The electron depolarization during dynamic nuclear polarization: measurements and simulations

Abstract

Dynamic nuclear polarization is typically explained either using microscopic systems, such as in the solid effect and cross effect mechanisms, or using the macroscopic formalism of spin temperature which assumes that the state of the electrons can be described using temperature coefficients, giving rise to the thermal mixing mechanism. The distinction between these mechanisms is typically made by measuring the DNP spectrum – i.e. the nuclear enhancement profile as a function of irradiation frequency. In particular, we have previously used the solid effect and cross effect mechanisms to explain temperature dependent DNP spectra. Our past analysis has however neglected the effect of depolarization of the electrons resulting from the microwave (MW) irradiation. In this work we concentrate on this electron depolarization process and perform electron–electron double resonance (ELDOR) experiments on TEMPOL and trityl frozen solutions, using a 3.34 Tesla magnet and at 2.7–30 K, in order to measure the state of the electron polarization during DNP. The experiments indicate that a significant part of the EPR line is affected by the irradiation due to spectral diffusion. Using a theoretical framework based on rate equations for the polarizations of the different electron spin packets and for those of the nuclei we simulated the various ELDOR line-shapes and reproduced the MW frequency and irradiation time dependence. The obtained electron polarization distribution cannot be described using temperature coefficients as required by the classical thermal mixing mechanism, and therefore the DNP mechanism cannot be described by thermal mixing. Instead, the theoretical framework presented here for the analysis of the ELDOR data forms a basis for future interpretation of DNP spectra in combination with EPR measurements.

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1 Introduction

The inherently low signal to noise ratio detected in NMR can be dramatically increased by the use of dynamic nuclear polarization (DNP). In this technique the large polarization of unpaired electrons, mainly of stable radicals, is transferred to their neighboring nuclei, typically using continuous wave (cw) microwave (MW) irradiation. This technique has been known for more than 60 years,^1–3 but it has only recently become of high interest to the NMR and MRI community due to the introduction of new technological and methodological developments. These enabled DNP in the solid state to be combined with magic angle spinning (MAS) NMR,^4–10 or with the dissolution DNP methodology,¹¹ which can be combined with high resolution liquid state NMR and MRI.

The fundamental mechanism of DNP is of interest both from the point of view of the basic spin physics, which differs from most pulsed magnetic resonance experiments due to its non-coherent nature, and from a practical point of view – allowing better understanding of the processes that can lead to higher NMR and MRI signals. Three main mechanisms were used over the years to explain DNP in non-conducting solid samples: the solid effect (SE),³ the cross effect (CE)^12–16 and the thermal mixing (TM)^17–19 mechanisms. The SE and CE mechanisms were originally explained based on rate equations for the electron and nuclear polarizations, derived from the microscopic quantum mechanical (QM) description of small spin systems, and taking into account the effects of irradiation using a perturbation approach²⁰ and the effects of relaxation. This QM description was recently extensively studied for DNP on static^21–27 and rotating^28–30 samples. Alternatively, the electron and nuclear polarizations can be considered using the macroscopic concept of spin temperature,^20,31–33 as used in the TM and SE^17,34,35 mechanisms in systems with EPR lines broader or narrower than the nuclear Larmor frequency, respectively. In this case, the state of the spin system is described using several temperature baths, each associated with a part of the system's Hamiltonian and characterized by its own spin temperature. In particular, the electron polarization in inhomogeneously broadened EPR lines is described using an electron Zeeman temperature and an electron non-Zeeman temperature, with the latter corresponding to the change in the electron polarization across the EPR line.

The DNP mechanism is typically studied based on measurements of the nuclear enhancement as a function of the MW frequency, termed the DNP spectrum. For radicals with an EPR line-width larger than the nuclear Larmor frequency these are interpreted either using the TM mechanism^18,35–44 or a combination of the SE and CE mechanisms.^45–47 There are only a few cases where the existence of spin temperature was directly detected experimentally.^48–50 In addition, the DNP results were only rarely analyzed together with EPR data.^49,51 In particular, while the SE and CE based analysis of the DNP line-shapes results in good fitting of the experimental line shapes, this model did not take into account the non-thermal distribution of the electron polarization within the EPR line-shape during DNP. This can influence the resulting DNP enhancement when different from its thermal value.²¹ As such, much insight into the DNP mechanism can be gained by first measuring the electron polarization distribution during MW irradiation (sometimes called the electron saturation profile).^51–55 The latter can be explored using electron–electron double resonance⁵⁶ (ELDOR) experiments, where the change in the EPR signal intensity due to a MW irradiation at different frequencies is observed. Such an experiment was performed by Granwehr and Köckenberger⁵² on DNP samples containing TEMPOL and trityl radicals, using longitudinally detected EPR at 1.5 K and 3.34 Tesla.

The electron polarization distribution is expected to change from its thermal value due to the MW irradiation on the electronic transitions, and because of the spread of polarization across the EPR line due to the electron spectral diffusion (eSD) process. The latter is used in the context of the TM mechanism to explain the formation of the electron non-Zeeman temperature, when its rate is faster than the electron spin lattice relaxation rate.^18,19,50,57 In addition, it is of much interest in EPR,^58–62 mainly in the context of time dependent echo detection, where it is explained based on electron spin flips, which induce stochastic fluctuating dipolar fields at the positions of their neighboring electrons. These electron spin flips are induced by dipolar flip-flop quantum fluctuations, the electron spin lattice relaxation mechanisms, or by hyperfine fluctuations induced by spin diffusion.^20,62

In this paper the electron polarization distribution of TEMPOL and trityl containing samples under DNP conditions is monitored at 2.7–30 K, using detailed ELDOR measurements performed on a 3.34 Tesla combined pulsed-EPR and NMR spectrometer.⁶³ The ELDOR line-shapes are analyzed using a theoretical model based on rate equations for the polarizations of the electrons and nuclei in the system, resulting in a good agreement between the simulations and the experimental line-shapes. This model includes a QM based description of the electron polarization loss due to (i) MW irradiation on the single quantum (SQ) electronic transitions, and due to (ii) SE DNP type of polarization transfer to the nuclei induced by irradiation on the electron-nucleus zero quantum (ZQ_en) and double quantum (DQ_en) transitions. In addition it includes the effects of (iii) spin–lattice relaxation as well as (iv) a phenomenological description of the spread of the electron polarization due to the eSD process. The influence of the electron polarization redistribution on the DNP mechanism and specifically on the DNP spectral profiles will be described in upcoming publications. Finally, we show that the measured electron polarization distribution cannot be described using the spin temperature coefficients, which forms the basis of the classical TM mechanism.^{18,35,44,50,64}

2 Experimental

2.1 Sample preparation

Samples were prepared using 15 mM trityl (OX63) radical and 40 mM TEMPOL radical, dissolved in 56/44 wt% mixtures of dimethyl sulfoxide (DMSO)/H₂O. 25 μl of each solution were placed in a Teflon sample holder for measurement. TEMPOL and DMSO were purchased from Sigma-Aldrich, and the trityl radical from Oxford Instruments.

