Dipolar spectroscopy and spin alignment in electron paramagnetic resonance

Dipolar spectroscopy and spin alignment in electron paramagnetic resonance

1 December 2000 Chemical Physics Letters 331 (2000) 243±252 www.elsevier.nl/locate/cplett Dipolar spectroscopy and spin alignment in electron param...

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1 December 2000

Chemical Physics Letters 331 (2000) 243±252

www.elsevier.nl/locate/cplett

Dipolar spectroscopy and spin alignment in electron paramagnetic resonance G. Jeschke *, M. Pannier, A. Godt, H.W. Spiess Max-Planck-Institut for Polymer Research, Postfach 3148, D-55021 Mainz, Germany Received 8 May 2000; in ®nal form 17 August 2000

Abstract Two single-frequency techniques for refocusing (SIFTER) dipolar couplings between electron spins are introduced. The experiments are based on the solid-echo and Jeener±Broekaert sequences, well established in dipolar NMR spectroscopy of solids, and open up new routes to high-resolution two-dimensional EPR spectroscopy with only moderate requirements on the spectrometer. For distances between paramagnetic centres larger than 3 nm, SIFTER provides better resolution than double electron±electron resonance (DEER). Good agreement between distances from SIFTER measurements and force-®eld computations is found for shape-persistent biradicals with distances up to 5.1 nm corresponding to a dipolar frequency of 390 kHz. Ó 2000 Elsevier Science B.V. All rights reserved.

1. Introduction The measurement of distances in the nanometer range in systems lacking long-range order is still a demanding task. The problem arises for instance for mesoscopically structured synthetic polymers like ionomers [1±3] as well as for biopolymers with large molecular weights that can often not be crystallized [4]. Magnetic resonance methods based on the measurement of dipole±dipole interactions between spins are well suited for such distance determinations, as they are sensitive only to local structure around the observer spins. In particular, electron paramagnetic resonance (EPR) is the method of choice for large distances because of the much larger magnetic moment of the electron spin compared with nuclear spins [5,6]. In

*

Corresponding author. Fax: +49-6131-379-100. E-mail address: [email protected] (G. Jeschke).

many cases, EPR distance measurements involve two nitroxide spin labels, because these can be conveniently attached to selected sites in proteins [4] as well as polymers [7]. To obtain utmost resolution and to access long distances, the dipole±dipole interaction has to be separated from all other interactions and ideally, relaxational broadening should be suppressed. A deadtime-free approach is required to access broad distributions of distances and to provide a faithful lineshape of dipolar spectra, which may be crucial for obtaining additional information. All these features are provided by the pulse double electron± electron resonance experiment (DEER) introduced by Milov et al. [8,9] and by a new double-quantum (DQ) EPR experiment introduced by Borbat and Freed [10]. It has been demonstrated recently that distances of nitroxide spin labels up to at least 2.8 nm can be measured by DEER and DQ EPR with good precision [10±12]. However, both these methods are technically demanding. While DEER

0009-2614/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 0 ) 0 1 1 7 1 - 4

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methods require an electron±electron double resonance (ELDOR) setup with a second microwave (m.w.) frequency source, the dead-time free DQ EPR experiment based on allowed coherence transfer pathways appeared to depend on large excitation bandwidths [10]. During the refereeing process we were alerted to simultaneous work by Borbat and Freed which shows that the excitation bandwidth requirements of DQ EPR decrease at longer distances. The `2+1' experiment [13] and earlier DQ EPR experiments based on forbidden coherence transfer pathways [14,15] are not free of deadtime. In this Letter, we introduce a new deadtime-free single-frequency technique for refocusing dipolar couplings (SIFTER) that is based on solid echo or Jeener±Broekaert echo refocusing. Deadtime-free single-quantum experiments developed simultaneously by Borbat and Freed, which were pointed out to us by the referee, are not based on the solid echo or Jeener±Broekaert echo. For distances larger than 3 nm our method does not require larger excitation bandwidths than DEER and provides complete separation of electron±electron couplings from other interactions. The partial refocusing of couplings to distant, homogeneously distributed spins leads to higher resolution than in DEER. The SIFTER method is easy to implement and requires only moderate phase cycling. On a series of shape-persistent monodisperse biradicals [16] we demonstrate that distances between electron spins in the range from 3 to 5 nm can be measured with SIFTER experiments with high precision.

