Hyperfine decoupling and pulsed dynamic nuclear polarization (DNP) are promising techniques to improve high field DNP NMR. We explore experimental and theoretical considerations to implement them with magic angle spinning (MAS). Microwave field simulations using the high frequency structural simulator (HFSS) software suite are performed to characterize the inhomogeneous phase independent microwave field throughout a 198 GHz MAS DNP probe. Our calculations show that a microwave power input of 17 W is required to generate an average EPR nutation frequency of 0.84 MHz. We also present a detailed calculation of microwave heating from the HFSS parameters and find that 7.1% of the incident microwave power contributes to dielectric sample heating. Voltage tunable gyrotron oscillators are proposed as a class of frequency agile microwave sources to generate microwave frequency sweeps required for the frequency modulated cross effect, electron spin inversions, and hyperfine decoupling. Electron spin inversions of stable organic radicals are simulated with SPINEVOLUTION using the inhomogeneous microwave fields calculated by HFSS. We calculate an electron spin inversion efficiency of 56% at a spinning frequency of 5 kHz. Finally, we demonstrate gyrotron acceleration potentials required to generate swept microwave frequency profiles for the frequency modulated cross effect and electron spin inversions.

Dynamic nuclear polarization (DNP) increases NMR sensitivity by transferring polarization from electron to nuclear spins [

Pulsed techniques yield significant advantages over continuous wave irradiation to control and measure interactions in the solid state NMR Hamiltonian. As Schaefer demonstrated in his development of double resonance MAS NMR, short, powerful pulses permit the measurement of weak heteronuclear dipolar couplings (REDOR) [

Performing high field DNP in the time domain can substantially improve electron to nuclear polarization transfer [

Another analogy between Schaefer's pioneering experiments in MAS NMR and current developments in DNP is the decoupling of spins with a larger magnetic moment after the polarization transfer period. Proton decoupling after cross polarization (CP) is often critical to extending the relaxation time of the observed nuclei and providing high-resolution NMR spectra, as was demonstrated in Schaefer's original CPMAS experiments [

Electron decoupling, also known as hyperfine decoupling and dynamic decoupling, was introduced by Jeschke and Schweiger in the context of pulsed EPR [

We discuss in the following sections how voltage tunable gyrotron oscillators could generate the microwave frequency profiles required for the inversion of electron spins and thus decoupling of hyperfine interactions. In particular, we show that EPR inversions are theoretically possible even with the inhomogeneous microwave field present within MAS rotors, and perform detailed electromagnetic simulations to characterize the microwave field intensity in a MAS DNP probe operating at 198 GHz. We also demonstrate modulations of the gyrotron acceleration potential suitable for the frequency modulated cross effect and electron spin inversions.

We used Ansys HFSS (high frequency structural simulator) to calculate the 198 GHz electromagnetic field structure within our custom MAS DNP probe [_{γS}_{1S} frequencies, where _{1S} represents the microwave field orthogonal to the main magnetic field.

We calculated the electromagnetic field structure for the sample geometry shown in

Our simulations are normalized to microwave power levels we expect to achieve with our custom 198 GHz voltage tunable gyrotron, designed in collaboration with Bridge12 Technologies (Waltham, MA). We modeled the microwave input into the stator as a pure Gaussian beam with a waist of 3.175 mm and power of 5.0 W.

HFSS solves the electromagnetic field structure with an initial excitation amplitude of |_{0}|. Only the initial electric field magnitude, |_{0}|, needs to be given because the magnetic component, |_{0}|, is proportional and orthogonal to |_{0}| for an electromagnetic wave traveling in free space. We calculated the initial electric field magnitude for our Gaussian beam propagating in free space and carrying 5 W of power (see _{0} is the permittivity of free space, _{0} = 3.175 mm is the beam waist of the Gaussian beam entering the stator.

To determine the _{γS}_{1S} within a specified volume, we first calculate the magnitude of the transverse components of the electromagnetic field. However, the HFSS calculation is not performed in the laboratory frame. We must account for this rotation of the HFSS coordinate axis from the laboratory frame to determine the magnitude of the magnetic field components orthogonal to the static field _{0} that contribute to the _{γS}_{1S}. Referring to

_{m} is the magic angle. Note that

We define the physically significant quantity for EPR and DNP to be the volume average of the peak transverse magnetic field similar to the treatment in Macor et al. [

Note, this differs slightly from the procedure outlined in Nanni et al. in which the volume root mean square (RMS) is computed [

