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

Distances between electron spins (and in particular distance distributions) are an important source of information for different research fields, ranging from structural biology of ordered and disordered proteins

With the advent of ultra-wideband EPR spectrometers as well as novel spin labels, in particular based on the trityl radical with its narrow EPR spectrum, single-frequency PDS techniques find broader applications

However, while contributions to the DEER signal including artefacts have been described theoretically

In this work, we derive a theoretical model for the intermolecular background of the SIFTER experiment based on dipolar terms and product operator formalism for the evolution of the spin density operator. After the mathematical derivation, we go on to compare this model to experimental data on nitroxides and trityls, both as monoradicals and biradicals.

The derivation section consists of three main parts. First, we briefly summarize the formation of the SIFTER signal in a sample consisting of isolated pairs of spins (intramolecular contributions only), according to the original derivation

Pulse sequences used in experiments are shown in

We restrict our derivation to the four-pulse SIFTER sequence shown in Fig.

All pulses are ideal (infinitely short with infinite excitation bandwidth).

We analyse SIFTER experiments on frozen solutions of biradicals prepared such that the intramolecular dipole–dipole interaction is much stronger than the intermolecular dipole–dipole interactions (high dilution).

We take into account only the secular part of the dipolar couplings, which for a pair of spins A and B is written as

Here,

Also, we assume for simplicity that all spin centres in the sample have the same EPR spectrum. This is, however, never explicitly used in the derivation, which therefore holds true also for SIFTER with heterogeneous spin pairs, should such an experiment appear to be of interest.

In the calculations here and in the sections below we can drop the electron Zeeman interaction, since it is refocused at the echo positions. Thus, in the case of an isolated spin pair, the spin Hamiltonian is simply given by Eq. (

The four-pulse SIFTER pulse sequence contains two

Initially, before the primary

The time

The first block

The term

Finally, the second evolution block

Note for later discussions that the term

The cosine term results from the evolution that takes place always on the same spin, while the sine term results from evolution on the first spin during the first refocusing block, coherence transfer and evolution on the second spin during the second refocusing block.

Before we make a detailed calculation in the following sections, let us discuss the overall topology of the density matrix propagation solution for the SIFTER experiment in the case of many weakly interacting biradicals (real frozen solution case). First, we note that the solution in Eq. (

Before we discuss the case of SIFTER experiment in the presence of intramolecular spin–spin distances, we need to discuss an important case of SIFTER experiment in a frozen solution of monoradicals. A monoradical-like signal also appears in the SIFTER experiment on biradicals because of incomplete excitation of paramagnetic species

To introduce abbreviations consistent between monoradical and biradical cases, we consider one spin centre, called A spin, which has an initial polarization of

Intra- and intermolecular electron–electron coupling frequencies for the case of a frozen solution of biradicals.

Most important terms describing the spin density matrix and the detected signal contributions.

With this notation, in the case of a monoradical solution, evolution during the first

Here, all terms that contain products with more than one

After the coherence transfer pulse, this transforms to

Here, the operator

The product of cosine contributions from all surrounding spins, after ensemble averaging, describes the detectable signal in a Hahn echo experiment. We will use the abbreviation

Note that this product centred at the A spin should not correlate with the analogous product centred at the

The function

Electron spins interact with surrounding nuclear spins and change their precession frequency according to the continuously ongoing configuration dynamics of the surrounding nuclei. It is common to call such a process spectral diffusion. By nature, this is a deterministic process, but due to the very large number of coupled nuclear spins and a random distribution of couplings, it demonstrates quasi-stochastic features. Accordingly, each pulse sequence can be seen as a path for a (quasi-)decay of electron coherence due to the nuclei-related dephasing, and at the same time it can be seen as a filter function, selecting certain electron–nuclear frequencies and suppressing others. The latter process is often called dynamical decoupling

Here, we consider a spin Hamiltonian that besides electron–electron couplings also takes into account the electron–nuclear (hyperfine) interactions, nuclear Zeeman and nuclear spin–spin couplings:

We can generally assume that during each echo refocusing block

The first term in Eq. (

Overview of the buildup of the non-modulated part of the SIFTER signal.

For the intermolecular coherence transfer term (second term in Eq.

In the situation of filtering, the two terms

Here, we used the following additional abbreviation for the ensemble-averaged square of the intermolecular dipolar frequency:

For a short overall length of the SIFTER trace, the filtration effects should be weak, and the two signals

In this section, we will analyse the frozen biradical solution, for which the spin Hamiltonian that includes electron–electron and electron–nuclear parts can be written in the form

In the term

In this and the next section we discuss in detail the evolution of the SIFTER signal in an ensemble of biradicals, i.e. in the case of a strong intramolecular coherence transfer. Let us define the density operator term

In order to describe the density matrix evolution upon all dipolar couplings between A spin and B spins, we will also use the abbreviation

After the

The second type of terms appears if we let only one

The third type of operator terms summed up in

Note again that the product

Here, it is obvious that the product

Next, we apply the

The operator

Additional terms appear in the above calculation (Eq.

