J. Phys. Chem. B 2006, 110, 19985-19989
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Excitation of Magnetization Using a Modulated Radiation Damping Field† Jamie D. Walls, Susie Y. Huang, and Yung-Ya Lin* Department of Chemistry and Biochemistry, UniVersity of California, Los Angeles, Los Angeles, California 90095 ReceiVed: April 14, 2006
In this work, pulsed-field gradients are used to modulate the radiation damping field generated by the detection coil in an NMR experiment in order that spins with significantly different chemical shifts can affect one another via the radiation damping field. Experiments performed on solutions of acetone/water and acetone/ DMSO/water demonstrate that spins with chemical shift differences much greater than the effective radiation damping field strength can still be coupled by modulating the radiation damping field. Implications for applications in high-field NMR and for developing sensitive magnetization detectors are discussed.
1. Introduction In NMR spectroscopy, radiation damping results from the back-reaction of the inductive coil upon the sample magnetization. After an initial excitation pulse, the precessing transverse magnetization generates a current in the detection coil (which is the basis for conventional NMR detection). This induced current in turn creates an additional radio frequency (RF) field, that is, the radiation damping field BRD(t), which acts back upon the sample and rotates the magnetization toward the equilibrium +zˆ direction (taken to be along the direction of the large static magnetic field used in the NMR experiment). This leads to a damping of the free induction decay (FID), which typically results in a broadened spectrum. Although radiation damping has been investigated since the early days of NMR,1-3 it is only with the advent of NMR spectrometers being routinely operated in fields greater than 10 T and with the use of more sensitive probes that radiation damping effects have started to “creep” into everyday experiments.4,5 Radiation damping has been the source of spectral artifacts in multidimensional experiments5-7 and has prompted the design of selective pulses that are compensated for radiation damping.8,9 The radiation damping field arises from the precession of the total transverse magnetization. If more than one chemical species is present, the magnetization of different species could couple to one another via the radiation damping field. This coupling has been used to perform selective excitation in solution NMR,10 and a sensitive magnetization detector utilizing the radiation damping coupling between spin species has been proposed.11 However, as has been noted before,5 if the chemical shift difference ∆ω between two spin species is much greater than the effective radiation damping field strength, i.e., |∆ω| . |γBRD| where γ is the gyromagnetic ratio, then the coupling between these species via the radiation damping field can be neglected. Note that experimentally controlled feedback between spins with large Zeeman shift differences has been performed in atomic systems.12 In this paper, a method for coupling spins via the radiation damping field is demonstrated for systems in which the chemical shift difference between the various spin species is greater than †
Part of the special issue “Charles B. Harris Festschrift”. *
[email protected].
the effective radiation damping field, i.e., |∆ω| . |γBRD|. This is accomplished by modulating the sample magnetization (and hence BRD(t)) using pulsed-field gradients in order to couple different spin species through the radiation damping field. Experiments performed on acetone/water and acetone/dimethyl sulfoxide (DMSO)/water solutions are presented which demonstrate the proposed methodology. Theory The Bloch equations in the rotating frame precessing with Larmor frequency ω0 ) γB0 in the presence of radiation damping2,4 are given by
My(t) Mz(t) - Meq Mx(t) dM(t) xˆ yˆ zˆ ) γM(t) × B(t) dt T2 T2 T1 (1) where T1 and T2 are the longitudinal and transverse relaxation times, Meq is the equilibrium magnetization, and B(t) is the effective magnetic field in the rotating frame. In the presence of radiation damping, B(t) can be written for a single isochromat (i.e., magnetization precessing at a single frequency, for example, ωi) as γB(t) ) δωizˆ + γBRD(t), where δωi ≡ ωi ω0, and the radiation damping field, BRD(t), is given by
γBRD(t) 1 ) {[Mx(t) sin(ψ) - My(t) cos(ψ)]xˆ + 2π M0τr [Mx(t) cos(ψ) + My(t) sin(ψ)]yˆ } (2) ψ is the phase of the radiation damping field,5,13,14 which depends on the tuning of the probe (for a perfectly tuned probe, ψ ) 0°, and the radiation damping field is perpendicular to the net transverse magnetization, which is the case considered here). The radiation damping time constant τr is defined here as (in CGS units)
1 ) M0Qγη τr
(3)
where Q is the coil quality factor, η is the filling factor, and M0 is some reference magnetization, often taken to be the equilibrium magnetization, Meq (in the rest of this paper, M0 will be
10.1021/jp062319x CCC: $33.50 © 2006 American Chemical Society Published on Web 06/06/2006
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Figure 1. Minimum value of [Mβz (t)/M0] as a function of chemical shift difference, ∆νRβ ≡ ∆ωRβ/2π, found by solving eq 1 with initial conditions Mβ(0) ) M0zˆ and MR(0) ) M0xˆ over an evolution time t ) 200 ms, with τr ) 10 ms, ψ ) 0°, and neglecting T1 and T2 relaxation. The initial magnitude of the radiation damping field, |γBRD(0)|/(2π), is equal to 100 Hz and is denoted by the dashed line. Mβ is only significantly affected by MR via BRD when |γBRD| g |∆ωRβ|.
