J. Phys. Chem. B 1998, 102, 8229-8238
8229
Three-Component Spin Echoes† Matthew P. Augustine* Department of Chemistry, One Shields AVenue, UniVersity of California, DaVis, California 95616
Erwin L. Hahn Department of Physics, UniVersity of California, Berkeley, California 94720 ReceiVed: March 5, 1998; In Final Form: June 9, 1998
Effects of radiation damping on nuclear magnetic resonance (NMR) pulse transient echo and free induction decay (FID) signals in the presence of inhomogeneous magnetic fields are measured and analyzed. An analytical theorem serves as a guide for computer simulations that confirm the experimental results. In comparison to the two-component refocusing of magnetization vectors typically observed as a spin echo in the limit of negligible radiation damping, a two-pulse π-π sequence produces an echo where all three components of the magnetization vary and refocus. The analytical theorem is extended to describe echo properties of various two-pulse and stimulated three-pulse echo sequences. Multiple echoes occur for a class of two-pulse sequences. Multiple rf-pulses interleaved with pulsed magnetic field gradients and the design of multiple pulse sequences for spectroscopic applications are discussed.
Introduction With the application of higher and more homogeneous polarizing magnetic fields, the reaction rf-field caused by the induced current in NMR receiver circuits produces significant radiation damping.1 During free precession, the coherent magnetization tends to tip back toward the polarizing field direction, often causing serious nonlinear distortions2 of highresolution NMR spectra in liquids. Special field gradient pulse sequences and feedback procedures have been developed to minimize these distortions.3 Although the radiation damping mechanism of a single spectrally sharp magnetization is understood, there has been little or no systematic understanding of the additional nonlinear effects that inhomogeneous broadening may impose on radiation-damped free precession signals. With the guidance of an analytical theorem, we present further measurements and confirmations with computer simulations of transient radiation damping introduced earlier4 that provide introductory information regarding the influence of inhomogeneous broadening. In the case of a single line specified by a homogeneous line width 1/T2 and in the limit of small tipping angles of magnetization M0, the extra exponential damping5 caused by transient radiation damping in the presence of inhomogeneous Lorentzian broadening characterized by T2* can be accounted for by adding extra decay terms 1/T2* + 1/TR to the Bloch equations. The single line radiation damping time is given by
TR ) (2πM0Qγξ)-1
(1)
where Q ) ωL/R is the LCR circuit figure of merit, ξ is the sample filling factor, and γ is the gyromagnetic ratio. The nonlinear problem to be solved for any tipping angle is how to * To whom correspondence should be addressed. † We dedicate this article to Mel Klein and Ken Sauer in recognition of their productive research careers and their partnership in the elucidation of the mechanism of oxygen production in photosynthesis.
account for the torques acting on all of the magnetization components in the inhomogeneous distribution at the same time. These torques are due to the combined action of the common time dependent rf-reaction field H1(t) and the inhomogeneous field offset δ/γ in the frame of reference rotating at the resonance frequency ω ) ω0. Three components of Bloch’s equations, u(δ,t), V(δ,t), and Mz(δ,t), are defined respectively as dispersive, absorptive, and longitudinal, and the relaxation times T1 and T2 are neglected. Attempts have been unsuccessful thus far to obtain analytical solutions of Bloch’s equations with radiation damping for a representative isochromat at any offset frequency δ in a normalized distribution g(δ) of inhomogeneous polarizing field. We gain a limited analytical insight however by being able to apply a theorem6,7 in conjunction with computer simulations to solve the Bloch equations. Total tipping angle changes ∆θ caused by radiation damping of the central magnetization vector at δ ) 0 in the g(δ) distribution are evaluated, and free precession signals are plotted, giving average values of V(δ,t) and Mz(δ,t) defined as 〈V(t)〉 and 〈Mz(t)〉 versus time. In the first part of this paper we review the derivation4 of ∆θ from the Bloch equations that couple the magnetization to the LCR detection circuit. The theorem is not restricted to refocusing prepared by rf-pulses and may apply as well to pulsed magnetic field gradients commonly used for phase cycling or magnetic resonance imaging. Unless otherwise stated, all computer simulations of radiation-damped transient signals in this paper assume a normalized symmetric Lorentzian distribution
g(δ) )
(
)
T*2 1 π 1 + δ2T*2 2
(2)
However, any symmetric function can be specified. It is shown that the usual spin echo is a limiting case in which radiation damping is negligible where the z-component of magnetization
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Mz does not change during free precession. A useful estimate of the size of the radiation damped free induction signal following an rf-pulse is easily related to the angle change ∆θ. Extension of the theorem to two or more rf-pulses provides a useful guide to the design of pulse sequences that involve three magnetization component variations in comparison to existing techniques that are limited to two, namely u and V. Analytical Theorem One Rf-Pulse. The analysis of the radiation-damped signal after a single pulse is first presented. The possibility of a timedependent phase shift φ of the reaction field H1(t) is included in situations where the inhomogeneous field distribution g(δ) may be asymmetric. When the g(δ) distribution is symmetric, there is zero phase shift in time. Coupling of the precessing magnetization to the LCR circuit1,8 will yield an expression for the slowly varying reaction field H1(t) in terms of the magnetization in the frame of reference rotating at frequency ω. The coupling between the reaction field due to circuit current I(t) and the magnetization Mx(t) in the laboratory frame is given by 2
I(t) d d d2 L 2I(t) + R I(t) + ) -4πξηA 2Mx(t) dt C dt dt
(3)
where η is the number of ampere turns and A is the cross sectional area of the coil. The current
I(t) )
xπLVcH (t) cos(ωt + φ) 1
(4)
Mx(δ,t) ) u(δ,t) cos ωt + V(δ,t) sin ωt
(5)
and magnetization
