149
Solid State Nuclear Magnetic Resonance, 1 (1992) 149-158
Elsevier Science Publishers B.V., Amsterdam
Study of a spin-3/2 system by a quadrupolar-echo suppression of spurious signals
sequence:
Pascal P. Man Laboratoire de Chimie des Surfaces, C.N.R.S. U.R.A. 1428, UniLlersitt Pierre et Marie Curie, 4 Place Jussieu, Tour 55, 75252 Paris Cedex 05. France
(Received 23 March 1992; accepted 15 April 1992)
Abstract The density matrix describing the evolution of a spin-3/2 system excited by a quadrupolar-echo sequence consistinb of two rf pulses in quadrature phase, {X}-r2-(Y}-r4-[acquisition(y)], is calculated from the equilibrium state to the acquisition period. The interactions involved are the first-order quadrupolar interaction throughout the experiment and a local heteronuclear magnetic dipolar term between the two pulses and during the acquisition period. Three echoes, one due to a satellite transition at 74 = 72 and two due to the central transition at 74 = 72 and 74 = 37*, are predicted. They have similar expressions than those obtained with two rf pulses of the same phase, (XJ-Q-{X)-TV-[acquisition(y)], except the signs. Moreover, it is shown experimentally that a combination of these two sequences, namely: (X}-To-{X)-T,+-[acquisition( y&recycle delay-(X)-~,-{ - X)-T4-[acquisition(y)]-recycle delay-(X)-T,-{Y)-T,-[acquisition( - y)]-recycle delay-(X]- T*-( - Y]-T~ -[acquisition( - y&recycle delay, cancels the spurious piezo-electric signals when studying a ferroelectric material in the single-crystal form. Keywords: NMR, quadrupolar echo; fictitious spin-l/&
spurious signal suppression
1. Introduction Recently, the time-domain response of a spin I = 3/2 system excited by two radio-frequency (rf) pulses of the same phase (IX)-T,-{XI-T,-[acquisition(y)] where y is the receiver phase) for any ratio of the quadrupolar coupling ho over the amplitude of the rf pulse wrf was carefully analyzed [l]. The interactions involved are the first-order quadrupolar interaction throughout the experiment and a local heteronucleur magnetic dipolar interaction (S’WI_sj = 41,) between the two pulses and during the acquisition period. These hypotheses are fulfilled when the homonuclear magnetic dipolar interaction is smaller than the heteronuclear magnetic dipolar interaction. This also means that 4 must be smaller than w,~ so that 4 can be neglected during the pulses. Three echoes were predicted: one due to a satellite transition at 74 = TV and two due to the central transition at TV= 72 and TV= 3~~. The main experimental restriction is that the interpulse delay T* must be twice as long as Tfid (duration of the fid) in order that the echoes and the fid following the second rf pulse are separated. As a result, the echoes are very small due to spin-spin relaxation. Fortunately, alternating the phase of the Correspondence to: Dr. P.P. Man, Laboratoire
de Chimie des Surfaces, C.N.R.S. U.R.A. 1428, UniversitC Pierre et Marie Curie, 4 Place Jussieu, Tour 55, 75252 Paris Cedex 05, France. 0926-2040/92/$05.00
0 1992 - Elsevier Science Publishers B.V. All rights reserved
150
P.P. Man /Solid
State Nucl. Magn. Ram.
I (1992) 149-158
second pulse and keeping the receiver phase unchanged, e.g., IX}-r,-(X}-r,-[acquisition(y)]-recycle delay-(Xj-r,-( -X)-r,-[acquisitiom y )I-recycle delay, cancel the fid following the second pulse without changing the echoes [2]. Therefore, one can reduce the interpulse delay r2 to one Tfid and the echoes appear much larger. However, this improvement cannot cancel either the spurious piezo-electric signals when studying a ferro-electric compound in the single-crystal form [3,41, or the spurious acoustic-ringing signals from the NMR probe when studying a low gyromagnetic ratio nucleus [5-71. The paper is organized as follows: In section 2.1, we calculate the density matrix p(t,, TV, t,, rJ, from the equilibrium state to the acquisition period, of a spin-3/2 system excited by a quadrupolar-echo sequence consisting of two rf pulses in quadrature phase (Fig. l), (Xj-r,-(Yj-r,-[acquisition(y)], using the fictitious spin-l/2 operator formalism [8,9]. The interactions involved are the same as for IX)-TV(X)--r,-[acquisition(y)] sequence [l]. We then introduce in section 2.2 the expression of the central- and a satellite-line intensities. In section 2.3 the echo amplitudes are deduced from the line intensities of the previous section. In section 3, we experimentally show that the sequence: (X)--72-(X)-~4[acquisition(y)]-recycle delay-(X)-r,-{ -X)-T,-[acquisition(y)]-recycle delay-(X)-r,-(Y)-r,[acquisitiom-y&recycle delay-{X)-r,-{ - Yj-r,-[acquisitiom-y)]-recycle delay, suppresses the spurious piezo-electric signals. This is illustrated with the lithium nucleus ‘Li in a single crystal of LiNbO,. Finally, section 4 is a brief conclusion,
2. Theory The Hamiltonians throughout the paper are defined in angular frequency units. Neglecting the relaxation phenomena and the second-order quadrupolar effects, the dynamics of a spin-3/2 system is described by the density matrix excited by a quadrupolar-echo sequence, {X)-T~-(Y]-T~, p(t,, 72, t,, TV) expressed in the rotating frame associated to the central transition: p( tl, 72, t3, 74) = eXp( -i%(')T4) Xp(0)
eXp(-iz’b’t3)
eXp(-ix(')T2)
exp( i2?‘“‘t,) exp( iJ?‘“)T,) exp( iSb’t,)
where the following quantities are written with the fictitious spin-l/2
eXp(-iz’a’tl) exp( iScb4)
(1)
operators:
p(0) = z, = 3z,‘,” + I,“,”
(2a)
Aq’
(2b)
= 5ua(3z2’
- Z( z + 1)) = 20Q( z,‘.’ - 12%“)
3e2qQ [3 cos2/3 - 1-t 77sin’/3 cos 2a] wQ = SZ(2Z - l)A
w
A?@) = -wrfZx
+ z#’
(2d)
c%@(b)= -wrfzy
+ A?p=
@'=Jg'+$z
z
= - aod(
I,‘,’ + Zx”‘“) - 2w,,z,2,3 + X#’
-l&orf(z;~2+z;~4)
- 2WrfZ,2,3 + GYy)
(2e) (29
A?$) is the first-order quadrupolar interaction, (Y and p represent Euler angles describing the orientation of the strong static magnetic field B, in the principal axis system of the electric field gradient (EFG) tensor and 77 the asymmetry parameter. t, and t, are the first and the second rf pulse lengths, respectively (Fig. 1). The matrix of A?@) and SSCb) expressed in the eigen-states of Z, are not diagonal.