chosen to be the equilibrium magnetization for pure water). As stated in the Introduction, BRD(t) tracks and rotates M(t) back to the equilibrium +zˆ direction for a single isochromat. Since BRD(t) depends on M(t) in eq 2, the resulting Bloch equations (eq 1) are nonlinear. As is often the case, a sample typically consists of more than one isochromat, M(t) ) ∑kMk(t), where each species k has a resonance frequency offset δωk ≡ ωk - ω0 in the rotating frame. Examples include a sample of a single chemical species in the presence of field inhomogeneities15-17 or a sample with more than one chemical species, each possessing a different chemical shift. The resulting radiation damping field is given by
γBRD(t)
)
1 M0τr
2π
∑k {[Mkx(t) sin(ψ) - Mky(t) cos(ψ)]xˆ + [Mkx(t) cos(ψ) + Mky(t) sin(ψ)]yˆ }
≡
γBRDk(t)
∑k
2π
(4)
where again the radiation damping field is “normalized” by a reference magnetization, M0. Eq 4 implies that the various magnetization isochromats, Mk(t), can couple to one another via BRD(t). To see this coupling more explicitly, consider the evolution under radiation damping (eq 1) for a solution containing an equal concentration of two isochromats, MR(t) and Mβ(t), which precess at ωR and ωβ, respectively. Following a selective 90°yˆ pulse on MR, the initial conditions are MR(0) ) M0xˆ and Mβ(0) ) M0zˆ. At t ) 0, BRD(0) only depends on MR(0) and begins to rotate MR back toward the +zˆ direction while tipping Mβ away from the zˆ axis. However, the degree of excitation of isochromat β depends on the chemical shift difference5 between the two isochromats, ∆ωRβ ) ωR - ωβ, which is demonstrated in Figure 1, where the minimum value of Mβz (t)/M0 over an evolution time t ) 200 ms is plotted as a function of ∆ωRβ. As can be seen, Mβ(t) remains closely aligned with the zˆ axis when ∆νRβ ≡ ∆ωRβ/2π . τr-1. Therefore, MR(t) and Mβ(t) can be considered to evolve independently of one another if the maximum value of |γBRRD(tRmax)| and |γBβRD(tβmax)| is much less R(β) than |∆ωRβ|, where tR(β) max is the time at which |BRD | reaches its maximum value. Under the condition |∆ωRβ| . γ|BRD|, the
Figure 2. (A) Pulse sequence for coupling different spin species via the radiation damping field. An initial selective pulse is used to excite one of the spin species. Pulsed-field gradients are then applied to periodically modulate the radiation damping field, BRD(t). If 2τ∆ωRβ ) 2πm, where m is an integer, then MR and Mβ can efficiently couple via BRD(t). After application of the pulsed-field gradients, the FID is then acquired. (B) Simulation of |γBRD(t)/(2π)| (using eq 1) for a twocomponent mixture of spin species with initial conditions MR(0) ) M0xˆ and Mβ(0) ) 0.10M0zˆ, with a chemical shift difference of ∆ωRβ/(2π) ) 1.5 kHz, τr ) 10 ms, and ψ ) 0°. This gives |γBRD(0)|/(2π) ) 100 Hz , ∆ωRβ/(2π). In the simulation, a sample length of x2 cm was divided into 1000 voxels, and a gradient strength of 5 × 10-4 T/cm was chosen. Application of the pulsed-field gradients periodically modulates BRD(t) at a frequency of (2τ)-1.
