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Extraordinary stimulated echoes in a large magnetic field gradient
26 February 2001
Physics Letters A 280 (2001) 221–226 www.elsevier.nl/locate/pla
Extraordinary stimulated echoes in a large magnetic field gradient Noriaki Okubo 1 Institute of Physics, University of Tsukuba, Tsukuba 305-8571, Japan Received 25 October 2000; received in revised form 20 December 2000; accepted 4 January 2001 Communicated by L.J. Sham
Abstract For a pulse sequence (π/2)x − τ1 − (π/2)x − τ2 − (π/2)x − t−(echo)y in a large magnetic field gradient, extraordinary stimulated echoes have been observed at t = i1 τ1 + i2 τ2 with integers i1 and i2 . The properties are described and possible mechanisms of the formation are discussed. 2001 Elsevier Science B.V. All rights reserved. PACS: 76.60.L Keywords: Stimulated echoes; Magnetic field gradient; Multiple spin echoes; NaCl; NaI; Na2 SO4
In the usual pulsed NMR two rf pulses are followed by one echo and the third pulse is followed by four echoes at most [1]. In a large magnetic field gradient, however, extraordinary echoes were found in addition to the ordinary ones. In this Letter the author reports the properties of the extraordinary echoes and discusses the mechanism of the occurrence. Detailed examination was done in a sample of polycrystalline NaCl, which is the same one as used in the preceding study [2]. The values of T1 and T2 for 23 Na nuclei in this sample are 16.5 s and 330 µs, respectively, at room temperature. In addition, two groups of samples were also examined. The one group consisted of samples prepared from NaCl in other states and the other consisted of those prepared from other substances. They were sealed in quartz tubes of 12 mm in o.d. and 20 mm in length. The tubes were placed in a superconducting magnet of 4.7 T with their axis perpendicular to the bore axis. The
E-mail address: [email protected] (N. Okubo). 1 Tel.: 81-298-53-4547, fax: 81-298-53-6618.
field distribution was examined using the 14 N NMR in liquid nitrogen. In the reference frame whose z axis is taken along the field, the rf field was applied along x axis and the y component of the magnetization was observed as usual. The calibration of the pulse width to the flip angle was made by comparing the width dependence of the height of the ordinary 2τ echo with the calculated flip angle dependence. The magnitude of the rf field H1 estimated from the pulse width was 2.2 mT. Echoes happened to overlap for some pulse timings, when the data of the echo height were discarded. Spin echoes were examined at room temperature unless otherwise described. Fig. 1(a) shows echoes of 23 Na nuclei in polycrystalline NaCl observed by applying three pulses. The sample was placed 10 cm away from the center and the field gradient at the position was 2 T/m. When the time intervals P1 P2 and P2 P3 in the figure are, respectively, denoted by τ1 and τ2 , these echoes appear later from the time P3 by i1 τ1 + i2 τ2 , where i1 and i2 are integers. In the following the echoes are referred to as Ei1 i2 . E1−1 is the primary echo, E10 the stimulated echo, and E01 and E11 are the secondary echoes.
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 0 2 7 - 5
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N. Okubo / Physics Letters A 280 (2001) 221–226
(a)
(b)
Fig. 1. 23 Na echoes in polycrystalline NaCl observed in a field gradient of 2 T/m. Following the first rf pulse P1 , two pulses P2 and P3 were applied with delay times τ1 and τ2 . Their leading edges are shown by the arrows. Frequency: 52.65 MHz, pulse widths: all 10 µs corresponding to a flip angle of about π/2, repeating time: 300 s, averaging times: 8. (a) Case of τ1 > τ2 . E1−2 , E12 , E13 , and E14 are extraordinary echoes. (b) Case of τ1 < τ2 . E−12 and E12 are extraordinary echoes. Data points are connected by continuous lines for eye-guide. Rather asymmetric shapes of the echoes are due to incomplete adjustment of the reference phase.
In addition to these ordinary spin echoes (OSE), extraordinary stimulated echoes (ESE), E1−2 , E12 , E13 , and E14 appear in the figure. Though this trace was recorded after averaging, the ESE cannot be attributed to the storage effect, because the repeating time was sufficiently long compared with T1 . As τ2 was decreased to zero, E01 was hidden by the third pulse and the other echoes gathered towards E10 eventually coalescing into one echo, as expected. The appearance of ESE is not restricted in the case of τ1 > τ2 . Fig. 1(b) shows the opposite case. When the sample was initially placed in the homogeneous field at the center of the magnet, the free induction decay and OSE signals were not separated from each other because of their extremely slow decay. As the sample was moved away from the center along the bore axis with tuning, they became progressively narrower and ESE appeared. Fig. 1 is just the pattern observed in a gradient of 2 T/m, but ESE could be observed even in one order of magnitude smaller gradient. On further moving away, however, the signal
became rapidly small. This is possibly because there the rf pulses used began to lack in the spreading of the Fourier components. The dependence of the heights of these echoes on τ1 and τ2 is shown in Figs. 2(a) and (b), respectively. As seen in Fig. 2(a), E01 is independent of τ1 while E10 decays with a time constant close to T2 /2 and E1−1 and E11 decay with a constant close to T2 /4 at large τ1 , whereas ESE all exhibit peaks. The positions where the peaks appear change with τ2 and the dependence seems to be different for each of ESE. In Fig. 2(b), on the other hand, OSE except E1−1 and E10 exhibit peaks, whereas ESE decay without peaking. Fig. 3 shows the dependence of the echo heights including signs on the width of the third pulse when the widths of the first and second pulses are fixed at values nearly corresponding to the angle π/2. As the width of the third pulse is increased, the shape of the echoes collapses and particularly E1−1 beats violently around its peak, so that the reliability of their heights in the figure lowers at large pulse widths.
