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Chirp Hyperfine Spectroscopy
JOURNAL OF MAGNETIC RESONANCE, ARTICLE NO.
Series A 119, 45–52 (1996)
0050
Chirp Hyperfine Spectroscopy* G. JESCHKE
AND
A. SCHWEIGER
Laboratorium fu¨r Physikalische Chemie, Eidgeno¨ssische Technische Hochschule, CH-8092 Zu¨rich, Switzerland Received August 14, 1995
A novel pulse ESR scheme for the direct measurement of hyperfine splittings is proposed. It is shown that this hyperfine spectroscopy reduces the number of the spectral lines considerably, overcomes assignment ambiguities inherent in electron-nuclear doubleresonance spectra, and allows one to separate signals that overlap in ENDOR spectra. The resolution enhancement of hyperfine spectroscopy is fully utilized since the hyperfine splittings are measured between nuclear-spin rather than electron-spin transitions. The pulse sequence for the experiment is based on the chirp ENDOR method and can yield one-dimensional as well as two-dimensional spectra that correlate the ENDOR and the hyperfine frequencies. Both experiments are discussed theoretically, and the first experimental spectra of a model system are presented. q 1996 Academic
equivalent nuclear spins is represented in the spectrum by only one peak at the hyperfine frequency, irrespective of the nuclear-spin quantum number. The two-dimensional stimulated-echo ENDOR experiment proposed by de Beer et al. (12) can be considered as the first realization of such a hyperfine spectroscopy. In this experiment, the hyperfine dimension is obtained by incrementing the interpulse delay between the first two microwave (MW) pulses. One then observes peaks at the frequency differences between allowed electron-spin transitions that correspond to the hyperfine couplings. Recently, another attempt to measure hyperfine frequencies, Fourier-transform hyperfine spectroscopy (FTHS), has been reported (13). In FT-HS, a transient hole burnt into the ESR line by a selective MW pulse is subsequently shifted by a selective radiofrequency pulse and finally detected via a free induction decay. This technique is distinguished by the multiplex advantage inherent in the FID detection of the hyperfine frequencies. Both stimulated ENDOR and FT-HS are basically 2D techniques correlating the hyperfine and nuclear transition (ENDOR) frequencies. In the hyperfine dimension, the linewidth is determined by the phase memory time Tm of the electron spins, since the hyperfine frequencies are measured as the differences between ESR transition frequencies. In the ENDOR dimension, the linewidth is governed by the phase memory time T 2n of the nuclear spins or by the power broadening caused by the RF pulse, whenever its length is shorter than T 2n . In this paper, we introduce a pulsed ESR method for the measurement of hyperfine spectra which is fundamentally different from 2D stimulated-echo ENDOR and FT-HS. This new type of HS features a much higher resolution than the former techniques since the frequencies in the hyperfine dimension are measured as differences or sums of nucleartransition frequencies (see below). The experimental scheme for the first time allows for the direct measurement of one-dimensional hyperfine spectra. The approach can also be extended to a second dimension, resulting in a new type of a hyperfine-correlated ENDOR spectroscopy. Here, the frequencies in both the hyperfine and ENDOR dimension are obtained in a time-domain rather than in a frequencydomain experiment; therefore, no power broadening is introduced. The time evolution of the spin system during the
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INTRODUCTION
The resolution of conventional solid-state electron-spin resonance spectroscopy which measures the electron-spin transition frequencies of a spin system is usually quite poor, even in single-crystal work. Hyperfine interactions between the unpaired electron and the nuclear spins, as well as small second-order shifts caused by the nuclear-quadrupole interactions are therefore often not resolved. The situation can be changed drastically by using electron-nuclear double-resonance (ENDOR) (1–9) or electron-spin-echo-envelope modulation (ESEEM) (10, 11) techniques. Since with these methods nuclear- rather than electron-spin transitions are measured, one can improve on the resolution of ESR by orders of magnitude. However, for systems with a large number of nuclear spins coupled to the electron spin, the number of nuclear-transition frequencies is again large, and, therefore, the resolution of the ENDOR spectrum may be still poor. This is particularly true for nuclear spins I ú 12, since in this case additional line splittings are introduced by the nuclear-quadrupole interaction. One can further improve on the resolution by using techniques that directly measure the frequencies representing the hyperfine couplings. In such an approach, each group of * Part of this work was presented at the DFG Rundgespra¨ch ‘‘Magnetische Resonanz,’’ Hirschegg, Austria, September 19–23, 1994. 45
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1064-1858/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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sequence consisting of selective and nonselective MW and chirped RF pulses will be discussed, and the first experimental examples will be presented. THEORY
Hyperfine Spectroscopy We first quantify the simplification of the spectrum achieved with HS by comparing the number of spectral lines observed with the different schemes for a system of n inequivalent nuclear spins Ii (i Å 1, . . . , n) coupled to one electron spin S Å 12. Considering the electron and nuclear Zeeman, the hyperfine, and the nuclear-quadrupole interactions, the number of ESR, ENDOR, and hyperfine lines is given by n
NESR Å
∏ (2Ii
/ 1),
suring frequency differences between two ESR transitions with DmI Å 1 or two ENDOR transitions with DmS Å 1, where mI and mS are the magnetic quantum numbers of the nuclear and electron spin, respectively. The transfer of polarization or coherence between appropriate ESR transitions can be achieved by flipping the nuclear spin with an RF p pulse (thereby inverting the hyperfine field seen by the electron spin), and between appropriate NMR transitions by flipping the electron spin with an MW p pulse (thereby inverting the hyperfine field seen by the nuclear spin). From a practical point of view, the two approaches differ mainly in the widths of the hyperfine lines, since the resolution of ESR transition frequencies is limited by the transverserelaxation time Tm of the electron spins, and that of NMR transition frequencies by the transverse-relaxation time T 2n of the nuclear spins. Since usually T 2n @ Tm , the second approach is superior and will be dealt with in the following.
[1] Chirp Hyperfine Spectroscopy
i Å1 n
NENDOR Å 4 ∑ Ii ,
[2]
i Å1
and NHS Å n,
[3]
respectively. For nuclei with spin I Å 12, Eqs. [1] – [3] reduce to NESR Å 2 n , NENDOR Å 2n, and NHS Å n. While, for large n, the density of the lines is reduced by orders of magnitude in going from ESR to ENDOR, it is further reduced by a factor of two in going from ENDOR to HS. The latter improvement is considerably increased for nuclei with I ú 1 2. For the frequently investigated nitrogen (I Å 1) or aluminum (I Å 52 ) nuclei, for example, the number of lines in HS compared to ENDOR is reduced by a factor of 4 and 10, respectively. This reduction is attained at the expense of the information content, namely through excluding information on the nuclear Zeeman and nuclear-quadrupole interactions. However, this information can be restored by correlating the hyperfine spectrum with the ENDOR spectrum in a 2D experiment. The assignment of the lines in an ENDOR spectrum may be troublesome even for well-resolved spectra. This is due to the fact that ENDOR lines of different nuclei often appear in the same spectral region and the hyperfine, the nuclear Zeeman, or the nuclear-quadrupole interaction may be the dominant one; ENDOR multiplets can therefore be centered about either of the interaction frequencies. These assignment problems can be solved by correlating the hyperfine spectrum with the ENDOR spectrum. In many cases it is, however, sufficient just to measure the 1D hyperfine spectrum. Since the hyperfine splittings are manifested in both the ESR and the NMR spectrum, HS can be performed by mea-
