- Email: [email protected]
Noise triggering of radiation damping from the inverted state
12 May 2000
Chemical Physics Letters 322 Ž2000. 111–118 www.elsevier.nlrlocatercplett
Noise triggering of radiation damping from the inverted state Matthew P. Augustine a
a,)
, Seth D. Bush b, Erwin L. Hahn
b
Department of Chemistry, One Shields AÕenue, UniÕersity of California, DaÕis, CA 95616, USA b Departments of Chemistry and Physics, UniÕersity of California, Berkeley, CA 94720, USA Received 14 March 2000; in final form 23 March 2000
Abstract Spontaneous radiation damping in nuclear magnetic resonance spectroscopy of sharp lines is commonly encountered in highly magnetized samples following magnetization inversion. The time delay for the maximum of the hyperbolic secant like signal to develop following inversion is measured and calculated. The distribution function of the measured delay times is Gaussian–Boltzmann and can be used to predict the sensitivity of both signal delay and phase to inversion errors. q 2000 Elsevier Science B.V. All rights reserved.
An inverted two-level system coupled to a cavity at resonance emits coherent radiation spontaneously above or at the maser threshold condition. This applies to nuclear magnetic resonance ŽNMR. as a special case w1x of radiation damping w2x for the condition T2rTR 0 1. The inverse line width is given by T2 , and the radiation damping time TR , is defined by T R s Ž 2 pg M 0 Q j .
y1
Ž 1.
where M0 is the equilibrium magnetization, g is the gyromagnetic ratio, Q is the cavity figure of merit, and j is the sample filling factor. In general, a u 0 pulse applied to the entire spin ensemble is followed by completion of a radiation damping signal, and the magnetization ends up along the polarizing field with a lesser final energy, having dissipated energy into
) Corresponding author. Fax: q1-530-752-8995; e-mail: [email protected]
the circuit resistance 1 w3,4x. From the initial u 0 orientation, the total angle D u through which the magnetization component at exact resonance tilts toward the ground state is given by w5x < Du < s
T2 TR
sin Ž u 0 y < D u < . .
Ž 2.
Here T2 is determined by symmetric inhomogeneous broadening. For an exact u 0 s p inversion pulse, the transcendental Eq. Ž2. reduces to < Du < s
T2 TR
sin < D u < .
Ž 3.
Eq. Ž3. specifies that the spin system may end up in either of two final states, as indicated in Fig. 1. For < D u < / 0, emission of radiation must occur, leading toward a final lower energy polarization state; or for
1 Analogous to coherent optical superradiant emission appearing in the form of propagation.
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 3 9 0 - 0
112
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
< D u < s 0, the exactly inverted polarization remains in a metastable inverted state and does not radiate even if the condition T2rTR 0 1 applies. The latter condition of trapping in the inverted state must always occur first if the magnetization has been prepared in an exact inverted state because there is no radiating polarization that can couple to the tuned circuit. After a time delay w6,7x the two-level system will escape from the metastable state and radiate coherently if a weak perturbation of some kind tilts the magnetization sufficiently away from its initial exact p alignment. Room temperature rf noise is the ever present perturbation of importance. After the circuit noise field H1 n initiates a small tilt angle away from the yz direction defined by uy, the radiation damping reaction field H1R comes into play and reinforces the increase in uy. As uy increases at random, both the mean square angle ² uy2 : and the H1R reaction field increase along with it. It will be seen that the combined effects of H1 n and H1R lead to the development of amplified spin noise in the inverted state and to the establishment of steady state spin noise w8x in the equilibrium ground state. The solid lines in Fig. 2 define both the transverse inphase u and quadrature out-of-phase Õ components of the signal following a p rf pulse. The dashed line corresponds to the envelope of the signal. The time t 0 is measured from the p rf pulse to the maximum of this signal. The time tdelay corresponds to the average value for t 0 following many measurements
Fig. 2. An example of the signal measured following a p rf pulse applied to a H 2 OrD 2 O Ž95:5%. solution at a 600 MHz 1 H Larmor frequency for TR s 7.1 ms. The two solid lines correspond to the in-phase u and out-of-phase Õ magnetization, while the dashed line labels the signal envelope. The time from completion of the rf pulse to the signal maximum defines t 0 as shown by the arrow in the figure. For a given TR , circuit noise will vary the time t 0 and the phase of the signal arctanŽ Õr u. at t 0 from experiment to experiment. Therefore, the time tdelay calculated here is the most probable value of t 0 extracted from many measurements.
of signals like those in Fig. 2. The signal phase at t 0 can also be extracted from these signals as arctanŽ Õru. as shown in Fig. 2. To evaluate the time tdelay and to introduce some nomenclature, consider first the equation for the tilt angle u of the magnetization from the qz direction in the sharp line limit T2 s `: du
sin u sy
dt
TR
.
Ž 4.
The hyperbolic secant solution w2x to this equation is proportional to the transverse magnetization Õ s M0 sin u s M0 sech Fig. 1. Graphical solutions to the transcendental Eq. Ž3. as a function of c n sT2 r TR . In all cases the metastable inverted state < D u < s 0 is a solution. However, when c n )1, < D u < / 0 values exist. Under these conditions, circuit noise stimulates the transition to the ground state.
t y t0
ž /
Ž 5.
TR
where t 0 s TR arcsech Ž sin u 0 . s TR ln
2
ž / tan u 0
Ž 6.
