THE JOURNAL OF CHEMICAL PHYSICS 122, 174504 ͑2005͒
A new mechanism for electron spin echo envelope modulation ͒ John J. L. Mortona Department of Materials, Oxford University, Oxford OX1 3PH, United Kingdom Alexei M. Tyryshkin Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544 Arzhang Ardavan Clarendon Laboratory, Department of Physics, Oxford University, Oxford OX1 3PU, United Kingdom Kyriakos Porfyrakis Department of Materials, Oxford University, Oxford OX1 3PH, United Kingdom S. A. Lyon Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544 G. Andrew D. Briggs Department of Materials, Oxford University, Oxford OX1 3PH, United Kingdom ͑Received 21 January 2005; accepted 18 February 2005; published online 2 May 2005͒
Electron spin echo envelope modulation ͑ESEEM͒ has been observed for the first time from a coupled heterospin pair of electron and nucleus in liquid solution. Previously, modulation effects in spin-echo experiments have only been described in liquid solutions for a coupled pair of homonuclear spins in nuclear magnetic resonance or a pair of resonant electron spins in electron paramagnetic resonance. We observe low-frequency ESEEM ͑26 and 52 kHz͒ due to a new mechanism present for any electron spin with SϾ1/2 that is hyperfine coupled to a nuclear spin. In ͑ ͒ ͑ ͒ our case these are electron spin S=3/2 and nuclear spin I=1 in the endohedral fullerene N@C60. The modulation is shown to arise from second-order effects in the isotropic hyperfine coupling of an electron and 14N nucleus. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1888585͔
I. INTRODUCTION electron spins with different Larmor frequencies͒ results in no modulation effects from this mechanism: The heterospin Measuring the modulation of a spin echo in pulsed coupling energy changes its sign upon application of the re- magnetic-resonance experiments has become a popular tech- focusing pulse ͑because only one spin flips͒, and the magne- nique for studying weak spin-spin couplings. It is used tization is thus fully refocused at the time of echo formation, extensively in the fields of chemistry, biochemistry, and in the same way as in the presence of any other inhomoge- materials science, both in liquids and solids, using nuclear neous magnetic fields. magnetic resonance ͑NMR͒ and electron para- A second electron spin echo envelope modulation ͑ES- magnetic resonance ͑EPR͒.1–3 Two distinct mechanisms EEM͒ mechanism, which does apply to coupled pairs of for spin-echo modulation have been identified in the litera- heterospins, has also been identified.2,3 This mechanism re- ͑ ture. quires anisotropic spin-spin interactions e.g., AzzSzIz ͒ In the first mechanism, a pair of spins, S and I, are +AzxSzIx and is therefore restricted to solids or high- coupled through an exchange or dipole-dipole interaction, viscosity liquids. The modulation arises as a result of ជ ជ “branching” of the spin transitions created by the refocusing J·S·I or J·SzIz, and the echo modulation arises for nonselec- tive refocusing pulses, which flip both coupled spins. The pulse. The resonant spin S precesses with a Larmor fre- magnitude of the echo signal oscillates as cos͑Jt͒, where t is quency that is different before and after the refocusing pulse, the interpulse delay time.4 This is most commonly observed and therefore accumulates an additional phase, which causes ͑ ͒ for coupled pairs of homonuclear spins,5 though it is also oscillations in the echo signal, as cos Ikt , where Ik is the known for pairs of coupled electron spins with identical or spin transition frequency of the nonresonant spin I. The am- similar Larmor frequencies.6,7 plitude of the oscillations depends on the magnitude of the ជ ជ anisotropic hyperfine component. It should be emphasised that a similar coupling, J·S·I, In this paper we demonstrate that contrary to previous between unlike spins ͑including coupling between hetero- belief, echo modulation effects can also be observed for a nuclear spins, hyperfine coupling between electron and heterospin pair coupled by a purely isotropic spin interaction, nuclear spins, and electron-electron coupling between two and we thus identify a new ESEEM mechanism. Our het-