ment can be performed by refocusing of the couplings between two excited spins which is achieved by the solid echo [17,18]. The original solid-echo sequence cannot be used for the problem at hand because it does not completely refocus inhomogeneities due to a distribution of g values or hyper®ne couplings. This problem can be solved by introducing two more refocusing p pulses (Fig. 1a). The sequence (p=2†x -s1 -…p†x -s1 -…p=2†y -s2 …p†x -s2 -echo generates a solid echo for s1 ˆ s2 . The g and hyper®ne inhomogeneities are refocused by the two p pulses for any choice of s1 and s2 , so that a variation of t ˆ s2 ÿ s1 results in an echo modulation that is exclusively due to electron±electron couplings. This variation is done by incrementing delay s2 by steps Ds and decrementing delay s1 by the same steps, hence the total length of the sequence 2…s1 ‡ s2 † is constant and there is no signal variation due to phase relaxation. For the theoretical treatment of this solid-echo SIFTER sequence we consider the action of ideally non-selective m.w. pulses on a system of two electron spins S1 ˆ 1=2; S2 ˆ 1=2 with the rotating frame spin Hamiltonian …1†

H ˆ XS S1z ‡ X…2† S2z ‡ xsec S1z S2z ‡ xff  …S1x S2x ‡ S1y S2y †; …1† XS

…1†

…2† XS

and are the resonance o€sets of the where two spins that exhibit inhomogeneous distributions due to g and hyper®ne anisotropy. The parameters xsec and xff characterize the secular and

2. Analytical treatment for ideal pulses High-resolution dipolar spectroscopy requires dead-time free acquisition of a free induction decay or echo with a modulation that is exclusively determined by electron±electron couplings. Because inverse phase memory times of electron spins are often of the same order of magnitude as the dipolar frequencies, it is also useful to exclude contributions of relaxation to the linewidth by performing constant-time experiments. A deadtime free acquisition in a single-frequency experi-

Fig. 1. Pulse sequences for SIFTER experiments. The echo amplitude is measured as a function of s2 ÿ s1 for constant s1 ‡ s2 . See Table 1 for the appropriate phase cycle: (a) Solidecho SIFTER. (b) Spin-alignment SIFTER. T > 2…s1 ‡ s2 † should be chosen to avoid echo crossings that are not eliminated by the phase cycle.

G. Jeschke et al. / Chemical Physics Letters 331 (2000) 243±252

¯ip-¯op part of the coupling between the two electron spins including dipole±dipole and possibly exchange contributions. The ¯ip-¯op part was neglected in previous treatments of pulse sequences for the separation of dipolar couplings [10,13]. In the high-®eld approximation, the equilibrium density matrix of this spin system can be written as req ˆ ÿS1z ÿ S2z neglecting constant factors and terms. We ®rst treat the case …1† …2† jxff j  jXS ÿ XS j where the ¯ip-¯op term can indeed be neglected. This is a good approximation for most spin packets contributing to the signal in situations where the excitation bandwidth of the m.w. pulses is much larger than xff . Straightforward application of product operator formalism [19] using the SOME software package [20] for Mathematica (Wolfram Research) yields for the density operator at time 2s1 immediately before the …p=2†y pulse r1 ˆ ÿ cos…xsec s1 †…S1y ‡ S2y † ‡ sin…xsec s1 †…2S1z S2x ‡ 2S1x S2z †:

…2†

The …p=2†y pulse inverts the sign of the second term in r1 (coherences in anti-phase) which results in the rephasing of the coupling contribution. The echo amplitude at time 2…s1 ‡ s2 † is given by rdet ˆ cos‰xsec …s2 ÿ s1 †Š…S1y ‡ S2y † ÿ sin‰xsec …s2 ÿ s1 †Š…2S1z S2x ‡ 2S1x S2z †:

