Available online at www.sciencedirect.com Journal of Magnetic Resonance 190 (2008) 233–247 www.elsevier.com/locate/jmr 5- and 6-pulse electron spin echo envelope modulation (ESEEM) of multi-nuclear spin systems B. Kasumaj *, S. Stoll 1 ETH Zurich, Laboratory of Physical Chemistry, 8093 Zurich, Switzerland Received 24 August 2007; revised 23 October 2007 Available online 6 November 2007 Abstract In 3-pulse ESEEM and the original 4-pulse HYSCORE, nuclei with large modulation depth (k 1) suppress spectral peaks from nuclei with weak modulations (k 0). This cross suppression can impede the detection of the latter nuclei, which are often the ones of interest. We show that two extended pulse sequences, 5-pulse ESEEM and 6-pulse HYSCORE, can be used as experimental alterna- tives that suffer less strongly from the cross suppression and allow to recover signals of k 0 nuclei in the presence of k 1 nuclei. In the extended sequences, modulations from k 0 nuclei are strongly enhanced. In addition, multi-quantum transitions are absent which sim- plifies the spectra. General analytical expressions for the modulation signals in these sequences are derived and discussed. Numerical simulations and experimental spectra that demonstrate the higher sensitivity of the extended pulse sequences are presented. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Pulse EPR; ESEEM; HYSCORE; Cross-suppression effect; Multi-nuclear spin system; Phase cycling 1. Introduction small modulation depth parameter k, i.e. a hyperfine inter- action with a small anisotropic component and a small Pulse electron paramagnetic resonance (EPR) tech- quadrupole coupling, this effect is negligible, the nuclei niques in general and electron spin echo envelope modula- do not affect each other, and the spectrum equals the super- tion (ESEEM) techniques in particular are widely used to position of the single-nucleus spectra. On the other hand, determine the electronic and geometric structure of para- the cross-suppression effect can have a strong impact in magnetic species [1]. The results lead to unique insights into the presence of strongly modulating nuclei, as they can the electronic and geometric structure of paramagnetic cen- cause partial or complete suppression of signals from ters in biological systems [2,3], as they allow for the deter- weakly modulating nuclei coupled to the same electron mination of small hyperfine and quadrupole couplings of spin. In some circumstances, this results in spectra where nuclear spins in the molecular environment of unpaired weakly coupled nuclei cannot be observed at all, although electrons. they have a sufficiently large hyperfine coupling to be Recent studies found that a cross-suppression effect [4,5] resolved. As these nuclei are often of structural or func- distorts signal intensities in standard ESEEM experiments tional interest, experimental ways to circumvent or allevi- (3-pulse ESEEM and 4-pulse HYSCORE) if more than ate the impact of the cross suppression are desirable. one nucleus contribute to the signal and can thus lead to In this work, we examine two extended ESEEM pulse misinterpretation of ESEEM spectra. If all nuclei have a sequences, 5-pulse ESEEM [6] and 6-pulse HYSCORE [7], that can recover signals from weakly modulating nuclei and yield spectra that are less affected by cross suppression. * Corresponding author. Fax: +41 44 632 1021. E-mail address: [email protected] (B. Kasumaj). These sequences give ESEEM spectra with peaks at the 1 Present address: Department of Chemistry, University of California, same positions as in 3-pulse ESEEM [8] and standard 4- One Shields Ave, Davis, CA 95616, USA. pulse HYSCORE [9], respectively. 1090-7807/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmr.2007.11.001 234 B. Kasumaj, S. Stoll / Journal of Magnetic Resonance 190 (2008) 233–247 There are two reasons why spectra obtained with these For an axially symmetric hyperfine interaction, Aq and Bq extended sequences are less affected by cross suppression are related to the principal values A^,q and Ai,q of the than the standard sequences. First, as we will show, they give hyperfine tensor and can be expressed as function of the substantially enhanced peak intensities for weakly modulat- isotropic and dipolar hyperfine coupling constants, aiso,q ing nuclei (those affected by cross suppression) and show and Tq,by decreased sensitivity towards strongly modulating nuclei 2 2 2 (which are responsible for the cross suppression). The second Aq ¼ Ak;q cos hq þ A?;q sin hq ¼ T qð3 cos hq À 1Þþaiso;q; reason is