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17910 EPR Spectroscopy10.1 Introduction to EPR SpectroscopyElectron paramagnetic resonance (EPR) shares the theoretical description and manyexperimental concepts with NMR spectroscopy. From a methodological point of view, themain difference is the much larger magnetic moment of electron spins, which exceeds the oneof protons by a factor of 660. Therefore EPR has a higher sensitivity at given spinconcentration, can observe spin-spin interactions over longer distances, and is sensitive tomolecular motion at shorter time scales. Resonance frequencies are in the microwave (mw)range (see Table 10.1). Typical pulse lengths and signal observation times are also shorter byabout a factor of 500. Spectrometer technology is thus more complicated and may differconsiderably between the different mw bands. In this lecture course, we shall refer to the mostcommon band, the X band, unless noted otherwise.Table 10.1: Microwave frequency bands used in EPR spectroscopy (band definition by Radio Societyof Great Britain)..From an application point of view almost all pure substances contain magnetic nucleiand are thus accessible to NMR spectroscopy, while only few pure substances containunpaired electrons and are thus accessible to EPR spectroscopy. This is because chemicalbinding is based on electron pair formation with spin cancellation. Most stable compounds areBand Frequency rangeTypical EPRFrequency [GHz]Typicalwavelength[mm]Typicala EPRField [mT]a. Assuming = 2.002319LSXKuQVWD1-22-48-1212-1830-5050-7575-110110-1701.53.09.51736709514020010030178432541103406001280250033905000vg ge~B0180 thus diamagnetic. The occurence of an EPR signal is thus often related to enhanced reactivity,as for instance in transition metal complexes and free radicals. Therefore, EPR is particularlywell suited for studies on synthetic and biological catalysts and for studies of radical-induceddegradation processes in organisms and materials. As an example consider the enzyme methyl-coenzyme M reductase (MCR), whichcatalyzes methane formation from methyl-coenzyme M (methyl-CoM) and coenzyme B.Biochemical studies do not reveal whether MCR catalyzes the reaction by attacking thethioether sulfur atom of methyl-CoM or the carbon atom of the CH3S group. This question wasaddressed by studying the reaction of MCR with the inhibitor 3-bromopropane sulfonate(BPS), which results in a stable paramagnetic species with a single electron in a orbitalon the nickel ion. By using BPS 13C labeled at the methyl group and the two-dimensionalHYSCORE experiment at X band (Section 10.4.4.4), it was possible to detect hyperfinecoupling between the unpaired electron and the methyl carbon atom (Figure 10.1). Thecouplings to the protons of BPS were obtained from Q-band HYSCORE and electron nucleardouble resonance (ENDOR) spectra. From this information a model for the binding modecould be proposed.Figure 10.1: Determination of the binding mode of the inhibitor 3-bromopropane sulfonate (BPS) tothe active center F430 of methyl-coenzyme M reductase (MCR). A) Structure of the active center. B)Structure of the inhibitor. C) X-band HYSCORE spectrum for the product obtained from F430 and BPS13C-labelled at the -position. The left quadrant contains the 13C peaks. D) Simulated HYSCOREspectrum. E) Structural model for BPS bound to F430 (from Hinderberger D. et al, Angew. Chem. Int.Ed. 2006, 45, 3602-3607).Some defects in inorganic materials, such as semiconductors, are also paramagnetic.These defects determine the electronic and optical properties of these materials. EPRdx2y2-30 -20 -10 0 10 20 300102030-30 -20 -10 0 10 20 300102030?2 /MHz?2 /MHz?1 / MHz?1 / MHzgzNiOGln?147H?H?1H? H?2SOSO33--NNNNNi*NNNNNin+HNH2NOCH3COOCOCH3COOCOOCOOOOCOHHOH2NGlna'147F430gba BPSBrA CDEB181spectroscopy provides more detailed information on their electronic and spatial structure thanelectrical or optical measurements. Furthermore, species with unpaired electrons are oftenobserved as reaction intermediates, in particular in one-electron transfer processes. Suchprocesses are the basis of energetics of living cells and occur in many redox reactions. Short-lived intermediates can be studied with special transient EPR experiments (photosyntheticreaction centers and light-sensitive proteins) or after spin trapping (Section 10.5.5). Forinstance, neurodegeneration during Alzheimers disease is related to an inflammatory processthat is in turn characterized by abnormally high levels of free radicals. EPR spectra recorded inthe presence of the spin trap N-tert-butyl-o-phenylnitrone (PBN) showed that toxic forms ofamyloid-| peptide lead to a formation of radicals. Such radical formation is not observed in theabsence of these peptides or in the presence of a peptide that is rendered nontoxic byreplacement of the methionine residue at position 35 by norleucine (Figure 10.2). Thisestablished the role of residue Met35 in the neurodegenerative process, the probableinvolvement of the sulfur atom of this residue in the chemistry of radical formation, and apotential protective effect of spin traps against Alzheimers disease. Meanwhile PBN andsimilar compounds are discussed as potential drugs.Figure 10.2: Demonstration of radical formation by amyloid-| peptide using spin trap EPR. A) Spintrap N-tert-butyl-o-phenylnitrone (PBN). B-E) EPR spectra for B) a control solution withoutA|-peptide, C) with A| 1-42 added, D) as before, but with residue Met35 replaced by anorleucine residue, E) with A| 25-35 added. Residues 1-24 and 36-42 appear to be notessential for radical formation (from: S. Varadarajan et al., J. Struct. Biol. 2000, 130, 184-208.For such applications EPR is better suited than NMR, mainly because of the lowconcentrations involved and the higher sensitivity of EPR. Unless paramagnetic centers relaxvery fast, they also cause strong broadening of NMR lines of nuclei in their vicinity. Activecenters of paramagnetic metalloproteins thus cannot be characterized by NMR. On the otherhand, the structure of domains remote from the active center are not accessible to EPR, but canbe studied by NMR. The two spectroscopies are thus often complementary.BCDEA182 Figure 10.3: Structural model of the dimer of the Na+/H+ antiporter of Escherichia coli determinedfrom the x-ray structure of the monomer by EPR distance measurements between spin-labeled residues.A) A selected residue is mutated to a cystein and the methanethiosulfonate spin label (MTSSL) isattached. B) For a homodimer with a C2 symmetry axis, four degrees of freedom (x, y, u, o) determinethe relative position of the two moieties. C) Nine distances were measured between spin labels in thetwo moieties of the dimer were measured (red lines). The dashed line marks a distance that was toolong for the applied technique. The eight known distances overdetermine the structure. D) Interactionbetween |-sheets in the two moieties contributes most to binding in the dimer. E) Another contributionto binding results from contacts between two helices (original publication: Hilger D. et al, Biophys. J.2007, 93, 3675-3683).EPR spectroscopy can also complement NMR spectroscopy on diamagnetic systems byproviding access to larger distances between sites and to dynamics on shorter time scales. Forthis purpose, stable free radicals have to be introduced in the system. EPR is then used as aprobe technique. Usually, nitroxides (Section 10.5.1) are introduced as non-covalentlyattached spin probes or covalently bound spin labels. For proteins site-directed spin labeling(SDSL) have been developed. This technique is based on mutation of an amino acid near thesite of interest to a cystein and covalent attachment of a thiol-specific spin marker. As is thecase with NMR, structure determination of biomacromolecules by EPR does not requirecrystallization.For instance, the dimer of the Na+/H+ antiporter NhaA of Escherichia coli is weaklybound and does not survive crystallization. A structural model could be determined from the x-ray structure of the monomer and measurements of nine distances between spin-labeledresidues in the two moieties of the dimer (Figure 10.3). If the two monomers in a homodimerare considered as rigid bodies, their relative position is fixed by only four degrees of freedom.Hence, the problem is overdetermined. The structural model shows that, to a large extent,binding between the monomers is due to interactions of two |-sheets and to a lesser extent byN OSSOON OSMTSSLC -SHbC -SbC2xy180fqZXYA B CD ER49Q47VIIAIXBR204V254L210W258183hydrophobic interactions between two helices. The |-sheet binding motif is common forsoluble proteins, but was found here for the first time for a membrane protein. Laterbiochemical experiments confirmed the importance of hydrogen-bonding residues in the |-sheets for dimer formation.10.2 Differences between EPR and NMR spectroscopyThe basics of the description of spin systems and spin dynamics are the same for EPRand NMR. Many of the experimental concepts from NMR can also be applied. In theremaining lectures we mostly discuss concepts that are particular to EPR. Such concepts resultfrom the following differences. First, the larger magnetic moment implies frequencies in themw range (Table 10.1) that pose more difficult problems for spectrometer technology. Largerexcitation bandwidths are required, which implies pulses with lengths in the nanosecond ratherthan microsecond range. For these reasons pulse techniques were developed later in EPR thanin NMR. The typical number of pulses in an experiment is smaller in EPR and pulse shape,phase, and frequency cannot be varied so easily. Experimental schemes are thus less complexin EPR compared to NMR.Due to the larger magnetic moment of the electron spin, typical hyperfine couplingsbetween an electron and a nuclear spin are by at least two orders of magnitude larger thantypical couplings between two nuclear spins. At the same time, nuclear Zeeman frequenciesare smaller than in NMR as external magnetic fields are usually smaller. Thus, hyperfinecouplings are often of the same order of magnitude as the nuclear Zeeman interaction and leadto mixing of nuclear spin states. The usual spectroscopic selection rules do no longer apply.Formally forbidden transitions can be excited by a single pulse or even by continuousirradiation. This allows for detection of nuclear frequencies in electron spin echo envelopemodulation (ESEEM) experiments (Section 10.4.4). The pulse sequences consist of only mwpulses that are far off-resonant for the nuclear