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Spin-locking in one pulse NMR experiment
Solid State Nuclear Magnetic Resonance 11 Ž1998. 225–230
Spin-locking in one pulse NMR experiment G.B. Furman, A.M. Panich ) , S.D. Goren Department of Physics, Ben-Gurion UniÕersity, Be’er-SheÕa, Israel Received 1 September 1997; accepted 15 November 1997
Abstract The response of a spin system to a long Žin comparison to spin–spin relaxation time T2 . radiofrequency pulse has been studied. We observed that the magnetization after the long pulse does not fall to zero at time t 4 T2 for both on-resonance and off-resonance conditions. The dependencies of the magnetization on frequency offset, linewidth and radiofrequency power are investigated, both theoretically and experimentally. The question of the effective field direction is also discussed. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Spin-locking; NMR
1. Introduction The response of the nuclear system to a radiofrequency Žr.f.. pulse is the essence of the NMR and NQR methods. One of them is the spin-locking scheme of excitation which is usually implemented by applying the phase shifted two pulse sequence w1x. The locking effect where the magnetization does not decay to zero after a long r.f. pulse of duration t, where T2 < t - T1 and T1 is spin-lattice relaxation time, is observed both in NMR w2–4x and NQR w5–7x. The primary aim of the experiments w2–4x was to check Redfield’s hypothesis w8x of spin-temperature in rotating frame. In the present paper, we study in more details both experimentally and theoretically the response of
)
Corresponding author. Fax: q972-7-6472-903; e-mail: [email protected].
a spin system to a long Žin comparison to spin–spin relaxation time T2 . r.f. pulse. We observe that the magnetization after the long pulse does not fall to zero at time t 4 T2 for both on-resonance and offresonance conditions. The dependencies of the magnetization on frequency offset, linewidth and r.f. power are investigated both theoretically and experimentally. In addition, we also obtain the direction and the value of the effective field, along which the magnetization is locked.
2. Theory 2.1. EffectiÕe field Let us consider a spin system consisting of nuclear spins of I™ s 1r2 subjected to a high external ™ magnetic field H0 s H0 ™ n, and an r.f. field H1Ž t . s ™ ™ ™ ™ H1Ž t . ™ m, where n and m are unit vectors along H0 and H1Ž t . directions, respectively. The evolution of
0926-2040r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 6 - 2 0 4 0 Ž 9 7 . 0 0 1 0 8 - 2
G.B. Furman et al.r Solid State Nuclear Magnetic Resonance 11 (1998) 225–230
226
the spin system can be described by the density matrix, r Ž t ., which satisfies the equation Ž " s 1.: drŽ t.
s HŽ t. ,rŽ t. dt with Hamiltonian:
i
Ž 1.
™
™
™ H Ž t . s w0 Ž ™ n P I . q 2 w1 Ž m P I . cos wrf t q Hdd Ž 2 .
where w0 s g H0 and w1 s g H1 , g is gyromagnetic ratio of the nuclei and Hdd is nuclear dipole–dipole interaction Hamiltonian, and wrf is the frequency of r.f. field. In the interaction representation, the density matrix takes the form: ™ ™
™ ™
r˜ Ž t . s e i wrf Ž nP I .tr eyi wrf Ž nP I .t and the equation for the r˜ Ž t . will be: d r˜ Ž t . i s H˜ Ž t . , r˜ Ž t . dt where: ™
H˜ Ž t . s D Ž ™ n P I . q 2 w1cos wrf t qisin wrf t ™
™
½
™
™
™
Ž 4.
n P I . cos 2 Ž™
™ n P I . ,Ž m PI . Ž™ ™
Ž 3.
q 4sin2
™ =Ž™ n P I .Ž m P I .Ž ™ n P I . q H˜dd Ž t .
5
wrf t 2
wrf t
™
where operator V Ž t . includes all time-dependent terms. Using Eq. Ž8. we can determine the angles ™ Žsee Fig. 1. between: Ža. dc field H0 and effective ™ field He : cos u 1 s Ž ™ n P™ e. s
2
Ž 5.
where D s w0 y wrf . Keeping only the time-independent part of Eq. Ž5., we obtain the effective Hamiltonian: He s we Ž ™ e P I . q Hd
Fig. 1. External magnetic field H0 , radiofrequency field H1 , and effective magnetic field vectors and corresponding angles Žschematic sketch..
Ž 6.
