## Dynamic nuclear polarization with an inhomogeneously broadened ESR line. I. Theory

##### Abstract

Theoretical investigations of the dynamic nuclear polarization (DNP) and the nuclear spin-lattice relaxation time Tn in diamagnetic dielectric crystals diluted with electron paramagnetic impurities are reported here. The steady-state DNP enhancement Ess and the DNP pump time τDNP are calculated for the solid effect (SE), the cross effect (CE), and DNP by the electron dipole-dipole reservoir (EDDR), assuming (i) a predominantly inhomogeneously broadened (IHB) ESR line with uniform spin-packet widths (for SE and CE, but not EDDR DNP), (ii) rapid nuclear spin diffusion, (iii) the electron and nuclear high-temperature limit, (iv) no phonon bottleneck, and (v) isotropic electron impurities or axially symmetric electrons whose symmetry axis (c axis) is aligned parallel to the applied magnetic field H⃗ 0 DNP results for IHB SE, which include DNP leakage factors, predict Ess comparable to the ideal enhancement in favorable cases, with Ess vs H0 curves not proportional to the ESR line-shape derivative at high microwave powers, contrary to early theories of IHB SE. The effects of electron jumps between spin packets are discussed. DNP and Tn are calculated for the cross effect for both the well-resolved and unresolved limits. We get larger SE DNP and smaller CE DNP for the unresolved CE than do Hwang and Hill, and consider CE DNP leakage factors they omitted. The inequality Tn2≤τDNP≤Tn is found for the well-resolved CE, while τDNP≈Tn is estimated for the unresolved CE, different from the general SE case where τDNP≪Tn often occurs. DNP by EDDR is extended from the pure homogeneously broadened (HB) ESR limit to the IHB case with fast spectral diffusion, as suggested by Abragam and Borghini, and is further extended from electronic isotropy to axially symmetric electrons with the c axis parallel to H⃗ 0 Various limits of Ess and τDNP are compared and contrasted for SE, CE, and EDDR DNP, to enable experimenters to differentiate these three DNP mechanisms, if possible. We propose a three-spectral-region model for wide ESR lines, which may exhibit IHB behavior in the wings of the line, HB EDDR effects near the ESR absoption peak, and CE effects in between.