PP Man /Solid
State Nucl. Magn. Ram.
-
I
I
1
z2
t1
151
1 (1992) 149-158
_I_ I-
t3
+
time
z4
_I_ I-
-I
Fig. 1, Hamiltonians and durations associated with a quadrupolar-echo
sequence.
The diagonalized forms A?.. and Xv, and the transformation respectively, were already defined 19-121:
T and I/ of GV’@)and .GVCb),
operators
A?r = T+ Xca)T = w13Z;,3 + 024Zz2,4+ wrf( Z,‘,” + Zz’.“)
(3a)
A?” = V+A?‘b’v=
P)
T=exp
i
w13z;1’2+ 024z;3’4 + W,f( z,‘,” + zz”q
i;(Zi34-Z;,3) ‘2,“)
exp( -i28,Z~~3) exp( -i282Zt,4)
(34
exp( -i28,Z,‘,2)
(34
exp(i282Z,3~4)
Eqn. (1) can be rewritten as: p( t,, TV, t,, 74) = exp( -i#“~,)V
exp( -iXvt3)V+
xexp( -iS#3Ttl)T+p(0)T X
2.1 Density matrix p(t,,
r2,
exp( -id?‘(C)~2)T exp(iL4?“b,)V
exp(iXTtl)Tf
exp( iA?vt3) V+ exp( iG+Pc’7,) t,,
(4)
r4)
This section is devoted to calculating, step by step, the density matrix [eqn. (411. The density matrix before the application of the second pulse has been obtained previously [ll. During the second pulse, the evolution of the spin system is described by A? @‘).So the density matrix becomes:
p(t,, TV)
p(t,,
TV, f3) = v exp( -izvt3)V+
P (t 1, T2)v
ew(i~vt3)~+=pI(f17
72y t3)
+
p2(t1, 72, t3)
(5)
with p,(t,,
TV, t3) = $C{i(U)O(n
P#,,
72,
f3) = %{$%)O(~(d2
+ i(e + d)P, sin +Q-~- +(e - d)P, sin -
e,P,
cos
4~~ + +(e, +
d,)P,
3+r2
cos
34~~
+
J3( r2))}K +
J4( T~))}K
(ha) (6b)
and J3( T)
=
[q, sin &r - q3 cos 2471 cos 2wo7 + [ri sin
J4( Q-)= - [q, cos v=(cos
4~ +
26,,cos 28,,
$7 -
r3 cos 2471 sin 2~0~7
q4 sin 2471 cos 2wQr - [r2 cos 47 + r4 sin 2471 sin 2~~7
sin 28,, sin 20,,1)
The matrices of spin operators Y and Y, are defined in Table 1. All the above parameters associated variables not explicitly defined here were already defined in refs. 1 and 10.
Pa)
(7b) and their
P.P. Man /Solid
152
State Nucl. Magn. Reson. I (1992) 149-158
TABLE 1 The hvo matrices of spin operators Y and YO of eqns. (6a) and (6b) /A Y=
- 4 -DI
E,
G,
-G,
-F
F
-D,
-C
-C
-B,
-B,
-G,
F
-F
A
-A
-E,
E,
-c
Bl
B,
H
H
-D,
-D,)
A H
-H
-G,
c
1 -G-C,
0
E+A,
0
0
0
(E + A,)/2
(E + A,)/2
0
0
G-C,
0
E-A,
(E - A,)/2
(E-
-E-A,
0
-G-C,
0
0
0
0
0
0
A,-E
0
0
- Fl - B
0
0
0 Y,,=
\
-C,+G
F, - B
-(CL
+ G)P
-CC,
+ G)/2
-(CL
- G)P -D
-CC,
- G)/2 D
0
B = I;.4 _ *;,2
A = Il.3 _ I?.4 1* I B = 13.4 + 11.2
C =
A = I’,4 _ 12.3
= 11.3 _
[I,3 _ x
12x4 x
c:
D = 12.3 _ Y
11.4 Y
D i = 1:’
E = 12.3 + 11.4 Y Y
E
F = 11.3 + 12.4 ? Y G = 11.2 + 13.4 Y Y
f?; = $3 G
-
A,)/2
1i.4 I;.”
= 12.3 + 11.4 x _
= I?.4 1 ,
_
I,‘,” II.2 Y
H = 11.4 + 12.3 i z
After the second pulse, the evolution of the spin system is described by SF’@).The density matrix becomes: p(r,, 72, t,, q) =P3(tr,
727 t,>
74)
+P&
72,
t3,
(84
74)
with P3(tl, r2, t,, r4) = exp( -i~‘C’r,)pr(tI,
72, t3) exp(i~‘“‘r,)
= $$( i( ([N - 1:,3P1 cos
+T~
-
f;*3P, sin
4~~
+Z,‘,4P3 cos 34r4 + Ii94P3 sin 34r4 + Z,( r4)]) xO[n p4(tl,
TV,
t,,
TV)
= exp( = $$(
+ i(e + d)P, sin
-i@c)74)P2(t,,
f(~[ArPs
-IiAP2
r2,
+ 1:,3P4 sin
i(e - d)P, sin 34r2 + J3( r2)])K
-
t3)
ew(ix@‘T,)
+T~
-
Ij?*3P4cos
(8b)
~$7~
sin 34~~ + Ii,4P2 cos 34r4 + Z,( r4)])
xO[ - i(e, - d,)P, z,(r)
4~~
cos
~$7~ + $(e, +
d,)P,
cos
34~~
+
J4( T~)])K
(8~)
= (Q, cos +T - Q, sin ~$7- Q3 cos 247 - Q, sin 247) cos 20~~7 + (R, cos
$7
-
R, sin C#I~ - R, cos 247 - R, sin 247) sin
20.~~7
(84
Z,( 7) = (X2 cos f&r - X, sin ~$7- X3 cos 247 - X4 sin 24~) cos 20~~7 +(Y2 cos $TThe matrices
P, (i =
Y, sin &r-Y,
cos 24~-Y~
sin 247) sin 2~~7
l-5), Qj, Rj, Xj and Yj (j = l-4) were defined in ref. 1.
