^{1,a)}, Galia T. Debelouchina

^{1}, Albert A. Smith

^{1}and Robert G. Griffin

^{1,b)}

### Abstract

Microwave driven dynamic nuclear polarization (DNP) is a process in which the large polarization present in an electron spin reservoir is transferred to nuclei, thereby enhancing NMR signal intensities. In solid dielectrics there are three mechanisms that mediate this transfer—the solid effect (SE), the cross effect (CE), and thermal mixing (TM). Historically these mechanisms have been discussed theoretically using thermodynamic parameters and average spin interactions. However, the SE and the CE can also be modeled quantum mechanically with a system consisting of a small number of spins and the results provide a foundation for the calculations involving TM. In the case of the SE, a single electron–nuclear spin pair is sufficient to explain the polarization mechanism, while the CE requires participation of two electrons and a nuclear spin, and can be used to understand the improved DNP enhancements observed using biradical polarizing agents. Calculations establish the relations among the electron paramagnetic resonance(EPR) and nuclear magnetic resonance(NMR) frequencies and the microwaveirradiation frequency that must be satisfied for polarization transfer via the SE or the CE. In particular, if δ, Δ < ω_{0I }, where δ and Δ are the homogeneous linewidth and inhomogeneous breadth of the EPRspectrum, respectively, we verify that the SE occurs when ω_{ M } = ω_{0S } ± ω_{0I }, where ω_{ M }, ω_{0S } and ω_{0I } are, respectively, the microwave, and the EPR and NMR frequencies. Alternatively, when Δ > ω_{0I } > δ, the CE dominates the polarization transfer. This two-electron process is optimized when and or , where and are the EPR Larmor frequencies of the two electrons. Using these matching conditions, we calculate the evolution of the density operator from electron Zeeman order to nuclear Zeeman order for both the SE and the CE. The results provide insights into the influence of the microwaveirradiation field, the external magnetic field, and the electron−electron and electron−nuclear interactions on DNP enhancements.

The authors thank Dr. Christian Farrar, Dr. Claudiu Filip, Dr. Dinu Iuga, Dr. Chan-Gyu Joo, Dr. Ramesh Ramachandran, Dr. Melanie Rosay, Dr. Vik Bajaj, Dr. Volker Weis, Dr. Gael de Paepe, Dr. Thorsten Maly, Dr. Marvin Bayro, Dr. Bjorn Corzilius, and Alexander Barnes for many enlightening discussions. The initial synthesis and characterization of biradical polarizing agents that stimulated much of this work was a collaboration with Dr. Bruce Yu, Dr. Changsik Song, Dr. Eric Dane, and Dr. Timothy M. Swager. We gratefully acknowledge their contributions. This research was supported by grants from the National Institutes of Health (EB-002804 and EB-002026).

I. INTRODUCTION

II. THEORY

A. Polarization transfer in electron–nuclear spin systems

B. The SE in an electron–nuclear spin system

1. Diagonalization of a two-spin Hamiltonian

2. The microwave Hamiltonian in the EBS

3. Polarization transfer during microwave excitation

C. The CE in an electron–electron–nuclear spin system

1. Diagonalization of a three-spin Hamiltonian

2. The microwave Hamiltonian in the EBS

3. Polarization transfer by the effective microwave excitation

III. EXPERIMENTAL DNP RESULTS

IV. DISCUSSION

A. Frequency matching conditions for DNP

B. Influence of the microwave field strength and the static magnetic field on DNP

C. Implications of quantum mechanical treatments on macroscopic rate equations

V. CONCLUSIONS

### Key Topics

- Polarization
- 93.0
- Microwaves
- 75.0
- Electron paramagnetic resonance spectroscopy
- 29.0
- Nuclear spin
- 18.0
- Magnetic fields
- 17.0

## Figures

A schematic representation of polarization transfer in DNP experiments. Region B represents the observable bulk nuclei. Region A represents the paramagnetic center and nuclei within the diffusion barrier. The electron polarization is transferred to a nuclear spin with a strong hyperfine interaction but residing outside region A. The enhanced polarization then propagates throughout the bulk nuclei via homonuclear spin diffusion.

