On the role of electron-nucleus contact and microwave saturation in Thermal Mixing DNP
We have explored the manifold physical scenario emerging from a model of Dynamic Nuclear Polarization (DNP) via thermal mixing under the hypothesis of highly effective electron-electron interaction. When the electron and nuclear reservoirs are also assumed to be in strong thermal contact and the microwave irradiation saturates the target electron transition, the enhancement of the nuclear polarization is expected to be considerably high even if the irradiation frequency is set far away from the centre of the ESR line (as already predicted by Borghini) and the typical polarization time is reduced on moving towards the boundaries of said line. More reasonable behaviours are obtained by reducing the level of microwave saturation or the contact between electrons and nuclei in presence of nuclear leakage. In both cases the function describing the dependency of the steady state nuclear polarization on the frequency of irradiation becomes sharper at the edges and the build up rate decreases on moving off-resonance. If qualitatively similar in terms of the effects produced on nuclear polarization, the degree of microwave saturation and of electron-nucleus contact has a totally different impact on electron polarization, which is of course strongly correlated to the effectiveness of saturation and almost insensitive, at the steady state, to the magnitude of the interactions between the two spin reservoirs. The likelihood of the different scenario is discussed in the light of the experimental data currently available in literature, to point out which aspects are suitably accounted and which are not by the declinations of thermal mixing DNP considered here.
In the last decade Dynamic Nuclear Polarization (DNP) has established itself as a powerful technique to overcome the limited sensitivity of Nuclear Magnetic Resonance (NMR) PNAS JHAL (). More recently, as a consequence of the impressive experimental results over a wide area of applications, ranging from analytical SS1 (); SS2 () to potentially diagnostic methods BA1 (); BA2 (); BA3 (), the scientific community has started to deepen the existing theoretical knowledge about the physics of the polarization process Vega1 (); kock (); Vega3 (). Depending on the specific conditions of the experiment, the transfer of magnetic order from the electron to the nuclear system occurs by different mechanisms, named Solid Effect Abragam e Goldman (); khut (), Cross Effect CE1 (); CE2 (); CE3 (); CE4 () and Thermal Mixing Borghini PRL (); Abragam e Goldman (); Weck (). The latter regime is believed to apply to those samples and experimental conditions typically exploited in biomedical applications JHAL2008 (), which nowadays are attracting a large interest.
The original theoretical description of low temperature DNP via Thermal Mixing (TM) is due to Borghini Borghini PRL () and re-proposed in a slighly different fashion by Abragam and Goldman in their famous review Abragam e Goldman (). The model is based on the hypothesis of (i) very efficient spectral diffusion, (ii) complete saturation of the irradiated Electron Spin Resonance (ESR) isocromate and (iii) existence of a perfect contact between electrons and nuclei. This latter forcing the establishment of a common temperature between nuclear and electron reservoirs at any time.
Despite the Borghini prediction qualitatively depicts some aspects of the experimental scenario, no information about the dynamics of the process is given, while the steady state nuclear polarization is always overestimated. The quantitative agreement is especially poor when moving from the centre to the edges of the ESR spectrum. In order to reduce the discrepancies between theory and experimental observations in the nuclear steady state behaviour, Jannin et al. JanninMW () has recently proposed a variant of the Borghini model where the irradiated ESR line portion is only partially saturated. Again, the dynamical problem has not been tackled.
A general methodology to compute the full time evolution of the nuclear polarization in the low temperature TM regime, relying on a mean field approach and based on a proper system of rate equations, has been described in nostroPCCP (). In this work we exploited that mathematical treatment (briefly recalled in Section II) for providing a comprehensive picture of the electron and nuclear polarization dynamics (including the relevant steady states), over the whole microwave spectrum, for different choices of the five time constants describing the basic interactions and relaxation mechanisms. In particular, under the assumption of an optimal electron spin-spin contact, the role of microwave power and electron-nucleus interaction was investigated.
The numerical results presented in Section III point out how the hypothesis of partial saturation introdocued in JanninMW (), not only improves the agreement between TM theory an the experimental data of steady state nuclear polarization, but also predicts a more realistic behaviour for the frequency dependence of the nuclear build up time. A similar qualitative agreement is obtained by mantaining the full saturation assumption included in the original Borghini model and relaxing the constrain of perfect electron-nucleus contact, in presence of a weak, electron independent, nuclear spin lattice relaxation term. The two options considered, both fairly good in accounting for the behaviour of the nuclear reservoir, generate completely different scenario in respect of the electron polarization, as widely discussed in Section IV in the light also of the experimental observations available in literature.