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The pulse sequence of the version of HS presented in this work (Fig. 1a) is based on the chirp ENDOR approach (14). We first give a qualitative description of chirp HS by using a simple model system consisting of one nuclear spin I Å 12 coupled to one electron spin S Å 12. The time evolution of the polarizations and coherences in the corresponding four-level diagram is illustrated in Fig. 1b. At time t 0 , the spin system is in thermal equilibrium. The selective MW p pulse of length t1 0 t 0 which is assumed to be on-resonance with the allowed ESR transition 1, 3 inverts the polarization of this transition and creates longitudinal two-spin order (9). The RF excitation with a linearly varying frequency (RF chirp pulse) applied during time tchirp Å t2 0 t1 transfers the two-spin order to nuclear coherence (NC) on the two transitions 1, 2 and 3, 4, which then evolves freely during time t with frequencies va and vb ( a, mS Å 12 state; b, mS Å 012 state). The nonselective MW p pulse transfers NC on transition 1, 2 to NC on transition 3, 4, and vice versa. The NC then evolves during time T 0 t with frequencies vb and va and is transferred to polarization by the RF detection chirp pulse applied during time tchirp Å t4 0 t3 . The polarization of transition 1, 3 at time t4 is finally detected by the two-pulse echo sequence p /2— t — p with selective MW pulses. The echo intensity as a function of t is then found to be modulated with the hyperfine frequency. The incrementation scheme with a fixed overall evolution time T, separated by the nonselective p pulse in two time intervals t and T 0 t (15) may be used in all types of pulsed ESR schemes where NC freely evolves (16). For a more stringent description of chirp HS, we consider an isotropic spin system consisting of n inequivalent nuclear spins I Å 12 coupled to one electron spin S Å 12 in a static magnetic field B0 . The effective Hamiltonian in the rotating frame (in angular frequencies) is given by
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FIG. 1. Chirp hyperfine spectroscopy. (a) Pulse sequence. (b) Four-level schemes visualizing the transfer of polarization and nuclear coherence for 1 1 an S Å 2, I Å 2 two-spin system. The selective MW pulses for preparation and detection are on-resonance with transition 1, 3. n
H0 Å VS Sz / ∑ ( v (I i ) / a ( i ) Sz )Iz ,
[4]
i Å1
where VS Å vS 0 vMW Å gebeDB0 / \ is the offset of the electron Zeeman frequency vS from the MW frequency vMW and v (I i ) Å 0 g (ni ) bn B0 / \ and a ( i ) are the nuclear Zeeman frequency and the isotropic hyperfine coupling of the ith nucleus, respectively. An idealization of a fast passage experiment is used to model the chirp pulse, by considering it as consisting of a sequence of selective RF pulses, each of them acting on a single transition at the time of passage through resonance (17). For simplicity, we further assume that the flip angle b of the RF chirp pulse is the same for each transition. For the calculation of the signal intensity, we made use of the product-operator formalism and the considerations presented in the Appendix of Ref. (14). We now focus on the pair of transitions of the jth nuclear spin ( j arbitrary) in the two mS manifolds with the frequencies v (aj ) Å Év (I j ) 0 a ( j ) /2É and v (bj ) Å Év (I j ) / a ( j ) /2É. In the following, we omit the index j for brevity. The sequences for chirp ENDOR and chirp HS differ in the way the time intervals are incremented and in the nonselective MW p pulse introduced in the latter to interchange the NC on the two transitions. Assume now that the NC with frequency va ( vb) is excited and detected at time ta (tb) after the start of the first and second chirp pulses, respectively. At time of excitation and detection of the NC, the phase is given by f0 / kt 2a ( f0 / kt 2b ), where f0 denotes the RF phase at time t1 and t3 , and k Å ( vmax 0 vmin )/tchirp is the sweep rate of the chirp pulse with the minimum frequency vmin and maximum frequency vmax .
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Assuming a uniform flip angle b for all nuclear transitions, the echo modulation caused by the nuclear spin j is given by Sj Å C0 {cos[ va((tchirp 0 ta) / t) { vb(T 0 t / tb) 0 k(t 2b 0 t 2a )] / cos[ vb((tchirp 0 tb) / t) { va(T 0 t / ta) 0 k(t 2a 0 t 2b )]},
[5]
with C0 Å C
1 2
n 01
sin 2b(1 / cos 2b ) n 01 ,
[6]
and a constant factor C (14). In Eq. [5], the upper sign holds for the weak-coupling case (Éa/2É õ ÉvIÉ), the lower sign for the strong-coupling case (Éa/2É ú ÉvIÉ). The different signs result from the fact that, in the strong-coupling case, the p pulse inverts the phase of the NC since the effective magnetic fields at the nucleus have then opposite directions in the two mS states. The factor C0 is the same as that found in chirp ENDOR and is derived in the Appendix of Ref. (14) by using the product-operator formalism. The first (second) term in Eq. [5] describes the signal obtained by exciting NC with frequency va ( vb) and detecting NC with frequency vb ( va). It is, therefore, necessary for the chirp pulse to cover the frequencies of the nuclear transitions
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in both mS states to observe this signal. The calculation of the optimum flip angle for the chirp pulse (i.e., the value of b for which C0 is maximum) and the considerations on the sensitivity of the experiment are the same as those for chirp ENDOR (14). Note that it is technically feasible to perform the experiment at the optimum flip angle and that the sensitivity of the experiment is comparable to or even higher than in the corresponding pulsed Davies–ENDOR experiment (18) which is based on polarization transfers. Using the relation ta,b Å ( va,b 0 vmin )/ k, Eq. [5] can be rearranged to
F
Sj Å 2C0 cos vI (T { tchirp ) |
va / b( va / b 0 vmin ) k
1 cos(at / f1 ),
G [7]
with the phase factor
f1 Å
1 [2a( vI 0 vmin ) 0 vb / a( vb / a 0 vmin )] k 0
a (T { tchirp ). 2
[8]
Equations [7] and [8] describe the echo modulation with the hyperfine frequency a and the phase f1 . The first index of the nuclear frequencies and the upper sign hold for the weak-coupling case where a Å Éva 0 vbÉ, the second index and the lower sign for the strong-coupling case where a Å va / vb . For the modulation of n nuclei, we find that n
S Å 2C0 ∑ Sj .
[9]
jÅ1
Fourier transformation with respect to t yields a spectrum consisting only of peaks at the hyperfine frequencies. The rather complicated expression for the phase (Eq. [8]) prevents phase correction after a cosine Fourier transformation; hence, magnitude spectra must be calculated. Since the amplitudes of the peaks depend on time T (Eq. [7]), particular values of T will cause blind spots in the spectrum. It is, therefore, necessary to repeat the experiment with different T values. The approach can easily be extended to a 2D experiment by incrementing time T. Another rearrangement of Eq. [5] yields the formula for the modulation in the 2D experiment Sj Å C0[cos( |at { vbT / f2 ) / cos(at { vaT / f3 )],
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with the phases f2 Å [ va( vmax 0 va) { vb( vb 0 vmin ) 0 2a( vI 0 vmin )]/ k,
[11a]
f3 Å [ vb( vmax 0 vb) { va( va 0 vmin ) 0 2a( vI 0 vmin )]/ k.