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
defines the duration of signal coherence extending from the application of any u 0 pulse at t s 0 to the maximum of the signal given by Eq. Ž5. at t s t 0 . Any fluctuations in the direction of the magnetization due to an incoherent circuit noise field H1 n introduces an additional term u˙n s g H1 n into Eq. Ž4.. A Langevin equation of the form du
u q
dt
TR
s u˙n
Ž 7.
is obtained in the small tip angle limit sin u f u near the ground state. We first apply Eq. Ž7. to evaluate the approach to thermal equilibrium when a balance in the ground state is reached between the emission of radiation damping and the circuit noise field stimulated absorption w8x. ŽFollowing this analysis, Eq. Ž7. will then be modified to deal with non-equilibrium time delay effects after inversion of the magnetization when the circuit noise field induces stimulated emission instead of absorption.. The time dependence of the average of the mean square deviation of the magnetization from the qz direction, given by d² u 2 :
2² u 2 : q
dt
TR
s 2² uu˙n :
Ž 8.
is obtained from Eq. Ž7. in the usual way by using the identity d u 2rd t s 2 u d urdt followed by application of an ensemble average as denoted by the paired brackets. To determine ² uu˙n :, consider Eq. Ž8. at steady state equilibrium where ² u 2 : s ² uu˙n :TR < 1. The equipartition theorem for the energy excess in the u degree of freedom is given by
™ ™
² M P H : y M0 H0 s M0 H0 Ž ²cos u : y 1 . f
M0 H0 2
² u 2 : s kTcrVs
Ž 9.
where k is Boltzmann’s constant, H0 is the magnetic field, Vs is the sample volume, and Tc is the temperature of the circuit resistance. Eq. Ž9. is then applied to Eq. Ž8. to evaluate the quantity ² uu˙n :, and the solution for ² u 2 : becomes ²u 2 : s
2 kTc M0 H0 Vs
1 y exp Ž y2 trTR . .
Ž 10 .
113
At t s ` this equation indicates that on average the single magnetization vector initially aligned along the qz direction will spread to a final mean square angular width ² u 2 :` due to the balance between absorption caused by H1 n and emission caused by H1R . This mean square angle established at equilibrium for I s 1r2 is given by ² u 2 :` s
2 kTc M0 H0 Vs
s
² M x2 : q ² M y2 : M02
Ng 2 " 2 s
2 M02
,
Ž 11 . identical to that anticipated for spin noise fluctuations w8x when the temperature Tc is equal to the lattice temperature Tlat . In the more general case when Tc / Tlat , the right-hand side of Eq. Ž11. must be multiplied by TcrTlat . More explicitly, it follows that ² u 2 :` s 2² 2 : 2 g H1R TR s g 2 ² H12n :TR Qrv defines the thermal 2 : equilibrium angle where ² H1R s ² H12n : QrTR v 2 and ² H1 n : s 4 p kTcrVc is obtained from the Nyquist theorem. Thus the average intensity of the radiation damping field emission is balanced by the fraction QrTR v of the circuit noise field intensity of absorption within the bandwidth 1rTR of the spins, neglecting relaxation. Data obtained at 600 MHz gives ² u 2 :` s 7 = 10y1 4 rad 2 and TR s 10 ms, yielding 2 :1r2 ² H1R f 1 nG. Before proceeding with the calculation of tdelay , one must be completely aware of the experimental conditions. To achieve essentially exact initial alignment in practice, care must be taken to avoid a small initial tilt angle of the inverted magnetization due to slight inaccuracies in pulse width, the inhomogeneity of the applied rf field over the sample, and excitation caused by insufficiently shielded rf leakage. This leakage is avoided by using both high isolation rf switches and incorporating additional crossed diodes between the rf-transmitter and NMR probehead. Unwanted macroscopic coherence due to pulse width errors and rf inhomogeneity are removed by the application of a sufficiently strong DC field gradient pulse just after the p rf pulse. Care must be exercised in the choice of field gradient pulse shapes. The eddy currents due to square pulses tend to produce transients that reintroduce small tipping of the inverted magnetization. These effects were mini-
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
114
mized in all measurements by using a 10 ms Gaussian gradient pulse shape having a 29 Grcm peak amplitude and 4 ms full width at half height. While radiation damping dominates the evolution with a line width of 10 Hz, during the 10 ms gradient pulse period the linewidth jumps from f 10 Hz to f 250 kHz and radiation damping is suppressed. This can be justified from Eq. Ž3. by comparing < D u < for both the 10 Hz and 250 kHz wide lines where T2 s 100 and 4 ms, respectively. Taking TR s 5 ms gives < D u < s 08 and 171.48 for the narrow line, indicating that both the metastable and ground state are accessible to the magnetization as shown in Fig. 1 for c 3 , c 4 , and c5 . On the other hand, for the same TR value, the wide line gives only < D u < s 0, suggesting that radiation damping does not effect a rotation of the magnetization towards the ground qz direction as shown in Fig. 1 for c1 and c 2 . Since application of the gradient has the dual effect of removing any unwanted tilt and suspending the effects of radiation damping, the development of ² uy2 : from zero due to nearly exact alignment along the yz direction immediately after the field gradient pulse can be evaluated. The solution for the time delay t 0 s TR ln
2
ž /
Ž 12 .
uy
may be expressed in terms of the small tip angle uy from the yz direction using Eq. Ž6., where u 0 s p y uy and tan uyf uy. In the inverted state, M0 changes sign and the sign of TR changes in Eq. Ž10. to give ² uy2 : s ² u 2 :` exp Ž 2 trTR . y 1 s uy2 s4 exp 2 Ž t y t 0 . rTR
Ž 13 .
after equating Eq. Ž10. to Eq. Ž5. in the small angle limit. Eq. Ž13. exhibits gain in the exponential regime of Eq. Ž6. but can only be applied for restricted times 0 - t - tdelay . Therefore ² uy2 : may be equated with the solution for uy2 from Eq. Ž5. in the exponential regime near t s 0, but where t s tX is large enough so that in Eq. Ž13. ² uy2 : f ² u 2 :`
X
exp w 2 t rTR x 4 ² u
2: `
Therefore a connection is made between the incoherent and coherent properties of the tipping magnetization for uy< 1. The time tX is chosen long enough so that the coherence of angle g H1R TR associated with H1R dominates over the smaller induced incoherent angle caused by H1 n field fluctuations from the circuit. Under these conditions, the solution for t 0 in Eq. Ž14. leads directly to the average time tdelay s
TR 2
Ž 14 .
ž
4 ² u 2 :`
/
.
Ž 15 .