N. Okubo / Physics Letters A 280 (2001) 221–226
(a)
223
(b)
Fig. 2. τ1 and τ2 dependences of the absolute echo height. The open circles stand for ordinary echoes and the closed ones for extraordinary echoes. No data point exists for cases where the echoes are supposed to precede the third pulse, that is, n1 τ1 + n2 τ2 < 0, except E1−1 . A part of data points is omitted because of overlapping with other echoes or the third pulse. (a) τ1 dependence. τ2 : 107 µs. (b) τ2 dependence. τ1 : 145 µs. Other conditions are the same as in Fig. 1.
The mechanism of the formation of OSE is accounted for by the vector model for the nuclear magnetization in the moderately homogeneous field [1]. Expressions for their heights can be obtained by multiplying the vector of the magnetization at thermal equilibrium by the matrices representing the motion during on-time and off-time of the rf pulse [2]. When the flip angle of the nth pulse is denoted by αn and the delay time of the following pulse by τn , the heights of the echoes in the pulse sequence (α1 )x − τ1 − (α2 )x − τ2 − (α3 )x − t − (echo)y are expressed as in Table 1, where the effect of diffusion is neglected. When α1 = α2 = α3 , they reduce to Hahn’s expressions [1], apart from the effect of diffusion. Comparing Fig. 2(a) with the calculated τ1 dependence, the following features are remarked: No depen-
dence of E01 and the decay of E10 with a decay constant close to T2 /2 are in accordance with the calculated one. However, the decays of E1−1 and E11 are about two times faster than the result of the calculation. Concerning τ2 dependence, on the other hand, no dependences of E1−1 and E10 and decays of E−11 , E01 , and E11 with constants close to T2 /2 at large τ2 agree with the calculation. Nevertheless, the growth of the height seen for E01 and E11 at small τ2 is not expected from the calculation. Thus, the large field gradient does not only create ESE but also causes the deviation of the τ1 and τ2 dependences of OSE from the calculation. The deviation of OSE is also seen in the dependence on the width of the third pulse. In Fig. 3 the dependence of E10 is close to the calculated one. However,
224
N. Okubo / Physics Letters A 280 (2001) 221–226 Table 1 Height of ordinary echoes following three rf pulses applied with delay times τ1 and τ2 Echo
Height
E1−1
cos2 (α3 /2) sin2 (α2 /2) sin α1 exp(−2τ1 /T2 )
E10
(1/2) sin α3 sin α2 sin α1 exp(−2τ1 /T2 − τ2 /T1 )
E−11
− sin2 (α3 /2) sin2 (α2 /2) sin α1 exp(−2τ2 /T2 )
E01
sin2 (α3 /2) sin α2 [Mz (τ1 )/M0 ] exp(−2τ2 /T2 )
E11
sin2 (α3 /2) cos2 (α2 /2) sin α1 exp[−2(τ1 + τ2 )/T2 ]
αn denotes the flipping angle of the nth pulse. The height is normalized by the thermal equilibrium value M0 . Mz (τ1 ) is expressed as M0 [1 − (1 − cos α1 ) exp(−τ1 /T1 )] for the case where spin-lattice relaxation is represented by a single relaxation time T1 . The expression for E1−1 is applied only to the case τ1 > τ2 and the factor cos2 (α3 /2) is replaced by unity for the case τ1 < τ2 .
