Quasi-multiple spin echoes in a large magnetic field gradient

16 November

1998

PHYSICS LETTERS A

Physics Letters A 248 ( 1998) 463467

Quasi-multiple spin echoes in a large magnetic field gradient Noriaki Okubo a, Takashi Suzuki a, Takayoshi Aoki b a Institute of Physics, University of Tsukuba, Tsukuba 305, Japan h Radio Isotope Center, University of Tsukuba, Tsukuba 305, Japan Received 29 April 1998; revised manuscript received 31 July 1998; accepted Communicated by L.J. Sham

for publication

19 August

1998

Abstract

The characteristics of multiple spin echoes observed in NaCl in a large magnetic field gradient are reported. By means of the vector model incorporating relaxation it is shown that this type of multiple echoes can be explained as an extension of Hahn’s secondary echo. @ 1998 Elsevier Science B.V. PAC.? 76.60.L Keywords: Multiple spin echoes; Large field gradient;

Secondary

In the usual pulsed NMR, when two rf pulses with resonance frequency are applied to the spin system at time t = 0 and t = 7, a single spin echo is observed at t = 27 [ 11. On the other hand, it is known that multiple spin echoes (MSE) appear in such cases as follows: in a system with quadrupolar interaction, Solomon echoes appear [ 21; in a magnetic system the echo enhanced by the domain wall acts effectively as an rf pulse, forming the next echo in succession [ 31; in systems represented by solid 3He the large nuclear magnetization modulates the Larmor frequency along the field gradient after the second pulse, forming MSE [ 4,5]. Recently, we found a different type of MSE in a tungsten bronze (Na,WOs ) placed accidentally in a large gradient of the magnetic field. Since the MSE were observed also in NaCl, it is not considered as a phenomenon restricted to special substances. This paper reports the characteristics of the MSE examined in NaCl and accounts for the mechanism of the formation. Fig. 1 shows MSE observed for 23Na nuclei in poly-

Hahn echo; Relaxation

times; NaCl

801

I

.I ? .

..

,

23Na in NaCl

I

-

. .’ 60-. Q

fl ..

TIME ( CIs) Fig. 1. MSE of 23Na in a field gradient of 10 ~~-10 ps (nearly ps, averaging: 1024

03759601/98/$ - see front matter @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00691-4

nuclei in polycrystalline NaCl at 52.70 MHz 2.0 T/m at room temperature. Pulse widths: corresponding to 90”-90”). delay time: 58 times, repetition time: 6 s.

464

N. Okubo et a/. /Physics

crystalline NaCl at 52.70 MHz in a field gradient of 2.0 T/m at room temperature. The polycrystals were prepared by crystallization from a saturated aqueous solution of NaCl, and they were dried and filled into a quartz ampoule of 12 mm in o.d. and 20 mm in length. The ampoule was placed in the field with its axis perpendicular to the field. The echoes appeared at times r = 117 (n = 2,3,4,. , .) decaying monotonically. The MSE were observed also in powder NaCI. However, further examination revealed that this multiplicity arises from applying the pulse train repeatedly for averaging the signal. Fig. 2 shows what happens in the system when the pulse train is applied successively. When a pulse train consisting of two pulses of the same width with delay time r is applied at t = 0 to the spin system at thermal equilibrium, only one echo appears at t = 27, as shown on the trace for the first repetition. Nevertheless, when the second pulse train of the same composition is applied at t = T (5 T, ), two echoes appear at time T + 27 and T + 37, as shown on the trace for the second repetition. In such a way, one pulse train applied a time T after the preceding one creates one extra echo at a time r later. Actually in Fig. 2, signal averaging was still employed for improvement of the signal-to-noise ratio, but since the next series of traces were acquired after waiting for a sufficiently long time (about 20 times Ti ), the effect of the repetition on the occurrence of the echoes is removed in the figure. Fig. 1 corresponds to the average of 1024 traces obtained in one series of acquisition. The pulse sequence used is expressed in the conventional form [6] as [(rr/2),r-(r/2j,-(T - T)-_I,,. Since the echo appended in every repetition of the pulse train is essential for these quasi-multiple spin echoes (QMSE), we focus our attention on it. The values of Ti and T2 measured in the homogeneous field of 4.7 T at the center of the magnet were 16.5 s and 330 ,us, respectively, at room temperature. Fig. 3 shows the T dependence of the absolute height of the nth echo in the nth repetition. The first echo is independent of T, while the other echoes decay exponentially with the time constant decreasing with n. Fig. 4 shows the r dependence of the absolute height of the nth echo. At large 7 all the echoes decay and the decay becomes faster with n. Since T2 is defined by putting the 27 dependence of the first echo in the homogeneous field as exp( -2r/Tz), the time constant

Letters A 248 (1998) 463-467

I

I

0

I

I

I

I

200 TIME

I

400 ( )-IS)

Fig. 2. Traces for the pulse trains repeated with a time interval of 6 s. A series of traces were acquired 16 times with a waiting time of 300 s. Each trace of the figure is the average of the corresponding 16 traces. The other conditions are the same as in Fig. I.

