required solution, degassed, and sealed. The samples, placed in the probe head, were then cooled to cryogenic temperatures, by inserting the probe head into a cryogen-free variable temperature system from Cryogenic Limited, UK. The DNP setup does not support inserting precooled samples.

2.2 EPR and ELDOR measurements

For all ELDOR measurements, the EPR signals were detected using a two-pulse echo detection sequence⁴⁹ α–τ–α–echo, with pulse durations of 400 ns and a delay, τ, of 1 μs, using a 180° phase cycle on the first pulse. The equal lengths of the pulses are dictated by the low power available in the DNP spectrometer, and were optimized to obtain maximal echo intensity.⁴⁹ As is also common in pulse EPR spectroscopy, the signal intensity was obtained by an integration window positioned around the half width of the echo peak. The ELDOR pulse sequence is shown in Fig. 1a. Experiments were conducted by applying a long MW excitation pulse at a frequency ν_exc for a duration of t_MW, followed by a delay t_relax = 10 μs, and then an EPR echo detection sequence applied at a detection frequency ν_det. The length of t_MW, required to achieve a steady-state ELDOR spectrum, depends on the value of T_1e. In the case of the EPR spectrometer, it was set at 20 ms which was sufficient at all temperatures. For the experiments on the DNP spectrometer, t_MW was equal to 40 ms at 20 K and 10 K and to 100 ms at 5 K and 3 K. For each ELDOR experiment, ν_det was kept fixed and ν_exc varied from 94.3 to 95.3 GHz, which is the active range of the spectrometer. In the EPR spectrometer, a series of ELDOR experiments were performed with different ν_det. The ELDOR spectra were normalized with respect to the thermal equilibrium EPR signal at each ν_det value, which was the intensity of the signal obtained after MW irradiation at a frequency ν_exc′, situated far outside the EPR spectrum. Such irradiation should not have any direct or indirect effect on the detected electron polarizations. The MW amplitude, ν₁, was 640 kHz for the DNP and 500 kHz for the EPR spectrometer. These values were determined by nutation experiments performed at 94.8 GHz, the frequency of the maximum of the EPR spectrum. For the DNP spectrometer, we assumed that due to the low Q of the MW setup, this ν₁ value is constant over the frequency range used in the ELDOR experiments. In the EPR spectrometer, a low-Q MW cavity was used, with a center frequency of 94.8 GHz, to minimize the variation in ν₁ over the same frequency range.

Fig. 1 (a) A schematic presentation of the ELDOR pulse sequence. This includes a MW irradiation pulse of length t_MW at frequency ν_exc followed by two short echo pulses at frequency ν_det with a delay τ. In this setup the frequencies ν_exc and ν_det can be varied independently. When desired a time delay t_relax can be inserted between the first MW pulse and the echo sequence. (b and c) Experimental steady-state E^ELDOR_exp(Δν_exc,Δν_det) 2D-ELDOR spectra of sample I⁴⁰ (b) and sample I²⁰ (c) recorded at 20 K. The contours are derived from 41 spectra E^ELDOR_exp(Δν_exc,[Δν_det]) recorded with fixed Δν_det values in the range −250 MHz < Δν_det < 150 MHz with a resolution of 10 MHz.

The temporal behavior of the electron depolarization during MW irradiation was measured by recording EPR echo signals after various values of t_MW. We also recorded the time-domain behavior of the electron polarization, following a short MW pulse of 100 μs, for monitoring the influence of the spectral diffusion process during the return of the EPR spectrum to its thermal equilibrium line shape.

2.3 Spin–lattice relaxation measurements

Spin–lattice relaxation times were measured, using a saturation recovery sequence which included a long MW pulse with a duration of 50–100 ms, followed by a variable delay and echo detection. The results were fitted using either a single exponent or, when required, using a double exponential function with the slow component considered as the true spin–lattice relaxation time.

2.4 Simulations of ELDOR profiles

We used the eSD model to calculate ELDOR spectra for different values of Λ^eSD and fit them to the experimental ELDOR spectra.³¹ The eSD model is based on approximating the EPR spectrum by a set of N discrete bins, with average electron polarizations P_e,j with j = 1…N and an average bin frequency of ω_j. The time-evolution under MW irradiation of the bin polarizations P_e,j is described by the following rate equation:Here the vector P⃑_e(t) of polarizations is defined by P⃑_e(t) = [1,P_e,1,P_e,2…P_e,N]. The first element in the vector will be given an index, P_e,0 = 1, and is necessary for the spin–lattice relaxation, as will be described. The rate matrices, R^b_1e and R^b_D, describe the action of the electron spin–lattice relaxation and the spectral diffusion, respectively, while the rate matrices, W^b_MW and W^b_SE, describe the effect of MW irradiation on the allowed and “forbidden” EPR transitions respectively. For completeness, we reintroduce the necessary expressions defining the matrix elements of these rate matrices:³¹

The spin–lattice relaxation of the jth bin is described by the equation:

where P_e,j(0) = P^eq_e,j, the polarization at thermal equilibrium of the jth bin. The element with the value of 1 ensures that the spin–lattice relaxation process eventually reaches thermal equilibrium.^50,51

The non-zero elements of the electron spin–lattice relaxation rate matrix, R^b_1e, are therefore given by:

The matrix, W^b_MW, describes the MW irradiation in the single quantum electron–spin transitions. Here we have only diagonal elements given by:where ω₁ is the MW amplitude, ω_MW is the irradiation frequency, and T_2e is the electron spin–spin relaxation time, influencing the range of possible off resonance excitation.

The electron–electron cross relaxation mechanism for any two bins, j and j′, with polarizations P_e,j(t) and P_e,j′(t), respectively, is described by the equation:

where the rate constants, , are defined byand

η_j,j′ = exp(−(ω_j − ω_j′)/k_BT) (4c)

The coefficients, f_j, are the relative number of electrons at each bin, j, extracted from the intensity of the EPR lineshape at each bin, with . The role of the η_j,j′ values is to assure that, at thermal equilibrium, each pair of polarizations P_e,j(t) and P_e,j′(t) maintain their Boltzmann ratio. The matrix elements of R^b_D will be a sum of the rate matrixes between pairs of bins, as described in detail elsewhere.³¹

The last rate matrix W^b_SE in eqn (1) represents electron depolarization due to the SE-DNP process. This is represented by an effective MW irradiation in the ZQ or DQ transitions, and again has only diagonal elements, given by:

where ω₁ is again the MW amplitude and ω_n is the nuclear Larmor frequency. A^SE is a parameter that represents the depolarization efficiency, determined by the pseudo-secular electron–nuclear anisotropic hyperfine interaction with coefficients A^±_e–n, in the electron–nuclear spin Hamiltonian, causing weak state mixing and leading to the SE-DNP mechanism.