2.2 The NMR/EPR spectrometer

The experiments were conducted in a 3.34 T field using a combined EPR (95 GHz) and NMR (144 MHz) spectrometer,⁶³ under experimental conditions which were previously used to study DNP enhancements and line-shapes of similar samples.^45–47,65 This setup is very flexible in terms of the range of possible sample temperatures and of the MW irradiation duration, and in terms of the range of available excitation and detection frequencies. MW excitation and detection were performed using a home-built bridge with two channels, controlled by the Specman4EPR software package.⁶⁶ The radio frequency (RF) excitation and detection are controlled by an APOLLO spectrometer from Tecmag Inc., externally triggered by a logic pulse generated by Specman4EPR. The probe-head and sample were located inside a Janis Research Inc. liquid helium flow cryostat, and experiments were performed in the temperature range of 2.7–30 K. Reaching 2.7 K required a reduced pressure in the cryostat, which was obtained by pumping.

ELDOR experiment. The ELDOR pulse sequence used is shown schematically in Fig. 1. The experiments were conducted by applying a cw-MW excitation pulse at a frequency ω_excite for a duration of t_excite using one of the two EPR channels. This was followed by an EPR echo detection using a pulse sequence, with pulses of 450 ns, produced by the second channel at ω_detect. Prior to each experiment a train of RF pulses was applied on the protons in order to remove their polarization between successive scans. As a result the delay between single experiments could be set to about 5T_1e, where T_1e is defined below. The MW amplitude, ω₁/2π, was 600 kHz, as measured by a nutation experiment. Due to the low Q of our resonator we assume that this value is uniform over the frequency range used in the experiments. For the experiments performed around 2.7 K the power was reduced, ω₁/2π = 30 kHz, in order to limit sample heating within a range of 0.5 K.

Fig. 1 A schematic representation of the ELDOR sequence used. This included a saturation pulse train on the ¹H nuclei, followed by MW irradiation at ω_excite for a duration of t_excite, and a detection of an echo at a frequency of ω_detect. This sequence was then repeated with a delay between scans of about 5T_1e or longer.

The detected signals for a given ω_excite and t_excite values can be expressed in terms of the electron polarization of the detected electrons prior to the application of the detection pulses, P_e(ω_excite,t_excite,ω_detect), using:

S(ω_excite,t_excite,ω_detect) = s(ω_detect)P_e(ω_excite,t_excite,ω_detect) + b(ω_detect), (1)

where b(ω_detect) is a baseline artifact contribution to the echo (which can also depend on t_excite), which probably originates from pulse ringing. s(ω_detect) represents the proportionality factor between the actual signal and the electron polarization, and depends on the electron concentration and volume, the fraction of electrons contributing to the detected signal (which is EPR line-shape dependent), and the Q factor of the probe. It is possible to normalize the detected signal with respect to the thermal equilibrium signals,⁵²S₀. These can be detected using the same experimental parameters but with excitation frequencies ω_excite′ that are situated far outside the EPR spectrum, such that the irradiation does not have any direct or indirect effect on the detected electrons. In practice, S₀(ω_excite′,t_excite,ω_detect) was evaluated using signal averaging over several ω_excite′ values. When b(ω_detect) is small compared to the signals, S₀ can be used to evaluate the relative saturation of the electrons, E_e = P_e(t)/P_e(0), where P_e(0) is the thermal electron polarization. For electrons positioned around ω_detect and MW irradiation at ω_excite for a duration of t_excite this is given by:

E_e(ω_excite,t_excite,ω_detect) = S(ω_excite,t_excite,ω_detect)/S₀(ω_excite′,t_excite,ω_detect). (2)

when E_e(ω_excite,t_excite,ω_detect) is one it means that the MW field at ω_excite does not affect the detected electrons, and when it is zero it indicates full saturation. When b(ω_detect) is non-negligible the estimation of S₀ in eqn (2) is not accurate, which will result in E_e(ω_excite,t_excite,ω_detect) ≠ 0 under full saturation. For clarity, we will define E_excite and E_detect as the E_e frequency profile obtained using a constant t_excite value for either a constant ω_detect or a constant ω_excite values, respectively. The former shows the dependence of the detected signals on ω_excite, as given in a regular echo detected ELDOR experiment, and the latter shows the polarization of the different electrons for a given ω_excite frequency. Note that E_detect requires measurements at both ω_excite and ω_excite′, for normalization.

The actual intensity of the EPR spectrum S*(ω_detect;ω_excite,t_excite) at the end of a MW irradiation of duration t_excite at frequency ω_excite can be calculated by multiplying E_detect(ω_detect;ω_excite,t_excite) by the EPR line intensity f(ω_detect) for each ω_detect:

S*(ω_detect;ω_excite,t_excite) = f(ω_detect) × E_detect(ω_detect;ω_excite,t_excite). (3)

This value is proportional to the total electron polarization around the detected frequency.

2.3 T ₁ measurements

The electron and nuclear relaxation times, T_1e and T_1n, were determined by analyzing the recovery of the echo intensity after signal saturation by a long cw MW pulse (using t_excite > T_1e) or a RF pulse train, respectively. The NMR recovery curves could be analyzed by single exponential functions with a time constant T_1n. The intensity of the EPR echo signals were recorded and fitted using either a single exponential or when required using a double exponential function with the slow (and major) component assigned to T_1e. The electron relaxation times were only collected around the frequencies of the maximum EPR line intensity.

3 Experimental results

In this section we show the results of ELDOR measurements on samples containing TEMPOL and trityl radicals as a function of t_excite, ω_excite, and ω_detect. In particular, E_detect profiles were measured using t_excite irradiation times of the order of T_1e in order to allow for a possible electron non-Zeeman temperature to form in the system. The spin–lattice relaxation rates of these samples were measured and are summarized in Table 1. For convenience, the values of ω_excite and ω_detect will be given in frequency units relative to a fixed reference frequency, ω_c, using δν_excite = (ω_excite − ω_c)/2π and δν_detect = (ω_detect − ω_c)/2π. The value of ω_c/2π was set to 95 GHz in the TEMPOL case and to 94.86 GHz in the trityl case, such that ω_c is about equal to the center frequency of the EPR line.

Table 1 The electron and nuclear relaxation times of the two samples measured at the different temperatures. The lengths of the saturation pulses, t_excite, used during the T_1e measurements are also given

Radical	TEMPOL			Trityl
Temperature	20 K	7 K	2.7 K	30 K	2.7 K
T 1e (ms)	5.5	50	240	32	270
T 1n (s)	13	30	220	14	250
t excite (ms)	20	100	1000	100	1000

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3.1 TEMPOL sample

The ELDOR measurements on a 40 mM TEMPOL solid solution were performed at 20 K, 7 K, and 2.7 K. Here the baseline distortions b(δν_detect) were significant with respect to the detected echo intensities, causing the detected echo signals to be different from zero even at full saturation conditions, resulting in a distortion of the E_e values calculated using eqn (2). Fig. 2a shows a contour of E_e(δν_excite,t_excite,δν_detect) obtained from ELDOR signals for different δν_detect and δν_excite frequencies at 20 K, using t_excite = 10 ms. Individual E_excite and E_detect profiles are plotted in Fig. 2b and c, respectively. The latter have a lower spectral resolution and are therefore more susceptible to noise. In addition they are affected by changes in the experimental b(δν_detect) values. The E_excite profiles show a sharp minimum at δν_excite = δν_detect, and a broad minimum around δν_excite = 0. The first minimum is a result of on-resonance MW saturation and the second minimum is attributed to the eSD mechanism. These broad minima appear in the E_detect profiles as an increase of their slopes around δν_excite = δν_detect when δν_excite is moving away from the center of the EPR line. In other words, irradiating at the center affects most of the electron packets composing the EPR line, while irradiation at the side affects only neighboring packets. In addition to these minima, some depolarization is observed for irradiation outside of the EPR line (see for example arrows in Fig. 2b). This is attributed to the polarization loss to the ¹H nuclei in the sample due to irradiation on DQ_en or ZQ_en transitions. The result of MW irradiation on the expected EPR line intensity S*(ω_detect;ω_excite,t_excite), defined in eqn (3), is plotted in Fig. 2d. As can be seen, irradiation close to the center of the EPR line results in a large reduction of the total electron polarization, whereas irradiation at the side of the spectrum has a relative small effect on it.

