Dynamic nuclear polarization

Dynamic nuclear polarization (DNP) is one of several hyperpolarization methods developed to enhance the sensitivity of nuclear magnetic resonance (NMR) spectroscopy. While an essential analytical tool with applications in several fields, NMR's low sensitivity poses major limitations to analyzing samples with low concentrations and limited masses and volumes. This low sensitivity is due to the relatively low nuclear gyromagnetic ratios (γ_n) of NMR active nuclei (¹H, ¹³C, ¹⁵N, etc.) as well as the low natural abundance of certain nuclei. Several techniques have been developed to address this limitation, including hardware adjustments to NMR instruments and equipment (e.g., NMR tubes), improvements to data processing methods, and polarization transfer methods to NMR active nuclei in a sample—under which DNP falls.

<math>{P_e \over P_n} \approx {\gamma_e \over \gamma_n} \approx \over {2.675221900 \times 10^8 s^{-1} \approx 658 </math>,

where

<math>P = \tanh() \approx </math>

is the Boltzmann equilibrium spin polarization. A larger gyromagnetic moment corresponds to a larger Boltzmann distribution of populations in spin states; through DNP, the larger population distribution in the electronic spin reservoir is transferred to the neighboring nuclear spin reservoir, leading to stronger NMR signal intensities. The larger γ and faster relaxation of electron spins also help shorten T₁ relaxation times of nearby nuclei, corresponding to stronger signal intensities. Some common PAs are shown.

Described below are the four different mechanisms by which the DNP effect operates: the Overhauser effect (OE), the solid effect (SE), the cross effect (CE), and thermal mixing (TM). The DNP effect is present in solids and liquids and has been utilized successfully in solid-state and solution-phase NMR experiments.

Mechanisms

Overhauser effect

DNP was first realized using the concept of the Overhauser effect, which is the perturbation of nuclear spin level populations observed in metals and free radicals when electron spin transitions are saturated by microwave irradiation. This effect relies on stochastic interactions between an electron and a nucleus. The "dynamic" initially meant to highlight the time-dependent and random interactions in this polarization transfer process.

The DNP phenomenon was theoretically predicted by Albert Overhauser in 1953

and initially drew some criticism from Norman Ramsey, Felix Bloch, and other renowned physicists of the time on the grounds of being "thermodynamically improbable". The experimental confirmation by Carver and Slichter as well as an apologetic letter from Ramsey both reached Overhauser in the same year.

The so-called electron-nucleus cross-relaxation, which is responsible for the DNP phenomenon is caused by rotational and translational modulation of the electron-nucleus hyperfine coupling. The theory of this process is based essentially on the second-order time-dependent perturbation theory solution of the von Neumann equation for the spin density matrix.

While the Overhauser effect relies on time-dependent electron-nuclear interactions, the remaining polarizing mechanisms rely on time-independent electron-nuclear and electron-electron interactions.

Solid effect

The simplest spin system exhibiting the SE DNP mechanism is an electron-nucleus spin pair. The Hamiltonian of the system can be written as:

:<math>H_0=\omega_eS_z+\omega_{\rm n}I_z+AS_zI_z+B\ S_zI_x</math>

These terms are referring respectively to the electron and nucleus Zeeman interaction with the external magnetic field, and the hyperfine interaction. S and I are the electron and nuclear spin operators in the Zeeman basis (spin considered for simplicity), ω_e and ω_n are the electron and nuclear Larmor frequencies, and A and B are the secular and pseudo-secular parts of the hyperfine interaction. For simplicity we will only consider the case of |A|,|B|<<|ω_n|. In such a case A has little effect on the evolution of the spin system. During DNP a MW irradiation is applied at a frequency ω_MW and intensity ω₁, resulting in a rotating frame Hamiltonian given by

:<math>H=\Delta\omega_e\;S_z+\omega_{\rm n}I_z+AS_zI_z+B\ S_zI_x+\omega_1 S_x</math>where <math>\Delta\omega_e=\omega_e-\omega_{\rm MW}</math>

The MW irradiation can excite the electron single quantum transitions ("allowed transitions") when ω_MW is close to ω_e, resulting in a loss of the electron polarization. In addition, due to the small state mixing caused by the B term of the hyperfine interaction, it is possible to irradiate on the electron-nucleus zero quantum or double quantum ("forbidden") transitions around ω_MW = ω_e ± ω_n, resulting in polarization transfer between the electrons and the nuclei. The effective MW irradiation on these transitions is approximately given by Bω₁/2ω_n.