Solving eqn (1) for a set of parameters results in a polarization profile determining the values of P_e,j as a function of their bin resonance frequencies ω_j. These profiles enable us to calculate the EPR spectra during MW irradiation, by taking the values of f_j into account:

E^EPR_sim(ω_j,t;ω_MW) = f_jP_e,j(t) (6)

For consistency, from here on, we will present all frequencies in units of Hertz as Δν_j = ω_j/2π − ν_e and Δν_MW = ω_MW/2π − ν_e, with ν_e = 94.8 GHz, and ν_n = ω_n/2π. The parameters necessary for fitting the data are: T_1e the electron spin–lattice relaxation, Λ^eSD the spectral diffusion parameter, ν₁ the MW irradiation intensity, ν_n the nuclear Larmor frequency, A^SE the SE depolarization parameter and T_2e the electron spin–spin relaxation. Before the data analysis we determined experimentally the shape of the EPR spectrum of the TEMPOL radical and its f_j values, the values of T_1e and ν₁, set ν_n equal to the proton Larmor frequency and chose a value for Δν_MW. We also measured phase memory times, T_M, which represent only a lower limit for T_2e. Thus using these values, the remaining parameters Λ^eSD, T_2e and A^SE are determined by a fitting procedure. The first parameter determines the overall depolarization profiles of the ELDOR spectra, the second defines the off-resonance excitation efficiency of the MW irradiation and the third is responsible for the spectral features induced by the SE-DNP process.

Using this rate equation we can now simulate ELDOR spectra with the following procedure: at first, after fixing the value for the widths of the frequency bins we define all bin frequencies ω_j in units of Hertz and with respect to ν_e = 94.8 GHz: Δν_j = (ω_j − 2πν_e)/2π. For each chosen set of parameters {Λ^eSD,A^SE(t_MW),T_2e} and MW irradiation at a specific bin frequency, Δν_exc,j, we can calculate a normalized steady state EPR spectrum of the form

for t_MW ≫ T_1e. Here E^EPR_eq(Δν_det,j′) is the pre-determined experimental equilibrium EPR spectrum, and Δν_det,j′ is the detection bin frequency with j′ = 1,…,N_b and N_b the number of bins. Repeating this for irradiation at all bin frequencies Δν_exc,jj = 1,…,N_b, we can extract the normalized ELDOR spectrum E^ELDOR_sim(Δν_exc,j,[Δν_det,j′]) corresponding to the parameters {Λ^eSD,A^SE(t_MW),T_2e}. This spectrum is then compared with the experimental spectra by manual inspection. This procedure for different sets of parameters must then be repeated until a satisfactory agreement is reached.

In all calculations discussed here, the bin-width was set equal to Δν_b = 2 MHz. This value was chosen because it is of the order of the average (e–e) dipolar interaction strength in the samples with the highest radical concentration. From our earlier calculations (not shown here), it follows that varying Δν_b between 2 and 0.5 MHz does not affect the best-fit values for Λ^eSD. Thus, to minimize the dimension of P⃑_e(t) in eqn (1), we choose a bin size of 2 MHz. During the calculations, the Δν_exc,j and Δν_det,j frequencies were always set at the center frequencies of one of the bins which, in turn, were fixed with respect to the EPR spectrum. This restriction is of course not valid for the experimental Δν_exc and Δν_det values. To compare the simulated and experimental ELDOR results, the detection frequencies Δν_det,j during the calculations were set to be as close as possible to the experimental Δν_det values, by choosing Δν_det − Δν_b/2 < Δν_det,j < Δν_det + Δν_b/2, and the same rule follows for Δν_exc.

3. Results

3.1 Relaxation time measurements

The T_1e values of all sample sets, I, II and III studied in this work (see Section 2.1), are listed in Table 1. We note that in contrast to recently published results^52–54 related to the anisotropy of T_1e, in all our samples the experimental values of T_1e (the slow component of the bi-exponential analysis of the spin–lattice experiments) were found to be only slightly dependent on the position of the detection frequency along the EPR line. In the ESI,† Fig S1, we show saturation recovery curves of sample III⁴⁰, of set III at 40 mM, recorded at Δν_det = −200, −100, 0 and 100 MHz. These curves show that the variation of T_1e is less than 10–20% along all the EPR line. The reported results, showing a frequency dependence of T_1e, were obtained from the sample at much higher temperatures than our samples^52,53 (>80 K), or at lower radical concentrations.⁵⁴ Under both these conditions, the spectral diffusion mechanism among radicals at different orientations is expected not to be very effective. Contrary to that, in our cases, the relatively high radical concentrations and the rather efficient cross relaxation are expected to quench the anisotropy in the experimentally determined T_1e.

Table 1 The measured values of T_1e of all samples studied, recorded at the maximum of the EPR line

Sample set	Temp. (K)	Conc. (mM)	T 1e (ms)	Sample set	Temp. (K)	Conc. (mM)	T 1e (ms)	Sample set	Temp. (K)	Conc. (mM)	T 1e (ms)
I	20	20	5.3	III	5	25	5.5	III	10	35	48
I	20	40	5.3	III	5	30	5.6	III	10	40	47
II	10	20	41.3	III	5	35	5.3	III	20	20	280
II	10	40	28.3	III	5	40	5.1	III	20	25	290
II	20	20	5.4	III	10	20	63	III	20	30	278
II	20	40	5.3	III	10	25	57	III	20	35	280
III	5	20	5.5	III	10	30	56	III	20	40	260

---

Table 2 The measured values of T_m of sample III as a function of concentration and temperature recorded on the DNP machine