D
™
Žb. r.f. field H1 and effective field: cos u 2 s
1
™ ™
DŽ n P m. q w w ½
1
2 1
½
™ ™ 2
we s D q w 1 y Ž n P m .
1r2
5
™ ™
Ž ePI .
Ž 7.
cos u 3 s
™ w1 1 y Ž ™ nPm .
and e is the unit vector along the effective field
1
2 1r2
we
Ž 12 .
2.2. Quasi-equilibrium magnetization
w ½
™ ™ ™ D y w1 Ž ™ nPm . n q w1 m
5
Ž 8.
and Hd is secular part of the dipole–dipole Hamiltonian. After the transformation ŽEq. Ž3.., the evolution of the density matrix is determined by the equation: d r˜ Ž t . dt
Ž 11 .
e:
g
e
i
5
we ™
™
He s es
2
Žc. effective field and x-axis of the laboratory frame, in which z-axis lies along the dc magnetic field:
™
™
™ 1y Ž™ nPm .
e
where: 2
Ž 10 .
we
s He q V Ž t . , r˜ Ž t .
Ž 9.
Now we consider the condition: 5 He 5 4 5 V Ž t . 5
Ž 13 .
a case which is experimentally realizable. Here 5 5 denotes value in units of frequence. From Eq. Ž13., it follows that there exists some t such that for t < t the term V Ž t . on the right-hand side of Eq. Ž9. may
G.B. Furman et al.r Solid State Nuclear Magnetic Resonance 11 (1998) 225–230
be neglected w1x, whereas not for t ) t . In MNR of solids, it is accepted to denote t as T2 . The problem to be solved is: if it is given a value M Ž0. s M0 for ™ the component of magnetization M along the direction ™ n, what is the value of Mk Žt . s M ŽT2 . for the ™ k components of the magnetization M ŽT2 . along the observed axis Ž k s m., along the effective field Ž k s e ., the dc field Ž k s n., and along the x-axis Ž k s x .? Since the effective Hamiltonian is time-independent, we can assume that during the time t ; T2 after beginning of the r.f. pulse the spin system will reach a quasi-equilibrium state w1x and the quasi-equilibrium density matrix takes the form Žin the high temperature approximation.:
re s 1 y be He for we ; wloc s
Tt Hd2
1r2
Tt I 2
Ž 14 .
and: ™
re s 1 y be we Ž ™ e P I . y bd Hd for we < wloc
Ž 15 .
where be and bd are inverse temperatures of the effective Zeeman and dipole subsystems, respectively. In the case of we – wloc and t ; T2 we can neglect the absorption of the r.f. field energy by the spin system and use the energy conservation law: Tt r Ž 0 . He s Tt w re He x
Ž 16 .
which gives:
where M0 is the initial value of magnetization; Žb. along effective field Ždirection along ™ e .:
we2 Ž ™ e P™ n. s 2 M0 we q w2loc Me
Ž 20 .
Žc. along dc field Ždirection along ™ n.: 2
we2 Ž ™ e P™ n. s 2 M0 we q w2loc Mn
Ž 21 .
Žd. along x-axis: ™
we2 Ž ™ e P™ n. Ž™ ePi. s 2 2 M0 we q wloc Mx
Ž 22 .
™
where i is unit vector along x-axis direction. Eq. Ž19. shows that there are two reasons for the decrease of the observed quasi-equilibrium magnetization. First, for the time t ; T2 due to an energy exchange between Zeeman and dipole–dipole reservoirs, and second because the magnetization becomes parallel to the effective field direction and the magnetization components which were previously perpendicular to the effective field direction disappear. In the on-resonance case, from Eqs. Ž10. and Ž19. follows that observed quasi-equilibrium magnetization is equal to zero. This result was obtained earlier w2x. In a more general case, i.e., taking into account the distribution of resonance NMR frequencies Ž Dt s w y w0 . over the real line with nonzero linewidth, d , Eq. Ž19. must be averaged over the line: Mm Mm Ž Dt . D1 D s Hy g Ž Dt . d Dt Ž 23 . D1 M0 M0 where g Ž Dt . is a normalized function of Dt ŽFig. 2a.. The numerical analysis of such averaging for the Gaussian w5,6x and Lorentzian w7x NQR lines has been already published. Let us here consider the Lorenzian distribution of resonance frequencies over a NMR line: 1 d g Ž Dt . s Ž 24 . 2 P d q Dt2
¦;
w0 we Ž ™ e P™ n. s 2 2 bL we q wloc be
Ž 17 .
where r Ž0. is the initial density matrix: ™
r Ž 0 . s 1 y b L w0 Ž ™ nPI .
