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Multiple pulse response in pure NQR of spin I = 32 nuclei in single crystals: response to phase-alternated pulse sequence (PAPS)
Journal of Molecular Structure, 192 (1989) 321-331 Elsevier Science Publishers B.V., Amsterdam - Printed
321 in The Netherlands
MULTIPLE PULSE RESPONSE IN PURE NQR OF SPIN I = 3/2 NUCLEI IN SINGLE CRYSTALS: RESPONSE TO PHASEALTERNATED PULSE SEQUENCE (PAPS)
A.K. DUBEY and P.T. NARASIMHAN* Department of Chemistry, Indian Institute of Technology, Kanpur 208 016 (Zndia) (Received
16 March 1988)
ABSTRACT The response of spin I = 3/2 nuclei in a single crystal to phase-alternated pulse sequence (PAPS) -2r-tTz-r), in pure NQR is theoretically investigated. Average Hamiltonian the(r-t;y ory is employed to calculate the spin-space averaging of the line-broadening internal interactions. The criterion employed for spin-space averaging is that the average Hamiltonian during the cycle for the interaction under consideration should commute with the initial density matrix prepared by the pulse t$ before the first cycle. The average Hamiltonian during PAPS for the main linebroadening interactions, namely electric field gradient (efg) inhomogeneity, torsional oscillations and heteronuclear dipolar interactions, is found to commute with the initial density matrix of the do not system when 2t,, = tw2.It is concluded that during PAPS these internal interactions contribute to the linewidth of the pure NQR spectrum of physically equivalent spin Z = 3/2 nuclei in a single crystal. These theoretical results are similar to those obtained earlier for spin Z = 1 system by Osokin. tow1 -
INTRODUCTION
Multiple-pulse sequences for suppressing the dipolar broadening and to obtain high resolution NMR spectra of spin I = l/2 in solids are now well established [ 11. It is therefore natural to ask the question whether a similar approach would be effective in pure NQR spectroscopy. In the case of pure NQR the spectrum is usually broadened due to the electric field gradient (efg) inhomogeneity, torsional oscillations of the units constituting the solid and heteronuclear dipolar interactions [ 21. The first experimental investigation using multiple-pulse sequences in pure NQR were those of Marino and Klainer [ 31 who employed a series of pulses with flip angle of 90’) phase shifted with respect to the first or preparatory pulse by 90”) the so-called Ostroff-Waugh (OW) sequence [4]. This sequence may be designated as t$- ( r-t$!-r)N. Surprisingly they observed the elongation of the echo train in the case of the *Author to whom correspondence
0022-2860/89/$03.50
should be addressed.
0 1989 Elsevier Science Publishers
B.V.
322
14N signal in a powder sample of NaN02. The spin-locking phenomenon observed by these authors is remarkably reminiscent of the NMR experiments in solids carried out earlier by Ostroff and Waugh [ 41. It has been pointed out by Marino and Klainer [3] that one can co-add the multiple echoes within a spin-locked spin echo train to enhance the S/N ratio as well. Multiple-pulse techniques can therefore be exploited to detect weak NQR signals. Subsequent to the work of Marino and Klainer [3] a number of authors have reported