Temperature	5 K	10 K	20 K
T m (20 mM)	0.53 μs	0.49 μs	0.46 μs
T m (40 mM)	0.35  μs	0.33 μs	0.28 μs

---

3.2 Steady-state ELDOR spectra and their analysis

Fig. 1 shows a contour plot of the steady-state 2D-ELDOR spectra of the 20 mM and 40 mM samples from set I, which we will name I⁴⁰ and I²⁰, respectively, measured on the EPR spectrometer at 20 K. The data were collected as described in Section 2.2, by performing a series of ELDOR measurements for different detection frequencies with a spectral resolution of 10 MHz on both axes. The axes of the plot are defined as Δν_det = ν_det − ν_e, Δν_exc = ν_exc − ν_e with ν_e = 94.8 GHz. The intensity of the 2D plot E^ELDOR_exp(Δν_exc,[Δν_det]) is normalized as described in Section 2.2 and is given by:with Δν_exc′ is situated far outside the EPR frequency range and E^ELDOR_exp(Δν_exc,[Δν_det]) is the echo intensity following irradiation. The 2D plot was generated by combining the 41 normalized ELDOR experiments (−250 MHz < Δν_det < 150 MHz). In all ELDOR experiments the length of irradiation was at least t_MW = 20  ms which is much longer than the electron spin relaxation times of the two samples, T_1e{I²⁰, 20 K} = 5.4 ms and T_1e{I⁴⁰, 20 K} = 5.3  ms (see Table 1). The differences between the spectra of the 40 and 20 mM samples are clear, while for I²⁰ the ¹H SE lines are resolved (see arrows in Fig. 1b), for I⁴⁰ they are buried within the broad ELDOR hole. As there is no difference between the T_1e values of these samples, it is clear that the difference is a result of a difference between their spectral diffusion parameters Λ^eSD.

To test the reproducibility of these ELDOR profiles, we prepared an additional pair of samples with 40 mM and 20 mM radical concentrations, samples II⁴⁰ and II²⁰ respectively, and measured their ELDOR spectra. For these samples, we chose only three detection frequencies, Δν_det = −100,  0,  and 100  MHz. A comparison of the detection frequency Δν_det = 0  MHz is shown in the ESI,† Fig. S2. While samples I⁴⁰ and II⁴⁰ give very similar ELDOR spectra, some differences are noted between samples I²⁰ and II²⁰.

All the steady-state ELDOR data collected were analyzed by the simulation procedure described in Section 2.3, resulting in the best fit values for A^SE and Λ^eSD. In practice, we first determined the values of the proton A^SE by reproducing the ELDOR spectra at frequencies outside the EPR spectrum. Manual eye inspection resulted in A^SE = 4 MHz for all samples and at both temperatures. During the determination of A^SE, the value for T_2e was set at 10 μs for all cases. This T_2e value determines also the off-resonance excitation efficiency of the MW irradiation and in particular, the shape of the ELDOR profiles after a short irradiation time, and is not necessarily equal to the phase-memory time T_m. The values of all Λ^eSD extracted from these best-fit simulations are given in Table 3, showing that the Λ^eSD values are basically the same for both sample sets within the uncertainty in the fitting parameters, and are generally temperature independent.

Table 3 The best-fit values of Λ^eSDof the sample sets I and II as a function of concentration and temperature

	I^20, 20 K	I^40, 20 K	II^20, 10 K	II^20, 20 K	II^40, 10 K	II^40, 20 K
Λ ^eSD (μs^−3)	500	2500	400 ± 50	400 ± 50	2500 ± 250	2500 ± 250

---

Fig. 2 shows the fitting of the ELDOR spectra of the II⁴⁰ (a–c) and II²⁰ (d–f) samples at 20 K (black) and 10 K (red) and at three detection frequencies, Δν_det = −100,  0,  and 100  MHz. We would like to emphasize that for all the fittings we have done in this work, the Λ^eSD values were independent of the detection frequency, i.e. we assume Λ^eSD is isotropic. This assumption is based on the understanding that the spectral diffusion is a result of a cross-relaxation process which we assume is isotropic. We notice that in particular at 20 K, we observe in the ELDOR spectra a broadening around Δν_det that originates from the ¹⁴N-SE depolarization. In the present simulations, these depolarization effects are not taken into account during the analysis and therefore the fit was based on reproducing the total width of the spectrum, relying less on the central part. The positions of the ¹⁴N-SE depolarizations and their effect on the ELDOR profiles determined by the hyperfine interaction with ¹⁴N will be treated in a separate publication.

Fig. 2 The experimental (dotted lines) and simulated (solid lines) steady state ELDOR spectra E^ELDOR_exp/sim(Δν_exc,[Δν_det]) are shown for sample II⁴⁰ in (a–c), and for sample II²⁰ in (d–f), for two temperatures, 10 K (red) and 20 K (black). The detection frequencies are Δν_det = −100 MHz in (a and d); Δν_det = 0 in (b and e); and Δν_det = 100 MHz in (c and f). In each case, the solid lines are the simulated spectra E^ELDOR_sim(Δν_exc,[Δν_det]) using the eSD model with the parameters listed in Table 1. The thermal equilibrium EPR spectrum is shown in green.

The estimated error of the best-fit Λ^eSD parameters was determined on the basis of the least squares fitting while keeping the parameters T_2e and A^SE fixed. To show this, we plot in Fig. S3 in the ESI† the sums of the squares of the differences for all Δν_exc,j, between the simulated and experimental ELDOR profiles of the II²⁰ and II⁴⁰ samples at 20 K, as a function of Λ^eSD for the fixed values Δν_det = −100, 0,  and 100  MHz. A similar procedure was performed for these samples at 10 K. The Λ^eSD values for each sample and temperature were set equal to the average of the best fit values for each Δν_det value, plus/minus their maximum deviations from this average value. These deviations are added to the Λ^eSD values in Table 3.