erospin pair is the endohedral fullerene N@C60 in CS2 ͒ a Electronic mail: [email protected] solution, with electron spin S=3/2 interacting through an
0021-9606/2005/122͑17͒/174504/7/$22.50122, 174504-1 © 2005 American Institute of Physics
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͑ ͒ FIG. 1. a EPR spectrum of N@C60 in CS2 at room temperature. Each line ͑ ͒ in the triplet signal is labeled with the corresponding projection MI of the FIG. 2. a Two-pulse ESE decays for N@C60 in CS2 at 190 K measured at 14 N nuclear spin. ͑b–d͒ Zoom-in for each line showing details of the line- the central MI =0 and the high-field MI =−1 hyperfine components of the are due to a hy- EPR spectrum. ͑b͒ The Fourier transform ͑FT͒ of the oscillatory echo decay ͒ء shape structure. The small satellite lines ͑marked with 13 perfine interaction with the natural abundance of C nuclei on the C60 cage. at MI =−1. Measurement parameters: microwave frequency, 9.67 GHz; microwave power, 0.5 W; modulation amplitude, 2 mG; modulation frequency, 1.6 kHz. frequency units͒ is
ជ ជ isotropic hyperfine coupling ͑a·S·I,a=15.8 MHz͒ to the nuclear spin I=1 of 14N. The isotropic hyperfine coupling ជ ជ H = S − I + a · S · I, ͑1͒ lifts the degeneracy of the electron-spin transitions, 0 e z I z leading to a profound modulation of the echo intensity at about 52 kHz. We shall show that this third modulation  ប  ប mechanism is only effective in high-spin electron systems where e =g B0 / and I =gI nB0 / are the electron and ͑ Ͼ ͒ 14 S 1/2 . The N@C60 molecule has an exceptionally long N nuclear Zeeman frequencies, g and gI are the electron ͑ ͒   electron-spin dephasing time T2 =210 s , enabling the ob- and nuclear g-factors, and n are the Bohr and nuclear ប servation of this low-frequency ESEEM for the first magnetons, is Planck’s constant, and B0 is the magnetic time. field applied along the z axis in the laboratory frame. Each ͓ ͑ ͒ ͔ hyperfine line marked in Fig. 1 a with MI =0 and ±1 in- volves the three allowed electron-spin transitions ⌬ II. MATERIALS AND METHODS MS =1 within the S=3/2 multiplet. These electron-spin transitions remain degenerate for M =0, as seen in Fig. 1͑c͒, 8 I High purity endohedral N@C60 was prepared, dissolved but split into three lines ͑with relative intensities 3:4:3͒ for in CS to a final concentration of 1ϫ1015–2ϫ1015/cm3, ͑ ͒ ͑ ͒ 2 MI = ±1, as seen in Figs. 1 b and 1 d . This additional split- freeze-pumped in three cycles to remove oxygen, and finally ting of 0.9 T originates from the second-order hyperfine sealed in a quartz EPR tube. The samples were 0.7–1.4-cm corrections a2 / =26 kHz, and its observation is only long, and contained approximately 5ϫ1013 N@C spins. e 60 possible because of the extremely narrow EPR linewidth Pulsed EPR measurements were done at 190 K using an Ͻ0.3 TinN@C . Similar second-order splittings have X-band Bruker Elexsys580e spectrometer, equipped with a 60 ͑ ͒ been reported for the related spin system of endohedral nitrogen-flow cryostat. In the two-pulse Hahn electron spin 31 ͑ ͒ fullerene P@C60, which has S=3/2 coupled with echo ESE experiments, /2− − − −echo, the /2 and 10 pulse durations were 56 and 112 ns, respectively. Phase I=1/2. ͑ ͒ cycling was used to eliminate the contribution of unwanted Figure 2 a shows two-pulse echo decays measured at free-induction decay ͑FID͒ signals. the central MI =0 and the high-field MI =−1 hyperfine lines. The decay is monotonic for MI =0 and has an exponential ͑ ͒ dependence exp −2 /T2 with T2 =210 s. However, the ͑ decay is oscillatory for MI =−1 and also for MI =+1, III. RESULTS AND DISCUSSION not shown͒—the Fourier transform of the decay reveals two Figure 1͑a͒ shows the continuous-wave EPR spectrum of peaks at frequencies 26 and 52 kHz, as seen in ͑ ͒ N@C60 in CS2 at room temperature. The spectrum is Fig. 2 b . These frequencies correlate closely to the splitting centered on the electron g-factor g=2.0036 and comprises of 26 kHz found in the EPR spectrum in three lines, resulting from the hyperfine coupling to 14N.9 Figs. 1͑b͒ and 1͑d͒, indicating that the two effects have the The relevant isotropic spin Hamiltonian ͑in angular same origin.