similar results for I = 1 [5], I = 3/2 [6] and I = 5/2 [7]. A theoretical treatment of the response to the OW sequence of 14N in NaN02 powder sample was attempted by Cantor and Waugh [ 81 who were able to successfully account for the short-time behaviour of the observed magnetization. However, the longtime behaviour was not accounted for. Hitrin et al. [9] employed a canonical transformation technique to account for the experimentally observed short and long-time behaviour. Maricq [lo] has employed Floquet theory to arrive at the correct experimental behaviour. Osokin has also theoretically investigated [5, 111 the spin I = 1 case subjected to PAPS and WAHUHA [l] sequences. Following the work of Waugh et al. [ 121 in NMR, Osokin [ 5,111 has shown that if the density matrix at the end of the preparatory pulse commutes with the average Hamiltonian in the presence of the rf pulses then a quasisteady state can be realized, and thus one can observe elongation of the echo train leading to line narrowing in NQR. Ainbinder and Furman [ 131 have given a theory of multiple-pulse averaging for spin systems with arbitrary nonequidistant spectra using the canonical transformation technique of Hitrin et al. [ 91 and the Krylov-Bogolyubov-Mitropol’skii averaging method [ 141. Detailed application of this theory has not yet been made in order to explain the experimental results of multiple pulse response in spin I&33/2 systems. We present in this paper a preliminary result of our theoretical investigation of response to phase-alternated pulse sequence (PAPS) t$- ( r-t~~00-2r-t~Z-r)N in pure NQR of physically equivalent spin I = 3/2 nuclei in single crystals. Our approach is akin to that of Osokin [ 51, and we have used the single transition operators [ 151 or generalized fictitious spin - l/2 operators [ 161. GENERAL
THEORY
OF LINE NARROWING
WITH
CYCLIC
MULTIPLE-PULSE
SEQUENCES
When the interaction
of a system is cyclic, the average Hamiltonian theory of the system. This theory proposes to replace the time-dependent Hamiltonian of the system over one cycle by an equivalent time-independent average Hamiltonian. It also proposes that it is enough to know the behaviour of the system over one cycle. For subsequent cycles the same behaviour is repeated so that [ 1 ] can be used to study the behaviour
0%)
=
bWIN
(1)
323
where p (t,) is the density matrix of the system after one cycle, t, is the cycle time and N a positive integer. The behaviour of the system is now described by the average Hamiltonian, 2, provided we observe the system only at specified “observation windows”, i.e. only after a cycle is complete. By making a judicious choice of cyclic multiple-pulse sequences we can introduce time dependence in the line-broadening internal interactions in such a way that its average Hamiltonian commutes with the initial density matrix p(O) prepared from the equilibrium density matrix pes by a preparatory pulse
tPw,m
(2)
=0