In Fig. 3a and b, we show a reconstruction of the steady-state EPR spectra of sample I⁴⁰ and sample I²⁰, respectively, as a function of Δν_det and for the fixed values Δν_exc = −200,  −100,  0,  and 100  MHz, derived from the 2D plot in Fig. 1. These spectra are obtained by extracting slices of the 2D plots, E^ELDOR_exp(Δν_exc,[Δν_det]), of constant excitation frequency and varying detection frequency, E^EPR_exp(Δν_det,[Δν_exc]), and then multiplying them with the EPR lineshape function, E^EPR_eq(Δν_det), of TEMPOL recorded with low concentration at thermal equilibrium.⁵⁵ Finally, these EPR spectra are presented by setting the maxima of E^EPR_eq(Δν_det) to unity. This way of presenting the data gives a better perspective on the effect of spectral diffusion when burning a hole. We can, of course, look also at the simulated EPR spectra, E^EPR_sim(Δν_det,[Δν_exc]), of the samples using the Λ^eSD parameters given in Table 3, as shown in Fig. 3c and d. As expected, we see in the simulated ELDOR spectra that the eSD model shows narrower MW excitation profiles around Δν_det than their corresponding experimental profiles, because ¹⁴N-SE effects are not taken into account.

Fig. 3 Experimental and simulated steady-state EPR spectra E^EPR_exp/sim(Δν_det,[Δν_exc]) derived from E^ELDOR_exp/sim(Δν_exc,[Δν_det]) × E^EPR_eq(Δν_det) obtained after a MW irradiation period t_MW = 20 ms of sample I⁴⁰ in (a and c), and of sample I²⁰ in (b and d). The experimental spectra in (a and b) are extracted from the 2D-ELDOR spectra shown in Fig. 1 at Δν_exc = −200 MHz (black), Δν_exc = −100 MHz (red), Δν_exc = 0 MHz (blue) and Δν_exc = 100 MHz (magenta). The spectra in (c and d) are colored accordingly. The EPR spectrum at thermal equilibrium, E^EPR_eq(Δ_det), is shown in green. The simulated EPR spectra obtained using the eSD model with parameters as in Table 3 are shown in (c and d).

Looking at the results in Table 3, we see that an increase of the radical concentration by a factor of 2 causes an increase of Λ^eSD by a factor of 6. This increase manifests itself as a profound expansion of the depolarization area when comparing the EPR spectra of I²⁰ and I⁴⁰. Another observation is that the Λ^eSD parameter stays constant when the temperature of the samples is varied between 10 and 20 K, while T_1e varies by a factor of 6–8 (see Table 1).

3.3 The time dependence of the electron depolarization

After reproducing the experimental results of the steady-state ELDOR profiles of the 40 mM and 20 mM samples with the eSD model, in this section, we show the temporal evolution of the electron depolarization by recording the ELDOR spectra of samples II⁴⁰ and II²⁰ at 10 K, after MW irradiation periods of different durations, t_MW. The purpose of these measurements is to further substantiate the eSD model and to show that the non-steady-state spectra can be reproduced by using the single Λ^eSD values derived from the steady-state spectra. In Fig. 4 the temporal dependences of the experimental ELDOR spectra, E^ELDOR_exp(Δν_exc,[t_MW,Δν_det]), are shown for two different detection frequencies, Δν_det = 0, 100 MHz, along with the simulation using the values T_2e and Λ^eSD, determined from the simulation of the ELDOR spectra under steady-state conditions. These profiles show clearly the effect of the ¹H-SE process on the electron polarizations, in the form of narrow holes for t_MW values less than 1 ms. To properly fit these ¹H-SE contributions, it was necessary to change the value of A^SE as a function of t_MW. To demonstrate why we should expect a t_MW dependence of A^SE, we present in the SI spin dynamics calculations on a small spin system composed of 9 coupled electrons and a nucleus, rationalizing the source of this dependence. The dependence on t_MW of the A^SE values used in the simulation is given in Fig. 4c and f, for samples II⁴⁰ and II²⁰ respectively, both showing a decrease in A^SE for longer t_MW. For simplicity, the values of A^SE were changed in discrete steps of 1 MHz. This still enabled reasonable fitting of the data, however, resulted in the step-like result in Fig. 4f. To show also the resemblance between the time constants characterizing the evolution of the experimental and simulated time dependent ELDOR profiles, we plot in Fig. 5 E^ELDOR_exp/sim(t_MW,[Δν_exc,Δν_det]) values for four different pairs of frequencies, Δν_det and Δν_exc, as mentioned in the figure caption. Fitting the resulting t_MW dependent functions to single exponential functions yield time constants τ_dep varying between 1 ms and 10 ms. The maximum deviation between simulations and experiments is found to be less than 10%. For a summary of all these τ_dep values, see Tables S1 and S2 in the ESI.†

Fig. 4 Experimental (dotted lines) and simulated (solid lines) ELDOR spectra, E^ELDOR_exp/sim(Δν_exc,[t_MW,Δν_det]), of sample II⁴⁰ at 10 K, are shown for 5 different values of t_MW with detection frequencies Δν_det = 0 MHz in (a), Δν_det = 100 MHz in (b), and of sample II²⁰ also for 5 values of t_MW, with detection frequencies Δν_det = 0 MHz in (d) and Δν_det = 100 MHz in (e). In (a and b) spectra are plotted for t_MW = 0.15, 0.6, 1, 4, 20 ms and in (d and e) for t_MW = 0.4, 1, 2, 8, 20 ms, in black, red, blue, green and magenta, respectively. The simulated solid lines are obtained using the eSD model with parameters listed in Table 1. The t_MW dependent values of A^SE used during the simulations are plotted for sample II⁴⁰ in (c) and for sample II²⁰ in (f). The frequency positions of the ¹H-SE features in the spectra are marked with arrows.

Fig. 5 Time-domain experimental (points connected by dotted lines) and simulated (points connected by solid lines) ELDOR signals, E^ELDOR_exp/sim(t_MW,[Δν_exc,Δν_det]), at 10 K are shown for sample II⁴⁰ in black and sample II²⁰ in red. The values of the pair of frequencies [Δν_exc,Δν_det] in MHz units are in (a) [0, −100], in (b) [0, 100], in (c) [100, 0] and in (d) [100, −200].

The τ_dep values are a complex function of the eSD parameter Λ^eSD, the MW amplitude ω₁ and the T_1e value and of course, the values of Δν_det and Δν_exc. If the electron depolarization would have been determined solely by the spectral diffusion mechanism, we would have expected the τ_dep times to be about inversely proportional to Λ^eSD. Inspection of the τ_dep values of sample II⁴⁰ and sample II²⁰, with the ratio between their Λ^eSD values equal to ∼6, reveals that the ratio between the τ_dep values at equal Δν_det and Δν_exc with different T_1e values varies between 3.5 and 5.5.