…3†

Only the ®rst term in rdet corresponds to observable transverse magnetization. Therefore, the echo is modulated with the secular coupling frequency between the two electron spins as a function of t ˆ s2 ÿ s1 . There is no term with a phase which depends on xsec …s2 ‡ s1 †, as in DEER, `2+1' and DQ EPR experiments, since the SIFTER experiment does not introduce additional decay due to the instantaneous di€usion described by such a term; instead it refocuses instantaneous di€usion. For this reason the SIFTER echo is expected to decay more slowly with increasing total duration of the pulse sequence than the echoes in the other experiments and resolution should thus be increased. …1† …2† In the other limiting case, jxff j  jXS ÿ XS j, we can diagonalize H by the unitary transformation U ˆ expfÿi…p=2†…S1x S2y ÿ S1y S2x †g. We obtain

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rdet ˆ U y eÿiHs2 U eÿipFx U y eÿiHs2 U eÿi…p=2†Fy U y eÿiHs1  U eÿipFx U y eÿiHs1 U eÿi…p=2†Fy req ei…p=2†Fy U y eiHs1  U eipFx U y eiHs1 U ei…p=2†Fy U y eiHs2 U eipFx U y eÿiHs2 U ˆ cos‰…xsec ÿ xff †…s2 ÿ s1 †Š…S1y ‡ S2y † ÿ sin‰…xsec ÿ xff †…s2 ÿ s1 †Š…2S1z S2x ‡ 2S1x S2z †: …4†

For distances r12 > 2 nm between the electron spins, the exchange coupling can be neglected and we have xsec ˆ ÿ2xff ˆ …3 cos2 h ÿ 1†

l2B l0 g1 g2 1 ; 3 h r12

…5†

where lB is the Bohr magneton, l0 the permeability of the vacuum, g1 and g2 are the g values of the two electrons, and h is the angle between the static ®eld direction and the axis connecting the loci of the two electrons. The ¯ip-¯op term therefore leads to an increase of the modulation frequency by a factor up to 3/2 for spin packets where the di€erence of the resonance frequencies of the two electron spins is of the order of xff or even smaller. Its in¯uence on distance measurements is addressed by numerical calculations below. Extensions of the experiment to two-dimensional spectroscopy may require storage of the magnetization at time 2s1 along the laboratory frame z axis where it relaxes with time constant T1 rather than T2 . Such a sequence can be based on the Jeener±Broekaert echo [21] rather than the solid echo. A state of dipolar order or two-spin alignment [22] is created by a …p=4†y pulse (Fig. 1b). After the decay of coherences during time T, the dipolar order can be converted back to anti-phase coherence. Neglecting ¯ip-¯op terms, an analogous calculation to the solid echo yields rxy ˆ 14‰cos…xsec …s2 ÿ s1 †† ÿ cos…xsec …s1 ‡ s2 ††Š  …S1y ‡ S2y † ÿ 14‰sin…xsec …s2 ÿ s1 †† ÿ sin…xsec …s1 ‡ s2 ††Š…S1x ‡ S2x †

…6†

for the terms of rdet corresponding to observable transverse magnetization. The sine terms vanish for symmetry reasons, and the cos…xsec …s1 ‡ s2 †† term is not refocused, so that the signal in this spin-alignment SIFTER experiment is by a factor