Fig. 3. Dependence of echo height including the sign on the width of the third pulse. The height of the downward echo is taken to be positive. τ1 : 245 µs, τ2 : 52 µs, first and second pulse widths: both 10 µs corresponding to π/2. Other conditions are the same as in Fig. 1.
the dependence of E1−1 and E11 is two or three times stronger than the calculated one even at small widths. Besides, the sign of E11 is reversed in conflict with the expression. On the other hand, since, for the first pulse with α1 = π/2, Mz (τ1 ) scarcely recovers at time much smaller compared with T1 , E01 should hardly be observed. Nevertheless, E01 was observed with a fairly strength at large widths of the third pulse. This means that the width of the first pulse is no longer sufficient as π/2 pulse in the gradient. To a rectangular line shape of a width 2ω in the frequency domain, an echo of the type of sine function, sin ωt/t, corresponds [1]. The oscillatory character of the echo is due to the singularity of the line shape, and the interval between the adjacent peak and bottom of the sine function is roughly given by π/ω. When a sample with an inherently sharp resonance is placed in a field gradient, the line shape is determined by
the cross section along the gradient. Since the cross section of a circle has also a singularity, it may be suspected that the present ESE originate from the oscillatory character of sine-like functions of the OSE. When the spreading of the field within the sample is denoted by H0 , the width of the line shape is given by γ H0 with gyromagnetic ratio γ . If ESE with a fixed i1 are attributed to the peaks and bottoms accompanying one of OSE, their separation should be roughly equal to 2π/γ H0 , which is about 5 µs for the present case, and should be independent of τ2 . This is not the case. Alternatively, if ESE are formed as a result of interference among sine-like functions accompanying some OSE, the height of ESE should exhibit an oscillatory behavior as τ2 is varied, because the separation between each OSE is τ2 for a fixed i1 . This also disagrees with the observation. In the preceding paper, we reported that in a large magnetic field gradient quasi multiple spin echoes (QMSE) appear following two pulses. The nature was found to be a storage of stimulated echoes which are successively created by signal averaging with an insufficient repetition time [2]. When T2∗ is so large as to be comparable with T2 they cannot be observed, whereas in a large field gradient T2∗ becomes short and they become visible, well separated each other and from the free induction decay. This situation is similar to the present case, but ESE differ from QMSE in that they appear even for one application of the sequence
N. Okubo / Physics Letters A 280 (2001) 221–226
of three pulses to a system where thermal equilibrium is established. Deville et al. [3] found multiple spin echoes (MSE) following two pulses in solid 3 He. Later, MSE were also reported for 1 H nuclei in hydrogen compounds [4,5] and for 115In in AuIn2 [6]. Ardelean et al. [7] treated MSE following three pulses and called them multiple nonlinear stimulated echoes (NOSE). Our ESE resemble to NOSE in some respects. Since MSE are created by modulation of the Larmor frequency due to a large nuclear magnetization M0 , their occurrence requires the condition 4πγ M0 T2 1.
(1)
For the reported nuclei this condition is certainly satisfied, whereas for 23 Na nuclei in NaCl the magnitude of the left hand amounts only to 5 × 10−5 . For the observation of MSE and NOSE a weak static field gradient is applied but the purpose is to make the analytical treatment feasible by avoiding the sample shape effect of the demagnetizing field of the nuclei, so that the gradient is not necessarily needed. On the contrary, a large gradient is indispensable for ESE to be created. Moreover, MSE appear even for application of only two rf pulses in the sample satisfying condition (1), whereas the present ESE do not appear until the third pulse is applied. The motion of the assembly of the nuclear spins is classically described in the rotating frame [8] as follows: In the present case, each isochromat represents the contribution from each portion of the sample with different z. During on-time of the rf pulse the isochromats rotate about the effective field as usual, but in the large field gradient the direction of the field is not x axis but inclined towards z axis by an angle corresponding to the difference of the field at the position from the resonance field. During off-time they are distributed in x–y plane more quickly than in the usual moderate inhomogeneity. If then a long pulse is used, an edge echo appears even after a single pulse, owing to the dephasing during the pulse and the following rephasing [9,10]. In the present case, however, such a signal was not observed with a significant level. Thus, the occurrence of ESE could not be accounted for by the known mechanisms. To obtain further information, other samples were also examined in the field gradient. The result is summarized in Table 2
225
together with the relevant values. For NaCl a pattern similar to Fig. 1 was also observed in the powder sample, and the pattern was not affected by the immersion into silicon oil (viscosity of 10 cSt). In the nearly saturated aqueous solution, however, only OSE were observed and none of ESE was observed. The values of diffusion constant in the table are those estimated from the primary echo in the gradient [8]. ESE must also decay owing to diffusion. However, their disappearance in the solution cannot be attributed to diffusion, unless ESE have a particularly strong dependence on it. No effect of immersing NaCl into silicon oil indicates that ESE have nothing to do with the surface of the sample. The possibility that ESE originate from piezoelectricity is excluded by the resonant character and the nonactive space group. In the table only the samples for which OSE could be observed with good signal-to-noise ratios are listed. It is distinct for them whether ESE are observed or not. ESE appeared always accompanying a series of OSE, whereas OSE were not always accompanied by ESE. The observation of ESE is so far limited within 23 Na nuclei in solid NaCl, NaI, and Na2 SO4 . Disappearance of ESE in the aqueous solution of NaCl and no appearance in NbS2 suggest that the dipolar interaction exceeding the quadrupolar one is needed for ESE to occur. Thus, the mechanism of the occurrence of ESE is not resolved for the present. However, closer examination of the time of the appearance of ESE revealed that the value of i1 is limited to 1 and −1. For i1 = 1 the observed i2 ranges from −4 to 6 including three OSE and for i1 = −1 the i2 ranges from 1 to 3 including one OSE. All echoes are therefore supposed to be formed centered at times τ1 before and after the third pulse with the interval τ2 , including virtual echoes in advance the third pulse. This suggests that ESE appear as a higher order effect of some, not necessarily restricted to one, interaction. The second derivative of the static field along the cross section of the sample may be required to be taken into account, as well as the inhomogeneity of the rf field. Also in the experiment made in a large field gradient, multi-pulse sequences such as (π/2)x − [τ − (π/2)x − τ − echo]n have often been employed [11]. Then, successively appearing ESE overlap successively appearing OSE. This is possibly the main reason why ESE have escaped one’s notice up to now, in spite of that they appear with fair
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N. Okubo / Physics Letters A 280 (2001) 221–226
Table 2 Examined samples Substance
State
Nucl.