To verify further the validity of the eSD model, we also followed the temporal evolution of the electron polarizations back to equilibrium after the MW irradiation, on sample II²⁰ at 10 K. The pulse sequence used for the evaluation consists of MW irradiation of fixed duration t_MW, followed by a varying time delay t_relax before the echo detection pulses. Here we chose a short MW irradiation time t_MW = 100 μs, preventing any significant cross-relaxation induced spectral diffusion effect during this pulse. Normalized EPR spectra E^EPR_exp(t_relax,Δν_det,[Δν_exc]) were constructed by performing a large set of ELDOR experiments within the range of −200 MHz < Δν_det < 100 MHz as a function of t_relax. The results are shown in Fig. 6a together with the thermal equilibrium EPR spectrum.

Fig. 6 Experimental (a) and simulated (b) EPR spectra, E^EPR_exp/sim(Δν_det,[t_relax,Δν_exc]), of sample II²⁰ at 10 K, obtained after a short MW pulse of t_MW = 100 μs and an increasing delay time, t_relax, shown in black (0.01 ms), red (0.2 ms), cyan (0.7 ms), green (5 ms), magenta (20 ms) and violet (200 ms). The simulated spectra in (b) are derived using the eSD model with parameters tabulated in Table 3. In (a) the ¹⁴N-SE depleted signals are marked with arrows.

The narrow hole in the spectrum created by the short MW irradiation broadens due to the spectral diffusion, and after 5 ms the normalized EPR spectrum reaches a flat profile that shows that the unnormalized EPR spectrum itself has reached the same shape as the thermal equilibrium EPR spectrum, but with a smaller overall intensity. This intensity grows to its equilibrium value with a time constant T_1e, that is the same for all Δν_det values. In addition to the hole at Δν_exc = 0, we observe the effect of SE processes of ¹H and ¹⁴N. In Fig. 6b we show the simulated normalized EPR spectra using again the value of Λ^eSD derived from the ELDOR analysis. The broadening of the initial hole and the equilibration of the intensity over the whole spectrum are clearly reproduced by the simulation in Fig. 6b. However, a comparison between the experimental and simulated spectra shows a significant deviation. The sharp features in the experiment are missing in the simulation and the level of equilibration is much closer to 1 than in Fig. 6a. The reason for this difference is the fact that the ¹⁴N-SE depolarization process is not taken into account in the calculations. A preliminary simulation taking the ¹⁴N-SE process into account is shown in Fig. S4 (ESI†). The evolution of E^EPR_exp/sim(t_relax,[Δν_detΔν_exc]) at detection frequencies that are not in the initial hole, burnt by the MW irradiation, during the initial 5 ms is shown in Fig. S4 (ESI†) for both samples. The initial decay can be fitted by single exponentials with a decay constant, τ_decay. The time constants of these fitting procedures are summarized in Tables S3 and S4 (ESI†), which show that their experimental and simulated values agree within 10%. As this decay process is a direct consequence of spectral diffusion effects, this agreement of experiments and simulations is another indication that the model describes properly the spectral diffusion process.

The experiments and their analysis in this section have shown that, for each sample, a single Λ^eSD value can explain the steady state ELDOR spectra, as well as the time behavior of the electron polarizations during and after MW irradiation. This shows that this empirical parameter indeed represents the dipolar interaction driven spectral diffusion mechanism. In the next section, we discuss in which manner Λ^eSD depends on the average electron–electron dipolar interaction in our samples.

4. Temperature and radical concentration dependence of Λ^eSD

In the above discussion, the eSD process was investigated on samples with two different TEMPOL concentrations, 20 mM and 40 mM, and at two temperatures, 10 K and 20 K. The eSD analysis showed that the Λ^eSD value changes with radical concentration and not with temperature. Realizing that the average electron–electron dipolar interaction is proportional to the concentration, it is of interest to determine the concentration dependence of Λ^eSD. To do so we prepared a new set of samples, III^c, with different concentrations c = 20, 25, 30, 35, and 40 mM. Steady-state ELDOR experiments on all these new samples were conducted at temperatures 5, 10 and 20 K and on sample III⁴⁰, also at 3 K. This time the data were recorded on the DNP spectrometer enabling data collection at 3 K. Except for the difference in the freezing procedure, described in the Experimental section, the experimental parameters were the same as for experiments on the EPR spectrometer reported above. The 20-to-40 mM range of concentrations was chosen because at lower values the spectral diffusion process is rather inactive. Above this range, the EPR spectrum is fully depolarized after a very short MW irradiation period. Typical ELDOR profiles for Δν_det = 0 MHz of samples III⁴⁰ and III²⁰ are shown in Fig. 7a and b. Additional profiles can be found in Fig. S5 in the ESI.† All profiles were analyzed as described above and the results are summarized in Table 4.

Fig. 7 Experimental (dotted lines) and simulated (solid lines) steady-state ELDOR spectra E^ELDOR_exp/sim(Δν_exc,[Δν_det]) of sample III⁴⁰ in (a) and sample III²⁰ in (b), with Δν_det = 0 MHz obtained at different temperatures are shown in magenta (3 K), blue (5 K), red (10 K) and black (20 K). The simulated spectra are derived using the eSD model with parameters tabulated in Table 3.