Ž 18 .
and b L is the inverse lattice temperature. Using Eqs. Ž8. – Ž12. and Ž15., we obtain the quasi-equilibrium magnetization: Ža. along observed ™ .: axis Ždirection along m ™ we2 Ž ™ e P™ n. Ž ™ ePm . s 2 2 M0 we q wloc
Mm
227
Ž 19 .
where: D1 y1 P s Hy D 1 g Ž Dt . d Dt s tan
and a s D1rd .
ž
2a 1ya 2
/
Ž 25 .
G.B. Furman et al.r Solid State Nuclear Magnetic Resonance 11 (1998) 225–230
228
In the case of on-resonance, D s 0, it follows from Eqs. Ž25. – Ž34. that the magnetization: Mm
¦; M0
Ds
™ ™ P n. Žm
1yb
1y
2
b P
tany1
ž
2a b 2
b ya 2
/ Ž 35 .
is not zero. In the limit a ™ ` from Eq. Ž26., we obtain: Mm
¦; M0
™ ™ Ds Ž m P n. 1 y
ž
q
w1 a d db
b Ž b q 1. d
a2 q Ž b y 1.
a2 q Ž b y 1. 2
2
™ ™ 1y Ž m P n.
/ 2
Ž 36 . Fig. 2. Ža. The Lorentzian distribution g Ž Dt . ŽEq. Ž24.. as a function of linewidth d and frequency deviation Dt . Žb. 19 F NMR spectrum of Teflon at ambient temperature. Dashed line is the simulated Lorentzian.
and in the on-resonance case, a ™ 0 from Eq. Ž36., we have: Mm
¦; M0
After averaging according to Eq. Ž23., we obtain: Mm
¦;
™ ™ Ds Ž m P n . Ž 1 y b 2 F1 .
M0
q
w1 d
2
™ ™ 1y Ž m P n . Ž aF1 q F2 .
Ž 26 .
where: F1,2 s
™ ™ Ds Ž m P n.
1
From Eq. Ž37., we can see that magnetization in on-resonance case is not equal to zero if the angle between dc and r.f. field is not equal pr2. This fact can be used to check the deviation angle from pr2. In the case we < wloc and t ; T2 , the spin system is characterized by two integrals of motion, namely ™ we Ž™ e P I . and Hd . Using conservation law: ™
1 d q
A1,2 q C1,2 P
ln
ž
B1,2 Pb
tany1
ž
2a b a2 q b 2 y a 2
2 Ž a q a . q b2 2 Ž a y a . q b2
/
Ž 37 .
bq1
™
Tt r Ž 0 . we Ž ™ e P I . s Tt re we Ž ™ ePI .
Ž 38 .
Tt r Ž 0 . Hd s Tt w re Hd x
Ž 39 .
we obtain:
/
Ž 27 .
be bL
A1 s 2C2 s a 2 q b 2 y 1
s
w0 ™ ™ Ž e P n. we
Ž 40 .
A 2 s 2C1 s 2 a
Ž 28 . Ž 29 .
B1 s a2 y b 2 q 1
Ž 30 .
From Eqs. Ž8., Ž15. and Ž38., we obtain the observed magnetization:
B2 s ya Ž a 2 q b 2 q 1 .
Ž 31 .
Mm
2
2
2
d s Ž a q b y 1. q 4 a
2
Ž 32 . Ž 33 .
a s Drd b2 s
1
d2
2 1
½w
™ ™ 1y Ž m P n.
2
q w2loc
5
Ž 34 .
bd ; 0.
M0
™ s Ž™ e P™ n. Ž ™ ePm .
Ž 41 .
Ž 42 .
This result shows that the decay of the magnetization is determined by its establishment along the direction of the effective field at the time t ; T2 . For the wide resonance line, the averaging of the Eq. Ž40. gives
G.B. Furman et al.r Solid State Nuclear Magnetic Resonance 11 (1998) 225–230
229
result similar to Eq. Ž26. with the substitution of 2 ™ ™2 b 2 s dw21 w1 y Ž m P n. x.