effective radiation damping field generated by species MR, BRRD(t), acts as a weak, off-resonance pulse that does not appreciably rotate Mβ. On the other hand, BRRD(t) appears to MR as a weak, on-resonance pulse that effectively tilts MR back to the +zˆ axis. However, if BRRD(t) is additionally modulated at the chemical shift difference, ∆ωRβ, then a part of BRRD(t) should be effectively “on-resonance" with Mβ and could therefore effectively excite Mβ. There are a variety of methods that can modulate BRD(t) at the desired frequencies in order to couple different spin species via BRD(t). Since BRD(t) is proportional to the Q of the detection coil (eq 3), the detection coil’s Q could be partially modulated at the frequency difference. Modulating the Q in this manner requires specially designed probes/ hardware18-20 and will not be considered further here. Since BRD(t) depends on M(t), an alternative method involves applying multiple RF pulses to modulate and control the magnetization.21 However, the “simplest” pulsed method of applying 180° pulses to effectively refocus the chemical shift difference between the spins also tends to refocus the radiation damping effects. An alternative to applying multiple RF pulses is using pulsedfield gradients to manipulate BRD(t). Pulsed-field gradients, Bgrad(r, t) ) [G(t)‚r]zˆ, are routinely used in high-field NMR in order to suppress radiation damping22,23 by dephasing the magnetization such that 〈Mx〉 ) 〈My〉 ) 0 and therefore BRD ) 0. A simple sequence utilizing pulsed-field gradients to effectively couple different spin species via the radiation damping field is illustrated in Figure 2A. The sequence works as follows: a selective 90° pulse is first applied to excite one of the spin species, for example, species R. A gradient pulse is then turned on that dephases the transverse magnetization, thereby turning off the radiation damping field. After a given time τ, the sign of the gradient is switched, which after an additional time τ refocuses the transverse magnetization, thereby turning on the radiation damping field again. However, the gradient continues to dephase the transverse magnetization, and this process is repeated. Figure
Coupled Spins via Radiation Damping Field
J. Phys. Chem. B, Vol. 110, No. 40, 2006 19987
2B shows a simulation (using eq 1) of |BRD(t)| for the sequence in Figure 2A. In this case, BRD(t) acts like a series of small flip-angle pulses separated by the time interval 2τ. If 2τ∆ωRβ ) 2πm, where m is an integer, then MR and Mβ can couple via the radiation damping field. This pulse sequence is analogous to the DANTE excitation sequence,24 although the tiny flipangle pulses in the present scheme are not hard pulses but rather arise from the radiation damping field. After repeating the pulsed-field gradients, the FID, 〈Mx(t) + iMy(t)〉, is acquired. If there is no coupling via the radiation damping field between different species, the only spectral resonances observed in the FID should come from the initially excited species. However, if radiation damping is able to couple different spin species in the sample, then other spectral resonances should appear in the resulting FID. Experimental Methods All experiments were performed on a 600 MHz AVANCE spectrometer (static magnetic field of 14.1 T and an operating frequency for 1H of 600.13 MHz), using a 5-mm Bruker TXI probe. The observed radiation damping time constant for the probe was τr ) 8 ms (where M0 in eq 3 was taken to be the total equilibrium magnetization for pure water in an equivalent sample volume). Gradient shimming was used to reduce field inhomogeneities. An acetone/water solution ([acetone] ) 1.6 M, giving the ratio of proton magnetizations to be |Macetone|/ |Mwater| ≈ 0.1) and an acetone/DMSO/water solution ([acetone] ) [DMSO] ) 1.2 M, giving the ratio of proton magnetizations to be |Macetone(DMSO)|/|Mwater| ≈ 0.08) were used to test the pulse sequence in Figure 2A. Rectangular gradients were used in all experiments, with a gradient strength of |G| ≈ 7.5 × 10-5 T/cm. A delay of τd ) 10 µs was placed between gradient-switching, thereby modulating BRD(t) at a frequency of (τeff)-1, where τeff ) 2τ + τd. The number of pulsed-field gradient repetitions n (given in Figure 2A) was set to n ) 12. For n > 12, the intensity of the magnetization tended to decrease, which is most likely due to incomplete