The paradox that t 0 s ` obtained from Eq. Ž12. when uys 0 is thus avoided when M0 is perfectly inverted after an ideal p pulse. Instead, after sufficient time tX the diffusion Eq. Ž13. develops an intermediate mean square angle ² uy2 Ž tX .: which defines tX s
TR 2
ln
ž
² uy Ž tX . 2 :
/
² u 2 :`
.
Ž 16 .
This means that ² uy Ž tX . 2 : s g 2 H12R TR2 4 ² u 2 :` specifies a coherent tipping angle much larger than any tipping caused by ² H12n : due to circuit noise. The coherent relation Eq. Ž12. then gives an additional delay time tX0 s
TR 2
ln
ž
4 ² uy Ž tX . 2 :
/
Ž 17 .
to be added to tX . The sum tX0 q tX s tdelay is again given by Eq. Ž15.. The essential assumption here is that while incoherence is dominant, the relation Eq. Ž15. which is obtained for a coherent model cannot be applied. It is only valid when initial angles uy in the distribution are large enough above the influence of circuit noise so that they appear equivalent to coherent initial tip angles u 0 in Eqs. Ž2. and Ž6.. The sum of the logarithms given by tX0 q tX reduces to a tdelay independent of t itself because ² uy Ž tX . 2 : cancels out leaving only the dependence on the equilibrium mean square angle ² u 2 :` . From knowledge of the mean square thermal equilibrium angle ² u 2 :` a Boltzmann probability distribution in uy may defined as P Ž uy . s
.
ln
1
(2 p ²u
2: `
exp yuy2 r Ž 2² u 2 :` .
Ž 18 .
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
115
that can be applied immediately following application of the field gradient pulse. The probability that M0 is oriented at any initial angle uy is determined by the distribution P Ž uy .. The average value of the time delay is then given by the normalized average
tdelay s ² t 0 :sTR p
=
H0
ln
( 2
2p ² u 2 :`
ž / uy
P Ž uy . sin uy d uy .
Ž 19 .
This expression incorporates the solid angle weighting factor Žsin uyr2. of M0 on the unit sphere about the yz direction. Note that the application of sin uy in Eq. Ž19. implies that for uys 0 the single vector magnetization is never exactly aligned along the z axis. It is in fact tipped slightly off axis in a direction representing an eigenstate of the magnetization because the spin system is coupled to the single mode of the cavity w8x as well as to the field H0 . It turns out that tdelay expressed by Eq. Ž15. depends upon the most probable Žpeak. value uy2 s ² u 2 :` of the weighted distribution sin uy P Ž uy . f uy P Ž uy . in Eq. Ž19., at which t 0 displays its average value. The validity of the expressions for tdelay in Eqs. Ž15. and Ž19. for the probability were tested experimentally using the 1 H signal in H 2 OrD 2 O mixtures at 600 MHz as shown in Fig. 3. For a specific TR value the histograms summarize 250 separate measurements of t 0 . A summary of experimental TR , ² u 2 :` , and tdelay values measured from the width and center of the hyperbolic secant bursts summarized in Fig. 3 is provided in Table 1 along with a comparison to theoretical estimates of these parameters from Eqs. Ž1., Ž11. and Ž15. with Q j s 9.3. Superimposed on each histogram for each choice of TR in Fig. 3 is a solid line representing the distribution of t 0 determined from the TR value calculated from Eq. Ž1. and the ² u 2 :` value calculated from Eq. Ž15. and tdelay . Here the abscissa is determined from Eq. Ž15. by scaling ² u 2 :`1r2 from its peak value by an amount "duy, while the ordinate is calculated from Eq. Ž18. using uys ² u 2 :`1r2 " duy. Returning to Table 1, both the experimental and theoretical TR and tdelay values are consistent for the three highest concentrations. The shape of the curve
Fig. 3. Summary of t 0 measurements for H 2 OrD 2 O Ž95:5%., H 2 OrD 2 O Ž75:25%., H 2 OrD 2 O Ž50:50%., and H 2 OrD 2 O Ž25:75%. solutions in Žb., Žc., Žd., and Že., respectively. Each histogram represents 250 separate determinations of t 0 at each concentration or equivalently each specific TR . The time tdelay at each TR corresponds to the average value of the distribution in Eq. Ž18. as seen in the compilation in Ža.. These values for tdelay along with theoretical estimates are provided in Table 1. The solid line in each graph corresponds to the distribution anticipated from Eq. Ž18..
predicted by Eq. Ž19. is similar to that predicted by Glauber w9x. The NMR measurements in Fig. 3 more literally conform to the theoretical predictions of Eq. Ž19. than do previous optical measurements by Vehren w10x in an inhomogeneously broadened optical two-level system with damping w11x. One source of error between experimental and theoretical ² u 2 :` values could lie in the assumption that the gradient pulse completely removes any initial tip angle, and therefore the Gaussian distribution P Ž uy, fy . in the angle uy becomes skewed. For perfect inversion, the distribution in signal phase fy is expected to be random since P Ž uy . is independent of fy. Fig. 4 summarizes the distribution of phases at t 0 extracted from the same 250 measurements used to determine tdelay in Fig. 3. Fig. 4a represent-
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
116
Table 1 Summary of theoretical and experimental values ² u 2 :`
H 2O
tR
Ž%.
Experiment Žms.
Theory Žms.
Experiment
Theory
Experiment Žms.
Theory Žms.
95 75 50 25
7.1 9.8 15.0 27.6
7.0 9.0 13.3 26.6
1.2 = 10 – 11 1.1 = 10 – 11 7.9 = 10 – 12 1.0 = 10 – 11
7.0 = 10 – 14 9.8 = 10 – 14 1.5 = 10 – 13 3.0 = 10 – 13
96.6 131.4 207.3 377.7
110.8 141.0 205.6 401.9
ing the highest H 2 O concentration is clearly a random phase distribution while the data for both the 75% and 25% H 2 O concentrations in Fig. 4b,d deviate from the completely random result. Most pronounced is the very tight distribution about 898 for the data in Fig. 4c. The data in Fig. 3a, Fig. 4a, and Table 1 indicate that the discrepancy between experimental and theoretical ² u 2 :` values persists even when inversion performance is nearly ideal. A likely cause of this error is linebroadening due to magnetic
Tdelay
field inhomogeneity, an effect not included in the current treatment. To account for the phase deviations for the three other 1 H concentrations studied here, the distribution in Eq. Ž18. must be adjusted to account for a small tip angle h remaining after application of the field gradient pulse. Here, the distribution is still independent of phase x about the axis tipped away from the yz direction by the angle h , but dependent on both polar angles uy and fy concentrated on the yz axis. Recasting the distribution in terms of the uy and fy polar angles as in shown in Fig. 5 gives
P Ž uy , fy . s
1
(2 p ²u
2: `
exp y Ž h 2 q uy2
y2huy cos fy . r Ž 2² u 2 :` .