3. Experiment 19
F NMR measurements in Teflon have been made with a Tecmag pulse NMR spectrometer at room temperature. Resonance frequency was 28.08 MHz. We applied the r.f. pulses of length from 2.5 m s Ž908-pulse. to 15 ms with the offset from y100 to 100 kHz. Magnetization values were obtained from the peak intensities of the absorption lines after Fourier transformation of the FID signals. The polymer polytetrafluoroethylene, or Teflon, is a chain of CF2-links with an extremely high molecular weight. The 19 F NMR line shape of Teflon in external magnetic field 0.7 T is nearly symmetric and is satisfactory described by the Lorentzian function Ž d ; 6 kHz. ŽFig. 2b.. The line broadening is mainly due to dipole–dipole coupling of nuclei; chemical shielding anisotropy is not pronounced. Nearly Lorentzian lineshape is likely caused by the reorientation mobility of the chains. Spin-lattice relaxation times in laboratory and rotating frames, T1 and T1 r , and spin–spin relaxation time T2 at ambient temperature have been measured using p – t – pr2 sequence, pr2-long Ž90. spin-locking sequence and pr2Ž0. – t – pr2Ž90. ‘solid echo’ sequence, respectively. The measurements yield T1 s 210 " 20 ms, T1 r s 12 " 3 ms, and T2 s 112 " 15 m s. One can see in Fig. 3 that the 19 F magnetization decay M Ž t . in Teflon exhibits two regions. First, the magnetization decays with the time constant of the
Fig. 3. Semilog plot of the 19 F magnetization vs. pulse length.
Fig. 4. Dependence of the 19 F magnetization vs. frequency offset observed after a long 250 m s r.f. pulse. Dashed curve is a result of fit with Eq. Ž36. for w1 s 34 kHz and the angle between the external magnetic field and r.f. field of 688. The values of magnetization are normalized to that of the 908 pulse.
order of T2 . Then, at t ) 120 m s, it decays with the time constant of 14 " 3 ms which is close to the spin-lattice relaxation time in rotating frame of the teflon sample, T1 r ŽFig. 3.. As well known, the rotating frame condition is usually achieved by applying a pr2Ž0.-long Ž90. phase-shifted two-pulse spin-locking sequence. The response of a spin system obtained in our experiment evidently suggests spin-locking behavior even after applying the only long r.f. pulse. The result of excitation of the spin system by a long off-resonance pulse with pulse length of 250 m s is shown in Fig. 4. One can see that the signal intensity Žor the value of magnetization. strongly depends on frequency offset and increases with the offset value up to D s "100 kHz. More of that, the magnetization measured after a long pulse is not equal to zero at the on-resonance condition, in agreement with the above theory ŽEqs. Ž36. and Ž37... The position of the minimum in the intensity was found to be shifted from the zero offset position. As it follows from Eqs. Ž36. and Ž37., this is evidently because the angle between directions of external magnetic field n and r.f. field m is not exactly equal to 908. These results are similar to the result recently obtained in 63 Cu NQR study of copper oxide w7x. The above theory predicts that the magnetization measured after a long pulse increases with the decrease of amplitude of r.f. field, if the external magnetic field and r.f. field vectors do not form the
230
G.B. Furman et al.r Solid State Nuclear Magnetic Resonance 11 (1998) 225–230
4. Conclusion We observed the response of a spin system to a long Žin comparison to T2 . r.f. pulse. Magnetization after the long pulse does not fall to zero at the time t 4 T2 for both on-resonance and off-resonance conditions and is shown to exhibit spin-locking behavior. The dependences of magnetization on frequency offset, linewidth and r.f. power are calculated and show good agreement with experimental data. Fig. 5. The dependence of the magnetization on the amplitude of r.f. field after applying a long 500 m s pulse at the on-resonance condition. Dashed curve is a result of fit with Eq. Ž37. for d s 37.5 kradrs, wloc s6.4 kradrs, and the angle between external magnetic field H0 and r.f. field H1 of 898.
References
right angle ŽEq. Ž37... The experimental dependence of the magnetization observed after a long 500 m s pulse vs. the applied r.f. field, v 1 , at the on-resonance condition is shown in Fig. 5. One can see that the growth of magnetization with reduced amplitude of r.f. field is in a good agreement with the above theory: fit with Eq. Ž37. yields the angle between external magnetic field H0 and r.f. field H1 of 898. The similar dependence has been recently obtained in one-pulse NQR experiment w7x.
w1x M. Goldman, Spin Temperature and Nuclear Magnetic Resonance in Solids, Clarendon Press, Oxford, 1970. w2x S.P. Slichter, W.C. Holton, Phys. Rev. 122 Ž1961. 1701. w3x W.L. Goldburg, Phys. Rev. 122 Ž1961. 831. w4x W.L. Goldburg, Phys. Rev. 128 Ž1962. 1554. w5x J.C. Pratt, P. Raghunatan, C.A. McDowell, J. Chem. Phys. 61 Ž1974. 1016. w6x J.C. Pratt, P. Raghunatan, C.A. McDowell, J. Magn. Res. 20 Ž1975. 313. w7x B. Bandyopadhyay, G.B. Furman, S.D. Goren, C. Korn, A.I. Shames, Z. Naturforsch. 51a Ž1996. 357. w8x A.G. Redfield, Phys. Rev. 98 Ž1955. 1787.