Table 4 The best-fitted values of A^SE and Λ^eSD as a function of concentration and temperature recorded on the DNP machine

Sample	III^20	III^25	III^30	III^35	III^40
A ^SE/2π (MHz)	1.5	1.5	2	2	2
Λ ^eSD	100 ± 20 μs^−3	200 ± 20 μs^−3	250 ± 25 μs^−3	400 ± 25 μs^−3	600 ± 50 μs^−3

---

For all concentrations c, the value of Λ^eSD was found to be temperature independent. The A^SE parameter under the steady state conditions needed to be varied from 2 to 1.5 MHz to reproduce the depolarization for Δν_exc values outside the EPR spectral range. The errors associated with Λ^eSD were evaluated in the same way as described in the previous section. The relevant least square analysis for the data of the III²⁰, III²⁵ and III³⁰ samples at 10 K is shown in Fig. S6 (ESI†). Another observation is the similarity between the T_1e temperature dependence for all samples, as shown in Table 1. We can, therefore, conclude that changes in the ELDOR profiles as a function of temperature for a particular concentration are all a result of the temperature dependence of T_1e. The dependence of Λ^eSD in μs⁻³ units on concentration in mM units is plotted in Fig. 8. The points in this figure can be analyzed by fitting them to a simple exponential function:

Λ^eSD(c) = λc^{[small script l]}, (9)

where Λ^eSD is measured in μs⁻³ units and c in units of mM. The best-fit parameters of the points are λ = 0.041 and [small script l] = 2.6. An important consequence of this result is that we can express the change in Λ^eSD as a result of a change in c from c₁ to c₂ as:Thus a twofold concentration increase causes the Λ^eSD value to increase by a factor of 2^2.6 = 6. As the strength of the average electron–electron dipolar interaction, D̄, between the radicals in the solid solutions should be proportional to the concentration, a similar dependence of Λ^eSD on D̄ is expected. A similar dependence was already suggested in ref. 47, where we determined Λ^eSD values from calculations on small electron spin systems. There we were able to describe the time domain and steady-state electron depolarisation profiles by introducing the zero quantum electron–electron cross-relaxation mechanism driven by electron flip–flip processes, governed by fluctuations of the zero-quantum D̄(S⁺_iS⁻_j + S⁻_iS⁺_j) terms.^35,38,56 An effort in analyzing a set of calculated EPR spectra by using the eSD model resulted in the following relationship between the Λ^eSD values for different D̄ values; Λ^eSD(D₁)/Λ^eSD(D₂) = (D₁/D₂)^3.1. This is explicitly shown in the ESI,† Section S3. Thus the experimental concentration dependence presented here suggests that the zero quantum electron–electron cross-relaxation should indeed be an important source of the spectral diffusion process.

Fig. 8 The concentration dependence of Λ^eSD derived from the ELDOR spectra of the III²⁰, III²⁵, III³⁰, III³⁵ and III⁴⁰ samples measured in the temperature range of 5–20 K. The solid black line is the best fitted curve for a function of the form Λ^eSD = λC^l with λ = 2.6 and l = 0.041.

The temperature independence of Λ^eSD may be related to the weak temperature dependence of the phase memory time (T_m), here observed for samples III²⁰ and III⁴⁰ and presented in Table 2. While the contribution of the dipolar flip–flop process on T_m^57–59 is expected to decrease its value at high concentration,^38,60 its temperature dependent influence is negligible. Similarly, the dipolar induced cross relaxation increases Λ^eSD with concentration and leaves it unchanged as a function of temperature. Naively, we can say that the weak temperature dependence of T_m is a consequence of the temperature-independent cross-relaxation process rate in this range, which also approximately justifies the temperature independent behavior of Λ^eSD obtained by ELDOR simulations.

We are aware of the fact that the Λ^eSD(c) values of the present samples, measured on the DNP spectrometer, are about four times smaller than what was obtained from the samples measured on the EPR spectrometer. Together with this, we observe that the T_1e values for each concentration measured using the DNP spectrometer are about 1.2 times longer than those measured using the EPR spectrometer. Whether these two differences are correlated is not clear at the moment. Changes in the distribution of the radicals caused by the sample preparation methods for the two spectrometers could be an origin for the differences in Λ^eSD and T_1e values. The “EPR” samples are frozen quickly by insertion into liquid nitrogen where, in the case of “DNP”, they are rather cooled slowly. Other differences may arise from the different B₁ distributions in the samples, knowing that the EPR spectrometer exhibits a better B₁ homogeneity at the position of the sample than the DNP spectrometer. We expect, however, that all these effects will not affect the concentration and temperature dependence of Λ^eSD.

Conclusions

We have succeeded in using the eSD model to fit a series of ELDOR spectra obtained from frozen solutions with different concentrations of TEMPOL radicals at different temperatures. The time-domain analysis substantiated the physical meaning of the parameter Λ^eSD as truly a rate constant for a spectral diffusion process. The temperature and concentration dependence studies show that it is insensitive to changes in temperature but concentration-dependent according to Λ^eSD(c) ∝ c^2.6. Comparing these results with model calculations on dipolar coupled electron spin systems can suggest that the spectral diffusion mechanism is at least partially governed by zero quantum cross-relaxation processes, which together with the static dipolar interaction, do not necessarily result in the standard TM model. Finally, we also showed that SE depolarization features in the ELDOR spectra, after the onset of the MW irradiation, can be accounted for by the effective ZQ and DQ rate matrices of the eSD model, only when the A^SE parameter is allowed to be dependent on the duration of the MW field t_MW.

Conflicts of interest

There is no conflicts to declare.

Acknowledgements

This work is supported by a grant of the Binational Science Foundation (Grant #2014149) and was made possible in part by the historic generosity of the Harold Perlman Family (D. G.). D. G. holds the Erich Klieger Professorial Chair in Chemical Physics.