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We shall use the spin-density operator formalism to de- Hamiltonian, correct to second order in a/ e, becomes rive the modulation effects for the spin system S=3/2, I=1. The spin-density matrix after our two-pulse echo experiment is given by 0000 3␦ 0 00 2 H = S ͑ + a͒ − I − , ͑4͒ ͑͒ ͑ x x͒ ͑ x x͒† ͑ ͒ 0 z e z I 002␦ 0 = UR2UR1 · 0 · UR2UR1 . 2 3␦ 00 0 Here, 0 is the density matrix at thermal equilibrium; in 2 the high-temperature approximation valid in our experi- 3 ments, 0 can be substituted with a spin operator Sz. The ͑ H ͒ which we can rearrange as evolution operator U =exp −i 0 describes a free evolu- tion of the spin system between the applied microwave pulses, and the spin rotation operators Rx describe spin rota- i ␦ 0 00 tion upon application of the two microwave pulses, i=1,2. ␦ 7␦ 0000 The measured echo intensity is H = S ͩ + a + ͪ − I ͩ + ͪ + I . 0 z e 2 z I 4 0000 z 000␦ V͑͒ =Tr͓͑͒ · D͔. ͑3͒ ͑5͒ ͑ ␦͒ The term −Iz I +7/4 represents a constant energy shift and can be ignored. We move into a resonant rotating frame ͑the coordinate system rotating with the microwave The detection operator D=S P involves the 14N y MI frequency around the laboratory z axis͒. In this frame P mw nuclear-spin projection operator M to selectively detect ͓ ͑ ͔͒ H I the spin Hamiltonian Eq. 5 transforms to − mwSz, such only those spin transitions associated with a specific nuclear- that spin projection MI. In a pulsed EPR experiment, this corre- sponds to performing measurements at the resolved hyper- fine line in the EPR spectrum of N@C and integrating over 3 60 ͩ ͪ⌬ + ␦ 0 00 the echo signal shape to average out oscillating signals from 2 other off-resonance hyperfine lines. The echo derives from a 1 sum of the single-quantum ͑SQ͒ coherences ͑represented by 0 ͩ ͪ⌬ 0 0 ͒ 2 the terms n,n+1 and n+1,n in the density matrix , weighted H = , 0 1 by factors from the detection operator D. 00− ͩ ͪ⌬ 0 H ͓ ͑ ͔͒ 2 Our Hamiltonian, 0 Eq. 1 , has small off-diagonal elements provided by the a͑I S +I S ͒ terms. These terms 3 x x y y 000− ͩ ͪ⌬ + ␦ are often omitted since they are known to contribute only 2 4,11 second-order energy corrections, but are, in fact, directly ͑6͒ responsible for the observed ESEEM. Diagonalisation of Eq. ͑1͒ yields the magnitude of these corrections to be of the ␦ 2 ⌬ ␦ order =a / e, consistent with the splitting observed in where = e +a+ /2− mw is the resonance offset fre- Figs. 1͑b͒ and 1͑d͒. In addition, the isotropic hyperfine inter- quency. The rotation operator Rx =exp͓−i͑H +H ͒t ͔ can be action introduces a small degree of mixing between Iz, Sz i 0 1 pi basis states; however, we find that this small mixing need not simplified by taking into account a finite excitation band- be considered to appreciate the origin of the observed ES- width of the microwave pulses, which are selective upon one hyperfine line in the EPR spectrum. H =gB S /ប, where EEM. With this assumption, our Hamiltonian and all other 1 1 x B is the microwave magnetic field applied along the x axis operators have block-diagonal structures, with nonzero ele- 1 in the rotating frame and t is the duration of the microwave ments only between states with the same M . Transitions pi I pulses. Furthermore, since the bandwidth of B Ӎ4.5 MHz is with simultaneous flip of both electron and nuclear spins are 1 12 large compared to both