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Quantum mechanical theory of dynamic nuclear polarization in solid dielectrics
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Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

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〉.

Image of FIG. 4.
FIG. 4.

Effective frequency vectors in the α and β subspaces of the PSB representation. A 1, B 1, and ω0I are assumed to be positive.

Image of FIG. 5.
FIG. 5.

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(dJ), and D o ≡ −d−2J.

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

(a) A typical DNP–MAS experimental pulse sequence. Continuous microwave irradiation is used to saturate the electrons in the sample, and the enhanced 1H polarization is transferred via a CP-MAS experiment to the 13C nuclei for observation. A train of saturating pulses on the 1H channel is used to ensure that all of the nuclear polarization arises from DNP. (b) 13C-urea spectra obtained from a sample doped with the biradical BT2E at a concentration of 5 mM (or 10 mM electrons), 90 K, ωr/2π = 3.5 kHz. The top row shows the enhanced signal (with microwave irradiation on) as a function of the irradiation time τDNP. The bottom row shows the 13C-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 1H T 1 curve, measured without microwave irradiation (empty circles).

Image of FIG. 8.
FIG. 8.

Experimental 1H DNP enhancement profiles for the SE and the CE mechanisms showing the positions of positive and negative enhancement and their dependence on ω0S and ω0I . (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.

Image of FIG. 9.
FIG. 9.

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.


Generic image for table
Table I.

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

Generic image for table
Table II.

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 ζα and ζβ (moderate electron–electron dipolar interaction) and ξ = 90 ° (full three-spin mixing). The asterisks denote the effective excitation selected by ω M .


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Quantum mechanical theory of dynamic nuclear polarization in solid dielectrics