A schematic representation of polarization transfer in DNP experiments. Region B represents the observable bulk nuclei. Region A represents the paramagnetic center and nuclei within the diffusion barrier. The electron polarization is transferred to a nuclear spin with a strong hyperfine interaction but residing outside region A. The enhanced polarization then propagates throughout the bulk nuclei via homonuclear spin diffusion.

Quantum mechanical picture of the electron–nuclear transitions (dashed arrows) in (a) the SE, (b) the CE and (c) TM mechanisms, which involve one, two or multiple electron spins, respectively. Note that the probabilities of electron–nuclear transitions are always small in the SE but could be large in the CE and TM when degeneracy exists between the states with alternating nuclear spin quantum numbers.

Quantum mechanical picture of the electron–nuclear transitions (dashed arrows) in (a) the SE, (b) the CE and (c) TM mechanisms, which involve one, two or multiple electron spins, respectively. Note that the probabilities of electron–nuclear transitions are always small in the SE but could be large in the CE and TM when degeneracy exists between the states with alternating nuclear spin quantum numbers.

A level diagram of an electron–nuclear system. Transition energies of EPR/NMR and couplings between product spin states are calculated to the first order. The DNP transitions leading to positive and negative enhancements are indicated by the solid and dashed arrows, respectively. Note that *E* _{ ij } = *E* _{ i }− *E* _{ j } and *H* _{ ij } = 〈*i*|*H*|*j*〉.

A level diagram of an electron–nuclear system. Transition energies of EPR/NMR and couplings between product spin states are calculated to the first order. The DNP transitions leading to positive and negative enhancements are indicated by the solid and dashed arrows, respectively. Note that *E* _{ ij } = *E* _{ i }− *E* _{ j } and *H* _{ ij } = 〈*i*|*H*|*j*〉.

Effective frequency vectors in the α and β subspaces of the PSB representation. *A* _{1}, *B* _{1}, and ω_{0I } are assumed to be positive.

Effective frequency vectors in the α and β subspaces of the PSB representation. *A* _{1}, *B* _{1}, and ω_{0I } are assumed to be positive.

A diagram of a three-spin electron–electron–nuclear system showing the energy levels associated with the various spin states. The diagonal and off-diagonal Hamiltonian terms correspond to the EPR/NMR transitions and the couplings between states, respectively. DNP transitions leading to positive and negative enhancements are indicated by the solid and dashed arrows, respectively. Note that *D* _{ d } ≡ 2(*d* − *J*), and *D* _{ o } ≡ −*d*−2*J*.

A diagram of a three-spin electron–electron–nuclear system showing the energy levels associated with the various spin states. The diagonal and off-diagonal Hamiltonian terms correspond to the EPR/NMR transitions and the couplings between states, respectively. DNP transitions leading to positive and negative enhancements are indicated by the solid and dashed arrows, respectively. Note that *D* _{ d } ≡ 2(*d* − *J*), and *D* _{ o } ≡ −*d*−2*J*.

The subspace of the electron–electron–nuclear spin system relevant for the CE mechanism. The tilde indicates the intermediate states that are different from the PSB.

The subspace of the electron–electron–nuclear spin system relevant for the CE mechanism. The tilde indicates the intermediate states that are different from the PSB.

(a) A typical DNP–MAS experimental pulse sequence. Continuous microwave irradiation is used to saturate the electrons in the sample, and the enhanced ^{1}H polarization is transferred via a CP-MAS experiment to the ^{13}C nuclei for observation. A train of saturating pulses on the ^{1}H channel is used to ensure that all of the nuclear polarization arises from DNP. (b) ^{13}C-urea spectra obtained from a sample doped with the biradical BT2E at a concentration of 5 mM (or 10 mM electrons), 90 K, ω_{r}/2π = 3.5 kHz. The top row shows the enhanced signal (with microwave irradiation on) as a function of the irradiation time τ_{DNP}. The bottom row shows the ^{13}C-urea signal in the absence of DNP (microwave irradiation off). The observed enhancement is 175 ± 25. (c) Buildup curve of the enhanced nuclear polarization (filled circles) with a time constant of 5 s, coinciding with the ^{1}H *T* _{ 1 } curve, measured without microwave irradiation (empty circles).