In order to make the reading of the manuscript more fluent, three Appendixes collecting most of the relevant mathematics have been added at the end of the main text.
Ii Model overview
A system of nuclear spin (, Larmor frequency ) and electron spins (, mean Larmor frequency () is considered. The electron frequency distribution (ESR line) is supposed to be inhomogeneously broadened 111The main contribution to the line broadening is assumed to be single ion anysotropy (i.e. spreading of -factors), while the influence of dipolar electron-electron interaction on the broadening is neglected. As far as a sample doped with 15 mM of trityl radical is concerned, the tipical ESR line width at T = 1.2 K and = 3.35 T is about 63 MHz, whereas the electron-electron dipolar interaction is about 2 MHz. Despite the electron dipolar interaction brings a negligible contribution to the line width, it plays an important role as source of spectral diffusion between different spin packets. and can be conveniently decomposed in a sequence of narrow individual spin packets of frequency , width and relative weight such that and . The system is assumed to be ruled by five processes: microwave irradiation (with a characteristic time ), spectral difusion (), ISS process (), electron spin-lattice relaxation () and nuclear spin-lattice relaxation () (see nostroPCCP () for detailed description).
The rate describes, as an effective parameter, both the nuclear spin diffusion process and the interaction of two generic electrons belonging to packets and (being the number of packets corresponding to ) and a generic nucleus . Electrons of the same packet are set identical by definition and characterized by a local polarization , whereas a unique polarization and inverse temperature is assigned to the whole nuclear system:
The system so defined is studied in the TM regime, where the spectral diffusion processes, mediated by the electron dipolar interaction, are far more efficient than any other process (). In this limit, even when the system is out of equilibrium because of the MW irradiation, a unique spin temperature is established at all times among the electron packets (Appendix A). The electron polarization can thus be written as:
where and the unique inverse temperature are time-dependent parameters.
The dynamics of and consequently of is determined not only by the highly efficient spectral diffusion but also by the remaining four processes. Their effect is described by the system of rate equations introduced in nostroPCCP () and here reported for convenience of the reader (under the assumption of finite rates for all the four processes).
where is a Kronecker delta and are given by the expressions:
For numerical computation a discrete time step is introduced:
where = , = , = and = . After each elementary evolution step according to Eq.(II), the effect of spectral diffusion (acting on a typical time scale ) is accounted by imposing that the polarizations satisfy Eq.(2) and the conservation of the energy and total polarization:
In this work we investigate three distinct regimes where, in addition to spectral diffusion, one or two more processes are assumed infinitely efficient.
ii.1 Regime I (‘Borghini’)
The evolution of the system is derived under the following assumptions:
: a perfect contact between the electron and the nuclear reservoirs which allows to establish a common electron-nucleus inverse temperature at all times (see Appendix A);
: a full saturation of the irradiated packet which corresponds to assume , so that .
The steady state solution , can be computed by solving numerically the known Borghini relation:
which can be easily obtained by solving the system of equations describing the time evolution of the two energy reservioirs (Zeeman electron and non-Zeeman plus Zeeman nuclear contributions, reported in Equation C4 of nostroPCCP ()) at the steady state, under the condition .
The dynamics of electron and nuclear polarizations can be obtained from the system of rate equations (II), conveniently adapted to this regime (Appendix B.1). Moreover, it is possible to write the rate equation for the inverse temperature (Appendix B.1), whose solution is not an exponential function.
ii.2 Regime II (‘partial MW saturation’)
The evolution of the system is derived under the following assumptions:
: a perfect contact between the electron and the nuclear reservoirs which imposes, as in regime I, ;
: an incomplete saturation of the irradiated packet .
The steady state solution is now function of two variables and and can be evaluated by numerically solving the following system of two equations:
which is a generalized version of the Borghini relation, again obtained as steady state solution of the system of rate equations reported in Equation c4 of nostroPCCP ().
ii.3 Regime III (poor electron-nucleus contact)
The evolution of the system is derived under the following assumptions:
: a full saturation of the irradiated packet which imposes :
: a poor contact between the electron and the nuclear reservoirs, modulated by the corresponding parameter , which leads (in presence of leakage) to two different inverse temperatures for electrons and nuclei, i.e. .