[11b]
Again, one obtains the modulation for all the nuclei by using Eq. [9]. Only cross peaks between the hyperfine frequency and the ENDOR frequencies of the same nucleus appear in this 2D hyperfine-correlated chirp ENDOR experiment. Equation [10] reveals that, in the weak-coupling case, the two cross peaks appear in different quadrants, while, in the strong-coupling case, they both share the second quadrant. For the derivation of Eqs. [5] – [11], we only considered the coherence-transfer pathway leading to the hyperfine frequencies. However, other pathways can also contribute to an echo modulation. For example, RF chirp pulses can excite and detect multiquantum NC (14). Moreover, in systems with anisotropic hyperfine couplings, the nonselective MW p pulse can transfer polarization to NC, and vice versa. In order to obtain spectra free of artifacts, the contributions of these pathways must be removed by a phase-cycling procedure. The most simple phase cycle is based on three assumptions. First, the excitation of multiquantum NC on the order of three or higher can be neglected because of the small flip angle b of the chirp pulse. Second, the selective MW pulses do not excite or detect NC. Third, Tm ! tchirp ; i.e., electron single-quantum and electron-nuclear zero- and double-quantum coherences decay completely during the chirp pulse. Violations of these conditions will be discussed below. If the first condition is fulfilled, the remaining zero- and double-quantum NC can be eliminated by the phase cycle (0, 0 0 p, 0) on the chirp pulses (14). The second and third conditions allow one to consider only pathways with a coherence order of zero for both electron and nuclear spins immediately before the first and after the second chirp pulse. The two-step phase cycle already forces a change in nuclear coherence order of one during the first chirp pulse. The exchange of polarization with NC by the nonselective MW p pulse can be prevented if we force an additional change of one in nuclear coherence order during the second chirp pulse. The complete phase cycle on the two chirp pulses is therefore (0, 0 0 p, 0 0 0, p / p, p ). Although Eqs. [5] – [11] were derived for a model system without pseudosecular hyperfine terms in the Hamiltonian (i.e., without forbidden transitions), they also represent a good approximation for the description of systems with weak forbidden transitions, provided this phase cycle is applied. Among the assumptions made above, the first is satisfied under all practical conditions. The second condition breaks down if ÉaÉ É 2ÉvIÉ. In this case, strong forbidden transi-
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tions are observed, and we can no longer neglect the pseudosecular hyperfine terms in the Hamiltonian. HS based on ESEEM (16) is then the more appropriate method. The third condition can be fulfilled whenever T 2n @ Tm by adjusting the length of the chirp pulse or even simpler by starting the incrementation with t § 5Tm 0 tchirp . Alternatively, the phase cycle can be extended to eight steps by applying a (0 / p ) phase cycle on the nonselective MW p pulse. This eightstep phase cycle relaxes the third assumption to Tm ! tchirp / T. If the MW p pulse is not sufficiently nonselective, an additional pathway must be considered that describes the NC remaining after the pulse in the same mS manifold. The contribution of this pathway to the signal cannot be eliminated by phase cycling since all changes of the coherence order are the same as in the wanted pathway. Fortunately, this pathway does not lead to an echo modulation in the 1D experiment, since the NC evolves with the same frequency before and after the nonselective MW pulse during the constant evolution time t / (T 0 t) Å T. The contribution of this pathway to the echo is therefore also constant and can be eliminated by a baseline correction of the time-domain data. The same procedure applied to each trace of the 2D experiment removes this contribution from the 2D spectrum. The fixed length of the whole pulse sequence also accounts for another peculiarity of chirp HS; there is no relaxational decay of the echo modulation. The actual resolution of the hyperfine spectrum is solely determined by the sweep range of time t. The width of the hyperfine lines therefore does not contain any information; in fact, there is no linewidth in the usual sense of the term. The trade-off between resolution and sensitivity of the experiment then becomes very transparent. The required resolution determines the sweep range of t; this in turn determines the minimum T and therefore the sensitivity of the experiment. As a rule of thumb, T / tchirp should be roughly equal to T * 2n , which characterizes the decay of the NC due to both transverse relaxation (T 2n ) and inhomogeneous line broadening. Obviously, the method is not well suited for systems with T * 2n ! tchirp . Since T 2n is usually sufficiently long, the latter condition applies mainly to disordered systems with a large inhomogeneous broadening of the ENDOR lines. Work on a 2D hyperfine spectroscopy more suitable for such systems is in progress. If the hyperfine coupling is anisotropic, also pseudo-secular hyperfine terms of the form BSz Ix must be considered in the effective Hamiltonian in the rotating frame given in Eq. [4]. The difference frequency (in the weak-coupling case) and the sum frequency (in the strong-coupling case) no longer represent the hyperfine coupling; they are, however, identical to the hyperfine splitting that would be observed in a sufficiently resolved ESR spectrum (4, 20). These hyperfine splittings differ somewhat from the hyperfine couplings because of a contribution from the nuclear Zeeman
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interaction. The differences become significant for the case ÉvIÉ É Éa/2É (intermediate coupling), where mS and mI are no longer good quantum numbers. For nuclear spins I ú 12, the nonselective MW p pulse transfers the coherence on the transition characterized by (mS Å 12, mI , mI / 1) to the coherence (mS Å 012, mI , mI / 1), and vice versa. Since the magnetic quantum numbers of the NC are not changed by the p pulse, there is no firstorder contribution of the nuclear-quadrupole interaction to hyperfine frequency. Therefore, to first order, the hyperfine spectrum of a nucleus with spin I again consists only of one peak at the hyperfine frequency. One can show, however, that the hyperfine frequencies depend slightly on mI if higher-order contributions are considered. For example, for a system with S Å 12, I (arbitrary) and assuming the strongcoupling case, the second-order contribution of an isotropic hyperfine coupling a to the frequency of the nuclear transition (mI } mI / 1) is given by (19) da É
0a 2 (mI / mS / 1/2) . 2v S
[12]
For I Å 12, this second-order contribution vanishes in the hyperfine spectra, whereas for I Å 1, da Å 0 for the (mS Å 1 1 2, 01 } 0) transition and the ( 02, 0 } 1) transition, da Å a 2 /2vS for the ( 12, 01 } 0) transition, and da Å 0a 2 /2vS for the ( 012, 0 } 1) transition. The peaks in the hyperfine spectrum are thus observed at a { a 2 /2vS . Similarly, second-order contributions of the nuclear-quadrupole interaction may result in small shifts of the hyperfine lines for nuclei with spin I ú 12. Finally, we note that also for electron spins S ú 12 the frequencies in the hyperfine spectra are given to first order by a and are thus independent of mS of the EPR observer transition. Here, second-order effects are manifest again as shifts of the hyperfine frequencies. A detailed calculation of second-order hyperfine frequencies in systems with I ú 12 and/or S ú 12 is beyond the scope of this paper and will be published elsewhere. EXPERIMENTAL
All experiments were performed with a homebuilt pulsed X-band spectrometer (21). The ENDOR probehead (22) is equipped with a p circuit for a rough matching of the impedance in the frequency range 5–30 MHz (14). Echo amplitudes have been measured by a boxcar averager (PAR, model 162 with gated integrator model 165). The chirp pulses were created by a LeCroy 9100 arbitrary function generator (AFG) with a digital time resolution of 5 ns. Shaped chirp pulses were used to avoid wiggles in the hyperfine spectrum. The envelope of the chirp pulses was a sine quarterwave in the first 25% of the RF pulse length, a cosine quarterwave in the last 25% of the pulse, and constant in
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between (compare Fig. 1a). However, it is also possible to use rectangular chirp pulses. In the Davies–ENDOR and the chirp HS experiments, the length of the first MW pulse was 100 ns; the pulse lengths in the echo-detection subsequence p /2— t – p were 100 and 200 ns. Interpulse delays t of 300 and 340 ns were used for the Davies–ENDOR and chirp HS experiments, respectively. The RF pulse length was 25 ms in the Davies–ENDOR experiment, and the RF was stepped with 50 kHz increments. In the chirp HS experiments, the RF was linearly varied from 3 to 23 MHz during the chirp pulse with a length of 5 ms. The length of the nonselective MW p pulse in chirp HS experiments was 20 ns. Time increments of 20 ns for t and 25 ns for T (compare Fig. 1) were used. The four-step phase cycle (0, 0 0 p, 0 0 0, p, /p, p ) of the chirp pulses was applied in the chirp HS experiments. The phase cycling was performed by recalling inverted waveforms from the AFG, an approach that requires only one RF channel. In the 1D chirp HS experiment, six traces with 600 data points each were sampled at different T values to test the blind-spot behavior. The time-domain data were apodized with a Hamming window, zero-filled to 2K data points, and Fourier transformed, and the magnitude spectra were calculated. Summation of the six traces yields a hyperfine spectrum which is virtually free of blind spots. A data array of 128 1 128 points was recorded in the 2D hyperfinecorrelated chirp ENDOR experiment. The raw data were baseline corrected by third-order polynomials in both dimensions and Fourier transformed, and the magnitude spectrum was calculated. All experiments were performed on a single crystal of triglycine sulfate (TGS) doped with 0.1 mass% copper(II) sulfate. A crystal orientation was used where the ESR lines of the two magnetically inequivalent sites were clearly separated from each other. All spectra were measured at a temperature of 15 K and a static magnetic field of 284.1 mT, corresponding to a proton Zeeman frequency of 12.10 MHz. RESULTS AND DISCUSSION
The two glycine molecules coordinated to the copper ion in Cu II-doped TGS are not equivalent, as detailed investigations of the orientation dependence of the proton ENDOR frequencies have shown (23). Therefore, for a single site, one expects eight nitrogen ENDOR lines, eight ENDOR lines each for the four NH2 and four CH2 protons, and a number of weakly coupled matrix protons. A Davies–ENDOR spectrum of Cu(II)TGS at arbitrary crystal orientation is shown in Fig. 2a. The RF pulse length of 25 ms is sufficient to avoid excessive power broadening. Nevertheless, only a few of the expected lines are resolved so that the spectrum does not allow for an unambiguous assignment of the lines and a precise measurement of the transition frequencies. A time-domain data trace of the chirp HS experiment is shown
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FIG. 2. Davies–ENDOR spectrum (a) and chirp hyperfine spectrum (sum of six traces with different interpulse delays T ) (b) of a Cu(II)doped TGS single crystal at arbitrary crystal orientation and a temperature of 15 K. Bars mark the assignment of the spectral lines to the different groups as obtained from the hyperfine spectrum.
in Fig. 3. As predicted by theory, no relaxational decay of the modulation is observed. The corresponding hyperfine spectrum shown in Fig. 2b is significantly better resolved than the ENDOR spectrum for four reasons: (1) The number of spectral lines is reduced by a factor of two (protons) and a factor of four (nitrogens). (2) The frequency differences between the lines in the hyperfine spectrum are twice as large as between the corresponding ENDOR transitions. (3) Power broadening is completely absent in HS. (4) The spectrum extends over a broader frequency range. Since, for the observation of a hyperfine line, the corresponding ENDOR frequencies in both mS states must lie within the excitation band of 3–23 MHz, the hyperfine lines at frequencies nh f ú 20 MHz can unambiguously be assigned to the nitrogens. A comparison of the remaining hyperfine frequencies with the ENDOR frequencies ( nh f õ 10 MHz) reveals that they all must be assigned to protons. The correlation between the hyperfine and the ENDOR spectra therefore can easily be established, and the evaluation of the parame-
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FIG. 3. Time-domain data trace of the hyperfine spectrum shown in Fig. 2b (T Å 7.865 ms). No relaxational decay of the oscillations is observed.