l

10-j

’ ’ ‘I20 10 REPETITION

2nd

L 30 TIME

’ 40 (s)

Fig. 3. Repetition time dependence of the absolute height of the )zth echo in the nth repetition. The best-fit lines for exponential decay are drawn with the time constant. Delay time: 4.5 ps.

of the decay of the first echo at large r in the figure should correspond to T2/2. In fact, the time constant is close to that. Instead, if for a given 7 the echo height is plotted versus the time that the nth echo appears on

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N. Okubo et al. /Physics Letters A 248 (1998) 463-467

1st 2nd a 3rd . 4th

0 l

0

DELAY TIME ( CIs) Fig. 4. Delay time dependence of the absolute height of the nth echo in the nth repetition. Repetition time: 4 s.

at t = T (> T),stimulated echoes appear at t = T + 7, 2T - 27, 2T - r, and 2T. He called the echo at T + 7 the secondary echo, because the others can be derived as primary echoes from combination of either two out of the three pulses and the echo at f = 27, which acts effectively as one pulse for the following pulse. In the following we show that the present QMSE can be derived as secondary echoes from the vector model [ 61. We inspect the time evolution of the nuclear magnetization vector M(t) in the reference frame which rotates about the z-axis along the magnetic field with an angular frequency yHe [ 61. y denotes the gyromagnetic ratio and HO the magnitude of the field. When the rf pulse with a negligibly small width is applied along the x-axis at time t= 0,the effect is represented as a rotation of M (O_ ) about the x-axis by an angle cy. After the rf field is turned off, the monochromats spread in the static field. The spreading during time t is represented as a rotation about the z-axis by an angle dwt, where AU denotes the deviation of the Larmor frequency of the monochromat from the center frequency of the resonance. The relaxation during t proceeds independently of the spreading. Therefore, the effect of spin-spin relaxation can be incorporated by multiplying the X- and y-components of M(t) following the rotations, M,(t)and M!,(t), by exp ( --t/T2). The effect of spin-lattice relaxation can be incorporated by replacing the z-component M, (r)

by Mo{~- [1 - MZ(t)/Molexp(-t/Tll)}, MObeing the magnitude at thermal equilibrium. erations can be formulated as follows, I

0

100

I

200

These op-

I 300

DELAY TIME (us) Fig. 5. Delay time dependence of time constant T~Q defined in the text. (Solid curve) Eq. (6) with T, = 16.5 s, T2 = 330~s. T = 4 sanda=90’.

M(r)

= P(cu,r)M(O-)

(1)

where P(a,t)

each trace, (n + 1)T, and if the change is fitted with a single exponential function exp( --t/T*o), a time constant T~Q is defined. The r dependence of T~Qis shown in Fig. 5. As r increases, TZQ increases linearly and levels off at 7 - 130 ps. Since the quadrupolar interaction is negligibly small in NaCl, the Solomon echo is not expected. In the case of a magnetic system or solid 3He, MSE appear even for one application of the pulse train [ 3-51. Therefore, our MSE are substantially different from these. Hahn [ 1 ] showed that, when one more pulse is applied

+ N(t),

= R(t)U,(Awf)U,(cu)

(2)

with e--IITl R(t) = (

0 0

0 e-tlrz

(3)

0

and

N(t)

=

MO

(4)

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A’. Okubo et al. /Physics

uX(a> and U, (Awt) are the unitary matrices for the rotations. Any pulse sequence without phase shift can be treated by applying Eq. (1) repeatedly. The usual echo at t = 27 appears as a term containing a factor of cos Ao( t - 27) in the y-component of M(t) . The stimulated echoes following one more pulse applied at t = T ( > T) appear as cos A?lw (t - T - I-), cosAw(t - 2T + 27), cosAw(t - 2T + 7) and cos Ao( t - 2T). In a similar way it can be shown that, when one more further pulse is applied at t = T + r, an extra echo is certainly created at t = T + 37 in addition to the usual one at t = T + 27. This is just the beginning of QMSE. When we repeat this procedure, we are forced to treat an increasing number of terms. However, the computation is simplified by noting the conditions T << T, and T >> T2 in the present case, and it can be shown successively that the nth pulse train creates the nthechoatt=(n-l)T+(n+l)rasinFig.2[7]. Though the expression for the height of the nth echo of the m( 2 n) th repetition, S,,,,, still consists of many terms, the height S,, can be written as S,, = (-;)“-I x

e -2m/Tze-(n-l)T/fi

sin2n-’ o sin2 ( $a).

(5)

This is a generalization of the expression which Hahn derived [ I] from Bloch’s equation, though diffusion is neglected. Eq. (5) shows that the nth echo is formed by the component which spends time (n - 1) T along the z-axis and time 2nr within the xy plane, and S,, can be regarded as decaying with the time constant T, /(n - 1) concerning T and with the time constant T2/2n concerning r. With the values of 16.5 s for TI and 330 pus for T2, Eq. (5) well accounts for the T dependence in Fig. 3 and the tendency at large r in Fig. 4. For fixed r and T, IS,,,,,] decays at every step of n by a factor of i exp (--2r/T2 - T/T,) sin2 LY.By the definition of T2o this factor is equal to exp( -7/T2~), so that -_=-+-

.