The steady state solution is now function of two variables and and can be evaluated by numerically solving a system composed by the Borghini relation (Eq.(7), which holds also in this regime, but it is not sufficient to determine unambiguously and ) and the rate equation for . This latter, once imposing the stationary condition, writes:
In the limit , the contact between the electrons and the lattice is more efficient than the contact between electrons and nuclei. The steady state polarization profile is then achieved in a typical time of the order of independentely from any feature of the nuclear reservoir. As a consequence, the nuclear system ‘sees’, through , an electron thermal bath at constant temperature and the rate equation for assumes the linear form:
where and are constant terms defined as:
being given by the Borghini relation (Eq.(7) in absence of nuclei). Its solution:
is an exponential function with a steady state and an exponential time constant equal to .
Iii Numerical Results
The three regimes introduced in Section II have been explored by computing a set of build up curves (i.e. polarization versus time) for different values of the significant parameters: , , and . All the other parameters of the rate equations, when not differently stated, have been set as follows: = 1000, = 1 s, = 15, = 3 and defined according to a Gaussian function with a full width at half maximum = 63 MHz and truncated at 3. This set of parameters is choosen to represent a sample of [1-13C]-pyruvic acid doped with 15 mM trityl radical in a magnetic field = 3.35 T, at temperature = 1.2 K . Such well known mixture is an ideal prototype to be tested against the outcome of our calulations, since it was argued to polarize via TM JHAL2008 () and has been studied experimentally in great detail JHAL2008 (); JHAL2010 (); JHALhighfield (). The build up curves obtained from the numerical simulation have been fitted by the phenomenological law:
where is the steady state value of the polarization, is the polarization time and is a stretching exponent. For = 1, the usual exponential function is recovered.
iii.1 Regime I: Borghini model
The physical scenario emerging under the assumptions defining the regime I is summarized in Figure 1. Panel A shows the steady state nuclear polarization as a function of the microwave frequency in absence of leakage and when = 10000 s. The same curve is obtained by solving Eq.(7). The calculated values are generally higher than those experimentally observed especially when moving from the centre to the edges of the ESR line. A maximum nuclear polarization of 0.85 is reached in absence of leakage when the microwave frequency is set to = - 43 MHz (corresponding to the irradiation of the packet = 4). Leakage has only a moderate effect on the curve, leading to a 13 reduction of the maximum polarization level for = 10000 s.
Another interesting quantity for comparison with experiments (see Section IV) is the average electron polarization , defined as:
The steady state value of this quantity is reported in panel B of Figure 1. When the irradiation frequency is close to , the ESR line is effectively saturated, i.e. . Conversely when the irradiation frequency is set at the edges of the ESR line, because of the low weigth of the side packets.
The dynamical evolution of the spin systems can be derived by means of Eq.(18) and Eq.(B.1). The behaviour of the nuclear polarization time as function of is shown in Figure 1, panel C. The larger is the shift between and , the shorter is or, in other words, the steady state is achieved faster when the edges of the ESR line are irradiated.
Finally, on panel D of Figure 1 the nuclear and electron build up curves are displayed for (i.e. with = 4) and no leakage starting from the thermal Boltzman equilibrium condition 0 and , . As introduced in the previous Section, the same inverse temperature (see inset) characterizes both nuclear and electron reservoir. The dynamics is non exponential as can be noticed by the mismatch between the best fitting curve and the simulated data if a stretching exponential = 1 is assumed. The function describing the dynamics of is computed in Appendix B.1. Despite nuclei and electrons share the same temperature over time, since the hyperbolic tangent is a non linear function, the nuclear and electron polarization build up times are slightly different, although both in the order of s. The initial condition originates as follows:
when MW are switched on the packet is immediately saturated () due to the assumption ;
the fast spectral diffusion () imposes: ;
the effective contact between electrons and nuclei gives: .