Fig. 2 Normalized electron polarization E_e(δν_excite,t_excite,δν_detect) measured for the 40 mM TEMPOL sample at 20 K, using t_excite = 10 ms. In (b) and (c) several E_excite and E_detect spectra are plotted, as taken from the full E_e contour shown in (a). This was done for several δν_detect and δν_excite values, respectively, with the values given by the color-code in (b). In (d) the expected signal intensity S* is plotted using the values obtained from (c), as given in eqn (3). The EPR line, taken from ref. 45, is plotted in gray in (b) and (d). The black arrows in (b) indicate depolarizations due to irradiation outside the EPR spectrum. All frequencies are given with respect to 95 GHz.

In Fig. 3E_excite spectra obtained for different t_excite values using δν_detect = 0 are plotted. For short irradiation times the MW irradiation results in a relatively narrow hole which is burned in the E_excite spectrum around δν_excite = δν_detect, due to MW irradiation on the SQ transitions. In addition to this some depolarization can be observed when irradiating around δν_excite = δν_detect ± 144 MHz, corresponding to polarization loss to the ¹H nuclei when irradiating on DQ_en and ZQ_en transitions of the detected electrons. For increasing t_excite values the widths of the profile increases and after 50 ms it decreases again to some extent. A steady state profile is reached in a time scale of about 1 s, which is much longer than T_1e, indicating that this narrowing effect involves polarization transfer to the ¹H or ¹⁴N nuclei in the sample.⁵³ This process is shorter than the nuclear polarization buildup time of the order of 13 s. The ELDOR results obtained at long t_excite times were recorded using a single scan, and therefore show a reduced signal to noise ratio (SNR).

Fig. 3 Measured E_excite(δν_excite) spectra at different t_excite values of the 40 mM TEMPOL sample at 20 K. A detection frequency of δν_detect = 0 was used. The t_excite values are defined by the color coding inside the figure, and all frequencies are given with respect to 95 GHz.

In Fig. 4E_excite spectra measured at 7 K (a, b) and 2.7 K (c, d) are plotted (a, c) for several δν_detect frequencies, with t_excite = 0.1 s at 7 K and 0.5 s at 2.7 K; and (b, d) for several t_excite values and δν_detect = 0. Once again, at 2.7 K the MW power was reduced and was set to about ω₁/2π = 30 kHz. As can be clearly observed, the width of the enhancement profiles broadens with the decrease in temperature, with the effect of the eSD process becoming strong enough to almost saturate the whole EPR line for MW fields applied at any δν_excite frequency inside the EPR line. As before, the saturation profiles reach their maximal width in a timescale comparable to T_1e, after which they show a slight narrowing.

Fig. 4 Measured E_excite(δν_excite) spectra of the 40 mM TEMPOL sample at 7 K and 2.7 K. In (a) the profiles obtained at 7 K using t_excite = 100 ms are plotted for different δν_detect values, defined by the color coding inside the figure. In (b) E_excite(δν_excite) profiles obtained at 7 K using δν_detect = 0 are plotted for increasing t_excite values, defined by the color coding inside the figure. In (c) and (d) similar profiles are plotted as in (a) and (b) but at 2.7 K, ω₁/2π = 30 kHz instead of 600 kHz, and using t_excite = 500 ms in (c). All frequencies are given with respect to 95 GHz.

3.2 Trityl sample

The ELDOR measurements on a 15 mM trityl solid solution were performed at 30 K and 2.7 K. The detected echoes were much higher than the baseline artifacts b(ν_detect), thus the latter could be neglected. In Fig. 5a the results of the ELDOR experiments at 30 K are summarized by plotting the contour E_e(δν_excite,t_excite,δν_detect) for t_excite = 0.1 s. The ELDOR results can be divided into three δν_excite frequency regions: the SQ center region, where the MW irradiation frequencies cover the frequency span of the EPR line, result in a direct saturation of the electrons; and the DQ_en and ZQ_en frequency regions, which are removed by 144 MHz from the center region, where irradiation on the electron – ¹H DQ_en and ZQ_en transitions results in electron polarization loss due to SE type of polarization transfer to the surrounding nuclei.⁵³ In addition to this, the polarization loss can spread through the EPR line due to eSD. Polarization loss in the SQ region can also occur due to irradiation on the electron – ¹³C DQ_en and ZQ_en transitions, influenced by the hyperfine interaction of the electrons with the natural abundance ¹³C nuclei of the trityl radicals themselves.⁶⁷ This will only be considered in the ESI.†

Fig. 5 Normalized electron polarization E_e(δν_excite,t_excite,δν_detect) measured for the 15 mM trityl sample at 30 K, using t_excite = 100 ms. In (b) and (c) several E_excite and E_detect spectra are plotted, as taken from the full E_e contour shown in (a). This was done for several δν_detect and δν_excite values, respectively, with the values given by the color-code in (b). In (d) the expected signal intensity S* is plotted using the values obtained from (c), as given in eqn (3). The EPR line, taken from ref. 46, is plotted in gray in (b) and (d). The arrows in (b) show the position of the electron hyper- and de-polarization features, which are not taken into account in the simulations, as explained in the text. All frequencies are given with respect to 94.86 GHz.

We first examine several E_excite profiles, as plotted in Fig. 5b. Starting from MW irradiation in the SQ region, clear minima are observed when δν_excite ≅ δν_detect and in addition broader minima appear at δν_excite values between δν_detect and zero. Small depolarization and hyperpolarization (E_detect > 1) effects are also observed when detecting around the edges of the EPR spectrum, as indicated by the black arrows in the figure. The obtained profiles are about antisymmetric with respect to δν_detect = 0, reflecting the low anisotropy of the EPR line. A similar behavior as in the SQ region can be seen when inspecting the smaller depolarization at the DQ_en and ZQ_en regions. The extra depolarization and hyperpolarization features are not observed here, possibly due to low SNR.

Several E_detect profiles are plotted in Fig. 5c, with δν_excite located inside the SQ region. Single minima are found at δν_excite ≅ δν_detect, and the shapes of these profiles around these minima have larger slopes towards the center EPR frequency than toward the edges of the EPR spectrum. In addition, the inward slopes increase when the |δν_excite| frequencies move away from zero. Once again, electron hyper-polarizations are observed for high |δν_detect| and |δν_excite| values, and the results show an inverse symmetry with respect to δν_excite = 0.

The expected EPR signal intensity S*(ω_detect;ω_excite,t_excite) as a function of δν_detect for different δν_excite values are plotted in Fig. 5d. In this non-normalized representations the electron hyperpolarization is hardly recognized. Thus, at this stage we ignore these small features, and discuss possible mechanisms leading to these effects in the ESI.†

In Fig. 6E_excite profiles are plotted for several t_excite values, with δν_detect equal to (a) 10 MHz or to (b) −30 MHz. In both cases the irradiation initially burns a narrow hole in the EPR line due to irradiation on the electron SQ transitions or the electron – ¹H DQ_en and ZQ_en transitions, which broadens with time. The steady state polarization distribution is reached in the timescale of the order of T_1e.