Ž 20 .
which can be used to determine just the distribution in fy by weighting P Ž uy, fy . by uy, normalizing, and integrating over the variable uy:
P Ž fy . s Fig. 4. Summary of signal phase measurements at t s t 0 for H 2 OrD 2 O Ž95:5%., H 2 OrD 2 O Ž75:25%., H 2 OrD 2 O Ž50:50%., and H 2 OrD 2 O Ž25:75%. solutions in Ža., Žb., Žc., and Žd., respectively. Each histogram represents 250 separate determinations of the phase at each TR value. The phase in Ža. is random as expected from the distribution in Eq. Ž18. but the spread of the data in Žb., Žc., and Žd. display a phase buildup at 2208, 898, and 08, respectively. The departure from random phase and its anisotropy caused by small tip angle h in Žc. is explained with the distribution function in Eq. Ž20. and does not generate appreciable error in the time tdelay .
(
2p ²u
2: `
q`
Hy` P Ž u
y , fy
. uy d uy
s exp y Ž h 2 sin2 fy . r Ž 2² u 2 :` .
ž
2p
( /
= 1 q h cos fy
² u 2 :`
.
Ž 21 .
For h s 0 the distribution centered around the yz direction is uniform from 0 ( fy( 2 p. As h is
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
117
integral which extracts the most probable value of uy will also depend on h and fy as tdelay s
f
TR 2 TR 2
Fig. 5. Graphical determination of the Gaussian probability distribution appropriate for tip angle error. The Gaussian is symmetric in the angle x about the axis inclined by angle h away from the y z direction. The displacement in terms of h , uy, and fy for small uy can be determined by the law of cosines.
increased, the distribution of phase begins to concentrate around the direction of h. As the term proportional to h cos fy in Eq. Ž21. becomes negative for pr2 ( fy( 3pr2 values and less than y1, P Ž fy . becomes a negative probability and the values of fy that satisfy this condition cannot occur. Very large values of h cause the phase distribution in the regions from 0 ( fy( pr2 and 3pr2 ( fy( 2 p to concentrate in a narrow region along the direction of h as the width of the distribution in Eq. Ž20. 2² u 2 :` rh 2 becomes smaller. In the limit that h is small, the full width at half height of the distribution in Eq. Ž 21 . is 2 ² u 2 :` 2 ln 2rh 2 and can be used to determine the sensitivity of the signal phase on h. For example, the full width at half height in Fig. 4c of 0.28 translates into a tip angle error of only h s 0.088. The remarkable feature of the data in Figs. 3 and 4 is the insensitivity of tdelay to tip angle error. The comparison of tdelay with theory for h s 0 in Table 1 suggests that the actual h s 0.088 error in the data sets in Fig. 4c does not appreciably affect tdelay . To appreciate this dependence on h consider the most probable value t delay from the integral of TR lnŽ2ruy . over the uy weighted distribution function in Eq. Ž20.. Since the distribution function in Eq. Ž20. depends on both h and fy, the resulting
(
4
ln
ž /
ln
ž
uy2
4 ²u
2: ` qh
(
cos fy ² u 2 :`
/
Ž 22 .
where quadratic and higher-order terms in h are neglected. Notice that for h 2 < ² u 2 :` , Eq. Ž22. reduces to Eq. Ž15.. This relation implies that unless h 2 4 ² u 2 :` , tdelay will not appreciably change from the h s 0 value. Also, since the h dependence following integration over uy still remains in the natural logarithm the overall effect of h on tdelay will be less pronounced. In summary, we have shown that a Gaussian– Boltzmann distribution in the tipping angle degree of freedom of a single vector magnetization M0 , governed by the circuit resistance temperature, accounts for the evolution in time of radiation damping emission of Zeeman energy from the inverted state, and also for the evolution toward equilibrium with the circuit near the ground state. Further refinement of the theory will take into account magnetic field inhomogeneity. The strong phase dependence of the hyperbolic secant radiation damping signal on tip angle error h may become a very sensitive detector of magnetic field. Any slight tip of the perfectly inverted magnetization from the yz direction by either increased rf noise, low-level coherent rf, the bulk demagnetizing field of dilute solutes, or the radiation damping signal for a dilute solute, will induce the phase transition from random to coherent as in Fig. 3. It is this transition that could form the basis for an extremely sensitive measurement of NMR in dilute solution, much like the performance of an injection seeded laser in optical spectroscopy.
Acknowledgements We thank Alex Pines for support and gratefully acknowledge the assistance of Thorsten Dieckmann in the course of data acquisition. M.P.A. thanks U.C. Davis and the Packard foundation for support during the course of this work.