the intrinsic EPR linewidth of 9 kHz thus forbidden and the evolution of electron spin can be and the second-order splitting of 26 kHz of the outer lines in treated individually for each nuclear-spin manifold. We can the EPR spectrum, all three spin transitions within the elec- therefore avoid the derivation in the full 12ϫ12 Hilbert tron S=3/2 multiplet are equally excited and the space in a general form, and instead reduce the dimension- x respective rotation operator can be approximated as Ri ality to 4ϫ4. The validity of this approximation is confirmed Ӎexp͑−iH t ͒=exp͑−i S ͒, where =gB t /ប is the mi- 1 pi i x i 1 pi below by a rigorous derivation using average Hamiltonian crowave pulse rotation angle. This results in the following theory. The reduced MI = +1 subspace of the diagonalised spin rotation operator for an on-resonance hyperfine line:
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i 3 i 1 3 cos3 i ͩsin3 i + sin ͪ − ͩcos3 i − cos i ͪ − i sin3 i 2 ͱ3 2 2 ͱ3 2 2 2 i 3 i 1 3 i 3 1 3 i ͩsin3 i + sin ͪ ͩcos3 i + 2 cos i ͪ − ͩsin3 i − 2 sin i ͪ − ͩcos3 i − cos ͪ ͱ3 2 2 3 2 2 3 2 2 ͱ3 2 2 Rx = . ͑7͒ i i i 3 i i 3 1 3 i i 3 i − ͱ ͩcos3 − cos ͪ − ͩsin3 i − 2 sin i ͪ ͩcos3 i + 2 cos i ͪ ͱ ͩsin3 + sin ͪ 3 2 2 3 2 2 3 2 2 3 2 2 1 3 i 3 i − i sin3 i − ͩcos3 i − cos i ͪ ͩsin3 i + sin ͪ cos3 i 2 ͱ3 2 2 ͱ3 2 2 2
We need not consider the rotation operator for off- to the measured echo intensity, with relative amplitudes resonance lines, as their excitation will lead to an oscillating 3:4:3. In other words, their sum will yield a constant com- echo signal which is averaged out by selective detection ͑i.e., ponent, and one oscillating with frequency 2␦, with respec- through the detection operator D͒, and therefore does not tive amplitudes 4:6. This is confirmed upon evaluation of contribute to the overall echo signal. Eq. ͑3͒, We evaluate Eq. ͑2͒ for the two-pulse echo experiment /2−−−−echo, and rearrange the terms for the pur- ͑͒ ␦ ͑ ͒ VM =±1 =2+3cos2 , 12 poses of the discussion which follows: I ͑͒ ͑ x ͒ ͑ x ͑ x ͒†͒ ͑ x ͒† ͑ ͒ = URU · R/2 · 0 · R/2 · URU . 8 and is consistent with the observed echo in Fig. 2͑b͒. The x modulation amplitude is deep, and the echo signal can Evaluating the heart of the echo sequence, UR U,is change its sign at the minima. instructive in understanding the source of the observed The effect is illustrated in Fig. 3͑a͒, which shows the modulation. From Eq. ͑7͒, a perfect rotation is phases gained during the “defocusing” period , i.e., free 000− i evolution after the initial pulse 1 and before the refocusing pulse 2. These phases derive from the differences between x 0 0 − i 0 R = , ͑9͒ adjacent elements along the diagonal of the Hamiltonian in 0 − i 0 0 Eq. ͑6͒͑the off-resonance ⌬ is ignored as it is fully canceled − i 0 00 upon echo formation͒. The six SQ coherence elements, shaded in the figure and responsible for echo formation, gain with the resulting echo sequence operator shown below. the phases 0 or ±␦. Upon application of the perfect refocus- 2i␦t 0 00e ing pulse with 2 = , each SQ coherence element uniquely transforms, as shown with arrows, and continues to evolve x 0 01 0 URU =−i ͑10͒ during the “refocusing” period to gain an additional phase 0100 which does not compensate, but instead doubles the initial ␦ e2i t 0 00 phase. Thus, at the time of echo formation the SQ coherences ␦ This indicates that over the course of the experiment, arrive with three different phases 0 and ±2 . Their vector states S=±3/2 pick up a phase of 2␦t with respect to the states S=±1/2, or in other words, the outer SQ coherences ͑ ͒ ␦ 1,2, 2,1, 3,4, and 4,3 oscillate with frequency 2 , while ͑ ͒ the phases of the inner SQ coherences 2,3 and 3,2 remain constant. The initial SQ coherences are provided by the first rota- tion ͑/2͒