ESE
T1 (s)
T2 (µs)
T2∗ (µs)
NaCl
polycrystal
23 Na
Obs
16.5
330
102
powder
23 Na
Obs
–
–
–
in oil
23 Na
Obs
–
–
–
aq. sol.
23 Na
Not
0.031
31000
510
powder
23 Na
Obs
7.1
530
130
powder
127 I
Not
0.0053
550
11
Na2 SO4
powder
23 Na
Obs
29, 4.6
690
26
e2 Qq = 2.6 MHza
Na0.9 WO3
powder
23 Na
Not
6.8, 1.3
1400
125, 63
e2 Qq = 1 MHzb
LiF
powder
7 Li
Not
9.0
56
19
H3 PO4
aq. sol.
31 P
Not
2.6
23000
400
D = 6.1 × 10−5 cm2 /s
NbS2
powder (77 K)
93 Nb
Not
∼ 0.01
800
2
e2 Qq = 59 MHzc
Nb
powder (77 K)
93 Nb
Not
∼ 0.001
48
7
NaI
D = 1.3 × 10−5 cm2 /s
The values of T1 and T2 for 23 Na and 127 I in NaI and 7 Li in LiF are those obtained from the measurement in a gradient of 0.2–2 T/m. Their significant dependence on the gradient was not observed. The relaxation in Na2 SO4 is governed by two T1 ’s. The NMR spectrum of Na0.9 WO3 consists of two components. No observation of ESE for 127 I in NaI, 23 Na in Na0.9 WO3 , and 31 P in H3 PO4 are the results in the gradient up to 0.5, 0.8, and 0.3 T/m, respectively. D represents diffusion constant and e2 Qq nuclear quadrupole coupling constant. All values listed are those obtained at room temperatures except NbS2 and Nb. a Ref. [12]. b Ref. [13]. c Ref. [14].
strengths solely by placing the sample, though limited, in a large field gradient. The resolution of the mechanism is an urgent problem, because the application of NMR in the field gradient is rapidly spreading. The construction of a model which can explain the above mentioned properties is desired.
Acknowledgement The author thanks Drs. T. Suzuki and K. Nojima for helpful discussions.
References [1] E.L. Hahn, Phys. Rev. 80 (1950) 580. [2] N. Okubo, T. Suzuki, T. Aoki, Phys. Lett. A 248 (1998) 463. [3] G. Deville, M. Bernier, J.M. Delrieux, Phys. Rev. 19 (1979) 5666.
[4] R. Bowtell, R.M. Bowley, P. Glover, J. Magn. Reson. 88 (1990) 643. [5] E.G. Kisvarsanyi, N.S. Sullivan, J. Low Temp. Phys. 101 (1995) 671. [6] Th. Wagner, S. Götz, N. Masuhara, G. Eska, J. Low Temp. Phys. 101 (1995) 657. [7] I. Ardelean, R. Kimmich, S. Stapf, D.E. Demco, J. Magn. Reson. 127 (1997) 217. [8] C.P. Slichter, Principles of Magnetic Resonance, 3rd ed., Springer Series in Solid-State Sciences, Vol. 1, Springer, Berlin, 1990. [9] A. Bloom, Phys. Rev. 98 (1955) 1105. [10] R. Kaiser, J. Magn. Reson. 42 (1981) 103. [11] For example, P. Bodart, T. Nunes, E.W. Randall, Solid State Nucl. Magn. Reson. 8 (1997) 257. [12] W. Gauss, S. Günther, A.R. Hasse, M. Kerber, D. Kessler, J. Kronenbitter, H. Krüger, O. Lutz, A. Nolle, P. Schrade, M. Schüle, G.E. Siegloch, Z. Naturforsch. A 33 (1978) 934. [13] G. Bonera, F. Borsa, M.L. Crippa, A. Rigamonti, Phys. Rev. B 5 (1971) 1708. [14] E. Ehrenfreund, A.C. Gossard, F.R. Gamble, Phys. Rev. B 5 (1972) 1708.