(se>
P.P. Man /Solid
State Nucl. Magn. Reson. 1 (1992)
153
149-158
2.2 Line intensities It has been repeatedly shown [lo,121 that the line intensity of a quadrupolar spin depends on the ratio of oo/w,r. As a result, the magnitude of ho can be determined in measuring the line intensity for a series of pulse length. Knowledge of the density matrix p(t,, T2. t3, TJ, Eqns. (8a)-(8e), allows the determination of the central-line intensity Fc(tl, r2, t,, TJ and its quadrature part GYt,, Q, t3, ~~1, and S-J and its quadrature part GYt,, TV, t,, TV). also the line intensity of a satellite transition FYt,, TV, t,, In this section, we focus mainly on the quadrature parts of the line intensity from where the amplitude of the bell-shaped echoes can be deduced 1131,whereas the quadrature part of the bell-shaped echoes, in other words the “sine-like” shaped echoes, are deduced from FYt,, TV, t,, TV) and FYt,, r2, t3, TJ: qt,,
72,
t,, rq) = Tr[p(t,.
r?, t,, 74)2z_?3]
G‘-(tl, 72, t,, TV) =Tr[p(l,, T2, t3,',1)24?'] Fs(tl, TV,
t,,
TV)
=Tr[p(t,,
7?,
t3,
T~)\/~I.J~*]
G”( t,, T?, t,,TV) =Tr[p(t,, 72, t,,T4)fili,*]
(9)
It is easy to show that
G’(t,,
72,
G’(t,,
Tz, t3,T4)=Tr[p,(t,, T2, t3)61;"]
t,, TV) =Tr[Pi(t,,
+
TV, t3)2Z.:,3] sin $~-~+Tr[~~(ti,
Tr[p,(t,,
+Tr[p2(t,, +
r2, tj)2z;‘3]
( 1Oa)
COs(2@~++)7~
T*, t3)vTZj,‘]
sin(2wo + 4)r4
T*,
t3)fiZ;*2]
COS(~~Q+C~)~~
t,)fiZ.J*‘]
sin(2wo + $J)T~
Tr[P2(f17r2,
cos $r4
( lob)
with
T+,(b
TV, t,)2Z2%3] = -{Au
-Zz,3)] --[Ah Tr[p(t,)(ZJ,4+~,‘,“)]
Tr[p(t,)(Zj”
- [A(e + d) Tr[ p( tl)Zz*‘] sin -
472
+
tA(e - d) Tr[ p( t,)Z;14] sin 34~~
{ except that A is replaced by - fir/2 and 830/2, respectively. In the same way Tr[p,(t,, Q-~,t3>61,‘y21and Tr[P2(tl, TV,t,)fiZi,‘] have a similar expression as eqn. (12a) except that A’ is substituted by - fir’/2 and fiC!‘/2, respectively. The two matrices LJ and r were defined previously [141 and I-1 r’=TR[Y0Z;,2]
=; \
0 0 0 0
0 0 0 0 0
‘0 0’ = -TRIYOZ;**] = $ : 0 ,O
0 0 0 0 -1
0000 0100 -1 0 0 0 0000 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 -1
0 1 0 0 0 0 0
0 0 -;;;
0 0 -;;;
0
0
0 0 0 0 0
0’ 0 0 0 01
(13)
TR means trace of each element of the matrix M,Z,“,” whereas Tr has the usual meanings. 2.3 Quadrupolar-echo amp&udes The quadrupolar-echo
sequence
is used for rephasing in the period 74 the coherences dephased the echoes are separated from the fid following the second rf pulse if 72 > 2Tfid. Knowledge of Gc(tl, r2, t3, TV) and Gs(tl, TV, t,, ~~1 allow the determination of ~~(t~, r2, t,, 74) and Es(tl, 72, t,, TV) the echo amplitude of the central transition and of a satellite transition, respectively. The expansion of eqns. (lOa> and (lob) yields lengthy expressions from which the bell-shaped echo terms are extracted: during the period 72 [15-211. Experimentally
2EC(t,, 729 t3, TV) = -eA(e
+ d) Tr[p(t,)Z;93]
+tA(e-d)
Tr[p(t,)ZiS4]
cos $(T~ - 74) cos +(37,-~))
+ vA’(e2 f d2) Tr[ p( t,)Z,?“] cos 4( 72 - 74) - vA’(e2 - d,) Tr[ p( 1,)Zi,4] cos 4(372 - 74)
(14)
P. P. Man /Solid
155
Stute Nucl. Magn. Reson. I (1992) IJQ-I58
and Lyt,, 6
'g, -
72, t,,~4)=(5f2g+5I%-vR
vr’b,)
Tr[ p( tr)( 1zS4+ I,‘,‘)]
xcOs[(2~,+~)(T2-T4)1
(15)
Two echoes appear at 74 = TV: mc(t,,
t,,T~=T~)
=
-[[A(e+d)
-vA’(e,+&)]
Tr[p(t,)1ix3]
( 16a)
~~‘(r,,t,,~~=i~)=(fnp+Erb-YR’g~-Yr’b~)Tr[~(l~)(l:.l+l:,~)] and another one at r4 = 2E”(t,,
t,,
( 16b)
3~~:
T~=~T~)
=
[[A(e-d)
-vA’(e,--d,)]
Tr[p(t,)1;“]
( 16c)
The echo occurring at r4 = 3~~ is due to the triple-quantum coherence developed during the first pulse. It has been observed in a ferro-magnetic material and explained by Abelyashev et al. [20]. The terms in eqns. (16a)-(16c) are defined in previous papers [10,14], except the following: 2vA’(e, + d,) = (sin 20,)2 cos w,$, + (sin 202)2 cos ~?~t, + (cos 20,)* + (cos 28,)* + 2 2vA’(e, - d2) = -(sin
28,)2 cos ~,~t, - (sin 20?)’ cos ti24t, - (cos 28,)2 - (cos 202)2 + 2
2tng
= (cos e_)2(cos
wi*t, + cos w34t3) +(sin
2[rb
= -(cos
2vT’b,
e+)‘(
Cos wt2t,
+ COS mj4t,)
-
e_)'(cos w,,t,+cos
(sin
e+)“(CoS
W23t3
OJ,~~,)
+ COS
w14t3)
= (COS 2e,)2 cos ~r$, + (COS 202)2 cos m14t, + (sin 28,)2 + (sin 202)2
21&‘g2 = -cos
W[jTj - cos wz4t7
(17)
The amplitudes of the echoes occurring at 74 = TV, eqns. (16a) and (16b), have the opposite sign of those of the {Xj-r,--{X1-T,-[acquisition(y)] sequence [ll. Consequently, all the remarks mentioned in the introduction section are also true for the 1X}--r,-{Yj-r,-[acquisition(y)] sequence. It must be emphasized that these two sequences are equivalent for the echoes occurring at r4 = r2 whereas early papers [16,17] argued that the sequence {Xj-r,-(Y}-r,-[acquisition(y)] provides larger echo amplitudes. As will be shown in the next section, the cancellation of spurious piezo-electric signals requires both sequences. In order to co-add the echoes, the receiver phase of one of these two sequences must change sign. On the other hand, the echo at 74 = 3r2 has the same sign in both sequences. We do not go further on the comments of these results as they have been discussed in some details in previous papers [1,2,111.