p(O) thus becomes a constant of motion during the cyclic interaction and the line-broadening internal interactions under consideration do not contribute to the evolution of the system. In effect we achieve line-narrowing. After describing the Hamiltonians of the system of physically equivalent spin I = 3/2 nuclei in a single crystal in a suitable representation (next section) we will next show that PAPS is a cyclic sequence. Following this, details of the evaluation of the initial density matrix p(O) following the preparatory pulse are given. We then calculate average Hamiltonians to zeroth order using Magnus expansion for the line-broadening interactions. The commutators of these average Hamiltonians with p(O) is then evaluated. HAMILTONIAN OF THE SYSTEM OF SPIN Z = 3/2 NUCLEI
The overall Hamiltonian of the system of spin I = 3/2 nuclei consists of several terms. The terms which are relevant to the particular problem at hand are included in (3)
Xo = F
[312z- 01+1)+V)(IL+12L)l
is the quadrupole tem ( QPAS ) . 3
Hamiltonian
expressed
in the quadrupole
principal
axis sys-
e2qQ
wQ= 2 1(2I-
1)fi
and r is the asymmetry parameter [ 21. We consider the rf field as being applied QPAS. Therefore X*f = - 2o41,
cos wt
in the direction
of x-axis of the
(5)
324
Here o1 = ?H, with Hi the rf field strength, y the magnetogyric ratio of the nuclei and o the frequency of the rf radiation. The last three terms in eqn. (3) are the main line-broadening interaction Hamiltonians. The Hamiltonian corresponding to the electric field gradient (efg) inhomogeneity is Azo = +[312z-1(1+1)+~)(I:,+12,)J
(6)
3 e"Q(&) where A.o, = 2 1(21-1)h’ ,&r is the secular part of the interaction of the spin system with the field gradient subject to torsional oscillations and is given by &
= 2
[2&V&&
I”,) -+(3+~)
(I&1$)
+E~(~--Y/) (I&I”Z)]
(7)
Here ei = @-(09). We choose the coordinate system such that the laboratory fixed axes make angles 8i (i = X, Y, 2) with respect to the corresponding QPAS of the unit which undergoes torsional oscillation. ( 0: ) is the thermal average of 0;. The last term J& in eqn. (3) is the Hamiltonian for heteronuclear dipoledipole interaction and is given by
Fi being the ith component of the field due to the non-resonant nuclei (S) at the site of the resonant nuclei (I). In order to simplify the evaluation of the expressions involving exponential operators we express 2 in a representation where &!o, the main term in the Hamiltonian, is diagonal. This representation is denoted throughout by a prime ( ’ ) as superscript and is realized by the unitary transformation
R=R-‘=
’ [ cos sin 0t/2 r/2
- cos r/2
0
sin t/2
sin 05/2
-cos sin 0512 512
cos 0t/2
where tan 5 = q/ (3 ) *12. Using eqn. (9) in eqn. (3 ) we obtain 2’
= %b+&+A~%!;,+xI,+%‘;s
1
(9)
(10)
where for example, %‘& represents R%$R-‘. We give here the form of the various terms in the right hand side of eqn. (10) in terms of the single transition operators [ 15, 161 J?& = K[I;-3-I;-4)
(11)
325
where K = T
(1+ $/3) ‘j2.
X& = -2~0, cos c&[ ((3)“”
cos {+sin 5) (I$“+I&“)
+ ((3)“” AS;,
sin <-cos
<) (I$“-I$“)+
(1$4+1$3)]
(12)
= AK(I&-3-I;-4)
(13)
J+o (1+$/3)l’2
(14)
where AK= 8$
= A(I;-2-I;-4)
+B(I$3-I$4)
(15)
where A = 9
(4(3)‘/2~8g
sin c- (3)1’2(3+q)62y(
(3)‘j2 cos r
-sin 0 - (3)“2(3-q)0$(
(3)“” cos t+sin r)}.
and
B=
-2
{4(3)“2@i COS<+ (3)1'2(3+Tj)@$(
(3)‘12 sin 5
+COs O+ (3)“2(3-?j)0:( Z’;s = F,{ ((3)“”
sin <-cos
(I$;“+Ik”,
0 (Ik”-1%“)
- ((3)“”
(3)“”
sin<-cos
0)
cos c+sin <)
+ <1k”+1$“,}+
FY{ ( (3)“2 sin r+cos <) (Ig4+Ig3)
(16)