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M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
References w1x P. Bosiger, E. Brun, D. Meier, Phys. Rev. Lett. 38 Ž1977. 602. w2x N. Bloembergen, R.V. Pound, Phys. Rev. 95 Ž1954. 8. w3x R.H. Dicke, Phys. Rev. 93 Ž1954. 99. w4x M.S. Feld, J.C. MacGillvary, in: M.S. Feld, V.S. Letokhov ŽEds.., Coherent Nonlinear Optics, Springer, Berlin, 1980. w5x M.P. Augustine, E.L. Hahn, J. Phys. Chem. 102 Ž1998. 8229.
w6x A. Sodickson, W.E. Maas, D.G. Cory, J. Mag. Res. B 110 Ž1991. 298. w7x J.H. Chen, X.A. Mao, J. Phys. D 32 Ž1999. 764. w8x T. Sleator, E.L. Hahn, C. Hilbert, J. Clarke, Phys. Rev. B 36 Ž1987. 1969. w9x R. Glauber, F. Haake, Phys. Lett. 68A Ž1978. 29. w10x Q.H.F. Vrehen, in: H. Walther, D.W. Rothe ŽEds.., Laser Spectroscopy IV, Springer, Berlin, 1979. w11x F. Haake, J.W. Haus, H. King, G. Schroder, R. Glauber, Phys. Rev. A 23 Ž1981. 1322.
Chemical Physics Letters 322 Ž2000. 111–118 www.elsevier.nlrlocatercplett
Noise triggering of radiation damping from the inverted state Matthew P. Augustine a
a,)
, Seth D. Bush b, Erwin L. Hahn
b
Department of Chemistry, One Shields AÕenue, UniÕersity of California, DaÕis, CA 95616, USA b Departments of Chemistry and Physics, UniÕersity of California, Berkeley, CA 94720, USA Received 14 March 2000; in final form 23 March 2000
Abstract Spontaneous radiation damping in nuclear magnetic resonance spectroscopy of sharp lines is commonly encountered in highly magnetized samples following magnetization inversion. The time delay for the maximum of the hyperbolic secant like signal to develop following inversion is measured and calculated. The distribution function of the measured delay times is Gaussian–Boltzmann and can be used to predict the sensitivity of both signal delay and phase to inversion errors. q 2000 Elsevier Science B.V. All rights reserved.
An inverted two-level system coupled to a cavity at resonance emits coherent radiation spontaneously above or at the maser threshold condition. This applies to nuclear magnetic resonance ŽNMR. as a special case w1x of radiation damping w2x for the condition T2rTR 0 1. The inverse line width is given by T2 , and the radiation damping time TR , is defined by T R s Ž 2 pg M 0 Q j .
y1
Ž 1.
where M0 is the equilibrium magnetization, g is the gyromagnetic ratio, Q is the cavity figure of merit, and j is the sample filling factor. In general, a u 0 pulse applied to the entire spin ensemble is followed by completion of a radiation damping signal, and the magnetization ends up along the polarizing field with a lesser final energy, having dissipated energy into
) Corresponding author. Fax: q1-530-752-8995; e-mail: [email protected]
the circuit resistance 1 w3,4x. From the initial u 0 orientation, the total angle D u through which the magnetization component at exact resonance tilts toward the ground state is given by w5x < Du < s
T2 TR
sin Ž u 0 y < D u < . .
Ž 2.
Here T2 is determined by symmetric inhomogeneous broadening. For an exact u 0 s p inversion pulse, the transcendental Eq. Ž2. reduces to < Du < s
T2 TR
sin < D u < .
Ž 3.
Eq. Ž3. specifies that the spin system may end up in either of two final states, as indicated in Fig. 1. For < D u < / 0, emission of radiation must occur, leading toward a final lower energy polarization state; or for
1 Analogous to coherent optical superradiant emission appearing in the form of propagation.
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 3 9 0 - 0
112
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
< D u < s 0, the exactly inverted polarization remains in a metastable inverted state and does not radiate even if the condition T2rTR 0 1 applies. The latter condition of trapping in the inverted state must always occur first if the magnetization has been prepared in an exact inverted state because there is no radiating polarization that can couple to the tuned circuit. After a time delay w6,7x the two-level system will escape from the metastable state and radiate coherently if a weak perturbation of some kind tilts the magnetization sufficiently away from its initial exact p alignment. Room temperature rf noise is the ever present perturbation of importance. After the circuit noise field H1 n initiates a small tilt angle away from the yz direction defined by uy, the radiation damping reaction field H1R comes into play and reinforces the increase in uy. As uy increases at random, both the mean square angle ² uy2 : and the H1R reaction field increase along with it. It will be seen that the combined effects of H1 n and H1R lead to the development of amplified spin noise in the inverted state and to the establishment of steady state spin noise w8x in the equilibrium ground state. The solid lines in Fig. 2 define both the transverse inphase u and quadrature out-of-phase Õ components of the signal following a p rf pulse. The dashed line corresponds to the envelope of the signal. The time t 0 is measured from the p rf pulse to the maximum of this signal. The time tdelay corresponds to the average value for t 0 following many measurements
Fig. 2. An example of the signal measured following a p rf pulse applied to a H 2 OrD 2 O Ž95:5%. solution at a 600 MHz 1 H Larmor frequency for TR s 7.1 ms. The two solid lines correspond to the in-phase u and out-of-phase Õ magnetization, while the dashed line labels the signal envelope. The time from completion of the rf pulse to the signal maximum defines t 0 as shown by the arrow in the figure. For a given TR , circuit noise will vary the time t 0 and the phase of the signal arctanŽ Õr u. at t 0 from experiment to experiment. Therefore, the time tdelay calculated here is the most probable value of t 0 extracted from many measurements.
of signals like those in Fig. 2. The signal phase at t 0 can also be extracted from these signals as arctanŽ Õru. as shown in Fig. 2. To evaluate the time tdelay and to introduce some nomenclature, consider first the equation for the tilt angle u of the magnetization from the qz direction in the sharp line limit T2 s `: du
sin u sy
dt
TR
.
Ž 4.
The hyperbolic secant solution w2x to this equation is proportional to the transverse magnetization Õ s M0 sin u s M0 sech Fig. 1. Graphical solutions to the transcendental Eq. Ž3. as a function of c n sT2 r TR . In all cases the metastable inverted state < D u < s 0 is a solution. However, when c n )1, < D u < / 0 values exist. Under these conditions, circuit noise stimulates the transition to the ground state.
t y t0
ž /
Ž 5.
TR
where t 0 s TR arcsech Ž sin u 0 . s TR ln
2
ž / tan u 0
Ž 6.