spins. Such experiments have no equivalent inNMR. Generally, the spin dynamics is more complex in EPR compared to NMR.In addition, the larger ratio between couplings and resonance frequencies than in NMRmakes relaxation rates faster compared to the resonance frequencies. As line widths areproportional to the transversal relaxation rate, this leads to a larger ratio between line widths184 and resonance frequencies, i.e. to lower resolution. To some extent this lower resolution iscompensated by a larger spread of magnetic parameters. The equivalent to the chemical shift inNMR is the relative deviation (g-ge)/ge of the electron g value from the ge value for the freeelectron. This deviation is typically three orders of magnitude larger than chemical shifts (pptinstead of ppm).Because of the larger frequeny shifts and the larger couplings, EPR spectra extend overa broader frequency range relative to the mean resonance frequency. This requires a largerrelative excitation bandwidth. However, the relative excitation bandwidths of pulses in theradio frequency (rf) and mw range are almost the same, if the same pulse power is available. Itis thus not usually possible in EPR spectroscopy to excite the whole spectrum by a singlepulse. This eliminates the sensitivity advantage of pulse Fourier transform experiments, so thatcontinuous-wave (cw) EPR is still the method of choice for recording basic EPR spectra(Section 10.7.1). Furthermore, cw techniques are applicable even if the transverse relaxationtime T2 is shorter than the shortest available pulses. Therefore, cw EPR is applicable in abroader temperature range than pulsed EPR.NMR spectroscopy relies strongly on the concept of separation of interactions bymultipulse sequences and 2D spectroscopy. These concepts can be transferred to pulse EPR.By using them, resolution can be improved tremendously and more information can beobtained. Therefore both cw and pulse EPR are commonly applied. Even in pulsed EPR theexternal magnetic field is varied to overcome the limitations in excitation bandwidth.Another difference arises from the much larger change of the EPR resonance frequencyassociated with a change of the orientation of the molecule in the magnetic field. Anisotropiesof EPR frequencies are on the order of tens of Megahertz to several Gigahertz. This matchesthe range of inverse rotational correlations times of molecules in fluid solution or soft matter.Hence, orientations of the molecule exchange on the time scale of cw EPR experiments. Theeffect on the spectral line shape can be treated in the same way as chemical-exchangephenomena in NMR were treated in Chapter 3. The rotational correlation time of the moleculecan then be inferered from the line shape (Section 10.5.2).Like NMR, EPR spectroscopy can be applied to fluid or solid samples. As the couplingbetween two electrons at the same distance is by a factor (ue/uH)2 = 436000 larger than185between two protons, the concentration of paramagnetic centers needs to be smaller to avoidexcessive line broadening by spin-spin interaction. Best resolution and sensitivity for liquidsamples is achieved at concentrations of of about 10-3-10-5 M. In crystalline samples theparamagnetic centers are often diluted by doping the paramagnetic compound into anisomorphous diamagnetic compound. Typical dilutions are 1:100 to 1:10000. If isomorphousdiamagnetic substitution is impossible, solid-state measurements are performed in frozensolution. Aggregation of the dissolved paramagnetic compound on freezing is avoided byusing glass-forming solvents. A selection of pure glas-forming solvents is given in Table 10.2.For amphiphilic compounds mixtures of polar and apolar solvents have to be used. Such glass-forming solvents are, for instance, isopentane/isopropanol in a 8/2 mixture, toluene/acetone ina 1/1 mixture or with an excess of toluene, and toluene/methanol in a 1/1 mixture or with anexcess of toluene. For biomacromolecules, water/glycerol mixtures are often used.Table 10.2: Glass-forming solvents with their melting temperatures Tm andglass transition temperatures Tg.Solvent Tm/K Tg/K2-methylbutane 113.3 68.22-methyltetrahydrofuran 137 91ethanol 155.7 97.2methanol 175.2 102.61-propanol 146.6 109toluene 178 117.2ethylene glycol 255.6 154.2di-1,2-n-butyl phthalate 238 179glycerol 291.2 190.9o-terphenyl 329.3 246poly(styrene) 513 373186 Another important difference between NMR and EPR is the appearance of the spectra.For technical reasons (see Section 10.7.1), best sensitivity and resolution in EPR is achievedwith cw experiments at fixed mw frequency. The magnetic field is swept and at the same timemodulated with a typical frequency of 100 kHz. Phase-sensitive detection of the signalmodulation at this frequency yields the derivative of the absorption spectrum (Fig. 10.4).Although the shape of a derivative Lorentzian absorption line looks like the shape of adispersion line, the derivative absorption line has a smaller linewidth.Figure 10.4: Detection of the derivative of the absorption spectrum by field modulation withmodulation amplitude AB0. The amplitude of the reflected mw AV is measured (A) providing aderivative lineshape (B). To avoid artificial line broadening the condition has to befulfilled. The peak-to-peak linewidth ABpp = I in field-swept spectra scales as 1/g (roughly with B0) ifthe transversal relaxation time T2 does not depend on g (C). 10.3 Interactions and EPR Hamiltonians 10.3.1 Classification of electron spin systemsTo avoid excessive relaxational line broadening, EPR spectra are usually measured in aconcentration regime where interactions between paramagnetic centers are negligible. Anexception are systems where the distance between two (or more) paramagnetic centers isrelatively well defined and is measured by EPR (Section 10.3.6). An isolated paramagneticcenter can contain a single unpaired electron (electron spin S=1/2, Section 10.3.3) or severaltightly coupled electron spins (group spin S>1/2, Section 10.3.4). The latter situation isDB0DBppDVDVABC= 2 hT2 g B3G .B0AB03ABpps187encountered in high-spin states of transition metal complexes or triplet states of organicmolecules. If the zero-field splitting in such a high-spin system exceeds the mw frequency,transitions can be induced only with pairs of levels with magnetic quantum numbers ofthe electron group spin S. Such a pair of levels is a Kramers doublet, meaning that the levelsare degenerate in the absence of an external magnetic field and are split by the field. TheKramers doublet can be treated as an effective spin S=1/2 (Section 10.3.5).Figure 10.5: Topology of spin systems. A) System consisting of seven nuclear spins. All magneticmoments and hence all 21 spin-spin couplings are comparable. B) System consisting of one electronspin with a large magnetic moment and six nuclear spins with much smaller moments. Only the sixhyperfine couplings between the electron spin and the nuclear spins are significant.In most paramagnetic centers the electron spin is coupled to nuclear spins. Because ofthe presence of one spin that has a much higher magnetic moment than all the other spins, suchelectron-nuclear spin systems have a different topology compared to the nuclear spin systemsobserved in NMR. In typical nuclear spin systems all spins exhibit significant couplings toseveral other spins (Fig. 10.5A), while in typical electron-nuclear spin systems the nuclear-nuclear couplings are negligible. Hence, the electron spin couples to many nuclei, but eachnucleus couples significantly only to the one electron spin (Fig. 10.5B). The spectrum of thenuclear spins is thus easier to analyze and better resolved. It can be observed by ENDOR(Section 10.4.3) or ESEEM (Section 10.4.4) experiments. Furthermore, spin evolution can bedescribed by considering only subsystems consisting of the electron spin and one nuclear spin,since the matrix representation of the spin Hamiltonian factorizes into Hamiltonians of two-spin systems. This is an important simplification, since two-body problems can be solvedexactly, while many-body problems cannot.mSA B188 10.3.2 Units of magnetic parameters in EPR spectroscopyIn NMR spectroscopy magnetic parameters are generally given in frequency units orare dimensionless. In EPR spectroscopy, magnetic field units are also used. Furthermore,wavenumbers are sometimes given for zero-field splittings and large hyperfine couplings thatmay be resolved in optical spectra.Frequency units and wavenumbers are energy units that can be interconverted withoutfurther knowledge (1 MHz = 3.33564110-5 cm-1). In contrast, the conversion betweenmagnetic field units and frequencies is not trivial, as it depends on the g value, and the g valueoften cannot directly be read off the spectrum. The conversion may thus require a full analysisof the spectrum. If the g value is known with sufficient precision, we have . [10.1]In particular, for (organic radicals), 1 MHz corresponds to 0.0356828 mT or 1mT to about 28 MHz. The cgs unit Gauss (G) is often used to specify the magnetic field. In factit is a unit of magnetic induction (SI unit Tesla (T)), with 10000 G = 1 T. In both NMR andEPR the magnetic induction is usually referred to as the magnetic field. More precisely, themagnetic field unit is A m-1 in SI units or Oersted in cgs units (1 A m-1 = 4t10-3 Oersted). Ifthe relative permeability of a material is unity, the magnetic induction in Gauss is the same asthe magnetic field in Oersted. This is a good approximation for diamagnetic and paramagneticsubstances, but not for ferromagnetic materials. Finally, when discussing spin Hamiltonians,we use angular frequency units for energies. Angular frequencies e are related to frequencies vby e=2tv, and to energies E by .10.3.3 One electron spin S=1/2 coupled to nuclear spinsThe spin Hamiltonian in the absence of mw or rf irradiation (static Hamiltonian) isgiven by , [10.2]AB0 Avh guB( ) =g ge~e E h =H0 HEZ HHF HNZ HNQ + + + =189where the terms are the electron Zeeman interaction , the hyperfine interaction , thenuclear Zeeman interaction , and the nuclear quadrupole interaction . In thefollowing they are discussed in turn.10.3.3.1 The electron Zeeman interactionThis interaction is formally equivalent to the (nuclear) Zeeman interaction treated inSections 5.1 and 5.4. It is given by , [10.3]where JT-1 is the Bohr magneton, Js Plancks quantum of action, the transpose (T)of the magnetic field vector , the g tensor, and the electron spinvector operator . In the PAS of the g tensor, the magnetic field vector