- ( (3)‘12 cos <-sin 5) (I&“I”;“) - (I&“-I”,-“)}+ FZ{2 cos <(I&4+Ig3)
+2 sin &I$“-Ig-“)
+ (I_&4-Ig3)}
We next take eqn. (10) to quadrupole interaction frame (QIF) by using the transformation ( 17 ) U = exp( -Mht)
(17)
In &IF X’ becomes j&f = U+X’U %f = U+Z’:,U+U+AZ’~U+U+~~U+U+~;sU ,&=
““‘+A&b+&+&
(18)
326
Note that we express all the terms in the Hamiltonian which have been transformed by the R matrix and subsequently by the U operator with a prime superscript plus a tilde. The non-zero commutators used to evaluate eqn. (18) are given in the Appendix. We have the following expressions for various terms on the right hand side of eqn. (18). 5Pcf = -2w,
cos wt[ ((3)“’
+ (I’,-2+I‘$3)sinKt}+ A*b
= AX;,
((3)“”
sin c-cos
(19)
5) (I_$“-I%-“) + (Ik4+Ig3)]
(29)
+B{ (I_$“+I$“)
= Fx{ (I$“+Ik”)
+ ((3)“”
cos Kt-
sin <-cos
(I’,-“-I”,-“)
+Fz{2 cos <(I;-4+I;-3) ((3)“”
+Fv(sin
sin Kt}
(21)
5) (I$“-1%“))
+FY{ ( (3)*12 sin r+cos 4) (Ik4+Ig3)
-FX
cos Kt
= AK(I;-3-1$-4)
G%‘$= A(I~2-I~4) 2is
cos l+sin LJ{ (I$“+I$“)
(22)
- (I&“-I”;“)}
+ (I;-4-I;-3)}
cos <+sin LJ{ (I$2+I$4)cos
Kt- (Ig2-I$4)sin
Kt}
5- (3)“” cos {>{ (Ik2+13;4)cos Kt+ (Ik2-Ig4)sin
Kt}
+ Fz2 sin <( (I$” - 1%“) cos Kt - (I$&+
Ik3) sin Kt}
In eqns. (19), (21) and (22) we truncate the time dependent part in the firstorder approximation [ 1 ] to obtain &,
= -L(I$2+1$4)
(23)
where L = a,( (3)“” cos r+sin <). j&
= A(I;-2-I;-4)
(24)
and 2is
= FX { (I_$“+I$“) +Fy{ ((3)“”
+ ( (3)‘12 sin <-cos
5) (I_$“-1%“))
sin <+cos <) (.Ik4+Ig3) - (I$“-I”;“)}
+FZ{2 cos &I;-4+I;-3)
(25)
+ (I;-“-I;-“)}
When an rf pulse is “on” we have the total Hamiltonian of the system as 2’
= &+A&+&+~;s~%‘;f
(18)
and when the pulse is “off” eqn. (18) reduces to 2’
= 2fnt = A*;,+&+%‘;s
(26)
since during the “off” period only the internal interactions are in operation.
321 THE CYCLICITY OF THE PHASE-ALTERNATED
This pulse sequence
is represented
PULSE SEQUENCE (PAPS)
schematically
tzl - (z-t~~-2Z-t~~-r)jv
as (271
Here the superscript represents the phase of the pulses and subscript the pulse pulse. After a width. The window is 22. The first pulse tzi is the preparatory time t, = 42 the events within the bracket repeat. In order that the above pulse sequence be cyclic, it must satisfy the condition [ 11 Ntc U,(Nt,)
= T exp[ -i
s
&&‘)dt’]
= 1
(28)
0
for N integer. T is the Dyson time ordering operator and %‘kfis given in eqn. (23). Urf is called the propagator. Using the Magnus expansion to express the solution of eqn. (28)) we see that a sufficient condition for the pulse sequence to be cyclic is - r(i) _ Xrf -0
(29)
for all i. Clearly if 2; is phase alternated such that for each pulse with rotation angle cy about the direction X corresponding to the single transition operators I$’ and I$* constituting the *kf, there is a subsequent pulse about -X with rotation angle ly so as to compensate for the first pulse, then 2:j”’ = 0 and to zeroth order Urf(Ntc) = 1. It is also easy to see that since %& contains only Ik” and 1%” operators and since I;* and 1%” commute 9kji) = 0. PAPS being cyclic we can use average Hamiltonian theory to study the response. EVALUATION OF THE INITIAL DENSITY MATRIX b(O) AT THE END OF FIRST PULSE, i.e. PREPARATORY PULSE