Spin-locking in one pulse NMR experiment G.B. Furman, A.M. Panich ) , S.D. Goren Department of Physics, Ben-Gurion UniÕersity, Be’er-SheÕa, Israel Received 1 September 1997; accepted 15 November 1997
Abstract The response of a spin system to a long Žin comparison to spin–spin relaxation time T2 . radiofrequency pulse has been studied. We observed that the magnetization after the long pulse does not fall to zero at time t 4 T2 for both on-resonance and off-resonance conditions. The dependencies of the magnetization on frequency offset, linewidth and radiofrequency power are investigated, both theoretically and experimentally. The question of the effective field direction is also discussed. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Spin-locking; NMR
1. Introduction The response of the nuclear system to a radiofrequency Žr.f.. pulse is the essence of the NMR and NQR methods. One of them is the spin-locking scheme of excitation which is usually implemented by applying the phase shifted two pulse sequence w1x. The locking effect where the magnetization does not decay to zero after a long r.f. pulse of duration t, where T2 < t - T1 and T1 is spin-lattice relaxation time, is observed both in NMR w2–4x and NQR w5–7x. The primary aim of the experiments w2–4x was to check Redfield’s hypothesis w8x of spin-temperature in rotating frame. In the present paper, we study in more details both experimentally and theoretically the response of
)
Corresponding author. Fax: q972-7-6472-903; e-mail: [email protected].
a spin system to a long Žin comparison to spin–spin relaxation time T2 . r.f. pulse. We observe that the magnetization after the long pulse does not fall to zero at time t 4 T2 for both on-resonance and offresonance conditions. The dependencies of the magnetization on frequency offset, linewidth and r.f. power are investigated both theoretically and experimentally. In addition, we also obtain the direction and the value of the effective field, along which the magnetization is locked.
2. Theory 2.1. EffectiÕe field Let us consider a spin system consisting of nuclear spins of I™ s 1r2 subjected to a high external ™ magnetic field H0 s H0 ™ n, and an r.f. field H1Ž t . s ™ ™ ™ ™ H1Ž t . ™ m, where n and m are unit vectors along H0 and H1Ž t . directions, respectively. The evolution of
0926-2040r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 6 - 2 0 4 0 Ž 9 7 . 0 0 1 0 8 - 2
G.B. Furman et al.r Solid State Nuclear Magnetic Resonance 11 (1998) 225–230
226
the spin system can be described by the density matrix, r Ž t ., which satisfies the equation Ž " s 1.: drŽ t.
s HŽ t. ,rŽ t. dt with Hamiltonian:
i
Ž 1.
™
™
™ H Ž t . s w0 Ž ™ n P I . q 2 w1 Ž m P I . cos wrf t q Hdd Ž 2 .
where w0 s g H0 and w1 s g H1 , g is gyromagnetic ratio of the nuclei and Hdd is nuclear dipole–dipole interaction Hamiltonian, and wrf is the frequency of r.f. field. In the interaction representation, the density matrix takes the form: ™ ™
™ ™
r˜ Ž t . s e i wrf Ž nP I .tr eyi wrf Ž nP I .t and the equation for the r˜ Ž t . will be: d r˜ Ž t . i s H˜ Ž t . , r˜ Ž t . dt where: ™
H˜ Ž t . s D Ž ™ n P I . q 2 w1cos wrf t qisin wrf t ™
™
½
™
™
™
Ž 4.
n P I . cos 2 Ž™
™ n P I . ,Ž m PI . Ž™ ™
Ž 3.
q 4sin2
™ =Ž™ n P I .Ž m P I .Ž ™ n P I . q H˜dd Ž t .
5
wrf t 2
wrf t
™
where operator V Ž t . includes all time-dependent terms. Using Eq. Ž8. we can determine the angles ™ Žsee Fig. 1. between: Ža. dc field H0 and effective ™ field He : cos u 1 s Ž ™ n P™ e. s
2
Ž 5.
where D s w0 y wrf . Keeping only the time-independent part of Eq. Ž5., we obtain the effective Hamiltonian: He s we Ž ™ e P I . q Hd
Fig. 1. External magnetic field H0 , radiofrequency field H1 , and effective magnetic field vectors and corresponding angles Žschematic sketch..
