73 Rotational-Echo, Double-Resonance NMR Terry Gullion Department of Chemistry, West Virginia University, Morgantown, WV 26506, USA Introduction Rotational-echo, double-resonance REDOR NMR is a high-resolution, solid-state NMR experiment for measuring the dipolar coupling between a heteronuclear spin pair [,2]. The /r 3 distance dependence of the dipolar coupling makes REDOR useful for the structural characterization of solids, and REDOR has become a valuable tool for characterizing a wide range of materials, including peptides and proteins, polymers, zeolites, guest host systems, glasses, and more. Since the REDOR experiment is based primarily on trains of π pulses, it has mostly been used to measure dipolar couplings between pairs of spin /2 nuclei. Under certain conditions, however, the REDOR experiment can also make effective use of quadrupolar nuclei as structural probes. In REDOR experiments, the NMR signal of the observed nucleus is attenuated when dipolar dephasing radio frequency rf pulses are applied to the non-observed nucleus. The dependence of this signal reduction on the dipolar evolution time provides a direct way to determine the dipolar coupling and obtain internuclear distances. Analysis of REDOR data involves the simple measurement of signal intensities and comparing the normalized dipolar dephased signal to a universal dipolar dephasing curve. The universal dipolar dephasing curve depends only on the dipolar coupling, and it is independent of all other NMR parameters such as chemical shift anisotropy and resonance offset. The ease of experimental implementation and straightforward data analysis are properties that make the REDOR experiment particularly attractive for structural characterization of complex molecular systems. Dipolar Recoupling With magic angle spinning MAS, the time-dependent dipolar Hamiltonian for a heteronuclear pair of spin /2 nuclei is [3 5] H = dts z I z, dt = d [ sin 2 β cos 2 ω r t α ] 2 sin 2β cos ω r t α. 2 The dipolar coupling, in SI units, is d = µ 0γ S γ I h, 3 πr 3 µ 0 is the permeability of free space, γ S and γ I are the magnetogyric ratios of the coupled spins, h is Planck s constant divided by, and r is the internuclear separation. The orientation of a vector directed between the S and I spins in the rotor frame is defined by the polar angle β and the azimuthal angle α. The sample spinning rate, ω r, is related to the period of the sample rotation, T r, through ω r = /Tr. Inspection of Equation 2 shows that the average heteronuclear dipolar interaction is zero, and the consequence of this averaging is that the dipolar interaction has little effect on observed spectra under typical high-resolution MAS conditions. By toggling the spin states of the S and I spins synchronously with the sample rotation, it is possible to produce an average dipolar Hamiltonian that is not equal to zero and, in effect, recouple the dipolar interaction. Synchronous toggling of spin states with the sample rotation is the principle behind the REDOR experiment. Two experiments are performed for REDOR measurements. An S-spin control signal is generated by omitting I -spin rf pulses. An S-spin dipolar dephased signal is produced by applying I -spin rf pulses. Comparison of the control and dipolar dephased signals provides a way to obtain the dipolar coupling. The concept of the control experiment is illustrated in Figure left, neither the S-channel nor the I -channel contains any rf pulses, and the S and I spin orientations magnetic quantum numbers remain unchanged during the entire rotor cycle. An I spin will generate a local magnetic field, B L, at the site of a neighboring S spin, and the S-spin transverse magnetization will precess at a rate ω S = γ S B L in the rotating frame defined by the Zeeman interaction. In the absence of rf pulses, the physical rotation of the sample about the sample rotor axis averages the local magnetic field and dipolar Hamiltonian to zero according to Equations and 2. The local field, B L,isnegative as often as it is positive. Hence, the dipolar interaction causes no net precession of the transverse component of the S-spin magnetization for the control experiment. The S-spin dipolar dephasing experiment shown in Figure right illustrates the principle of dipolar Graham A. Webb ed., Modern Magnetic Resonance, 73 78. C 2008 Springer.