3. Results and discussion Despite the fact that ‘Li in LiNbO, does not satisfy all the conditions involved in the introduction section, namely, the homonuclear magnetic dipolar interaction is more important than the heteronuclear one, we have still chosen a single crystal of this material since it has a strong ferro-electric property which is suitable to our proposal for cancelling spurious signals. LiNbO, crystallizes into the noncentrosymmetric space group R3c, with two molecules per rhombohedral unit cell. The lithium atoms have a
P.P. Man /Solid State Nucl. Magn. Ream. 1 (1992) 149-158
156
trigonal symmetry environment and the EFG tensor at these sites is axially symmetric: 77 is zero and = 55 kHz. 7Li echoes were recorded at room temperature with a Bruker MSL 400 multinuclear high power pulsed NMR spectrometer. The high power static probehead was equipped with a 10 mm diameter horizontal solenoid coil. w,/27~ = 13.9 kHz and the associated 7r/2 pulse length t,,. = 18 ps were determined using a lithium chloride solution. The acquisition parameters were: t, = 15 ps, TV= 300 pus, t, = 10 ~LS,an acquisition delay of 20 ps, a recycle delay of 100 s and a spectral width of 125 kHz. The 7Li echoes acquired with the same number of scans and the following sequences: (a) {X}-7,-IX}-T,-[acquisition(y)]-recycle delay (b) {X}-T,-{Yj-T,-[acquisition(y)]-recycle delay (c) (X}-T,-{X}-T,-[aCqUiSitiOU(y)]-recycle delay-{X}-T2-{ -X}-T,-[aCqUiSitiOn(y rCCyCk delay (d) {X}-T,-(Y}-T,-[aCqUiSitiOn( y)]-recycle delay-{X)-T,-{ - Y}-T,-[aCqUiSitiOn(yrecycle delay (e) {X}-T,-{X}-T,-[acquisition(y)]-recycle delay-(X)-T,-{ -Xj-T,-[acquisition(y)]-recycle delay(X}-T,-{Y}-T,-[aCqUiSitiOn( -y)]-rCCyCk delay-{X)-T,-{ - Yl- T,-[acquisition( - y)]-recycle delay are shown in Figs. 2a, 2b, 2c, 2d and 2e, respectively. In Figs. 2a and 2b, the spurious signals are as important as those of the 7Li echo. Furthermore, these two figures support our previous remark concerning the equivalence of these two sequences: the echoes have the same amplitude and the same e2qQ/h
!n;. b
Fig. 2. ‘Li echoes in a single crystal of LiNbO, acquired with the following sequences: (a) (Xl- TV-(X)-74-[acquisition(y)]-redelay-(Xl-T,cycle delay, (b) (XJ-T*--(Y)-T.,-[acquisition(y)]-recycle delay, Cc) (XI- 72-(X}-T4-[acquisition(y)]-recycle (d) (X}-TV--(Y)-~Jacquisition(y)]-recycle delay-(XI-7,-I-Y}--T~(- X}- T4-[acquisition(y)]-recycle delay, 74-[acquisition(y)]-recycle delay-(X)-7,-(X)-~4-[acquisition(y)]-recycle [acquisition(y)]-recycle delay, (e) (XF-T~-(XIdelay-(X)-~2-(Y)-~,-[acquisition( - y)]-recycle delay-(X)-7,-( - Y)-TV-[acquisition(y)l-recycle delay.
P.1’. Mm /Solid
State Nucl. Magn. Reson. I (1992) 149-158
157
shape, except of course for the sign (the receiver phase is the same in both sequence). Alternating the phase of the second pulse reduces the spurious signals a little (Figs. 2c and 2d), but only the application of sequence (e) cancels them completely. In fact, sequence (e) combined with CYCLOPS phase cycling has already been proposed and illustrated by Rance and Byrd [22], Kunwar et al. [23], and Barriquand et al. [24] for suppressing the spurious acoustic-ringing signals from the NMR probe. In the early days of solid-state NMR, the piezo-electric signals were cancelled by immersing the single crystal in paraffin [3] or shielding the sample with an extra coil inside the rf coil [4]. Later when the spectrometers were equipped with computer and versatile phase cycling possibilities, several rf pulse sequences with r and r/2 pulse lengths were proposed for cancelling spurious signals [5-71; but for quadrupolar spins in solid samples, these pulse lengths [25,26] depend on the strength of the EFG tensor around the nuclei and so varied from sample to sample. The main advantage of sequence (e) is that the spurious signals are cancelled whatever the two rf pulse lengths t, and t,.
4. Conclusion The echo amplitudes of a spin-3/2 system excited by two rf pulses in quadrature phase have been obtained, with some assumptions, for any ratio of wo/w,r and any pulse lengths t, and t,. The results show that the two sequences, (Xl-r,-{Xl-r,-[acquisition(y)] and {Xl-r,-{Yl-r,-[acquisition(are equivalent for the echoes occurring at r2 = TVwhereas early papers argued that the second sequence provided larger echo amplitudes. A combination of these two sequences, designed in this paper by sequence (e>, has proved to be effective for cancelling not only the spurious acoustic ringing from the probe, but also the spurious piezo-electric signals when studying a ferro-electric material in the single-crystal form. Further progress on the theory of quadrupolar echo formation should include the homonuclear magnetic dipolar interaction. This case is well documented with the so-called solid-echo sequences.
Acknowledgements
The author thanks Prof. J. Leblond for providing the single crystal of LiNbO, and Dr. A. Trokiner for a critical reading of the manuscript.
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P.P. Man /Solid
State Nucl. Magn. Ram
I. Solomon, Whys.Reu., 110 (1958) 61. J. Butterworth, Proc. Phys. Sot. London, 86 (19651 297. A.M. Flett and J.C.S. Richards, Proc. Phys. Sot. London, 86 (1965) 171. G. Bonera and M. Galimberti, Solid State Commun., 4 (1966) 589. M. Suemitsu and N. Nakajo, .L Appl. Phys., 66 (1989) 3178. G.N. Abelyashev, V.N. Berzhanskii, N.A. Sergeev and Yu.V. Fedotov, Sou. Phys. JETP, 67 (1988) 127. T.K. Halstead, P.A. Osment, B.C. Sanctuary, J. Tegenfeldt, and I.J. Lowe, J. Magn. Reson., 67 (1986) 267. M. Rance and R.A. Byrd, J. Magn. Reson., 52 (1983) 221. AC. Kunwar, G.L. Turner and E. Oldfield, J. Magn. Reson., 69 (1986) 124. F. Barriquand, P. Odier and D. Jerome, Physica, Cl77 (1991) 230. P.P. Man, R. Couty and J. Fraissard, J. Mugn. Reson., 86 (1990) 613. J.A.M. der Mijden, R. Janssen and W.S. Veeman, Mol. Phys., 69 (1990) 53.