magnetization refocusing due to diffusion during the pulsed-field gradients. A Gaussian-shaped pulse sampled by 1000 points (total pulse duration of 15 ms and with pulse power at 50 dB attenuation) was used to selectively excite the water magnetization in all experiments. Results and Discussion Excitation of magnetization using a modulated radiation damping field was first tested on a two-component solution of acetone/water. Figure 3A gives the spectrum following a hard 90° pulse. The observed frequency difference between the acetone and water resonances was ∆νwater,acetone ) 1527 Hz. The spectrum obtained after selective excitation of the water resonance is shown in Figure 3B. Very little acetone was excited by the selective pulse. Furthermore, |γBRD(0)/(2π)| ≈ 111 Hz, which was much smaller than ∆νwater,acetone; thus, acetone was not efficiently excited by the radiation damping field generated by the water magnetization (Figure 1). Using the sequence in Figure 2A, however, BRD(t) could be modulated in order to excite the acetone magnetization. Figure 3C shows the resulting spectrum after applying the sequence in Figure 2A for τ ) 324 µs, which effectively modulated BRD(t) at a frequency of (τeff)-1 ) 1520 Hz (where τeff ) 2τ + τd). In Figure 3C, about 44% of the acetone magnetization was excited. In principle, |Mwater|/
x|Mwater|2+|Macetone|2
≈ 99% of the acetone magnetization should be able to be excited; however, field inhomogeneities, diffusion, and relaxation tend to decrease the maximum achiev-
Figure 3. Spectra demonstrating excitation of acetone by modulating the radiation damping field in a 1.6 M aqueous solution of acetone. (A) Spectrum following a hard 90° pulse exciting all the spins. The observed frequency difference between the water and acetone resonances was ∆νwater,acetone ) 1.527 kHz. (B) Spectrum obtained after applying a selective Gaussian pulse (∼90°) to excite the water magnetization. (C) Excitation of the acetone magnetization by modulating the radiation damping field of water using the pulse sequence in Figure 2A. In this sequence, τ ) 324 µs, thereby modulating BRD(t) at a frequency of (τeff)-1 ) 1.520 kHz, where τeff ) 2τ + τd. Since ∆νwater,acetone ≈ (τeff)-1, about 44% of the acetone magnetization was excited. (D) Application of the pulse sequence in Figure 2A, but with τ ) 312 µs, thereby modulating BRD(t) at a frequency of (τeff)-1 ) 1.577 kHz. Since the pulse was effectively “off-resonance” by about 50 Hz, the acetone magnetization was not excited by BRD(t).
able excitation. The sequence was also repeated using τ ) 312 µs, giving a modulation frequency of (τeff)-1 ) 1577 Hz, which was “off-resonance” by 50 Hz. In this case, essentially no acetone magnetization was excited (Figure 3D). This indicates that the modulated radiation damping field can be made to selectively couple different spin species. Further demonstrations of selective excitation using BRD(t) were carried out on an aqueous solution of acetone and DMSO (each with a concentration of 1.2 M). The initial radiation damping field generated by water was manipulated to excite either the acetone or the DMSO magnetization. Figure 4A gives the spectrum obtained after a hard 90° pulse was applied. The observed frequency difference between the acetone and water resonances was still ∆νwater,acetone ) 1527 Hz, whereas the observed frequency difference between the DMSO and water was ∆νwater,DMSO ) 1230 Hz. Following the selective excitation of the water magnetization, the spectrum in Figure 4B was obtained. |γBRD(0)/(2π)| ≈ 103 Hz, which was much smaller than either ∆νwater,acetone or ∆νwater,DMSO; therefore, neither the acetone or DMSO were efficiently excited by the radiation damping field of the water (Figure 1). BRD(t) was then modulated to excite the acetone magnetization (Figure 4C) using the sequence in Figure 2A for τ ) 324 µs, which effectively modulated BRD(t) at a frequency of (τeff)-1 ) 1520 Hz. In Figure 4C, about 33% of the acetone magnetization was excited, while practically none of the DMSO magnetization was excited. To selectively excite the DMSO, the sequence was also repeated using τ ) 404.5 µs, giving a modulation frequency of (τeff)-1 ) 1221 Hz, which was approximately on resonance with the DMSO magnetization. In this case, 27% of the DMSO was excited, and essentially no acetone magnetization was excited (Figure 4D).