We also assume μ is constant throughout the entire sample and is approximated by the permeability of free space, μ_{0} = 1.257 × 10^{−6} H/m. The permeability, μ, is a measure of the response of the material to an applied magnetic field in the similar way that the dielectric constant, ε, is the response of a material to an applied electric field. The detailed calculations including all numerical values and screen-shots from HFSS are given in the

When we integrate over the entire sample volume of 18.6 μL we find that 5.0 W of power gives an average γ_{S}_{1S} of 0.46 MHz. 17 W of input power into the stator is required to generate the average γ_{S}_{1S} of 0.84 MHz which is now commonly used in theoretical treatments of MAS DNP [

The microwave beam emitted from gyrotrons is linearly polarized, however electron spins only strongly interact with circularly polarized light. We can decompose the linearly oscillating field into two circularly polarized components that each have half the amplitude of the original beam as shown in _{1S} over the sample given is _{⊥}/2. It should be noted that if _{⊥} is given as the RMS, _{⊥,RMS} of a linearly oscillating field then,

A concern often voiced about MAS DNP using high power microwaves is heating of the sample. This has already been calculated from similar electromagnetic field simulations [

Using HFSS we solved this integral and found that 7.1% of the power goes into heating the sample for our 198 GHz system. As previously discussed, the ~1.0 W of power that contributes to dielectric heating is effectively dissipated in MAS DNP experiments by the sapphire rotor which is cooled by a high flow of cryogenic gas [

To determine the effect of geometry and frequency on the microwave field structure we applied our same analysis to the 4.0 mm DNP NMR probe modeled by Nanni et al. at 250 GHz as shown in

The microwave intensities in the 4 mm rotor at 250 GHz are qualitatively similar to the structures seen previously, although we cannot directly compare the values because the stator geometries are not identical. The values in Nanni et al. include a silver coating applied to the inside of the stator that we have not modeled. The analysis given by Nanni et al. reports γ_{S}_{1S} values of 0.84 MHz for a silver coated stator, and 0.70 MHz without the silver coating.

With 5 W of input power, we found an average value γ_{s}_{1S} of 0.37 MHz within the sample at 250 GHz. However, if we take a volume RMS, as outlined by Nanni et al., we determine a RMS value of 0.46 MHz within the sample volume. This agrees well with the value of

We calculated the standard deviation of the peak transverse magnetic field normalized to the mean transverse magnetic field within the sample as:

The more homogenous γ_{S}_{1S} distribution will have a standard deviation closer to 0. For our 198 GHz system we found

The instantaneous microwave intensities shown in _{S}_{1S} distribution. This is because the microwave field intensity is time dependent and follows the 5 ps oscillation period of the 198 GHz microwaves. _{S}_{1S} distribution in _{S}_{1S} distribution is much more homogeneous than the instantaneous intensity of the microwaves. However, we note there is still a considerable inhomogeneity in the γ_{S}_{1S} frequencies, which we account for in our simulations of EPR inversions in Section 4.

To give insight into the experimental requirements for performing MAS pulsed DNP, we calculated the amount of microwave power necessary to generate a π/2 pulse on the electron spin manifold. At 7 T, the inhomogeneously broadened nitroxide lineshape simulated with EasySpin [_{S}_{1S} frequency required to control this lineshape with a short (~1 ns) hard pulse is thus ~750 MHz and would require an |_{0}| value of 4.5 × 10^{7} V/m, which corresponds to a microwave input power of 13 MW. Impressively, gyrotrons built for fusion research approach such power levels but do not generate more than 1.3 MW [

One strategy for MAS pulsed DNP at high magnetic fields is to implement a microwave resonant structure with a high quality factor similar to high field non-spinning DNP probes [

Stable organic radicals with more symmetric g-tensors, such as 1,3-bis(diphenylene)-2-phenylallyl (BDPA) and trityl [_{1S}) [

Again, we apply the results from our HFSS microwave field calculations and determine that 21 kW of microwave power are required to generate a 30 MHz γ_{S}_{1S} field to effectively excite the BDPA lineshape with a hard microwave pulse of ~33 ns. Similar to the previous discussion, 110 W of power would be required if a microwave cavity with a quality factor of 200 was implemented. Our laboratory is engaged in designing such MAS microwave resonators. In the following sections, we propose strategies that can be implemented with the inhomogeneous γ_{S}_{1S} frequencies currently available to perform time domain DNP and hyperfine decoupling with MAS.