1 (1992) 149-158
Solid State Nuclear Magnetic Resonance, 1 (1992) 149-158
Elsevier Science Publishers B.V., Amsterdam
Study of a spin-3/2 system by a quadrupolar-echo suppression of spurious signals
sequence:
Pascal P. Man Laboratoire de Chimie des Surfaces, C.N.R.S. U.R.A. 1428, UniLlersitt Pierre et Marie Curie, 4 Place Jussieu, Tour 55, 75252 Paris Cedex 05. France
(Received 23 March 1992; accepted 15 April 1992)
Abstract The density matrix describing the evolution of a spin-3/2 system excited by a quadrupolar-echo sequence consistinb of two rf pulses in quadrature phase, {X}-r2-(Y}-r4-[acquisition(y)], is calculated from the equilibrium state to the acquisition period. The interactions involved are the first-order quadrupolar interaction throughout the experiment and a local heteronuclear magnetic dipolar term between the two pulses and during the acquisition period. Three echoes, one due to a satellite transition at 74 = 72 and two due to the central transition at 74 = 72 and 74 = 37*, are predicted. They have similar expressions than those obtained with two rf pulses of the same phase, (XJ-Q-{X)-TV-[acquisition(y)], except the signs. Moreover, it is shown experimentally that a combination of these two sequences, namely: (X}-To-{X)-T,+-[acquisition( y&recycle delay-(X)-~,-{ - X)-T4-[acquisition(y)]-recycle delay-(X)-T,-{Y)-T,-[acquisition( - y)]-recycle delay-(X]- T*-( - Y]-T~ -[acquisition( - y&recycle delay, cancels the spurious piezo-electric signals when studying a ferroelectric material in the single-crystal form. Keywords: NMR, quadrupolar echo; fictitious spin-l/&
spurious signal suppression
1. Introduction Recently, the time-domain response of a spin I = 3/2 system excited by two radio-frequency (rf) pulses of the same phase (IX)-T,-{XI-T,-[acquisition(y)] where y is the receiver phase) for any ratio of the quadrupolar coupling ho over the amplitude of the rf pulse wrf was carefully analyzed [l]. The interactions involved are the first-order quadrupolar interaction throughout the experiment and a local heteronucleur magnetic dipolar interaction (S’WI_sj = 41,) between the two pulses and during the acquisition period. These hypotheses are fulfilled when the homonuclear magnetic dipolar interaction is smaller than the heteronuclear magnetic dipolar interaction. This also means that 4 must be smaller than w,~ so that 4 can be neglected during the pulses. Three echoes were predicted: one due to a satellite transition at 74 = TV and two due to the central transition at TV= 72 and TV= 3~~. The main experimental restriction is that the interpulse delay T* must be twice as long as Tfid (duration of the fid) in order that the echoes and the fid following the second rf pulse are separated. As a result, the echoes are very small due to spin-spin relaxation. Fortunately, alternating the phase of the Correspondence to: Dr. P.P. Man, Laboratoire
de Chimie des Surfaces, C.N.R.S. U.R.A. 1428, UniversitC Pierre et Marie Curie, 4 Place Jussieu, Tour 55, 75252 Paris Cedex 05, France. 0926-2040/92/$05.00
0 1992 - Elsevier Science Publishers B.V. All rights reserved
150
P.P. Man /Solid
State Nucl. Magn. Ram.
I (1992) 149-158
second pulse and keeping the receiver phase unchanged, e.g., IX}-r,-(X}-r,-[acquisition(y)]-recycle delay-(Xj-r,-( -X)-r,-[acquisitiom y )I-recycle delay, cancel the fid following the second pulse without changing the echoes [2]. Therefore, one can reduce the interpulse delay r2 to one Tfid and the echoes appear much larger. However, this improvement cannot cancel either the spurious piezo-electric signals when studying a ferro-electric compound in the single-crystal form [3,41, or the spurious acoustic-ringing signals from the NMR probe when studying a low gyromagnetic ratio nucleus [5-71. The paper is organized as follows: In section 2.1, we calculate the density matrix p(t,, TV, t,, rJ, from the equilibrium state to the acquisition period, of a spin-3/2 system excited by a quadrupolar-echo sequence consisting of two rf pulses in quadrature phase (Fig. l), (Xj-r,-(Yj-r,-[acquisition(y)], using the fictitious spin-l/2 operator formalism [8,9]. The interactions involved are the same as for IX)-TV(X)--r,-[acquisition(y)] sequence [l]. We then introduce in section 2.2 the expression of the central- and a satellite-line intensities. In section 2.3 the echo amplitudes are deduced from the line intensities of the previous section. In section 3, we experimentally show that the sequence: (X)--72-(X)-~4[acquisition(y)]-recycle delay-(X)-r,-{ -X)-T,-[acquisition(y)]-recycle delay-(X)-r,-(Y)-r,[acquisitiom-y&recycle delay-{X)-r,-{ - Yj-r,-[acquisitiom-y)]-recycle delay, suppresses the spurious piezo-electric signals. This is illustrated with the lithium nucleus ‘Li in a single crystal of LiNbO,. Finally, section 4 is a brief conclusion,
2. Theory The Hamiltonians throughout the paper are defined in angular frequency units. Neglecting the relaxation phenomena and the second-order quadrupolar effects, the dynamics of a spin-3/2 system is described by the density matrix excited by a quadrupolar-echo sequence, {X)-T~-(Y]-T~, p(t,, 72, t,, TV) expressed in the rotating frame associated to the central transition: p( tl, 72, t3, 74) = eXp( -i%(')T4) Xp(0)
eXp(-iz’b’t3)
eXp(-ix(')T2)
exp( i2?‘“‘t,) exp( iJ?‘“)T,) exp( iSb’t,)
where the following quantities are written with the fictitious spin-l/2
eXp(-iz’a’tl) exp( iScb4)
(1)
operators:
p(0) = z, = 3z,‘,” + I,“,”
(2a)
Aq’
(2b)
= 5ua(3z2’
- Z( z + 1)) = 20Q( z,‘.’ - 12%“)
3e2qQ [3 cos2/3 - 1-t 77sin’/3 cos 2a] wQ = SZ(2Z - l)A
w
A?@) = -wrfZx
+ z#’
(2d)
c%@(b)= -wrfzy
+ A?p=
@'=Jg'+$z
z
= - aod(
I,‘,’ + Zx”‘“) - 2w,,z,2,3 + X#’
-l&orf(z;~2+z;~4)
- 2WrfZ,2,3 + GYy)
(2e) (29
A?$) is the first-order quadrupolar interaction, (Y and p represent Euler angles describing the orientation of the strong static magnetic field B, in the principal axis system of the electric field gradient (EFG) tensor and 77 the asymmetry parameter. t, and t, are the first and the second rf pulse lengths, respectively (Fig. 1). The matrix of A?@) and SSCb) expressed in the eigen-states of Z, are not diagonal.