iii.2 Regime II: partial MW saturation
An overview of the regime characterized by a partial saturation of the ESR line is presented in Figure 2. In panel A and B, and as function of are shown for = 0, 0.1 and 1 s, i.e. moving from high to low MW power. The effect of a partial saturation is twofold: on one side a significant reduction of the maximum nuclear and electron polarization values is observed. On the other side a clipping of the wings of the steady state nuclear polarization curve occurs, making this latter more similar to the DNP spectrum experimentally observed in the prototype trityl doped sample studied in JHAL2008 (). That uncomplete MW saturation can be invoked to better account for nuclear steady state data was pointed out previously in JanninMW (). Thank to this assumption the authors succeeded in fitting the DNP spectrum of a [1-13C]-sodium acetate sample doped with TEMPO, a free radical characterized by a much shorter with respect to trityls and by a higher anisotropy of the -tensor, resulting in turn in a wider ESR spectrum ( 200 MHz vs 60 MHz of trityls).
The behaviour of the polarization time versus (panel C) is, interestingly, completely different from regime I. As long as the irradiation frequency is close to , is relatively short, becoming longer and longer on moving towards the edges of the ESR line (from 1800 to 6800 s when = 0.1 s and from 4800 to 15000 s when = 1 s). As expected, longer is , less effective the polarization mechanism is.
The time evolution of represented in panel D, as well as the build up curve of (inset) are similar to those obtained in regime I, with a non exponential behaviour (a rigorous demonstration is not reported in this case) and a typical time constant in the order of s. The build up of the electron polarization instead is somehow more complex and characterized by two different time scales. The detail of such behaviour are analyzed in Appendix C.
iii.3 Regime III: poor electron-nucleus contact
The third regime is characterized by a finite contact rate between nuclei and electrons. Nuclear and electron polarization were computed for = 0.1 and 1 s both in absence of leakage and with = 10000 s. The main results are shown in Figure 3, following the same scheme of the regimes discussed above.
Panel A shows the dependence of the nuclear polarization on . In absence of leakage, as already discussed in nostroPCCP (), the contact rate does not affect the steady state but only the dynamics and thus the two curves at different overlap. In presence of leakage is reduced. The longer is the higher the decrease is. Moreover, the reduction is more significant at the edges of the DNP spectrum, a behaviour that becomes clear in the light of panel C, where is shown to be strongly increased at the wings of the spectrum. As a consequence, nuclear relaxation (with rate = 10000 s) becomes a strong competing mechanism with respect to the ISS process, forcing towards a lower steady state value.
Panel B highlights a rather interesting feature of regime III: the steady state electron polarization is almost unaffected either by and . This indicates that the nuclear system, for sufficiently high values of , is only a spectator of the electrons re-arrangement under MW irradiation, playing no active roles in the evolution of the electron systems towards its equilibrium. Evolution that proceeds through a two-step process is discussed in Appendix C. Nuclei have a ‘delayed response’ characterized by a time constant in the order of (about - s for the set of parameters used here) and by an exponential shape as confirmed by the good match between the fitting and the simulated data in panel D and as demonstrated in Section II. The exponential time course of stems from the linear rate equation (10), that further remarks the passive role of the nuclear reservoir in the polarization process of electrons. Correspondingly the nuclear inverse temperaure builds up (inset of panel D) towards a steady state value that, in presence of leakage, is substantially different from the end value of . One has, for instance, x s vs x s for s and s.
Iv Discussion and Conclusions
The original description of the TM mechanism proposed by Borghini and here analyzed in depth in terms of electron and nuclear polarization, polarization times and dynamics of the inverse spin temperatures has only a partial qualitative overlap with the experimental observations reported in literature. As, for example, pointed out in Figure 8 of reference JHAL2008 (), the Borghini model overestimates the final values of especially at the edges of the ESR line, leading to an unsatisfactory shape of the DNP spectrum ( versus ). Similarly, our computation of the model, even when the MW frequency is 3 times lower than (approx 80 MHz with our choice of parameters) and consequently the electron population of the corresponding energy levels is very low, predicts a very high enhancement of the nuclear polarization which is quite unrealistic and - more important - it is not experimentally observed. Conversely, by relaxing either the constraint of a complete MW saturation or the constraint of a perfect electron-nucleus contact, lower values and sharper DNP spectrum are obtained (panels A in Figure 2, 3).
