We calculated the electromagnetic field structure for the sample geometry shown in Fig. 1a. Inside the stator is a 5-turn RF coil with a wire diameter of 0.686 mm. The rotor sheath is a sapphire cylinder with an outside diameter of 3.175 mm and has a wall thickness of 0.55 mm. Further details of the model geometry are given in the supplementary information. We assumed that all microwave power leaving the volume shown in Fig. 1a was lost to its surroundings. Our simulation models the sample as a frozen glycerol and water mixture. We simulate the effect of the sample on the 198 GHz field structure by setting the real part of the dielectric constant (εr′) to 3.5 and the loss tangent to 0.007, as previously determined by Nanni et al. [39]. The relative dielectric constant is defined as: (1)εr=ε′+iε′′ε0

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 |E₀|. Only the initial electric field magnitude, |E₀|, needs to be given because the magnetic component, |H₀|, is proportional and orthogonal to |E₀| 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 Supplementary information): (2)∣E0∣=4P0πε0cw02=15,425V∕m where ε₀ is the permittivity of free space, c is the speed of light, and w₀ = 3.175 mm is the beam waist of the Gaussian beam entering the stator.

To determine the _γSB_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 B₀ that contribute to the _γSB_1S. Referring to Fig. 2a we see that in the primed coordinate frame (HFSS simulation axis): (3)y^=(0sin(θm)cos(θm))

x^′=−x^, and θ_m is the magic angle. Note that z^′ is the incident axis of the Gaussian beam and y^′ is the long rotor axis of the sample (see Fig. 2a).

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. [40]: (4)〈B⊥〉=1V∫V∣B⊥∣dV=μV∫V∣H⋅x^∣2+∣H⋅y^∣2dV

Note, this differs slightly from the procedure outlined in Nanni et al. in which the volume root mean square (RMS) is computed [39].

We also assume μ is constant throughout the entire sample and is approximated by the permeability of free space, μ₀ = 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 supplementary information.

When we integrate over the entire sample volume of 18.6 μL we find that 5.0 W of power gives an average γ_SB_1S of 0.46 MHz. 17 W of input power into the stator is required to generate the average γ_SB_1S of 0.84 MHz which is now commonly used in theoretical treatments of MAS DNP [41,42].

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 Fig. 2b. The component rotating with the spins contributes to the nutation; the other can be neglected. This means that the average B_1S over the sample given is B_⊥/2. It should be noted that if B_⊥ is given as the RMS, B_⊥,RMS of a linearly oscillating field then, B1S=B⊥,RMS∕2. This is the treatment given in Nanni et al. [39] and Macor et al. [40].

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 [39,40]. For dielectric heating the generated power density per volume is: (5)W=12ω∊r′′∊0∣E∣2 where ω is the angular frequency of the incoming linearly polarized light. The power transferred to heat for a given volume of dielectric material is the integral: (6)∫VWdV=12ω∊r′′∊0∫V∣E∣2dV

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 [13,39,43].

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 Fig. 1f. The field structures are considerably different with the 3.2 mm rotor at 198 GHz, and the 4 mm rotor at 250 GHz. Calculating the microwave structure at the appropriate geometry and frequency are important aspects to understanding and optimizing MAS DNP performance.

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_SB_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_sB_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 0.49=0.70∕2 MHz, which accounts for the circularly polarized component at 250 GHz.

We calculated the standard deviation of the peak transverse magnetic field normalized to the mean transverse magnetic field within the sample as: (7)σ^=〈B⊥2〉−〈B⊥〉2〈B⊥〉

The more homogenous γ_SB_1S distribution will have a standard deviation closer to 0. For our 198 GHz system we found σ^=0.28. For the 250 GHz system outlined we found σ^=0.75. This suggests that the transverse magnetic field is more homogenous within the sample for our 198 GHz system than the 250 GHz/4 mm probe absent of a silver coating applied to the inside of the stator.

The instantaneous microwave intensities shown in Fig. 1 do not accurately represent the γ_SB_1S distribution. This is because the microwave field intensity is time dependent and follows the 5 ps oscillation period of the 198 GHz microwaves. Fig. 3a clearly shows the microwave field intensity within the sample is strongly dependent on the phase of the microwaves as they propagate. We show a time dependent microwave field structure and the time independent γ_SB_1S distribution in Fig. 3b. The time (and phase) independent field structure is similar to the field structures shown in Nanni et al. [39]. Clearly the γ_SB_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_SB_1S frequencies, which we account for in our simulations of EPR inversions in Section 4.