References

1. J. G. Krummenacker, V. P. Denysenkov, M. Terekhov, L. M. Schreiber and T. F. Prisner, DNP in MRI: An in-bore approach at 1.5 T, J. Magn. Reson., 2011, 215, 94–99.
2. K. Golman, R. i. Zandt, M. Lerche, R. Pehrson and J. H. Ardenkjaer-Larsen, Metabolic Imaging by Hyperpolarized 13C Magnetic Resonance Imaging for In vivo Tumor Diagnosis, Cancer Res., 2006, 66, 10855–10860.
3. A. B. Barnes, G. De Paëpe, P. C. A. Van Der Wel, K. N. Hu, C. G. Joo, V. S. Bajaj, M. L. Mak-Jurkauskas, J. R. Sirigiri, J. Herzfeld, R. J. Temkin and R. G. Griffin, High-field dynamic nuclear polarization for solid and solution biological NMR, Appl. Magn. Reson., 2008, 34, 237–263.
4. R. G. Griffin and T. F. Prisner, High Field Dynamic Nuclear Polarization-The Renaissance, Phys. Chem. Chem. Phys., 2010, 12, 5737–5740.
5. P. Berruyer, M. Lelli, M. P. Conley, D. L. Silverio, C. M. Widdifield, G. Siddiqi, D. Gajan, A. Lesage, C. Copéret and L. Emsley, Three-dimensional structure determination of surface sites, J. Am. Chem. Soc., 2017, 139, 849–855.
6. C. D. Jeffries, Dynamic orientation of nuclei by forbidden transitions in paramagnetic resonance, Phys. Rev., 1960, 117, 1056–1069.
7. T. J. Schmugge and C. D. Jeffries, High dynamic polarization of protons, Phys. Rev., 1965, 138, A1785–A1801.
8. A. V. Kessenikh, V. I. Lushchikov, A. A. Manenkov and Y. V. Taran, Proton Polarization In Irradiated Polyethylenes, Phys. Solid State, 1963, 5, 321–329.
9. C. F. Hwang and D. A. Hill, New effect in dynamic polarization, Phys. Rev. Lett., 1967, 18, 110–112.
10. C. F. Hwang and D. A. Hill, Phenomenological Model for the New Effect in Dynamic Polarisation, Phys. Rev. Lett., 1967, 19, 1011–1014.
11. A. V. Kessenikh, A. A. Manenkov and G. I. Pyatnitskii, On Explanation of Experimental Data on Dynamic Polarization of Protons in Irradiate Polyethylenes, Phys. Solid State, 1964, 6, 641–643.
12. D. S. Wollan, Dynamic nuclear polarization with an inhomogeneously broadened ESR line. I. Theory, Phys. Rev. B: Solid State, 1976, 13, 3671–3685.
13. D. S. Wollan, Dynamic nuclear polarization with an inhomogeneously broadened ESR line. II. Experiment, Phys. Rev. B: Solid State, 1976, 13, 3686–3696.
14. B. N. Provotorov, Magnetic resonance saturation in crystals, J. Exp. Theor. Phys., 1962, 14, 1126–1131.
15. M. Borghini, Spin-temperature model of nuclear dynamic polarization using free radicals, Phys. Rev. Lett., 1968, 20, 419–421.
16. A. Abragam and M. Goldman, Principles of dynamic nuclear polarisation, Rep. Prog. Phys., 1978, 41, 395–467.
17. A. Abragam and M. Borghini, Dynamic Polarization of Nuclear Targets, Prog. Low Temp. Phys., 1964, 4, 384–449.
18. M. Goldman, Spin temperature and magnetic resonance in solids, London, 1970.
19. A. G. Redfield, Nuclear Magnetic Resonance Saturation and Rotary Saturation in Solids, Phys. Rev., 1955, 98, 1787–1809.
20. A. Karabanov, G. Kwiatkowski, C. Perotto, D. Wisniewski, J. McMaster, I. Lesanovsky and W. Kockenberger, Dynamic nuclear polarisation by thermal mixing: quantum theory and macroscopic simulations, Phys. Chem. Chem. Phys., 2016, 18, 30093–30104.
21. F. Caracciolo, M. Filibian, P. Carretta, A. Rosso and A. De Luca, Evidences of spin-temperature in Dynamic Nuclear Polarization: an exact computation of the EPR spectrum, Phys. Chem. Chem. Phys., 2016, 18, 25655–25662.
22. A. De Luca and A. Rosso, Dynamic Nuclear Polarization and the Paradox of Quantum Thermalization, Phys. Rev. Lett., 2015, 115, 1–5.
23. W. T. Wenckebach, Dynamic nuclear polarization via thermal mixing: Beyond the high temperature approximation, J. Magn. Reson., 2017, 277, 68–78.
24. V. A. Atsarkin and A. V. Kessenikh, Dynamic Nuclear Polarization in Solids: The Birth and Development of the Many-Particle Concept, Appl. Magn. Reson., 2012, 43, 7–19.
25. V. A. Atsarkin and F. S. Dzheparov, Spin Dynamics and Establishing of Internal Quasi-Equilibrium in Dilute Paramagnetic Solids, Z. Phys. Chem., 2017, 231, 545–560.
26. D. Guarin, S. Marhabaie, A. Rosso, D. Abergel, G. Bodenhausen, K. L. Ivanov and D. Kurzbach, Characterizing Thermal Mixing DNP via Cross-Talk between Spin Reservoirs, J. Phys. Chem. Lett., 2017, 8, 5531–5536.
27. V. A. Atsarkin, Experimental Investigation of the Manifestations of the Spin-spin Interaction Reservoir in a System of EPR Lines Connected with Cross Relaxation, J. Exp. Theor. Phys., 1971, 32, 421–425.
28. S. Jannin, A. Comment and J. J. van der Klink, Dynamic Nuclear Polarization by Thermal Mixing Under Partial Saturation, Appl. Magn. Reson., 2012, 43, 59–68.
29. V. A. Atsarkin and M. I. Rodak, Temperature of Spin-Spin Interactions in Electron Spin Resonance, Phys.-Usp., 1972, 15, 251–265.
30. V. A. Atsarkin, Verification of the Spin-spin Temperature Concept in Experiments on Saturation of Electron Paramagnetic Resonance, J. Exp. Theor. Phys., 1970, 31, 1012–1018.
31. Y. Hovav, I. Kaminker, D. Shimon, A. Feintuch, D. Goldfarb and S. Vega, The electron depolarization during dynamic nuclear polarization: measurements and simulations, Phys. Chem. Chem. Phys., 2015, 17, 226–244.
32. J. Granwehr and W. Köckenberger, Multidimensional low-power pulse EPR under DNP conditions, Appl. Magn. Reson., 2008, 34, 355–378.