PP Man /Solid
State Nucl. Magn. Ram.
-
I
I
1
z2
t1
151
1 (1992) 149-158
_I_ I-
t3
+
time
z4
_I_ I-
-I
Fig. 1, Hamiltonians and durations associated with a quadrupolar-echo
sequence.
The diagonalized forms A?.. and Xv, and the transformation respectively, were already defined 19-121:
T and I/ of GV’@)and .GVCb),
operators
A?r = T+ Xca)T = w13Z;,3 + 024Zz2,4+ wrf( Z,‘,” + Zz’.“)
(3a)
A?” = V+A?‘b’v=
P)
T=exp
i
w13z;1’2+ 024z;3’4 + W,f( z,‘,” + zz”q
i;(Zi34-Z;,3) ‘2,“)
exp( -i28,Z~~3) exp( -i282Zt,4)
(34
exp( -i28,Z,‘,2)
(34
exp(i282Z,3~4)
Eqn. (1) can be rewritten as: p( t,, TV, t,, 74) = exp( -i#“~,)V
exp( -iXvt3)V+
xexp( -iS#3Ttl)T+p(0)T X
2.1 Density matrix p(t,,
r2,
exp( -id?‘(C)~2)T exp(iL4?“b,)V
exp(iXTtl)Tf
exp( iA?vt3) V+ exp( iG+Pc’7,) t,,
(4)
r4)
This section is devoted to calculating, step by step, the density matrix [eqn. (411. The density matrix before the application of the second pulse has been obtained previously [ll. During the second pulse, the evolution of the spin system is described by A? @‘).So the density matrix becomes:
p(t,, TV)
p(t,,
TV, f3) = v exp( -izvt3)V+
P (t 1, T2)v
ew(i~vt3)~+=pI(f17
72y t3)
+
p2(t1, 72, t3)
(5)
with p,(t,,
TV, t3) = $C{i(U)O(n
P#,,
72,
f3) = %{$%)O(~(d2
+ i(e + d)P, sin +Q-~- +(e - d)P, sin -
e,P,
cos
4~~ + +(e, +
d,)P,
3+r2
cos
34~~
+
J3( r2))}K +
J4( T~))}K
(ha) (6b)
and J3( T)
=
[q, sin &r - q3 cos 2471 cos 2wo7 + [ri sin
J4( Q-)= - [q, cos v=(cos
4~ +
26,,cos 28,,
$7 -
r3 cos 2471 sin 2~0~7
q4 sin 2471 cos 2wQr - [r2 cos 47 + r4 sin 2471 sin 2~~7
sin 28,, sin 20,,1)
The matrices of spin operators Y and Y, are defined in Table 1. All the above parameters associated variables not explicitly defined here were already defined in refs. 1 and 10.
Pa)
(7b) and their
P.P. Man /Solid
152
State Nucl. Magn. Reson. I (1992) 149-158
TABLE 1 The hvo matrices of spin operators Y and YO of eqns. (6a) and (6b) /A Y=
- 4 -DI
E,
G,
-G,
-F
F
-D,
-C
-C
-B,
-B,
-G,
F
-F
A
-A
-E,
E,
-c
Bl
B,
H
H
-D,
-D,)
A H
-H
-G,
c
1 -G-C,
0
E+A,
0
0
0
(E + A,)/2
(E + A,)/2
0
0
G-C,
0
E-A,
(E - A,)/2
(E-
-E-A,
0
-G-C,
0
0
0
0
0
0
A,-E
0
0
- Fl - B
0
0
0 Y,,=
\
-C,+G
F, - B
-(CL
+ G)P
-CC,
+ G)/2
-(CL
- G)P -D
-CC,
- G)/2 D
0
B = I;.4 _ *;,2
A = Il.3 _ I?.4 1* I B = 13.4 + 11.2
C =
A = I’,4 _ 12.3
= 11.3 _
[I,3 _ x
12x4 x
c:
D = 12.3 _ Y
11.4 Y
D i = 1:’
E = 12.3 + 11.4 Y Y
E
F = 11.3 + 12.4 ? Y G = 11.2 + 13.4 Y Y
f?; = $3 G
-
A,)/2
1i.4 I;.”
= 12.3 + 11.4 x _
= I?.4 1 ,
_
I,‘,” II.2 Y
H = 11.4 + 12.3 i z
After the second pulse, the evolution of the spin system is described by SF’@).The density matrix becomes: p(r,, 72, t,, q) =P3(tr,
727 t,>
74)
+P&
72,
t3,
(84
74)
with P3(tl, r2, t,, r4) = exp( -i~‘C’r,)pr(tI,
72, t3) exp(i~‘“‘r,)
= $$( i( ([N - 1:,3P1 cos
+T~
-
f;*3P, sin
4~~
+Z,‘,4P3 cos 34r4 + Ii94P3 sin 34r4 + Z,( r4)]) xO[n p4(tl,
TV,
t,,
TV)
= exp( = $$(
+ i(e + d)P, sin
-i@c)74)P2(t,,
f(~[ArPs
-IiAP2
r2,
+ 1:,3P4 sin
i(e - d)P, sin 34r2 + J3( r2)])K
-
t3)
ew(ix@‘T,)
+T~
-
Ij?*3P4cos
(8b)
~$7~
sin 34~~ + Ii,4P2 cos 34r4 + Z,( r4)])
xO[ - i(e, - d,)P, z,(r)
4~~
cos
~$7~ + $(e, +
d,)P,
cos
34~~
+
J4( T~)])K
(8~)
= (Q, cos +T - Q, sin ~$7- Q3 cos 247 - Q, sin 247) cos 20~~7 + (R, cos
$7
-
R, sin C#I~ - R, cos 247 - R, sin 247) sin
20.~~7
(84
Z,( 7) = (X2 cos f&r - X, sin ~$7- X3 cos 247 - X4 sin 24~) cos 20~~7 +(Y2 cos $TThe matrices
P, (i =
Y, sin &r-Y,
cos 24~-Y~
sin 247) sin 2~~7
l-5), Qj, Rj, Xj and Yj (j = l-4) were defined in ref. 1.