Physics Letters A 280 (2001) 221–226 www.elsevier.nl/locate/pla
Extraordinary stimulated echoes in a large magnetic field gradient Noriaki Okubo 1 Institute of Physics, University of Tsukuba, Tsukuba 305-8571, Japan Received 25 October 2000; received in revised form 20 December 2000; accepted 4 January 2001 Communicated by L.J. Sham
Abstract For a pulse sequence (π/2)x − τ1 − (π/2)x − τ2 − (π/2)x − t−(echo)y in a large magnetic field gradient, extraordinary stimulated echoes have been observed at t = i1 τ1 + i2 τ2 with integers i1 and i2 . The properties are described and possible mechanisms of the formation are discussed. 2001 Elsevier Science B.V. All rights reserved. PACS: 76.60.L Keywords: Stimulated echoes; Magnetic field gradient; Multiple spin echoes; NaCl; NaI; Na2 SO4
In the usual pulsed NMR two rf pulses are followed by one echo and the third pulse is followed by four echoes at most [1]. In a large magnetic field gradient, however, extraordinary echoes were found in addition to the ordinary ones. In this Letter the author reports the properties of the extraordinary echoes and discusses the mechanism of the occurrence. Detailed examination was done in a sample of polycrystalline NaCl, which is the same one as used in the preceding study [2]. The values of T1 and T2 for 23 Na nuclei in this sample are 16.5 s and 330 µs, respectively, at room temperature. In addition, two groups of samples were also examined. The one group consisted of samples prepared from NaCl in other states and the other consisted of those prepared from other substances. They were sealed in quartz tubes of 12 mm in o.d. and 20 mm in length. The tubes were placed in a superconducting magnet of 4.7 T with their axis perpendicular to the bore axis. The
E-mail address: [email protected] (N. Okubo). 1 Tel.: 81-298-53-4547, fax: 81-298-53-6618.
field distribution was examined using the 14 N NMR in liquid nitrogen. In the reference frame whose z axis is taken along the field, the rf field was applied along x axis and the y component of the magnetization was observed as usual. The calibration of the pulse width to the flip angle was made by comparing the width dependence of the height of the ordinary 2τ echo with the calculated flip angle dependence. The magnitude of the rf field H1 estimated from the pulse width was 2.2 mT. Echoes happened to overlap for some pulse timings, when the data of the echo height were discarded. Spin echoes were examined at room temperature unless otherwise described. Fig. 1(a) shows echoes of 23 Na nuclei in polycrystalline NaCl observed by applying three pulses. The sample was placed 10 cm away from the center and the field gradient at the position was 2 T/m. When the time intervals P1 P2 and P2 P3 in the figure are, respectively, denoted by τ1 and τ2 , these echoes appear later from the time P3 by i1 τ1 + i2 τ2 , where i1 and i2 are integers. In the following the echoes are referred to as Ei1 i2 . E1−1 is the primary echo, E10 the stimulated echo, and E01 and E11 are the secondary echoes.
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 0 2 7 - 5
222
N. Okubo / Physics Letters A 280 (2001) 221–226
(a)
(b)
Fig. 1. 23 Na echoes in polycrystalline NaCl observed in a field gradient of 2 T/m. Following the first rf pulse P1 , two pulses P2 and P3 were applied with delay times τ1 and τ2 . Their leading edges are shown by the arrows. Frequency: 52.65 MHz, pulse widths: all 10 µs corresponding to a flip angle of about π/2, repeating time: 300 s, averaging times: 8. (a) Case of τ1 > τ2 . E1−2 , E12 , E13 , and E14 are extraordinary echoes. (b) Case of τ1 < τ2 . E−12 and E12 are extraordinary echoes. Data points are connected by continuous lines for eye-guide. Rather asymmetric shapes of the echoes are due to incomplete adjustment of the reference phase.
In addition to these ordinary spin echoes (OSE), extraordinary stimulated echoes (ESE), E1−2 , E12 , E13 , and E14 appear in the figure. Though this trace was recorded after averaging, the ESE cannot be attributed to the storage effect, because the repeating time was sufficiently long compared with T1 . As τ2 was decreased to zero, E01 was hidden by the third pulse and the other echoes gathered towards E10 eventually coalescing into one echo, as expected. The appearance of ESE is not restricted in the case of τ1 > τ2 . Fig. 1(b) shows the opposite case. When the sample was initially placed in the homogeneous field at the center of the magnet, the free induction decay and OSE signals were not separated from each other because of their extremely slow decay. As the sample was moved away from the center along the bore axis with tuning, they became progressively narrower and ESE appeared. Fig. 1 is just the pattern observed in a gradient of 2 T/m, but ESE could be observed even in one order of magnitude smaller gradient. On further moving away, however, the signal
became rapidly small. This is possibly because there the rf pulses used began to lack in the spreading of the Fourier components. The dependence of the heights of these echoes on τ1 and τ2 is shown in Figs. 2(a) and (b), respectively. As seen in Fig. 2(a), E01 is independent of τ1 while E10 decays with a time constant close to T2 /2 and E1−1 and E11 decay with a constant close to T2 /4 at large τ1 , whereas ESE all exhibit peaks. The positions where the peaks appear change with τ2 and the dependence seems to be different for each of ESE. In Fig. 2(b), on the other hand, OSE except E1−1 and E10 exhibit peaks, whereas ESE decay without peaking. Fig. 3 shows the dependence of the echo heights including signs on the width of the third pulse when the widths of the first and second pulses are fixed at values nearly corresponding to the angle π/2. As the width of the third pulse is increased, the shape of the echoes collapses and particularly E1−1 beats violently around its peak, so that the reliability of their heights in the figure lowers at large pulse widths.