33. L. L. Buishvili, M. D. Zviadadze and G. R. Khutsishvili, Role of Spectral Diffusion and Dipole-dipole Reserviour in the Saturation of an Inhomogeneously Broadened Line, J. Exp. Theor. Phys., 1969, 29, 159–164.
34. V. A. Skrebnev, The Role of Reservoirs of Dipole–Dipole Interactions in Cross-Relaxation, Phys. Status Solidi, 1973, 60, 51–57.
35. M. K. Bowman and J. R. Norris, Cross relaxation of free radicals in partially ordered solids, J. Phys. Chem., 1982, 86, 3385–3390.
36. N. Bloembergen, S. Shapiro, P. S. Pershan and J. O. Artman, Cross-relaxation in spin systems, Phys. Rev., 1959, 114, 445–459.
37. V. G. Kustarev, Contribution to spin-spin cross relaxation theory, 1977, vol. 44, pp. 802–807.
38. K. M. Salikov and Y. D. Tsvetkov, in Time domain electron spin rsonance, ed. L. Kevan and R. Schwartz, New York, 1979, pp. 231–277.
39. Y. Hovav, A. Feintuch and S. Vega, Dynamic nuclear polarization assisted spin diffusion for the solid effect case, J. Chem. Phys., 2011, 134, 074509.
40. A. S. Lilly Thankamony, J. J. Wittmann, M. Kaushik and B. Corzilius, Dynamic nuclear polarization for sensitivity enhancement in modern solid-state NMR, Prog. Nucl. Magn. Reson. Spectrosc., 2017, 102–103, 120–195.
41. T. V. Can, Q. Z. Ni and R. G. Griffin, Mechanisms of dynamic nuclear polarization in insulating solids, J. Magn. Reson., 2015, 253, 23–35.
42. C. Ramanathan, Dynamic nuclear polarization and spin-diffusion in non-conducting solids, Appl. Magn. Reson., 2008, 34, 409–421.
43. P. Schosseler, T. Wacker and A. Schweiger, Pulsed ELDOR detected NMR, Chem. Phys. Lett., 1994, 224, 319–324.
44. J. S. Hyde, R. C. Sneed and G. H. Rist, Frequency-swept electron-electron double resonance: DPPH in liquid and frozen solution, J. Chem. Phys., 1969, 51, 1404–1416.
45. A. Leavesley, D. Shimon, T. A. Siaw, A. Feintuch, D. Goldfarb, S. Vega, I. Kaminker and S. Han, Effect of electron spectral diffusion on static dynamic nuclear polarization at 7 Tesla, Phys. Chem. Chem. Phys., 2017, 19, 3596–3605.
46. Y. Hovav, D. Shimon, I. Kaminker, A. Feintuch, D. Goldfarb and S. Vega, Effects of the electron polarization on dynamic nuclear polarization in solids, Phys. Chem. Chem. Phys., 2015, 17, 6053–6065.
47. K. Kundu, A. Feintuch and S. Vega, Electron–Electron Cross-Relaxation and Spectral Diffusion during Dynamic Nuclear Polarization Experiments on Solids, J. Phys. Chem. Lett., 2018, 9, 1793–1802.
48. D. Goldfarb, Y. Lipkin, A. Potapov, Y. Gorodetsky, B. Epel, A. M. Raitsimring, M. Radoul and I. Kaminker, HYSCORE and DEER with an upgraded 95 GHz pulse EPR spectrometer, J. Magn. Reson., 2008, 194, 8–15.
49. A. Feintuch, D. Shimon, Y. Hovav, D. Banerjee, I. Kaminker, Y. Lipkin, K. Zibzener, B. Epel, S. Vega and D. Goldfarb, A Dynamic Nuclear Polarization spectrometer at 95 GHz/144 MHz with EPR and NMR excitation and detection capabilities, J. Magn. Reson., 2011, 209, 136–141.
50. J. Jeener, Superoperators in Magnetic Resonance, Academic Press, Inc., 1982, vol. 10.
51. M. H. Levitt and L. Di Bari, Steady state in magnetic resonance pulse experiments, Phys. Rev. Lett., 1992, 69, 3124–3127.
52. M.-A. Geiger, A. Jagtap, M. Kaushik, H. Sun, D. Stöppler, S. Sigurdsson, B. Corzilius and H. Oschkinat, Efficiency of water-soluble nitroxide biradicals for dynamic nuclear polarization in rotating solids at 9.4 T: bcTol-M and cyolyl-TOTAPOL as new polarizing agents, Chem. – Eur. J., 2018, 24, 13485–13494.
53. M.-A. Geiger, M. Orwick-Rydmark, K. Märker, W. T. Franks, D. Akhmetzyanov, D. Stöppler, M. Zinke, E. Specker, M. Nazaré, A. Diehl, B.-J. van Rossum, F. Aussenac, T. Prisner, Ü. Akbey and H. Oschkinat, Temperature dependence of cross-effect dynamic nuclear polarization in rotating solids: advantages of elevated temperatures, Phys. Chem. Chem. Phys., 2016, 18, 30696–30704.
54. E. M. M. Weber, H. Vezin, J. G. Kempf, G. Bodenhausen, D. Abergl and D. Kurzbach, Anisotropic longitudinal electronic relaxation affects DNP at cryogenic temperatures, Phys. Chem. Chem. Phys., 2017, 19, 16087–16094.
55. D. Shimon, Y. Hovav, A. Feintuch, D. Goldfarb and S. Vega, Dynamic nuclear polarization in the solid state: a transition between the cross effect and the solid effect, Phys. Chem. Chem. Phys., 2012, 14, 5729–5743.
56. D. P. Lin, D. F. Feng, F. Q. H. Ngo and L. Kevan, Electron-electron double resonance study of magnetic energy transfer between trapped electrons and radicals in organic glasses: Relation between dipolar cross relaxation times and dipolar interaction distances, J. Chem. Phys., 1976, 65, 3994–4000.
57. M. Romanelli and L. Kevan, Evaluation and interpretation of electron spin∼echo decay part I: Rigid samples, Concepts Magn. Reson., 1997, 9, 403–430.
58. A. D. Milov, K. M. Salikhov and Y. D. Tsvetkov, Effect of Spin Dipole-Dipole Interaction on Phase Relaxation in Magnetically Dilute Solid Bodies, Zh. Eksp. Teor. Fiz., 1972, 63, 2329–2335.
59. J. Klauder and P. Anderson, Spectral Diffusion Decay in Spin Resonance Experiments, Phys. Rev., 1962, 125, 912–932.
60. D. T. Edwards, S. Takahashi, M. S. Sherwin and S. Han, Distance measurements across randomly distributed nitroxide probes from the temperature dependence of the electron spin phase memory time at 240 GHz, J. Magn. Reson., 2012, 223, 198–206.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8cp05930f

---