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of four smaller than in solid-echo SIFTER disregarding signal losses by relaxation during time T. However, spin-alignment SIFTER may feature larger modulation depths than solid-echo SIFTER. This is because for incomplete excitation of the EPR spectrum, there are constant contributions to the signal in solid-echo SIFTER due to coherences which are not in¯uenced by the y pulses. Such contributions cannot be removed by phase cycling since the y pulse does not induce a change in the coherence order for the wanted contributions. On the other hand, in spin-alignment SIFTER coherence is only stored by the …p=4†y pulses if it has defocused by the dipolar coupling, otherwise it decays during the interpulse delay T  T2 . The …p=4†y -T -…p=4†y sequence therefore acts as a ®lter that suppresses unmodulated contributions. Spin-alignment SIFTER is thus related to the DQ EPR approach where a ®lter e€ect is obtained by phase cycling [10]. While the spin-alignment ®lter is less e€ective than the double-quantum ®lter employed by Borbat and Freed, our approach avoids the requirement for refocusing of the double-quantum coherence. This reduces excitation bandwidth requirements considerably and minimizes relaxational losses. Additional echo crossings as compared to solid-echo SIFTER require an extension of the phase cycle only if T 6 2…s1 ‡ s2 †. Since T1  T2 such a choice of T can usually be avoided without sacri®cing much sensitivity. Multi-spin e€ects are beyond the scope of this Letter. They can be handled in analogy to solid state NMR [17,21]. 3. Numerical simulations for non-ideal pulses An analytical treatment of the SIFTER experiments considering non-ideal pulses and the full Hamiltonian in Eq. (1) leads to expressions unsuitable for discussion. Instead, we used numerical simulations for estimating modulation depths for typical experimental situations and for studying the in¯uence of the ¯ip-¯op term on dipolar spectra. The computation was performed with the authors' own implementation of density operator formalism for MATLAB (The Math Works) and a

Monte-Carlo approach for the variation of …1† …2† XS ; XS , r12 , and h. The coupling parameters xsec and xff were calculated by Eq. (5) using the free electron g value for both spins. As the distances for the model computation, we used r12 ˆ 2:84 nm (Gaussian distribution with a variance r ˆ 0.05 nm) corresponding to radical 1 below, and r12 ˆ 3:68 nm, r ˆ 0:10 nm (radical 2), and r12 ˆ 5:10 nm, r ˆ 0:10 nm (radical 3). Uncorre…1† …2† lated Gaussian distributions of XS and XS with the same variance of 33.6 MHz were assumed to model the mI …14 N† ˆ 0 component of the nitroxide spectrum, while h was uniformly distributed between 0° and 90° with a weighting factor of sin h for proper powder averaging. Pulse lengths were 24 and 48 ns for the p=2 and p pulses, respectively, as in the experiments described below. Time-domain data traces with 256 data points were computed by adding 18 000 computer experiments with di€erent random parameters. After removing the unmodulated signal part by a baseline correction, dipolar spectra were obtained by sine bell apodization, zero-®lling to 1024 data points and Fourier transformation. For the shorter distance, r12 ˆ 2:84 nm, the solid-echo SIFTER time-domain data displayed in Fig. 2a clearly exhibit a modulation with the frequency of 2.27 MHz expected for the singularities of the dipolar Pake pattern from only the secular part of the coupling. The corresponding spectrum in Fig. 2c is somewhat distorted and contains additional singularities at half of the expected frequency. We assign this e€ect to pathways where one of the two p pulses does not ¯ip both spins, which is in agreement with the observation that these modulations have their time origin at s1 ˆ 0 and s2 ˆ 0 rather than at s1 ˆ s2 . To check if these e€ects are due to the ¯ip-¯op term of the dipolar coupling, we performed the same simulation without the ¯ip-¯op term in parallel, that is, using the same values for the random variables. The time-domain trace shown in Fig. 2b has a somewhat larger modulation amplitude and slightly di€erent shape. The dipolar spectrum computed from this trace (Fig. 2d) is indeed closer to the ideal Pake pattern, but the additional singularities persist with only somewhat smaller relative amplitude. On the

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Fig. 2. Monte-Carlo simulations of SIFTER experiments for isolated spin pairs with distances of 2.84 nm (a±d) and 5.1 nm (e±h) assuming an EPR linewidth as for the centre line (mI …14 N† ˆ 0) in the nitroxide spectrum: (a) Time-domain data for solid-echo SIFTER computed with the full Hamiltonian. (b) Time-domain data for solid-echo SIFTER neglecting the ¯ip-¯op term. (c) Spectrum corresponding to (a). (d) Spectrum corresponding to (b). (e) Time-domain data for solid-echo SIFTER after addition of the unmodulated background and multiplication with a Gaussian function. (f) Time-domain data for spin-alignment SIFTER considering unmodulated background and decay. (g) Spectrum corresponding to (e). (h) Spectrum corresponding to (f).