0 − iͱ3/2 00 iͱ3/2 0 − i 0 x ͑ x ͒† R/2 · 0 · R/2 = ͱ , 0 i 0 − i 3/2 FIG. 3. Phases gained by the density matrix elements during the two-pulse 00iͱ3/2 0 ESE experiment at MI = +1. Phases caused by off-resonance refocus fully, ͑11͒ and are therefore omitted for clarity. The shaded off-diagonal elements rep- resent single-quantum ͑SQ͒ coherences which generate the echo signal upon refocusing. In ͑a͒ the arrows indicate the transition between the SQ coher- and the measuring weighting factors for each coherence are ͑ ͒ ͱ ͱ ͑ ences caused by a perfect refocusing pulse with 2 = . b shows all pos- 3/2, 1, and 3/2 associated with the detection operator sible transitions ͑spin coherence branching͒ for one SQ coherence element, D͒ Þ . Together these imply that the three coherences contribute caused by an imperfect refocusing pulse with 2 .
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x sum interferes destructively to produce an echo signal the erator Ri and detection operator D have the same block di- magnitude of which oscillates as 2␦ in accordance with Eq. agonal structures described above ͑nonzero elements only ͑ ͒ ͒ 12 . within the same MI subspace . This allows us again to reduce Þ ϫ The case of an imperfect refocusing pulse with 2 is the dimensionality of the Hilbert space to 4 4. ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ shown in Fig. 3 b . In contrast to 2 = with a one-to-one Substituting Eqs. 7 , 14 , and 15 into Eq. 3 , and transformation of each density matrix element, the nonideal after some manipulation, we find the following expressions pulse generates branching of the electron-spin transitions. for the echo amplitude in a general two-pulse sequence 1 Therefore, the SQ coherence element which initially gained − − 2 − −echo for the S=3/2, I=1 spin system. The echo ␦ the phase + during the defocusing period, refocuses into modulation is identical for the two outer MI = ±1 hyperfine three SQ coherences ͑shown with the arrows͒, each accumu- lines, lating different phases during the refocusing period. Thus, at ͑͒ 2 2 ͓ ͑ ͒ ͑ ͒ ␦ the time of echo formation the accumulated phases are 0, VM =±1 = 2 sin 1 sin A0 2 + A1 2 cos +␦, and +2␦. The vector sum of these and other SQ coher- I 2 ences produces a complex interference, with the echo signal ͑ ͒ ␦͔ ͑ ͒ + A2 2 cos 2 , 16 oscillating with two frequencies ␦ and 2␦ as observed in Fig. 2. Thus, the second harmonic ␦ found in the echo modulation where is the result of an imperfect refocusing pulse. 