based on the occurence of blind spots [10,11]:In3- Bq ¼ 3T q sin hq cos hq; pulse ESEEM and standard HYSCORE, a fixed inter-pulse ð3Þ delay of s causes selective suppression of peaks at some fre- quencies, called blind spots. The cross-suppression effect where hq is the angle between the external magnetic field can be counteracted by choosing s so that peaks from and the electron–nucleus axis [1]. strongly modulating nuclei that cause cross suppression fall The resonance frequencies of nucleus q associated with 1 1 on blind spots. The two extended pulse sequences have two the aðmS ¼þ2Þ and bðmS ¼2Þ electron spin manifolds experimentally adjustable fixed delays, s and s , and they are 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi thus offer more flexibility in choosing blind spots. 2 2 This article is structured as follows: The theoretical back- xa;b;q ¼ ðxI;q Æ Aq=2Þ þðÆBq=2Þ : ð4Þ ground needed to derive and discuss the analytical expres- The pseudo-secular part B of the hyperfine tensor tilts the sions for the two extended sequences is summarized in q quantization axes of the nuclear spins for each m manifold Section 2. Section 3 gives details of the system and experi- S by an angle ments used to illustrate our findings. The main part of the article is contained in Sections 5 and 6, where echo modula- ÆBq=2 ga;b;q ¼ arctan ð5Þ tion signals for 2-, 3- and 5-pulse ESEEM and 4- and 6-pulse xI;q Æ Aq=2 HYSCORE are compared theoretically and illustrated experimentally. Finally, Section 6 summarizes the insight with respect to B0. The half-angle between the two nuclear obtained. quantization axes is given by gq =(ga,q À gb,q)/2. The mod- ulation depth parameter kq, a fundamental quantity of 2. Theoretical background ESEEM, is 2 2 BqxI;q The rotating-frame spin Hamiltonian, in angular fre- kq ¼ sin ð2gqÞ¼ ð6Þ 1 xa;qxb;q quency units, of an electron spin ðS ¼ 2Þ coupled to NI 1 nuclei ðI1 ¼ÁÁÁ¼I N I ¼ 2Þ using the high-field approxima- with 0 6 kq 6 1. It is a measure of the degree of dissimilar- tion is given by [1,12] ity of the nuclear sublevels in the a and the b electron XN I manifolds. H 0 ¼ XS Sz þ ðxI;qI z;q þ AqSzIz;q þ BqSzIx;qÞ; ð1Þ The behavior of the spin system during a pulse sequence q¼1 is described by the evolution of the density operator r [1,14–16]. The density operator at a time t is given by where XS = xS À xmw is the offset between the electron res- 1 1 1 onance frequency xS and the frequency of the applied rðtÞ¼...P Q P r P À QÀ P À ... ð7Þ 2 t1 1 0 1 t1 2 microwave (mw) pulse, xmw. xI,q = Àgn,qlnB0/⁄ denotes the Larmor frequency of nucleus q. where Pj represent pulse propagators and Qtj free evolu- In general, the hyperfine part of the Hamiltonian is related tions. In the ideal pulse approximation, the static Hamilto- to the principal values A11,q, A22,q and A33,q of the ortho- nian H0 is neglected during the microwave pulse, so that rhombic hyperfine interaction matrix Aq, and two polar ÀibjSxðyÞ P j ¼ e ð8Þ angles hq and /q describing the orientation of the external magnetic field B0 = (0,0,B0) with respect to the frame where is the propagator for a pulse with flip angle bj and phase the hyperfine interaction matrix is diagonal [13]: x(y). The propagator during free evolution is 2 2 2 2 iH diagt Aq ¼ A33;q cos hq þ sin hqðA11;q cos /q þ A22;q sin /qÞ; y À 0 j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qtj ¼ U e U ð9Þ B ¼ B2 þ B2 ; q x;q y;q with [1] 1 B A A A YN I x;q ¼ sin hq cos hq½ð2 33;q À 11;q À 22;qÞ diag 2 y Àiðga;qSaþgb;qSbÞIy;q H 0 ¼ UH 0U U ¼ e : ð10Þ þ cosð2/qÞðA22;q À A11;qÞ; q¼1 1 B ¼ sin h sinð2/ ÞðA À A Þ: At the beginning of the pulse sequence, the system is at y;q 2 q q 22;q 11;q thermal equilibrium. Conventionally,Q this is represented ð2Þ by r0 = ÀSz/M, where M ¼ qð2Iq þ 1Þ is the total num- B. Kasumaj, S. Stoll / Journal of Magnetic Resonance 190 (2008) 233–247 235 ber of nuclear sub-states per electron manifold. In order to mation, each density matrix corresponding to a particular simplify our calculations we use pathway can be written as a tensor product of density sub- 1 1 1 matrices of single-nucleus subsystems, so that the eCTP r ¼ S ¼ I À S ð11Þ signal from a multi-nuclear spin system is a product of sig- 0 M b M 2 z nals from spin systems with one nucleus each. Summing instead. Adding to r0 a term proportional to the unity over all contributing eCTPs, we get operator I does not affect the expectation value of S as + XN r YN I the unity operator is invariant during the pulse sequence ðiÞ E ¼ fi Eq ; ð14Þ and tr½SþðcI þ rÞ ¼ tr½Sþr is satisfied.