Furthermore, in the Borghini regime, the dependence of the efficiency of the polarization transfer on the microwave frequency ( vs , panel C in Figure 1) disagrees with the experimental observations reported for a sample of [1-13C]-pyruvic acid doped with 10 mM of trityl. In Figure 5 of ref. JHAL2010 () in fact Macholl et al. showed that is relatively short as long as is set between the two values corresponding to the positive and negative maximum of nuclear polarization (DNP optimum frequencies) whilst becoming longer and longer on moving towards the edges of the ESR line. Remarkably, the correct qualitative behaviour of the polarization time is recovered under the assuptions underlying both regime II and regime III, as shown in panels B of Figure 2 and Figure 3. This type of dependence of on looks not restricted to the reference sample and magnetic field value considered so far, but rather general for DNP experiments performed at very low temperature. Similar behaviours have been reported in fact for a sample of [1-13C]-pyruvic acid doped with trityl 18.5 mM at 1.2 K and 4.64 T, corresponding to an electron Larmor frequency of 130 GHz (Figure 3 in reference JHALhighfield ()), as well as for [1-13C]-labelled acetate doped with TEMPO 50 mM at both 3.35 T (Figure 1 of Jannin2008 ()) and at 5 T (corresponding to 140 GHz, Figure 2 of Jannin2008 ()) at a temperature of 1.2 K. The robustnees of the predictions of the model analyzed here against magnetic field strength has been verified for both regime II (data not shown) and regime III (Figure 4) by repeating the computation for our reference sample at higher field (parameters were set as follow, according to reference JHALhighfield (): = 4.64 T, = 1.2 K, = 1 s, = 63 MHz). The extension of our calculations to a model representing different radicals and eventually higher temperatures, for comparison with the experimental observations achieved under such conditions Jannin2008 (); Griffin1 (); Griffin2 (); Griffin3 (), will be faced in a next dedicated study.
Up to here, as long as the nuclear parameters only ( and ) are considered, regime II and III are both in qualitative agreement with the experimental observations and nearly superimposable Jannin2008 (); JHAL2010 (); JHALhighfield (). Actually the two regimes are very different, as can be understood from panels B and D of Figure 2 and Figure 3. For limited MW power and = 0, the nuclear system and the electron one share always the same inverse temperature, generally lower than the achieved in case of full saturation. In fact, as shown in panel B of Figure 2, tends to the frequency-independent equilibrium value when increases, as the competition between the MW pumping and the electron spin-lattice relaxation unbalances the steady state towards the Boltzman equilibrium. Being the electron system weakly affected by MW irradiation, it is not anymore a forceful source of polarization for nuclei.
On the other hand, in case of finite electron-nucleus contact and 0, the electron system under the effect of the saturating MW pumping reaches in a short time a -stationary polarization profile, characterized by an inverse temperature that slowly evolves while cooling the nuclear reservoir. In absence of leakage the nuclear system sees only the pre-thermalized electron reservoir and, with a characteristic time dependent on the contact ratio reaches a final inverse temperature . In presence of leakage the nuclear reservoir is on one side in thermal exchange with the electron system at and, on the other side, with the lattice at . The final nuclear inverse temperature is a trade-off value between and and, as well as nuclear build up time, depends on the two contact parameters and .
In order to discriminate which scenario fits better with the experimental observations, data about the behaviour of electrons must be considered. A valid attempt to characterize the electron system was made by Ardenkjaer-Larsen and collaborators and it is reported in JHAL2008 (); JHALhighfield (). By measuring the shift of the C resonance line () caused mainly by the dipolar fields associated to the polarized paramagnetic centres, the authors indirectly estimated the average electron polarization according to Abragam e Goldman ():
where is a coefficient which depends on the shape of the sample, is the electron gyromagnetic ratio, is the nuclear gyromagnetic ratio and is the number of electrons per unit volume.
In particular, in JHALhighfield () (Figure 4-6) the dependence of the nuclear shift, and thus indirectly of , on the MW frequency and power was measured 222The experiments in JHALhighfield () have been performed at 4.64 T while the results of our calculation presented in Section III have been obtained at 3.35 T. We have however repeated the calculation of at 4.64 T and found that its qualitative behaviour is not affcted by the intensity of , as shown in Figure 4 panel C..