27 ͑ ͒ 2 2 4 2 The preceding physical description provides an intuitive A0 2 =1−6cos + cos , view of the new ESEEM effect. For it to be fully rigorous, 2 2 2 however, we should consider the effect of the mixing of the 2 2 Iz, Sz basis states. This necessarily involves the full 12- ͑ ͒ 2 ͩ 2 ͪ A1 2 = 6 cos 2−3cos , dimensional Hilbert space, and the argument rapidly be- 2 2 comes opaque. Fortunately, the same results can be rigor- ously obtained from the first-order correction in average 3 2 2 2 2 13 A ͑ ͒ = sin ͩ1−3cos ͪ. ͑17͒ Hamiltonian theory ͑AHT͒, which is equivalent to a stan- 2 2 2 2 2 dard perturbation theory approach in the rotating frame. We begin by transforming the original Hamiltonian ͓Eq. The signal is modulated with frequencies ␦ and 2␦, consis- ͑ ͔͒ tent with the experimental observations in Fig. 2. The modu- 1 into a rotating frame of angular frequency mw, defining ⍀ lation amplitudes in Eq. ͑17͒ depend strongly on the rotation e as the deviation from the electron Larmor frequency, ⍀ angles of the microwave pulses. At optimal rotation angles, e = e − mw, =/2, =, Eq. ͑12͒ is recovered. H ⍀ ͓ ͑ ͒ 1 2 0 = eSz − IIz + a SzIz + SxIx cos mwt In contrast to the two outer lines, the echo signal at the ͑ ͔͒ ͑ ͒ central M =0 hyperfine line shows no modulation effects, − SyIy sin mwt . 13 I The oscillatory terms in Eq. ͑13͒ are averaged out in the ͑͒ 2 2 ͑ ͒ VM =0 = 2 sin 1 sin . 18 zeroth-order average Hamiltonian I 2
͑ ͒ H¯ 0 = ⍀ S − I + aS I . ͑14͒ It is instructive to consider which terms in the average 0 e z I z z z H¯ ͑1͒ Hamiltonian 0 give rise to the modulation effects. The H ͑ ͒ ͑ ͒ Here, the bar over 0 refers to an average over one period of terms I I+1 Sz and S S+1 Iz are not responsible as they pro- ͑ ͒ the oscillation mw. As the Hamiltonian in Eq. 14 results in duce only a constant shift to the electron and nuclear Zeeman 2 no modulation of the echo signal for a coupled heterospin frequencies, respectively. The term Iz Sz is also irrelevant be- pair, the higher order terms of the average Hamiltonian must cause electron-nuclear flip-flop transitions are forbidden, ͑ ͒ be included see, for example, Ref. 3 . The first-order correc- hence MI stays invariant during the experiment. This term tion is changes its sign, but not its magnitude, during the refocusing 2 ␦ pulse, and thus fully refocuses. Therefore, Sz Iz is solely re- H¯ ͑1͒ ͓͑ ͑ ͒ 2͒ ͑ ͑ ͒ 2͒ ͔ ͑ ͒ 0 = I I +1 − Iz Sz − S S +1 − Sz Iz . 15 sponsible for the modulation effects. For MI =0, this last 2 term becomes zero, and consequently there are no modula- ͑ ͒ ͑ ͒ tion effects produced at the central hyperfine line in the EPR We find that the average Hamiltonian H¯ =H¯ 0 +H¯ 1 is suf- 0 0 0 spectrum. Note that this term is responsible for the spin- ficient to describe the modulation effects. In this approach dependent shifts to the energies of the spin states used in the the time-dependent mixing terms in the spin eigenstates have ͓ ͑ ͒ ͑ ͔͒ ͑ earlier derivation see Eqs. 4 and 5 . been properly accounted for to produce after time averag- ¯ ͑1͒ The effect of a non- refocusing pulse angle is shown in ing͒ the second-order energy corrections in H . However, ͑ ͒ 0 Fig. 4. As predicted by Eq. 16 , we observe that when 2 these mixing terms appear to average to zero, with the result =2/3, the modulation effects are dominated by the low- H¯ that the effective Hamiltonian 0 is a diagonal matrix in the frequency ␦, rather than the high-frequency 2␦ found when I ,S basis, thus validating our earlier qualitative approach. z z 2 = . It can also be verified that in the presence of the applied Equation ͑16͒ also confirms that a perfect refocusing microwave field, the average Hamiltonian is, to first order, pulse yields only a 2␦ frequency component in the modula- H¯  ប ␦ the simple sum 0 +g B1Sx / . Therefore, the rotation op- tion, however, a component is clearly observed in Fig. 2.