are substituted into eq 3, terms proportional to ω-n where n > 1 are dropped, and the low Q limit is assumed. The central vector is defined at ω - ω0 ) 0 ) δ, where the circuit response is flat at ω ) 1/(LC)1/2 and ω/Q . γH1(t). The problem of off-resonance tuning of the LCR circuit, which reduces the effective Q, is not considered here. Equating the resultant real and imaginary parts and integrating over the distribution g(δ) gives
∫
1 V(δ,t)g(δ) dδ θ˙ cos φ ) M0TR
(6)
direction of u by the time-dependent phase angle φ and a more complicated relationship among H1(t), V(t), and u(t) results. In that case the on-resonant central magnetization vector at δ ) 0 in the distribution g(δ) will not remain at resonance as a function of time. Since we treat only the case of symmetric g(δ), the average quantity 〈u(t)〉 ) 0 at all times and eq 6 can be used to derive the analytical theorem, beginning with the time integral
∫0tf∞θ˙ dt′ ) ∆θ1 ) -M01TR∫0tf∞∫V(δ,t′)g(δ) dδ dt′
where ∆θ1 ) θ∞ - θ1. The initial angle that the total magnetization M0 makes with respect to the z-axis after the first pulse is given by θ1, the final angle is given by θ1 + ∆θ1, and the radiation damping time constant TR is given by eq 1. A manipulation used to explain the phenomenon of self-induced transparency9 is now applied to transform eq 9. After substituting the Bloch equation u3 (δ,t) ) δV(δ,t) from eq 8 into the integrand of eq 9 and integrating over t′, one obtains
u(δ,t f ∞) - u(δ,t ) 0) 1 ∆θ1 ) g(δ) dδ (10) M0TR δ
∫
After a θ1 rf-pulse the initial value u1 ) u(δ,t ) 0) ) 0, and at t ) ∞ the freely precessing final value u∞ is given by u∞ ) u(δ,t ) T + t′′) ) u(δ,T) cos δt′′ + V(δ,T) sin δt′′. The time T ) ∞ is chosen when the initial conditions for freely precessing u and V modes can be specified and H1(t) has vanished. As u(t) and V(t) precess freely during time t′′ beyond T, integration over the inhomogeneous distribution involves a δ function which selects only the final value of the central V(δ ) 0,T) mode that evolved at exact resonance (δ ) 0) in the past after H1(t) disappeared. Of course all the inhomogeneous spin vectors mutually cancel so that the integrated average of V(δ,t) given by 〈V(t)〉 over the distribution is zero. Therefore ∆θ1 represents the total angle through which the central vector M(0,T) ) [v2(0,T) + Mz2(0,T)]1/2 is tipped toward the +z axis from its original angle θ1. The angle change ∆θ1 is independent of the functional behavior of H1(t) over time. Henceforth all angle changes of ∆θ1 are expressed in terms of the absolute values ∆θ1 ) -|∆θ1|, and all tipping angles prepared by a pulse or by effects of radiation damping are confined between zero and π. Using these definitions in the case of a Lorentzian g(δ), it follows that g(0) ) T2*/π and g(0)V(0,T) ) (T2*/π)M0 sin(θ1 - |∆θ1|), giving
|∆θ1| )
and
∫
1 θ˙ sin φ ) u(δ,t)g(δ) dδ M0TR
(7)
where θ˙ ) γH1(t). The Bloch equations that we apply are
d M (δ,t) ) -θ˙ cos φ V(δ,t) - θ˙ sin φ u(δ,t) dt z d V(δ,t) ) θ˙ cos φ Mz(δ,t) - δu(δ,t) dt
(8)
d u(δ,t) ) θ˙ sin φ Mz(δ,t) + δV(δ,t) dt The components u and V are respectively odd and even functions for φ ) 0. If g(δ) is a symmetric distribution about δ ) 0, then the average 〈u(t)〉 ) 0 and H1(t) is fixed along the u direction. If g(δ) is asymmetric, H1(t) swings away from the
(9)
∫
1 sinδt′′ g(0)V(0,T) | d(δt′′) M0TR δt′′ δ)0
T*2 ) sin(θ1 - |∆θ1|) TR
(11)
The usefulness of this transcendental equation can be seen from its graphical solution for θ1 ) π/2 shown in Figure 1a. Here y1 ) |∆θ1| and y2 ) cn sin(θ1 - |∆θ1|) are ordinates and |∆θ1| is the abscissa. When y1 is written in this way, it is clear that the constants cn represent different possible choices for T2*/TR. The intersection of y1 with y2 in this figure defines abscissa values of |∆θ1| acquired at t ) ∞ for various ratios of T2*/TR. Physically the |∆θ1| obtained from the plot in Figure 1a is the rotation angle of the on-resonance isochromat following a π/2pulse during radiation damping. The more interesting case is when θ1 ) π as shown in Figure 1b. Not all values of T2*/TR in this case give a nonzero |∆θ1|. When the maser condition T2*/TR g 1 applies and the π pulse is slightly inaccurate (π ( ), an initial transverse V polarization can seed the start of the
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Figure 1. Dependence of the angle changes |∆θ1| of magnetization for different values of cn ) T2*/TR. Here y1 is given by |∆θ1| and y2 by cn sin(θ1 - |∆θ1|), where initial pulse tipping angles are θ1 ) π/2 in (a) and θ1 ) π in (b). When the condition T2* < TR applies for an inverted system the only solution for |∆θ1| is zero, as shown for the c1 curve in (b).
TABLE 1: Summary of Normalized g(δ) Values g(δ)
normalized g(0)
Lorentzian Gaussian triangle sphere box
T2*/π xln(2)/πT2* T2*/2 T2*/2 x3T2*
damping signal. This manifests itself graphically by shifting the y2 curve slightly to the left and permitting only one solution for |∆θ1|. Below the maser condition T2*/TR < 1, radiation damping is negligible, |∆θ1| ) 0, and the magnetization decays by relaxation processes toward the +z axis as shown for the c1 and c2 curves in Figure 1b. When the π-pulse is precise and the maser condition is fulfilled, the damping signal can be initiated by noise that would manifest itself as a random shift of the y2 curve slightly to the left. Although this noise effect is a separate interesting problem, the experiments reported in this paper do not rely on spontaneous signal generation initiated by noise. Although the derivation of the analytical theorem in eq 11 and its related graphical solutions apply only to a Lorentzian distribution function, any symmetric g(δ) is permissable. Unfortunately it is difficult or impossible to realize assumed types of bell-shaped inhomogeneous distributions in the laboratory. In reality one typically deals with linear field gradients and either cylindrical, spherical, or other shaped sample containers that may define a g(δ) that has a box, disk, triangular, or perhaps spherical distribution. In any case the transformation used to generate eq 11 will select only the on-resonance central vector component g(0)M0 dδ. In practice one also must be aware that if the reaction field H1(t) is to be reasonably uniform over the sample, the sample volume must be confined well within the volume of the inductive detection coil.
Figure 2. (a) Different symmetric distribution functions examined in this study. The peak intensities of these distributions g(0) are given in Table 1. (b) Comparison of |∆θ1| obtained from the analytical theorem (solid line) and by computer integration of Bloch’s equations corrected for radiation damping (crosses) appropriate for θ1 ) π/2.