ters (hyperfine and nuclear-quadrupole couplings) becomes straightforward. Since the range of the hyperfine frequencies is known in our experimental example (nitrogens, 13–24 MHz; NH2 protons, 5–15 MHz; CH2 protons, 1–8 MHz) (23, 24), the assignment of the hyperfine lines to protons in the particular groups in the structure is also known (horizontal bars in Fig. 2). In the hyperfine spectrum in Fig. 2b, the two nitrogen lines (21–22 MHz) and the four lines of the NH2 protons (6.5–8 MHz) are nicely resolved. Note that the nitrogen line at 21.3 MHz shows an additional splitting. For an isotropic hyperfine coupling of É21 MHz, a second-order splitting of about 50 kHz is predicted (Eq. [12]) in agreement with the observation. The hyperfine lines of the CH2 protons observed in the frequency range 2–3 MHz are of rather low intensity. This is due to the fact that, in the Davies-based version of the chirp HS experiment, the first MW p pulse with a length of 100 ns cannot create sufficient polarization on the nuclear transitions of weakly coupled nuclei. This problem may be overcome by using Mims-type HS (14), which however suffers from additional blind spots. In our experimental demonstration of HS, the blind spots resulting from the fixed interpulse delay T (Eq. [7]) have been removed by measuring six time-domain traces at different T values and displaying the sum of the six magnitude spectra in Fig. 2b. Three of the single traces are shown in Fig. 4. Although none of the hyperfine lines vanish completely in any of these traces, the relative intensities show a strong dependence on T. This blind-spot behavior may actually result in a resolution of the single traces apparently higher than that of the sum spectrum, since, due to the long interpulse delay T, the blind spots are rather dense and may selectively suppress some of the lines. To check the correlations between the frequencies in the hyperfine and the ENDOR spectrum and to demonstrate the gain in information obtained with a 2D measurement, we performed a hyperfine-correlated chirp ENDOR experiment. The resulting 2D plot is shown in Fig. 5a. As expected from Eq. [11a], the cross peaks of the NH2 protons (weakcoupling case) appear in both quadrants at (7.8, 9.0 MHz)
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and at (7.8, 016.5 MHz). All the cross peaks of the nitrogen nuclei (strong-coupling case) are observed in the second quadrant (negative ENDOR frequencies). Single traces parallel to the ENDOR axis represent ENDOR spectra of only those nuclear transitions that correspond to a given hyperfine splitting: The nuclear-transition frequencies are hyperfine correlated. The traces at the hyperfine frequencies 7.8 MHz (NH2 protons) and 21.9 MHz ( 14N) are shown in Figs. 5b and 5c, respectively. In the latter case, the four-line structure resulting from the nuclear Zeeman and the nuclear-quadrupole interaction can clearly be seen. The 2D spectrum therefore fully supports the assignment of the ENDOR frequencies given above. CONCLUSION
A new type of hyperfine spectroscopy based on the chirp ENDOR approach (14) has been introduced, which allows
FIG. 4. Chirp hyperfine spectra of Cu(II)-doped TGS at the same crystal orientation and temperature as described in the legend to Fig. 2 for different chirp-interpulse delays T (see Fig. 1). (a) T Å 7.815 ms. (b) T Å 7.840 ms. (c) T Å 7.865 ms.
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of the nuclear transitions. We expect that, with the further development of HS, investigations of paramagnetic compounds that cannot be studied with known techniques due to the excessive complexity of their spectra will become feasible. ACKNOWLEDGMENTS The authors thank Professor Rolf Bo¨ttcher for the gift of the copper-doped triglycine sulfate crystal. Technical assistance by Jo¨rg Forrer is gratefully acknowledged. This research has been supported by the Swiss National Science Foundation. G.J. thanks the Stiftung Stipendien-Fonds des Verbandes der Chemischen Industrie (Germany) for a Kekule´ grant.
REFERENCES 1. G. Feher, Phys. Rev. 103, 834 (1956). 2. L. Kevan and L. D. Kispert, ‘‘Electron Spin Double Resonance Spectroscopy,’’ Wiley–Interscience, New York, 1976. 3. M. Dorio and J. H. Freed (Eds.), ‘‘Multiple Electron Resonance Spectroscopy,’’ Plenum, New York, 1979. 4. A. Schweiger, Struct. Bonding (Berlin) 51, 1 (1982). 5. H. Kurreck, B. Kirste, and W. Lubitz, ‘‘Electron Nuclear Double Resonance Spectroscopy in Solution,’’ VCH, New York, 1988. 6. W. B. Mims, Proc. R. Soc. London 283, 452 (1965). 7. P. K. Dinse, in ‘‘Advanced EPR’’ (A. J. Hoff, Ed.), Chap. 17, Elsevier, Amsterdam, 1989. 8. A. Grupp and M. Mehring, in ‘‘Modern Pulsed and ContinuousWave Electron Spin Resonance’’ (L. Kevan and M. Bowman, Eds.), Chap. 4, Wiley, New York, 1990. 9. C. Gemperle and A. Schweiger, Chem. Rev. 91, 1481 (1991). 10. S. A. Dikanov and Yu. D. Tsvetkov, ‘‘Electron Spin Echo Envelope Modulation (ESEEM) Spectroscopy, CRC Press, Boca Raton, 1992. FIG. 5. Two-dimensional hyperfine-correlated chirp ENDOR spectroscopy of Cu(II)-doped TGS at the same crystal orientation and temperature as described in the legend to Fig. 2. (a) Contour plot. (b,c) ENDOR traces at the hyperfine frequencies nh f Å 7.8 MHz (NH2 protons) (b), and nh f Å 21.9 MHz (nitrogen) (c).
one to measure the complete hyperfine spectrum in an 1D experiment. It has been demonstrated by theory and experiment that this technique is distinguished by a drastic increase in resolution compared to ENDOR spectroscopy, which is caused by a reduction in the number of spectral lines. Moreover, ENDOR suffers from assignment ambiguities because of the dependence of the ENDOR frequencies on mS and the hyperfine, the nuclear Zeeman, and the nuclear-quadrupole interactions. In HS, the dependence on mS is eliminated, the dependence on the nuclear Zeeman interaction is greatly reduced, and there is only a second-order contribution of the nuclear-quadrupole interaction to the frequencies. HS therefore directly measures the hyperfine splittings of the nuclear spins with a resolution determined by the linewidth
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11. A. Schweiger, in ‘‘Modern Pulsed and Continuous-Wave Electron Spin Resonance’’ (L. Kevan and M. K. Bowman, Eds.), Chap. 2, Wiley, New York, 1990. 12. R. de Beer, H. Barkhuijsen, E. L. de Wild, and R. P. J. Merks, Bull. Magn. Reson. 2, 420 (1981). 13. Th. Wacker and A. Schweiger, Chem. Phys. Lett. 191, 136 (1992). 14. G. Jeschke and A. Schweiger, J. Chem. Phys. 103, 8329 (1995). 15. G. Jeschke and A. Schweiger, Proceedings of the EMARDIS Workshop 1995, Sofia. 16. M. Hubrich, G. Jeschke, and A. Schweiger, Proceedings of the Chianti Conference on Magnetic Resonance, 1995. 17. J. A. Ferretti and R. R. Ernst, J. Chem. Phys. 65, 4283 (1976). 18. E. R. Davies, Phys. Lett. A 47, 1 (1974). 19. N. M. Atherton, ‘‘Principles of Electron Spin Resonance,’’ Chap. 3.11, Ellis Horwood, New York, 1993. 20. A. Ponti and A. Schweiger, J. Chem. Phys. 102, 5207 (1995). 21. Th. Wacker, Ph.D. thesis, ETH Zu¨rich, No. 9913, 1992. 22. J. Forrer, S. Pfenninger, J. Eisenegger, and A. Schweiger, Rev. Sci. Instrum. 61, 3360 (1990). 23. R. Bo¨ttcher, D. Heinhold, and W. Windsch, Chem. Phys. Lett. 49, 148 (1977). 24. R. Bo¨ttcher, D. Heinhold, S. Wartewig, and W. Windsch, J. Mol. Struct. 46, 363 (1978).