Before the application of the preparatory pulse the system is assumed to be at equilibrium under the influence of the Hamiltonian 2 = ~?b + interactions. JY;,t Z &Yb where Zint is the sum of the internal line-broadening Using the high temperature approximation [ 171 we obtain ~~~;~~~[exp(-a)l)-‘exp(-a);[Tirl)l’(~-~)
(30)
The first term is unaffected by any evolution of the system, so we follow the evolution of the second term only which is proportional to the reduced density matrix p defined by setting Q=
[D(l)]-'
[
1
-(+I
328
Thus, pes = %;a = K(I&3-1;-4)
(32)
In QIF, it is Pep = K(Ii-3--1;-4)
(33)
when we apply a pulse of duration twl along the x axis; the density immediately after the pulse is given by p(O) = exp( -&&,)p,,
matrix
exp(i%&,)
(34)
Substituting eqn. (23) for *& and using the commutators pendix, we get the initial density matrix
given in the Ap-
cos cyi - (12;1+13;4) sin vi}
P(O) = K{ @“-I;-:“,
(35)
where w1 = ry, ( (3) “’ cos <+ sin <) &. ZEROTH
ORDER
AVERAGE
HAMILTONIANS
FOR
THE
LINE-BROADENING
INTERACTIONS
The zeroth order average Hamiltonians for the above mentioned line-broadening interactions during PAPS can now be calculated. We give here only the final results. The commutators needed to evaluate them are given in the Appendix. Firstly, A2’$n
= AK
(I’-2z-13-4z)cos
@+ (12-1y+13-1y)sin 2
[
1
v/z cos fi 2 2
where y/, = o1 ( (3 ) ‘I2 cos lj+ sin 5) tw2
(36)
secondly, j@“)
= A
(I’-2z-
13-4z)cos F+
(12-1y+13-4y)sin
5
1
cos 5
(37)
and thirdly,
2 &” = kl(Ik4+Ik3)
(I$“-I$“)cos
+k2
$-
(Ig4+13;l)sin
+k3 1 (I&4-12J3)cos
+k4(Ik4+I$3)
F-
(I$3-I$4)sin
(I~4-I~3)C0s
+k6 i
+ lz, (I;-” + Ii-“)
5
cos F 1
( F 1 cos 7
F+
(Ik2+Ig4)sin
F
cos y 1 (33)
Here k, = F,; k, = Fx( (3)“” sin {-cos r+cosl); k5 = F,; and k, = BF+os r. COMMUTATION
c); k, = -F,,; k4 = FY( (3)‘j2 sin
OF j?(O) WITH THE AVERAGE HAMILTONIANS
OF THE LINE-
BROADENING INTERACTIONS
Using eqn. (35) along with eqns. (36)- (38) and the necessary in the Appendix, it can easily be shown that [p(0),A.%‘ti”)] [p(o),.R$O’]
= 0 = 0
[p(o),G?yp] =
commutators
(39) (40)
0
(41)
= 0
(42)
when v2 = 2~~. Therefore, [b(O),$,!,“,‘] where .g;;i’
= Aj$9’$) + j@“’ +2;&0’
(43)
CONCLUSION
Since [p(O),~~~~‘] = 0, p(O) is a constant of motion during PAPS with respect to the three line-broadening interactions, namely, efg inhomogeneity, torsional oscillations of the units constituting the solid and heteronuclear dipolar interaction. Therefore, these interactions do not contribute to the linewidth during PAPS. We therefore conclude that PAPS will narrow the spectral line of a pure NQR signal from physically equivalent spin I = 3/2 nuclei in a single crystal sample when the rf pulses are applied along the x-axis with the pulse width of tw2 = 2t,,. A similar condition on pulse width was obtained earlier by Osokin [ 51 for spin I = 1 nuclei. This condition on the pulse widths therefore appears to be a general one. APPENDIX
Wokaun and Ernst [ 151 and Vega [ 161 have developed the single transition operator formalism or generalized fictitious spin-l/2 operator formalism which has been used in this paper. They give the form and properties of these operators in detail. We give here the non-zero commutators used in this paper to evaluate expressions involving exponential operators at various places mentioned.
