Advantages of electron-spin based detection of nuclear frequency spectra Nuclear frequency spectra in the liquid (Section 4.3.2) and solid states (4.3.4) exhibit much better hyperfine resolution than EPR spectra, because the former spectra feature fewer and narrower lines. In fact, small hyperfine couplings to ligand nuclei in metal complexes are not usually resolved in EPR spectra and only the largest hyperfine couplings may be resolved in solid-state EPR spectra. The nuclear frequency spectra cannot be measured by a dedicated NMR spectrometer because they extend over several Megahertz to several tens of Megahertz, whereas NMR spectrometers are designed for excitation and detection bandwidths of a few tens of kilohertz. Furthermore, electron spin transitions have 660 times more polarization than proton transitions and more than that for other nuclei. Their larger magnetic moment also leads to higher detection sensitivity. It is thus advantageous to transfer polarization from electron spins to nuclear spins and to backtransfer the response of the nuclear spins to the electron spins for detection. Two classes of experiments can achieve this, electron nuclear double resonance (ENDOR) experiments, discussed in Section \(8.1\) and electron spin echo envelope modulation (ESEEM) experiments discussed in Section 8.2. Types of ENDOR experiments An ENDOR experiment can be performed with strong CW irradiation of both electron and nuclear spins. In this CW ENDOR experiment, an electron spin transition is partially saturated, \(S \gg 1\) in Eq. (7.5). By driving a nuclear spin transition that shares an energy level with the saturated transition, additional relaxation pathways are opened up. The electron spin transition under observation is thus partially desaturated, and an increase in the EPR signal is observed. The experiment is performed at constant magnetic field with strong microwave irradiation at a maximum of the first-derivative absorption spectrum (i.e. the CW EPR spectrum) and the EPR signal is recorded as a function of the frequency of additional radiofrequency irradiation, which must fulfill the saturation condition \(S \gg 1\) for the nuclear spins. Usually, the radiofrequency irradiation is frequency modulated and the response is detected with another phase-sensitive detector, which leads to observation of the first derivative of the nuclear frequency spectrum. The CW ENDOR experiment depends critically on a balance of relaxation times, so that in the solid state sufficient sensitivity may only be achieved in a certain temperature range. Furthermore, simultaneous strong continuous irradiation by both microwave and radiofrequency, while keeping resonator frequency and temperature constant, is experimentally challenging. Therefore, CW ENDOR has been largely replaced by pulsed ENDOR techniques. However, for liquid solution samples CW ENDOR is usually the only applicable ENDOR technique. The conceptually simplest pulsed ENDOR experiment is Davies ENDOR (Section 8.1.3), where saturation of the EPR transition is replaced by inversion by a \(\pi\) pulse (Fig. 8.1(a)). A subsequent radiofrequency \(\pi\) pulse, which is on-resonant with a transition that shares a level with the inverted EPR transition, changes population of this level and thus polarization of the EPR observer transition. This polarization change as a function of the radiofrequency is observed by a Hahn echo experiment on the observer transition. The approach works well for moderately large hyperfine couplings \((>3 \mathrm{MHz})\) , in particular for \({ }^{14} \mathrm{~N}\) nuclei directly coordinated to a transition metal ion or for protons at hydrogen-bonding distance or distances up to about \(4 \AA\) . As we shall see in Section 8.1.3, the experiment is rather insensitive for very small hyperfine couplings. The smallest hyperfine couplings can be detected with the Mims ENDOR experiment that is based on the stimulated echo sequence \((\pi / 2)-\tau-(\pi / 2)-T-(\pi / 2)-\tau-e c h o(\) Fig. \(8.1(\) b) \()\) ). The preparation block \((\pi / 2)-\tau-(\pi / 2)\) creates a polarization grating of the functional form \(A\left(\Omega_{S}\right) \cos \left(\Omega_{S} \tau\right)\) , where \(A\left(\Omega_{S}\right)\) is the EPR absorption spectrum as a function of the resonance offset \(\Omega_{S}\) and \(\tau\) is the delay between the two \(\pi / 2\) microwave pulses. A radiofrequency \(\pi\) pulse with variable frequency is applied during time \(T\) when the electron spin magnetization is aligned with the \(z\) axis. If this pulse is on resonant with a nuclear transition that shares a level with the observer EPR transition, half of the polarization grating is shifted by the hyperfine splitting \(A_{\text {eff }}\) , as will also become apparent in Section 8.1.3. For \(A_{\text {eff }} \tau=2(k+1) \pi\) with integer \(k\) the polarization grating is destroyed by destructive interference. Since the stimulated echo is the free induction decay (FID) of this polarization grating, it is canceled by a radiofrequency pulse that is on resonant with a nuclear transition. It is apparent that the radiofrequency \(\pi\) pulse has no effect for \(A_{\text {eff }} \tau=2 k \pi\) with integer \(k\) , where the original and frequency-shifted gratings interfere constructively. Hence, the Mims ENDOR experiment features blind spots as a function of interpulse delay \(\tau\) . These blind spots are not a serious problem for very small hyperfine couplings \(A_{\mathrm{eff}} \ll \pi / \tau\) . Note however that the first blind spot corresponds to \(A_{\mathrm{eff}}=0\) . Hence, long interpulse delays \(\tau\) are required in order to detect very small hyperfine couplings, and this leads to strong echo attenuation by a factor \(\exp \left(-2 \tau / T_{2}\right)\) due to electron spin transverse relaxation. It can be shown that maximum sensitivity for very small couplings is attained approximately at \(\tau=T_{2}\) . Davies ENDOR The Davies ENDOR experiment is most easily understood by looking at the polarization transfers. At thermal equilibrium the electron spin transitions (red and pale red) are much more strongly polarized than the nuclear spin transitions (Fig. 8.2(a)). Their frequencies differ by an effective hyperfine splitting \(A_{\text {eff }}\) to a nuclear spin \(I=1 / 2\) that is color-coded blue. The first microwave \(\pi\) pulse is transition-selective, i.e., it has an excitation bandwidth that is smaller than \(A_{\text {eff }}\) . Accordingly, it inverts only one of the two electron spin transitions. We assume that the \(|\beta \alpha\rangle \leftrightarrow|\alpha \alpha\rangle\) transition (red) is inverted and the \(|\beta \beta\rangle \leftrightarrow|\alpha \beta\rangle\) transition (pale red) is not inverted; the other case is analogous. Such transition-selective inversion leads to a state of two-spin order, where all individual transitions in the two-spin system are polarized. However, the two electron spin transitions are polarized with opposite sign and the two nuclear transitions are also polarized with opposite sign (Fig. 8.2(b)). Now a radiofrequency \(\pi\) pulse is applied. If this pulse is not resonant with a nuclear transition, the state of two-spin order persists and the observer electron spin transition (red) is still inverted. The radiofrequency pulse is also transition-selective. We now assume that this pulse inverts the \(|\alpha \alpha\rangle \leftrightarrow|\alpha \beta\rangle\) transition (blue); the other case is again analogous. After such a resonant radiofrequency pulse, the two nuclear transitions are polarized with equal sign and the two electron spin transitions are saturated with no polarization existing on them (Fig. 8.2(c)). After the radiofrequency \(\pi\) pulse a microwave Hahn echo sequence is applied resonant with the observer transition (Fig. 8.1(a)). If the radiofrequency pulse was off resonant (situation as in Fig. 8.2(b)), an inverted echo is observed. If, on the other hand, the radiofrequency pulse was on resonant (situation as in Fig. 8.2(c)) no echo is observed. In practice, polarization transfers are not complete and a weak echo is still observed. However, an on-resonant radiofrequency pulse causes some recovery of the inverted echo. If the radiofrequency is varied, recovery of the inverted echo is observed at all frequencies where the radiofrequency pulse is resonant with a nuclear transition. Further understanding of Davies ENDOR is gained by considering an inhomogeneously broadened EPR line (Fig. 8.3). In such a line with width \(\Gamma_{\text {inhom }}\) , each individual spin packet with much narrower width \(\Gamma_{\text {hom }}\) can, in principle, be selectively excited. A long rectangular \(\pi\) pulse inverts the on-resonant spin packet and partially inverts spin packets roughly over a bandwidth corresponding to the inverse length of the pulse. In Davies ENDOR, pulse lengths between 50 and \(400 \mathrm{~ns}\) , corresponding to excitation bandwidths between 20 and \(2.5 \mathrm{MHz}\) are typical. Such a pulse creates an inversion hole centered at the microwave frequency \(\omega_{\text {mw }}\) . In an \(S=1 / 2, I=1 / 2\) electron-nuclear spin system, two on-resonant spin packets exist, those where \(\omega_{\mathrm{mw}}\) is the frequency of the \(|\beta \alpha\rangle \leftrightarrow|\alpha \alpha\rangle\) transition and those where it is the frequency of the \(|\beta \beta\rangle \leftrightarrow|\alpha \beta\rangle\) transition. For the former spin packet, inversion of the nuclear spin from the \(|\beta\rangle\) to the \(|\alpha\rangle\) state increases the EPR frequency by the effective hyperfine splitting \(A_{\text {eff }}\) , whereas for the latter packet, inversion from the \(|\alpha\rangle\) to the \(|\beta\rangle\) state decreases it by \(A_{\text {eff }}\) . In both cases half of the inversion hole is shifted to a side hole, leaving a saturation hole at \(\omega_{m w}\) and creating a saturation side hole. The saturation center holes of the two spin packets coincide in frequency and combine to a saturation hole in the inhomogeneously broadened line. At the side hole frequencies \(\omega_{\mathrm{mw}} \pm A_{\mathrm{eff}}\) , only one of the two spin packets contributes to the hole, so that the side holes are only half as deep. The Hahn echo subsequence in the Davies ENDOR sequence must have a detection bandwidth that covers only the central hole (pale red in Fig. \(8.3(\mathrm{~d})\) ), since no ENDOR effect would be observed if the side hole would also be covered. For this purpose, the detection bandwidth of the Hahn echo sequence can be limited either by using sufficiently long microwave pulses or by using a sufficiently long integration gate for the inverted echo. In any case, a Davies ENDOR effect is only be observed if \(A_{\text {eff }}\) exceeds the width of the original inversion hole. The smaller \(A_{\mathrm{eff}}\) , the longer the first inversion pulse needs to be and the fewer spin packets contribute to the signal. In general, hyperfine splittings much smaller than the homogeneous linewidth \(\Gamma_{\text {hom }}=1 / T_{2}\) in the EPR spectrum cannot be detected. In practice, Davies ENDOR becomes very insensitive for \(\pi\) pulse lengths exceeding 400 ns. If broadening of the inversion hole by electron spin relaxation is negligible, the suppression of signals with small hyperfine couplings in Davies ENDOR can be described by a selectivity parameter \[\eta_{\mathrm{S}}=\frac{A_{\mathrm{eff}} t_{\pi}^{(1)}}{2 \pi}\] where \(t_{\pi}^{(1)}\) the length of the first mw \(\pi\) pulse. Maximum absolute ENDOR intensity \(V_{\max }\) is obtained for \(\eta_{\mathrm{S}}=\sqrt{2} / 2\) . As a function of \(\eta_{\mathrm{S}}\) , the absolute ENDOR intensity is given by \[V\left(\eta_{\mathrm{S}}\right)=V_{\max }\left(\frac{\sqrt{2} \eta_{\mathrm{S}}}{\eta_{\mathrm{S}}^{2}+1 / 2}\right)\] The hyperfine contrast selectivity described by Eq. (8.2) can be used for spectral editing. For instance, \({ }^{14} \mathrm{~N}\) ENDOR signals of directly coordinated ligand nitrogen atoms in transition metal complexes with \(A_{\text {eff }}\) of the order of \(20-40 \mathrm{MHz}\) overlap with \({ }^{1} \mathrm{H}\) ENDOR signals of weakly coupled ligand protons at X-band frequencies. At an inversion pulse length of about \(50 \mathrm{~ns}^{1} \mathrm{H}\) ENDOR signals are largely suppressed. The sensitivity advantage of Mims ENDOR for very small hyperfine couplings can also be understood in the hole burning picture. Instead of a single center hole, a preparation block \((\pi / 2)-\tau-(\pi / 2)\) with nonselective microwave pulses creates a polarization grating that can be imagined as a comb of many holes that are spaced by frequency difference \(1 / \tau\) . The width of each hole is approximately \(1 / 2 \tau\) . The width of the comb of holes is determined by the inverse length of the non-selective \(\pi / 2\) pulses, which are typically \(10 \mathrm{~ns}\) long. For small couplings, where \(t_{\pi}^{(1)}\) in Davies ENDOR needs to be very long, more than an order of magnitude more spin packets take part in a Mims ENDOR experiment than in a Davies ENDOR experiment. The Mims ENDOR effect arises from the shift of one quarter of the polarization grating by frequency difference \(+A_{\text {eff }}\) and one quarter of the grating by \(-A_{\text {eff }}\) . The shifted gratings interfer with the grating at the center frequency. Depending on \(A_{\text {eff }}\) and on the periodicity \(1 / \tau\) of the grating, this interference is destructive (ENDOR effect) or constructive (no ENDOR effect).