can bewritten as, [10.4]where u and o are polar angles that characterize the direction of this vector in the PAS.Expansion of Eq. [10.3] in this coordinate system yields . [10.5]For an anisotropic g tensor,1 the direction of the effective field depends on the orientation ofthe molecule in the magnetic field. In other words, the quantization axis of the electron spin isnot necessarily parallel to the magnetic field and can have different directions for differentorientations of the molecule. The effective g value is thus given by . [10.6]1Strictly speaking, does not have the transformation properties of a tensor, as it connects two independentspaces (spin space and laboratory space). Therefore it is sometimes called an interaction matrix. In this lec-ture, we call all interaction matrices tensors, as most magnetic resonance literature does.HEZ HHFHNZ HNQHEZuBh------B0TgS=uB |e9.27400899 37 ( )24 10 = =hh 2t ( ) 1.0545715968234 10 = = B0TB0 B0xB0yB0z. . ( ) = g SSx Sy Sz . . ( )B0 B0 u o u o u cos . sin sin . cos sin ( ) =HEZuBh------B0gx u oSx gy u oSy sin sin gz uSz cos + + cos sin ( ) =ggeffgx2u sin2o cos2gy2u sin2o sin2gz2u cos2+ + =190 In the high-field approximation, where the electron Zeeman interaction dominates over all theother interactions, its contribution to the energy of the spin states is proportional to ,where mS is the magnetic quantum number of the electron spin. The resonance field at thisorientation of the molecule with respect to the magnetic field is then . [10.7]The deviation of the g value of bound electrons from the ge value of the free electron ismostly due to spin-orbit coupling. As orbital angular momentum is quenched for non-degenerate ground states, spin-orbit coupling is due to admixture of excited states to theground state by the orbital angluar momentum operator . This admixture is usually small andcan thus be treated by perturbation theory. To second order the g tensor is given by , [10.8]where indices i and j run over the Cartesian directions x, y, and z and is the Kroneckerdelta. The outer sum with index k runs over all atoms in the molecules, where is the spin-orbit coupling constant for the kth atom and and are Cartesian components of the orbitalangular momentum operator of this atom. As spin-orbit coupling is a relativistic effect, increases with the mass of the atomic nucleus. The inner sum with index m runs over themolecular orbitals with index 0 designating the singly occupied molecular orbital (SOMO),where the unpaired electron resides in the ground state. Note that this index can be bothnegative (occupied orbitals) and positive (empty orbitals). Accordingly, the energy difference in the denominator can also be positive or negative. Spin-orbit coupling with emptyp-, d-, or f-orbitals thus leads to negative deviations of g from ge (high-field shifts), while spin-orbit coupling with occupied orbitals leads to positive deviations (low-field shifts). The lattercase is more often encountered. Spin density in s orbitals does not contribute to deviations of gfrom ge.The g tensor is thus a complicated function of the frontier orbitals of the molecule. It isused as fingerprint information for the class of paramagnetic centers. Since light elements (firstand second period) have a small spin-orbit coupling constant, deviations of g from ge are onlymSgeffB0 res .hvmwgeffuB--------------- =lgijgecij2 km lki0 ( , 0 lkjm ( ,E0Em----------------------------------------------m 0 =k+ gecij2 k Ak( )ijk+ = =cijklkilkjkE0Em191of the order of for organic radicals, unless there exist near degenerate frontierorbitals ( ). Typical cases of orbital degeneracy are the hydroxyl radical OH,alkoxy radicals RO, and thiyl radicals RS in the gas phase. In condensed phase, degeneracy islifted by interaction with neighboring molecules. For these radicals, the g tensor thus stronglydepends on intermolecular interactions. For first row transition metal ions deviations of g fromge are of the order of . They are dominated by contributions from orbitals on thetransition metal ion, so that we find , [10.9]where is the spin-orbit coupling constant of the transition metal.10.3.3.2 The hyperfine interactionMore detailed information on the electronic and spatial structure of the paramagneticcenter can be obtained from the hyperfine couplings. The hyperfine interaction term of the spinHamiltonian is given by , [10.10]where k runs over all magnetic nuclei (Ik >0) in the molecule that have hyperfinecouplings larger than the linewidths in the spectrum under discussion. The are nuclear spinvector operators for these nuclei and the are hyperfine tensors. When analyzing liquid-stateEPR spectra, hyperfine couplings exceeding 3 MHz have to be considered. In solid-state EPRspectra, such couplings are usually resolved only when they exceed 15 MHz. When analyzingENDOR, ESEEM, or hyperfine sublevel correlation (HYSCORE) spectra the resolution limitis at 0.2-0.5 MHz. Within the high-field approximation the contribution of the hyperfinecoupling to the energies of the spin states is , where Aeff is an effective hyperfinecoupling.Hyperfine coupling of the electron spin to a nuclear spin comes about by through-spacedipole-dipole coupling of the two magnetic moments and by the Fermi contact interaction.The Fermi contact interaction is due to the non-zero probability to find the electron spin at the103 .102 E0Em 0 ~102 .101 g ge1 2m lki0 ( , 0 lkjm ( ,E0Em----------------------------------------------m 0 =+ ge1 2A + = =HHF STAkIkk=IkAkmSmIAeff192 same point in space as the nuclear spin. This happens only when the unpaired electron is in ans orbital. The Fermi contact interaction leads to a purely isotropic coupling with , [10.11]where ps is the spin density in the s orbital under consideration, gn is the nuclear g value and JT-1 the nuclear magneton ( ). The factor is the probability to find the electron at this nucleus in the ground state withwavefunction .1 The magnetic moment of the electron is fully characterized by the ge valueof the free electron, as there is no orbital angular momentum in s orbitals.Unpaired electrons in p-, d, and f- orbitals do not contribute to Fermi contactinteraction, as these orbitals have a node at the nucleus. However, due to the non-sphericalsymmetry of these orbitals, dipole-dipole coupling between the magnetic moments of theelectron and nuclear spin does not average. Spin density in such orbitals thus gives rise topurely anisotropic couplings.2 In general, the matrix elements Tij of the total anisotropichyperfine coupling tensor of a given nucleus are computed from the ground statewavefunction by . [10.12]Quantum chemistry programs such as ORCA, ADF, or Gaussian can compute as well asaiso.A special situation applies to protons, alkali metals and earth alkaline metals, whichhave no significant spin densities in p-, d-, or f-orbitals. In this case, the anisotropiccontribution can only arise from through-space dipole-dipole coupling to other centers of spindensity. In a point-dipole approximation the hyperfine tensor is then given by1Tabulated in: J. R. Morton, K. F. Preston, J. Magn. Reson. 1978, 30, 577.2For given spin densities these couplings can be computed from parameters given in: J. R. Morton, K. F.Preston, J. Magn. Reson. 1978, 30, 577. aisoSTIaiso ps23---u0h-----geuBgnun v00 ( )2 =un |n5.05078317 20 ( )27 10 = = gnun nh =v00 ( )2v0TTiju04th----------geuBgnun v0( '3rirj cijr2r5--------------------------- v0' , =T193 , [10.13]where the sum runs over all centers with significant spin density pj (summed over all orbitals atthis center), the Rj are distances between the nucleus under consideration and the centers ofspin density, and the are unit vectors along the direction from the considered nucleus to thecenter of spin density. For protons in transition metal complexes it is often a goodapproximation to consider spin density only at the central ion. The distance R from the protonto the central ion can then be directly inferred from the anisotropy of the hyperfine coupling.Admixed orbital angular momentum also contributes to dipole-dipole coupling. Thiscan be considered by a simple correction, giving the dipole-dipole hyperfine tensor . [10.14]We thus obtain for the total hyperfine tensor in Eq. [10.10] . [10.15]Note that the product may have an isotropic part, although is purely anisotropic. Thispseudocontact contribution depends on the relative orientation of the g tensor and the spin-only dipole-dipole hyperfine tensor .The hyperfine couplings can be used to map the SOMO in terms of a linearcombination of atomic orbitals. For catalytic species this can provide insight into reactivity andthus into the mechanism of catalysis. For organic compounds that form reasonably stableradical anions or cations the SOMO of the radical anion corresponds to the LUMO of theparent compound and the SOMO of the radical cation to the HOMO of the parent compound.The contributions discussed so far can be understood in a single-electronrepresentation. A further contribution arises from correlation between electrons in themolecule. Assume that the pz orbital on a carbon atom contributes to the SOMO, so that the ospin state of the electron is preferred in that orbital (Fig. 10.6). Electrons in other orbitals onthe same atom will then also have a slight preference for the o state, as electrons with the sameTku04th----------geuBgnun pj3njnjT1 Rj3-----------------------j k ==njAddgTge------- =AkAkaiso k .1gTkge--------- + =gTkTkTk194 spin tend to avoid each other and thus have less electrostatic repulsion.1 In particular, thismeans that the spin configuration in Fig. 10.6A is slightly more preferable than the one in Fig.10.6B. According to the Pauli principle the two electrons that share the o bond orbital of the C-H bond must have antiparallel spin. Thus, the electron in the s orbital of the hydrogen atomthat is bound to the spin-carrying carbon atom has a slight preference for the | state. Thiscorresponds to a negative isotropic hyperfine coupling of the directly bound o proton, which isinduced by the positive hyperfine coupling of the adjacent carbon atom. The effect is termedspin polarization, although it is entirely different from the polarization of electron spintransitions in an external magnetic field.Figure 10.6: Spin polarization on an adjacent hydrogen atom (o proton) induced by spin density in a pzorbital on carbon. The spin configuration in A) is slightly favored with respect to the one in B).Figure 10.7: Isotropic hyperfine coupling of a next-neighbor hydrogen atom (| proton) induced byhyperconjugation. The overlap of the carbon pz orbital and the hydrogen s orbital depends on dihedralangle ;.Spin polarization is observed both in o radicals, where the spin density is confined to asingle pz orbital, and in t radicals, where the spin density is delocalized in a t system. In t1This preference for electrons on the same atom to have parallel spin is also the basis ofHunds rule.C C H Hs spz pzA BCHHcA B195radicals spin polarization is the only contribution to the isotropic hyperfine coupling of aproton that is directly bound to a carbon atom of the conjugated system. The isotropichyperfine coupling of such an o proton can thus be predicted by the McConnell equation , [10.16]where pt is the spin density at the adjacent carbon atom and QH is a parameter of the order of mT, which slightly depends on the structure of the t system.The isotropic hyperfine coupling of | protons in o radicals is caused by spindelocalization from the spin-carrying pz orbital to the s orbitals on the | hydrogen atoms.These orbitals are sufficiently close in space to allow for overlap. The extent of thishyperconjugation depends on the dihedral angle between the pz orbital lobes and the C-H|bond. When the C| group rotates freely with respect to the spin-carrying orbital, an averagehyperfine coupling is observed. Unlike spin polarization, hyperconjugation does not depend onelectron correlation, but can rather be seen as spin density transfer.10.3.3.3 Nuclear Zeeman interactionThe nuclear Zeeman interaction can be neglected in the analysis of EPR spectra, unlessit is of the same order of magnitude as a resolved hyperfine coupling of the same nucleus. Inanalysis of ENDOR, ESEEM, and HYSCORE spectra, the nuclear Zeeman interaction is usedto assign hyperfine couplings to elements (isotopes). This information is missing in EPRspectra. Chemical shift information cannot be obtained from EPR, ENDOR, ESEEM, orHYSCORE spectra, as chemical shifts are smaller than the linewidths for nuclear spins withsignificant hyperfine couplings. Chemical shift is thus neglected in the nuclear ZeemanHamiltonian, which is given by . [10.17]In the laboratory frame this simplifies to , [10.18]Aiso H .QHpt=2.5 HNZgnunh-----------B0TI =HNZ eIIz=196 with eI = -nB0. Within the high-field approximation the nuclear Zeeman interactioncontributes an energy mI eI to the spin states. At S-band to Q-band frequencies the nuclear Zeeman frequency is often comparable tohyperfine couplings or smaller. This leads to state mixing by anisotropic hyperfineinteractions, which is the basis of ESEEM and HYSCORE experiments. At W-bandfrequencies and above this phenomenon is observed only for a few low- nuclei with largehyperfine couplings (large spin densities), such as for nitrogen atoms that are directlycoordinated to a transition metal ion. 10.3.3.4 Nuclear quadrupole interactionThe nuclear quadrupole interaction (Section 5.9) does not contribute to EPR spectraunless it is of the same order of magnitude as the hyperfine coupling of the same nucleus andthe hyperfine coupling of this nucleus is resolved. Usually this happens only for heavyelements from the third row of the periodic table onwards. In analyzing ENDOR, ESEEM, andHYSCORE spectra of nuclei with spin Ik>1/2 the nuclear quadrupole interaction has to beconsidered. In an EPR context, the nuclear quadrupole Hamiltonian is often written as , [10.19]where the principal values of the nuclear quadrupole tensor are given by , [10.20] , [10.21]and . [10.22]The principal values are thus directly related to the field gradient components Vxx, Vyy, and Vzzdefined in Section 5.9. Within the high-field approximation the contribution of the nuclearquadrupole interactions to the energies of the spin states is proportional to , where Peffis an effective nuclear quadrupole interaction. To first order, the transition energy for a pair ofHNQ ITPI=Pze2qQ2I 2I 1 ( )h---------------------------- =PxPy nPz=PxPyPz+ + 0 =mI2Peff197levels thus does not depend on the nuclear quadrupole interaction. In particular, thisapplies to the allowed transition .10.3.4 One electron group spin S>1/2 coupled to nuclear spinsIf m electrons are distributed indegenerate orbitals, each orbital is first singly occupied(Fig. 10.8A, Hunds rule). This is because electron pairing in the same orbital would lead toincreased Coulomb repulsion. In the presence of a ligand field that removes orbitaldegeneracy, pairing is preferred if the energy difference between the upper and lower set oforbitals exceeds electron pairing energy (Fig. 10.8B,C). Depending on the strength of theligand field, several electrons may thus be unpaired (high-spin state) or paired as far aspossible (low-spin state).Figure 10.8: High-spin and low-spin configurations for a d5 transition ion, such as Fe3+. A) In theabsnece of a ligand field all unpaired electrons in the degenerate d orbitals have parallel spin (high spinS= 5/2). B) In a weak ligand field, where the splitting between the eg and t2g levels is smaller than theelectron pairing energy, all spins are parallel (high spin, S= 5/2). C) In a strong ligand field, where thelevel splitting exceeds the electron pairing energy, all electrons occupy the set of lowest-lying orbitals,where they pair (low spin, S= 1/2).In high spin states of transition metal ions all the unpaired electrons reside mainly inorbitals on the same atom. They are so close in space that they couple strongly and cannot beexcited separately from each other. Hence, it is appropriate to describe them as a single groupspin S = m/2. For instance, Fe3+ with a 3d5 configuration in weak ligand fields assumes an S =5/2 high-spin state, while in strong ligand fields it assumes an S = 1/2 low-spin state.On first sight one might expect that the distribution of five unpaired electrons on allfive 3d orbitals is spherically symmetric so that no equivalent to the nuclear quadrupolemImI1 2 1 2 ( ) =A B Cdegt2gt2geg198 interaction would result. However, spin-orbit coupling breaks this symmetry. Electron-groupspins S > 1/2 thus have an additional interaction that is formally analogous to the nuclearquadrupole interaction. This zero-field splitting1 contribution to the spin Hamiltonian iswritten as , [10.23]where is the traceless zero-field splitting tensor. In the principal axes frame of this tensor,the Hamiltonian can be written as . [10.24]with D = 3Dz/2 and E = (Dx-Dy)/2. By convention, Dz is the principal value with the largestmagnitude, so that E cannot exceed D/3. For axial symmetry, E = 0. In cubic symmetry,quadrupolar zero-field splitting vanishes (D= E = 0). In that case and for S > 3/2, the muchsmaller hexadecapolar contribution may become observable. For transition metal ions, g shifts with respect to ge (Eq. [10.9]) and zero-field splittingare related, . [10.25]Within the high-field approximation the contribution of the zero-field splitting to the energylevels is proportional to . Thus the transition that is allowed inEPR spectra of half-integer group spins S is not affected to first order by zero-field splitting.10.3.5 Effective spin S=1/2 in a high-spin system S>1/2If the zero-field splitting is much larger than the mw quantum and the electron Zeemaninteraction at accessible magnetic fields, only part of the transitions of a spin S > 1/2 areaccessible for EPR. In conventional EPR at X band, the limit is at about 20 GHz. For integergroup spins S, none of the transitions is accessible unless the zero-field splitting tensor has at1The term of the Hamiltonian is sometimes called fine structure term instead of zero-field splitting.HZF STDS=DHZFPAS ( )DxSx2DySy2DzSz2+ + =D Sz213---S S 1 + ( ) E Sx2Sy2 ( ) + =D 2A =mS2DeffmS1 2 1 2 ( ) =199least axial symmetry. Such integer-spin systems are termed EPR silent, although they areparamagnetic and can be studied at higher fields and frequencies. For half-integer group spinsthere is at least one pair of low-lying levels that are degenerate in the absence of a magneticfield and are split in its presence. Such a Kramers doublet can be described as an effective spinS = 1/2. For instance, for high-spin Fe3+ (S = 5/2) in non-cubic symmetry, EPR transitions canbe observed only within the three Kramers doublets, but not between them (Fig. 10.9). Thespin Hamiltonian for each Kramers doublet is equivalent to the spin Hamiltonian of a spin S =1/2 (Section 10.3.3), except that the effective g values depend on the zero-field splittingparameters D and E and are not strictly field-independent.Figure 10.9: Kramers doublets with effective spin S=1/2 for high-spin Fe3+ (3d5) with E/D = 1/3. Themw quantum (vertical bars) is too small to excite transitions between different Kramers doublets. Thethree Kramers doublets have different effective g values.10.3.6 A pair of weakly coupled electron spins10.3.6.1 Exchange couplingConsider two electron spins S1 = 1/2 and S2 = 1/2 as individual paramagnetic centers. Atdistances up to at least 15 there is still significant overlap of the two SOMOs, so that the twoelectrons can exchange. This exchange interaction leads to a splitting between the singlet state(group spin S = 0) and the triplet state (group spin S = 1) of the coupled system. If this splittingB00g = 9.67g = 4.3g = 0.6200 is so small that transitions between the singlet and triplet state can be excited in an EPRexperiment, it is more convenient to treat the system in terms of two individual spins that arecoupled to each other. Typically at distances larger than 5-10 the individual spin treatment isapplied.The energy difference between the singlet and triplet state is the exchange integral , [10.26]where v1 and v2 are the wavefunctions of the two unpaired electrons.1 For positive J, thesinglet state is lower in energy, i.e., the orbital overlap is bonding and the interaction isantiferromagnetic. Negative J correspond to a lower-lying triplet state, i.e., antibonding orbitaloverlap and a ferromagnetic interaction.The exchange term of the spin Hamiltonian is given by . [10.27]This term describes a scalar (purely isotropic) coupling. Anisotropic contributions to theexchange coupling may occur for species involving heavy elements, but can be neglected fororganic radicals. Within the high-field approximation the contribution of the exchangeinteraction to energy levels is proportional to mS,1mS,2J. To a good approximation the exchange interaction decays exponentially with distance rbetween the paramagnetic centers. The decay rate depends on the conductivity of the mediumbetween the centers and on the presence or absence of a conjugated network of bonds betweenthem. If conjugation is weak and the matrix isolating, exchange coupling is much smaller thanthe dipole-dipole coupling through space at distances longer than 15 . 