The general analytical theorem given by eq 11 can be written to pertain to any symmetric g(δ) as |∆θ1| ) g(0)π sin(θ1 |∆θ1|)/TR. Table 1 lists g(0) for some distributions where g(δ) is normalized to unity and 1/T2* defines the half-width of the appropriate g(δ) pictured in Figure 2a. A computer check of the validity of eq 11 is shown in Figure 2b for the case of θ1 ) π/2. The solid line represents a plot of the transcendental value of |∆θ1| versus log(g(0)π/T2*). Numerical integration of Bloch’s equations corrected for radiation damping combined with eq 9 yields a point (crosses) falling on the transcendental curve. Any particular point of which four are chosen represents the coordinates predicted by computer evaluation of eq 9. The choice of a particular value of the abscissa is true for any g(0) from the list in Table 1. Computer integration of Bloch’s equations (eq 8) was accomplished in a stepwise fashion using a Fortran program optimized to perform on a Unix workstation. Typical time steps varied between 1 ns and 1 µs depending on the particular choice of T2*. The angle change |∆θ1| relates not only to the orientation of M(0,T) as seen from graphs in Figures 1 and 2 but also to experimentally observable quantities. The area under the observed free induction signal S(t) is related to the average value
〈V(t)〉 ) kS(t) )
∫V(δ,t)g(δ) dδ
(12)
by a proportionality constant k that can be measured by a small tipping angle rf-pulse on samples with known spin concentration. Combination of this relationship with eq 11 shows that
|∆θ1| )
∫0tf∞S(t′) dt′
k M0TR
(13)
expresses the area under the radiation-damped free induction signal extended to t ) ∞. The difference between the average initial and final z-components of magnetization can also be obtained and related to experiment in a similar way by
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Figure 3. Numerical simulation of the free induction signal following nearly a π-pulse. Here TR ) 5 ms was chosen so that the maximum of 〈V(t)〉 (solid line) in the narrow line limit occurs at t ) 50 ms and 〈Mz(t)〉 (dashed line) nearly recovers to full alignment along the +z direction following the π-pulse. The field homogeneity was adjusted to give T2* ) 25 ms producing T2*/TR ) 5.0 and 〈Mz(t)〉 ) 0.5 at 200 ms. Note the low-frequency oscillation following the signal maximum. These oscillations have been observed experimentally in the literature following one π-pulse, revealed by computer analysis here, and warrant further study.
integrating the Bloch eq 8 M˙ z(δ,t) ) θ˙ V(δ,t) over both t and δ so that
〈Mz(t ) ∞)〉 - 〈Mz(t ) 0)〉 ) )
∫0tf∞∫M˙ z(δ,t′)g(δ) dδ dt′ k2 M0TR
∫0tf∞S2(t′) dt′
(14)
This expression is proportional to the area under the square of the free induction signal extended to t ) ∞. The change in Zeeman energy in eq 14, proportional to the integral of S2(t) or I2(t) over all time, is dissipated in the circuit resistance as Joule heat. From a measurement of just one free induction decay, a test of eq 11 can be made using eqs 12-14. The above relations for |∆θ1| and 〈Mz(t)〉 obtained from the analytical theorem, together with the rule of angle changes θ1 - |∆θ1| of the central vector with pulse tipping angle θ1, now connect experiment to computer simulations. Starting from thermal equilibrium and following the chain connecting final to initial conditions, any pulse sequence can be simulated. The simulation following a single θ1 ) π-pulse is shown in Figure 3. The inverted polarization M0 is the initial condition for the phenomenon of “superradiance” first obtained in optical twolevel systems,10,11 where the polarization dumps to the ground state. Bo¨siger et al.12 first observed this phenomenon for nuclear spins without consideration of inhomogeneous broadening. In our case, the final average polarization 〈Mz(∞)〉 ) 0.5M0 does not convert completely to the ground state because of inhomogeneous broadening, where TR ) 5 ms is chosen so that the maximum of 〈V(t)〉 in the narrow line limit occurs at t ) 50 ms and the ratio T2*/TR ) 5. Therefore, |∆θ1| ) 8π/10 is evaluated from either eq 11 or from integration of 〈V(t)〉 in Figure 3. The simple form of eq 11 can in fact predict the integrated signal after the complicated phase evolution of the isochromats in the inhomogeneous distribution is completed. During the phase evolution generated by computer simulation, the spatial distribution of the end points of isochromats projected onto a unit sphere is shown in Figure 4 for T2*/TR ) 5 at different times following a π-pulse. The maximum of 〈V(t)〉 in Figure 3 corresponds approximately to a circle in the southern hemisphere in Figure 4b at t ) 57 ms. At later times the signal decreases due to cancellation of the isochromats on the unit sphere. An initial glance at Figure 4d suggests that the disordered distribution of
Figure 4. Spatial distribution of the end points of isochromats projected onto a unit sphere (solid black line) at given times from (a) to (d) following nearly a π-pulse for T2*/TR ) 5.00. The sequence and parameters apply to those of Figure 3. The integrals of the x, y, and z components of these isochromats weighted by g(δ) indicate 〈V(t)〉, 〈u(t)〉, and 〈Mz(t)〉, respectively. Immediately following the first π pulse all of the isochromats are aligned along the -z direction as shown both by a dot at the south pole and by a downward pointing arrow in (a). As time proceeds, an observable signal develops and the magnetization vectors spread out in a circular pattern in three dimensions on the unit sphere. The diameter of this pattern continues to increase to a maximum, as shown in (b) at a time t ) 57 ms (Figure 3) corresponding to the maximum of the damping signal. As each isochromat tends to precess about the changing effective field direction, the circular pattern evolves into a figure eight and then into two connected orbits seen in (c) for t ) 75 ms. Finally, when H1(t) vanishes, an extremely complex array of connected orbits spreads over the surface of the unit sphere, averaging the magnetization to produce zero signal and resulting in a value for 〈Mz(t)〉. This state of free precession at t ) 200 ms is shown in (d).