7 Chemistry then = and the toggling frame Hamiltonian for the second half of the rotor cycle is also H = S z I z. Hence, the average dipolar Hamiltonian for the dipolar dephasing experiment is not equal to zero and is H = S z I z. 0 Fig.. Illustration of the control and dipolar dephasing experiment. Representative spin states are indicated by arrows for each half of the rotor cycle. For the control experiment, no change in spin states occurs. The local field, B L, is that generated by the I spin and appearing at the S spin. Toggling frame Hamiltonians are shown for the first and second half of the rotor cycle for the dipolar dephasing experiment. recoupling. A π pulse is applied to the I -channel at the midpoint of the rotor cycle. From time 0 to T r /2, the toggling frame Hamiltonian is [6] H = S z I z, [ Tr/2 ]/[ Tr/2 ] = dtdt dt 0 0 which becomes = 2 2 d sin 2β sin α. 6 π The toggling frame Hamiltonian during the time following the I -channel π pulse is Since 5 H = S z I z, 7 [ Tr [Tr / ] = dtdt]/ 2. 8 T r/2 Tr 0 dtdt = 0, 9 The consequence of toggling the I spin from the perspective of the S spin is also illustrated by the local field, B L. During the second half of the rotor cycle, the sign of B L is opposite to that found in the control experiment for the same time period because of the sudden change in the spin state of the I spin caused by the π pulse. Consequently, the average local field dashed line experienced by the S spin is no longer zero, as it was in the control experiment, and the S-spin transverse magnetization undergoes a net nonzero dephasing leading to signal attenuation. Each orientation of the S I spin pairs in the powder sample contributes a dipolar dephased signal, s d,tothe total powder signal. This contribution is [,2,7] s d = cos ω d τ, ω d = 2 2D sin 2β sin α, 2 and D = d/ has units of Hz and τ is the dipolar evolution time in seconds. A powder sum over the angles α and β provides the normalized dipolar dephased signal intensity, S d, which is S d = Practical Details β β α α / cos ω d τ sin β dα dβ sin β dα dβ 3 Figure 2 shows three practical REDOR pulse sequences designed to increase the dipolar evolution period over multiple rotor cycles [8]. In order to maintain the average dipolar Hamiltonian provided by Equation 0, two π pulses per rotor cycle are required and the timing between any two adjacent π pulses is T r /2. The dipolar evolution time, τ, is the rotor period multiplied by the number of rotor cycles, N c, during the dipolar evolution period: τ = N c T r. If the pulse spacing between adjacent π pulses differs from T r /2, then Equation 3 is still applicable but Equation 2 takes on a slightly different form. For most work, the best pulse spacing is T r /2.
Rotational-Echo, Double-Resonance NMR Practical Details 75 Table : Phases of the xy- cycle and its supercycles Cycle xy- xy-8 xy-6 rf phases xyxy xyxy yxyx xyxy yxyx xyxy yxyx Fig. 2. Three commonly used REDOR pulse sequences. Each is shown for a dipolar evolution period of 0 rotor cycles. If protons are not available for cross-polarization CP, then the S-channel CP pulse is replaced with a simple π/2 pulse. The original REDOR pulse sequence is shown in Figure 2 top. Protons, if present, are used to enhance the S-spin magnetization through cross-polarization CP, and the protons are decoupled thereafter. A single S-spin π pulse is applied at the midpoint of the dipolar evolution period to refocus isotropic chemical shifts and produce an echo at the start of data acquisition. If no I -channel pulses are applied, then the S spins undergo no net dipolar dephasing and the detected S-spin signal serves as a control signal, S, which is used to account for T 2 type losses. The control signal is also referred to as the full signal. Application of the I -channel π pulse train recouples the dipolar interaction and produces dipolar dephasing of the S-spins according to Equation 3. The experimental dipolar dephased signal, S r, also referred to as the reduced signal will be less intense than the control signal. The dipolar coupling can be determined by the measured ratio S/S m = Sr. S The REDOR pulse sequence shown in Figure 2 middle provides better overall performance for S-spin refocusing and dipolar dephasing since it is more tolerant to rf pulse imperfections. Experimental problems with rf pulses arise from limited rf power, rise and fall times of pulses, phase glitches, and resonance offsets. The xy-, xy-8, or xy-6 phase cycling schemes should be applied to the π pulses for all three pulse sequences to eliminate problems associated with pulse imperfections [9,0]. In most cases, the xy-8 