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A.
Blank

Schulich Faculty of Chemistry, Technion – Israel Institute of Technology, Haifa, 3200003, Israel. E-mail: ab359@technion.ac.il

Received
7th November 2016
, Accepted 27th January 2017

First published on 27th January 2017

ESR spectroscopy can be efficiently used to acquire the distance between two spin labels placed on a macromolecule by measuring their mutual dipolar interaction frequency, as long as the distance is not greater than ∼10 nm. Any hope to significantly increase this figure is hampered by the fact that all available spin labels have a phase memory time (T_{m}), restricted to the microseconds range, which provides a limited window during which the dipolar interaction frequency can be measured. Thus, due to the inverse cubic dependence of the dipolar frequency over the labels' separation distance, evaluating much larger distances, e.g. 20 nm, would require to have a T_{m} that is ∼200 microsecond, clearly beyond any hope. Here we propose a new approach to greatly enhancing the maximum measured distance available by relying on another type of dipole interaction-mediated mechanism called spin diffusion. This mechanism operates and can be evaluated during the spin lattice relaxation time, T_{1} (commonly in the milliseconds range), rather than only during T_{m}. Up until recently, the observation of spin diffusion in solid electron spin systems was considered experimentally impractical. However, recent developments have enabled its direct measurement by means of high sensitivity pulsed ESR that employs intense short magnetic field gradients, thus opening the door to the subsequent utilization of these capabilities. The manuscript presents the subject of spin diffusion, the ways it can be directly measured, and a theoretical discussion on how intramolecular spin-pair distance, even in the range of 20–30 nm, could be accurately extracted from spin diffusion measurements.

(1) |

Here we present and theoretically analyse an alternative approach that has the potential to significantly increase the upper measurable distance limitation to the range of a few tens of nanometers. This approach, as in the case of DEER, is also based on measuring processes that are mediated by the dipolar interaction between the two spin labels. However, the main difference is that it examines processes that occur within T_{1} and not in the T_{m} time scale, which for most systems of relevance (e.g., nitroxides, Gd ions, trityls) can be more than 3–4 orders of magnitude longer at the cryogenic temperatures commonly employed in DEER and DQC.^{19} The physical phenomena that can be monitored and relied upon is spin diffusion. The next section will provide more details about spin diffusion and how it can be measured (relying on our recent work, where for the first time we experimentally measured the spin diffusion of electron spins in a solid sample^{20}). Following this, it will be shown how the information on spin diffusion can directly lead to finding the distance between two spin labels. Several quantitative simulated data are provided to support these claims, as well as a description of the experimental capabilities required to enable spin diffusion and intramolecular spin distance measurements in the range of a few tens of nanometers.

(2) |

W_{jk} = K_{ex}^{2}(3cos^{2}θ_{jk} − 1)^{2}/d_{jk}^{6} | (3) |

For the sake of completeness of presentation, we briefly provide here the approximate analytical expression relating W to the spin diffusion coefficient, D_{s}, (using the approach of Bloembergen^{21} and those who followed his work). We assume a sample with S = 1/2 spins that are located on a cubic lattice with equal spacing a, and have an equal nearest neighbor flip-flop rate W = W_{jk} between spins j and k (assuming no other flip-flop events). We denote the polarization p(x,t) = P_{+}(x,t) − P_{−}(x,t), where P_{+(−)}(x,t) is the probability of finding at x and at time t, a |+1/2〉 (|−1/2〉) state. Thus, based on the definition of W, it is possible to write that:

(4) |

(5) |

(6) |

In order to qualitatively understand the reasons for deviations from the Stekel–Tanner prediction when treating samples of relevance to our present theme, let us assume the case of a solid solution of doubly-labeled macromolecules, that have a fixed intramolecular spin-pair distance of 20 nm and a mean intermolecular distance of 135 nm (corresponding 4 × 10^{14} molecules per cm^{3}, or to a concentration of ∼0.66 μM). We now assume that we apply the pulse sequence of Fig. 1b and observe the echo decay due to spin diffusion as a function of the evolution time, Δ, for two types of gradient pulses, both of them with a duration of δ = 2 μs, but the first one with g = 290 T m^{−1} and the second one with g = 1460 T m^{−1}. These gradient pulses correspond to values of λ = 2π/q ≈ 100 and 20 nm, respectively (where q is defined as q = γδg^{29}). Based on the predictions of eqn (6), it seems that as we step up Δ, we should observe a meaningful exponential decay of the echo signal. In practice, due to the discreteness of the spins’ locations, and the fact that there are two very different scales of distances in the sample (the intramolecular spin-pair distance of 20 nm and the intermolecular distance of 135 nm) the behavior of the decay curve will be non-exponential, as can be seen in Fig. 2. The results of Fig. 2 are based on a numerical simulation whose details are provided in the Appendix. The “noise” is due to the relatively small number of molecules used in the simulation (100 spin pairs, but with each calculation repeated 1000 times) to keep the calculation time reasonable (about 1 h for each evolution graph in Fig. 2 on an Intel i7 2.7 GHz machine).