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
defines the duration of signal coherence extending from the application of any u 0 pulse at t s 0 to the maximum of the signal given by Eq. Ž5. at t s t 0 . Any fluctuations in the direction of the magnetization due to an incoherent circuit noise field H1 n introduces an additional term u˙n s g H1 n into Eq. Ž4.. A Langevin equation of the form du
u q
dt
TR
s u˙n
Ž 7.
is obtained in the small tip angle limit sin u f u near the ground state. We first apply Eq. Ž7. to evaluate the approach to thermal equilibrium when a balance in the ground state is reached between the emission of radiation damping and the circuit noise field stimulated absorption w8x. ŽFollowing this analysis, Eq. Ž7. will then be modified to deal with non-equilibrium time delay effects after inversion of the magnetization when the circuit noise field induces stimulated emission instead of absorption.. The time dependence of the average of the mean square deviation of the magnetization from the qz direction, given by d² u 2 :
2² u 2 : q
dt
TR
s 2² uu˙n :
Ž 8.
is obtained from Eq. Ž7. in the usual way by using the identity d u 2rd t s 2 u d urdt followed by application of an ensemble average as denoted by the paired brackets. To determine ² uu˙n :, consider Eq. Ž8. at steady state equilibrium where ² u 2 : s ² uu˙n :TR < 1. The equipartition theorem for the energy excess in the u degree of freedom is given by
™ ™
² M P H : y M0 H0 s M0 H0 Ž ²cos u : y 1 . f
M0 H0 2
² u 2 : s kTcrVs
Ž 9.
where k is Boltzmann’s constant, H0 is the magnetic field, Vs is the sample volume, and Tc is the temperature of the circuit resistance. Eq. Ž9. is then applied to Eq. Ž8. to evaluate the quantity ² uu˙n :, and the solution for ² u 2 : becomes ²u 2 : s
2 kTc M0 H0 Vs
1 y exp Ž y2 trTR . .
Ž 10 .
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At t s ` this equation indicates that on average the single magnetization vector initially aligned along the qz direction will spread to a final mean square angular width ² u 2 :` due to the balance between absorption caused by H1 n and emission caused by H1R . This mean square angle established at equilibrium for I s 1r2 is given by ² u 2 :` s
2 kTc M0 H0 Vs
s
² M x2 : q ² M y2 : M02
Ng 2 " 2 s
2 M02
,
Ž 11 . identical to that anticipated for spin noise fluctuations w8x when the temperature Tc is equal to the lattice temperature Tlat . In the more general case when Tc / Tlat , the right-hand side of Eq. Ž11. must be multiplied by TcrTlat . More explicitly, it follows that ² u 2 :` s 2² 2 : 2 g H1R TR s g 2 ² H12n :TR Qrv defines the thermal 2 : equilibrium angle where ² H1R s ² H12n : QrTR v 2 and ² H1 n : s 4 p kTcrVc is obtained from the Nyquist theorem. Thus the average intensity of the radiation damping field emission is balanced by the fraction QrTR v of the circuit noise field intensity of absorption within the bandwidth 1rTR of the spins, neglecting relaxation. Data obtained at 600 MHz gives ² u 2 :` s 7 = 10y1 4 rad 2 and TR s 10 ms, yielding 2 :1r2 ² H1R f 1 nG. Before proceeding with the calculation of tdelay , one must be completely aware of the experimental conditions. To achieve essentially exact initial alignment in practice, care must be taken to avoid a small initial tilt angle of the inverted magnetization due to slight inaccuracies in pulse width, the inhomogeneity of the applied rf field over the sample, and excitation caused by insufficiently shielded rf leakage. This leakage is avoided by using both high isolation rf switches and incorporating additional crossed diodes between the rf-transmitter and NMR probehead. Unwanted macroscopic coherence due to pulse width errors and rf inhomogeneity are removed by the application of a sufficiently strong DC field gradient pulse just after the p rf pulse. Care must be exercised in the choice of field gradient pulse shapes. The eddy currents due to square pulses tend to produce transients that reintroduce small tipping of the inverted magnetization. These effects were mini-
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
114
mized in all measurements by using a 10 ms Gaussian gradient pulse shape having a 29 Grcm peak amplitude and 4 ms full width at half height. While radiation damping dominates the evolution with a line width of 10 Hz, during the 10 ms gradient pulse period the linewidth jumps from f 10 Hz to f 250 kHz and radiation damping is suppressed. This can be justified from Eq. Ž3. by comparing < D u < for both the 10 Hz and 250 kHz wide lines where T2 s 100 and 4 ms, respectively. Taking TR s 5 ms gives < D u < s 08 and 171.48 for the narrow line, indicating that both the metastable and ground state are accessible to the magnetization as shown in Fig. 1 for c 3 , c 4 , and c5 . On the other hand, for the same TR value, the wide line gives only < D u < s 0, suggesting that radiation damping does not effect a rotation of the magnetization towards the ground qz direction as shown in Fig. 1 for c1 and c 2 . Since application of the gradient has the dual effect of removing any unwanted tilt and suspending the effects of radiation damping, the development of ² uy2 : from zero due to nearly exact alignment along the yz direction immediately after the field gradient pulse can be evaluated. The solution for the time delay t 0 s TR ln
2
ž /
Ž 12 .
uy
may be expressed in terms of the small tip angle uy from the yz direction using Eq. Ž6., where u 0 s p y uy and tan uyf uy. In the inverted state, M0 changes sign and the sign of TR changes in Eq. Ž10. to give ² uy2 : s ² u 2 :` exp Ž 2 trTR . y 1 s uy2 s4 exp 2 Ž t y t 0 . rTR
Ž 13 .
after equating Eq. Ž10. to Eq. Ž5. in the small angle limit. Eq. Ž13. exhibits gain in the exponential regime of Eq. Ž6. but can only be applied for restricted times 0 - t - tdelay . Therefore ² uy2 : may be equated with the solution for uy2 from Eq. Ž5. in the exponential regime near t s 0, but where t s tX is large enough so that in Eq. Ž13. ² uy2 : f ² u 2 :`
X
exp w 2 t rTR x 4 ² u
2: `
Therefore a connection is made between the incoherent and coherent properties of the tipping magnetization for uy< 1. The time tX is chosen long enough so that the coherence of angle g H1R TR associated with H1R dominates over the smaller induced incoherent angle caused by H1 n field fluctuations from the circuit. Under these conditions, the solution for t 0 in Eq. Ž14. leads directly to the average time tdelay s
TR 2
Ž 14 .