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Figure 5. Using radiation damping to create a sensitive magnetization detector.11 The solvent magnetization is inverted along the unstable -zˆ direction. Any small deviation of Msolvent away from the zˆ axis will tilt the solvent magnetization back to the +zˆ direction due to radiation damping. The detector works by utilizing the radiation damping field generated by some small solute magnetization, which can then cause a small deviation of the solvent magnetization away from the -zˆ direction. Figure 4. Spectra demonstrating selective excitation of acetone and DMSO by modulating the radiation damping field in an aqueous solution of acetone [1.2 M] and DMSO [1.2 M]. (A) Spectrum following a hard 90° pulse exciting all of the spins. The observed frequency difference between the water and acetone resonances was ∆νwater,acetone ) 1.527 kHz, and the observed frequency difference between the water and DMSO resonances was ∆νwater,DMSO ) 1.23 kHz. (B) Spectrum obtained after applying a selective Gaussian pulse (∼90°) to excite the water magnetization. (C) Selective excitation of the acetone magnetization by modulating the radiation damping field using the pulse sequence in Figure 2A for τ ) 324 µs, thereby modulating BRD(t) at a frequency given by (τeff)-1 ) 1.520 kHz. Since ∆νwater,acetone ≈ (τeff)-1, about 33% of the acetone magnetization was excited, while almost none of the DMSO was excited. (D) Selective excitation of the DMSO magnetization by modulating the radiation damping field using the pulse sequence in Figure 2A for τ ) 404.5 µs, thereby modulating BRD(t) at a frequency given by (τeff)-1 ) 1.221 kHz. Since ∆νwater,DMSO ≈ (τeff)-1, about 27% of the DMSO magnetization was excited, and practically none of the acetone was excited.
Since pulsed-field gradients are routinely used to suppress radiation damping in high-field NMR22,23 and serve as essential components in fast multidimensional NMR25,26 and imaging techniques, radiation damping induced artifacts between spins with large chemical shift differences could appear if the repetition rate of the pulsed-field gradients matched a multiple of the shift difference between the spins. Care must thus be taken in setting experimental parameters to avoid such spectral artifacts. Besides being a potential source of artifacts, enhancing the radiation damping between different spin species could be used to realize a sensitive NMR detector, as depicted in Figure 5. The abundant solvent magnetization, Msolvent, is excited to the -zˆ direction. This configuration is highly unstable; if Msolvent deviates slightly from the -zˆ direction, Msolvent will begin to rotate toward the +zˆ direction under radiation damping.11,27 As shown in Figure 5, the radiation damping field generated by some small solute magnetization would begin to rotate the large, unstable solvent magnetization away from the -zˆ direction. Monitoring the evolution of the solvent magnetization would thus provide information about the initial strength of the solute’s radiation damping field11 and hence its concentration. The method demonstrated here for enhancing the coupling of chemically shifted species using pulsed-field gradients could be applied to help realize a magnetization detector based on this idea; results will be presented elsewhere. Conclusion A new approach was presented whereby the radiation field BRD(t) was modulated using pulsed-field gradients in order to