The adiabatic full passage, or adiabatic inversion is a technique in which the magnetization of the sample is slowly swept 180 degrees from +_{z}_{z}_{1} frequency is much less than the spin resonance linewidth, as is currently the case for EPR in MAS DNP experiments. Adiabatic spin inversions can be accomplished by modulating the frequency of the RF (or microwave) field according to linear, tangential, and hyperbolic secant functions, among others. During the frequency sweep, the magnetization is largely locked along the direction of the effective field, provided the sweep rate is optimized, the γ_{S}_{1S} is the appropriate frequency, and the relaxation times are long enough. One major advantage of adiabatic sweeps over pulsed magnetic resonance is that given a sufficiently strong microwave field, the inversion efficiency is essentially independent of _{1} inhomogeneities [

When performing an adiabatic inversion, one comes up against two competing phenomena as described by Kupce and Freeman [

Here ω_{eff} (_{eff} (_{1}(

θ (

Differentiating with respect to time yields:

If we now rewrite _{1}(

In the special case of a linear sweep, which is the sweep function simulated in Section 4.3, the frequency offset is modulated as a function of time according to the following equation:

Here _{1}/_{1}(

The larger the adiabaticity factor, the better the inversion, so we need only concern ourselves with the minimum value of this function, which occurs when

They suggest keeping the adiabaticity factor at a value of at least 1 for NMR. This is difficult in EPR, however, because the relaxation times for an electron are substantially shorter than for a nucleus, which we take into account in our simulations.

The inversion of SA-BDPA EPR spectrum was calculated using the SPINEVOLUTION simulation package [_{8}-glycerol/H_{2}O/D_{2}O (60/30/10)% glassy matrix at 80 K determined by Smith and his colleagues [_{1S} was 56 ms, and the phase memory time (homogeneous transverse relaxation) was 3.3 μs. The majority of the SA-BDPA linewidth arises from intramolecular hyperfine couplings.

In order to calculate the hyperfine dipolar interactions, the computational package GAMESS was used to obtain the relative unpaired spin densities and the atomic coordinates for each of the spins [

The unpaired electron of SA-BDPA is highly delocalized over a large pi network. This delocalization was approximated by repeating the simulation for 33 different electron positions. Specifically, the atomic coordinates and relative unpaired spin densities at the nuclei of each of the 33 carbons were obtained from a density functional theory calculation of SA-BDPA. The calculation was performed using the GAMESS program, with a B3LYP functional, and a 6-31G** basis set [

To invert the electron spins of SA-BDPA, we simulated a linear frequency sweep from 20.5 MHz below the center of the EPR resonance to 20.5 MHz above the resonance, over a period of 13.75 μs. With a _{γ1}_{1S} of 0.84 MHz, this yields an adiabaticity factor of 1.51 (from _{γ1}_{1S} values. Also, more efficient sweep schemes exist than the linear sweep which yield better adiabaticity factors. These could also be implemented with frequency agile gyrotrons.

Using HFSS we computed the _{γS}_{1S} in each of 75,295 voxels in the sample. The average _{γS}_{1S} value over the entire sample was determined by averaging the values of all the voxels. This came out to a value of 0.84 MHz with 17 W of power input into the MAS stator and was used to optimize the sweep parameters.

The tip of the magnetization vector's trajectory is shown in _{γ1}_{1S} of 0.84 MHz were binned into groups. Electron spin inversions were calculated on each bin, and the results were weighted according to the population distribution shown in

_{γ1}_{1S}=0.84 MHz.

There are some factors that will attenuate the inversion percentages reported here. We first consider the dipolar interaction between electrons on nearby SA-BDPA molecules. In order to investigate this effect, inversion simulations on two SA-BDPA molecules as a function of inter-electron distance were performed. In the simulations, we also included the isotropic and dipolar hyperfine interactions of each electron with two protons within its own molecule. All proton-proton dipolar couplings were neglected. The effect of the inter-electron distance up to 150 Å on the electron spin inversion efficiency is shown in

Another parameter that can affect the inversion efficiency is the phase memory relaxation time. _{2.} A substantial amount of inversion is achieved at 3.3 μs.

Magic angle spinning can also affect the efficiency of the inversion. Simulations were conducted between 0 kHz and 30 kHz in steps of 2 kHz and are shown in

Gyrotrons can provide continuous wave output powers of 10–100 W and are now commonly employed in high field MAS DNP experiments [

In gyrotron oscillators, an electron beam generated by an electron gun deposits microwave energy into an interaction cavity. A heated filament located beneath the surface of an annular barium emitter and a large (~10 to 20 kV) potential overcome the work function of electrons on the surface of the barium emitter. Electrons ejected from the emitter accelerate under the electric potential between the cathode and the anode (_{C} is the cyclotron resonant frequency, _{e} is the rest mass of the electron.

