Relaxation enhancement by microwave irradiation may limit dynamic nuclear polarization

Dynamic nuclear polarization enables the hyperpolarization of nuclear spins beyond the thermal-equilibrium Boltzmann distribution. However, it is often unclear why the experimentally measured hyperpolarization is below the theoretically achievable maximum polarization. We report a (near-) resonant relaxation enhancement by microwave (MW) irradiation, leading to a significant increase in the nuclear polarization decay compared to measurements without MW irradiation. For example, the increased nuclear relaxation limits the achievable polarization levels to around 35% instead of hypothetical 60%, measured in the DNP material TEMPO in 1H glassy matrices at 3.3 K and 7 T. Applying rate-equation models to published build-up and decay data indicates that such relaxation enhancement is a common issue in many samples when using different radicals at low sample temperatures and high Boltzmann polarizations of the electrons. Accordingly, quantification and a better understanding of the relaxation processes under MW irradiation might help to design samples and processes towards achieving higher nuclear hyperpolarization levels.


Introduction
Methods to generate hyperpolarization on nuclei have a broad field of application in nuclear magnetic resonance (NMR) 1,2 , magneticresonance imaging (MRI) [3][4][5][6][7] , particle physics 8,9 and quantum science [10][11][12][13][14][15] .Hyperpolarization can be generated by many different methods 16 although reaching the theoretical maximum for a given method remains experimentally challenging.The polarizationtransfer process is a poorly understood problem and it is difficult to assess what actually limits the enhancements.In many cases, hyperpolarization is generated by a small but continuous injection of polarization which, in the absence of relaxation, should enable polarization levels up to the theoretical limit.
Dynamic nuclear polarization (DNP) is a hyperpolarization method that is based on the transfer of polarization (spin magnetic moment) from unbound valence electrons (radicals, defects, paramagnetic centers) to hyperfine-coupled nuclear spins under microwave (MW) irradiation.DNP experiments at cryogenic temperatures offer the advantages of long electron relaxation time and increased Boltzmann polarization of the electrons to near unity.Typically, the electron spins are very dilute in large nuclear spins systems with on average hundreds of nuclear spins per electron.Such a configuration leads to a slow build up of hyperpolarization as successively more and more nuclear spins are polarized either directly from the electrons or indirectly through nuclear spin diffusion.Eventually a steady-state polarization (P 0 ) is reached where nuclear relaxation and polarization injection compensate each other.The build-up of the polarization can often be described by a characteristic time constant τ bup .Upon switching off the MW irradiation, the hyperpolarized state relaxes back to the thermal equilibrium, commonly called polarization decay and described by a Institute for Biomedical Engineering, University and ETH Zurich, 8092 Zurich, Switzerland b Department of Chemistry and Applied Biosciences, ETH Zurich, 8093 Zurich, Switzerland * Corresponding author: maer@ethz.cha characteristic time constant τ decay .Rate equation or kinetic models offer the possibility to qualitatively and even quantitatively describe the balance between hyperpolarization injection and relaxation 17 .
In this paper we use such rate equation models to determine the relaxation-rate constant of the nuclei during the DNP polarization build-up process.We find that in many cases the nuclear relaxation-rate constants during the hyperpolarization buildup under MW irradiation are much larger than the ones obtained without MW irradiation.Microwave irradiation was previously shown to shorten rotating-frame relaxation times (T 1ρ ) in crosspolarization experiments 18 and also nuclear coherence life times (T ′ 2 ) 19 at liquid helium temperatures and 6.7 T field strength.In this work, the increased nuclear relaxation-rate constants under microwave irradiation during the build up of hyperpolarization is found to limit the achievable maximum polarization compared to the prediction when using the electron equilibrium polarization, nuclear relaxation-rate constants (without MW irradiation) and polarization injection rates.

Samples
All measurements were performed with 50 mM 4-oxo-TEMPO in water/ glycerol mixtures and DNP performed on the 1 H spins.In particular, we compare two different sample formulations with (i) TEMPO in a (6:3:1) V mixture of glycerol-d 8 , D 2 O and H 2 O (DNP juice) and (ii) TEMPO in (1/1) V H 2 O/ glycerol (no deuteration, all natural abundance).After mixing the ingredients, the filled sample container was frozen in liquid nitrogen before being transferred to the cryogenically pre-cooled polarizer (cryostat temperature during the transfer below 20K).All chemicals were purchased commercially and used without further purification.To check the reproducibility of the results, we tested two independently formulated natural abundance samples.The measurements were performed together with those reported in Ref. 17 .