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0050
Chirp Hyperfine Spectroscopy* G. JESCHKE
AND
A. SCHWEIGER
Laboratorium fu¨r Physikalische Chemie, Eidgeno¨ssische Technische Hochschule, CH-8092 Zu¨rich, Switzerland Received August 14, 1995
A novel pulse ESR scheme for the direct measurement of hyperfine splittings is proposed. It is shown that this hyperfine spectroscopy reduces the number of the spectral lines considerably, overcomes assignment ambiguities inherent in electron-nuclear doubleresonance spectra, and allows one to separate signals that overlap in ENDOR spectra. The resolution enhancement of hyperfine spectroscopy is fully utilized since the hyperfine splittings are measured between nuclear-spin rather than electron-spin transitions. The pulse sequence for the experiment is based on the chirp ENDOR method and can yield one-dimensional as well as two-dimensional spectra that correlate the ENDOR and the hyperfine frequencies. Both experiments are discussed theoretically, and the first experimental spectra of a model system are presented. q 1996 Academic
equivalent nuclear spins is represented in the spectrum by only one peak at the hyperfine frequency, irrespective of the nuclear-spin quantum number. The two-dimensional stimulated-echo ENDOR experiment proposed by de Beer et al. (12) can be considered as the first realization of such a hyperfine spectroscopy. In this experiment, the hyperfine dimension is obtained by incrementing the interpulse delay between the first two microwave (MW) pulses. One then observes peaks at the frequency differences between allowed electron-spin transitions that correspond to the hyperfine couplings. Recently, another attempt to measure hyperfine frequencies, Fourier-transform hyperfine spectroscopy (FTHS), has been reported (13). In FT-HS, a transient hole burnt into the ESR line by a selective MW pulse is subsequently shifted by a selective radiofrequency pulse and finally detected via a free induction decay. This technique is distinguished by the multiplex advantage inherent in the FID detection of the hyperfine frequencies. Both stimulated ENDOR and FT-HS are basically 2D techniques correlating the hyperfine and nuclear transition (ENDOR) frequencies. In the hyperfine dimension, the linewidth is determined by the phase memory time Tm of the electron spins, since the hyperfine frequencies are measured as the differences between ESR transition frequencies. In the ENDOR dimension, the linewidth is governed by the phase memory time T 2n of the nuclear spins or by the power broadening caused by the RF pulse, whenever its length is shorter than T 2n . In this paper, we introduce a pulsed ESR method for the measurement of hyperfine spectra which is fundamentally different from 2D stimulated-echo ENDOR and FT-HS. This new type of HS features a much higher resolution than the former techniques since the frequencies in the hyperfine dimension are measured as differences or sums of nucleartransition frequencies (see below). The experimental scheme for the first time allows for the direct measurement of one-dimensional hyperfine spectra. The approach can also be extended to a second dimension, resulting in a new type of a hyperfine-correlated ENDOR spectroscopy. Here, the frequencies in both the hyperfine and ENDOR dimension are obtained in a time-domain rather than in a frequencydomain experiment; therefore, no power broadening is introduced. The time evolution of the spin system during the
Press, Inc.
INTRODUCTION
The resolution of conventional solid-state electron-spin resonance spectroscopy which measures the electron-spin transition frequencies of a spin system is usually quite poor, even in single-crystal work. Hyperfine interactions between the unpaired electron and the nuclear spins, as well as small second-order shifts caused by the nuclear-quadrupole interactions are therefore often not resolved. The situation can be changed drastically by using electron-nuclear double-resonance (ENDOR) (1–9) or electron-spin-echo-envelope modulation (ESEEM) (10, 11) techniques. Since with these methods nuclear- rather than electron-spin transitions are measured, one can improve on the resolution of ESR by orders of magnitude. However, for systems with a large number of nuclear spins coupled to the electron spin, the number of nuclear-transition frequencies is again large, and, therefore, the resolution of the ENDOR spectrum may be still poor. This is particularly true for nuclear spins I ú 12, since in this case additional line splittings are introduced by the nuclear-quadrupole interaction. One can further improve on the resolution by using techniques that directly measure the frequencies representing the hyperfine couplings. In such an approach, each group of * Part of this work was presented at the DFG Rundgespra¨ch ‘‘Magnetische Resonanz,’’ Hirschegg, Austria, September 19–23, 1994. 45
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1064-1858/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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sequence consisting of selective and nonselective MW and chirped RF pulses will be discussed, and the first experimental examples will be presented. THEORY
Hyperfine Spectroscopy We first quantify the simplification of the spectrum achieved with HS by comparing the number of spectral lines observed with the different schemes for a system of n inequivalent nuclear spins Ii (i Å 1, . . . , n) coupled to one electron spin S Å 12. Considering the electron and nuclear Zeeman, the hyperfine, and the nuclear-quadrupole interactions, the number of ESR, ENDOR, and hyperfine lines is given by n
NESR Å
∏ (2Ii
/ 1),
suring frequency differences between two ESR transitions with DmI Å 1 or two ENDOR transitions with DmS Å 1, where mI and mS are the magnetic quantum numbers of the nuclear and electron spin, respectively. The transfer of polarization or coherence between appropriate ESR transitions can be achieved by flipping the nuclear spin with an RF p pulse (thereby inverting the hyperfine field seen by the electron spin), and between appropriate NMR transitions by flipping the electron spin with an MW p pulse (thereby inverting the hyperfine field seen by the nuclear spin). From a practical point of view, the two approaches differ mainly in the widths of the hyperfine lines, since the resolution of ESR transition frequencies is limited by the transverserelaxation time Tm of the electron spins, and that of NMR transition frequencies by the transverse-relaxation time T 2n of the nuclear spins. Since usually T 2n @ Tm , the second approach is superior and will be dealt with in the following.
[1] Chirp Hyperfine Spectroscopy
i Å1 n
NENDOR Å 4 ∑ Ii ,
[2]
i Å1
and NHS Å n,
[3]
respectively. For nuclei with spin I Å 12, Eqs. [1] – [3] reduce to NESR Å 2 n , NENDOR Å 2n, and NHS Å n. While, for large n, the density of the lines is reduced by orders of magnitude in going from ESR to ENDOR, it is further reduced by a factor of two in going from ENDOR to HS. The latter improvement is considerably increased for nuclei with I ú 1 2. For the frequently investigated nitrogen (I Å 1) or aluminum (I Å 52 ) nuclei, for example, the number of lines in HS compared to ENDOR is reduced by a factor of 4 and 10, respectively. This reduction is attained at the expense of the information content, namely through excluding information on the nuclear Zeeman and nuclear-quadrupole interactions. However, this information can be restored by correlating the hyperfine spectrum with the ENDOR spectrum in a 2D experiment. The assignment of the lines in an ENDOR spectrum may be troublesome even for well-resolved spectra. This is due to the fact that ENDOR lines of different nuclei often appear in the same spectral region and the hyperfine, the nuclear Zeeman, or the nuclear-quadrupole interaction may be the dominant one; ENDOR multiplets can therefore be centered about either of the interaction frequencies. These assignment problems can be solved by correlating the hyperfine spectrum with the ENDOR spectrum. In many cases it is, however, sufficient just to measure the 1D hyperfine spectrum. Since the hyperfine splittings are manifested in both the ESR and the NMR spectrum, HS can be performed by mea-
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The pulse sequence of the version of HS presented in this work (Fig. 1a) is based on the chirp ENDOR approach (14). We first give a qualitative description of chirp HS by using a simple model system consisting of one nuclear spin I Å 12 coupled to one electron spin S Å 12. The time evolution of the polarizations and coherences in the corresponding four-level diagram is illustrated in Fig. 1b. At time t 0 , the spin system is in thermal equilibrium. The selective MW p pulse of length t1 0 t 0 which is assumed to be on-resonance with the allowed ESR transition 1, 3 inverts the polarization of this transition and creates longitudinal two-spin order (9). The RF excitation with a linearly varying frequency (RF chirp pulse) applied during time tchirp Å t2 0 t1 transfers the two-spin order to nuclear coherence (NC) on the two transitions 1, 2 and 3, 4, which then evolves freely during time t with frequencies va and vb ( a, mS Å 12 state; b, mS Å 012 state). The nonselective MW p pulse transfers NC on transition 1, 2 to NC on transition 3, 4, and vice versa. The NC then evolves during time T 0 t with frequencies vb and va and is transferred to polarization by the RF detection chirp pulse applied during time tchirp Å t4 0 t3 . The polarization of transition 1, 3 at time t4 is finally detected by the two-pulse echo sequence p /2— t — p with selective MW pulses. The echo intensity as a function of t is then found to be modulated with the hyperfine frequency. The incrementation scheme with a fixed overall evolution time T, separated by the nonselective p pulse in two time intervals t and T 0 t (15) may be used in all types of pulsed ESR schemes where NC freely evolves (16). For a more stringent description of chirp HS, we consider an isotropic spin system consisting of n inequivalent nuclear spins I Å 12 coupled to one electron spin S Å 12 in a static magnetic field B0 . The effective Hamiltonian in the rotating frame (in angular frequencies) is given by