1There exist different conventions for the sign of J and the factor 2 may be missing. One should alwaysascertain which convention a certain author is adhering to.J 2e2 v1-r1( )v2-r2( )v1r2( )v2r1( )r1r2------------------------------------------------------------------------ r1d r2d =HEX JS1TS2 J S1xS2x S1yS2y S1zS2z + + ( ) = =20110.3.6.2 Dipole-dipole couplingThe dipole-dipole coupling between two electron spins is analogous to that betweentwo nuclear spins (Section 5.8). For transition metal and rare earth ions the g anisotropy is solarge that the two magnetic moments are not parallel to the magnetic field. The interactionenergy according to Eq. [5.74] is then parametrized by three angles u1, u2, and o (Fig. 10.10)and is given by . [10.28]The dipole-dipole coupling term of the spin Hamiltonian assumes the form . [10.29]Figure 10.10: Coupling between two magnetic moments and in a general orientation withrespect to the external magnetic field and definition of the angles u1, u2, and o that parametrize thisinteraction. Note that these angles implicitly depend on the orientation of the molecule in the magneticfield.In the special case of two parallel magnetic moments, which is a good approximationfor two organic radicals, the dipole-dipole coupling tensor has the principal values -edd, -edd, 2edd with . [10.30]The orientation of the molecule can then be characterized by a single angle u between themagnetic field axis and the spin-spin vector. The orientation-dependent dipolar splitting thusassumes the form . [10.31]E u04t------ u1u21r3---- 2 u1 u2cos cos u1 u2 o cos sin sin ( ) =Hdd S1TDS21r3---- u04th---------- g1g2uB2S1S23r2---- S1r ( ) S2r ( ) = =r0B12q1q2fu1 u2B0Dedd1r3---- u04th---------- g1g2uB2 =d u ( ) 3 u cos21 ( )edd=202 For organic radicals, is a good approximation, so that the dipole-dipolecoupling vdd=edd/2t in frequency units is given by . [10.32]With typical transversal relaxation times T2 of electron spins of a few microseconds,dipole-dipole couplings can be measured down to about 100 kHz in favourable cases,corresponding to distances of 8 nm. For soluble proteins and membrane proteins reconstitutedinto detergent micelles this upper distance limit may reduce to about 6 nm and for membraneproteins reconstituted into liposomes to about 5 nm. These limits are comparable to thediameter of protein molecules.Within the high-field approximation the contribution of the dipole-dipole interaction toenergy levels is proportional to mS,1mS,2 edd,eff. This is the same dependence on magneticquantum numbers as for the exchange coupling. The two contributions to electron-electroncoupling thus cannot be separated by spin manipulation.Figure 10.11: Transversal and longitudinal relaxation of the electron spin of a nitroxide molecule by acollsion with triplet dioxygen. (a) The molecules are separated and diffuse towards each other. Forexample, both unpaired electrons of the oxygen molecule are in the | state, while the unpaired electronof the nitroxide molecule is in the o state. (b) The molceules collide, their orbitals overlap, and the threeunpaired electrons are no longer distinguishable. (c) The molecules have separated again. With aprobability of 2/3 the nitroxide molecule is now left with one of the unpaired electrons that originallybelonged to the oxygen molecule (and vice versa). The electron spin of the nitroxide thus has lost phasememory and has changed its spin state.10.4 Measurement of hyperfine couplings10.4.1 cw and echo-detected EPR spectroscopyHyperfine couplings manifest in the spectra of both electron and nuclear spins. Largehyperfine couplings (>20...50 MHz or 0.7...2 mT at g = ge) can often be determined withg1g2ge~ ~vdd52.04 MHzr3nm3 ----------------------------- =N NNO OOO=O O=O O=O a b c203sufficient precision from EPR spectra even in the solid state. For small radicals in solutionswith low viscosity, precision of an EPR measurement may even be sufficient for hyperfinecouplings as small as 3 MHz. In these situations, cw EPR or field-swept echo-detected EPR isthe method of choice, as it is more sensitive and technically easier than measurement of thenuclear spectrum. To obtain utmost hyperfine resolution, line broadening due to couplingsbetween electron spins has to be avoided. In the solid state this requires concentrations of 1mmol L-1 or less. .Figure 10.12: Line broadening due to collisonal exchange. 1,8-Dimethyl naphthaline radical anion at aconcentration of a) 10-3 M, b) 10-5 M. c) Wursters blue without (top) and with (bottom) oxygen in thesolution.In the liquid state linewidths are smaller. Furthermore, exchange broadening can alsoarise due to collisions between paramagnetic molecules. During such a collision the orbitalsoverlap, the unpaired electrons of both molecules become indistinguishable and may beexchanged when the molecules separate again (Fig. 10.11). This leads to sudden changes inresonance frequency and thus to phase relaxation. The transversal relaxation time T2 isshortened and lines are broadened. In exceptional cases, concentrations down to 200 umol L-1may be required to avoid such broadening (Fig. 10.12A). Exchange broadening is also caused by dissolved oxygen, as dioxygen has a tripletground state and is thus paramagnetic. The effect is often tolerable for measurements inaqueous solution, but highly detrimental for measurements in unpolar solvents, where oxygensolubility is much higher (Fig. 10.12B).c)204 .Figure 10.13: Simulated EPR spectrum of the phenyl radical (bottom) and schematic drawing of howthe splitting pattern arises (top).The hyperfine splitting pattern of radicals in solution is generated in the same way asdiscussed in Section 7.1 for J-coupled spectra of weakly-coupled nuclear spins in NMR, exceptthat only the spectrum of one of the spins (the electron spin is observed). Furthermore, thenumber of nuclear spins with significant hyperfine coupling to the electron spin is usuallylarger than the corresponding number in J-coupled NMR spectra. As a particularly simple case,the EPR spectrum of the phenyl radical is shown in Fig. 10.13. This radical can be generatedfor instance by photolysis of phenylbromide in solution. Due to the low natural abundance of13C, carbon hyperfine splittings are apparent only in weak satellite lines that do not concern ushere. The splitting pattern is thus entirely due to proton hyperfine couplings. Among theprotons, the ones in positions 2 and 6 are magnetically equivalent with a coupling of 1.743 mT,the ones in positions 3 and 5 with a coupling of 0.625 mT and the proton in position 4 has a2H(1.743 mT)2H(0.625 mT)1H(0.204 mT)375 380 385B0 (mT)H HHHH2 6354205coupling of 0.204 mT. The resulting spectrum, a triplet of triplets of doublets has 18 lines. Asthe pattern is clearly apparent, the hyperfine couplings can be read off directly.Figure 10.14: Experimental EPR spectrum of the paracyclophan free radical.Such a simple analysis may no longer be possible if the subpatterns overlap. Thishappens almost with certainty when the number of hyperfine coupled nuclei increases, as thenumber of lines in the EPR spectrum increases multiplicatively , [10.33]where index i runs over the groups of equivalent nuclei, the ki are the numbers of nuclei withineach group, and the Ii their nuclear spin quantum numbers. For instance, for the free cyclophanradical in solution with 4 equivalent aromatic protons, 9 equivalent exo, and 9 equivalent endoprotons, lines arise (Fig. 10.14). Coordination with a K+ ion (nuclear spin I =3/2 for 39K, quadruplet splitting) removes the equivalence between the two moieties of themolecule, so that each group of equivalent protons splits into two inequivalent subgroups. Thenumber of lines increases to .Spectra like this are difficult to analyse. Due to the special topology of electron-nuclearspin systems (Fig. 10.5B), the nuclear spectrum in the liquid state has a much smaller numberof linesnEPR2kiIi1 + ( )iI=5 9 9 405 =3 3 5 5 5 5 4 22500 =206 , [10.34]where i again runs over the groups of equivalent nuclei. For instance, the nuclear spectra ofphenyl radicals, uncoordinated paracyclophan radicals, and K+-coordinated paracyclophaneradicals have only 6, 6, and 14 lines, respectively. In the solid state, additional splitting due tonuclear quadrupole couplings leads to a number of lines . [10.35]Among our examples, this would change the number of lines only for K+-coordinatedparacyclophane radicals to 18.Figure 10.15: Typical S = 1/2 EPR spectra in the absence of hyperfine couplings. The frequencydispersion is caused only by a g tensor with axial symmetry (A,B) or orthorhombic symmetry (C,D).The low-field and high-field edges of the absorption spectra (A,C) correspond to selection of a smallsubset of orientations near the x or z axis of the g tensor PAS. The first derivative lineshape, as detectedin cw EPR spectroscopy, is dominated by the singularities that arise at principal axis directions.nNMR liq .2S 1 +i=nNMR sol .2Ii2S 1 + ( )i=300 320 340h g n mmw B/ ||h g n mmw B/ zh g n mmw B/ ||h g n mmw B/ zh g n mmw B/ xh g n mmw B/ ^ h g n mmw B/ yh g n mmw B/ ^h g n mmw B/ xh g n mmw B/ y300 320 340B0 (mT) B0 (mT)q = 0q = 90300 320 340A CB D300 320 340207The nuclear spin spectra cannot be measured with an NMR spectrometer, as thehyperfine splittings exceed the bandwidth of NMR probeheads by orders of magnitude.Furthermore, detection at the low NMR frequency insted of the high EPR frequency wouldlead to a drastic loss in sensitivity. For these reasons spectra of hyperfine coupled nuclei aredetected by EPR-based methods such as ENDOR or ESEEM.The dispersion of electron spin resonance frequencies due to g anisotropy for powdersor frozen solutions causes a substantial drop in hyperfine resolution compared to the liquidstate. Even for transtion metal ions with only moderate g anisotropy, such as V(IV), Cu(II), orCo(II) the spectrum in the absence of any hyperfine couplings extends over tens of mT (Fig.10.15).Except for the singularities corresponding to principal axis directions, the firstderivative of the absorption lineshape is barely detectable. If hyperfine anisotropy is muchsmaller than g ansiotropy or if the PAS of the g and hyperfine tensors coincide, observablehyperfine splittings thus correspond to the principal axis directions of the g tensor. If g andhyperfine anisotropy are of the same order of magnitude and the PAS are non-coincident,spectrum analysis requires lineshape simulations.Figure 10.16: Hyperfine splittings in solid state EPR spectra of powders, glasses or frozen solutions.