For a particular B spin with the index

This can be ensemble averaged and projected onto the detection operator

Note that here we approximated

The two products in this equation correspond to the initial molecule's surrounding, for which the intermolecular dipolar frequencies are marked as

The term

Finally, we need to demonstrate that all density operator terms, which include more than one

Here, as in the monoradical case, we will have to consider pathways with and without coherence transfer, as depicted in Fig.

In the situation of filtering, the two terms

Here, we used an abbreviation

Using the trigonometric relation

Note that here the first term is equal to the normal SIFTER intramolecular signal, and the second term is constant at any time point for a given total length of the SIFTER trace. Equation (

The first term in this equation can be further rewritten as

To summarize, the detected SIFTER signal contains a modulated and a non-modulated part:

Nitroxide biradical (3)

EPR measurements were performed on a home-built high-power (150 W traveling-wave tube amplifier) Q-band spectrometer

In this work we derived analytic equations for the SIFTER signal in frozen glassy solutions of monoradicals and biradicals. Importantly, in this analysis we obtained the SIFTER signal for monoradicals as a sum of two well defined contributions that can be also independently determined in auxiliary measurements. Also for biradicals, we determined the dipolar modulated part of the SIFTER signal to consist of two terms, each presented as a product of an intramolecular contribution and an intermolecular contribution. Moreover, the analysis suggests that the main signal, which represents the classical intramolecular dipolar evolution signal, has a well defined intermolecular contribution that can be determined by the SIDRE experiment and variable delay Hahn echo experiments. This signal (first term in Eq.

Note also that the relative contributions of the artefact term should not depend on the thermal Boltzmann polarization of the spins, since this only affects the initial polarization of the spin system, but it does not influence any steps in the presented density matrix propagation. Thus, intensities of all terms in the final equations would simply scale linearly with the Boltzmann polarization, and their ratios would remain unaffected.

There is, also, another important effect that makes the amplitude of the artefact signal

The quantitative analysis of the structure of the intramolecular SIFTER signal, and validation of the presented analytical solution requires substantial effort and needs good quality reference data on the “true distance distribution” in the sample under study (e.g. measured by DEER). Here, we will concentrate on the analysis of the intermolecular SIFTER signal in monoradical solutions and the non-modulated part of the SIFTER signal of biradicals. These contributions should be described by Eqs. (

For our current purpose of validation of theory, however, it is more convenient to divide both monoradical and biradical SIFTER data by the corresponding SIDRE traces and compare the obtained shapes with the shapes of the division traces

The similarity in the shapes in two such series would, first, confirm the above assumption of the uncorrelated intermolecular contributions from the dipolar coupled spins in the coherence transfer terms in the SIFTER signal. Second, in the case of biradical SIFTER traces, such a comparison would also confirm our result related to the composition of the SIFTER signal as a sum of “monoradical-like” and “biradical-like” contributions.

Experimental SIFTER traces exhibit a characteristic dependence of their background shape on the trace length (see Fig.

Analysis of data from SIFTER and SIDRE at various trace lengths on 50

Observing a two-pulse echo in comparison to the refocused echo, we predominantly find an upward curving of the divided traces (Fig.

Analysis of two-pulse decay and refocussed echo data on 50

Stepping away from the monoradicals, we perform the same analysis for biradicals (Figs.

Analysis of data from SIFTER and SIDRE at various trace lengths on 50

Analysis of two-pulse decay and refocussed echo data on 50

While many of the trends just described for nitroxides remain identical to what we have described for monoradicals we observed prominent additional effects that are not covered by our model. We will attempt to ignore the prominent oscillations visible in the SIFTER traces, especially of nitroxide (Fig.

Overall, there is a good match between the shapes of SIFTER data divided by the SIDRE traces (

Also, as predicted by the analytic equations, while some dipolar evolution artefacts must be present in SIFTER data, their relative contributions are very weak for most of the practically important cases. This prediction matches with the presented experimental SIFTER data, where such artefacts were not observed. Thus, the analytic approach proposed here, appears to be accurate to a good approximation. This opens up the possibility of a more detailed analysis of intramolecular SIFTER data, and quantitative evaluation and accuracy estimates of the distance distributions obtained from SIFTER measurements. Due to the complexity of the background problem outlined here, concomitant fitting of the modulated SIFTER signal and background will be an advantage, as recently shown for DEER

Experimental data as well as scripts for data processing are made available via Zenodo with the following DOI:

AV performed experimental work and data processing; HH and MS synthesized the biradical compounds under supervision of AG. JS, FDB and YP conducted initial experimental studies, which guided the theory derivation. MY derived the theoretical model. DK, GJ and MY designed the research. AV, MY and DK analysed the data and wrote the paper with contributions from all authors.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Financial support by the SNSF (200020_188467) to Gunnar Jeschke and by Deutsche Forschungsgemeinschaft (GO 555/4-3) to Adelheid Godt is gratefully acknowledged. Janne Soetbeer thanks the Günthard Foundation for a scholarship, and Muhammad Sajid thanks the Higher Education Commission of Pakistan for a fellowship.

This research has been supported by the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (grant no. 200020_188467) and the Deutsche Forschungsgemeinschaft (grant no. GO 555/4-3).

This paper was edited by Stefan Stoll and reviewed by Frédéric Mentink-Vigier.