Ž 6.
D
™
Žb. r.f. field H1 and effective field: cos u 2 s
1
™ ™
DŽ n P m. q w w ½
1
2 1
½
™ ™ 2
we s D q w 1 y Ž n P m .
1r2
5
™ ™
Ž ePI .
Ž 7.
cos u 3 s
™ w1 1 y Ž ™ nPm .
and e is the unit vector along the effective field
1
2 1r2
we
Ž 12 .
2.2. Quasi-equilibrium magnetization
w ½
™ ™ ™ D y w1 Ž ™ nPm . n q w1 m
5
Ž 8.
and Hd is secular part of the dipole–dipole Hamiltonian. After the transformation ŽEq. Ž3.., the evolution of the density matrix is determined by the equation: d r˜ Ž t . dt
Ž 11 .
e:
g
e
i
5
we ™
™
He s es
2
Žc. effective field and x-axis of the laboratory frame, in which z-axis lies along the dc magnetic field:
™
™
™ 1y Ž™ nPm .
e
where: 2
Ž 10 .
we
s He q V Ž t . , r˜ Ž t .
Ž 9.
Now we consider the condition: 5 He 5 4 5 V Ž t . 5
Ž 13 .
a case which is experimentally realizable. Here 5 5 denotes value in units of frequence. From Eq. Ž13., it follows that there exists some t such that for t < t the term V Ž t . on the right-hand side of Eq. Ž9. may
G.B. Furman et al.r Solid State Nuclear Magnetic Resonance 11 (1998) 225–230
be neglected w1x, whereas not for t ) t . In MNR of solids, it is accepted to denote t as T2 . The problem to be solved is: if it is given a value M Ž0. s M0 for ™ the component of magnetization M along the direction ™ n, what is the value of Mk Žt . s M ŽT2 . for the ™ k components of the magnetization M ŽT2 . along the observed axis Ž k s m., along the effective field Ž k s e ., the dc field Ž k s n., and along the x-axis Ž k s x .? Since the effective Hamiltonian is time-independent, we can assume that during the time t ; T2 after beginning of the r.f. pulse the spin system will reach a quasi-equilibrium state w1x and the quasi-equilibrium density matrix takes the form Žin the high temperature approximation.:
re s 1 y be He for we ; wloc s
Tt Hd2
1r2
Tt I 2
Ž 14 .
and: ™
re s 1 y be we Ž ™ e P I . y bd Hd for we < wloc
Ž 15 .
where be and bd are inverse temperatures of the effective Zeeman and dipole subsystems, respectively. In the case of we – wloc and t ; T2 we can neglect the absorption of the r.f. field energy by the spin system and use the energy conservation law: Tt r Ž 0 . He s Tt w re He x
Ž 16 .
which gives:
where M0 is the initial value of magnetization; Žb. along effective field Ždirection along ™ e .:
we2 Ž ™ e P™ n. s 2 M0 we q w2loc Me
Ž 20 .
Žc. along dc field Ždirection along ™ n.: 2
we2 Ž ™ e P™ n. s 2 M0 we q w2loc Mn
Ž 21 .
Žd. along x-axis: ™
we2 Ž ™ e P™ n. Ž™ ePi. s 2 2 M0 we q wloc Mx
Ž 22 .
™
where i is unit vector along x-axis direction. Eq. Ž19. shows that there are two reasons for the decrease of the observed quasi-equilibrium magnetization. First, for the time t ; T2 due to an energy exchange between Zeeman and dipole–dipole reservoirs, and second because the magnetization becomes parallel to the effective field direction and the magnetization components which were previously perpendicular to the effective field direction disappear. In the on-resonance case, from Eqs. Ž10. and Ž19. follows that observed quasi-equilibrium magnetization is equal to zero. This result was obtained earlier w2x. In a more general case, i.e., taking into account the distribution of resonance NMR frequencies Ž Dt s w y w0 . over the real line with nonzero linewidth, d , Eq. Ž19. must be averaged over the line: Mm Mm Ž Dt . D1 D s Hy g Ž Dt . d Dt Ž 23 . D1 M0 M0 where g Ž Dt . is a normalized function of Dt ŽFig. 2a.. The numerical analysis of such averaging for the Gaussian w5,6x and Lorentzian w7x NQR lines has been already published. Let us here consider the Lorenzian distribution of resonance frequencies over a NMR line: 1 d g Ž Dt . s Ž 24 . 2 P d q Dt2
¦;
w0 we Ž ™ e P™ n. s 2 2 bL we q wloc be
Ž 17 .
where r Ž0. is the initial density matrix: ™
r Ž 0 . s 1 y b L w0 Ž ™ nPI .