isochromats in three dimensions may not be reversible according to the conventional two-pulse spin echo. However, the symmetry of free precession phase cancellation in the rotating frame holds although radiation damping implies a unidirectional loss of spin energy toward the ground state. As we shall see in the next section, much like the usual spin echo,13 any combination of two rf-pulses of tipping angles θ1 and θ2 will lead to at least partial refocusing of the magnetization, now due to the combined torque of the static inhomogeneous and time-dependent reaction field. Two Rf-Pulses. In the case of a purely homogeneously broadened system, the disappearing free induction signal during radiation damping produces a corresponding decrease in H1(t), reflecting the special fact that ultimately 〈V(t)〉 ) V(δ,∞) ) 0. With inhomogeneous broadening, as seen in Figure 4, the signal after the first θ1 pulse subsides when H1(t) ) 0, but all the V(δ,t) isochromats remain active and precessing while their average 〈V(t)〉 over the inhomogeneous distribution is zero. This zero value may apply at any arbitrary time t > T when H1(t > T) ) 0, during which time a second θ2 pulse may be applied. The final u mode that occurs after the first θ1 pulse is altered by the second θ2 pulse and becomes the initial u mode for application again of the analytical theorem, eq 10, to determine the second
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Figure 5. Vector diagram showing the relationship between the central vector M(0,t) at different times t during a two-pulse sequence when an equilibrium characteristic of H1(t) ) 0 is present.
angle change |∆θ2| after the signal averages to zero. The computer simulation is adjusted to avoid the overlap of echo radiation damping signals with “FID” signals that occur directly after pulses. Note that the analytical theorem itself cannot predict in any way the occurrence time or shape of the free induction signals. In the previous section for the evaluation of |∆θ1| we defined u(initial) ) u(δ,t ) 0) ) 0 and u(final) ) u(δ,T + t′′) ) u(δ,T) cos δt′′ + V(δ,T) sin δt′′, where V(0,T) ) M(0,T) sin(θ1 - |∆θ1|). Analogously, to evaluate ∆θ2 after the second θ2 pulse having the same phase as the first rf-pulse, we define u(initial) ) M(0,T) sin[|(θ1 - |∆θ1|) - θ2|] sin δt′′′ and u(final) ) u(δ,T′ + t′′′) ) u(δ,T′) cos δt′′′ + V(δ,T′) sin δt′′′. Since the δ function integration discards all terms containing cosines, only the sine terms need to be retained. These terms are given by u(final) ) M(0,T) sin(|Θ2| - |∆θ2|) sin(δt′′′) and u(initial) ) M(0,T) sin Θ2 sin δt′′′, where Θ2 ) (θ1 - |∆θ1|) - θ2. Therefore the analytical theorem extended to two rf-pulses indicates that |∆θ2| is proportional to the difference between the final values V(0,T) and V(0,T′) as
|∆θ2| )
T*2 [sin(|Θ2| - |∆θ2|) - sin Θ2] TR
(15)
It is important to note that the relationship between |∆θ2| and the integrated signal following the θ2 pulse will be identical to that shown in eq 13 with |∆θ1| replaced by |∆θ2|. The absolute value signs in eq 15 stem from the constraint that all angles of M(0,t) with respect to the z direction must lie between 0 and π as seen from Figure 5. The first rf-pulse rotates all of the isochromats M(δ,0) to an angle θ1 with respect to +z in the Mz-V plane. Equation 11 dictates that the torque due to both H1(t) and g(δ) will cause M(0,0) to rotate toward +z by an amount |∆θ1|. Following this pulse and subsequent evolution, M(0,T) is inclined away from +z by an angle θ1 - |∆θ1|. A second rf-pulse transforms M(0,T) toward +z if (θ1 - |∆θ1|) - θ2 ) Θ2 < π. If Θ2 > π, the rotation of M(0,T) by θ2 overshoots the +z direction and falls in the second quadrant of the Mz-V plane as shown by the dashed arrow in Figure 5. The analytical theorem of eq 15 is now applied to the design of two-pulse experiments, the simplest of which prove to be the π/2-π and the π-π sequences. Applying the π/2-π sequence to a system having TR ) 50 s and T2* ) 5 ms gives T2*/TR ) 10-4 and the analytical theorems in both eqs 11 and 15 provide |∆θ1| ) |∆θ2| ) 0. The corresponding simulation shown in Figure 6a demonstrates the property that during free precession 〈V(t)〉 * 0 while 〈Mz(t)〉 ) 0 at all times. Equivalently, M(0,t) never departs from the transverse plane during evolution and refocusing of the magnetization, consistent with the conventional interpretation of the spin echo. Adjusting the radiation damping time constant to TR ) 5 ms for the same π/2-π sequence produces the signal 〈V(t)〉 shown as a solid line in Figure 6b. Equations 11 and 15 yield |∆θ1| ) 1.31 and
Figure 6. Simulation of two- and three-component refocusing as a function of TR, θ1, and θ2 for T2* ) 25 ms, a Lorentzian g(δ), and a pulse separation of T ) 130 ms. The usual two component spin echo having θ1 ) π/2 and θ2 ) π is shown in (a) for 〈V(t)〉 (solid line) with TR ) 50 s. Since T2*/TR ) 5 × 10-4, |∆θ1| ) |∆θ2| ) 0 and 〈Mz(t)〉 is always zero. Decreasing TR to 5 ms and leaving θ1 and θ2 fixed at π/2 and π, respectively, brings Mz(δ,t) into the refocusing process as shown by the dashed line for 〈Mz(t)〉 in (b). Since |∆θ2| ) 2|∆θ1| the echo at t ) 260 ms has twice the integrated signal as the initial free induction signal after the first θ1 ) π/2 pulse. Leaving T2*/TR ) 5.0 and changing the tipping angle of the two rf-pulses to θ1 ) 160° and θ2 ) 135° produces the multiple echo signal after the second pulse shown in (c).
|∆θ2| ) 2.62 implying that the magnetization will be rotated away from the transverse u-V plane due to radiation damping. This is manifest in Figure 6b by the averaged z-component 〈Mz(t)〉 shown as a dashed line changing from zero to ∼0.8 M0, by the apparent increase in decay rate following the θ1 ) π/2pulse, and by the decreased width of the echo in time following the θ2 ) π-pulse. Note that the analytical theorem predicts that the echo contains twice as much integrated signal as the free induction decay following a θ1 ) π/2 pulse, resulting in |∆θ2| ) 2 |∆θ1|. The application of a θ1 ) θ2 ) π two-pulse sequence reduces eq 15 to
T*2 |∆θ2| - |∆θ1| ) - sin(|∆θ2| - |∆θ1|) TR
(16)
indicating that |∆θ1| ) |∆θ2|. Partial refocusing of the V-mode
8234 J. Phys. Chem. B, Vol. 102, No. 42, 1998
Figure 7. Spatial distribution of the end points of isochromats projected onto a unit sphere (solid black line) at given times from (a) to (d) following a second π-pulse for T2*/TR ) 5.00. The sequence and parameters apply to those of Figure 8. The effect of the second π-pulse at 200 ms is to invert the sphere in Figure 4d to the configuration in (a). The isochromats evolve with a reversal of the phase accumulated in the sequence shown in Figure 4a-d. This is seen in (b) at t ) 325 ms and (c) at the echo maximum occurring at t ) 343 ms. Finally the magnetization refocuses along the +z direction at t ) 400 ms after H1(t) has vanished.