DNP and EPR
The DNP measurements were performed on a home-built polarizer 20 at a temperature of 3.3 K unless otherwise indicated.The system was equipped with a Bruker Avance III (Bruker BioSpin AG, Switzerland) spectrometer at 7 T (299 MHz 1 H Larmor frequency).The natural-abundance sample formulation was reported to build up mono-exponentially under the same conditions before 21 .Compared to the previously published work on this material, the polarizer was upgraded to a more powerful MW source (200 mW, Virginia Diodes Inc., USA) and in-house electroplated low-loss wave guides, giving us approximately eight times more MW power as before at the sample space (around 65 mW at the sample) 22 .The other details of the set-up are described elsewhere and were unchanged 20,22 .The absolute values of the polarization were calculated on a single thermal equilibrium measurement per sample as reported and discussed previously 17 .
Electron paramagnetic resonance (EPR) spectra were acquired with our in-house developed longitudinal-detection (LOD) EPR set-up at 7 T at a temperature of 5 K 22 .
For all build-up curves (except where otherwise stated to check the off-resonant irradiation), the MW frequency was set to 197.05 GHz without using any MW frequency modulation.The MW power was set to the maximum output unless explicitly stated.For decay curves under MW irradiation, the MW frequency was switched to 197.28 GHz (zero crossing of the DNP profile) after the sample (nearly) reached its steady-state polarization using maximum output power unless otherwise stated.
All data processing was performed with in-house developed MATLAB (MathWorks Inc., Natick, Massachusetts, U.S.A) scripts.All experimental uncertainties in the processed experimental data result from the 95% fit intervals.Experimental instabilities like slight changes in the MW output or minor temperature fluctuations as well as uncertainties in the thermal equilibrium measurement were not included in the analysis.

Hyperpolarization decays under microwave irradiation
Figure 1 shows polarization build-up (a) and decay (b) curves for the natural-abundance sample at two different temperatures, 3.3 K (red and yellow lines) and 2.0 K (blue lines) corresponding to 89 and 99% thermal electron polarization at 7 T.The two sets of build-up curves (Fig. 1a) were recorded under identical conditions and show the good reproducibility of the measurements.Due to the higher thermal polarization at 2 K, the build-up curves at lower temperature (blue curves) reach a higher steady-state polarization.
The corresponding decays shown in Fig. 1b were recorded with the MW source switched off (dark blue and red curves) as well as with the MW source turned on (light blue and yellow curves) but tuned to the frequency of the zero crossing of the DNP profile (197.28GHz, see Fig. 1d), minimizing the DNP enhancement during the decay measurements.There are several effects visible when comparing the decay curves with and without MW irradiation: (i) Without MW irradiation, the decay of the nuclear polarization is slower (dark blue and red line).Switching the MW irradiation to the frequency of the zero crossing of the DNP profile significantly speeds up the nuclear spin-lattice (T 1 ) relaxation (light blue and yellow line in Fig. 1b).
(ii) The decay curves with on-resonance MW irradiation (light blue and yellow lines in Fig. 1b) start at a higher value, the ones without MW irradiation (dark blue and red lines in Fig. 1b) at a lower value than the end point of the build-up curves.This can be explained by the presence of strongly hyperfine-coupled and, therefore, invisible (hidden or quenched) nuclear spins.Microwave irradiation will decouple some of these spins and make them visible and contribute to the observable nuclear signal intensity 23,24 .
(iii) In the low temperature (2.0 K), MW-off decays corrected for the perturbations by RF pulses shown in Fig. ?? (dark blue), the observed signal first increases slightly, in contrast to the expected decay as observed in the uncorrected data (Fig. 1b and Fig. ??).This can again be explained by invisible strongly hyperfine-shifted spins that are coupled to the visible spins by slow spin diffusion 25 .
In the following, we will not discuss the latter two points but focus on the first point.Such a dependence of nuclear decay rates on the MW irradiation at low temperatures has not yet been reported to the best of our knowledge.Similar and related effects, i.e., a decrease of rotating-frame relaxation times (T 1ρ ) in cross-polarization experiments under MW irradiation 18 and a decrease of transverse nuclear coherence life times (T ′ 2 ) in echo experiments 19 with MW irradiation at liquid-He temperatures have been reported recently.
Fig. 1c shows the fitted decay-rate constants for three different samples with and without MW irradiation at three different temperatures.In all cases, a significant increase of the nuclear decayrate constant is observed under MW irradiation.Under MW irradiation, the relaxation is nearly independent of the temperature between 2.0 and 3.9 K, contrasting the nearly one order of magnitude difference in decay-rate constants without MW irradiation over this range of temperatures.
To investigate this further, we measured the decay-rate constants as a function of the MW settings (power and frequency) which is summarized in Fig. 2a and b.The relaxation enhancement shows a frequency-dependent resonant behavior with stronger relaxation enhancement close to the zero crossing of the DNP profile which corresponds to the center of the EPR spectrum (Fig. 1d).For a DNP frequency of 197.28 GHz (zero crossing of the DNP profile), the power dependence of the relaxation enhancement can be described with a square-root scaling (Fig. 2b) and scales in the same way as the build-up time constant (see Fig. 2c) as a function of MW power.The steady-state polarization P 0 , however, shows a different behavior as shown in Fig. 2d: It is nearly independent of the power between 20 and 100% of the MW output.
To understand the almost MW-power independent steady-state polarization while the decay-rate as well as build-up time constant τ bup follow a square root scaling, we use a single compartment model 17 .For a single homogeneous compartment, the hyperpolarization build-up can be described using a first-order differential equation with a hyperpolarization injection rate constant k W and a