Ž 18 .
and b L is the inverse lattice temperature. Using Eqs. Ž8. – Ž12. and Ž15., we obtain the quasi-equilibrium magnetization: Ža. along observed ™ .: axis Ždirection along m ™ we2 Ž ™ e P™ n. Ž ™ ePm . s 2 2 M0 we q wloc
Mm
227
Ž 19 .
where: D1 y1 P s Hy D 1 g Ž Dt . d Dt s tan
and a s D1rd .
ž
2a 1ya 2
/
Ž 25 .
G.B. Furman et al.r Solid State Nuclear Magnetic Resonance 11 (1998) 225–230
228
In the case of on-resonance, D s 0, it follows from Eqs. Ž25. – Ž34. that the magnetization: Mm
¦; M0
Ds
™ ™ P n. Žm
1yb
1y
2
b P
tany1
ž
2a b 2
b ya 2
/ Ž 35 .
is not zero. In the limit a ™ ` from Eq. Ž26., we obtain: Mm
¦; M0
™ ™ Ds Ž m P n. 1 y
ž
q
w1 a d db
b Ž b q 1. d
a2 q Ž b y 1.
a2 q Ž b y 1. 2
2
™ ™ 1y Ž m P n.
/ 2
Ž 36 . Fig. 2. Ža. The Lorentzian distribution g Ž Dt . ŽEq. Ž24.. as a function of linewidth d and frequency deviation Dt . Žb. 19 F NMR spectrum of Teflon at ambient temperature. Dashed line is the simulated Lorentzian.
and in the on-resonance case, a ™ 0 from Eq. Ž36., we have: Mm
¦; M0
After averaging according to Eq. Ž23., we obtain: Mm
¦;
™ ™ Ds Ž m P n . Ž 1 y b 2 F1 .
M0
q
w1 d
2
™ ™ 1y Ž m P n . Ž aF1 q F2 .
Ž 26 .
where: F1,2 s
™ ™ Ds Ž m P n.
1
From Eq. Ž37., we can see that magnetization in on-resonance case is not equal to zero if the angle between dc and r.f. field is not equal pr2. This fact can be used to check the deviation angle from pr2. In the case we < wloc and t ; T2 , the spin system is characterized by two integrals of motion, namely ™ we Ž™ e P I . and Hd . Using conservation law: ™
1 d q
A1,2 q C1,2 P
ln
ž
B1,2 Pb
tany1
ž
2a b a2 q b 2 y a 2
2 Ž a q a . q b2 2 Ž a y a . q b2
/
Ž 37 .
bq1
™
Tt r Ž 0 . we Ž ™ e P I . s Tt re we Ž ™ ePI .
Ž 38 .
Tt r Ž 0 . Hd s Tt w re Hd x
Ž 39 .
we obtain:
/
Ž 27 .
be bL
A1 s 2C2 s a 2 q b 2 y 1
s
w0 ™ ™ Ž e P n. we
Ž 40 .
A 2 s 2C1 s 2 a
Ž 28 . Ž 29 .
B1 s a2 y b 2 q 1
Ž 30 .
From Eqs. Ž8., Ž15. and Ž38., we obtain the observed magnetization:
B2 s ya Ž a 2 q b 2 q 1 .
Ž 31 .
Mm
2
2
2
d s Ž a q b y 1. q 4 a
2
Ž 32 . Ž 33 .
a s Drd b2 s
1
d2
2 1
½w
™ ™ 1y Ž m P n.
2
q w2loc
5
Ž 34 .
bd ; 0.
M0
™ s Ž™ e P™ n. Ž ™ ePm .
Ž 41 .
Ž 42 .
This result shows that the decay of the magnetization is determined by its establishment along the direction of the effective field at the time t ; T2 . For the wide resonance line, the averaging of the Eq. Ž40. gives
G.B. Furman et al.r Solid State Nuclear Magnetic Resonance 11 (1998) 225–230
229
result similar to Eq. Ž26. with the substitution of 2 ™ ™2 b 2 s dw21 w1 y Ž m P n. x.