(a) A typical DNP–MAS experimental pulse sequence. Continuous microwave irradiation is used to saturate the electrons in the sample, and the enhanced ^{1}H polarization is transferred via a CP-MAS experiment to the ^{13}C nuclei for observation. A train of saturating pulses on the ^{1}H channel is used to ensure that all of the nuclear polarization arises from DNP. (b) ^{13}C-urea spectra obtained from a sample doped with the biradical BT2E at a concentration of 5 mM (or 10 mM electrons), 90 K, ω_{r}/2π = 3.5 kHz. The top row shows the enhanced signal (with microwave irradiation on) as a function of the irradiation time τ_{DNP}. The bottom row shows the ^{13}C-urea signal in the absence of DNP (microwave irradiation off). The observed enhancement is 175 ± 25. (c) Buildup curve of the enhanced nuclear polarization (filled circles) with a time constant of 5 s, coinciding with the ^{1}H *T* _{ 1 } curve, measured without microwave irradiation (empty circles).

Experimental ^{1}H DNP enhancement profiles for the SE and the CE mechanisms showing the positions of positive and negative enhancement and their dependence on ω_{0S } and ω_{0I }. (a) A typical SE enhancement profile obtained with 40 mM trityl. (b) A CE enhancement profile obtained with 10 mM TOTAPOL (20 mM electrons). Samples were prepared as described in the supporting information, and the position of the EPR spectrum of each radical is shown on the top. The lines connecting the data points are to guide the eye.

Experimental ^{1}H DNP enhancement profiles for the SE and the CE mechanisms showing the positions of positive and negative enhancement and their dependence on ω_{0S } and ω_{0I }. (a) A typical SE enhancement profile obtained with 40 mM trityl. (b) A CE enhancement profile obtained with 10 mM TOTAPOL (20 mM electrons). Samples were prepared as described in the supporting information, and the position of the EPR spectrum of each radical is shown on the top. The lines connecting the data points are to guide the eye.

A level diagram for a triplet-singlet-nuclear spin system. The variables used in the expressions of the transitions and the coupling energies are defined in Sec. II. Because of the forbidden EPR transitions associated with the singlet state, only half of the DNP transitions are allowed. They are marked by the solid and dashed arrows for positive and negative DNP enhancement, respectively.

A level diagram for a triplet-singlet-nuclear spin system. The variables used in the expressions of the transitions and the coupling energies are defined in Sec. II. Because of the forbidden EPR transitions associated with the singlet state, only half of the DNP transitions are allowed. They are marked by the solid and dashed arrows for positive and negative DNP enhancement, respectively.

## Tables

Definitions of the operators used in the diagonalization of the CE Hamiltonian [Eqs. (30), (42), and (47)].

Definitions of the operators used in the diagonalization of the CE Hamiltonian [Eqs. (30), (42), and (47)].

The effective excitation Hamiltonians and the corresponding microwave frequencies that produce positive DNP enhancements are listed. The selection of excitation depends on the microwave bandwidth and amplitude. The maximum enhancement and the Rabi oscillation characterize the time dependence of the nuclear polarization. The results are based on small ζ_{α} and ζ_{β} (moderate electron–electron dipolar interaction) and ξ = 90 ° (full three-spin mixing). The asterisks denote the effective excitation selected by ω_{ M }.

The effective excitation Hamiltonians and the corresponding microwave frequencies that produce positive DNP enhancements are listed. The selection of excitation depends on the microwave bandwidth and amplitude. The maximum enhancement and the Rabi oscillation characterize the time dependence of the nuclear polarization. The results are based on small ζ_{α} and ζ_{β} (moderate electron–electron dipolar interaction) and ξ = 90 ° (full three-spin mixing). The asterisks denote the effective excitation selected by ω_{ M }.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content