produces a spin echo, followed by the startling result that 〈Mz(0)〉 ) -M0 at t ) 0 returns to the +M0 ground state at t ) T′. This complete refocusing is shown by computer simulation and by experiment. The analytical theorem here shows only that the angle ∆θ1 of M(0,T) with respect to -z, acquired after the first pulse, is reflected by the second pulse to be the same angle through which M(0,T) rotates into alignment with +z. Although the phase relations that develop among the inhomogeneous isochromats are quite complicated and nonlinear, the distribution refocuses. The ensemble reverts to a new final energy state, in this case from -M0H0 to +M0H0. Power is transferred collectively from the spin ensemble to the circuit resistance, while u and V yield echo and free induction signals. As seen from Figures 3 and 4, with the condition that T2*/TR ) 5, the signal maximum occurs at 57 ms after the θ1 ) π-pulse. At t ) T, when H1(t) ) 0, the angle |∆θ1| ) 8π/10 and 〈Mz(t)〉 ) 0.5M0. The second θ2 ) π-pulse applied at 200 ms is followed by the spin echo, occurring not at twice the pulse spacing but lesser by the 57 ms time given above. The time sequence for the refocusing process is shown in Figure 4 for the signal immediately following the θ1 ) π-pulse. After the θ2 ) π-pulse, the inverse and symmetric sequence is shown in Figure 7. Figure 8a,b shows the time plot of 〈V(t)〉 and 〈Mz(t)〉 when 〈Mz(t)〉 ) 0.5M0 and 0 for T2*/TR ) 5 and T2*/TR ) 2.56, respectively. Experimental comparison between homogeneous and inhomogeneous broadening is shown in Figure 9 for the π-π pulse sequence applied to water at 11.74 T. The signal in Figure 9a shows the expected result for water in a well-shimmed magnet. Radiation damping quickly restores the inverted magnetization fully to the +z direction following the first pulse. Thus, the
Augustine and Hahn
Figure 8. Numerical simulation of complete three-component refocusing by applying two π-pulses having the same phase and separated by 200 ms. Again TR ) 5 ms was chosen so that the maximum of 〈V(t)〉 (solid line) in the narrow line limit occurs at t ) 50 ms and 〈Mz(t)〉 (dashed line) nearly recovers to full alignment along the +z direction following the first π-pulse. The conditions in (a) are identical to those used in Figure 3. Application of a second π-pulse at t ) 200 ms flips the transverse magnetization as discussed in the text, and an echo maximum appears ∼ 150 ms later. In (b) the field homogeneity was adjusted to give T2* ) 12.8 ms producing T2*/TR ) 2.56 and 〈Mz〉 ) 0 at 200 ms, prior to the second π-pulse.
behavior of the magnetization following the second π-pulse at t ) 600 ms is identical to that after the first π-pulse. In Figure 9b a different effect occurs when the water line is homogeneously broadened from 21 to 27 Hz by adding the paramagnetic relaxation agent manganese(II) chloride to the sample. The additional line width in this case does not allow the magnetization to fully recover along the +z direction due to either radiation damping or T1 relaxation before the application of a second π-pulse. Therefore, only a small fraction of the full magnetization M0 contributes to TR following the second π-pulse. Since M0 is not large enough to satisfy the maser criteria cn g 1, the radiation damping becomes negligible and minimal signal is seen after the second π-pulse applied at 600 ms. The results of the π-π pulse sequence on the same water sample used for Figure 9a are shown in Figure 9c. Here additional line width is introduced by partially deshimming the magnet to provide a 29 Hz inhomogeneously broadened water line of nearly the same width as the 27 Hz homogeneously broadened line in Figure 9b. The maximum amplitude of the signal following the second π-pulse does not occur at 50 ms as in Figure 9a after the first π-pulse but at a time proportional to the pulse spacing of 600 ms. We attribute this signal to a spontaneously generated threecomponent spin echo. The decreased echo amplitude in comparison to the signal following the first π-pulse results from T1 and T2 relaxation and diffusion on the time scale of the experiment, 2 s. The experimental proof that the full distribution completely refocuses along the +z direction was carried out by first
Three-Component Spin Echoes
Figure 9. Experimental verification of three-component refocusing observed at B0 ) 11.74 T. The free induction decays shown here correspond to an offset frequency of 50 Hz for a well-shimmed water line having a full width at half-maximum of 21 Hz. In (a) the maximum of the signal following the second π-pulse applied at 600 ms is consistent with the magnetization having nearly reoriented completely along the +z direction after the first π-pulse. This corresponds to T2*/ TR . 1 as shown in Figure 1b for T2*/TR > c5 so that |∆θ1| ∼ π. In (b) the homogeneous line width of the sample is increased to a broadening of ∼27 Hz by adding a small amount of paramagnetic manganese(II) chloride (0.1 mM). This increases the time at which the radiation-damping signal appears from 50 ms in (a) to 100 ms in (b) and eliminates the radiation-damping signal following the second pulse. Increasing the inhomogeneous line width of the original water sample to 29 Hz by adjusting the magnetic field shims produces the anticipated three-component refocusing at ∼1.15 s, as shown in (c).
