Intermolecular contributions, filtration effects and signal composition of SIFTER (single-frequency technique for refocusing)

Abstract

To characterize structure and molecular order in the nanometre range, distances between electron spins and their distributions can be measured via dipolar spin-spin interactions by different pulsed electron paramagnetic resonance experiments. Here, for the single-frequency technique for refocusing dipolar couplings (SIFTER), the buildup of dipolar modulation signal and intermolecular contributions is analysed for a uniform random distribution of monoradicals and biradicals in frozen glassy solvent by using the product operator formalism for electron spin $S=1/2$. A dipolar oscillation artefact appearing at both ends of the SIFTER time trace is predicted, which originates from the weak coherence transfer between biradicals. The relative intensity of this artefact is predicted to be temperature independent but to increase with the spin concentration in the sample. Different compositions of the intermolecular background are predicted in the case of biradicals and in the case of monoradicals. Our theoretical account suggests that the appropriate procedure of extracting the intramolecular dipolar contribution (form factor) requires fitting and subtracting the unmodulated part, followed by division by an intermolecular background function that is different in shape. This scheme differs from the previously used heuristic background division approach. We compare our theoretical derivations to experimental SIFTER traces for nitroxide and trityl monoradicals and biradicals. Our analysis demonstrates a good qualitative match with the proposed theoretical description. The resulting perspectives for a quantitative analysis of SIFTER data are discussed.

Figure 1

Pulse sequences used in experiments are shown in (a). The time axis in SIFTER traces is $t={\mathit{\tau }}_{\mathrm{1}}-{\mathit{\tau }}_{\mathrm{2}}$ , and the interpulse delays are incremented/decremented by d $t$ to keep the total transverse evolution time $\mathrm{2}{\mathit{\tau }}_{\mathrm{0}}=\mathrm{2}({\mathit{\tau }}_{\mathrm{1}}+{\mathit{\tau }}_{\mathrm{2}})$ constant. The chemical structures of the studied compounds are depicted in (b). Nitroxide monoradical (TEMPO) (1), trityl monoradical (2), nitroxide biradical (3) and trityl biradical (4). TIPS is triisopropylsilyl.

Figure 2

Intra- and intermolecular electron–electron coupling frequencies for the case of a frozen solution of biradicals. (a) Intramolecular dipolar frequency for the target biradical, containing the A spin ( ${\mathit{\omega }}_{\mathrm{0}}$ ) and the intramolecular dipolar frequencies in other biradicals ( ${\stackrel{\mathrm{~}}{\mathit{\omega }}}_i$ ), both spins in these biradicals are B spins. (b) Intermolecular dipolar frequencies ${\mathit{\omega }}_i$ describe couplings of A spin to remote B spins. (c) The dipolar frequencies ${\stackrel{\mathrm{~}}{\mathit{\omega }}}_{i,j}$ describe intermolecular dipolar couplings between B spins. The case of a frozen solution of monoradicals is obtained by keeping only the intermolecular dipolar frequencies and dropping the intramolecular ones. Only a selection of intermolecular couplings is shown in the figures to reduce visual crowding.

Figure 3

Overview of the buildup of the non-modulated part of the SIFTER signal. (a) SIDRE pulse sequence, which omits coherence transfer by the central $(\mathit{\pi }/\mathrm{2}{)}_y$ pulse. (b) SIFTER pulse sequence with coherence transfer. The two respective pulse sequence blocks show a difference in evolution with and without coherence transfer by the central $(\mathit{\pi }/\mathrm{2}{)}_y$ pulse. (c) Sketch of the SIDRE signal $B_{\mathrm{S}}({\mathit{\tau }}_{\mathrm{1}},{\mathit{\tau }}_{\mathrm{2}})$ with the point of optimal dynamic decoupling indicated by the arrow. (d) Sketch of two variable-time Hahn echo traces, which are combined into the second type of background term, valid in the case of coherence transfer as shown in (b). (e) Comparison sketch of the SIDRE signal (black), combination of two variable-time Hahn echo decays (solid red line) and the double-Hahn echo decay signal rescaled to the same amplitude as the SIDRE signal at the time point $t=\mathrm{0}$ (dashed red line). (f) Intermolecular coherence transfer factor $D{\mathit{\tau }}_{\mathrm{1}}{\mathit{\tau }}_{\mathrm{2}}$ as a function of the SIFTER time $t$ . (g) Sketch of the divided trace $B_t\left({\mathit{\tau }}_{\mathrm{1}}\right)B_t\left({\mathit{\tau }}_{\mathrm{2}}\right)/B_{\mathrm{S}}({\mathit{\tau }}_{\mathrm{1}},{\mathit{\tau }}_{\mathrm{2}})$ (violet) and this signal multiplied by the intermolecular coherence transfer factor $D{\mathit{\tau }}_{\mathrm{1}}{\mathit{\tau }}_{\mathrm{2}}$ (blue). Note that ${\mathit{\tau }}_{\mathrm{0}}={\mathit{\tau }}_{\mathrm{1}}+{\mathit{\tau }}_{\mathrm{2}}$ , and for the SIFTER time $t={\mathit{\tau }}_{\mathrm{1}}-{\mathit{\tau }}_{\mathrm{2}}$ .

Figure 4

Analysis of data from SIFTER and SIDRE at various trace lengths on 50  $\mathrm{\mu }$ M monoradicals in OTP. Panels (a)–(c) show data of nitroxide monoradical (1); (d)–(f) show data of trityl monoradical (2). The recorded traces of SIFTER (a, d) and SIDRE (b, e) are shown. The remaining panels (c, f) show the result from division of SIFTER by the corresponding SIDRE trace. Traces in (c) and (f) are displayed in stack plots at arbitrary offset.

Figure 5

Analysis of two-pulse decay and refocussed echo data on 50  $\mathrm{\mu }$ M monoradicals in OTP. Panels (a)–(c) show data of nitroxide monoradical (1); (d)–(f) show data of trityl monoradical (2). two-pulse decays with corresponding SSE fits (grey), mirrored and aligned to reflect offsets ${\mathit{\tau }}_{\mathrm{1}}$ and ${\mathit{\tau }}_{\mathrm{2}}$ in the SIFTER experiment (a, d), product of the fits of the aligned decay traces (b, e) and result from division by corresponding SIDRE traces (c, f) (reflecting $B_{\mathrm{S}}$ ) are shown. Traces in (c) and (f) are displayed in stack plots at arbitrary offset.

Figure 6

Analysis of data from SIFTER and SIDRE at various trace lengths on 50  $\mathrm{\mu }$ M biradicals in OTP. Panels (a)–(c) show data of nitroxide biradical 3; (d)–(f) show data of trityl biradical 4. The recorded traces of SIFTER (a, d), and SIDRE (b, e) are shown. The remaining panels (c, f) show the result from division of SIFTER by the corresponding SIDRE traces. Traces in (c) and (f) are displayed in stack plots at arbitrary offset.

Figure 7

Analysis of two-pulse decay and refocussed echo data on 50  $\mathrm{\mu }$ M biradicals in OTP. Panels (a)–(c) show data of nitroxide biradical (3); (d)–(f) show data of trityl biradical (4). Two-pulse decays with corresponding SSE fits (grey), mirrored and aligned to reflect offsets ${\mathit{\tau }}_{\mathrm{1}}$ and ${\mathit{\tau }}_{\mathrm{2}}$ in SIFTER experiment (a, d), product of the fits of the aligned decay traces (b, e) and result from division by corresponding SIDRE trace (c, f) (reflecting $B_{\mathrm{S}}$ ) are shown. Traces in (c) and (f) are displayed in stack plots at arbitrary offset.