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FIG. 1. Chirp hyperfine spectroscopy. (a) Pulse sequence. (b) Four-level schemes visualizing the transfer of polarization and nuclear coherence for 1 1 an S Å 2, I Å 2 two-spin system. The selective MW pulses for preparation and detection are on-resonance with transition 1, 3. n
H0 Å VS Sz / ∑ ( v (I i ) / a ( i ) Sz )Iz ,
[4]
i Å1
where VS Å vS 0 vMW Å gebeDB0 / \ is the offset of the electron Zeeman frequency vS from the MW frequency vMW and v (I i ) Å 0 g (ni ) bn B0 / \ and a ( i ) are the nuclear Zeeman frequency and the isotropic hyperfine coupling of the ith nucleus, respectively. An idealization of a fast passage experiment is used to model the chirp pulse, by considering it as consisting of a sequence of selective RF pulses, each of them acting on a single transition at the time of passage through resonance (17). For simplicity, we further assume that the flip angle b of the RF chirp pulse is the same for each transition. For the calculation of the signal intensity, we made use of the product-operator formalism and the considerations presented in the Appendix of Ref. (14). We now focus on the pair of transitions of the jth nuclear spin ( j arbitrary) in the two mS manifolds with the frequencies v (aj ) Å Év (I j ) 0 a ( j ) /2É and v (bj ) Å Év (I j ) / a ( j ) /2É. In the following, we omit the index j for brevity. The sequences for chirp ENDOR and chirp HS differ in the way the time intervals are incremented and in the nonselective MW p pulse introduced in the latter to interchange the NC on the two transitions. Assume now that the NC with frequency va ( vb) is excited and detected at time ta (tb) after the start of the first and second chirp pulses, respectively. At time of excitation and detection of the NC, the phase is given by f0 / kt 2a ( f0 / kt 2b ), where f0 denotes the RF phase at time t1 and t3 , and k Å ( vmax 0 vmin )/tchirp is the sweep rate of the chirp pulse with the minimum frequency vmin and maximum frequency vmax .
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Assuming a uniform flip angle b for all nuclear transitions, the echo modulation caused by the nuclear spin j is given by Sj Å C0 {cos[ va((tchirp 0 ta) / t) { vb(T 0 t / tb) 0 k(t 2b 0 t 2a )] / cos[ vb((tchirp 0 tb) / t) { va(T 0 t / ta) 0 k(t 2a 0 t 2b )]},
[5]
with C0 Å C
1 2
n 01
sin 2b(1 / cos 2b ) n 01 ,
[6]
and a constant factor C (14). In Eq. [5], the upper sign holds for the weak-coupling case (Éa/2É õ ÉvIÉ), the lower sign for the strong-coupling case (Éa/2É ú ÉvIÉ). The different signs result from the fact that, in the strong-coupling case, the p pulse inverts the phase of the NC since the effective magnetic fields at the nucleus have then opposite directions in the two mS states. The factor C0 is the same as that found in chirp ENDOR and is derived in the Appendix of Ref. (14) by using the product-operator formalism. The first (second) term in Eq. [5] describes the signal obtained by exciting NC with frequency va ( vb) and detecting NC with frequency vb ( va). It is, therefore, necessary for the chirp pulse to cover the frequencies of the nuclear transitions
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in both mS states to observe this signal. The calculation of the optimum flip angle for the chirp pulse (i.e., the value of b for which C0 is maximum) and the considerations on the sensitivity of the experiment are the same as those for chirp ENDOR (14). Note that it is technically feasible to perform the experiment at the optimum flip angle and that the sensitivity of the experiment is comparable to or even higher than in the corresponding pulsed Davies–ENDOR experiment (18) which is based on polarization transfers. Using the relation ta,b Å ( va,b 0 vmin )/ k, Eq. [5] can be rearranged to
F
Sj Å 2C0 cos vI (T { tchirp ) |
va / b( va / b 0 vmin ) k
1 cos(at / f1 ),
G [7]
with the phase factor
f1 Å
1 [2a( vI 0 vmin ) 0 vb / a( vb / a 0 vmin )] k 0
a (T { tchirp ). 2
[8]
Equations [7] and [8] describe the echo modulation with the hyperfine frequency a and the phase f1 . The first index of the nuclear frequencies and the upper sign hold for the weak-coupling case where a Å Éva 0 vbÉ, the second index and the lower sign for the strong-coupling case where a Å va / vb . For the modulation of n nuclei, we find that n
S Å 2C0 ∑ Sj .
[9]
jÅ1
Fourier transformation with respect to t yields a spectrum consisting only of peaks at the hyperfine frequencies. The rather complicated expression for the phase (Eq. [8]) prevents phase correction after a cosine Fourier transformation; hence, magnitude spectra must be calculated. Since the amplitudes of the peaks depend on time T (Eq. [7]), particular values of T will cause blind spots in the spectrum. It is, therefore, necessary to repeat the experiment with different T values. The approach can easily be extended to a 2D experiment by incrementing time T. Another rearrangement of Eq. [5] yields the formula for the modulation in the 2D experiment Sj Å C0[cos( |at { vbT / f2 ) / cos(at { vaT / f3 )],
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with the phases f2 Å [ va( vmax 0 va) { vb( vb 0 vmin ) 0 2a( vI 0 vmin )]/ k,
[11a]
f3 Å [ vb( vmax 0 vb) { va( va 0 vmin ) 0 2a( vI 0 vmin )]/ k.
[11b]
Again, one obtains the modulation for all the nuclei by using Eq. [9]. Only cross peaks between the hyperfine frequency and the ENDOR frequencies of the same nucleus appear in this 2D hyperfine-correlated chirp ENDOR experiment. Equation [10] reveals that, in the weak-coupling case, the two cross peaks appear in different quadrants, while, in the strong-coupling case, they both share the second quadrant. For the derivation of Eqs. [5] – [11], we only considered the coherence-transfer pathway leading to the hyperfine frequencies. However, other pathways can also contribute to an echo modulation. For example, RF chirp pulses can excite and detect multiquantum NC (14). Moreover, in systems with anisotropic hyperfine couplings, the nonselective MW p pulse can transfer polarization to NC, and vice versa. In order to obtain spectra free of artifacts, the contributions of these pathways must be removed by a phase-cycling procedure. The most simple phase cycle is based on three assumptions. First, the excitation of multiquantum NC on the order of three or higher can be neglected because of the small flip angle b of the chirp pulse. Second, the selective MW pulses do not excite or detect NC. Third, Tm ! tchirp ; i.e., electron single-quantum and electron-nuclear zero- and double-quantum coherences decay completely during the chirp pulse. Violations of these conditions will be discussed below. If the first condition is fulfilled, the remaining zero- and double-quantum NC can be eliminated by the phase cycle (0, 0 0 p, 0) on the chirp pulses (14). The second and third conditions allow one to consider only pathways with a coherence order of zero for both electron and nuclear spins immediately before the first and after the second chirp pulse. The two-step phase cycle already forces a change in nuclear coherence order of one during the first chirp pulse. The exchange of polarization with NC by the nonselective MW p pulse can be prevented if we force an additional change of one in nuclear coherence order during the second chirp pulse. The complete phase cycle on the two chirp pulses is therefore (0, 0 0 p, 0 0 0, p / p, p ). Although Eqs. [5] – [11] were derived for a model system without pseudosecular hyperfine terms in the Hamiltonian (i.e., without forbidden transitions), they also represent a good approximation for the description of systems with weak forbidden transitions, provided this phase cycle is applied. Among the assumptions made above, the first is satisfied under all practical conditions. The second condition breaks down if ÉaÉ É 2ÉvIÉ. In this case, strong forbidden transi-