As the electron beam passes through the cavity of the gyrotron, the high energy electrons deposit energy into the interaction cavity at the cyclotron resonant frequency. The velocity in

Alberti et al. demonstrated fast frequency control and an instantaneous bandwidth of > 2 GHz from a voltage tunable gyrotron oscillator designed for DNP experiments at 263 GHz [

We used a frequency dependence on the gyrotron acceleration voltage of 460 MHz/1.2 kV to determine the potentials shown in

We simulated the spatial intensity of microwave fields within a 198 GHz MAS DNP sample. Given these field strengths, we calculated that adiabatic EPR inversions should be possible with currently accessible DNP instrumentation, providing an immediate avenue for broadband hyperfine decoupling in rotating solids. We also show modulations of gyrotron acceleration potentials required to perform both hyperfine decoupling and the frequency modulated cross effect. All together, these results suggest a promising strategy for the implementation of time-domain DNP and hyperfine decoupling in rotating solids.

Research reported in this publication was supported by the National Institute of General Medical Sciences of the National Institutes of Health under Award number DP2GM119131, the National Cancer Institute of the National Institutes of Health under Award number P50CA094056, and the National Science Foundation under Award number STTR-1521314. We thank Mark Conradi, Jagadishwar Sirigiri, Albert Smith, and Marc Caporini for helpful discussions. Computations were performed using the facilities of the Washington University Center for High Performance Computing, which were partially funded by NIH Grants 1S10RR022984-01A1 and 1S10OD018091-01.

^{13}C Nuclear magnetic resonance of polymers spinning at the magic angle

Simulations of the instantaneous magnetic field magnitude in the y′–z′ plane. These intensities are not representative of the _{γS}_{1S} inhomogeneity as they are dependent on time. (a) A computer assisted design (CAD) representation of our MAS DNP probe. (b) A simulation with a Gaussian beam incident on the sample. The mesh in this simulation has 15,000 tetrahedra within the sample volume. (c) The same simulation as (b) but with a higher mesh of 48,000 tetrahedra in the sample. The microwave distribution is almost identical, suggesting the calculation has converged with respect to tetrahedral density within the mesh. (d) A scale with the magnetic field magnitude normalized to 5 W of input power. (e) An expansion of the simulation of (c) shows considerable fine structure within the sapphire. (f) Simulation of a 250 GHz microwave field structure on the model geometry given in Nanni et al. [

(a) Coordinate axis of simulation relative to the main magnetic field. _{m} = 54.74°. (b) Decomposition of a linearly polarized magnetic vector into two oppositely rotating vectors with 1/2 the magnitude.

(a) Phase dependence of the microwave fields within the sample (b) comparison of a time dependent microwave field structure (top) and time independent _{γS}_{1S} distribution (bottom).

Sulfonated BDPA (a) chemical structure and (b) energy minimized structure from GAMESS [

Adiabatic EPR inversions for MAS DNP. (a) Simulation of the EPR magnetization during an adiabatic inversion with a homogeneous _{γS}_{1S}=0.84 MHz. (b) Histogram indicating the distribution of _{γS}_{1S} frequencies in the 3.2 mm rotor at 200 GHz as determined by the HFSS model. (c) The longitudinal magnetization during adiabatic inversions with a homogeneous _{γS}_{1S} (black line) and the inhomogeneous _{γS}_{1S} calculated from the HFSS model (dotted line). (d) Voltage trace from an oscilloscope showing the estimated gyrotron potential required for the electron spin inversions.

The effect of various parameters on electron spin inversion efficiencies. (a) Inter-electron spin separation (b) transverse homogenous electron relaxation time (c) spinning frequency.

High power microwave frequency agility scheme. (a) Flow chart showing setup connectivity from the software input to gyrotron connection for frequency modulation. (b) CAD solid model of the electron gun (purchased from Bridge12) indicating the electron beam, anode, cathode housing and the beam emitter.

High power microwave frequency modulation scheme for the cross effect (CE) under MAS. (a) Anode driving potential modulations at selected frequencies of 10 kHz (black), 100 kHz (blue), and 500 kHz (red). (b) Predicted bandwidth of the microwave frequency output arising from modulating the anode potential overlaid on the nitroxide EPR lineshape at 7 T. The EPR spectrum is from EasySpin [^{14}N hyperfine couplings. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)