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measurement of the hyperfine constant in a continuous-wave ͑ EPR measurement is subject to B0 field instability typically Ͼ10 mG͒. However, the ESEEM frequency can be measured accurately, given a sufficiently long dephasing time, poten- tially providing a more precise measurement. In this case accuracy may be improved by moving to lower applied mag- netic fields ͑lower EPR frequency͒. In contrast with other types of ESEEM, this would also lead to a higher modulation frequency. Finally, we note that given appropriate electron- nuclear spin coupling energies and decoherence times, this effect will also lead to a modulation in nuclear-spin echo experiments. Note added in proof: It has come to our attention that similar results were observed by C. Knapp and described in a thesis which can be obtained from K.-P. Dinse at TU Darm- stadt, Germany.
FIG. 4. Two-pulse ESE decays ͑a͒ and their Fourier transform spectra ͑b͒ measured at hyperfine line MI =−1 of the EPR spectrum, using the refocus- ing pulse 2 = and 2 =2 /3. Other experimental conditions are the same as in Fig. 2. ACKNOWLEDGMENTS The imperfection is explained by the inhomogeneity of the We would like to thank Wolfgang Harneit’s group at the microwave magnetic field B1 in the resonator cavity, which results in a distribution of spin rotation angles 2 across the Hahn–Meitner Institute for providing nitrogen-doped ensemble. If we assume a Gaussian distribution of rotation fullerenes, John Dennis at Queen Mary’s College, London, angles, the relative intensities of the low- and high-frequency and Martin Austwick and Gavin Morley for the purification
components I1 /I2 =0.17 in the experimental spectrum corre- of N@C60. A Foresight LINK grant Nanoelectronics at the sponds to a standard deviation of =0.31 rad. This corre- quantum edge, an EPSRC grant, and the Oxford-Princeton sponds to a 10% error in a rotation angle, consistent with Link fund supported this project. We thank Brendon Lovett previously reported values for B1 field inhomogeneity in this 14 for valuable discussions. A.A. is supported by the Royal So- resonator cavity. To verify that the B1 field inhomogeneity ciety. Work at Princeton was supported by the NSF Interna- is the source of the low-frequency component, we applied an tional Office through the Princeton MRSEC Grant No. error-correcting composite -pulse ͑type BB115͒ as the refo- DMR-0213706 and by the ARO and ARDA under Contract cusing pulse. The resulting ESEEM contained only the single No. DAAD19-02-1-0040. 52-kHz frequency component ͑the 26-kHz component was at least a factor of 0.03 smaller and fell below the noise level͒. 1R. R. Ernst, G. Bodenhausen, and A. Wokaun, in Principles of Nuclear The derivation of the modulation is easily generalised to Magnetic Resonance in One and Two Dimensions, The International Se- the case of an arbitrary electron spin SϾ1/2 coupled ries of Monographs on Chemistry ͑Clarendon, Oxford University Press, through an isotropic hyperfine interaction to a magnetic New York, 1987͒, Vol. 14. nucleus. Using an approach similar to that described in Eqs. 2S. A. Dikanov and Y. D. Tsvetkov, Electron Spin Echo Envelope Modu- ͑ ͒ ͑ ͒ lation (ESEEM) Spectroscopy ͑CRC, Boca Raton, FL, 1992͒. 