When performing an adiabatic inversion, one comes up against two competing phenomena as described by Kupce and Freeman [57]. The effective field must be swept slowly enough that the magnetization of the sample is locked along it. However, if it is varied too slowly the extent of relaxation, particularly via spin– spin relaxation, becomes prohibitively large. The first condition is commonly described by a parameter known as the “adiabaticity factor” and is defined according to the following equation [57]: (8)Q(t)=∣ωeff(t)dθ(t)dt∣

Here ω_eff (t) is the effective field in units of angular frequency, and dθ (t)/dt is the rate at which the angle of the effective field with the x–y plane is changing with time. A useful form of this equation can be realized by recognizing that ω_eff (t), the frequency offset, Δω(t, and the strength of the microwave field, ω₁(t), are related in the following manner: (9)ωeff(t)=[Δω(t)2+ω1(t)2]1∕2

θ (t) can also be written in terms of the frequency offset and microwave strength: (10)θ(t)=tan−1(Δω(t)ω1(t))

Differentiating with respect to time yields: (11)dθ(t)dt=ω1(t)dΔω(t)dt−Δω(t)dω1(t)dtω1(t)2+Δω(t)2

If we now rewrite Q (t) in terms of Δω(t) and ω₁(t), we obtain a general formula for the adiabaticity factor in terms of experimental variables: (12)Q(t)=∣[Δω(t)2+ω1(t)2]32ω1(t)dΔω(t)dt−Δω(t)dω1(t)dt∣

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: (13)Δω(t)=±k(t−12τ)

Here k is the sweep rate, τ is the total time of the sweep and t runs from 0 to τ. Additionally, the strength of the microwave field is held constant, so dω₁/dt = 0 and the time dependence of ω₁(t) can be dropped. Although there could be a substantial voltage and frequency dependence on the microwave power over a sweep range of 750 MHz that would be required to invert a nitroxide lineshape, the microwave power over a 40 MHz band, which is all that needed to invert a SA-BDPA resonance, is nearly constant during the frequency sweep. So for the inversion of SA-BDPA, the strength of the microwave field is assumed to be held constant. Putting these values into Eq. (12) renders the following for a linear sweep of the frequency offset: (14)Q(t)=∣[k2(t−τ∕2)2+ω12]32±ω1k∣

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 t=12τ. Here the adiabaticity factor is equal to the expression given by Kupce and Freeman [57]: (15)Qminimum=ω12k

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 [64]. We used EPR parameters previously determined at 140 GHz of 40 mM SA-BDPA frozen in a d₈-glycerol/H₂O/D₂O (60/30/10)% glassy matrix at 80 K determined by Smith and his colleagues [48,56,65]. The inhomogeneous linewidth was 30 MHz, the spin lattice T_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.

Fig. 4a shows the 4 protons that were included in the simulations. There are 8 strongly coupled protons, each with a isotropic hyperfine interaction of ~5.1 MHz. We included the 4 protons shown in Fig. 4a in the spin simulations as a representative sample due to the computational demand of including more spins. An isotropic hyperfine interaction of 5.1 MHz was used in the simulations according to Haze et al. [56], and the dipolar hyperfine interaction was calculated explicitly.

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 [63]. The geometry of the energy minimized SA-BDPA structure is shown in Fig. 4b. Dipolar coupling between the hydrogen spins were neglected to shorten the calculation time.

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 [63]. Spin densities were obtained via Lowdin population analysis. Simulations were carried out for each electron position, weighted by the normalized spin density, and then summed.

Fig. 4c shows results from SPINEVOLUTION compared to those from SPINACH [66]. Both results approximate the Lorentzian profile of experimental results close enough for our theoretical treatment considering our goal is to demonstrate that broadband electron spin inversions at 198 GHz are possible with currently available microwave sources. Including more protons on the SA-BDPA molecule, along with those of the ice surrounding the molecule, would improve this agreement. Ultimately, the choice of simulation package comes to one of computation time. A five-spin system in SPINEVOLUTION is much faster with reasonable accuracy and thus allows for the exploration of more inversion parameters.

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 _γ1B_1S of 0.84 MHz, this yields an adiabaticity factor of 1.51 (from Eq. (15)). This adiabaticity factor can be improved with higher _γ1B_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 _γSB_1S in each of 75,295 voxels in the sample. The average _γSB_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 Fig. 5a. The inversion efficiency achieved with an entirely homogenous field of 0.84 MHz was 57%. In order to simulate the inhomogeneity of the microwave field, the microwave strengths generated in HFSS with an average _γ1B_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 Fig. 5b. This process was done using both 50 and 100 bins, both of which resulted in 46% spin inversion, indicating the binning of the microwave field inhomogeneity converges at less than 50 bins.

Fig. 5c shows the time dependencies of the z components of the SA-BDPA magnetization under an adiabatic inversion in both the homogeneous, average field, and the real, inhomogeneous field. The similar magnetization profiles demonstrate that frequency swept microwave pulses are a robust strategy to invert electron spins in MAS experiments. To connect our theoretical analysis to experiments, we demonstrate in Fig. 5d the gyrotron acceleration profile required to produce the frequency sweep and electron spin inversion. All subsequent simulations discussed from here employ a homogeneous, average field of _γ1B_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 Fig. 6a. It can be seen that there is no inversion up to about 12 Å of separation, at which point the inversion efficiency achieves its maximum value rather quickly. Using a simple cubic lattice of electrons as a model, the average distance between an electron and its nearest neighbor at 40 mM is about 35 Å. As can be seen from the simulation, at this distance the inversion is essentially the same as the isolated case.

Another parameter that can affect the inversion efficiency is the phase memory relaxation time. Fig. 6b shows the dependence of the inversion efficiency on T_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 Fig. 6c. We calculated an inversion efficiency of 56% at 5 kHz and 30 kHz. Slow to moderate spinning frequencies seem to have a negligible impact on the inversion efficiency.

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 (Fig. 7b) and approach relativistic speeds. In the field of the gyrotron's super-conducting magnet, the electrons follow a tight helical path about the magnetic field lines with a cyclotron resonant frequency of: (16)Ωc=eB0∕m where Ω_C is the cyclotron resonant frequency, e is the charge of an electron, and m is the relativistic mass of an electron. The relativistic mass is: (17)m=me1−ν2∕c2 where v is the velocity of the electron, c is the speed of light, and m_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 Eq. (17) is a function of both the magnetic field and the electric field potential applied across the electron gun to accelerate the electron beam. Therefore, a modulation of the electric potential across the electron gun will lead to a change in the cyclotron resonant frequency and thus provide tuning of the microwave frequency, given the interaction cavity can support continuous frequency tuning [15].

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 [68,69]. Idehara et al. recently demonstrated high power levels (> 100 W) during frequency modulations for DNP experiments at 460 GHz [70,71]. Following these demonstrations, we have integrated the control of the gyrotron anode potential into the NMR-EPR console (TecMag, TX) to allow for flexible control and synchronization of NMR pulse sequence elements and the rotor period with microwave pulses and EPR control. The NMR console has an arbitrary waveform generator (AWG) with a 10 ns step size that drives a high voltage amplifier. The high voltage amplifier (TREK, model 5/80) can generate ± 5 kV potential with ± 80 mA current and has a slew rate of 1 kV/μs. Fig. 7a shows how the anode potential is controlled from the NMR EPR console. This integrated EPR control gives us flexibility to implement the swept voltage profiles shown in Fig. 5d for adiabatic EPR spin inversion of the SA-BDPA line-shape and voltage modulations shown in Fig. 8a for the frequency modulated cross effect.

We used a frequency dependence on the gyrotron acceleration voltage of 460 MHz/1.2 kV to determine the potentials shown in Figs. 5d and 8a [72]. Following Han and colleagues [73], we expect to correct for deviations from this linear response by altering the controlling voltage waveform. Fig. 8a shows the estimated voltage modulations required for the frequency modulated cross effect. Fig. 8b overlays predicted microwave frequency bandwidths of different modulation frequencies on the nitroxide lineshape at 7 T calculated with EasySpin [44]. The smaller voltage magnitude of the 500 KHz modulation compared to 10 kHz and 100 kHz reflects the limitation of the microwave frequency bandwidth at higher modulation frequencies due to the slew rate of the TREK HV amplifier.