relaxation-rate constant of the build-up
with A describing the theoretical maximum of hyperpolarization achievable, i.e., the thermal electron polarization in DNP.The solution of Eq. 1 is a mono-exponential curve which can be compared with the phenomenological description of the build-up curve using P(t) = P 0 (1 − e −t/τ bup ) to express the experimental parameters in terms of model parameters.Here, P 0 is the steady-state polariza-tion and τ bup the build-up time.
For the decay, k W would be set to zero (MW off), leading to . The model can reproduce the experimental build-up curve as shown in Fig. 1a (black line).A different approach to measure the rate-equation parameters is discussed in Sec.?? of the supplementary information.
From the definition of the build-up time in Eq. 2a and the observed square root scaling of τ decay and τ bup with MW power, the injection rate constant, k W will obey a power scaling similar to a square root.Since all rates in the steady-state polarization (cf.Eq. 2b) follow roughly a square root scaling, a power-independent steady-state polarization is expected.Therefore, higher MW power will lead to a faster build-up but not to a higher steady-state polarization.
However, this does not hold true for low MW powers as in this case the relaxation enhancement under MW irradiation is small compared to the thermal relaxation and the total relaxation-rate constant will not scale with the square root of the MW power.Hence, the total relaxation in the system competing with the DNP injection can be described as k R = k decay R

+k MW
R sB 1,MW with the MWfield amplitude B 1,MW ∝ √ P MW (MW power P MW ) and a scaling factor s.For small MW powers, the total relaxation in the system is dominated by thermal relaxation, explaining the increase of the steady-state polarization with MW power as only the injectionrate constant grows with the square root of the MW power (linear in B 1,MW ).For larger MW powers, MW induced relaxation governs the total relaxation and no further improvements of the steady-state polarization and only a faster build-up is expected (see Fig. 2).A shortening of the build-up time with a preserved enhancement might help to improve the NMR signal per unit time 26 .
Extracting the rate-equation parameters k R and k W directly from the experimental build-up time constant and steady-state polarization value using Eq. 2 leads to relaxation-rate constants three to four times larger than the decay-rate constant measured without MW irradiation (compare Figs. 1c and ??c).Using the injectionrate constant from the highest applied MW power and assuming the decay-rate constant measured without MW irradiation, the steady-state polarization would be around 60-70% instead of the experimentally measured 30-40% as shown in Fig. ??.
The injection rate as used in Eq. 1 refers to the creation of polarization (normalized magnetization) and not to absolute magnetization which would describe the absolute number of effective electron-nuclear polarization transfers.In our experiments.all samples contained the same concentration of radicals (50 mM).The approximately 90% deuteration reduced the number of protons per radical.Hence, the similar (polarization) injection rates for deuterated and natural abundance samples (cf.Fig. ??c) indicate that DNP (at least in the partially deuterated sample) is limited by transport from the strongly hyperfine coupled spins into the bulk rather than by the initial electron-to-nucleus polar-ization transfer: Similar steady-state polarizations are found for natural abundance and partially deuterated samples (cf.Fig. ??) which requires around ten times more nuclear spins to be polarized per TEMPO radical in the the natural abundance sample.If the electron-to-nuclear polarization transfer would be the DNP limiting step, a ten times higher (polarization) injection rate would be expected for the deuterated sample.
Similar conclusions have been made for partially deuterated samples 27 , termed "spin-diffusion limited" samples, with a somewhat more complicated model.For high 1 H concentrations, Ref. 27 ascribes the polarization to be "spin-exchange limited" as too many nuclei per radical need to be polarized, possibly supported by the slight decrease in k W for the natural abundance samples compared to the partially deuterated sample (cf.Fig. ??c).
To investigate whether an enhancement of the apparent decay rates under MW irradiation is a particular property of our experimental sample composition, we have reevaluated data available in the literature using the one-compartment model.Using experimental build-up and decay times as well as steady-state polarization levels allows us to calculate and compare the relaxation-rate constants during build up and decay.