Fig. 6 Measured E_excite(δν_excite) spectra at different t_excite values of the 15 mM trityl sample at 30 K. The detection was performed at a δν_detect of (a) 10 MHz and (b) −30 MHz. All frequencies are given with respect to 94.86 GHz.

In Fig. 7a the experimental E_e(δν_excite,t_excite,δν_detect) values obtained at 2.7 K are shown, with t_excite = 1 s. Once again, here the MW intensity was much lower (30 kHz) than during the measurement at 30 K. The E_excite profiles (Fig. 7b) again show a minimum at δν_excite = δν_detect and an additional broad minimum closer to δν_excite = 0. Some hyperpolarization can be seen when ν_detect is close to zero, as indicated by the arrows in the figure, which goes together with a narrowing of the profile. The minima in the ZQ and DQ regions are all at about the same frequency, with increasing electron depolarization for larger |δν_detect| values.

Fig. 7 Normalized electron polarization E_e(δν_excite,t_excite,δν_detect) measured for the 15 mM trityl sample at 2.7 K, using t_excite = 1 s. In (b) and (c) several E_excite and E_detect spectra are plotted, as taken from the full E_e contour shown in (a). This was done for several δν_detect and δν_excite values, respectively, with the values given by the color-code in (b). In (d) the expected signal intensity S* is plotted using the values obtained from (c), as given in eqn (3). The EPR line, taken from ref. 46, is plotted in gray in (b) and (d). The arrows in (b) show the position of the electron hyperpolarization. All frequencies are given with respect to 94.86 GHz.

The E_detect profiles, plotted in Fig. 7c, again show global minima around δν_detect = δν_excite; however here there is also a local maximum around δν_detect = 0, with lower electron polarization around it. This may indicate that electron packets which are removed in frequency are connected to one another by both eSD and another mechanism. Possible explanations for this are proposed in the ESI.† The virtual symmetry axis of these profiles is slightly shifted with respect to the 30 K profiles, and is at about δν_excite = 5 MHz. The real magnitudes of these effects are relatively small when considering the actual signal, as seen in Fig. 7d.

4 Spin temperature and the experimental results

As mentioned in the Introduction, one of the fundamental assumptions of the TM mechanism is that the distribution of the electron polarizations can be described using a Zeeman temperature, T_eZ, associated with the weight averaged center electron frequency ω_e,0, and an additional electron non-Zeeman spin temperature, T_enZ, for the parts of the Hamiltonian which contribute to the inhomogeneous EPR line shape, namely the hyperfine with strongly coupled nuclei and the g-anisotropy.^{18,19,35,38,44,50,57,64} The direct contribution of the dipolar interaction to T_enZ is assumed to be negligible. However, it plays an important indirect role as the source of the eSD process, which enables the creation of the T_neZ temperature in the system when its rate is higher than T_1e⁻¹.^18,19,50,57 Under these assumptions, for a long enough MW irradiation the polarization of the electrons P_e(ω_j) at a frequency ω_j = ω_e,0 + δω_e,j can be described in the ω_excite MW rotating frame as:where ℏ and k_B are the Planck and Boltzmann constants, and Δω_e = ω_e,0 − ω_excite. These P_e(ω_j) values result in a monotonic change in E_detect, with a slope proportional to T_enZ⁻¹ that can be positive or negative depending on the value of ω_excite.

We can next compare this prediction to the experimentally detected E_detect profiles. These profiles, obtained for the TEMPOL sample at 20 K and for the trityl sample at 30 K and 2.7 K as shown in Fig. 2c, 5c and 7c, respectively, do not follow the expected monotonic change in P_e(ω_j). Instead they show a clear minimum around ω_detect = ω_excite and an increase of the polarizations on both sides of these conditions. In particular, the results shown in Fig. 2d, 5d and 7d can be compared with the measurements shown in ref. 48–50, where spin temperature was observed. The E_excite contours detected at 7 K and 2.7 K and plotted in Fig. 4a and c show only very little dependence on δν_detect, which could perhaps justify the introduction of very large T_enZ values. Yet in the ESI† we show that at 7 K such a T_enZ does not exist. At 2.7 K further measurements are needed to verify the existence of T_enZ. Inspection of the ELDOR results obtained by Granwehr and Köckenberger⁵² at 1.5 K leads to the same conclusion, namely that the system cannot be described by using T_eZ and T_enZ temperatures for describing the E_detect profiles.

5 Theoretical analysis of the ELDOR spectra

In this section we present a model for the time evolution and frequency dependence of the electron polarization that accounts for the major parts of the experimental ELDOR results. It takes into account the effects of the spin–lattice relaxation, MW irradiation (including off-resonance effects), and eSD. In addition, the influence of the nuclei on the electrons, as was seen in the experiment, is taken into account by relying on the QM descriptions of the SE DNP processes in small spin systems.²¹ A justification for relying on the spin dynamics of small model spin systems to capture the physics of large and complex spin systems can be found in ref. 45–47. Although the CE mechanism can have a large influence on the nuclear polarization^45–47 we did not take it into account in this model. This can be justified by the fact that during ELDOR measurements the electron polarizations do not show significant changes in timescale of the DNP enhancement buildups, and since only a small fraction of the electrons satisfy the CE condition.^25,26 The model presented here does not rely on spin thermodynamics related parameters, such as spin temperature, energy conservation, and heat capacities of spin baths.

5.1 Theoretical model

The model presented here describes the temporal evolution of the electron and nuclear polarizations in our systems. The electron spins in these systems are assumed to have an inhomogeneous broadened spectrum due to the anisotropy of the g-tensor. As such, in the present discussion we ignore for simplicity the fact that some of the EPR features can originate from hyperfine interactions, as in the TEMPOL case. As in previous theoretical models^{16,18,44,51,64,68–70} we divide the electrons into a set of electron packets with constant average resonance frequencies, ω_j, and define an average electron polarization for each packet, P_e,j. The relative weights of the packets, f_j, are determined by normalization of the integral of the EPR line, ∑f_j = 1. As such, the detected EPR signal at a frequency ω_j is proportional to N_ef_jP_e,j, where N_e is the number of unpaired electrons in the system. To add the effect of the nuclei on the electrons we distinguish between two nuclear types:⁴⁷ (i) the “local” nuclei, with average polarizations P_n,j, which are directly or indirectly dipolar coupled to the electrons at ω_j, and which get polarized due to irradiation on the DQ_en and ZQ_en transitions; and (ii) the remote “bulk” nuclei, with an average polarization of P_bulk, which get polarized due to spin diffusion with the local nuclei. The latter determine the signals during DNP experiments.