other hand, the deviations between the lineshapes in Fig. 2c and d are only small, hence for distances r12 P 2:8 nm analysis of SIFTER data with neglect of ¯ip-¯op terms is suciently precise. With increased excitation bandwidth this is also true for shorter distances. The lineshape of the dipolar spectrum improves for r12 ˆ 3:68 nm (data not shown) and resembles a broadened Pake pattern very closely at r12 ˆ 5:10 nm (Fig. 2g). The decreasing in¯uence of the ¯ip-¯op terms is also the reason that the modulation depth increases with decreasing dipolar coupling, that is, increasing distance. To estimate the modulation depth for the longer distance, an unmodulated contribution with three times the signal amplitude of the modulated component at s1 ˆ s2 was added. This accounts for spins whose coupling partners have a mI …14 N† 6ˆ 0 component and for those spins that

are excited at the observer position but do not have mI …14 N† ˆ 0. This signal was multiplied with a Gaussian decay function centred at s1 ˆ s2 to model the decay due to intermolecular contributions and is shown in Fig. 2e. Although the lineshape is clearly di€erent from a Gaussian, the modulation can hardly be recognized. A parallel simulation of spin-alignment SIFTER with the same post-processing yields time-domain data with larger modulation depth as anticipated in Section 2. Except for the lower signal-to-noise ratio, the corresponding dipolar spectrum shown in Fig. 2h is very similar to the one computed for solid-echo SIFTER. Note that the modulation depth increases with increasing excitation bandwidth, that is, decreasing pulse lengths, since the proportion of excited spin pairs to single spins then increases. This has been con®rmed by numerical simulations (data not shown).

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4. Results and discussion

alignment SIFTER. Solutions in toluene were prepared from the biradicals shown in Fig. 3 and from 4-hydroxy-2,2,6,6,-tetramethyl-piperidine-1oxyl (TEMPOL). The concentration of nitroxide groups was 1 mM except for biradical 2 and 3 where it was 2 mM, and the samples were measured in a glassy state obtained by shock freezing. On the biradicals 2 and 3 DEER measurements as described in Ref. [11] were performed for comparison. At these long distances we found it necessary to use an asymmetric four-pulse DEER sequence …p=2†m1 -s1 -…p†m1 -t-…p†m2 -s1 ‡ s2 -t-…p=2†m1 s2 -echo that allows for longer dipolar evolution intervals t at given total length of the sequence. This version of four-pulse DEER with s1 ˆ 200 ns features approximately the same sensitivity and resolution as three-pulse DEER and is free of deadtime. Theoretical estimates of the distances between nitroxide moieties in biradicals (Fig. 3) were

All experiments were performed on a Bruker ESP 380E pulse EPR spectrometer operating at X band frequencies using a Bruker EN4118X-MD4 probehead. The length of p=2 and p pulses was tp ˆ 24 and 48 ns, respectively. For the pulses with nominal ¯ip angle p=4 in spin alignment experiments, tp ˆ 8 ns was found to yield better results than tp ˆ 16 ns; only multiples of 8 ns are available as pulse lengths. A time increment for s2 of 8 ns corresponding to an increment for s2 ÿ s1 of 16 ns was used for the measurement on biradical 1, in all other cases the time increment for s2 ÿ s1 was 32 ns. Data traces with 256 data points were recorded. Experiments were performed at 80 K with liquid nitrogen cooling at an EPR observer ®eld corresponding to the maximum of the nitroxide spectrum. The phase cycle given in Table 1 was applied for both solid-echo SIFTER and spin-

Table 1 Phase cycle for SIFTER experimentsa /1 /2 /3 /det a

‡x ‡y ‡x ‡y

‡y ‡y ‡x ÿy

ÿx ‡y ‡x ‡y

ÿy ‡y ‡x ÿy

‡x ÿy ‡x ‡y

‡y ÿy ‡x ÿy

ÿx ÿy ‡x ‡y

ÿy ÿy ‡x ÿy

‡x ‡y ÿx ‡y

‡y ‡y ÿx ÿy

ÿx ‡y ÿx ‡y

ÿy ‡y ÿx ÿy

The phase /2 refers to the dipolar refocusing pulse labelled y in Fig. 1.