Relaxation enhancement by MW irradiation across samples and DNP conditions
Trityl-based dissolution DNP (dDNP) of pyruvic acid is one of the most important medical applications of dissolution DNP while TEMPO and other related aminoxyl-radicals are typically used in NMR applications of DNP.Thus, we focus on these materials in the following discussion.The available data in the literature is summarized in Tab. 1.To be able to analyze the data, the following information was extracted from the literature: (i) temperature and static magnetic field to calculate the electron polarization, (ii) absolute steady-state polarization, (iii) build-up (τ bup ) and (iv) decay-time (τ decay ) constants.
The relaxation-rate constant extracted from the build-up k bup R (compare Eqs. 2) and decay k decay R are compared to estimate the relaxation enhancement by MW irradiation with a ratio of k bup R /k decay R = 1 indicating no relaxation enhancement in the sample.The estimated ratios for the relaxation enhancements are sensitive to inaccuracies in the thermal equilibrium measurement as this affects the steady-state polarization quoted.The perturbing effects of monitoring RF pulses can be minimized by using few and small flip-angle pulses or correcting for the pulses 17 .
In the following, relaxation enhancement by MW irradiation depending on the experimental conditions (temperature and field) as well as sample formulation (radical, concentration and target spins) are discussed.Trends are first discussed for trityl-based and then for TEMPO-based samples.
For trityl radicals and direct DNP to 13 C nuclei, the relaxation enhancement is more pronounced at lower temperatures and lower fields.At 3.35 T, the relaxation enhancement is nearly absent at 3.5 K, grows to more than fourfold at 1.3 K and sixfold at 1.0 K. Increasing the field strength reduces the relaxation enhancement in the range of 3.35 and 10.1 T and temperatures of 1.0-1.3K. Higher trityl (electron) concentrations lead to pronounced = 1 indicates the absence of relaxation enhancement by MW irradiation.b The sample formulation was broken down to the radical together with its target spins and possible gadolinium (Gd) doping, although the exact sample formulations vary between the different works.The first three samples from this work refer to the natural abundance sample and the second three to the partially deuterated (DNP juice) sample (cf.Figs.1,2).RF correction was applied by the authors to the decay of Ref. 29 but not to any of the other reported measurements according to our knowledge.The TEMPO data set from Ref. 21 (same sample formulation as for our natural abundance samples apart from the [ 13 C]urea) was recorded in our lab previously although with an older MW set-up with an approximately eight times lower MW power at the sample (see Methods), possibly explaining the discrepancy to the measurements reported herein.
increases of the relaxation enhancement.For example, at 10.1 T and 1.3 K, relaxation enhancement is absent for a 15 mM sample while a twelvefold relaxation enhancement is found for 45 mM of trityl -resulting in twelvefold shortening of τ bup while preserving steady-state polarization P 0 .The relaxation enhancement at higher electron concentrations can be reduced by sample doping with gadolinium (Gd).
Raising the temperature to 20 K (and DNP to the 1 H nuclei), the relaxation enhancement is nearly completely suppressed or masked by increased thermal relaxation even at very high trityl concentrations of 100 mM.
For TEMPO with DNP performed to 1 H or 13 C and 4.2 K, no field dependence is evident between measurement at 3.35 and 7 T.The elevated temperature complicates discrimination between temperature-related and radical-related (DNP mechanism) changes.The relaxation enhancement appears slightly larger for 1 H compared to 13 C although it is not increasing beyond a factor of 2.2.
Two general trends may be deduced from Tab. 1: (i) The relaxation enhancement by MW irradiation requires elevated electron concentrations and electron dipolar couplings causing electron spectral diffusion (eSD).Thus, suppression of eSD should reduce the relaxation enhancement.This is in agreement with the results of Gd doping, shortening the electronic relaxation times 33,34 , or higher magnetic fields, broadening the EPR line and shortening the electronic relaxation 34 .
(ii) Lower temperatures increase the relaxation enhancement.Lower temperatures prolong the electronic relaxation times and increase the electron polarization.Together, these determine the nuclear relaxation times near liquid helium temperature as all nuclear relaxation can be assumed to originate from paramagnetic relaxation.With lower thermal relaxation, the relaxation enhancement by MW irradiation contributes increasingly more to the total relaxation.
In the following we will discuss possible explanations for the relaxation enhancement by MW irradiation.MW irradiation saturates parts of the electron line, creating a polarization difference between different parts of the electron line.At liquid-He temperatures, nuclear relaxation is most likely due to electron flip-flops 35 .Cross effect and thermal mixing DNP convert this polarization difference into nuclear hyperpolarization.Assuming that the nonirradiated part of the electron line remains at the thermal electron polarization (A = P thermal e ) while the irradiated part of the electron line is saturated to P MW e , nuclear relaxation for electrons with different polarization is proportional to 1 − P thermal e P MW e 35,36 (a similar expression is derived for direct nuclear (paramagnetic) relaxation in Ref. 35 ).Hence, nuclear relaxation and DNP injection scale with the electron saturation, leading to a MW power independent steady-state polarization (for small enough thermal relaxation, cf.Fig. 2d).For MW irradiation at the center of the electron line, the higher electron spectral density and relative ease to fulfill the matching condition for triple spin flips results in a frequency dependence with the higher nuclear relaxation at the center of the electron line.Increased electron spectral diffusion increases the probability for electron flip-flops which cause nuclear relaxation.
Additionally, sample heating due to MW absorption would appear (nearly) frequency independent over the narrow experimental frequency regime 37 .A higher sample temperature reduces the thermal electron polarization which increases the relaxation scaling as 1 − (P thermal e ) 2 .This might explain the slightly decreasing steady-state polarization for the highest MW powers (see Fig. 2d) and enhanced relaxation for far off-resonant MW irradiation (see Fig. 2a) compared to the MW off situation.For elevated temperature, the thermal relaxation following a 1 − (P thermal e ) 2 scaling is large such that a relaxation enhancement by MW irradiation is difficult to observe.