330
For taking A?& to QIF and to evaluate p”(0)
[ (I&3-Ig4),
(Ik;2+I$4)]
= i(I$‘+Ig3)
(I;“-I$“),
(1$2+1$3)]
= -i(I$2+I$4)
[
To evaluate pes and A2a”)
[ (I$“+I$“), [
(I$“+I$“),
(Ik3-Ig4)]
= i(12;‘+Ig4)
(12;‘+13;4)] = -i(I&3-Iim4)
For taking Xk, i.e. quadrupolar Hamiltonian due to torsional oscillations of units in the solid, to &IF
[ (Ik”-I%“), (I’-3x+I$4)] [(I&I;-;“,,
(I:_“- Iy)]
= i(I&3-12;4) = -i(Iy+Iy)
For taking Y&, i.e. heteronuclear dipole-dipole interaction Hamiltonian, to QIF
[ (Ig,-“-I%;“), (Ik2+Ig4)]
= i(I$2-I$4)
[
(I~z-I~4),
(Ik2-I3;4)]
= -i(Ijf2+I$f4)
[
(Ig2-Iff4),
(I$‘+13J4)] = -;(I_$“-1%“)
[
(Ig2-Ig4),
(I$“-I$“)]
[
(I~m2-I~4), (I&“-1$“)]
[
(Ik2-Ig4),
= i(Ik2+IG4) = i(Ig4+Ik3)
(12;4+Ik3)] = -i(I$3-I$4)
To evaluate &Lo’
[ (I$“+IG”), (I$“-I$“)]
= i(Ig4+Ig1)
[
(I$‘+I$“),
(12;“+13Jl)] = -i(I$‘-1%‘)
[
(I_t+j”+I$-“),(Ig4-12;3)]
= ;(I_$“-1%“)
[
(Ik”+I$“),
= -i(Ik4-12J3)
(I$“-I:“)]
331
To evaluate [p(O), 2&O) ]
REFERENCES 1 2
7 8 9 10 11 12 13 14 15 16 17
(a) U. Haeberlen, High Resolution NMR in Solids, Academic Press, New York, 1976. (b) M. Mehring, Principles of High Resolution NMR in Solids, Springer Verlag, Berlin, 1983. T.P. Das and E.L. Hahn, Nuclear Quadrupole Resonance Spectroscopy, Academic Press, New York, 1958. R.A. Marino and SM. Klainer, J. Chem. Phys., 67 (1977) 3388. E.D. Ostroff and J.S. Waugh, Phys. Rev. Lett., 16 (1966) 1097. D.Ya. Osokin, J. Mol. Struct., 83 (1982) 243 and references cited therein. (a) N.E. Ainbinder, G.B. Furman, G.E. Kibrik, A.Ya. Poljakov and I.G. Shaposhnikov, Z. Naturforsch., Teil A, 41 (1986) 366. (b) N. Reddy, Ph.D. Thesis, Indian Institute of Technology, Kanpur, India, 1987 (unpublished). (c) N. Reddy and P.T. Narasimhan, J. Mol. Struct., 192 (1989) 345. R. Reddy and P.T. Narasimhan, Ninth Int. Symp. Nuclear Quadrupole Resonance, Kanpur, India, Jan. 1988, to be published. R.S. Cantor and J.S. Waugh, J. Chem. Phys., 73 (1980) 1054. A.K. Hitrin, G.E. Karnaukh and B.N. Provotorov, J. Mol. Struct., 83 (1982) 269. M. Matti Maricq, Phys. Rev. B, 33 (1986) 4501. V.L. Ermakov and D.Ya. Osokin, Mol. Phys., 53 (1984) 1335. J.S. Waugh, C.H. Wang, L.M. Huber and R.L. Vold, J. Chem. Phys., 48 ( 1968) 662. N.E. Ainbinder and G.B. Furman, Sov. Phys-JETP, 58(3) (1983) 575. N.N. Bogolyubov and Yu.A. Mitropol’skii, Analytic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1964. A. Wokaun and R.R. Ernst, J. Chem. Phys., 67 (1977) 1752. S. Vega, J. Chem. Phys., 68 (1978) 5518. M. Goldman, Spin Temperature and Nuclear Magnetic Resonance in Solids, Oxford University Press, London, 1970.