ž
4 ² u 2 :`
/
.
Ž 15 .
The paradox that t 0 s ` obtained from Eq. Ž12. when uys 0 is thus avoided when M0 is perfectly inverted after an ideal p pulse. Instead, after sufficient time tX the diffusion Eq. Ž13. develops an intermediate mean square angle ² uy2 Ž tX .: which defines tX s
TR 2
ln
ž
² uy Ž tX . 2 :
/
² u 2 :`
.
Ž 16 .
This means that ² uy Ž tX . 2 : s g 2 H12R TR2 4 ² u 2 :` specifies a coherent tipping angle much larger than any tipping caused by ² H12n : due to circuit noise. The coherent relation Eq. Ž12. then gives an additional delay time tX0 s
TR 2
ln
ž
4 ² uy Ž tX . 2 :
/
Ž 17 .
to be added to tX . The sum tX0 q tX s tdelay is again given by Eq. Ž15.. The essential assumption here is that while incoherence is dominant, the relation Eq. Ž15. which is obtained for a coherent model cannot be applied. It is only valid when initial angles uy in the distribution are large enough above the influence of circuit noise so that they appear equivalent to coherent initial tip angles u 0 in Eqs. Ž2. and Ž6.. The sum of the logarithms given by tX0 q tX reduces to a tdelay independent of t itself because ² uy Ž tX . 2 : cancels out leaving only the dependence on the equilibrium mean square angle ² u 2 :` . From knowledge of the mean square thermal equilibrium angle ² u 2 :` a Boltzmann probability distribution in uy may defined as P Ž uy . s
.
ln
1
(2 p ²u
2: `
exp yuy2 r Ž 2² u 2 :` .
Ž 18 .
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
115
that can be applied immediately following application of the field gradient pulse. The probability that M0 is oriented at any initial angle uy is determined by the distribution P Ž uy .. The average value of the time delay is then given by the normalized average
tdelay s ² t 0 :sTR p
=
H0
ln
( 2
2p ² u 2 :`
ž / uy
P Ž uy . sin uy d uy .
Ž 19 .
This expression incorporates the solid angle weighting factor Žsin uyr2. of M0 on the unit sphere about the yz direction. Note that the application of sin uy in Eq. Ž19. implies that for uys 0 the single vector magnetization is never exactly aligned along the z axis. It is in fact tipped slightly off axis in a direction representing an eigenstate of the magnetization because the spin system is coupled to the single mode of the cavity w8x as well as to the field H0 . It turns out that tdelay expressed by Eq. Ž15. depends upon the most probable Žpeak. value uy2 s ² u 2 :` of the weighted distribution sin uy P Ž uy . f uy P Ž uy . in Eq. Ž19., at which t 0 displays its average value. The validity of the expressions for tdelay in Eqs. Ž15. and Ž19. for the probability were tested experimentally using the 1 H signal in H 2 OrD 2 O mixtures at 600 MHz as shown in Fig. 3. For a specific TR value the histograms summarize 250 separate measurements of t 0 . A summary of experimental TR , ² u 2 :` , and tdelay values measured from the width and center of the hyperbolic secant bursts summarized in Fig. 3 is provided in Table 1 along with a comparison to theoretical estimates of these parameters from Eqs. Ž1., Ž11. and Ž15. with Q j s 9.3. Superimposed on each histogram for each choice of TR in Fig. 3 is a solid line representing the distribution of t 0 determined from the TR value calculated from Eq. Ž1. and the ² u 2 :` value calculated from Eq. Ž15. and tdelay . Here the abscissa is determined from Eq. Ž15. by scaling ² u 2 :`1r2 from its peak value by an amount "duy, while the ordinate is calculated from Eq. Ž18. using uys ² u 2 :`1r2 " duy. Returning to Table 1, both the experimental and theoretical TR and tdelay values are consistent for the three highest concentrations. The shape of the curve
Fig. 3. Summary of t 0 measurements for H 2 OrD 2 O Ž95:5%., H 2 OrD 2 O Ž75:25%., H 2 OrD 2 O Ž50:50%., and H 2 OrD 2 O Ž25:75%. solutions in Žb., Žc., Žd., and Že., respectively. Each histogram represents 250 separate determinations of t 0 at each concentration or equivalently each specific TR . The time tdelay at each TR corresponds to the average value of the distribution in Eq. Ž18. as seen in the compilation in Ža.. These values for tdelay along with theoretical estimates are provided in Table 1. The solid line in each graph corresponds to the distribution anticipated from Eq. Ž18..
predicted by Eq. Ž19. is similar to that predicted by Glauber w9x. The NMR measurements in Fig. 3 more literally conform to the theoretical predictions of Eq. Ž19. than do previous optical measurements by Vehren w10x in an inhomogeneously broadened optical two-level system with damping w11x. One source of error between experimental and theoretical ² u 2 :` values could lie in the assumption that the gradient pulse completely removes any initial tip angle, and therefore the Gaussian distribution P Ž uy, fy . in the angle uy becomes skewed. For perfect inversion, the distribution in signal phase fy is expected to be random since P Ž uy . is independent of fy. Fig. 4 summarizes the distribution of phases at t 0 extracted from the same 250 measurements used to determine tdelay in Fig. 3. Fig. 4a represent-
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
116
Table 1 Summary of theoretical and experimental values ² u 2 :`
H 2O
tR
Ž%.
Experiment Žms.
Theory Žms.
Experiment
Theory
Experiment Žms.
Theory Žms.