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tions are observed, and we can no longer neglect the pseudosecular hyperfine terms in the Hamiltonian. HS based on ESEEM (16) is then the more appropriate method. The third condition can be fulfilled whenever T 2n @ Tm by adjusting the length of the chirp pulse or even simpler by starting the incrementation with t § 5Tm 0 tchirp . Alternatively, the phase cycle can be extended to eight steps by applying a (0 / p ) phase cycle on the nonselective MW p pulse. This eightstep phase cycle relaxes the third assumption to Tm ! tchirp / T. If the MW p pulse is not sufficiently nonselective, an additional pathway must be considered that describes the NC remaining after the pulse in the same mS manifold. The contribution of this pathway to the signal cannot be eliminated by phase cycling since all changes of the coherence order are the same as in the wanted pathway. Fortunately, this pathway does not lead to an echo modulation in the 1D experiment, since the NC evolves with the same frequency before and after the nonselective MW pulse during the constant evolution time t / (T 0 t) Å T. The contribution of this pathway to the echo is therefore also constant and can be eliminated by a baseline correction of the time-domain data. The same procedure applied to each trace of the 2D experiment removes this contribution from the 2D spectrum. The fixed length of the whole pulse sequence also accounts for another peculiarity of chirp HS; there is no relaxational decay of the echo modulation. The actual resolution of the hyperfine spectrum is solely determined by the sweep range of time t. The width of the hyperfine lines therefore does not contain any information; in fact, there is no linewidth in the usual sense of the term. The trade-off between resolution and sensitivity of the experiment then becomes very transparent. The required resolution determines the sweep range of t; this in turn determines the minimum T and therefore the sensitivity of the experiment. As a rule of thumb, T / tchirp should be roughly equal to T * 2n , which characterizes the decay of the NC due to both transverse relaxation (T 2n ) and inhomogeneous line broadening. Obviously, the method is not well suited for systems with T * 2n ! tchirp . Since T 2n is usually sufficiently long, the latter condition applies mainly to disordered systems with a large inhomogeneous broadening of the ENDOR lines. Work on a 2D hyperfine spectroscopy more suitable for such systems is in progress. If the hyperfine coupling is anisotropic, also pseudo-secular hyperfine terms of the form BSz Ix must be considered in the effective Hamiltonian in the rotating frame given in Eq. [4]. The difference frequency (in the weak-coupling case) and the sum frequency (in the strong-coupling case) no longer represent the hyperfine coupling; they are, however, identical to the hyperfine splitting that would be observed in a sufficiently resolved ESR spectrum (4, 20). These hyperfine splittings differ somewhat from the hyperfine couplings because of a contribution from the nuclear Zeeman
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interaction. The differences become significant for the case ÉvIÉ É Éa/2É (intermediate coupling), where mS and mI are no longer good quantum numbers. For nuclear spins I ú 12, the nonselective MW p pulse transfers the coherence on the transition characterized by (mS Å 12, mI , mI / 1) to the coherence (mS Å 012, mI , mI / 1), and vice versa. Since the magnetic quantum numbers of the NC are not changed by the p pulse, there is no firstorder contribution of the nuclear-quadrupole interaction to hyperfine frequency. Therefore, to first order, the hyperfine spectrum of a nucleus with spin I again consists only of one peak at the hyperfine frequency. One can show, however, that the hyperfine frequencies depend slightly on mI if higher-order contributions are considered. For example, for a system with S Å 12, I (arbitrary) and assuming the strongcoupling case, the second-order contribution of an isotropic hyperfine coupling a to the frequency of the nuclear transition (mI } mI / 1) is given by (19) da É
0a 2 (mI / mS / 1/2) . 2v S
[12]
For I Å 12, this second-order contribution vanishes in the hyperfine spectra, whereas for I Å 1, da Å 0 for the (mS Å 1 1 2, 01 } 0) transition and the ( 02, 0 } 1) transition, da Å a 2 /2vS for the ( 12, 01 } 0) transition, and da Å 0a 2 /2vS for the ( 012, 0 } 1) transition. The peaks in the hyperfine spectrum are thus observed at a { a 2 /2vS . Similarly, second-order contributions of the nuclear-quadrupole interaction may result in small shifts of the hyperfine lines for nuclei with spin I ú 12. Finally, we note that also for electron spins S ú 12 the frequencies in the hyperfine spectra are given to first order by a and are thus independent of mS of the EPR observer transition. Here, second-order effects are manifest again as shifts of the hyperfine frequencies. A detailed calculation of second-order hyperfine frequencies in systems with I ú 12 and/or S ú 12 is beyond the scope of this paper and will be published elsewhere. EXPERIMENTAL
All experiments were performed with a homebuilt pulsed X-band spectrometer (21). The ENDOR probehead (22) is equipped with a p circuit for a rough matching of the impedance in the frequency range 5–30 MHz (14). Echo amplitudes have been measured by a boxcar averager (PAR, model 162 with gated integrator model 165). The chirp pulses were created by a LeCroy 9100 arbitrary function generator (AFG) with a digital time resolution of 5 ns. Shaped chirp pulses were used to avoid wiggles in the hyperfine spectrum. The envelope of the chirp pulses was a sine quarterwave in the first 25% of the RF pulse length, a cosine quarterwave in the last 25% of the pulse, and constant in
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between (compare Fig. 1a). However, it is also possible to use rectangular chirp pulses. In the Davies–ENDOR and the chirp HS experiments, the length of the first MW pulse was 100 ns; the pulse lengths in the echo-detection subsequence p /2— t – p were 100 and 200 ns. Interpulse delays t of 300 and 340 ns were used for the Davies–ENDOR and chirp HS experiments, respectively. The RF pulse length was 25 ms in the Davies–ENDOR experiment, and the RF was stepped with 50 kHz increments. In the chirp HS experiments, the RF was linearly varied from 3 to 23 MHz during the chirp pulse with a length of 5 ms. The length of the nonselective MW p pulse in chirp HS experiments was 20 ns. Time increments of 20 ns for t and 25 ns for T (compare Fig. 1) were used. The four-step phase cycle (0, 0 0 p, 0 0 0, p, /p, p ) of the chirp pulses was applied in the chirp HS experiments. The phase cycling was performed by recalling inverted waveforms from the AFG, an approach that requires only one RF channel. In the 1D chirp HS experiment, six traces with 600 data points each were sampled at different T values to test the blind-spot behavior. The time-domain data were apodized with a Hamming window, zero-filled to 2K data points, and Fourier transformed, and the magnitude spectra were calculated. Summation of the six traces yields a hyperfine spectrum which is virtually free of blind spots. A data array of 128 1 128 points was recorded in the 2D hyperfinecorrelated chirp ENDOR experiment. The raw data were baseline corrected by third-order polynomials in both dimensions and Fourier transformed, and the magnitude spectrum was calculated. All experiments were performed on a single crystal of triglycine sulfate (TGS) doped with 0.1 mass% copper(II) sulfate. A crystal orientation was used where the ESR lines of the two magnetically inequivalent sites were clearly separated from each other. All spectra were measured at a temperature of 15 K and a static magnetic field of 284.1 mT, corresponding to a proton Zeeman frequency of 12.10 MHz. RESULTS AND DISCUSSION
The two glycine molecules coordinated to the copper ion in Cu II-doped TGS are not equivalent, as detailed investigations of the orientation dependence of the proton ENDOR frequencies have shown (23). Therefore, for a single site, one expects eight nitrogen ENDOR lines, eight ENDOR lines each for the four NH2 and four CH2 protons, and a number of weakly coupled matrix protons. A Davies–ENDOR spectrum of Cu(II)TGS at arbitrary crystal orientation is shown in Fig. 2a. The RF pulse length of 25 ms is sufficient to avoid excessive power broadening. Nevertheless, only a few of the expected lines are resolved so that the spectrum does not allow for an unambiguous assignment of the lines and a precise measurement of the transition frequencies. A time-domain data trace of the chirp HS experiment is shown
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FIG. 2. Davies–ENDOR spectrum (a) and chirp hyperfine spectrum (sum of six traces with different interpulse delays T ) (b) of a Cu(II)doped TGS single crystal at arbitrary crystal orientation and a temperature of 15 K. Bars mark the assignment of the spectral lines to the different groups as obtained from the hyperfine spectrum.
in Fig. 3. As predicted by theory, no relaxational decay of the modulation is observed. The corresponding hyperfine spectrum shown in Fig. 2b is significantly better resolved than the ENDOR spectrum for four reasons: (1) The number of spectral lines is reduced by a factor of two (protons) and a factor of four (nitrogens). (2) The frequency differences between the lines in the hyperfine spectrum are twice as large as between the corresponding ENDOR transitions. (3) Power broadening is completely absent in HS. (4) The spectrum extends over a broader frequency range. Since, for the observation of a hyperfine line, the corresponding ENDOR frequencies in both mS states must lie within the excitation band of 3–23 MHz, the hyperfine lines at frequencies nh f ú 20 MHz can unambiguously be assigned to the nitrogens. A comparison of the remaining hyperfine frequencies with the ENDOR frequencies ( nh f õ 10 MHz) reveals that they all must be assigned to protons. The correlation between the hyperfine and the ENDOR spectra therefore can easily be established, and the evaluation of the parame-
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FIG. 3. Time-domain data trace of the hyperfine spectrum shown in Fig. 2b (T Å 7.865 ms). No relaxational decay of the oscillations is observed.