321 in The Netherlands
MULTIPLE PULSE RESPONSE IN PURE NQR OF SPIN I = 3/2 NUCLEI IN SINGLE CRYSTALS: RESPONSE TO PHASEALTERNATED PULSE SEQUENCE (PAPS)
A.K. DUBEY and P.T. NARASIMHAN* Department of Chemistry, Indian Institute of Technology, Kanpur 208 016 (Zndia) (Received
16 March 1988)
ABSTRACT The response of spin I = 3/2 nuclei in a single crystal to phase-alternated pulse sequence (PAPS) -2r-tTz-r), in pure NQR is theoretically investigated. Average Hamiltonian the(r-t;y ory is employed to calculate the spin-space averaging of the line-broadening internal interactions. The criterion employed for spin-space averaging is that the average Hamiltonian during the cycle for the interaction under consideration should commute with the initial density matrix prepared by the pulse t$ before the first cycle. The average Hamiltonian during PAPS for the main linebroadening interactions, namely electric field gradient (efg) inhomogeneity, torsional oscillations and heteronuclear dipolar interactions, is found to commute with the initial density matrix of the do not system when 2t,, = tw2.It is concluded that during PAPS these internal interactions contribute to the linewidth of the pure NQR spectrum of physically equivalent spin Z = 3/2 nuclei in a single crystal. These theoretical results are similar to those obtained earlier for spin Z = 1 system by Osokin. tow1 -
INTRODUCTION
Multiple-pulse sequences for suppressing the dipolar broadening and to obtain high resolution NMR spectra of spin I = l/2 in solids are now well established [ 11. It is therefore natural to ask the question whether a similar approach would be effective in pure NQR spectroscopy. In the case of pure NQR the spectrum is usually broadened due to the electric field gradient (efg) inhomogeneity, torsional oscillations of the units constituting the solid and heteronuclear dipolar interactions [ 21. The first experimental investigation using multiple-pulse sequences in pure NQR were those of Marino and Klainer [ 31 who employed a series of pulses with flip angle of 90’) phase shifted with respect to the first or preparatory pulse by 90”) the so-called Ostroff-Waugh (OW) sequence [4]. This sequence may be designated as t$- ( r-t$!-r)N. Surprisingly they observed the elongation of the echo train in the case of the *Author to whom correspondence
0022-2860/89/$03.50
should be addressed.
0 1989 Elsevier Science Publishers
B.V.
322
14N signal in a powder sample of NaN02. The spin-locking phenomenon observed by these authors is remarkably reminiscent of the NMR experiments in solids carried out earlier by Ostroff and Waugh [ 41. It has been pointed out by Marino and Klainer [3] that one can co-add the multiple echoes within a spin-locked spin echo train to enhance the S/N ratio as well. Multiple-pulse techniques can therefore be exploited to detect weak NQR signals. Subsequent to the work of Marino and Klainer [3] a number of authors have reported similar results for I = 1 [5], I = 3/2 [6] and I = 5/2 [7]. A theoretical treatment of the response to the OW sequence of 14N in NaN02 powder sample was attempted by Cantor and Waugh [ 81 who were able to successfully account for the short-time behaviour of the observed magnetization. However, the longtime behaviour was not accounted for. Hitrin et al. [9] employed a canonical transformation technique to account for the experimentally observed short and long-time behaviour. Maricq [lo] has employed Floquet theory to arrive at the correct experimental behaviour. Osokin has also theoretically investigated [5, 111 the spin I = 1 case subjected to PAPS and WAHUHA [l] sequences. Following the work of Waugh et al. [ 121 in NMR, Osokin [ 5,111 has shown that if the density matrix at the end of the preparatory pulse commutes with the average Hamiltonian in the presence of the rf pulses then a quasisteady state can be realized, and thus one can observe elongation