The average electron polarization at a fixed MW power (Figure 5 and 6 of JHALhighfield ()) was found to depend on with a behaviour similar to that reported in panel B of Figures 1, 2, 3, where the degree of electron saturation is higher for and lower when moving towards the edges of the ESR line and, as expected, when the MW power is reduced. It is worth to notice that increases rapidly at low microwave power and then reaches a plateau for power on the order of 40 - 60 mW. For a direct comparison of experimental (Figure 4 of JHALhighfield ()) and computational data, the dependence of simulated levels of and as function of the MW power, expressed by , for different values of is reported in Figure 5. The same qualitative behaviour is obtained in experimental and calculated data. In numerical simulations the plateau is reached for 0.05 - 0.1 s, whereas experimentally a plateau of is reached above few tens of mW. Such values are lower than the power level commonly used in DNP experiments at low temperature ( 1.2 K), thus suggesting that the assumption of full saturation is more appropriate than the hypothesis of partial saturation in interpreting DNP results collected on trityl doped samples in this temperature range. In JanninMW (), Jannin et. al. argued that the increase of the microwave power could lead, in the TEMPO doped sample considered, to a heating of the thermal bath which competes with the polarizing action of the MW themselves, affecting the equilibrium nuclear polarization and ending up in lower steady state values. Such an argument can not be extended to explain the observations on trityl doped samples, as the heating effect would also affect the equlibrium electron polarization, contributing positively to the total saturation of the ESR line. In that case, should go to zero on increasing the MW power (1/), instead of going to the low temperature plateau observed in JHALhighfield (). Thus, the assumption of partial MW saturation, although successful in improving the description of DNP from the nuclear point of view, shows an intrinsic weakness in accounting for the electron behaviour of trityl doped samples (different conclusions may apply to DNP samples doped with different radicals, such as TEMPO, provided that 0 on increasing the irradiation power). The model of finite electron-nucleus contact on the other hand, has similar capability in describing the nuclear system, but without explicitly contradicting the experimental behaviour of electrons . Overall, given the low temperature DNP experimental data available so far on the target compound considered here, relaxing the condition = 0 appears more promising than removing the saturation condition = 0.
An elegant experimental test for better judging the physical meaningfulness of regime II and III would consist in measuring the electron polarization profiles . The expected trends for the two regimes are shown in Figure 6 for = (panel A and C) and for (panel B and D) and described by Eq.(2): under the assumption of regime II and no packets are fully saturated, whereas in regime III in absence of leakage and the irradiated packet is characterized by . Especially when is set close to the electron profile of the two regimes are considerably different.
In summary we have presented the articulated picture of thermal mixing DNP generated by the five parameters model introduced in nostroPCCP (), in the limit where = 0. Three cases in particular have been discussed in detail: the Borghini regime, characterized by a strong saturation of the ESR line and by a perfect contact between electrons and nuclei ( and = 0), the regime of partial saturation of the ESR line ( 0 and = 0) and the regime of finite electron-nucleus contact ( = 0 and 0). The former regime has been shown to be less accurate in accounting for the available experimental observations, whereas the latter two are both capable of properly capturing more features of the nuclear spin dynamics, whilst predicting different behaviour for the electron system. Additional dedicated experiments would be desirable in order to clarify which of the two predictions gives a better picture of the physical reality, although the finite electron-nucleus contact regime looks more consistent than partial saturation in describing the behaviour of trityl doped samples on varying the MW irradiation power.
The theoretical picture exposed in this work cannot capture by definition those polarization phenomena driven by the Solid Effect or by the Cross Effect. Moreover it is still unable to describe some facts observed in experiments where the thermal mixing mechanism is expected to dominate, such as the inverse dependence of the nuclear steady state polarization on electron concentration, observed systematically when the ratio exceeds a certain value. The statistical approach introduced in nostroPCCP () however, can be extended to explore regimes with limited efficiency of the electron-electron interaction or, in other words, with limited thermal contact between different electronic packets. Moreover, the approach is flexible enough to allow the introduction of additional interaction terms. By exploiting these residual opportunities, we are confident that also the still unexplained behaviours will find suitable interpretation within the general framework of thermal mixing.