95 75 50 25
7.1 9.8 15.0 27.6
7.0 9.0 13.3 26.6
1.2 = 10 – 11 1.1 = 10 – 11 7.9 = 10 – 12 1.0 = 10 – 11
7.0 = 10 – 14 9.8 = 10 – 14 1.5 = 10 – 13 3.0 = 10 – 13
96.6 131.4 207.3 377.7
110.8 141.0 205.6 401.9
ing the highest H 2 O concentration is clearly a random phase distribution while the data for both the 75% and 25% H 2 O concentrations in Fig. 4b,d deviate from the completely random result. Most pronounced is the very tight distribution about 898 for the data in Fig. 4c. The data in Fig. 3a, Fig. 4a, and Table 1 indicate that the discrepancy between experimental and theoretical ² u 2 :` values persists even when inversion performance is nearly ideal. A likely cause of this error is linebroadening due to magnetic
Tdelay
field inhomogeneity, an effect not included in the current treatment. To account for the phase deviations for the three other 1 H concentrations studied here, the distribution in Eq. Ž18. must be adjusted to account for a small tip angle h remaining after application of the field gradient pulse. Here, the distribution is still independent of phase x about the axis tipped away from the yz direction by the angle h , but dependent on both polar angles uy and fy concentrated on the yz axis. Recasting the distribution in terms of the uy and fy polar angles as in shown in Fig. 5 gives
P Ž uy , fy . s
1
(2 p ²u
2: `
exp y Ž h 2 q uy2
y2huy cos fy . r Ž 2² u 2 :` .
Ž 20 .
which can be used to determine just the distribution in fy by weighting P Ž uy, fy . by uy, normalizing, and integrating over the variable uy:
P Ž fy . s Fig. 4. Summary of signal phase measurements at t s t 0 for H 2 OrD 2 O Ž95:5%., H 2 OrD 2 O Ž75:25%., H 2 OrD 2 O Ž50:50%., and H 2 OrD 2 O Ž25:75%. solutions in Ža., Žb., Žc., and Žd., respectively. Each histogram represents 250 separate determinations of the phase at each TR value. The phase in Ža. is random as expected from the distribution in Eq. Ž18. but the spread of the data in Žb., Žc., and Žd. display a phase buildup at 2208, 898, and 08, respectively. The departure from random phase and its anisotropy caused by small tip angle h in Žc. is explained with the distribution function in Eq. Ž20. and does not generate appreciable error in the time tdelay .
(
2p ²u
2: `
q`
Hy` P Ž u
y , fy
. uy d uy
s exp y Ž h 2 sin2 fy . r Ž 2² u 2 :` .
ž
2p
( /
= 1 q h cos fy
² u 2 :`
.
Ž 21 .
For h s 0 the distribution centered around the yz direction is uniform from 0 ( fy( 2 p. As h is
M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
117
integral which extracts the most probable value of uy will also depend on h and fy as tdelay s
f
TR 2 TR 2
Fig. 5. Graphical determination of the Gaussian probability distribution appropriate for tip angle error. The Gaussian is symmetric in the angle x about the axis inclined by angle h away from the y z direction. The displacement in terms of h , uy, and fy for small uy can be determined by the law of cosines.
increased, the distribution of phase begins to concentrate around the direction of h. As the term proportional to h cos fy in Eq. Ž21. becomes negative for pr2 ( fy( 3pr2 values and less than y1, P Ž fy . becomes a negative probability and the values of fy that satisfy this condition cannot occur. Very large values of h cause the phase distribution in the regions from 0 ( fy( pr2 and 3pr2 ( fy( 2 p to concentrate in a narrow region along the direction of h as the width of the distribution in Eq. Ž20. 2² u 2 :` rh 2 becomes smaller. In the limit that h is small, the full width at half height of the distribution in Eq. Ž 21 . is 2 ² u 2 :` 2 ln 2rh 2 and can be used to determine the sensitivity of the signal phase on h. For example, the full width at half height in Fig. 4c of 0.28 translates into a tip angle error of only h s 0.088. The remarkable feature of the data in Figs. 3 and 4 is the insensitivity of tdelay to tip angle error. The comparison of tdelay with theory for h s 0 in Table 1 suggests that the actual h s 0.088 error in the data sets in Fig. 4c does not appreciably affect tdelay . To appreciate this dependence on h consider the most probable value t delay from the integral of TR lnŽ2ruy . over the uy weighted distribution function in Eq. Ž20.. Since the distribution function in Eq. Ž20. depends on both h and fy, the resulting
(
4
ln
ž /
ln
ž
uy2
4 ²u
2: ` qh
(
cos fy ² u 2 :`
/
Ž 22 .
where quadratic and higher-order terms in h are neglected. Notice that for h 2 < ² u 2 :` , Eq. Ž22. reduces to Eq. Ž15.. This relation implies that unless h 2 4 ² u 2 :` , tdelay will not appreciably change from the h s 0 value. Also, since the h dependence following integration over uy still remains in the natural logarithm the overall effect of h on tdelay will be less pronounced. In summary, we have shown that a Gaussian– Boltzmann distribution in the tipping angle degree of freedom of a single vector magnetization M0 , governed by the circuit resistance temperature, accounts for the evolution in time of radiation damping emission of Zeeman energy from the inverted state, and also for the evolution toward equilibrium with the circuit near the ground state. Further refinement of the theory will take into account magnetic field inhomogeneity. The strong phase dependence of the hyperbolic secant radiation damping signal on tip angle error h may become a very sensitive detector of magnetic field. Any slight tip of the perfectly inverted magnetization from the yz direction by either increased rf noise, low-level coherent rf, the bulk demagnetizing field of dilute solutes, or the radiation damping signal for a dilute solute, will induce the phase transition from random to coherent as in Fig. 3. It is this transition that could form the basis for an extremely sensitive measurement of NMR in dilute solution, much like the performance of an injection seeded laser in optical spectroscopy.
Acknowledgements We thank Alex Pines for support and gratefully acknowledge the assistance of Thorsten Dieckmann in the course of data acquisition. M.P.A. thanks U.C. Davis and the Packard foundation for support during the course of this work.
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M.P. Augustine et al.r Chemical Physics Letters 322 (2000) 111–118
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