measuring the signal following a small tipping angle rf-pulse applied to a sample having field homogeneity identical to Figure 9c at thermal equilibrium. After the π-π sequence which should provide complete refocusing for any pulse spacing as discussed above, the same probe pulse was applied to the sample. For a pulse spacing of 78 ms a signal having approximately 96% of the thermal equilibrium signal was observed. This 4% difference can be attributed to relaxation damping. When θ1 is different from θ2 in the two-pulse sequence, the usefulness of the transcendental relation shown in eq 15 can be seen from its graphical solutions shown in Figure 10. Now define y1 ) |∆θ2| and y2 ) cn{sin[|(θ1 - |∆θ1|) - θ2 | - |∆θ2|] - sin[(θ1 - |∆θ1|) - θ2]} as ordinates and |∆θ2| as the abscissa. Clearly there will be a family of solutions for |∆θ2| depending on cn for particular choices of the initial angle Θ2 ) (θ1 |∆θ1|) - θ2 which lies between -π and +π. Figure 10a shows the graphical solution for eq 15 for Θ2 ) -π/2 and various cn. The intersection of y1 with y2 gives the |∆θ2| when H1(T′) ) 0 following the second rf-pulse. Choosing other negative initial rotation angles Θ2 corresponds to shifting the y2 curve maximum either to the left or right in Figure 10a. As long as Θ2 < 0 and |∆θ1| * 0, y1 will intersect y2 and there will be a nonzero solution for |∆θ2| for any cn. When Θ2 > 0, curves such as
J. Phys. Chem. B, Vol. 102, No. 42, 1998 8235
Figure 10. Dependence of the angle changes |∆θ2| of magnetization following a two-pulse sequence for different values of cn ) T2*/TR. Here y1 is given by |∆θ2| and y2 by cn{sin[|(θ1 - |∆θ1|) - θ2 | |∆θ2|] - sin[(θ1 - |∆θ1|) - θ2]}. The angle of the magnetization immediately following the second pulse is chosen as Θ2 ) (θ1 - |∆θ1|) - θ2 ) -π/2 in (a) and Θ2 ) 8π/10 in (b).
those in Figure 10b for Θ2 ) +8π/10 result. In this case y2 is always zero when |∆θ2| ) 0, and modifications of M(0,T) or nonzero values of |∆θ2| are observed only when the initial slope of y2 is greater than the initial slope of y1. Taking the derivative of eq 15 with respect to |∆θ2| at |∆θ2| ) 0 sets boundary conditions for choices of Θ2 such that |∆θ2| is nonzero. The possible values of Θ2 that lead to a nonzero |∆θ2| using a Lorentzian g(δ) are
()
|Θ2| < π - arccos
TR T*2
(17)
This emphasizes that |∆θ1| * 0 and that T2*/TR must be greater than 1 for the arc cosine in eq 17 to be defined. Graphically, as Θ2 tends toward the upper limit in eq 17, the |∆θ2| ) 0 abscissa intercept shifts to zero making zero the only possible intersection point of y1 and y2. Computer simulations indicate that multiple echoes appear for any choice of θ1 and θ2 except when θ2 ) π. Figure 6c shows a computer simulation of 〈V(t)〉 and 〈Mz(t)〉 appropriate for θ1 ) 160° and θ2 ) 135° and T2*/TR ) 5.0. Since the analytical theorem cannot predict when and how many echoes appear, the values of ∆θ1 ) 2.31 and ∆θ2 ) 2.38 in this case represent only the integrated changes in the angle of the central vector. The hint as to a phenomenological explanation for the appearance of multiple echoes comes from the fact that they do not occur when θ2 ) π for any choice of θ1. In these cases the second π-pulse reorients the inhomogeneous isochromats so that any average 〈Mz〉 is reversed in sign and does not project any component of net 〈V〉 magnetization after the π-pulse. If θ2 is other than π, a net average 〈V〉 component appears in the transverse plane. In ordinary spin echoes, if a third pulse is applied after a two-pulse sequence, an echo will result from the second and third pulse, provided that the second pulse is
8236 J. Phys. Chem. B, Vol. 102, No. 42, 1998
Augustine and Hahn
TABLE 2: Summary of Possible Θn Values n
Θn
2 3 4 m
(θ1 - |∆θ1|) - θ2 |(θ1 - |∆θ1|) - θ2| - |∆θ2| - θ3 ||(θ1 - |∆θ1|) - θ2| - |∆θ2| - θ3|- |∆θ3| - θ4 |Θm-1| - |∆θm-1| - θm
not π, and converts any z-component into some V-component. In the situation here the reaction field H1(t) of the first echo is apparently large enough and lasts a sufficient length time (∆t ≈ 50 ms) to act as a third pulse to produce a second echo. One would expect that some fraction of the tipping angle during the first echo |Θ2| - |∆θ2| would characterize a “hypothetical third pulse”. The phase characteristics of these multiple echoes are consistent with such a model. For example a y-y pulse sequence produces multiple echoes having the same phase while y-yj and yj-y sequences cause the even and odd ordered echoes to oscillate in phase between zero and π as seen in Figure 6c for y-yj. Unfortunately the phase properties of these echoes are identical to those produced by sample shape or demagnetization effects.14 Therefore the echoes noticed in Figure 6c for the first time will be difficult to experimentally measure unless spherical sample containers that eliminate sample shape demagnetization factors are used. n Rf-Pulses. An analytical theorem describing the tipping angle of ∆θn of M(0,∞) after n-rf θn pulses can be derived in much the same way that |∆θ2| was determined for the arbitrary θ1-θ2 two-pulse sequence. Again the spacing between the θn pulses must be sufficiently long so that H1(t) averages to zero. The n-pulse case could be analyzed by determining un and u∞ in terms of δ and T and from eq 10 leading to a relationship between |∆θn|, θn, and values of |∆θm| and θm at earlier times, where 1 < m < n. The problem simplifies from the special property that, regardless of the number of rf-pulses n, the accumulated angle |∆θn| will always be related geometrically to the initial angle Θn and the angle to which it evolves |Θn| |∆θn| as
|∆θn| )
T*2 [sin(|Θn| - |∆θn|) - sin Θn] TR
(18)
for a Lorentzian g(δ). Actual values of the Θn can be obtained from simple vector diagrams such as Figure 5 by making the following replacements: θ1 - |∆θ1| f Θn-1 and |(θ1 - |∆θ1|) - θ2| f |Θn|. Note again that the absolute value operation emphasizes that all tipping angles due to pulses and evolutions are constrained to lie between zero and π. Table 2 lists values of Θn up to n ) 4 and also shows the generic recursion formula for Θn in terms of earlier |Θn-1|, |∆θn-1|, and θn values. In direct analogy to the one and two rf-pulse experiments, the analytical theorem in eq 18 combined with Table 2 can serve as a guide to the design of multiple pulse sequences during radiation damping in an inhomogeneous magnetic field. The usefulness of the transcendental relationship in eq 18 again comes from its graphical solutions for various initial conditions Θn and line width factors cn. Because of the similarity between eqs 15 and 18, the graphs in Figure 10 represent not only the solution for |∆θ2| after two pulses but also the solution for |∆θn| after n rf-pulses. The correspondence can be made by replacing |∆θ2| with |∆θn| for Θn ) -π/2 in Figure 10a and for Θn ) +8π/10 in Figure 10b. The experimental results shown in Figure 11 demonstrate the usefulness of the analytical theorem in eq 18 for a simple threepulse experiment in H2O at B0 ) 14 T. Here θ1 ) π, θ2 )
Figure 11. Three-component stimulated echoes in H2O at B0 ) 14 T with a spacing between the first two pulses of 200 ms and a delay between the last two pulses of 1.5 s. In (a) θ1 ) θ3 ) π and θ2 ) π/2, while in (b) θ3 is changed to π/2. The decreased signal amplitude in (b) is consistent with both the analytical theorem and the idea of threecomponent evolution during phase accumulation after the first pulse and refocusing following the third pulse.