Conclusions
Microwave (MW) irradiation in DNP generates not only hyperpolarization of the nuclei but can cause a relaxation enhancement of the nuclei by more than a factor of ten at temperatures of a few Kelvin compared to the decay of nuclear polarization without MW irradiation.This is similar to the experimentally observed shortened relaxation times T 1ρ 18 and T ′ 2 19 under MW irradiation at liquid-helium temperatures.The relaxation enhancement was measured on 1 H in samples consisting of a glassy matrix containing TEMPO radicals.A literature survey shows that the phenomenon is more general and can also be observed in trityl-based samples, suggesting a radical independent origin.The largest relaxation enhancement was obtained with the microwave (MW) irradiation on the EPR resonance frequency, suggesting a resonant relaxation mechanism beyond possible sample heating.
Build-up and decay data with and without RF correction

Experimental validation of the one compartment model
In order to verify the validity of the extracted model parameters, namely k W and k R , from our measurements, we performed build-up experiments with amplitude modulation.In our case, we switched the MW between zero and full output power in a fixed cycle.For the choice cycle time, we have two different time scales to consider: (i) It should be much longer than the thermal T 1,e (around 500 µs under our DNP conditions measured with longitudinal detected (LOD) electron paramagnetic resonance (EPR) [1], probably dominated by electron spectral diffusion) such that we can consider the MW irradiation blocks as continuous wave irradiation and (ii) much shorter than the build-up time such that we can average our modulated irradiation over time and consider it to be a constant irradiation on the time scale of the build-up.Based on these consideration, we chose a cycle time of 1 s.This would even be enough if the LOD EPR-based T 1,e is two orders of magnitude shorter than the "true" T 1,e .Upon varying the time with full MW output power per cycle, we can translate this into a linear variation of our model parameters, eventually enabling us to disentangle the MW-induced relaxation from the DNP injection.
For the amplitude modulated build-ups (MW switched between full and zero power) as shown in Fig. S3, the large error bars are a result of electronics communications problems leading to clear bumps in the data (not shown), visible during processing.The affected data sets were cut but often resulted in short, usable build-up durations, leading to relatively large uncertainties in the fit results.
The estimated values of the thermal electron polarization (A) appear below the thermal electron polarization based on the experimental conditions (89% at 3.3 K and 7 T) which might be an indication of sample heating from MW absorption as discussed in the main text.