N. Okubo / Physics Letters A 280 (2001) 221–226
(a)
223
(b)
Fig. 2. τ1 and τ2 dependences of the absolute echo height. The open circles stand for ordinary echoes and the closed ones for extraordinary echoes. No data point exists for cases where the echoes are supposed to precede the third pulse, that is, n1 τ1 + n2 τ2 < 0, except E1−1 . A part of data points is omitted because of overlapping with other echoes or the third pulse. (a) τ1 dependence. τ2 : 107 µs. (b) τ2 dependence. τ1 : 145 µs. Other conditions are the same as in Fig. 1.
The mechanism of the formation of OSE is accounted for by the vector model for the nuclear magnetization in the moderately homogeneous field [1]. Expressions for their heights can be obtained by multiplying the vector of the magnetization at thermal equilibrium by the matrices representing the motion during on-time and off-time of the rf pulse [2]. When the flip angle of the nth pulse is denoted by αn and the delay time of the following pulse by τn , the heights of the echoes in the pulse sequence (α1 )x − τ1 − (α2 )x − τ2 − (α3 )x − t − (echo)y are expressed as in Table 1, where the effect of diffusion is neglected. When α1 = α2 = α3 , they reduce to Hahn’s expressions [1], apart from the effect of diffusion. Comparing Fig. 2(a) with the calculated τ1 dependence, the following features are remarked: No depen-
dence of E01 and the decay of E10 with a decay constant close to T2 /2 are in accordance with the calculated one. However, the decays of E1−1 and E11 are about two times faster than the result of the calculation. Concerning τ2 dependence, on the other hand, no dependences of E1−1 and E10 and decays of E−11 , E01 , and E11 with constants close to T2 /2 at large τ2 agree with the calculation. Nevertheless, the growth of the height seen for E01 and E11 at small τ2 is not expected from the calculation. Thus, the large field gradient does not only create ESE but also causes the deviation of the τ1 and τ2 dependences of OSE from the calculation. The deviation of OSE is also seen in the dependence on the width of the third pulse. In Fig. 3 the dependence of E10 is close to the calculated one. However,
224
N. Okubo / Physics Letters A 280 (2001) 221–226 Table 1 Height of ordinary echoes following three rf pulses applied with delay times τ1 and τ2 Echo
Height
E1−1
cos2 (α3 /2) sin2 (α2 /2) sin α1 exp(−2τ1 /T2 )
E10
(1/2) sin α3 sin α2 sin α1 exp(−2τ1 /T2 − τ2 /T1 )
E−11
− sin2 (α3 /2) sin2 (α2 /2) sin α1 exp(−2τ2 /T2 )
E01
sin2 (α3 /2) sin α2 [Mz (τ1 )/M0 ] exp(−2τ2 /T2 )
E11
sin2 (α3 /2) cos2 (α2 /2) sin α1 exp[−2(τ1 + τ2 )/T2 ]
αn denotes the flipping angle of the nth pulse. The height is normalized by the thermal equilibrium value M0 . Mz (τ1 ) is expressed as M0 [1 − (1 − cos α1 ) exp(−τ1 /T1 )] for the case where spin-lattice relaxation is represented by a single relaxation time T1 . The expression for E1−1 is applied only to the case τ1 > τ2 and the factor cos2 (α3 /2) is replaced by unity for the case τ1 < τ2 .
Fig. 3. Dependence of echo height including the sign on the width of the third pulse. The height of the downward echo is taken to be positive. τ1 : 245 µs, τ2 : 52 µs, first and second pulse widths: both 10 µs corresponding to π/2. Other conditions are the same as in Fig. 1.
the dependence of E1−1 and E11 is two or three times stronger than the calculated one even at small widths. Besides, the sign of E11 is reversed in conflict with the expression. On the other hand, since, for the first pulse with α1 = π/2, Mz (τ1 ) scarcely recovers at time much smaller compared with T1 , E01 should hardly be observed. Nevertheless, E01 was observed with a fairly strength at large widths of the third pulse. This means that the width of the first pulse is no longer sufficient as π/2 pulse in the gradient. To a rectangular line shape of a width 2ω in the frequency domain, an echo of the type of sine function, sin ωt/t, corresponds [1]. The oscillatory character of the echo is due to the singularity of the line shape, and the interval between the adjacent peak and bottom of the sine function is roughly given by π/ω. When a sample with an inherently sharp resonance is placed in a field gradient, the line shape is determined by
the cross section along the gradient. Since the cross section of a circle has also a singularity, it may be suspected that the present ESE originate from the oscillatory character of sine-like functions of the OSE. When the spreading of the field within the sample is denoted by H0 , the width of the line shape is given by γ H0 with gyromagnetic ratio γ . If ESE with a fixed i1 are attributed to the peaks and bottoms accompanying one of OSE, their separation should be roughly equal to 2π/γ H0 , which is about 5 µs for the present case, and should be independent of τ2 . This is not the case. Alternatively, if ESE are formed as a result of interference among sine-like functions accompanying some OSE, the height of ESE should exhibit an oscillatory behavior as τ2 is varied, because the separation between each OSE is τ2 for a fixed i1 . This also disagrees with the observation. In the preceding paper, we reported that in a large magnetic field gradient quasi multiple spin echoes (QMSE) appear following two pulses. The nature was found to be a storage of stimulated echoes which are successively created by signal averaging with an insufficient repetition time [2]. When T2∗ is so large as to be comparable with T2 they cannot be observed, whereas in a large field gradient T2∗ becomes short and they become visible, well separated each other and from the free induction decay. This situation is similar to the present case, but ESE differ from QMSE in that they appear even for one application of the sequence
N. Okubo / Physics Letters A 280 (2001) 221–226
of three pulses to a system where thermal equilibrium is established. Deville et al. [3] found multiple spin echoes (MSE) following two pulses in solid 3 He. Later, MSE were also reported for 1 H nuclei in hydrogen compounds [4,5] and for 115In in AuIn2 [6]. Ardelean et al. [7] treated MSE following three pulses and called them multiple nonlinear stimulated echoes (NOSE). Our ESE resemble to NOSE in some respects. Since MSE are created by modulation of the Larmor frequency due to a large nuclear magnetization M0 , their occurrence requires the condition 4πγ M0 T2 1.
(1)
For the reported nuclei this condition is certainly satisfied, whereas for 23 Na nuclei in NaCl the magnitude of the left hand amounts only to 5 × 10−5 . For the observation of MSE and NOSE a weak static field gradient is applied but the purpose is to make the analytical treatment feasible by avoiding the sample shape effect of the demagnetizing field of the nuclei, so that the gradient is not necessarily needed. On the contrary, a large gradient is indispensable for ESE to be created. Moreover, MSE appear even for application of only two rf pulses in the sample satisfying condition (1), whereas the present ESE do not appear until the third pulse is applied. The motion of the assembly of the nuclear spins is classically described in the rotating frame [8] as follows: In the present case, each isochromat represents the contribution from each portion of the sample with different z. During on-time of the rf pulse the isochromats rotate about the effective field as usual, but in the large field gradient the direction of the field is not x axis but inclined towards z axis by an angle corresponding to the difference of the field at the position from the resonance field. During off-time they are distributed in x–y plane more quickly than in the usual moderate inhomogeneity. If then a long pulse is used, an edge echo appears even after a single pulse, owing to the dephasing during the pulse and the following rephasing [9,10]. In the present case, however, such a signal was not observed with a significant level. Thus, the occurrence of ESE could not be accounted for by the known mechanisms. To obtain further information, other samples were also examined in the field gradient. The result is summarized in Table 2
225
together with the relevant values. For NaCl a pattern similar to Fig. 1 was also observed in the powder sample, and the pattern was not affected by the immersion into silicon oil (viscosity of 10 cSt). In the nearly saturated aqueous solution, however, only OSE were observed and none of ESE was observed. The values of diffusion constant in the table are those estimated from the primary echo in the gradient [8]. ESE must also decay owing to diffusion. However, their disappearance in the solution cannot be attributed to diffusion, unless ESE have a particularly strong dependence on it. No effect of immersing NaCl into silicon oil indicates that ESE have nothing to do with the surface of the sample. The possibility that ESE originate from piezoelectricity is excluded by the resonant character and the nonactive space group. In the table only the samples for which OSE could be observed with good signal-to-noise ratios are listed. It is distinct for them whether ESE are observed or not. ESE appeared always accompanying a series of OSE, whereas OSE were not always accompanied by ESE. The observation of ESE is so far limited within 23 Na nuclei in solid NaCl, NaI, and Na2 SO4 . Disappearance of ESE in the aqueous solution of NaCl and no appearance in NbS2 suggest that the dipolar interaction exceeding the quadrupolar one is needed for ESE to occur. Thus, the mechanism of the occurrence of ESE is not resolved for the present. However, closer examination of the time of the appearance of ESE revealed that the value of i1 is limited to 1 and −1. For i1 = 1 the observed i2 ranges from −4 to 6 including three OSE and for i1 = −1 the i2 ranges from 1 to 3 including one OSE. All echoes are therefore supposed to be formed centered at times τ1 before and after the third pulse with the interval τ2 , including virtual echoes in advance the third pulse. This suggests that ESE appear as a higher order effect of some, not necessarily restricted to one, interaction. The second derivative of the static field along the cross section of the sample may be required to be taken into account, as well as the inhomogeneity of the rf field. Also in the experiment made in a large field gradient, multi-pulse sequences such as (π/2)x − [τ − (π/2)x − τ − echo]n have often been employed [11]. Then, successively appearing ESE overlap successively appearing OSE. This is possibly the main reason why ESE have escaped one’s notice up to now, in spite of that they appear with fair
226
N. Okubo / Physics Letters A 280 (2001) 221–226
Table 2 Examined samples Substance
State
Nucl.