Fig. 3. Structures of the used shape-persistent monodisperse biradicals.

‡x ÿy ÿx ‡y

‡y ÿy ÿx ÿy

ÿx ÿy ÿx ‡y

ÿy ÿy ÿx ÿy

G. Jeschke et al. / Chemical Physics Letters 331 (2000) 243±252

obtained from force ®eld computations using the Merck Molecular Force Field as implemented in the Titan software package (Wavefunction Inc. and Schr odinger Inc.). The N±O groups were replaced by keto groups to avoid problems with force ®eld parametrization for radicals [24], and the mean values of the C±C and O±O distances of the keto groups were taken as the theoretical distances. To assess the in¯uence of conformational distributions, we performed a Monte-Carlo conformational search with 256 trials for each biradical and compared the distance for the conformation with minimum energy to the average distance of the 100 conformations with lowest energy (see Table 2). Nuclear modulation, in particular from matrix protons, is a common problem in EPR pulse sequences for separation of dipolar interactions [10,11,23]. Such modulations cannot be suppressed by phase cycling, since the phase of nuclear coherences is not in¯uenced by the phase of the m.w. pulses. In the four-pulse DEER experiment, nuclear modulations could be strongly diminished by setting a ®xed interpulse delay to a modulation blind spot [11]. Both analytical calculations and experiments show that in solid-echo SIFTER some nuclear modulations persist for any particular choice of s1 ‡ s2 . However, the phase of the modulation contributions depends on this choice, so that suppression can be achieved by averaging over several values of s1 ‡ s2 . At the same time, dipolar modulations from spin pairs depend only on s1 ÿ s2 , see Eq. (3), and are thus not averaged. We used this to suppress nuclear modulations in all measurements by adding signals from 25 experiments with incrementation of both interpulse

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delays by Ds1 ˆ Ds2 ˆ 8 ns between the experiments. This technique works only for experiments where all contributions to the nuclear modulation exhibit blind spots with respect to the total duration of the constant time experiment. The two SIFTER experiments ful®ll this requirement, but four-pulse DEER, for example, does not. In particular the measurement of long distances (r > 3 nm) of spin pairs requires careful correction of the background from a homogeneous spatial distribution of radicals in the sample [5,11]. We have determined this background experimentally by measurements on glassy frozen solutions of TEMPOL in toluene and have found that it is almost perfectly approximated by a Gaussian function; small deviations are observed for s2 ÿ s1 < 200 ns. This decay is due to homogeneously distributed distant electron spins. It is non-exponential in contrast to the decay in DEER, `2+1', and multiple-quantum experiments because of the refocusing properties of the solid echo for small couplings. The exponential background in these experiments as well as the Gaussian decay in SIFTER experiments introduces one additional ®tting parameter. The initial decay is also signi®cantly slower than with the other experiments, which contributes to the better resolution of SIFTER. Solid-echo SIFTER time-domain data of biradicals were processed by a least-square ®t of the experimental decay function of TEMPOL to the data and subsequent subtraction of the ®tted decay. The amplitude and a linear scaling factor for the time axis were varied in the ®ts. For spinalignment SIFTER, the background correction was performed by ®tting and subtracting a Gaussian function. This approach is sucient for

Table 2 Theoretical and experimental distances in nm between the nitroxide moieties of the biradicals shown in Fig. 3 Radical

Force ®elda

SIFTER

DEER

X-ray di€raction

1 2 3

2.91/2.84 3.86/3.79 5.23/5.15

2:84  0:03 3:68  0:06 5:10  0:12 5:00  0:20d

2:83  0:05b 3:63  0:10 4:92  0:20

2.784b; c ± ±

a

Conformation with minimum energy/ average of 100 conformations with lowest energy. Taken from Ref. [11]. c Single crystal of the pure radical. d Spin-alignment SIFTER. b