(simulations for a square planar copper(II) complex with four equal ligands L). A) Each nuclear spinstate gives rise to a separate axial powder pattern. B) The absorption spectrum, as detected by field-swept echo-detected EPR is the sum of the four separate patterns. C) The derivative of the absorptionspectrum, as detected by cw EPR, exhibits hyperfine splitting along g||. The inset shows a stereoview ofthe complex with the unique axis of the g and copper hyperfine tensor.0.3 0.32 0.34 0.3 0.32 0.34ABCmI = |-3/2mI = |-1/2mI = |+1/2mI = |+3/2B0 (T) B0 (T)h g n mmw B/ ||hfi at g||A||h g n mmw B/ ^hfi at g^CuLL LLg||,A||208 A simple case is illustrated in Fig. 10.16 on the example of a square planar coppercomplex with four equal ligands L. This species has a C4 symmetry axis. A Cn symmetry axiswith implies an axial g tensor with its unique axis coinciding with the symmetry axis. Asthe SOMO also has C4 point symmetry at the Cu2+ ion, the copper hyperfine tensor is alsoaxial and has the same unique axis. For each magnetic quantum number mI = -3/2, -1/2, +1/2,and +3/2 the two tensors for the nuclear Zeeman and copper hyperfine interaction add to a totalaxial tensor that describes the anisotropy of the resonance frequency.1 Each nuclear spin statethus gives rise to a separate axial powder pattern (Fig. 10.16A). The edges of the patterns areshifted with respect to each other by multiples of the hyperfine coupling A|| (low-field edge)and (high-field edge). The small splitting is usually unresolved in the EPR spectrum(Fig. 10.16B). The parameters g||, , and A|| can be directly read off the cw EPR spectrum(Fig. 10.16C).10.4.2 Nuclear spin spectraAllowed transitions in nuclear spin spectra involve a change of the magnetic quantumnumber of the nuclear spin mI by unity while the magnetic quantum number of the electronspin mS remains constant, i.e., they are of the type . If the high-fieldapproximation applies to both the electron and nuclear spin, the angular frequencies of suchtransitions are given by . [10.36]For simplicity, the following considerations are restricted to a system consisting of a nuclearspin I=1/2 coupled to an electron spin S=1/2. This system has two nuclear transitions with theangular frequencies . [10.37]Depending on the sign of the gyromagnetic ratio of the nuclear spin, the sign of the hyperfinecoupling, and the relative magnitudes of the two interactions, the transition frequencies can be1The argument applies strictly only in frequency domain, but the qualitative conclusions are true also infield domain.n 3 >AAgmImI 1 +Ae eImSAeff2mI1 + ( )Peff+ + =Ae eIAeff2---------- =209either negative or positive and they can have either the same or a different sign. The absolutesign of the frequency can only be detected if the nuclear transitions are directly excited withcircularly polarized radiofrequency (rf) irradiation. As linearly polarized irradiation is usedthis information is lost. The relative sign can be detected in experiments that correlate the twotransitions, such as in the hyperfine sublevel correlation (HYSCORE) experiment (Section10.4.2.3).The sign uncertainty complicates assignment and interpretation of the spectra. This isfirst discussed for the case of isotropic hyperfine coupling. The same considerations apply tospectra of single crystals.10.4.2.1 Isotropic caseConsider the case of the phenyl radical whose EPR spectrum is shown in Fig. 10.13. AtX-band frequencies the nuclear Zeeman frequency vI=eI/2t is about -15 MHz. The hyperfinecouplings are A1/2t = 5.7 MHz for the para proton H4, A2/2t = 17.5 MHz for the meta protonsH3 and H5, and A3/2t = 48.8 MHz for the ortho protons H2 and H6. The couplings of the paraand meta protons are in the weak coupling regime (weak coupling) [10.38]while the couplings of the ortho protons are in the strong coupling regime (strong coupling) . [10.39]In the weak coupling case the doublet is centered at |vI| and split by |A|/2t (Fig. 10.17A,B).1 Inthe strong coupling case, one of the frequencies has opposite sign, corresponding to precessionof the magnetization with an opposite sense of rotation. As the sign is lost, the correspondingline is mirrored to positive frequencies. The doublet is thus centered at |A|/4t (|A|/2 on anangular frequency scale) and split by 2|vI|. When different nuclear isotopes contribute to thespectrum, transitions of strongly coupled low- nuclei, e.g. 14N, may thus overlap withtransitions of weakly coupled high- nuclei, e.g. 1H. This problem is common at X-band1In the literature, A is often used as an angular frequency in equations but as a frequency in spectra ortables. Beware!A 2 eI210 frequencies and below. The overlap may be eliminated by going to higher frequencies or byHYSCORE.10.4.2.2 Anisotropic caseIn the regime of very weak coupling , the high-field approximation is validfor the nuclear spins. The nuclear spins are then quantized along the direction of the externalmagnetic field. The hyperfine interaction perpendicular to this direction is truncated, and Eq.[10.37] applies with Aeff = Az being the hyperfine coupling along the external field direction.Figure 10.17: Schematic nuclear spectrum of the phenyl radical. A) Subspectrum of the para proton(weak coupling). B) Subspectrum of the meta protons (weak coupling). C) Subspectrum of the orthoprotons (strong coupling). The line at (relative) negative frequencies is mirrored to positivefrequencies. D) Total spectrum.If the nuclear Zeeman and hyperfine interaction are of the same order of magnitude, thehyperfine field at the nucleus perpendicular to the external field (terms andof the Hamiltonian) can no longer be neglected. Instead of being non-secular, as in the high-field approximation, these terms are now pseudosecular. Together they contribute an effectiveinteraction in the xy plane of the laboratory frame. The corresponding fieldA eIw( H)12w( H)1w( H)1w( H)1w( H)10000CBADA3A1A2A3/2wnuclearwnuclearwnuclearwnuclearAzxSzIx AzySzIyB Azx2Azy2+ =211at the electron spin (terms and of the Hamiltonian) is usually still negligibleas are the remaining off-diagonal terms, since the high-field approximation still applies to theelectron spin with its much larger Zeeman interaction. Thus, the truncated Hamiltonian for theS=1/2, I=1/2 spin system becomes . [10.40]A new x direction was chosen for the nuclear spin frame to simplify the Hamiltonian.1 TheHamiltonian is written in a singly rotating frame, where the electron spin space rotates with themw frequency while the nuclear spin space is fixed with respect to the laboratory frame.Hence, OS is a resonance offset, while eI is the total nuclear Zeeman frequency.Figure 10.18: Local fields on the nucleus in the mS=1/2 (o) and mS=-1/2 (|) states. The effective fielddirections for the two states differ; they are tilted by angles no and n| with respect to the external fielddirection z.When divided by the gyromagnetic ratio n, the Hamiltonian terms containing I spinoperators can be interpreted as local fields at the nuclear spin (Fig. 10.12). The pseudosecularcontribution causes a tilt of the effective field away from the external field direction z. As thepseudosecular contribution has different signs for the o and | states of the electron spin, theeffective field axis ist tilted by angles no and n| in different directions. As the effective fieldcomponent along z is once the sum and once the difference of the nuclear Zeeman field and thesecular hyperfine field, the magnitudes of no and n| also differ. The quantization axis of thenuclear spin thus is no longer well defined and mI is no longer a good quantum number.1This is convenient, unless the nuclear spins are directly irradiated by rf.AxzSxIz AyzSyIzH0 OSSz eIIz ASzIz BSzIx + + + =zxA/2-A/2- /2 B B/2wI2hahhbwwbaabb a212 This has two important consequences. First, an mw pulse that excites the electron spinwill also have an influence on nuclear spin state, as the direction of the local field at thenucleus is changed. To some extent the pulse thus excites forbidden transitions in which bothmS and mI change. This is the basis of ESEEM experiments (Section 10.4.4). Second, Eq.[10.37] for the nuclear frequencies does no longer apply. The two nuclear frequencies are nowgiven by[10.41]and , [10.42]as can be inferred by vector addition (Fig. 10.18). The doublet is now centered at a frequencythat is slightly higher than eI while the splitting is slightly smaller than A.For the hyperfine and nuclear Zeeman field along z cancel exactly for oneof the two electron spin states, either o or |. In this situation of exact cancellation the effectivefield is within the xy plane, the nuclear frequency is B/2, and the transition moments ofallowed and forbidden transitions are nearly the same. For quadrupolar nuclei (I>1/2) withsmall hyperfine anisotropy the nucleus at exact cancellation experiences a near zero-fieldsituation. Narrow lines at the zero-field nuclear quadrupole resonance (NQR) frequencies arethen observed.In macroscopically disordered systems, such as powders, glasses, and frozen solutions,A and B depend on the relative orientation of the hyperfine tensor PAS and the magnetic field.For a hyperfine tensor with axial symmetry and principal values (aiso+2T, aiso-T, aiso-T), theorientation dependence is fully described by the angle u between the unique axis of thehyperfine tensor and the magnetic field, , [10.43] . [10.44]eo eIA2---- +\ .| |2B24------ + =e| eIA2---- \ .