ESE
T1 (s)
T2 (µs)
T2∗ (µs)
NaCl
polycrystal
23 Na
Obs
16.5
330
102
powder
23 Na
Obs
–
–
–
in oil
23 Na
Obs
–
–
–
aq. sol.
23 Na
Not
0.031
31000
510
powder
23 Na
Obs
7.1
530
130
powder
127 I
Not
0.0053
550
11
Na2 SO4
powder
23 Na
Obs
29, 4.6
690
26
e2 Qq = 2.6 MHza
Na0.9 WO3
powder
23 Na
Not
6.8, 1.3
1400
125, 63
e2 Qq = 1 MHzb
LiF
powder
7 Li
Not
9.0
56
19
H3 PO4
aq. sol.
31 P
Not
2.6
23000
400
D = 6.1 × 10−5 cm2 /s
NbS2
powder (77 K)
93 Nb
Not
∼ 0.01
800
2
e2 Qq = 59 MHzc
Nb
powder (77 K)
93 Nb
Not
∼ 0.001
48
7
NaI
D = 1.3 × 10−5 cm2 /s
The values of T1 and T2 for 23 Na and 127 I in NaI and 7 Li in LiF are those obtained from the measurement in a gradient of 0.2–2 T/m. Their significant dependence on the gradient was not observed. The relaxation in Na2 SO4 is governed by two T1 ’s. The NMR spectrum of Na0.9 WO3 consists of two components. No observation of ESE for 127 I in NaI, 23 Na in Na0.9 WO3 , and 31 P in H3 PO4 are the results in the gradient up to 0.5, 0.8, and 0.3 T/m, respectively. D represents diffusion constant and e2 Qq nuclear quadrupole coupling constant. All values listed are those obtained at room temperatures except NbS2 and Nb. a Ref. [12]. b Ref. [13]. c Ref. [14].
strengths solely by placing the sample, though limited, in a large field gradient. The resolution of the mechanism is an urgent problem, because the application of NMR in the field gradient is rapidly spreading. The construction of a model which can explain the above mentioned properties is desired.
Acknowledgement The author thanks Drs. T. Suzuki and K. Nojima for helpful discussions.
References [1] E.L. Hahn, Phys. Rev. 80 (1950) 580. [2] N. Okubo, T. Suzuki, T. Aoki, Phys. Lett. A 248 (1998) 463. [3] G. Deville, M. Bernier, J.M. Delrieux, Phys. Rev. 19 (1979) 5666.
[4] R. Bowtell, R.M. Bowley, P. Glover, J. Magn. Reson. 88 (1990) 643. [5] E.G. Kisvarsanyi, N.S. Sullivan, J. Low Temp. Phys. 101 (1995) 671. [6] Th. Wagner, S. Götz, N. Masuhara, G. Eska, J. Low Temp. Phys. 101 (1995) 657. [7] I. Ardelean, R. Kimmich, S. Stapf, D.E. Demco, J. Magn. Reson. 127 (1997) 217. [8] C.P. Slichter, Principles of Magnetic Resonance, 3rd ed., Springer Series in Solid-State Sciences, Vol. 1, Springer, Berlin, 1990. [9] A. Bloom, Phys. Rev. 98 (1955) 1105. [10] R. Kaiser, J. Magn. Reson. 42 (1981) 103. [11] For example, P. Bodart, T. Nunes, E.W. Randall, Solid State Nucl. Magn. Reson. 8 (1997) 257. [12] W. Gauss, S. Günther, A.R. Hasse, M. Kerber, D. Kessler, J. Kronenbitter, H. Krüger, O. Lutz, A. Nolle, P. Schrade, M. Schüle, G.E. Siegloch, Z. Naturforsch. A 33 (1978) 934. [13] G. Bonera, F. Borsa, M.L. Crippa, A. Rigamonti, Phys. Rev. B 5 (1971) 1708. [14] E. Ehrenfreund, A.C. Gossard, F.R. Gamble, Phys. Rev. B 5 (1972) 1708.