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distance determination from the singularities of the dipolar spectrum, but the spectrum is somewhat distorted close to zero frequency. Solid-echo SIFTER was applied to a series of three shape-persistent monodisperse biradicals with distances between the nitrogen moieties in the range from 2.8 to 5.2 nm (Fig. 3). For biradicals 1 and 2 the dipolar modulation is clearly visible in the time-domain data shown in Fig. 4a±d. After background correction of these time-domain signals with 256 data points, data processing as described in Section 3 yields the dipolar spectra displayed in Fig. 4e±h. Note that absolute-value spectra can be computed without resolution loss because the dispersion contribution cancels for the FT of an echo signal. For biradical 1 (Fig. 4e), the feet of the Pake pattern are not well de®ned, which is probably due to the pseudo-secular contributions as discussed in the section on numerical

computations. Both lineshape and modulation depth are expected to improve at larger excitation bandwidths. Also in agreement with the numerical simulations, we ®nd a second set of singularities at half the dipolar frequency. From the position of the outer singularities we obtain a distance of 2.84 nm, which is close to theoretical estimates (see Table 2) and earlier DEER results [11]. Note that the broadening of the dipolar spectrum has to be considered in the determination of the frequency of the singularities since it leads to a shift of the maximum of the asymmetric singularity. For the longer rod-like biradicals 2 and 3 we obtain dipolar spectra that are fairly close to the expected Pake patterns (see Fig. 4g, f). The pattern for radical 2 is better resolved than the one we obtained with DEER. Again the experimental distances of 3.68 and 5.10 nm, respectively, are in good agreement with force ®eld computations and

Fig. 4. Experimental SIFTER time-domain data (a±d) and dipolar spectra (e±h). (a), (e) Solid-echo SIFTER on biradical 1 (2.84 nm). The arrows in (e) mark the singularities corresponding to this distance. (b), (f) Solid-echo SIFTER on biradical 2 (3.68 nm). The upper trace in (f) shows the four-pulse DEER spectrum for comparison. (c), (g) Solid-echo SIFTER on biradical 3 (5.10 nm). (d), (h) Spinalignment SIFTER on biradical 3.

G. Jeschke et al. / Chemical Physics Letters 331 (2000) 243±252

DEER measurements. In the DEER experiment for the longest biradical, the distance could only be obtained by ®tting the time-domain data, no resolved Pake pattern was obtained after Fourier transformation (data not shown). Such time-domain ®tting depends critically on the assumption of an exponential decay due to all other interactions than the dipolar coupling of the spin pair. It becomes precarious and ®nally impossible for increasing distances. In contrast, SIFTER yields a nicely resolved Pake pattern for biradical 3, although the frequency of the singularity is only 390 kHz. This resolution advantage is a direct consequence of the solid-echo refocusing which allows us to use a longer total duration of this constanttime experiment and which leads to a slower decay of the signal due to the small couplings of distant electron spins. To the best of our knowledge, such a large well-de®ned distance has not yet been measured by EPR techniques. For biradical 3 we have also performed a spin-alignment SIFTER experiment to enhance the modulation depth. The slow dipolar modulation can now already be recognized in the time-domain data as a marked deviation from a Gaussian decay (see Fig. 4d). As expected, the dipolar spectra obtained by solidecho SIFTER and spin-alignment SIFTER for this radical are quite similar (see Fig. 4g,h). The main di€erence is distortions at very low frequencies that are due to the background correction with a Gaussian decay rather than an experimental SIFTER decay for the case of spin-alignment SIFTER. 5. Conclusion Solid-echo refocusing of dipolar couplings between electron spins has been achieved in a new single-frequency technique for distance determination in the nanometer range. The SIFTER experiments are particularly well suited for distances above 3 nm where the whole dipolar spectrum can be uniformly excited by m.w. pulses with moderate excitation bandwidth. A well resolved dipolar spectrum was obtained for a rod-like biradical with a distance of 5.1 nm between the nitroxide moieties corresponding to a frequency of the sin-

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gularity of less than 400 kHz. This high resolution is due to the fact that SIFTER refocuses instantaneous di€usion while existing techniques introduce an additional decay of the part of the signal which is not already lost by instantaneous di€usion. Spin-alignment SIFTER allows for the application of additional perturbations on the time scale of longitudinal relaxation of the electron spins and can thus be considered as a basic sequence for two-dimensional experiments that correlate the dipolar coupling between electron spins to other interactions.

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