9 – 11 , the general expression for two-pulse ESEEM with a 3 perfect refocusing pulse can be shown to be A. Schweiger and G. Jeschke, Principles of Pulse Electron Paramagnetic Resonance ͑Oxford University Press, London, 2001͒. S 4A. Abragam, The Principles of Nuclear Magnetism ͑Clarendon, Oxford, ͑ ͒ ␦ ͑͒ ͑ ͒͑ ͒ i 1+2MS MI ͑ ͒ 1961͒. V = ͚ S − MS S + MS +1 e . 19 5E. L. Hahn and D. E. Maxwell, Phys. Rev. 88, 1070 ͑1952͒. MS=−S 6V. F. Yudanov, K. M. Salikhov, G. M. Zhidomirov, and Y. D. Tsvetkov, ͑ ͒ The summation is over electron-spin projections MS, while Teor. Eksp. Khim. 5, 663 1969 . 7A. D. Milov and Y. D. Tsvetkov, Dokl. Akad. Nauk SSSR 288,924 the nuclear-spin projection MI identifies the hyperfine line of ͑1986͒. the EPR spectrum in which the modulation effects are ob- 8 served. M. Kanai, K. Porfyrakis, G. A. D. Briggs, and T. J. S. Dennis, Chem. Commun. ͑Cambridge͒ (2004) pp. 210–211. 9T. Almeida-Murphy, T. Pawlik, A. Weidinger, M. Hohne, R. Alcala, and J. M. Spaeth, Phys. Rev. Lett. 77,1075͑1996͒. 10 IV. CONCLUSIONS C. Knapp, N. Weiden, K. Kass, K. P. Dinse, B. Pietzak, M. Waiblinger, and A. Weidinger, Mol. Phys. 95, 999 ͑1998͒. 11 ͑ Potential applications of this new mechanism include C. P. Slichter, Principles of Magnetic Resonance, 3rd ed. Springer, Berlin, 1996͒. measuring the hyperfine coupling constant and determining 12These flip-flop transitions are allowed only to third order—the transition the electron-spin number; it may also be relevant to certain probability is proportional to a2 /3, and thus is negligibly small at a 16 e quantum information processing schemes. The accurate =15.8 MHz for N@C60.
Downloaded 18 Nov 2005 to 129.67.85.146. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 174504-7 Electron spin echo envelope modulation J. Chem. Phys. 122, 174504 ͑2005͒
13U. Haeberlen and J. Waugh, Phys. Rev. 175, 453 ͑1968͒. responsible for the observed ESEEM must be taken into account when 14J. J. L. Morton, A. M. Tyryshkin, A. Ardavan, K. Porfyrakis, S. A. Lyon, designing pulse sequences to perform a quantum algorithm. However, it and G. A. D. Briggs, Phys. Rev. A 71, 012332 ͑2005͒. could also be exploited to provide a separate family of gates for perform- 15 H. K. Cummins, G. Llewellyn, and J. A. Jones, Phys. Rev. A 67, 042308 ing operations between sublevels, increasing the potential of N@C60 as a ͑2003͒. single quantum bit. 16 17 ͑ ͒ The N@C60 molecule has been proposed as an electron spin-based qubit W. Harneit, Phys. Rev. A 65, 032322 2002 . in several quantum information processing schemes.17,18 At the very least, 18A. Ardavan, M. Austwick, S. C. Benjamin et al., Philos. Trans. R. Soc. the slow evolution within the sublevels of the S=3/2 system, which is London 361, 1473 ͑2003͒.
Downloaded 18 Nov 2005 to 129.67.85.146. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp