Saturation factor

The importance of the saturation factor is evident from the discussion of the previous section: if coupling factors are to be extracted from DNP enhancements,smust be determined. Furthermore, increasing saturation to its maximum value of s = 1

improves sensitivity. However, this situation is more complicated for radicals such as TEMPO because the EPR spectrum typically displays several (two or three) hyperfine lines as a result of strong hyperfine coupling of the unpaired electron with the nearby nitrogen nucleus (15N or14N). Therefore, only partial saturation of non- resonant lines can be achieved, though this improves as the radical concentration is increased.

It was initially assumed that saturation of one resonance of a hyperfine- split EPR spectrum would leave the populations of the other levels unaffected [20], resulting in a saturation factor ofs= 1/3 ors= 1/2 for nitroxides with14N and15N, respectively. This assumption has been made as recently as 1994 when TEMPOL was used in a study of proton-electron double-resonance imaging (PEDRI) [123]. However, since the 1970s it has been realised that this is far too pessimistic a view of electron spin saturation.

In 1977, Bates and Drozdoski [124] developed a mathematical model based on Heisenberg spin exchange which allowed the electron spins of two nitrogen nuclei in different spin states to be exchanged. This was compared with experiments of the radical HPNO (4-hydroxy-2,2,6,6-tetramethyl piperid-1-yloxy) in benzene, and found to be in good agreement. Little work was done in this area and saturation factors for nitroxide radicals were consistently underestimated until Armstrong and Han [111] improved the spin exchange-based model that had been proposed 30 years earlier. They presented a new model which, in addition to spin exchange, incorporated intramolecular nitrogen spin relaxation by explicitly accounting for the populations of the 12 energy levels in the proton-electron-nitrogen system. With this treatment, they derived equations for the maximum saturation in nitroxide radicals with14N (Equation 3.1) and15N (Equation 3.2) which are dependent on the intrinsic electron relaxation ratep(neglecting dipolar contributions from solvent protons and intermolecular dipolar interactions of electrons), the nitrogen relaxation ratewNand

the (concentration dependent) electron exchange rateκ:

smax= 1 3 (2 +wN/p+ 6κ/p)(2 + 3wN/p+ 6κ/p) 4 + (wN/p+ 2κ/p)(wN/p+ 6κ/p) + 2(3wN/p+ 8κ/p) (3.1) smax= 1 2 1 +wN/p+ 2κ/p 1 +wN/2p+κ/p (3.2) In their analysis they noted that, whilst electron spin exchange is only really signifi- cant at higher radical concentrations, nitrogen spin relaxation effects are concentra- tion independent and significant even when little radical is used. They found their theory to be consistent with data they collected at 0.35 T.

An alternative for calculating the electron spin saturation in a DNP experi- ment was proposed in 2009 by Sezer et al. [125]. Here, semi-classical theory was used to treat the electron-nitrogen interaction, whilst ignoring the relaxation effects of coupling to the proton spin. This method, for the first time, explicitly accounted for spin-spin relaxation of the electron and obtained DNP enhancement values (usingξ from Reference [40]) as a function of frequency that closely replicate experimental values at 9.2 T. This approach has been shown to describe experiments well pro- vided the input parameters can be determined: the frequency and strength of the microwave field, the electron spin-lattice and spin-spin relaxation times, the Heisen- berg exchange rate and the magnetic tensors. Independent measurements can often be made of these parameters using EPR, however, the information is not always obtainable for a given setup, especially at high-field. In the example given by Sezer et al., approximations had to be made forT1S and the microwave field strength. The

usefulness of this as a general approach is further brought into question by the fact that it is only valid in the regime of fast tumbling. This is the case for most high- field liquid DNP experiments at present, which often utilise small nitroxide radicals, but the calculations may break down for radicals tethered to macromolecules or larger polarising agents (such as the biradicals proposed to induce a cross-effect, see Section 3.4).

Finally, after years of theoretical debate in the literature, T¨urke et al. [31] applied an experimental strategy for determinings— using pulsed ELDOR (electron double resonance) to pump one hyperfine EPR line whilst monitoring a coupled hyperfine line. This was demonstrated with TEMPONE-D, 15_{N at 0.35 T and}

found to give a coupling factor ofξ = 0.33±0.02 consistent with NMRD and MD calculations (see Section 3.5.1), providing further validation for those approaches.

As demonstrated by the above discussion, determining electron spin satu- ration for nitroxide radicals is non-trivial, and far from the simplistic view of the early 1960s. The work of T¨urke, Sezer and co-workers presents a very strong case for determination of s through either NMRD, MD or pulsed ELDOR, at least at the field strengths currently available. However, avoiding the issue of saturation (by comparing distinguishable nuclei within the same sample and calculating ξ ratios) may be a necessary tactic for situations where such measurements and calculations are not readily available. This has been demonstrated by Krummenacker et al. [98] and is adopted by the author in Chapter 6.

Other than extracting precise values ofsfrom theory and experiment, simply maximising it is also an important goal. To this end, H¨ofer et al. [13] attempted a ‘dual irradiation’ DNP experiment with 2.5 mM 2H/15N TEMPONE in water

at 0.35 T, where irradiation of both hyperfine lines was attempted simultaneously in an effort to boost the NMR signal. This was accomplished with two separate microwave sources and two separate amplifiers combining to give a maximum total power of 1.5 W. However, they found no benefit to this method, providing yet more data supporting the mixing of hyperfine lines via electron spin exchange and nuclear spin relaxation allowing s to approach 1. A similar approach has been attempted at 3.4 T for the first time, and is presented in Chapter 7.