supercycle is adequate, but the xy-6 and xy-32 supercycles may offer minor improvements. The phases of the xy-, -8, and -6 supercycles are shown in Table. The REDOR pulse sequence shown in Figure 2 bottom has been used whenever the I spin has a very large anisotropic interaction either very large chemical shift anisotropies or modest quadrupolar interactions. It has been particularly useful for I = 2 H, especially when the lone I -spin pulse is a composite π/2 pulse []. This particular pulse sequence is very sensitive to the spinning rate, which should be stabilized to within a fraction of a Hz [2,3]. The other two pulse sequences do not require such a high degree of control of the spinning rate. Examples of REDOR spectra are shown in Figure 3, and full experimental details can be found in Ref. []. The observed spin is 3 C and the dipolar dephasing spin is 2 H. The sample is a 30-residue AlaGly 5 peptide having repeated β turns. The carbonyl carbon of Gly is 3 C enriched and a deuteron is attached to the C α carbon of Ala7. The 3 C 2 H REDOR full spectrum bottom was obtained by omitting the 2 H rf pulse and the carbonyl 3 C resonance intensity provides the signal S. The reduced, or dipolar dephased, 3 C spectrum is shown in the middle. It is evident that the signal intensity at the carbonyl 3 C resonance position has been attenuated, and the measured intensity of the carbonyl 3 C resonance produces S r. The difference spectrum full minus reduced is shown at the top and the measured signal intensity of the carbonyl 3 C resonance position produces S. The measured ratio S/S m is obtained by the simple measurement of signal intensities S and S r or S. Dipolar couplings are obtained by comparing the measured S/S m to S/S = S d from average Hamiltonian theory. Equations 2 and 3 provide S d for S = /2, I = /2 spin pairs, and the ideal S/S is shown in Figure it is plotted against the dimensionless
76 Chemistry Ala5 Gly6 Gly A Ala7 difference S/S reduced full 300 200 00 0 00 ν 3 C ppm Fig. 3. Example of full, reduced, and difference 3 C 2 H REDOR spectra. Experimental details can be found in Ref. []. parameter λ = N c T r D. There are two easy ways to generate S/S plots. The first is to perform a powder sum over all s d to produce S d, which is Equation 3. The second method is to calculate S d using Bessel functions of the first kind, and for S = /2, I = /2 spin pairs [5] S d = J / J /. The Bessel function method has the advantage of speed of calculation and, if the data set is large enough, may be used to produce dipolar frequencies via the REDOR transform. REDOR experiments can also be performed with quadrupolar nuclei, I > /2 [6 22]. Table 2 provides a summary of normalized dipolar dephased signals, S d, in terms of Bessel functions and s d in terms of cosine functions for ideal REDOR behavior for S = /2, I > /2 spin pairs. The overall width of the lineshape of a quadrupolar nucleus can be very large, depending on the size of the quadrupolar coupling constant, χ in Fig.. Universal dipolar dephasing curve for an S = /2, I = /2 spin pair. frequency units, χ = e 2 qq/h. Hence, it is difficult to uniformly irradiate the broad I -spin spectrum with typical rf field strengths that are available experimentally. For each value of I in Table 2, an approximate upper limit of χ is provided which was determined by assuming an I -channel rf field strength of 50 khz. As long as χ is less than the indicated value, the experimental data should obey the REDOR dipolar dephasing equations given in Table 2 reasonably well and provide a distance within % of the expected value. Higher values of χ will result in REDOR dephasing of the S spins, but numerical simulations that take into account the rf field strength, resonance offsets, χ, and quadrupolar asymmetry parameter will be necessary in order to extract the dipolar coupling. The functions for I = intable2are primarily for 2 H and using the pulse sequence shown in Figure 2 bottom. In particular, the case the I -spin pulse is a composite π/2 pulse instead of a π pulse has been shown to provide good interatomic distances even for rigid deuterons, which have χ = 67kHz []. For typical rf field intensities, a π pulse on the 2 H channel does not provide ideal dipolar dephasing except for cases there is significant motional narrowing of the 2 H lineshape, such as for the case of methyl rotation χ = 55kHz. This chapter provided an introduction to the REDOR experiment. A more complete description of the theory can be found in Refs. [2,7]. A discussion of experimental effects, such as pulse imperfections and resonance offset, can be found in Ref. [23].
Rotational-Echo, Double-Resonance NMR Practical Details 77 Table 2: Dipolar dephasing functions /2 3/2 χ<70 khz I SdJ sdcos 2 J/ [ J/ J / J / J/ 3 J / 3 cos ωdτ ] 2 [cos ω dτ cos 3 ωdτ] 5/2 χ<0 khz 3 [ J/ J/ 5 J / J / 5 ] J/ 3 J / 3 3 [cos ω dτ cos 3 ωdτ cos 5 ωdτ] 7/2 χ<85 khz [ J/ J/ 5 J / J / 5 J/ 3 J/ J / 7 3 J / 7 ] [cos ω dτ cos 3 ωdτ cos 5 ωdτ cos 7 ωdτ] 2 H; π 3 [ 2 J/ 2 J / 2 ] 3 [ 2 cos 2 ω dτ] 2 H; π/2 6 [ J/ J / J/ 2 J / 2 ] 6 [ cos ω dτ cos 2 ωdτ]
78 Chemistry References. Gullion T, Schaefer J. J. Magn. Reson. 989;8:96. 2. Gullion T, Schaefer J. In: WS Warren Ed. Advances in Magnetic Resonance, Vol. 3. Academic Press: San Diego, 989, p 57. 3. Maricq MM, Waugh JS. J. Chem. Phys. 979;70:3300.. Munowitz MG, Griffin RG. J. Chem. Phys. 982;76:288. 5. Slichter CP. Principles of Magnetic Resonance, 3rd ed. Springer: New York, 989. 6. Ernst RR, Bodenhausen G, Wokaun A. Principles of Nuclear Magnetic Resonance in One and Two Dimensions. Oxford University Press: Oxford, 990. 7. Gullion T. Magn. Reson. Rev. 997;2:83. 8. Garbow JR, Gullion T. Chem. Phys. Lett. 992;92:7. 9. Gullion T, Baker DB, Conradi MS. J. Magn. Reson. 990;89:79. 0. Gullion T, Schaefer J. J. Magn. Reson. 99;92:39.. Gullion T. J. Magn. Reson. 2000;6:220. 2. Chopin L, Rosanske R, Gullion T. J. Magn. Reson. A. 996; 22:237. 3. Hughes E, Gullion T. Solid State Nucl. Magn. Reson. 200;26:6.. Gullion T, Kishore R, Asakura T. J. Am. Chem. Soc. 2003;25:750. 5. Mueller KT, Jarvie TP, Aurentz DJ, Roberts BW. Chem. Phys. Lett. 995;22:535. 6. Sack I, Balazs YS, Rahimipour S, Vega S. J. Am. Chem. Soc. 2000;22:2263. 7. Sandstrom D, Hong M, Schmidt-Rohr K. Chem. Phys. Lett. 999;300:23. 8. Schmidt A, Mackay R, Schaefer J. J. Magn. Reson. 992;96:6. 9. Fyfe C, Mueller KT, Grondey H, Wongmoon KC. Chem. Phys. Lett. 992;99:98. 20. Fernandez C, Lang D, Amoureux JP, Pruski M. J. Am. Chem. Soc. 998;20:2672. 2. Hughes E, Jordan J, Gullion T. J. Phys. Chem. B. 200;05:5887 9. 22. Reichert D, Pascui O, Judeinstein P, Gullion T. Chem. Phys. Lett. 2005;02:3 7. 23. Gullion T. Concepts Magn. Reson. 998;0:277 89.