Fig. 2 The ratio of the stimulated echo acquired by the sequence in Fig. 1b with gradient E^{g}, to that without gradient E^{0}, as a function of the evolution time, Δ, for two values of gradients (corresponding to two different λ values). Results are based on numerical simulation (Appendix). The simulation assumes a sample with a molecular concentration of 4 × 10^{14} molecules per cm^{3} (0.66 μM), an intramolecular spin-pair distance of d = 20 nm, and K_{ex} = 3.35 × 10^{5} Hz nm^{3}. |

The behavior seen in Fig. 2 can be explained in the following manner: the λ value of the gradient pulses implies that spins must diffuse to a distance of ∼λ/2 so that their phases, due to the gradient pulse, would be different enough to affect the echo signal's magnitude. The first λ = 100 value that we chose is well above 20 nm, meaning that diffusion can be observed only if it is over distances of ∼50 nm or more. Namely, a diffusion to a distance of only 20 nm (as is the spin-pair distance of the doubly spin-labeled molecule) would not generate any appreciable echo decay. Thus, a drop of only ∼5% in the echo signal is observed as a rapid exponential decay during relatively short values of Δ. Furthermore, in solid-state samples (e.g., frozen solutions), spin diffusion occurs only through discrete “jumps” between spin to spin. The rate of these jumps, W, is proportional to 1/r^{6} (see eqn (2)), and at a spin–spin distance of r = 135 nm it is expected to be much less than 1 Hz. (Please note that our recent results show that W(r = 46 nm) ∼ 10 Hz.^{20}) Thus, during a relatively short evolution time (Δ ≪ 1/W(r = 135 nm)), it is highly unlikely that spins would “jump” all the way to the “extra-molecular” space between molecules. Clearly, under the conditions and gradient values we specified above, it would not be possible to observe any appreciable echo decay in our sample, even for Δ of ∼10 ms. In longer evolution times, the slow exponential decay would be seen because of intermolecular spin diffusion (the molecules are randomly distributed with a mean distance of 135 nm).

Contrary to what occurs at λ ∼ 100 nm, if we increase the value of g, making λ ∼ 20 nm, then even for Δ of ∼10 ms we should be seeing an appreciable echo decay – evidence of the spin diffusion phenomenon (Fig. 2, red curve). In longer evolution times, we experience a similar slow exponential decay as observed at the lower gradient value, since both values of λ ∼ 100 and λ ∼ 20 are smaller than the mean intermolecular distance. This anomalous diffusion behavior (bi-exponential decay) is exactly the phenomena we can make use of to find the intramolecular distance between the spin pairs. Namely, if we choose to use a moderate evolution time 1/W(r = 20 nm) ≪ Δ ≪ 1/W(r = 135 nm), and then start a series of experiments while increasing the gradients g (or δ), at some point λ will start to be in the order of the intramolecular spin-pair distance and we would see a dramatic decrease in the echo magnitude. Based on this observation, we could extract the intramolecular spin-pair distance, d.

This λ-dependent anomalous onset behavior is depicted in Fig. 3, which shows the results of the numerical simulation for another sample concentration, also in the case of a doubly spin-labeled molecule with a fixed intramolecular spin-pair distance d = 20 nm. The results show a clear decay of the echo signal due to spin diffusion. It is evident that at λ ≫ d the gradients have a relatively small effect on the echo signal. When approaching the range of λ ∼ 2d the signal drops in a much more pronounced manner. However, a further increase in the gradients (corresponding to smaller λ values) does not change much the decay curve, with the initial decay limited to E^{g}/E^{0} of ∼0.5. This kind of behavior basically conforms to the description provided above, where a relatively short 10 ms evolution period is not enough to allow spin diffusion to cross the relatively large intermolecular distance of ∼200 nm in this example. This prevents any appreciable echo decay during this time frame from intermolecular sources at both small and also large λ values. However, at small λ values there is a significant rapid decay from intramolecular sources with an initial decay down to E^{g}/E^{0} of ∼0.5, but not to a lower value, because spin diffusion within the closely-spaced pair is almost completely reversible (flip-flops to both directions), as they are almost completely isolated with polarization leaking out slowly to the intermolecular space. A further reduction in λ cannot change the echo signal significantly since the maximum effects of spin diffusion (distribution of initial spin polarization equally between the two closely-spaced spins) have already been reached with λ ∼ d, and there are no spins that are closer than d and would be further affected by λ reduction.