π/2, and θ3 ) π in Figure 11a, θ3 ) π/2 in Figure 11b, and the delay between the first and second pulse is 200 ms while the delay between the second and third pulse is 1.5 s. A 50 ms 5 G/cm field gradient pulse placed midway between the second and third rf-pulses removes unwanted refocusing commonly encountered in three pulse experiments. The signal following the π-π/2-π sequence in Figure 11a shows the echo formation at t ) 150 ms as expected due to the spacing between the first two rf-pulses t ) 200 ms. The decrease in signal noticed in Figure 11b upon going to a π-π/2-π/2 sequence can be explained using the analytical theorem. A good choice for T2*/ TR is 6.0 because TR is typically between 1 and 5 ms in highresolution liquid NMR probes and typical damped line widths are usually between 15 and 25 Hz producing a range of T2* values from 12 to 23 ms. For T2*/TR ) 6.0 the analytical theorem dictates that, immediately after the final π-pulse of a π-π/2-π sequence, the central vector of g(δ), M(0,t) is directed at an angle of Θ3 ) 143°. Due to the combined effect of the reaction and inhomogeneous magnetic fields the central vector reorients to an angle |Θ3 - |∆θ3|| ) 9°. In direct analogy to the signals produced by one- and two-pulse sequences, the signal here will be proportional to |∆θ3| ) 134°. Applying a similar analysis to the π-π/2-π/2 sequence leads to |∆θ3| ) 43°. The ratio of the |∆θ3| values for both sequences χ ) 43°/ 134° ) 0.32 can be used to directly compare experimental signals using eq 13. Integration of the magnitude of the three component stimulated echoes in Figure 11 as a function of time along with eq 13 produces χ ) 0.318, consistent with the value expected on the basis of the analytical theorem alone. More detailed discussion of this relationship between the analytical theorem and experimental signals will be provided in a future publication.15 A more physical interpretation of the three component stimulated echo in Figure 11a can be obtained by recalling the
Three-Component Spin Echoes standard two component stimulated echo following a π/2-π/ 2-π/2 three-pulse sequence. The second π/2 pulse stores the magnetization along +z that dephased in the u-V plane following the first π/2 pulse. The third π/2 pulse applied a time t < T1 rotates this magnetization back into the u-V plane and refocusing occurs. Two- and three-component stimulated echoes serve to encode and decode g(δ) along +z while, for the three-component refocusing, H1(t) also participates. The evolution of M(δ,t) occurs in a three-dimensional fashion following the first π-pulse in Figure 4. A portion of this complicated array of isochromats is then stored along +z with a π/2-pulse. Maximum refocusing is established by re-inverting the magnetization stored along +z with a second π-pulse. Pulsed Field Gradients. The analytical theorem is not restricted to the application of rf-pulses. In the same way that there is a similarity between two- and three-component echoes produced by rf-pulses, three-component refocusing can be obtained by switching the magnitude and sign of field inhomogeneity across a sample. The analogous two component experiment is the gradient refocused echo. An analytical theorem can be deduced that describes the tipping angle |∆θ2| of the central vector of g(δ) during such a refocusing. In this experiment one first applies an rf-pulse of tipping angle θ1. M(0,0) adjusts in three dimensions to a final tipping angle θ1 - |∆θ1| with respect to the +z direction. The transcendental relation shown in eq 11 can be used to calculate |∆θ1| for a given Lorentzian distribution having width 1/T2*. Let the Lorentzian field inhomogeneity be switched in sign and in magnitude from T2* to T2*′ at a time t > T after H1(t) following the θ1 pulse has vanished. This causes the frequency labeling of g(δ) to also change in sign and in magnitude from δ to -δ′, where δ′ ) δ(T2*/ T2*′). The central vector M(0,T) will adjust from θ1 - |∆θ1| by an amount |∆θ2| that will reflect the changes in both the size and sign of the altered field gradient across the sample. To calculate |∆θ2| one simply replaces δ with -δ′ in eq 10 and uses u∞ ) u(δ′,T′) cos(-δ′t′′′) + V(δ′,T′) sin(-δ′t′′′) ) u(δ′,T′) cos δ′t′′′ - V(δ′,T′) sin δ′t′′′ along with u2 ) u(δ′,T) cos δ′t′′ + V(δ′,T) sin δ′t′′. An identical analysis to that used for two rf-pulses naturally leads to the angle change
|∆θ2| )
T*2′ [sin(|θ1 - |∆θ1|| - |∆θ2|) + sin(θ1 - |∆θ1|)] TR (19)
When T2*′ ) T2* eq 19 is identical to the general n ) 2 rfpulse result in eq 18 for Θ2 < 0. The graphical solution of the transcendental form in eq 19 for θ1 - |∆θ1| ) π/2 or equivalently Θ2 ) -π/2 is shown for various cn ) T2*′/TR in Figure 10a. Plots such as these can be used to determine |∆θ2| due to gradient switching at t > T for any choice of θ1 - |∆θ1|. Just like the two-pulse result, the only difference is a shifting of the maximum of the y2 curve to different |∆θ2| values as a function of θ1 - |∆θ1|. The analytical theorem in eq 19 can be used to determine the feasibility of using gradient switching to enable full threecomponent refocusing from -Mz to +Mz. Here the magnetization is prepared by applying a θ1 ) π-pulse. The central vector M(0,0) reorients and at a time t > T is inclined at an angle π |∆θ1| with respect to +z. For two π-pulses we saw that full refocusing occurred when |∆θ2| ) |∆θ1| and M(0,T′), as well as the full distribution, aligned along +z. That analysis demonstrated that M(0,T′) could be used as a measure of the behavior of the full distribution when H1(t) following the second θ2 ) π-pulse disappeared. Continuing in this vein, one might anticipate that for full refocusing of all three magnetization
J. Phys. Chem. B, Vol. 102, No. 42, 1998 8237 components, there exists a special angle |∆θ2| after the gradient switch that would allow restoration of M(0,T) to the +z direction. Inserting this angle as |∆θ2| ) π - |∆θ1| into eq 19, and realizing that 0 < θ1 - |∆θ1| < π, indicates that the absolute value can be dropped. Therefore the condition