| |2B24------ + =A 2 eI=A aisoT 3 u 1 cos2( ) + =B 3T u u cos sin =213Powder patterns computed from Eq. [10.41-10.44] are shown for weak coupling, exactcancellation, and strong coupling in Fig. A, B, and C, respectively.Figure 10.19: Simulated anisotropic nuclear frequency spectra for an S=1/2, I=1/2 system with weakcoupling (A), at exact cancellation (B), and with strong coupling (C).The same sign is assumed for aisoand eI.10.4.2.3 Hyperfine sublevel correlation (HYSCORE)The frequencies of a pair of transitions of the same nucleus in the electron spin o and |manifold can be correlated in a 2D experiment by using a mw pulse for mixing. To keepwith usual notation in EPR literature flip angles are given in radians from now on. A pulse is thus a t pulse and a pulse a t/2 pulse. The correlation peaks created by mixingwith the t pulse appear in the quadrant ( , ) for weakly coupled nuclei, while theyappear in the quadrant ( , ) for strongly coupled nuclei. A schematic spectrum forthe phenyl radical is shown in Fig. 10.20 and can be compared to the EPR spectrum in Fig.10.13 and the one-dimensional nuclear spectrum in Fig. 10.17. For macroscopically disorderedsystems with anisotropic hyperfine coupling, curved ridges result. The anisotropy of thehyperfine coupling can be extracted from their curvature and their shift with respect to theantidiagonal at eI, even if only part of the correlation ridge can be observed (Fig. 10.21).ABCwIwI000aiso/2B/2wawawbwb18018090e10 > e20 >e10 > e20 2/T2W Wnmw0A BIhom2 T2 =|o ' , oo ' , || ' , o| ' , 219resonance offset, while a quarter is shifted by Aeff and another quarter by -Aeff (Fig. 10.26).This shift of the hole can only occur if Aeff is larger than the width I of the hole.Figure 10.26: An on-resonant rf t pulse in Davies ENDOR shifts intensity from a hole burnt into theinhomogeneous EPR line to side holes. A) Situation before the rf pulse (point 1 in Fig. 10.22). B)Situation after the rf pulse (point 2).10.4.3.2 Mims ENDORThe inversion of spin packets can be performed in a more controlled way by splittingthe mw t pulse in two t/2 pulses that are separated by an interpulse delay t. This can be seenby computing the effect of such a subsequence (t/2)x-t-(t/2)x on an inhomogeneouslybroadened line with product operator formalism. For brevity, only the electron spin S isconsidered so that the rotating-frame Hamiltonian is . The experiment starts atthermal equlibrium with density operator .1 Application of an ideal mw t/2 pulsealong x, , [10.47]results in . [10.48]After subsequent evolution for time t under the density operator is given by . [10.49]1The negative sign is a consequence of the negative electron charge that leads to preferential population of| states of electron spins with their spin antiparallel to the external field.A BAeff AeffWWG p 2 /tpH0 OSSz =oeqSz =oeq o1t 2 Sxo1Sy =H0o2 OSt ( )Sy cos OSt ( ) sin Sx =220 The following t/2 pulse along x leaves the second term on the right-hand side of Eq. [10.49]unaffected, while the first term is converted to negative polarization . [10.50]. The second term on the right-hand side of Eq. [10.50] decays by transversal relaxation or canbe removed by a phase cycle. The first term corresponds to a polarization grid in theinhomogeneously broadened EPR line whose resolution depends on interpulse delay t, but noton the excitation bandwidth (Fig. 10.27). The excitation bandwidth determines the envelope ofthe grating. It is thus possible to create a finely spaced grid to which many spin packetscontribute.Figure 10.27: Polarization grating in an inhomogeneous line created by a (t/2)x-t-(t/2)x subsequence(numerical simulation). Unlike in the product operator treatment, the limited excitation bandwidth ofthe mw pulses is considered in this simulation.The polarization grid can be detected by applying another mw t/2 pulse. This createsan FID signal of the oscillatory grating. As the Fourier transform of a cosine function is a Diracdelta function, the FID signal corresponding to the first term on the right-hand side of Eq.[10.50] is confined at time t. It is a stimulated echo signal. The width of the echo is determinedby the excitation bandwidth of the t/2 pulses and is thus comparable to the pulse length.In the Mims ENDOR experiment the effect of an rf pulse with variable frequency onthe intensity of the stimulated echo is observed (Fig. 10.28). An on-resonant rf t pulse shiftstwo quarters of the polarization grating by Aeff and -Aeff, respectively, just as the rf t pulse inDavies ENDOR shifts two quarters of the burnt hole (compare Fig. 10.26). The unshifted halfof the grating and the two shifted quarters interfer. For with interference is destructive. The stimulated echo is canceled. For an on-resonant rf pulse and, the shifted gratings interfer constructively and the stimulated echo is nearlyo3 OSt ( )Sz cos OSt ( ) sin Sx =DW1/tAefft 2k 1 + ( )t = k 0 1 . . . =Aefft 2kt =221unaffected. This situation cannot be distinguished from the one for an off-resonant rf pulse.The Mims ENDOR experiment thus features blind spots at ..Figure 10.28: Pulse sequence for the Mims ENDOR experiment. The intensity of the stimulated echois observed as a function of the frequency of the rf t pulse. The density operators are given inEq. [10.47-10.50] with .The ENDOR efficiency[10.51]quantifies this blindspot behavior. The total intensity of the stimulated echo scales with. By considering this and Eq. [10.51], it can be shown that very small hyperfinecouplings are detected with highest sensitivity at t = T2. To safely detect all large couplings,Mims ENDOR spectra have to be recorded at different t and added. This averages the blindspots.10.4.4 ESEEM spectroscopy and the HYSCORE experimentIn NMR spectroscopy spin echo signals usually decay smoothly as a function of theinterpulse delays. In EPR spectroscopy, the decay is often superimposed by modulations withnuclear frequencies. This electron spin echo envelope modulation (ESEEM) effect allows for adetection of nuclear spectra without directly exciting the nuclear spins. The indirect excitationof the nuclear spins arises from the tilt of the effective field at the nucleus with respect to theexternal field (Fig. 10.18). Excitation of coherence on formally forbidden transitions can beunderstood by transforming the mw Hamiltonian to the eigensystem of the static Hamiltoniangiven in Eq. [10.40].Aefft 2kt =p/2 p/2 p/2t t Tpm.w.r.f.s0^s1^s2^s3^o1.o3o0 oeq=FENDOR14--- 1 Aefft ( ) cos ( ) =e2t T2

222 10.4.4.1 Transition moments for allowed and forbidden transitionsIn Eq. [10.40] the static Hamiltonian of the S=1/2, I=1/2 spin system is given in asystem where the z axes of both the electron and nuclear spin frame are parallel to the externalmagnetic field and perpendicular to the linearly polarized mw field. In this frame theoscillatory mw Hamiltonian, which describes the excitation, is given by , [10.52]with . The effective g value geff,x pertains to the x direction of the mwfield. The spin transitions are, however, defined between eigenstates of the spin system. Tocompute the excitation strengths for these transitions, Eq. [10.52] thus has to be transformed tothe eigenframe of . The necessary transformations can be read off directly from Fig. 10.18.To bring the effective field axis to the z axis, the o subspace of the electron spin S has to berotated clockwise about the y axis of the nuclear frame by angle no. This is described by aunitary transformation , [10.53]where no is defined as . [10.54]The polarization operator is given by . The negative sign of the argumentof the arcustangens arises since a clockwise rotation is opposite to a rotation in mathematicalsense. A clockwise rotation is required since the y axis of a right-handed frame points into thepaper plane in Fig. 10.18. Likewise, the subspace corresponding to the | state of S has to berotated anticlockwise about the y axis of the nuclear frame by angle n|. This is described by aunitary transformation[10.55]withH0H1 e1Sx =e1geff x . uBB1h =H0UoinoSoIy ( ) exp =noarcB 2eIA +-------------------\ .| |tan =SoSoE2 Sz + =U|in|S|Iy ( ) exp =223 , [10.56]and the polarization operator . In this frame the static Hamiltonian for theweak coupling case1 takes the form . [10.57]Since and commute, the transformation from Eq. [10.40] to Eq. [10.57] can bedescribed by a single unitary transformation . [10.58]The transformation of Eq. [10.52] to the eigensystem of is more simple withCartesian operators. By substituting the polarization operators, UEB takes the form , [10.59]with , . [10.60]Since , the second term in the argument on the right-hand side of Eq. [10.59] hasno effect on . In the eigenbasis, the oscillatory Hamiltonian takes the form . [10.61]The first term on the right hand side of Eq. [10.61] describes excitation of the allowedtransitions. The second term is more easily interpreted when the product operator is written interms of ladder operators, . [10.62]1In the strong coupling case, the sign of either the eo or the e| term changes. Which sign changes dependson the relative signs of eI and A.n|arcB2eIA -------------------\ .| |tan =S|E2 Sz =H0 OSSz eoSoIz e|S|Iz + + =SoIy S|IyUEBUoU|U|Uoi noSoIy n|S|Iy + ( ) ( ) exp = = =H0UEBi n2SzIy cIy + ( ) ( ) exp =n no n|2------------------- = c no n|+2------------------- =Iy Sx [ , ] 0 =H1H1EB ( )UEBH1UEBn ( )e1Sxcos n ( ) sin e1SyIy + = =SyIy14--- S+I-S-I+S+I+ S-I- + ( ) =224 The terms on the right hand side of Eq. [10.62] drive the forbidden electron-nuclear transitions (first two terms) and (last two terms) as indicated in Fig. 10.29A.Figure 10.29: Level scheme and schematic EPR spectrum of the S=1/2, I=1/2 system. A) Level schemewith allowed electron (green), forbidden electron-nuclear (red), and nuclear (blue) transitions. B)Spectrum with allowe (green) and forbidden (red) transitions, where andThe transition moment of allowed transitions thus scales with cosn, while the transitionmoment of forbidden transitions scales with sinn. Spectral intensities are proportional to thetransition probability, which is the square of the transition moment. This is because thetransition moment applies to both excitation and detection. The EPR spectrum of the S=1/2,I=1/2 system with anisotropic hyperfine interaction thus has the appearance shown in Fig.10.29B. The hyperfine splitting is given by the difference frequency[10.63]and the splitting of the forbidden transitions by the sum frequency . [10.64]10.4.4.2 Two-pulse ESEEMTwo-pulse ESEEM is observed by measuring the amplitude of a (t/2)-t-(t)-t echo as afunction of the interpulse delay t. The first t/2 pulse excites coherence with amplitude cosn onthe two allowed transitions and coherence with amplitude sinn on the two forbiddentransitions. The coherence evolves during the first interpulse delay t, defocuses due to thedistribution of resonance offsets OS, and decays with the transversal relaxation times T2a or T2fof the respective transitions. The t pulse inverts the phase of all coherences and thus leads to ao| ' , |o ' , oo ' , || ' , 1234wawbw24 w13w14 w23wwWs+aaabbabbA Bcos2hsin2he- eo e| =e+ eo e|+ =Aeff e- eo e| = =e+ eo e|+ =225refocusing of magnetization after another delay t. Those pathways, where the coherenceremains on the same transition during the transfer by the t pulse, contribute two-pulse echoesthat smoothly decay with the respective relaxation rates.Figure 10.30: Branching of magnetization (coherence