efficiently couple chemically different spin species. By modulating the radiation damping field at a frequency given by the chemical shift difference between any two spin species, the spins can affect one another via radiation damping, even when |∆ω| . |γBRD|. Experiments performed on acetone/water and acetone/ DMSO/water solutions demonstrate that the radiation damping field from water can selectively and substantially excite either the acetone or DMSO peaks. Future applications using the radiation damping field generated by the solute in order to trigger the evolution of the solvent magnetization under radiation damping will be presented soon. Acknowledgment. We thank Dr. J. W. Logan for reading the manuscript. J.D.W. also thanks Dr. Jane Strouse and Dr. Robert E. Taylor at UCLA for discussions about Q-modulation which helped to inspire this work. This work was supported by the Camille and Henry Dreyfus Foundation (NF-01-078 and TC05-053), Research Corporation (RI0781), NSF (CHE-0349362 and CHE-0116853), donors of the Petroleum Research Fund (ACS-PRF 41355-G6), and NSF Graduate Research Fellowship Program (S.Y.H). References and Notes (1) Bloembergen, N.; Pound, R. V. Phys. ReV. 1954, 95, 8-12. (2) Bloom, S. J. Appl. Phys. 1957, 28, 800-805. (3) Szoke, A.; Meiboom, S. Phys. ReV. 1959, 113, 585-586. (4) Warren, W. S.; Hammes, S. L.; Bates, J. L. J. Chem. Phys. 1989, 91, 5895-5904. (5) Vlassenbroek, A.; Jeener, J.; Broekaert, P. J. Chem. Phys. 1995, 103, 5886-5897. (6) Ball, G. E.; Bowden, G. J.; Heseltine, T. H.; Prandolini, M. J.; Bermel, W. Chem. Phys. Lett. 1996, 261, 421-424. (7) Chen, J. H.; Mao, X. A. Chem. Phys. Lett. 1997, 274, 549-553. (8) Chen, J. H.; Jerschow, A.; Bodenhausen, G. Chem. Phys. Lett. 1999, 308, 397-402. (9) Rourke, D. E.; Augustine, M. P. Phys. ReV. Lett. 2000, 84, 16851688. (10) Miao, X.; Chen, J. H.; Mao, X. A. Chem. Phys. Lett. 1999, 304, 45-50. (11) Augustine, M. P.; Bush, S. D.; Hahn, E. L. Chem. Phys. Lett. 2000, 322, 111-118. (12) Stoner, R. E.; Rosenberry, M. A.; Wright, J. T.; Chupp, T. E.; Oteiza, E. R.; Walsworth, R. L. Phys. ReV. Lett. 1996, 77, 3971-3974. (13) Barjat, H.; Chadwick, G. P.; Morris, G. A.; Swanson, A. G. J. Magn. Reson., Ser. A 1995, 117, 109-112. (14) Huang, S. Y.; Anklin, C.; Walls, J. D.; Lin, Y.-Y. J. Am. Chem. Soc. 2004, 126, 15936-15937. (15) Augustine, M. P.; Hahn, E. L. Concepts Magn. Reson. 2001, 13, 1-7. (16) Augustine, M. P. Prog. Nucl. Magn. Reson. Spectrosc. 2002, 40, 111-150. (17) Louis-Joseph, A.; Lallemand, J. Y.; Abergel, D. C. R. Chim. 2004, 7, 329-333.
Coupled Spins via Radiation Damping Field (18) Anklin, C.; Rindlisbacher, M.; Otting, G.; Laukien, F. H. J. Magn. Reson., Ser. B 1995, 106, 199-201. (19) Broekaert, P.; Jeener, J. J. Magn. Reson., Ser. A 1995, 113, 6064. (20) Barjat, H.; Mattiello, D. L.; Freeman, R. J. Magn. Reson. 1999, 113, 114-117. (21) Louis-Joseph, A.; Abergel, D.; Lallemand, J. Y. J. Biomol. NMR 1995, 5, 212-216. (22) Sklenar, V. J. Magn. Reson., Ser. A 1995, 114, 132-135.
J. Phys. Chem. B, Vol. 110, No. 40, 2006 19989 (23) Zhang, S.; Gorenstein, D. G. J. Magn. Reson., Ser. A 1996, 118, 291-294. (24) Bodenhausen, G.; Freeman, R.; Morris, G. A. J. Magn. Reson. 1976, 23, 171-175. (25) Frydman, L.; Sherf, T.; Lupulescu, A. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 15858-15862. (26) Shrot, Y.; Frydman, L. J. Am. Chem. Soc. 2003, 125, 11385-11396. (27) Sodickson, A.; Maas, W. E.; Cory, D. G. J. Magn. Reson., Ser. B 1996, 110, 298-303.