This type of behavior can be exploited in order to deduce the intramolecular spin-pair distance, d, from the experimental data. One possible way to approach this it is to acquire the E^{g}/E^{0} data for various values of g (which is inversely proportional to λ), but just for a specific, yet still plausible evolution period (e.g., 10 ms in our present example), and then plot the level of the signal at this evolution time point as a function of λ. This form of data analysis leads to the curves shown in Fig. 4. It is clear that as the intramolecular spin-pair distance on a molecule, d, is reduced, larger gradients (smaller λ values) are required to reach a regime of an appreciable decay.

The nature and the form of the type of E^{g}/E^{0}vs. λ plots shown in Fig. 4 depend only on three parameters: the intramolecular spin-pair distance, d, the sample concentration, C, and the flip-flop rate constant, K_{ex}. This dependence is depicted in Fig. 5 and 6. In a typical experimental procedure the E^{g}/E^{0} data would be collected at a fixed Δ value of ∼10 ms, for several values of g(λ), and then the experimental curve of E^{g}/E^{0}vs. λ would be fitted to the simulated data of Fig. 4. The calculation of such theoretical curves using current-day PCs can take several hours, meaning that the total fitting procedure can take around a day.

In order to properly extract d based on an experimentally-measured curve, it is therefore necessary to have a good knowledge of C and K_{ex}. The knowledge of C (for a homogenous sample) is trivial, and furthermore, if C is small enough its exact value is of negligible importance (see Fig. 5). On the other hand, the value of K_{ex} cannot be calculated accurately^{22,24} and, as can be seen in Fig. 6, it may affect the nature of the measured plot. Our recent experiments with phosphorus-doped ^{28}Si:P have managed to measure K_{ex} for the first time, and this type of work can be also employed for the present challenge of evaluating d. For example, K_{ex} could be extracted from the decay curves of known samples with known d and C values, and then assumed not to change when switching to the sample with an unknown d, if the same solvent and the same temperature are used for the measurement. Alternatively, it is possible to use experimental E^{g}/E^{0}vs. λ data on the molecules of interest with known concentration, but with single rather than two spin labels, and subsequently find K_{ex} which provides the best fit of the theoretical curve to the measured data. Clearly, however, much more work is required along this line to develop a better understanding of the nature of this parameter and its dependence on the sample and environmental conditions.

Overall, it can be summarized that the accuracy of the method for determining d relies on two main parameters: the signal-to-noise-ratio (SNR) of the measured E^{g}/E^{0} plots vs. λ for a given sample, and the a priori knowledge of K_{ex}. Based on the results shown in Fig. 4, it is clear that an SNR of at least 100 is required to make possible a good differentiation between the plots for two different d values, varying by 1 nm one from the other. This, however, assumes that K_{ex} is well-known for the type of sample measured, or that its values are acquired using the procedure just described above. Any uncertainly in K_{ex} will be translated to uncertainly in the fitting of d. However, based on the results of Fig. 4 and 6 it is evident that even 100% uncertainly in K_{ex} would lead to only ∼1–2 nm uncertainly in the fitted d value, so clearly this aspect is not very critical to the accuracy of the method.

One additional issue to consider is the possibility of having a distribution of several distances, and how this might affect the measured results. Fig. 7 addresses this question, simulating the same conditions as shown in Fig. 4, but assuming that d has Gaussian distribution around its mean value with standard deviation of 1, and 5 nm (Fig. 7a and b, respectively). It is evident that for a small standard deviation (1 nm, Fig. 7a), there are very small differences compared to the original plots, for single d value (Fig. 4). However, for a relatively large standard deviation, of 5 nm (Fig. 7b), there are clear differences, with the larger average d values having smaller signal reduction than in the original plots of Fig. 4. The implication of this behavior is that it would probably be difficult to get accurate (better than ∼10%) readings of the average d in such cases of molecules having very broad distance distribution of the intramolecular spin-pair distance.

Fig. 7 Numerical simulation of E^{g}/E^{0} with the same parameters as in Fig. 4, just assuming random distance Gaussian distribution around mean intramolecular spin-pair distance, d, with standard deviation of 1 nm (a) and 5 nm (b). |

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