transfers) during the t pulse in the two-pulseESEEM experiment. A) Transfer of coherence on the allowed transition to the two allowedand two forbidden transitions combined with a phase inversion. B) Transfer of coherence on theforbidden transition to the two allowed and two forbidden transitions combined with a phaseinversion.However, the t pulse also changes the nuclear spin states with a probability sin2n, andthis corresponds to a transfer of coherence between the different transitions. This branching ofmagnetization is indicated in a density matrix representation in Fig. 10.30. For instance,coherence on the allowed transition is merely phase-inverted with probabilitycos2n and transferred with probability sin(n)cos(n) to the forbidden transition (Fig.10.30A). In contrast, coherence on the forbbiden transition is phase-inverted onlywith the small probability sin2n and transferred to the forbidden transition with thelarger probability cos2n (Fig. 10.30B).All transferred coherences also experience phase inversion. Thus they also formechoes. These coherence transfer echoes oscillate with the difference of the transitionfrequencies before and after the t pulse. From Fig. 10.29 and Eqs. [10.63] and [10.64] it can beverified that the possible difference frequencies are eo, e|, e, and e+. Each of the basicnuclear frequencies eo and e| appears in two coherence transfer pathways, and each of thecombination frequencies e and e+ appears in only one pathway. Furthermore, the product ofthe excitation probability (transition moment for excited coherence), transfer probability(branching factor), and detection probability (transition moment for detected coherence) is theA Bcos hcos hcos h cos hcos h-sin h sin hcos h-sin hsin h sin hsin h2 222|aaaa|ab|ba|bb||aa |ab |ab |ba |ba |bb |bboo |o oo || oo |o oo || oo || o| |o 226 same for each pathway, it takes the value sin2n cos2n = sin2(2n)/4. The amplitude of allnuclear modulations thus scales with the modulation depth parameter . [10.65]With proper bookeeping of the coherence transfer pathways, the two-pulse ESEEM formula[10.66]results. A schematic spectrum for the weak coupling case is shown in Fig. 10.31. Note that hecombination frequencies are close to A and 2eI, but are slightly shifted towards lower andhigher frequencies, respectively. The phase information (positive basic frequency peaks andnegative combination peaks) is often hard to access, as spectrometer dead time preventsobservation of the signal at short t values. Usually phase correction fails and magnitudespectra are displayed, where all peaks are positive.Figure 10.31: Schematic two-pulse ESEEM spectrum for a weakly coupled S=1/2, I=1/2 system.10.4.4.3 Three-pulse ESEEMNuclear frequencies are measured by two-pulse ESEEM as differences between thefrequencies of electron spin transitions. Hence, the linewidth in the spectra is twice thehomogeneous EPR linewidth, which is much larger than the natural linewidth for nucleark 2n sin2BeIeoe|--------------\ .| |2= =V2p t ( ) 1k4--- 2 2 eot ( ) cos 2 e|t ( ) cos e-t ( ) cos e+t ( ) cos + + ( ) =AwI2wI227transitions. The resolution can thus be drastically improved by observing the evolution ofcoherence on nuclear transitions.Figure 10.32: Magnetization branching during the second t/2 pulse in a mw pulse subsequence. A) Coherence on the allowed electron spin transition is transferredto nuclear coherences with probability sinn. B) Coherence on the forbidden transition istransferred to nuclear coherences with probability cosn.A single, ideal mw pulse does not excite nuclear coherence. However, an mw pulsesubsequence (t/2)-t-(t/2) does generate such coherence due to magnetization branchingduring the second t/2 pulse (Fig. 10.32). This nuclear coherence evolves during delay T and isthen transferred to observable electron coherence by another t/2 pulse. After another delay t,the dispersion of electron spin resonance offsets OS is refocused. The coherence transferpathways that involve nuclear coherence contribute signals that oscillate with frequencies eoand e| as a function of time T. Coherence transfer pathways that do not involve nuclearcoherence give rise to a smoothly decaying stimulated echo. The total echo signal as a functionof the fixed interpulse delay t and the variable delay T, neglecting relaxation, is given by . [10.67]No oscillation with combination frequencies is observed. This simplifies spectra and is anadditional advantage of three-pulse ESEEM compared to two-pulse ESEEM. However, theamplitude of the nuclear oscillation now depends on the nuclear frequency in the other electronspin state and on the fixed delay t. This leads to blind spots, which are a disadvantage of three-A B|aaaa|ab|ba|bb||aa |ab |ab |ba |ba |bb |bbcoshcoshcoshsinhcoshi2i2sinhsinhi2i2i2i2-i2-sinhi2-t 2 ( ) t t 2 ( ) oo |o oo || V3p t T . ( ) 1k4--- 1 e|t ( ) cos | | 1 eoT t + ( ) ( ) cos | | { =1 eot ( ) cos | | 1 e|T t + ( ) ( ) cos | | } +228 pulse ESEEM. To avoid suppression of peaks, the experiment thus has to be performed atseveral t values and the magnitude spectra have to be added.Figure 10.33: Pulse sequence for three-pulse ESEEM. The echo amplitude is observed as a function ofinterpulse delay T for fixed delays t.10.4.4.4 HYSCOREHyperfine sublevel correlation spectra, as introduced in Section 10.4.2.3, can bemeasured by a two-dimensional extension of the three-pulse ESEEM experiment. For thatpurpose, an mw t pulse is introduced at a variable delay t1 after the second t/2 pulse (Fig.10.34). This t pulse transfers nuclear coherence between the two manifolds corresponding tothe electron spin states o and |. Thus it correlates frequency eo of a given nucleus withfrequency e| of the same nucleus and vice versa. The new frequency is measured byintroducing another variable delay t2 between the t pulse and the final t/2 pulse thatreconverts the nuclear coherence to observable electron coherence.Figure 10.34: Pulse sequence of the two-dimensional HYSCORE experiment. Echo amplitude isobserved as a function of the two variable delays t1 and t2 for fixed delays t.Forbidden transitions during the t pulse convert some nuclear coherence to electronpolarization and vice versa. This leads to axial peaks at e1 = 0 and e2 = 0. These peaks areunwanted, as they do not contain correlation information. They can be removed by baselinecorrection of the time-domain data before Fourier transformation. Because of its limitedexcitation bandwidth, the nominal t pulse does not fully invert the electron spin for largeresonance offsets OS. This leads to diagonal peaks e1 = e2 = eo and e1 = e2 = e|. Thesep/2 p/2 p/2t t Tp/2 p/2 p/2pt t t1 t2229diagonal peaks cannot reliably be removed by data processing and may obscure cross peaks forsmall hyperfine couplings. Their contribution scales with the ratio between the excitationbandwidths of the t/2 pulses and the t pulse. Therefore, it is advantageous to use the full mwpower for the t pulse and less power for the t/2 pulses, as indicated in Fig. 10.34.The modulation in the HYSCORE experiment corresponding to the cross peaks isdescribed by[10.68]with,[10.69].[10.70]In this representation with unsigned nuclear frequencies, the weak coupling case correspondsto and the strong coupling case to . In the former case, the crosspeaks are thus stronger in the quadrant (e1>0, e2>0) and in the latter case in the quadrant(e1>0, e2 n sin2n cos2>eot 2 ( ) sin e|t 2 ( ) sin230 spectroscopic response of the spin probe is interpreted in terms of structure and dynamics ofthe system. Ideal probes do not perturb the system under investigation, can be directed at willto sites of interest, and give a spectroscopic response that strongly depends on theirenvironment.The EPR spin probe technique is particularly suitable for soft organic matter, as it doesnot rely on long-range order and as the spectra of typical spin probes are very sensitive tomolecular dynamics on time scales between 10 ps and 1 us. This time range corresponds tomotion on length scales longer than a chemical bond, but smaller than colloidal dimensionsthat start at about 100 nm. On these length scales supramolecular interactions determine theself-organization, stability, and functionality of soft matter systems. Distances between spinprobes can be measured on length scales between about 0.8 and 8 nm.Figure 10.35: Structures of the most common spin probes and labels. R is a functional group used fordierctiong the probe or attaching the label to the site of interest. 1 4-oxo-2,2,6,6-tetramethyl-piperidin.2 2,2,6,6-tetramethyl-piperidin-1-oxyls, TEMPO derivatives. 3 2,2,5,5-tetramethyl-pyrrolydine-1-oxyls, PROXYL derivatives. 4 2,2,5,5-tetramethyl-pyrroline-1-oxyls. 5 4,4,-dimethyl-3-oxazolidinyloxy probes, DOXYL derivatives. 6 Methanethiosulfonate spin label, MTSSL.The most widely applied class of spin probes are nitroxides (Fig. 10.35). They areoriginally derived from compound 1, 4-oxo-2,2,6,6-tetramethyl-piperidin, which is the basiscompound of hindered amine light stabilizers. This compound is formed in a one-pot reaction fromacetone and ammonia and is easily oxidized to TEMPON, 4-oxo-2,2,6,6-tetramethyl-piperidin-1-oxyl(compound 2 with -R being =O). The carbonyl group can be converted to other functional groups (e.g. -OH, -NH2, -N+(CH3)3). Such groups direct site attachment via hydrogen bonds or electrostaticinteractions (spin probes in a strict sense). Reactive groups R can be used for covalent attachment to thesite of interest (spin labels). Somewhat smaller probes or labels are obtained by reducing the ring size toNORNHONORNORO N ON OSSOOH3C15 62 3 4231PROXYL derivatives 3 or 3,4-dehydro-PROXYL derivatives 4. The most important label is themethanethiosulfonate spin label (MTSSL) 6 that attaches with high selectivity under very mildconditions to cysteine residues in proteins. The size of the MTSSL-derived sidechain is similar to theone of an aromatic amino acid residue. The SDSL technique for proteins involves mutation of an aminoacid at the site of interest to cystein and subsequent reaction with a thiol-selective spin labels such asMTSSL, iodoacetamido-PROXYL, or maleimido-PROXYL. For distance measurements two selectedsites are labeled.Synthesis of DOXYL derivatives 5 is more difficult and proceeds with lower yields.Their advantage is a very rigid attachment to alkyl chains that allows to probe motion andliquid crystalline ordering of surfactant molecules, lipids, and steroids in more detail.The unpaired electron in nitroxides is s

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