Static sample case

In a simple picture of an electron-nucleus two-spin system, the solid effect occurs when a transition involving an electron-nucleus mutual flip (called zero quantum or double quantum) is excited by a microwave irradiation, in the presence of relaxation. This kind of transition is in general weakly allowed, meaning that the transition moment for the above microwave excitation results from a second-order effect of the electron-nuclear interactions and thus requires stronger microwave power to be significant, and its intensity is decreased by an increase of the external magnetic field B₀. As a result, the DNP enhancement from the solid effect scales as B₀⁻² when all the relaxation parameters are kept constant. Once this transition is excited and the relaxation is acting, the magnetization is spread over the "bulk" nuclei (the major part of the detected nuclei in an NMR experiment) via the nuclear dipole network.

This polarizing mechanism is optimal when the exciting microwave frequency shifts up or down by the nuclear Larmor frequency from the electron Larmor frequency in the discussed two-spin system. The direction of frequency shifts corresponds to the sign of DNP enhancements.

Solid effect exist in most cases but is more easily observed if the linewidth of the EPR spectrum of involved unpaired electrons is smaller than the nuclear Larmor frequency of the corresponding nuclei.

Magic angle spinning case

In the case of magic angle spinning DNP (MAS-DNP), the mechanism is different but to understand it, a two spins system can still be used. The polarization process of the nucleus still occurs when the microwave irradiation excites the double quantum or zero quantum transition, but due to the fact that the sample is spinning, this condition is only met for a short time at each rotor cycle (which makes it periodical). The DNP process in that case happens step by step and not continuously as in the static case.

Cross effect

Static case

The cross effect requires two unpaired electrons as the source of high polarization. Without special condition, such a three spins system can only generate a solid effect type of polarization. However, when the resonance frequency of each electron is separated by the nuclear Larmor frequency, and when the two electrons are dipolar coupled, another mechanism occurs: the cross-effect. In that case, the DNP process is the result of irradiation of an allowed transition (called single quantum) as a result the strength of microwave irradiation is less demanded than that in the solid effect. In practice, the correct EPR frequency separation is accomplished through random orientation of paramagnetic species with g-anisotropy. Since the "frequency" distance between the two electrons should be equal to the Larmor frequency of the targeted nucleus, cross-effect can only occur if the inhomogeneously broadened EPR lineshape has a linewidth broader than the nuclear Larmor frequency. Therefore, as this linewidth is proportional to external magnetic field B₀, the overall DNP efficiency (or the enhancement of nuclear polarization) scales as B₀⁻¹. This remains true as long as the relaxation times remain constant. Usually going to higher field leads to longer nuclear relaxation times and this may partially compensate for the line broadening reduction.

In practice, in a glassy sample, the probability of having two dipolarly coupled electrons separated by the Larmor frequency is very scarce. Nonetheless, this mechanism is so efficient that it can be experimentally observed alone or in addition to the solid-effect.

Magic angle spinning case

As in the static case, the MAS-DNP mechanism of cross effect is deeply modified due to the time dependent energy level. By taking a simple three spin system, it has been demonstrated that the cross-effect mechanism is different in the Static and MAS case. The cross effect is the result of very fast multi-step process involving EPR single quantum transition, electron dipolar anti-crossing and cross effect degeneracy conditions.

In the most simple case the MAS-DNP mechanism can be explained by the combination of a single quantum transition followed by the cross-effect degeneracy condition, or by the electron-dipolar anti-crossing followed by the cross-effect degeneracy condition.

This in turn change dramatically the CE dependence over the static magnetic field which does not scale like B₀⁻¹ and makes it much more efficient than the solid effect.

DNP-NMR

DNP can be performed to enhance NMR signals but also to introduce an inherent spatial dependence: the magnetization enhancement takes place in the vicinity of the irradiated electrons and propagates throughout the sample. Spatial selectivity can finally be obtained using magnetic resonance imaging (MRI) techniques, so that signals from similar parts can be separated based on their location in the sample.

DNP has triggered enthusiasm in the NMR community because it can enhance sensitivity in solid-state NMR. In DNP, a large electronic spin polarization is transferred onto the nuclear spins of interest using a microwave source. There are two main DNP approaches for solids. If the material does not contain suitable unpaired electrons, exogenous DNP is applied: the material is impregnated by a solution containing a specific radical. When possible, endogenous DNP is performed using the electrons in transition metal ions (metal-ion dynamic nuclear polarization, MIDNP) or conduction electrons. The experiments usually need to be performed at low temperatures with magic angle spinning. It is important to note that DNP was only performed ex situ as it usually requires low temperature to lower electronic relaxation.