(6)

For large 7, T2o approaches T2/2, whereas for small r it decreases toward zero along a line with a slope of l/[T/Tl - ln( i sin2 LY)]. The pulse width used was calibrated to be 90 * 10” by comparing the LYdependence of the echo height with Eq. (5). The solid curve

Letters A 248 (1998) 463-467

in Fig. 5 is drawn for T~Q calculated from Eq. (6) using cy = 90’. Though the accuracy of the value of LYis not high, the cxdependence of T~Q is as weak as f4 ps even for f20° of a. The agreement with experiment is fairly good for r 5 150 pus. Usually, sufficiently slow repetition is employed to avoid saturation. However, even if fast repetition is employed, in the homogeneous field it is difficult to observe the individual echoes composing QMSE separately, because QMSE decay with a time constant T2/2 even for r 2 T2 while the component echoes expand their wings to a time of the order of T; comparable to T2. The main role of the large gradient is to make the echoes be observed as MSE by shortening their T;. Therefore, the condition for QMSE to be observable is that 1/T; (N 1/T2 + yAHo/2r) is sufficiently large compared with 2/T2; that is, yAHoT2 > 2~,

(7)

where AH0 is the inhomogeneity of the field within the sample. In fact, for NaCl in the homogeneous field it was not easy to observe even the first echo properly owing to the interference with the large free induction, but when yA HoT2 exceeded about 15, QMSE became observable. For smaller T2 larger A HO are required and consequently the pulses of smaller widths are needed to flip the monochromats spreading over the larger Aw. However, it is not difficult for MO in a solid to satisfy the condition (7). On the contrary, it is difficult to satisfy 4yr&loT2 > HIT,

(8)

one of the conditions for the observation of MSE originating from the modulation of the Larmor frequency due to MO [4], except at exceedingly low temperature, because T2 is generally not so long in the solid. In the case of NaCl the value of 4y7rMeT2 is 5 x IOF in 4.7 T at room temperature. The advantage of the use of the large gradient has been taken extensively in the stray field imaging (STRAFI) [ 81, where in most cases the pulse sequence (r/2) --7-[ (n-/2) -r-echo-r-] n including phase shift is employed and the analysis is made using density matrix [ 91. In STRAFI the fringe field of the magnet is utilized intentionally and the gradient is one order of magnitude larger than our case, where the sample was placed only 10 cm away from the

N. Okubo et al./Physics

center of the magnet. Moreover, Randall et al. [lo] observed MSE in various solids even by one application of the pulse train composed of two rf pulses as primary Hahn echoes of high order, referring to MSE in solid 3He. On the other hand, our QMSE were observed by repeating the pulse train and the occurrence as well as the main features have been accounted for as an extension of the secondary Hahn echoes. The large gradient is not required for the creation of the extra secondary echoes, but only supports the observation of QMSE. However, this does not exclude the possibility that the large gradient might play some substantial roles. There are some discrepancies between the theory and the experiment. The most significant is the deviation of the echo height from the single exponential decay in Fig. 4. The decay becomes progressively fast with 7. Also in Fig. 5, T~Q suddenly ceases to increase near r = 130 /.LSwith a value much smaller than T2/2. In the derivation of Eq. (5) all the monochromats have been assumed to rotate about an identical (x-) axis and by an identical angle (cu) . This approximation deteriorates when approaching the upper and the lower ends of the sample in the large gradient. In addition, closer

Letters A 248 (1998) 463-467

46-l

inspection of Fig. 2 shows that even with an extraordinarily long recycling time (2OTi) there remains a slight dip at time 37 for the first repetition. This may also imply that the role of the large gradient is not only confined to narrowing the component echoes.

References [ I] EL. Hahn, Phys. Rev. 80 ( 1950) 580. [21 I. Solomon, Phys. Rev. 110 (1958) 61. [3 I H. Abe, H. Yasuoka. A. Hirai, J. Phys. Sot. Japan 21 ( 1966) II. [4] G. Deville. M. Bemier, J.M. Dehieux, Phys. Rev. B 19 ( 1979) 5666. 151 A. Abragam, M. Goldman, Nuclear Magnetism: Order and disorder (Oxford Univ. Press, Oxford, 1982). [6] C.P. Slichter, Principles of magnetic resonance. 3rd Ed. Springer Series in Solid-State Sciences, Vol. 1. ed. P. Fulde (Springer, Berlin, 1990) [71 N. Okubo et al., to be published. 181 P.J. McDonald, Prog. Nucl. Magn. Reson. Spec. 30 (1997) 69. L9j A.D. Bain, E.W. Randall, J. Magn. Reson. A 123 (1996) 49. [ 101 E.W. Randall, A.A. Samoilenko, T. Nunes, J. Magn. Reson. A I16 (1995) 259.