1 ]( 1 : 1 )Fig. 1
Fig.1Nuclear relaxation depending on the temperature and MW irradiation.Build-ups (a) and the respective decays (b) of the (second) natural abundance sample at 3.3 K (red and orange) and 2.0 K (light and darker blue) without correcting for the effects of radio-frequency (RF) pulses.The MW is set to the same power and frequency (197.05GHz) during all build-ups.During the decays depicted in darker blue and red, the MW is switched off.For the decays depcited in orange (3.3 K) and light blue (2.0 K), the MW is switched to the central EPR frequency (197.28GHz) resulting in zero DNP, as can be seen in (d).The measured polarization in (a) can be simulated (solid black line) with a single compartment rate equation model17 .The only model input parameters are the thermal electron polarization, measured build-up time and steady-state polarization.(c) Relaxation rates during MW-on decays (dots) and MW-off decays (diamonds) for the natural abundance samples (green) and partially deuterated (DNP juice) sample (brown-gray/taupe, see Methods for details).The data shown are without RF correction (explained in the main text) (d) Overlay of the EPR line at 5 K (blue) from22 and a DNP profile.For simplicity, all measurements presented in this work were recorded with continuous wave (CW) irradiation without MW modulation.

( 1 :Fig. 2
Fig. 2 Microwave-dependence of the time constants and polarization.Frequency (a) and power dependence (b) of the relaxation enhancement during MW irradiation in decay experiments.For the frequency dependence, the MW was set to maximum power.All power dependencies were recorded at the central zero-DNP frequency (197.28GHz).The data in (a) is fitted with τ ν MW =0 R − m/(c + |197.28 − ν MW |).Build-up time (c) and steady-state polarization (d) show the same power-dependence for natural abundance (green) and partial deuteration (brown-gray/ taupe).The steady-state polarization is relatively independent of the MW-power except for small powers.Data in (b) and (c) were both fitted with a square-root ansatz (∝ P −1/2 MW with the microwave power P MW ).

4 ( 1 : 1 ) 2 3 4 ( 6 : 3 : 1 )Figure S1 . 4 ( 1 : 1 ) 2 3 4 ( 6 : 3 : 1 )Figure S2 .
Figure S1.Build-up and decay with RF correction Build-ups (a) and the respective decays (b) at 3.3 K (red and orange) and 2.0 K (light and darker blue) with correcting for the effects of radio-frequency (RF) pulses.The MW is set to the same power and frequency (197.05GHz)during all build-ups.During the darker decays (darker blue and red), the MW is switched off.For the lighter decays (3.3 K in orange and 2.0 K in light blue), the MW is switched to the central EPR frequency (197.28GHz) resulting in zero DNP, as displayed in ??d.

( 1 :Figure S3 .
Figure S3.Measuring the one compartment model parameters.(a) We fit the build-up times for different fractions of the MW being switched on to full power (amplitude modulation with full and zero power) with (k R,res + b • x) −1 .k R,res is a fitted residual relaxation rate (in good agreement with the thermal relaxation rate in MW-off decays, see Fig. ??c or [2], b = k W + k R,MW and x being the MW-on fraction.The green data sets belong to natural abundance and brown-gray (taupe) to the partially deuterated sample as throughout this paper.(b) P 0 /τ bup shows a linear behaviour as expected from Eq. ?? with the slope being given by Ak W . (c) Comparison of k bup R and k W through two different analysis approaches: (i, full color) Inserting the measured steady-state polarization and build-up time into Eqs.??, (ii, half transparent) The relaxation rate k bup R can be estimated from the fit shown in Fig. ??a.With k bup R at hand, k W can be extracted from the fit shown in part (a) of this figure.Finally, A can be deduced from the linear fit shown in (b).

Table 1
Summary of relaxation enhancements a by MW irradiation in different samples b and conditions.