(
)
T*2′ T*2 π ) -1 TR |∆θ1| TR
(20)
is required to rotate M(0,T) from -z to the +z direction. Using θ1 ) π in eq 11 indicates that for T2*/TR ) 5.0, one obtains |∆θ1| ) 2.596 ≈ 8π/10 and T2*′/TR ) 1.05 from eq 20. By the choice of the field inhomogeneities in this way, M(0,0) should rotate from the -z to the +z direction due to the gradient switch. Numerical integration of Bloch’s equations corrected for radiation damping agrees with the predictions of the analytical theorem in eq 19 and the conditions set by eq 20. In contrast to the π-π sequence, these calculations show that the vector M(0,0) can only be transformed from the -z to the +z direction when the line width restrictions set by eq 20 are met. What the theorem does not predict is the dependence of the full integrated z-component of magnetization on T2* and T2*′. Computer calculations show that even when M(0,0) is transformed from the -z to the +z direction, 〈Mz(∞)〉 never fully refocuses. This is presumably due to the special fact that the π-π sequence effectively inverts the reaction field by rotating the isochromats while gradient switching only modifies the distribution of the isochromats g(δ). Clearly, there is a complicated interplay between field inhomogeneity and H1(t) that is not contained or predicted by the analytical theorem. The only hope of deducing this dependence is in the complete analytical solution to Bloch’s equations corrected for radiation damping in an inhomogeneous system. Unfortunately, to date such a solution has not been possible. Conclusions By taking into account the unusual property of threecomponent free induction signals, one may conceive of pulse sequence and field gradient manipulations very much like those that have been developed for pulsed NMR. The analytical theorem developed here for one, two, and many rf-pulses with and without pulsed field gradients will be central to the design of such experiments. By using the theorem alone, one can determine properties of the three-component refocusing effect that are strangely different from the ordinary two-component case. The ordinary stimulated echo following a three-pulse sequence (e.g. the π/2-π/2-π/2 sequence) is independent of the defocusing and refocusing of u and V components prior to the third pulse because Mz remains constant. In the threecomponent case, however, Mz is also changing with u and V during this time manifesting itself in the analytical theorem as a nonzero tipping angle for 〈M(t)〉. As an example of these effects a radiation-damped stimulated echo was considered. The signal following a π-π/2-π/2 sequence was small in comparison to a substantially larger stimulated echo following a π-π/2-π sequence. These observations were explained by examining the tipping angle following the third pulse as predicted by the theorem for both of these experiments. In virtually all documented radiation-damping studies, the effect is considered to be a nuisance and the focus is on its elimination. By including field inhomogeneity, one may realize new applications where radiation damping is actually a critical factor to the success of the measurement. Consider a double
8238 J. Phys. Chem. B, Vol. 102, No. 42, 1998 resonance experiment in which an abundant spin species is refocused by radiation damping. This damping signal can serve as a detector of a resonant dipole-dipole or scalar-coupled second spin species. Even if only a fraction of the inhomogeneous spectrum of the abundant spins is coupled to the second spin, the perturbation of that fraction should be amplified by the radiation-damping signal because of coupling through the tuned circuit to all parts of the inhomogeneous line. Finally we note an unusual property relating the lifetime of the radiation-damping signal to the ratio T2*/TR. In standard spin echo experiments the echo lifetime is proportional to the inverse line width. Here, the lifetime of the radiation-damping signal lengthens for broader inhomogeneous lines. One would think that a greater spread of isochromats over an inhomogeneous spectrum should cause the signal to disappear faster due to dephasing. However, the smaller coherent polarization that results develops a weaker reaction field H1(t). Hence the rate slows down and H1(t) lasts a longer time despite dephasing. This property suggests a means for high-resolution NMR in very high field magnets where it is difficult to obtain purely homogeneously broadened narrow lines. Acknowledgment. The authors are indebted to Alex Pines for fruitful discussions and Dave Wemmer, Ho Cho, and Seth
Augustine and Hahn Bush for aid in the course of data acquisition. E.L.H. thanks Alex Pines for support during the course of this work, and M.P.A. gratefully acknowledges the National Science Foundation under Grant No. CHE-9504655. References and Notes (1) Bloembergen, N.; Pound, R. V. Phys. ReV. 1954, 95, 8. (2) Mao, X. A.; Ye, C. H. J. Chem. Phys. 1993, 99, 7455. (3) Broekaert, P.; Jeener, J J. Magn. Reson. Ser. A 1995, 113, 60. (4) Augustine, M. P.; Hahn, E. L. J. Chem. Phys. 1997, 107, 3324. (5) Szo¨ke, A.; Meiboom, S. Phys. ReV. 1959, 93, 99. (6) Hahn, E. L. In NMR and More-A Festschrift in Honor of Anatole Abragam; Goldman, M., Porneuf, M., Eds.; Les Editions de Physique: Paris, 1994. (7) Hahn, E. L. Concepts Magn. Reson. 1997, 9, 65. (8) Bloom, S. J. Appl. Phys. 1957, 28, 800. (9) McCall, S. L.; Hahn, E. L. Phys. ReV. 1969, 183, 457. (10) Dicke, R. H. Phys. ReV. 1954, 93, 99. (11) Feld, M. S.; MacGillivary, J. C. Coherent Nonlinear Optics; Feld, M. S., Letokhov, V. S., Eds.; Springer: Berlin, 1980. (12) Bo¨siger, P.; Brun, E.; Meier D. Phys. ReV. Lett. 1977, 38, 602. (13) Hahn, E. L. Phys. ReV. 1950, 80, 580. (14) Augustine, M. P.; Zilm, K. W. J. Magn. Reson., Ser. A 1996, 123, 145. (15) Augustine, M. P.; Hahn, E. L. Mol. Phys., in press.