The general form of the set of homogeneous rate equations for the electron and nuclear polarizations, for a system with N frequency bins (j = 1,…,N) can be represented as:

The rate matrix R in these coupled equations contains the actions of the electron (R_1e) and nuclear (R_1n) spin–lattice relaxation, the MW irradiation (R_MW), the nuclear spin diffusion (R_nSD), and the eSD (R_eSD) processes:

R = (R_1e + R_1n + R_MW + R_nSD + R_eSD). (6)

The resulting polarizations can then be calculated by solving these equations for a given MW irradiation frequency ω_excite and time, t_excite,

The explicit form of this set of equations is derived in what follows by summing up rate equations affecting one or two of these populations, derived from the rate equations for the populations in small systems, as described in ref. 21 and 22. To transfer such rate equations on the p_α and p_β populations of a single spin ½ to its corresponding polarization, P = p_β − p_α, we use the transformation:

which conserves the total population of the system, p_α + p_β = 1. In analogy, the populations of an electron–nuclear two-spin system, p_αe,αn, p_αe,βn, p_βe,αn, p_βe,βn, (assuming a nucleus with spin ½) can be transferred to the P_e = −p_αe,αn − p_αe,βn + p_βe,αn + p_βe,βn and P_n = p_αe,αn − p_αe,βn + p_βe,αn − p_βe,βn polarizations, the difference polarization P_en = p_αe,αn − p_αe,βn − p_βe,αn + p_βe,βn and the conserved sum polarization 1 = p_αe,αn + p_αe,βn + p_βe,αn + p_βe,βn:

Here we have neglected the effect of the small state mixing that is present in a electron–nuclear system due to the non-secular hyperfine interaction. This effect is compensated by the introduction of the effective DQ_en and ZQ_en irradiation derived from the MW irradiation in the case of the SE. The transformation results in two time independent polarization components 1 and P_en and the latter can therefore be neglected. This approach is obviously a simplified version of the calculations of the quantum dynamics in a many-spin system.^{21,23,25,26,71}

5.1.1 The spin–lattice relaxation rates. The electron spin–lattice relaxation rate matrix is given by R_1,e = ∑R^j_1e, where R^j_1e are matrices with non zero-elements of the form in the subspace,^72,73 where P⁰_e,j = tanh(ω_e,jℏ/2k_BT_L) are the thermal equilibrium polarizations of the electrons, and T_L is the lattice temperature. In analogy, the nuclear spin–lattice rate matrix of the nuclei is defined as R_1,n = ∑R_1n,j + R_1bulk, where R_1n,j are the non-zero elements given by in the subspace, and the bulk relaxation rate matrix R_1bulk has the form in the subspace. The equilibrium polarization of the nuclei equals P⁰_n = tanh(ω_nℏ/2k_BT_L), where ω_n is the nuclear Larmor frequency, and for simplicity we assume that the nuclei have the same sign of the gyromagnetic ratio as for ¹H nuclei.

In the present model we do not explicitly consider electron–nuclear cross-relaxation processes. We realize, however, that during calculations the effects of cross-relaxation processes can in many instances be mimicked by combining electron and nuclear relaxation processes.²¹

5.1.2 The MW irradiation. The MW irradiation field, of strength ω₁ and applied at a frequency ω_excite, can excite in our model the SQ transitions, the DQ_en, and the ZQ_en transitions. To make a distinction between the three types of contributions to the rate matrix R_MW representing the MW irradiation effect we write:

R_MW = ∑R^j_SQ + ∑R^j_DQ + ∑R^j_ZQ. (10)

The R^j_SQ matrices have a single diagonal element operating on P_e,j with a value

where T_2e is the electron spin–spin relaxation time of the system (rather than the phase memory time measured in time domain EPR) and Δω_j = ω_j − ω_excite is the off resonance value of the electron packet at ω_j. The expressions for these rates rely on the assumption that |ω₁| ≪ |T⁻¹_2e + iΔω|.^21,22 When this does not hold this will result in a saturation time constant that is larger than T_2e.

The MW irradiation on the DQ_en or ZQ_en transitions of an electron coupled to neighboring nuclei becomes allowed due to the state mixing induced by the pseudo-secular parts of their hyperfine interactions. This irradiation results in a SE process polarizing the nuclei and is represented by the matrix elements of R^j_ZQ/DQ. This DQ_en or ZQ_en irradiation results in an equilibration of the polarizations of the electron and the nuclei.²¹ If on average an electron in bin j transfers its polarization to η local nuclei, and assuming the polarizations of these nuclei become all equal due to fast spin diffusion rates or dipolar state mixing, the MW irradiation will result in the ideal case in a steady state polarization of the electron given by P_e,j = ±P_n,j = P⁰_e,j/(η + 1). To accomplish this, the sub-matrices R^j_DQ and R^j_ZQ in the subspace have the general form and , respectively. The actual steady state polarization of the electrons and nuclei during the MW irradiation will be influenced not only by the action of these DQ and ZQ effective irradiation, but also by the relaxation times of the system.²¹ The values of the w^DQ/ZQ_j rates can be estimated using perturbation theory on the Bloch equations^21,22 as

where Ā^± is some effective hyperfine coefficient that corresponds to the average root mean square sum of the pseudo-secular hyperfine term of the nuclei polarized by each electron.

Although here the bulk nuclei are polarized via the local nuclei (i.e. their hyperfine interaction value is taken to be zero), it is possible to conduct the simulations by considering both of these nuclei together. This is justified for fast spin diffusion rates. In such a case only the bulk polarization should be taken into account in eqn (5), and R_DQ/ZQ in eqn (10) should be substituted by R^bulk_DQ/ZQ = ∑R^j,bulk_DQ/ZQ with rate matrices R^j,bulk_DQ/ZQ with elements and in the subspace . The expressions for w^DQ/ZQ_j,bulk are the same as in eqn (12) by replacing Ā^± by an effective bulk hyperfine coefficient Ā^±_bulk. The η^b_j parameter must be adjusted to the average number of polarized bulk nuclei per electron, which can be defined by η^b_j = C_n/f_e,jC_e, where C_e and C_n are the concentrations of the electrons and nuclei, respectively, and f_e,jC_e is the concentration of the electrons with a frequency ω_j. Once again, here we will consider the bulk nuclei to be polarized via spin diffusion by the local nuclei, as explained in what follows.

5.1.3 Spin diffusion. The spin diffusion process, which transfers polarization between the local and bulk nuclei, is introduced using a rate matrix R_nSD. This matrix is given by R_nSD = ∑R^j_nSD, with R^j_nSD having non-zero elements in the subspace,^47,74 where w^nSD is the effective nuclear spin diffusion rate between neighboring nuclei and η^b_j/η is the ratio between the number of bulk and local nuclei.

5.1.4 Spectral diffusion. Finally, the spectral diffusion rate matrix R_eSD must be added to the rate equations. Deriving the general form of this matrix, we make the following assumptions: (i) the eSD process can be described by introducing effective (single) rate constants defining the polarization transfer process between pairs of bins. (ii) At thermal equilibrium the polarizations satisfy the Boltzmann ratio, . And (iii), the eSD process conserves the total polarization of the electron spin system, . As a result our eSD model is not energy conserving, as is required in the spectral diffusion process considered in the theory leading to the TM mechanism.⁷⁵ Based on these assumptions, the polarization exchange process described by the R_eSD matrix can be expressed as a sum of rate matrices, , which have non-zero elements of the formin the subspace. For the polarization exchange rate, and in particular its dependence on the frequency difference between the bins, we choose the from:where Λ^eSD is a fitting parameter. A justification for expressing the rates in this form is given in Appendix A. As will be shown in the next section, this form enabled us to analyze all detected ELDOR spectra, indicating the validity of this (ω_j − ω_j′) dependence.