(se>
P.P. Man /Solid
State Nucl. Magn. Reson. 1 (1992)
153
149-158
2.2 Line intensities It has been repeatedly shown [lo,121 that the line intensity of a quadrupolar spin depends on the ratio of oo/w,r. As a result, the magnitude of ho can be determined in measuring the line intensity for a series of pulse length. Knowledge of the density matrix p(t,, T2. t3, TJ, Eqns. (8a)-(8e), allows the determination of the central-line intensity Fc(tl, r2, t,, TJ and its quadrature part GYt,, Q, t3, ~~1, and S-J and its quadrature part GYt,, TV, t,, TV). also the line intensity of a satellite transition FYt,, TV, t,, In this section, we focus mainly on the quadrature parts of the line intensity from where the amplitude of the bell-shaped echoes can be deduced 1131,whereas the quadrature part of the bell-shaped echoes, in other words the “sine-like” shaped echoes, are deduced from FYt,, TV, t,, TV) and FYt,, r2, t3, TJ: qt,,
72,
t,, rq) = Tr[p(t,.
r?, t,, 74)2z_?3]
G‘-(tl, 72, t,, TV) =Tr[p(l,, T2, t3,',1)24?'] Fs(tl, TV,
t,,
TV)
=Tr[p(t,,
7?,
t3,
T~)\/~I.J~*]
G”( t,, T?, t,,TV) =Tr[p(t,, 72, t,,T4)fili,*]
(9)
It is easy to show that
G’(t,,
72,
G’(t,,
Tz, t3,T4)=Tr[p,(t,, T2, t3)61;"]
t,, TV) =Tr[Pi(t,,
+
TV, t3)2Z.:,3] sin $~-~+Tr[~~(ti,
Tr[p,(t,,
+Tr[p2(t,, +
r2, tj)2z;‘3]
( 1Oa)
COs(2@~++)7~
T*, t3)vTZj,‘]
sin(2wo + 4)r4
T*,
t3)fiZ;*2]
COS(~~Q+C~)~~
t,)fiZ.J*‘]
sin(2wo + $J)T~
Tr[P2(f17r2,
cos $r4
( lob)
with
T+,(b
TV, t,)2Z2%3] = -{Au
-Zz,3)] --[Ah Tr[p(t,)(ZJ,4+~,‘,“)]
Tr[p(t,)(Zj”
- [A(e + d) Tr[ p( tl)Zz*‘] sin -
472
+
tA(e - d) Tr[ p( t,)Z;14] sin 34~~
{ except that A is replaced by - fir/2 and 830/2, respectively. In the same way Tr[p,(t,, Q-~,t3>61,‘y21and Tr[P2(tl, TV,t,)fiZi,‘] have a similar expression as eqn. (12a) except that A’ is substituted by - fir’/2 and fiC!‘/2, respectively. The two matrices LJ and r were defined previously [141 and I-1 r’=TR[Y0Z;,2]
=; \
0 0 0 0
0 0 0 0 0
‘0 0’ = -TRIYOZ;**] = $ : 0 ,O
0 0 0 0 -1
0000 0100 -1 0 0 0 0000 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 -1
0 1 0 0 0 0 0
0 0 -;;;
0 0 -;;;
0
0
0 0 0 0 0
0’ 0 0 0 01
(13)
TR means trace of each element of the matrix M,Z,“,” whereas Tr has the usual meanings. 2.3 Quadrupolar-echo amp&udes The quadrupolar-echo
sequence
is used for rephasing in the period 74 the coherences dephased the echoes are separated from the fid following the second rf pulse if 72 > 2Tfid. Knowledge of Gc(tl, r2, t3, TV) and Gs(tl, TV, t,, ~~1 allow the determination of ~~(t~, r2, t,, 74) and Es(tl, 72, t,, TV) the echo amplitude of the central transition and of a satellite transition, respectively. The expansion of eqns. (lOa> and (lob) yields lengthy expressions from which the bell-shaped echo terms are extracted: during the period 72 [15-211. Experimentally
2EC(t,, 729 t3, TV) = -eA(e
+ d) Tr[p(t,)Z;93]
+tA(e-d)
Tr[p(t,)ZiS4]
cos $(T~ - 74) cos +(37,-~))
+ vA’(e2 f d2) Tr[ p( t,)Z,?“] cos 4( 72 - 74) - vA’(e2 - d,) Tr[ p( 1,)Zi,4] cos 4(372 - 74)
(14)
P. P. Man /Solid
155
Stute Nucl. Magn. Reson. I (1992) IJQ-I58
and Lyt,, 6
'g, -
72, t,,~4)=(5f2g+5I%-vR
vr’b,)
Tr[ p( tr)( 1zS4+ I,‘,‘)]
xcOs[(2~,+~)(T2-T4)1
(15)
Two echoes appear at 74 = TV: mc(t,,
t,,T~=T~)
=
-[[A(e+d)
-vA’(e,+&)]
Tr[p(t,)1ix3]
( 16a)
~~‘(r,,t,,~~=i~)=(fnp+Erb-YR’g~-Yr’b~)Tr[~(l~)(l:.l+l:,~)] and another one at r4 = 2E”(t,,
t,,
( 16b)
3~~:
T~=~T~)
=
[[A(e-d)
-vA’(e,--d,)]
Tr[p(t,)1;“]
( 16c)
The echo occurring at r4 = 3~~ is due to the triple-quantum coherence developed during the first pulse. It has been observed in a ferro-magnetic material and explained by Abelyashev et al. [20]. The terms in eqns. (16a)-(16c) are defined in previous papers [10,14], except the following: 2vA’(e, + d,) = (sin 20,)2 cos w,$, + (sin 202)2 cos ~?~t, + (cos 20,)* + (cos 28,)* + 2 2vA’(e, - d2) = -(sin
28,)2 cos ~,~t, - (sin 20?)’ cos ti24t, - (cos 28,)2 - (cos 202)2 + 2
2tng
= (cos e_)2(cos
wi*t, + cos w34t3) +(sin
2[rb
= -(cos
2vT’b,
e+)‘(
Cos wt2t,
+ COS mj4t,)
-
e_)'(cos w,,t,+cos
(sin
e+)“(CoS
W23t3
OJ,~~,)
+ COS
w14t3)
= (COS 2e,)2 cos ~r$, + (COS 202)2 cos m14t, + (sin 28,)2 + (sin 202)2
21&‘g2 = -cos
W[jTj - cos wz4t7
(17)
The amplitudes of the echoes occurring at 74 = TV, eqns. (16a) and (16b), have the opposite sign of those of the {Xj-r,--{X1-T,-[acquisition(y)] sequence [ll. Consequently, all the remarks mentioned in the introduction section are also true for the 1X}--r,-{Yj-r,-[acquisition(y)] sequence. It must be emphasized that these two sequences are equivalent for the echoes occurring at r4 = r2 whereas early papers [16,17] argued that the sequence {Xj-r,-(Y}-r,-[acquisition(y)] provides larger echo amplitudes. As will be shown in the next section, the cancellation of spurious piezo-electric signals requires both sequences. In order to co-add the echoes, the receiver phase of one of these two sequences must change sign. On the other hand, the echo at 74 = 3r2 has the same sign in both sequences. We do not go further on the comments of these results as they have been discussed in some details in previous papers [1,2,111.