This study has been supported in part by Regione Piemonte (POR FESR 2007/2013, line I.1.1), by the COST Action TD1103 (European Network for Hyperpolarization Physics and Methodology in NMR and MRI) and by ANR grant 09-BLAN-0097-02.
Appendix A Spin temperature in the Thermal Mixing regime
Abragam and Goldman Abragam e Goldman () gave a description of TM DNP based on the separation between electron Zeeman and non-Zeeman contributions in the magnetic Hamiltonian of the system.
Starting from such hamiltonian one may derive the energy of a single electron spin (belonging to packet ) associated to its two possible states: up () of energy and down (), ). When MW are off the system is at thermal equilibrium with the lattice, at an inverse temperature (where is the Boltzman constant) and the probability for the spin to be in the state up is given by the Boltzman weight:
When MW are on, the system is out of equilibrium. If now the existence of a unique temperature among the different packets is postulated as in Abragam e Goldman (), the probability can be expressed in terms of a generalized Boltzman weight:
where the two parameters and are normally referred as Zeeman and non Zeeman inverse temperature respectively. The polarization of the spin can be then written as:
The same espression can be derived by observing that, whenever a process much faster than the other events ruling the system exists, the detailed balance for such a process must be satisfied at any point in time. In all the TM scenario considered in this work, the ‘spectral diffusion’ mechanism depicted here below
has been always assumed to be a fast process. Its corresponding detailed balance condition can be written in terms of the fraction of electrons up - - and of the fraction of electrons down - - at time :
Then, by using the relation
one comes to an equation for the electron polarization
which is satisfied if
Repeating the same procedure for the ISS process one obtains the following equation for the nuclear polarization:
which, being , can be rewritten as:
Eq.(16), valid when is as fast as spectral diffusion, defines the existence of a unique common temperature between nuclear spin system and electron non Zeeman reservoir.
Appendix B Electron and nuclear spin dynamics
The numerical procedure described in the main text to evaluate and , when all the processes but the spectral diffusion have a finite transition rate, has been conveniently adapted for the three regimes considered. The strategy consists in using conservation laws to manage all mechanisms assumed to be infinitely efficient, while computing rate equations only for processes with finite rate.
b.1 Regime I: ‘Borghini’ ( and )
Rate equations are used to account only for the effect of the electron and nuclear spin-lattice relaxation:
Fast processes (spectral diffusion, electron-nucleus contact and MW saturation) are accounted by the conservation of the total polarization (for variations induced by spectral diffusion or ISS) and of the total electron non Zeeman plus nuclear Zeeman energies:
where indicates the variation due to MW irradiation and the time step being the characteristic time of the transitions , and . These equations are conveniently written as:
so that, by means of simple algebra, the condition:
is obtained. By solving this equation one derives and computes and .
It is interesting to study also the evolution of the inverse temperature that in regime I, as demonstrated in the Appendix A, is the same for both the electron non Zeeman and the nuclear Zeeman reservoirs: . Moreover, since full saturation imposes , is the only unknown variable of the problem. Hence, by means of Eq.(18) and Eq.(B.1), it is possible to describe analitically the time behaviour of . At a generic time , by assuming , Eq.(18) writes:
Now, using Eq.(B.1) for replacing and one obtains:
Then, with the first order expansions:
and some algebric calculations, the following equation for is achieved:
Eq.(B.1) is conveniently rewritten as:
after negleting with good approximation the term .
Although no attemp to solve analically Eq.(20) is done here, it is clear that the solution can not be an exponential function, as already anticipated in the main text.
b.2 Regime II: partial MW saturation (, )
The system of rate equations is used to describe the effect of partial MW saturation as well as electron and nuclear spin-lattice relaxation:
whereas spectral diffusion and electron-nucleus interaction are accounted by the following conservation laws:
with being the characteristic time of the transitions and . By solving this system one obtains and and computes and .
b.3 Regime III: poor electron-nucleus contact ( and )
The system of rate equations takes into account the effect of the electron-nucleus contact and of the electron and nuclear spin-lattice relaxation:
The effect of the other processes (spectral diffusion and full MW saturation) is accounted by the conservation of the total polarization (when the variation is induced by spectral diffusion) and of the total electron non Zeeman plus nuclear Zeeman energies:
where indicates the variation due to MW irradiation and the time step being the characteristic time of the transitions and . These equations are conveniently written as: