The decay of the refocused Hahn echo in double electron-electron resonance (DEER) experiments

Abstract

Double electron-electron resonance (DEER) is a pulse electron paramagnetic resonance (EPR) technique that measures distances between paramagnetic centres. It utilizes a four-pulse sequence based on the refocused Hahn spin echo. The echo decays with increasing pulse sequence length $2({\tau }_1+{\tau }_2)$, where ${\tau }_1$ and ${\tau }_2$ are the two time delays. In DEER, the value of ${\tau }_2$ is determined by the longest inter-spin distance that needs to be resolved, and ${\tau }_1$ is adjusted to maximize the echo amplitude and, thus, sensitivity. We show experimentally that, for typical spin centres (nitroxyl, trityl, and Gd(III)) diluted in frozen protonated solvents, the largest refocused echo amplitude for a given ${\tau }_2$ is obtained neither at very short ${\tau }_1$ (which minimizes the pulse sequence length) nor at ${\tau }_1={\tau }_2$ (which maximizes dynamic decoupling for a given total sequence length) but rather at ${\tau }_1$ values smaller than ${\tau }_2$. Large-scale spin dynamics simulations based on the coupled cluster expansion (CCE), including the electron spin and several hundred neighbouring protons, reproduce the experimentally observed behaviour almost quantitatively. They show that electron spin dephasing is driven by solvent protons via the flip-flop coupling among themselves and their hyperfine couplings to the electron spin.

Figure 1

Schematics of (a) the three-pulse DEER sequence and (b) the dead-time-free four-pulse DEER sequence. Observer pulses are in blue and pump pulses in orange. The dashed line indicates $t=\mathrm{0}$ .

Figure 2

Chemical structures of the paramagnetic centres studied in this work, namely (a) 3-maleimido-proxyl, (b) trityl OXO63, and (c) Gd-C2.

Figure 3

Refocused echo decay for 100  $\mathrm{\mu }\mathrm{M}$ 3-maleimido-proxyl in ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$   $/$  glycerol ( $\mathrm{80}:\mathrm{20}$ $v/v$ ) at 25 K. Panel (a) shows the echo amplitude as a function of ${\mathit{\tau }}_{\mathrm{1}}$ and ${\mathit{\tau }}_{\mathrm{2}}$ , and panel (b) shows the same data after normalization of each slice along ${\mathit{\tau }}_{\mathrm{1}}$ . The red lines (a, b) indicate the location of the maxima along ${\mathit{\tau }}_{\mathrm{1}}$ for fixed ${\mathit{\tau }}_{\mathrm{2}}$ (upper line) and vice versa (lower line; only in a). Panel (c) shows slices along ${\mathit{\tau }}_{\mathrm{1}}$ for several ${\mathit{\tau }}_{\mathrm{2}}$ values (indicated by grey arrows in a, b), together with a Hahn echo decay.

Figure 4

Refocused echo decay for 100  $\mathrm{\mu }\mathrm{M}$ 3-maleimido-proxyl in D ${}_{\mathrm{2}}$ O  $/$  glycerol-d ${}_{\mathrm{8}}$ ( $\mathrm{80}:\mathrm{20}$ $v/v$ ) at 25 K. Panel (a) shows the echo amplitude as a function of ${\mathit{\tau }}_{\mathrm{1}}$ and ${\mathit{\tau }}_{\mathrm{2}}$ , and panel (b) shows the same data after normalization of each slice along ${\mathit{\tau }}_{\mathrm{1}}$ . Panel (c) shows slices along ${\mathit{\tau }}_{\mathrm{1}}$ for several ${\mathit{\tau }}_{\mathrm{2}}$ values (indicated by grey arrows in a, b), together with a Hahn echo decay.

Figure 5

Refocused echo decay for 100  $\mathrm{\mu }\mathrm{M}$ 3-maleimido-proxyl in ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$   $/$  glycerol solvents ( $\mathrm{80}:\mathrm{20}$ $v/v$ ) with varying degrees of solvent protonation at 25 K, i.e. (a) 25 %, (b) 50 %, and (c) 75 %. The red lines indicate the location of the maxima along ${\mathit{\tau }}_{\mathrm{1}}$ for fixed ${\mathit{\tau }}_{\mathrm{2}}$ (upper line) and vice versa (lower line). Plots for 100 % and 0 % solvent protonation are shown in Figs. 3a and 4a, respectively.

Figure 6

Refocused echo decays for 100  $\mathrm{\mu }\mathrm{M}$ trityl OXO63 in (a)  ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$   $/$  glycerol ( $\mathrm{80}:\mathrm{20}$ $v/v$ ) and (b)  ${\mathrm{D}}_{\mathrm{2}}\mathrm{O}$   $/$  glycerol-d ${}_{\mathrm{8}}$ at 25 K; in panel (a), the red lines show the position of the echo maximum along ${\mathit{\tau }}_{\mathrm{1}}$ for each constant ${\mathit{\tau }}_{\mathrm{2}}$ (upper line) and vice versa (lower line). Panel (c) shows slices of the data in panel (b) along ${\mathit{\tau }}_{\mathrm{1}}$ for several ${\mathit{\tau }}_{\mathrm{2}}$ values (indicated by grey arrows in b), together with a Hahn echo decay.

Figure 7

Refocused echo decays for 100  $\mathrm{\mu }\mathrm{M}$ GdCl ${}_{\mathrm{3}}$ in (a)  ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$   $/$  glycerol ( $\mathrm{80}:\mathrm{20}$ $v/v$ ) and (b)  ${\mathrm{D}}_{\mathrm{2}}\mathrm{O}$   $/$  glycerol-d ${}_{\mathrm{8}}$ at 10 K; in panel (a), the red lines show the position of the echo maximum along ${\mathit{\tau }}_{\mathrm{1}}$ for each constant ${\mathit{\tau }}_{\mathrm{2}}$ (upper line) and vice versa (lower line). Grey arrows in (b) indicate the ${\mathit{\tau }}_{\mathrm{2}}$ values of slices shown in (c), together with a Hahn echo decay.

Figure 8

Refocused echo decay experiments (constant ${\mathit{\tau }}_{\mathrm{2}}$ ; variable ${\mathit{\tau }}_{\mathrm{1}}$ ) on 25  $\mathrm{\mu }\mathrm{M}$ MdfA V44C/V307C, doubly labelled with Gd-C2, at 10 K. Selected values for ${\mathit{\tau }}_{\mathrm{2}}$ in nanoseconds are colour coded. Data taken from the Supplement of Yardeni et al. (2019).

Figure 9

Simulation of the refocused echo decay for 3-maleimido-proxyl in ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$   $/$  glycerol ( $\mathrm{80}:\mathrm{20}$ $v/v$ ). Panel (a) shows the simulated refocused echo amplitude as a function of ${\mathit{\tau }}_{\mathrm{1}}$ and ${\mathit{\tau }}_{\mathrm{2}}$ , using 3-CCE. The ${\mathit{\tau }}_{\mathrm{1}}={\mathit{\tau }}_{\mathrm{2}}$ line is shown in black. The upper red curve in panel (a) indicates the ridge of panel (b), which normalizes each slice along ${\mathit{\tau }}_{\mathrm{1}}$ (with constant ${\mathit{\tau }}_{\mathrm{2}}$ ) to unit maximal amplitude. The lower red curve in panel (a) is the analogous ridge for normalization along ${\mathit{\tau }}_{\mathrm{2}}$ . Panel (c) shows the simulated refocused echo decay for ${\mathit{\tau }}_{\mathrm{1}}={\mathit{\tau }}_{\mathrm{2}}$ as a function of cluster truncation level (1-CCE through 4-CCE).

Figure 10

Simulated refocused echo decay of 3-maleimido-proxyl. (a) Factorization of the four-cluster decay into a second-order term in $b_{nm}$ (top) and all higher-order terms (bottom). (b) Simulated refocused echo decays at two-, three-, and four-cluster level (2-CCE, 3-CCE, and 4-CCE; top), with the corresponding slice-wise normalized decays (bottom). For this simulation, a single orientation of the radical was used.