In order to characterize the effect of the R_eSD rate matrix we can define a characteristic polarization exchange rate constant by considering the exchange rate between the polarizations of the electrons in neighboring bins j_max and j_max−1 with the largest f_j values, f_max and f_max−1:

5.2 Simulations of the experimental data

In order to compare the simulations with the experimental results, the electron polarizations were calculated using eqn (7), with the initial values of the polarization set to their thermal values for the electrons and set to zero for the nuclei, where the latter mimics the effect of ideal saturation of the nuclei. The simulated saturation values of the different electrons can then be obtained using

E_e(ω_excite,t_excite,ω_j) = P_e,j(ω_excite,t_excite)/P_e,j(0), (16)

in analogy to eqn (2) with ω_detect = ω_j. The E_excite and E_detect profiles are defined, in analogy to the experimental profiles, as the E_e value of the single electron packet at ω_j = ω_detect as a function of the excitation frequency ω_excite, and as the E_e profile of the different electron packets obtained using a single ω_excite value, respectively, and for a given t_excite value. The simulated profiles were compared with experimental E_e profiles, and the fitted values were determined using a visual fitting procedure, by varying the parameters defining the matrix elements of the rate matrix in eqn (7). In addition, the expected EPR signal can be calculated using

S*(ω_j;ω_excite,t_excite) = f_j·E_detect(ω_j;ω_excite,t_excite), (17)

in analogy with eqn (3).

The parameters entering the total rate matrix can be divided into two groups: those that are experimentally determined – i.e. the T_1e and T_1n,bulk relaxation times, the values of f_j, the MW irradiation strength ω₁ and the concentrations of the electrons and bulk nuclei, C_e and C_n; and those that we cannot determine experimentally – i.e. the T_2e relaxation time (which is longer than the phase memory time of the system), the Ā^± value, the w^nSD rate and the Λ^eSD coefficient, the T_1n,j relaxation time of the local nuclei, and their concentration ηC_e. In addition, we must choose the width of the frequency bins, ω_j − ω_j−1. These parameters depend on one another, and in particular on the bin size, which changes the frequency range on which the averaging is performed. As there are many unknown parameters the fit to the data was done manually and the parameter set chosen is not necessarily the only possibility to fit the data. The significance of the parameter values used in our simulations, as summarized in Table 2, is then in their consistency for different experimental conditions, rather than in their absolute value. In order to simplify the problem, a single T_1e,j time was used for all j and was set equal to the experimental T_1e value, the T_2e value used was not changed with temperature, and ω₁ was assumed to be frequency independent. A bin width of (ω_j − ω_j−1)/2π = 2 MHz was used due to computation limitations when considering the TEMPOL sample, and was kept the same in the case of the trityl sample for comparison, although in the latter a smaller value could be used.

Table 2 Parameter values used in the simulation of the E_e profiles. The values of T_1e and T_1n were taken from Table 1

Radical	TEMPOL	Trityl
a ω 1/2π = 0.03 MHz was used at 2.7 K.
ω 1H/2π (MHz)	144
ω e/2π (GHz)	∼95
^1H concentration (M)	100
Radical concentration (M)	0.04	0.015
(ωj − ωj−1)/2π (MHz)	2
T 2,e (μs)	10	100
Λ ^eSD (μs^−3)	2000	10
w ^nSD (μs^−1)	0.001
Ā ^± (MHz)	10	0.1
ω 1/2π^a (MHz)	0.6

---

The fitting of the experimental data was performed based on the E_excite profiles in two steps. At first only the electron system was considered, by removing the R_DQ/ZQ irradiation matrices (eqn (10)) and by varying ω_excite only in the frequency range of the EPR line. The shape of the simulated ELDOR spectrum then depends on the T_2e and Λ^eSD fitting parameters, with the former influencing the MW saturation efficiency as given in eqn (11). For the TEMPOL sample, where the MW irradiation can directly saturate only a small fraction of the EPR line and the spectral diffusion was dominant for the given experimental conditions, T_2e had a relatively small effect on the ELDOR profile. The measured ELDOR spectra were then fitted by varying Λ^eSD and using T_2e = 10 μs, which is a reasonable guess. A ten-fold increase in T_2e had a negligible effect and a small effect could be seen for a twofold change of the bin width. In the trityl sample the direct MW excitation can have a large influence on the entire EPR line, and a value of T_2e = 100 μs had to be used to reduce this effect and allow for the experimental ELDOR profiles to be fitted. Λ^eSD was then determined for this particular choice of T_2e and bin width.

In the second step the R_DQ/ZQ rates were reintroduced. In the trityl case, where the EPR line is much narrower than the ¹H Larmor frequency, this step only affects the ELDOR results for irradiation outside of the EPR line, and therefore has no effect on the results of the first step within physical limits. In the TEMPOL case the MW irradiation can excite SQ, DQ and ZQ transitions simultaneously. However, for the parameters used in the first step, the R_DQ/ZQ rates had little influence on the ELDOR profiles for ω_excite within the EPR line. This may not be the case when the influence of the spectral diffusion is reduced. Therefore, in both cases the main influence of these rates was for the case of irradiation outside of the EPR line. The ELDOR spectrum in this region was then influenced by Ā^±, η, w^nSD, and the T_1n parameters, as well as by the T_2e and Λ^eSD parameters. The latter two were chosen based on the values used in the first step. The ELDOR profiles were hardly influenced by the loss of polarization to the local nuclei and was strongly influenced by the polarization transfer to the bulk nuclei. As such T_1n,j were chosen to be equal to the experimentally measured T_1,bulk. The value of w^nSD was chosen to be of the order of the nuclear dipolar interaction, 1000 s⁻¹, resulting in a strong coupling between the local and bulk nuclei. The polarization transfer between the electrons to the bulk nuclei then depended on Ā^± and η parameters, with a faster polarization transfer for larger values of the two. Here the experimental ELDOR spectra were fitted by varying Ā^± while using η = 4 as a reasonable guess.

5.2.1 TEMPOL sample. In order to fit the experimental data shown in Fig. 2–4 the TEMPOL EPR line-shape shown in ref. 45 (gray line in Fig. 8b and d) was used. The ¹⁴N nuclei of TEMPOL have a large influence on this line-shape, through their hyperfine interactions, and their relaxation may have a significant effect on the ELDOR line-shapes.^52,76–78 However, for simplicity their presence was not taken explicitly into account in the simulations, except for their influence on the EPR line-shape itself. The results of these simulations for the measurements at 20 K are shown in Fig. 8, 9, and in Fig. 10 for the 7 K and 2.7 K measurements. The parameters used in these simulations are summarized in Table 2. The fits were obtained using an eSD coefficient constant Λ^eSD = 2000 (μs)⁻³, resulting in a characteristic T^eSD_max time of about 10 μs (with f_max + f_max−1 = 0.017 in eqn (15)). This rate is much faster than T_1e⁻¹, even at 20 K, and therefore has a significant influence on the electron polarization distribution.

Fig. 8 Normalized electron polarization E_e(δν_excite,t_excite,δν_detect) simulated for the 40 mM TEMPOL sample at 20 K, using t_excite = 10 ms. In (b) and (c) several E_excite and E_detect spectra are plotted, as taken from the full E_e contour shown in (a). This was done for several δν_detect and δν_excite values, respectively, with the values given by the color-code in (b). In (d) the expected signal intensity S* is plotted using the values obtained from (c), as given in eqn (17). The EPR line, taken from ref. 45, is plotted in gray in (b) and (d). The parameters used in the simulations are given in Table 2. All frequencies are given with respect to 95 GHz.