3. Results and discussion Despite the fact that ‘Li in LiNbO, does not satisfy all the conditions involved in the introduction section, namely, the homonuclear magnetic dipolar interaction is more important than the heteronuclear one, we have still chosen a single crystal of this material since it has a strong ferro-electric property which is suitable to our proposal for cancelling spurious signals. LiNbO, crystallizes into the noncentrosymmetric space group R3c, with two molecules per rhombohedral unit cell. The lithium atoms have a
P.P. Man /Solid State Nucl. Magn. Ream. 1 (1992) 149-158
156
trigonal symmetry environment and the EFG tensor at these sites is axially symmetric: 77 is zero and = 55 kHz. 7Li echoes were recorded at room temperature with a Bruker MSL 400 multinuclear high power pulsed NMR spectrometer. The high power static probehead was equipped with a 10 mm diameter horizontal solenoid coil. w,/27~ = 13.9 kHz and the associated 7r/2 pulse length t,,. = 18 ps were determined using a lithium chloride solution. The acquisition parameters were: t, = 15 ps, TV= 300 pus, t, = 10 ~LS,an acquisition delay of 20 ps, a recycle delay of 100 s and a spectral width of 125 kHz. The 7Li echoes acquired with the same number of scans and the following sequences: (a) {X}-7,-IX}-T,-[acquisition(y)]-recycle delay (b) {X}-T,-{Yj-T,-[acquisition(y)]-recycle delay (c) (X}-T,-{X}-T,-[aCqUiSitiOU(y)]-recycle delay-{X}-T2-{ -X}-T,-[aCqUiSitiOn(y rCCyCk delay (d) {X}-T,-(Y}-T,-[aCqUiSitiOn( y)]-recycle delay-{X)-T,-{ - Y}-T,-[aCqUiSitiOn(yrecycle delay (e) {X}-T,-{X}-T,-[acquisition(y)]-recycle delay-(X)-T,-{ -Xj-T,-[acquisition(y)]-recycle delay(X}-T,-{Y}-T,-[aCqUiSitiOn( -y)]-rCCyCk delay-{X)-T,-{ - Yl- T,-[acquisition( - y)]-recycle delay are shown in Figs. 2a, 2b, 2c, 2d and 2e, respectively. In Figs. 2a and 2b, the spurious signals are as important as those of the 7Li echo. Furthermore, these two figures support our previous remark concerning the equivalence of these two sequences: the echoes have the same amplitude and the same e2qQ/h
!n;. b
Fig. 2. ‘Li echoes in a single crystal of LiNbO, acquired with the following sequences: (a) (Xl- TV-(X)-74-[acquisition(y)]-redelay-(Xl-T,cycle delay, (b) (XJ-T*--(Y)-T.,-[acquisition(y)]-recycle delay, Cc) (XI- 72-(X}-T4-[acquisition(y)]-recycle (d) (X}-TV--(Y)-~Jacquisition(y)]-recycle delay-(XI-7,-I-Y}--T~(- X}- T4-[acquisition(y)]-recycle delay, 74-[acquisition(y)]-recycle delay-(X)-7,-(X)-~4-[acquisition(y)]-recycle [acquisition(y)]-recycle delay, (e) (XF-T~-(XIdelay-(X)-~2-(Y)-~,-[acquisition( - y)]-recycle delay-(X)-7,-( - Y)-TV-[acquisition(y)l-recycle delay.
P.1’. Mm /Solid
State Nucl. Magn. Reson. I (1992) 149-158
157
shape, except of course for the sign (the receiver phase is the same in both sequence). Alternating the phase of the second pulse reduces the spurious signals a little (Figs. 2c and 2d), but only the application of sequence (e) cancels them completely. In fact, sequence (e) combined with CYCLOPS phase cycling has already been proposed and illustrated by Rance and Byrd [22], Kunwar et al. [23], and Barriquand et al. [24] for suppressing the spurious acoustic-ringing signals from the NMR probe. In the early days of solid-state NMR, the piezo-electric signals were cancelled by immersing the single crystal in paraffin [3] or shielding the sample with an extra coil inside the rf coil [4]. Later when the spectrometers were equipped with computer and versatile phase cycling possibilities, several rf pulse sequences with r and r/2 pulse lengths were proposed for cancelling spurious signals [5-71; but for quadrupolar spins in solid samples, these pulse lengths [25,26] depend on the strength of the EFG tensor around the nuclei and so varied from sample to sample. The main advantage of sequence (e) is that the spurious signals are cancelled whatever the two rf pulse lengths t, and t,.
4. Conclusion The echo amplitudes of a spin-3/2 system excited by two rf pulses in quadrature phase have been obtained, with some assumptions, for any ratio of wo/w,r and any pulse lengths t, and t,. The results show that the two sequences, (Xl-r,-{Xl-r,-[acquisition(y)] and {Xl-r,-{Yl-r,-[acquisition(are equivalent for the echoes occurring at r2 = TVwhereas early papers argued that the second sequence provided larger echo amplitudes. A combination of these two sequences, designed in this paper by sequence (e>, has proved to be effective for cancelling not only the spurious acoustic ringing from the probe, but also the spurious piezo-electric signals when studying a ferro-electric material in the single-crystal form. Further progress on the theory of quadrupolar echo formation should include the homonuclear magnetic dipolar interaction. This case is well documented with the so-called solid-echo sequences.
Acknowledgements
The author thanks Prof. J. Leblond for providing the single crystal of LiNbO, and Dr. A. Trokiner for a critical reading of the manuscript.
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