3. Experiment 19
F NMR measurements in Teflon have been made with a Tecmag pulse NMR spectrometer at room temperature. Resonance frequency was 28.08 MHz. We applied the r.f. pulses of length from 2.5 m s Ž908-pulse. to 15 ms with the offset from y100 to 100 kHz. Magnetization values were obtained from the peak intensities of the absorption lines after Fourier transformation of the FID signals. The polymer polytetrafluoroethylene, or Teflon, is a chain of CF2-links with an extremely high molecular weight. The 19 F NMR line shape of Teflon in external magnetic field 0.7 T is nearly symmetric and is satisfactory described by the Lorentzian function Ž d ; 6 kHz. ŽFig. 2b.. The line broadening is mainly due to dipole–dipole coupling of nuclei; chemical shielding anisotropy is not pronounced. Nearly Lorentzian lineshape is likely caused by the reorientation mobility of the chains. Spin-lattice relaxation times in laboratory and rotating frames, T1 and T1 r , and spin–spin relaxation time T2 at ambient temperature have been measured using p – t – pr2 sequence, pr2-long Ž90. spin-locking sequence and pr2Ž0. – t – pr2Ž90. ‘solid echo’ sequence, respectively. The measurements yield T1 s 210 " 20 ms, T1 r s 12 " 3 ms, and T2 s 112 " 15 m s. One can see in Fig. 3 that the 19 F magnetization decay M Ž t . in Teflon exhibits two regions. First, the magnetization decays with the time constant of the
Fig. 3. Semilog plot of the 19 F magnetization vs. pulse length.
Fig. 4. Dependence of the 19 F magnetization vs. frequency offset observed after a long 250 m s r.f. pulse. Dashed curve is a result of fit with Eq. Ž36. for w1 s 34 kHz and the angle between the external magnetic field and r.f. field of 688. The values of magnetization are normalized to that of the 908 pulse.
order of T2 . Then, at t ) 120 m s, it decays with the time constant of 14 " 3 ms which is close to the spin-lattice relaxation time in rotating frame of the teflon sample, T1 r ŽFig. 3.. As well known, the rotating frame condition is usually achieved by applying a pr2Ž0.-long Ž90. phase-shifted two-pulse spin-locking sequence. The response of a spin system obtained in our experiment evidently suggests spin-locking behavior even after applying the only long r.f. pulse. The result of excitation of the spin system by a long off-resonance pulse with pulse length of 250 m s is shown in Fig. 4. One can see that the signal intensity Žor the value of magnetization. strongly depends on frequency offset and increases with the offset value up to D s "100 kHz. More of that, the magnetization measured after a long pulse is not equal to zero at the on-resonance condition, in agreement with the above theory ŽEqs. Ž36. and Ž37... The position of the minimum in the intensity was found to be shifted from the zero offset position. As it follows from Eqs. Ž36. and Ž37., this is evidently because the angle between directions of external magnetic field n and r.f. field m is not exactly equal to 908. These results are similar to the result recently obtained in 63 Cu NQR study of copper oxide w7x. The above theory predicts that the magnetization measured after a long pulse increases with the decrease of amplitude of r.f. field, if the external magnetic field and r.f. field vectors do not form the
230
G.B. Furman et al.r Solid State Nuclear Magnetic Resonance 11 (1998) 225–230
4. Conclusion We observed the response of a spin system to a long Žin comparison to T2 . r.f. pulse. Magnetization after the long pulse does not fall to zero at the time t 4 T2 for both on-resonance and off-resonance conditions and is shown to exhibit spin-locking behavior. The dependences of magnetization on frequency offset, linewidth and r.f. power are calculated and show good agreement with experimental data. Fig. 5. The dependence of the magnetization on the amplitude of r.f. field after applying a long 500 m s pulse at the on-resonance condition. Dashed curve is a result of fit with Eq. Ž37. for d s 37.5 kradrs, wloc s6.4 kradrs, and the angle between external magnetic field H0 and r.f. field H1 of 898.
References
right angle ŽEq. Ž37... The experimental dependence of the magnetization observed after a long 500 m s pulse vs. the applied r.f. field, v 1 , at the on-resonance condition is shown in Fig. 5. One can see that the growth of magnetization with reduced amplitude of r.f. field is in a good agreement with the above theory: fit with Eq. Ž37. yields the angle between external magnetic field H0 and r.f. field H1 of 898. The similar dependence has been recently obtained in one-pulse NQR experiment w7x.
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