Fig. 9 Simulated E_excite(δν_excite) spectra at different t_excite values of the 40 mM TEMPOL sample at 20 K. A detection frequency of δν_detect = 0 was used. The t_excite values are defined by the color coding inside the figure, and all frequencies are given with respect to 95 GHz. The parameters used in the simulations are given in Table 2.

Fig. 10 Simulated E_excite(δν_excite) spectra of the 40 mM TEMPOL sample at 7 K and 2.7 K. In (a) the 7 K profiles obtained using t_excite = 100 ms are plotted for different δν_detect values, defined by the color coding inside the figure. In (b) the 7 K E_excite(δν_excite) profiles obtained using δν_detect = 0 are plotted for increasing t_excite values, defined by the color coding inside the figure. In (c) and (d) similar profiles are plotted as in (a) and (b) but for 2.7 K, using ω₁/2π = 30 kHz instead of 600 kHz, and using t_excite = 500 ms in (c). The parameters used in the simulations are given in Table 2. All frequencies are given with respect to 95 GHz.

Starting with the data obtained at 20 K it can be seen that a good agreement is obtained between the experimental and simulated results (Fig. 2 and 3), with the simulations showing all main features of the E_excite and E_detect profiles (as well as the corresponding S* profiles). This includes both the effects of direct irradiation on the detected electrons close to δν_detect = δν_excite, the polarization spread due to the eSD mechanism, and the effect of irradiation on the DQ_en and ZQ_en transitions. There are however some differences that will be discussed now.

First, the temporal dependence of the experimental E_excite profiles shows some narrowing of its shape at relatively long times. This is not fully reproduced by the simulations. The source of this effect can be due to an oversimplification of the nuclear saturation process, or due to the presence of the ¹⁴N nuclear relaxation that was not taken into account in the rate equations. In addition, the polarizations at δν_detect = δν_excite is lower in the simulation than observed experimentally. This is a result of an overestimation of the SQ MW representation in eqn (11), in particular since a single off resonance value is used for all electrons in each bin. These differences could have been corrected by applying a simple convolution procedure on the simulated profiles, mimicking the action of the detection sequence. This was not done in order to demonstrate the sharp saturation features resulting from this model. An average off resonance value could also have been added to the calculation to reduce these differences, which would also have resulted in an increase of the Ā^± value needed in order to fit the experimental results.

The ELDOR data recorded at 7 K and 2.7 K (Fig. 4) are again in good agreement with the simulated results. In particular, it is apparent that the broadening of the E_excite ELDOR profiles is a consequence of the lengthening of T_1e. The simulated spectra reach values that are somewhat lower than the experimental ones, and which could be a result of the finite baseline artifact, b, as described around eqn (2). The influence of the nuclei on the ELDOR data manifests itself in the form of sharp features that are more pronounced in the simulated profiles than in the experimental ones. Possible changes in the EPR line-shape at low temperatures are not investigated and are left for future studies, together with the influence of the instantaneous diffusion and eSD on the spin relaxation measurements. It should be noted that some mismatches between the simulations and the experiments at different temperatures can be reduced when the individual ELDOR spectra are fitted independently. In addition, the influence of the nuclei on the ELDOR data could also have been calculated by considering both the local and bulk nuclei together, as explained in Section 5.1.2.

5.2.2 Trityl sample. The experimental profiles in Fig. 5 and 6 can also be analyzed by simulating similar saturation profiles using the coupled rate equations in eqn (5). Relying on the EPR line-shape reported in ref. 46, the simulated profiles are shown in Fig. 11 and 12, where we used the parameters summarized in Table 2. Once again, a relatively long T_2,e value had to be used during these simulations in order to reduce the electron depolarization originating from off resonance irradiation on the SQ transitions. The fitted value of Λ^eSD results in a characteristic T^eSD_max time of about 250 μs (with f_max + f_max−1 = 0.125 in eqn (15)), which is again much faster than T_1e.

Fig. 11 Normalized electron polarization E_e(δν_excite,t_excite,δν_detect) simulated on the 15 mM trityl sample at 30 K, using t_excite = 100 ms. In (b) and (c) several E_excite and E_detect spectra are plotted, as taken from the full E_e contour shown in (a). This was done for several δν_detect and δν_excite values, respectively, with the values given by the color-code in (b). In (d) the expected signal intensity S* is plotted using the values obtained from (c), as given in eqn (17). The EPR line, taken from ref. 46, is plotted in gray in (b) and (d). The parameters used in the simulations are given in Table 2. All frequencies are given with respect to 94.86 GHz.

Fig. 12 Simulated E_excite(δν_excite) spectra at different t_excite values for the 15 mM trityl sample at 30 K. A detection frequency of δν_detect = 0 was used. The t_excite values are defined by the color coding inside the figure, and all frequencies are given with respect to 94.86 GHz. The parameters used in the simulations are given in Table 2.

The shapes of the simulated E_excite profiles, as plotted in Fig. 11b, show many of the experimental features, but do not capture the change in the position of the broad minimum exactly. As expected, the extra depolarization and hyperpolarization effects are not reproduced, because the source of these effects is not present in our rate equations. The individual simulated E_detect shapes in 11c have the same general features as in the experiment, with a minimum around δν_excite = δν_detect with a larger recovery towards the thermal value for irradiation farther away from δν_excite = 0. However, the experimental E_detect curves obtained for high |δν_excite| values are less symmetrical with respect to δν_excite = δν_detect than the simulated ones. These differences are much less pronounced when comparing the expected EPR signal intensity S*, as shown in Fig. 11d. The simulated time dependence of the E_excite profiles (Fig. 12), showing the gradual change from a narrow hole in the electron polarization to broad eSD mediated features, are in agreement with the experimental results.

Some specific features appearing in 2.7 K measurements (Fig. 7) cannot be captured using our present model, and we therefore leave their interpretation for a later study. Whether or not their appearance is correlated with the unique extra features observed at 30 K must still be studied. Possible sources for these observations are large dipolar or hyperfine interactions, which cause large splittings in the SQ spectra of electrons such that only one transition is detected.⁷⁹ This is illustrated in the ESI,† based on QM calculations performed on small model spin systems.

6 Discussion

In this study we have focused on the electron depolarization during static ELDOR experiments in the solid state. From the ELDOR data it immediately follows that for both the trityl sample, at 30 K and 2.7 K, and the TEMPOL sample, at 20, 7, and 2.7 K, the electron polarization is highly affected by the electron spectral diffusion (eSD) process. The ELDOR results were used to monitor the polarization of the electrons during MW irradiation at a fixed frequency, offering a way to infer whether a spin temperature was created in the electron non-Zeeman manifold, in analogy to the cw EPR detected results shown in ref. 48–50. Although an energy conserving eSD mechanism was postulated to lead to the creation of such a temperature, our measurements on the TEMPOL sample at 20 K (and the 7 K results to a lesser extent) and the trityl containing sample at 30 and 2.7 K, show clearly that the distribution of the electron polarization cannot be described using the two spin temperatures coefficients appearing in eqn (4). We conclude that the TM mechanism, as described by the Provotorov based high temperature rate equations or the low temperature Borghini based model,^{18,35,44,50,64} cannot be used to analyze DNP data under the conditions used here. However, these results do not show whether a common dipolar temperature⁵⁰ exists in our samples, and does not directly contradict the assumptions of the bin-based TM models presented in ref. 51 and 68–70.