ters (hyperfine and nuclear-quadrupole couplings) becomes straightforward. Since the range of the hyperfine frequencies is known in our experimental example (nitrogens, 13–24 MHz; NH2 protons, 5–15 MHz; CH2 protons, 1–8 MHz) (23, 24), the assignment of the hyperfine lines to protons in the particular groups in the structure is also known (horizontal bars in Fig. 2). In the hyperfine spectrum in Fig. 2b, the two nitrogen lines (21–22 MHz) and the four lines of the NH2 protons (6.5–8 MHz) are nicely resolved. Note that the nitrogen line at 21.3 MHz shows an additional splitting. For an isotropic hyperfine coupling of É21 MHz, a second-order splitting of about 50 kHz is predicted (Eq. [12]) in agreement with the observation. The hyperfine lines of the CH2 protons observed in the frequency range 2–3 MHz are of rather low intensity. This is due to the fact that, in the Davies-based version of the chirp HS experiment, the first MW p pulse with a length of 100 ns cannot create sufficient polarization on the nuclear transitions of weakly coupled nuclei. This problem may be overcome by using Mims-type HS (14), which however suffers from additional blind spots. In our experimental demonstration of HS, the blind spots resulting from the fixed interpulse delay T (Eq. [7]) have been removed by measuring six time-domain traces at different T values and displaying the sum of the six magnitude spectra in Fig. 2b. Three of the single traces are shown in Fig. 4. Although none of the hyperfine lines vanish completely in any of these traces, the relative intensities show a strong dependence on T. This blind-spot behavior may actually result in a resolution of the single traces apparently higher than that of the sum spectrum, since, due to the long interpulse delay T, the blind spots are rather dense and may selectively suppress some of the lines. To check the correlations between the frequencies in the hyperfine and the ENDOR spectrum and to demonstrate the gain in information obtained with a 2D measurement, we performed a hyperfine-correlated chirp ENDOR experiment. The resulting 2D plot is shown in Fig. 5a. As expected from Eq. [11a], the cross peaks of the NH2 protons (weakcoupling case) appear in both quadrants at (7.8, 9.0 MHz)
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and at (7.8, 016.5 MHz). All the cross peaks of the nitrogen nuclei (strong-coupling case) are observed in the second quadrant (negative ENDOR frequencies). Single traces parallel to the ENDOR axis represent ENDOR spectra of only those nuclear transitions that correspond to a given hyperfine splitting: The nuclear-transition frequencies are hyperfine correlated. The traces at the hyperfine frequencies 7.8 MHz (NH2 protons) and 21.9 MHz ( 14N) are shown in Figs. 5b and 5c, respectively. In the latter case, the four-line structure resulting from the nuclear Zeeman and the nuclear-quadrupole interaction can clearly be seen. The 2D spectrum therefore fully supports the assignment of the ENDOR frequencies given above. CONCLUSION
A new type of hyperfine spectroscopy based on the chirp ENDOR approach (14) has been introduced, which allows
FIG. 4. Chirp hyperfine spectra of Cu(II)-doped TGS at the same crystal orientation and temperature as described in the legend to Fig. 2 for different chirp-interpulse delays T (see Fig. 1). (a) T Å 7.815 ms. (b) T Å 7.840 ms. (c) T Å 7.865 ms.
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of the nuclear transitions. We expect that, with the further development of HS, investigations of paramagnetic compounds that cannot be studied with known techniques due to the excessive complexity of their spectra will become feasible. ACKNOWLEDGMENTS The authors thank Professor Rolf Bo¨ttcher for the gift of the copper-doped triglycine sulfate crystal. Technical assistance by Jo¨rg Forrer is gratefully acknowledged. This research has been supported by the Swiss National Science Foundation. G.J. thanks the Stiftung Stipendien-Fonds des Verbandes der Chemischen Industrie (Germany) for a Kekule´ grant.
REFERENCES 1. G. Feher, Phys. Rev. 103, 834 (1956). 2. L. Kevan and L. D. Kispert, ‘‘Electron Spin Double Resonance Spectroscopy,’’ Wiley–Interscience, New York, 1976. 3. M. Dorio and J. H. Freed (Eds.), ‘‘Multiple Electron Resonance Spectroscopy,’’ Plenum, New York, 1979. 4. A. Schweiger, Struct. Bonding (Berlin) 51, 1 (1982). 5. H. Kurreck, B. Kirste, and W. Lubitz, ‘‘Electron Nuclear Double Resonance Spectroscopy in Solution,’’ VCH, New York, 1988. 6. W. B. Mims, Proc. R. Soc. London 283, 452 (1965). 7. P. K. Dinse, in ‘‘Advanced EPR’’ (A. J. Hoff, Ed.), Chap. 17, Elsevier, Amsterdam, 1989. 8. A. Grupp and M. Mehring, in ‘‘Modern Pulsed and ContinuousWave Electron Spin Resonance’’ (L. Kevan and M. Bowman, Eds.), Chap. 4, Wiley, New York, 1990. 9. C. Gemperle and A. Schweiger, Chem. Rev. 91, 1481 (1991). 10. S. A. Dikanov and Yu. D. Tsvetkov, ‘‘Electron Spin Echo Envelope Modulation (ESEEM) Spectroscopy, CRC Press, Boca Raton, 1992. FIG. 5. Two-dimensional hyperfine-correlated chirp ENDOR spectroscopy of Cu(II)-doped TGS at the same crystal orientation and temperature as described in the legend to Fig. 2. (a) Contour plot. (b,c) ENDOR traces at the hyperfine frequencies nh f Å 7.8 MHz (NH2 protons) (b), and nh f Å 21.9 MHz (nitrogen) (c).
one to measure the complete hyperfine spectrum in an 1D experiment. It has been demonstrated by theory and experiment that this technique is distinguished by a drastic increase in resolution compared to ENDOR spectroscopy, which is caused by a reduction in the number of spectral lines. Moreover, ENDOR suffers from assignment ambiguities because of the dependence of the ENDOR frequencies on mS and the hyperfine, the nuclear Zeeman, and the nuclear-quadrupole interactions. In HS, the dependence on mS is eliminated, the dependence on the nuclear Zeeman interaction is greatly reduced, and there is only a second-order contribution of the nuclear-quadrupole interaction to the frequencies. HS therefore directly measures the hyperfine splittings of the nuclear spins with a resolution determined by the linewidth
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11. A. Schweiger, in ‘‘Modern Pulsed and Continuous-Wave Electron Spin Resonance’’ (L. Kevan and M. K. Bowman, Eds.), Chap. 2, Wiley, New York, 1990. 12. R. de Beer, H. Barkhuijsen, E. L. de Wild, and R. P. J. Merks, Bull. Magn. Reson. 2, 420 (1981). 13. Th. Wacker and A. Schweiger, Chem. Phys. Lett. 191, 136 (1992). 14. G. Jeschke and A. Schweiger, J. Chem. Phys. 103, 8329 (1995). 15. G. Jeschke and A. Schweiger, Proceedings of the EMARDIS Workshop 1995, Sofia. 16. M. Hubrich, G. Jeschke, and A. Schweiger, Proceedings of the Chianti Conference on Magnetic Resonance, 1995. 17. J. A. Ferretti and R. R. Ernst, J. Chem. Phys. 65, 4283 (1976). 18. E. R. Davies, Phys. Lett. A 47, 1 (1974). 19. N. M. Atherton, ‘‘Principles of Electron Spin Resonance,’’ Chap. 3.11, Ellis Horwood, New York, 1993. 20. A. Ponti and A. Schweiger, J. Chem. Phys. 102, 5207 (1995). 21. Th. Wacker, Ph.D. thesis, ETH Zu¨rich, No. 9913, 1992. 22. J. Forrer, S. Pfenninger, J. Eisenegger, and A. Schweiger, Rev. Sci. Instrum. 61, 3360 (1990). 23. R. Bo¨ttcher, D. Heinhold, and W. Windsch, Chem. Phys. Lett. 49, 148 (1977). 24. R. Bo¨ttcher, D. Heinhold, S. Wartewig, and W. Windsch, J. Mol. Struct. 46, 363 (1978).
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