The experimental results presented here differ from those of Granwehr and Köckenberger,⁵² where the double minimum in the ELDOR spectra was not observed. This may be due to the difference in samples, measurement conditions, or detection schemes and has to be addressed elsewhere. However, in both cases the eSD has a large influence on the electron polarization. Analyzing their results in the same manner as in Section 4 does again not show the formation of the above mentioned spin temperatures.

The theoretical model describing the electron depolarization presented here captures almost all of the experimental ELDOR features in the TEMPOL and the trityl case, and especially the main features due to the eSD mechanism in the TEMPOL sample. This can next be used to study the effects of the eSD on the shapes of DNP spectra, which will enable us to improve the SE and CE based analysis used in ref. 45–47 and our understanding of the electron and nuclear spin dynamics during MW irradiation. In particular, the electron depolarization is expected to result in a reduction of the DNP efficiency,^21,55 which can lead to a change in the expected SE and CE line shapes and may explain the decrease of the DNP efficiency at low temperatures reported earlier.^45–47

The eSD rates used here do not depend on temperature, and resulted in Λ^eSD and T^eSD parameters of 10 (μs)⁻³ and 250 μs for the trityl sample and 2000 (μs)⁻³ and 10 μs for the TEMPOL sample (as defined in eqn (15)). As the magnitude of Λ^eSD is expected to scale with the cube of the average dipolar interaction (see Appendix A), it can be expected that Λ^eSD for 15 mM trityl will be about an order of magnitude smaller than for 40 mM TEMPOL. That the actual ratio is still another order of magnitude smaller can be attributed to model limitations or to physical differences between the radicals. The former includes effects of the finite bin size and uncertainty in the simulation parameters – such as the value of T_2e. Physical reasons include non-eSD electron polarization transfer mechanisms, such as polarization transfer to the nuclei and electron-nucleus relaxation, and in particular to the ¹⁴N nuclei in the TEMPOL sample.^52,76–78 Such non-eSD mechanisms can explain the narrowing of the E_excite profile of the TEMPOL sample at long MW irradiation times (Fig. 3 and 4). As a result of this discussion we must conclude that at this point the fitted values of Λ^eSD cannot yet be straightforwardly correlated to the network of interacting electrons and the fluctuating dipolar interactions between them. More experimental results are necessary to enable us to interpret the Λ^eSD values and to investigate their temperature dependence. These experiments must include measurements at different electron concentrations on each of the radical types and replacing ¹H nuclei by ²H, and in nitroxide based radicals ¹⁴N by ¹⁵N. In addition, a better knowledge of the mechanism can be gained by measuring the electron polarization distribution as a function of time.

To conclude, in this articles we have shown that the electron spectral diffusion mechanism has a large influence on the electron polarization distributions in samples presently used during DNP experiments. These distributions can be explained using a set of rate equation for the electron and nuclear polarizations, and introducing a model for taking the eSD process into account. In future studies this model will be used to explain the influence of the electron polarization distribution on the DNP enhancements, and can also be beneficial in the context of EPR spectroscopy. The resulting distribution profiles cannot be described relying on the classical TM mechanism and this should therefore not be used during the interpretation of the DNP induced nuclear enhancements observed in our samples.

Appendix A: derivation of the electron spectral diffusion rate

In this Appendix we discuss the assumptions and some considerations leading to the form of the polarization exchange rates between the frequency bins given in eqn (14). The dependence of these rates on the interaction parameters of the spin system is difficult to derive; however a dependence on the frequency differences between the electron bins, |ω_j − ω_j′|, the EPR line-shape, and on the bin size, |ω_j − ω_j−1| can be suggested. In order to propose such a dependence we first consider the polarization exchange between two single electrons at frequencies ω_j′ and ω_j′′ in the system. This exchange process, ignoring the effect of all other interactions, depends on the magnitude of the off-diagonal dipolar flip-flop terms in the two-electron spin Hamiltonian. Electron spin flips modulating this dipolar interaction are the source of the eSD exchange process.^20,60,62 For a fixed electron pair the efficiency of this process is largest when the resonance frequencies of the electrons are equal and decreases for increasing frequency difference |ω_j − ω_j′′|. Thus, the dependence of the exchange rate between these electrons, , on the frequency difference must have some bell shaped form. Without deriving an explicit expression for the exchange rate between the electrons that can directly be transferred to the exchange rates between the bins, we assume a Lorenzian line-shape dependence of the formwhere 2δ_j,j′ is the width of the Lorenzian and Δ_j,j′ is a coefficient determined by the magnitude of the dipolar interaction between the electrons. Considering that the value of δ_j,j′ is determined by the dipolar spin fluctuations, we expect that δ_j,j′ < Δ_j,j′. The transformation of this expression to the polarization exchange rate between bins in real samples is not straightforward and we therefore assume that the expression for the average (macroscopic) exchange rate between the polarizations of electrons in bin j and bin j′ has the same form as eqn (18):

Here the value of the dipolar parameter [capital Delta, Greek, macron] is expected to be smaller than the dipolar interaction between neighboring electrons, and the width coefficient [small delta, Greek, macron] is again smaller than this value. The frequency differences become a multiple of the frequency width of the bins |ω_j − ω_j−1|. Eqn (19) can be simplified if [small delta, Greek, macron] ≪ |ω_j − ω_j′|, resulting in

with . This assumption is justified for the systems considered in the present study, where the value of [capital Delta, Greek, macron] and the bin width are both of the order of a couple of MHz.

We next consider the bin structure of our macroscopic system composed of N_e electrons, with N_j = N_ef_j electrons in each of the ω_j bins. The rate of polarization exchange between the average polarizations P_j in bin j and P_j′ in bin j′ depends on the number of electrons in each of these bins, N_j and N_j′, the average exchange rate between pairs of electrons, , and the number of such electron pairs, n_j,j′. Realizing that we average over all possible polarization distributions inside each bin and that the average exchange rates inside the bins is faster that the inter-bin polarization exchange, such that at all time each bin can be described by an average P_j polarization, the macroscopic exchange rates between the pairs of bins can be estimated to have the following dependence:

Assuming that each electron in bin j has only a single directly exchanging electron in bin j′, the parameter n_j,j′ can be expressed by

n_j,j′ = N_ef_jf_j′ (22)

and eqn (21) can be rewritten as:

Thus, without taking polarization conservation into account but leaving the Boltzmann steady state conditions, we get that the exchange dynamics between pairs of polarizations is given by

This form is identical to the form used to describe the effect of the matrices on the polarization distribution, as given in eqn (13), with eqn (14) given by eqn (20).

In future studies eqn (18) and (19) must be justified theoretically, and the exact dependence of the parameters in the latter (and therefore also in eqn (14)) on the electron concentration and the bin size must be derived.

Acknowledgements

We thank Dr Arnold Raitsimring for helpful discussions regarding the origins of electron spectral diffusion. This work was supported by the German-Israeli Project Cooperation of the DFG through a special allotment by the Ministry of Education and Research (BMBF) of the Federal Republic of Germany. This research was also made possible in part by the historic generosity of the Harold Perlman Family. D.G. holds the Erich Klieger Professorial Chair in Chemical Physics.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c4cp03825h

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