of the echo train leading to line narrowing in NQR. Ainbinder and Furman [ 131 have given a theory of multiple-pulse averaging for spin systems with arbitrary nonequidistant spectra using the canonical transformation technique of Hitrin et al. [ 91 and the Krylov-Bogolyubov-Mitropol’skii averaging method [ 141. Detailed application of this theory has not yet been made in order to explain the experimental results of multiple pulse response in spin I&33/2 systems. We present in this paper a preliminary result of our theoretical investigation of response to phase-alternated pulse sequence (PAPS) t$- ( r-t~~00-2r-t~Z-r)N in pure NQR of physically equivalent spin I = 3/2 nuclei in single crystals. Our approach is akin to that of Osokin [ 51, and we have used the single transition operators [ 151 or generalized fictitious spin - l/2 operators [ 161. GENERAL
THEORY
OF LINE NARROWING
WITH
CYCLIC
MULTIPLE-PULSE
SEQUENCES
When the interaction
of a system is cyclic, the average Hamiltonian theory of the system. This theory proposes to replace the time-dependent Hamiltonian of the system over one cycle by an equivalent time-independent average Hamiltonian. It also proposes that it is enough to know the behaviour of the system over one cycle. For subsequent cycles the same behaviour is repeated so that [ 1 ] can be used to study the behaviour
0%)
=
bWIN
(1)
323
where p (t,) is the density matrix of the system after one cycle, t, is the cycle time and N a positive integer. The behaviour of the system is now described by the average Hamiltonian, 2, provided we observe the system only at specified “observation windows”, i.e. only after a cycle is complete. By making a judicious choice of cyclic multiple-pulse sequences we can introduce time dependence in the line-broadening internal interactions in such a way that its average Hamiltonian commutes with the initial density matrix p(O) prepared from the equilibrium density matrix pes by a preparatory pulse
tPw,m
(2)
=0
p(O) thus becomes a constant of motion during the cyclic interaction and the line-broadening internal interactions under consideration do not contribute to the evolution of the system. In effect we achieve line-narrowing. After describing the Hamiltonians of the system of physically equivalent spin I = 3/2 nuclei in a single crystal in a suitable representation (next section) we will next show that PAPS is a cyclic sequence. Following this, details of the evaluation of the initial density matrix p(O) following the preparatory pulse are given. We then calculate average Hamiltonians to zeroth order using Magnus expansion for the line-broadening interactions. The commutators of these average Hamiltonians with p(O) is then evaluated. HAMILTONIAN OF THE SYSTEM OF SPIN Z = 3/2 NUCLEI
The overall Hamiltonian of the system of spin I = 3/2 nuclei consists of several terms. The terms which are relevant to the particular problem at hand are included in (3)
Xo = F
[312z- 01+1)+V)(IL+12L)l
is the quadrupole tem ( QPAS ) . 3
Hamiltonian
expressed
in the quadrupole
principal
axis sys-
e2qQ
wQ= 2 1(2I-
1)fi
and r is the asymmetry parameter [ 21. We consider the rf field as being applied QPAS. Therefore X*f = - 2o41,
cos wt
in the direction
of x-axis of the
(5)
324
Here o1 = ?H, with Hi the rf field strength, y the magnetogyric ratio of the nuclei and o the frequency of the rf radiation. The last three terms in eqn. (3) are the main line-broadening interaction Hamiltonians. The Hamiltonian corresponding to the electric field gradient (efg) inhomogeneity is Azo = +[312z-1(1+1)+~)(I:,+12,)J
(6)
3 e"Q(&) where A.o, = 2 1(21-1)h’ ,&r is the secular part of the interaction of the spin system with the field gradient subject to torsional oscillations and is given by &
= 2
[2&V&&
I”,) -+(3+~)
(I&1$)
+E~(~--Y/) (I&I”Z)]
(7)
Here ei = @-(09). We choose the coordinate system such that the laboratory fixed axes make angles 8i (i = X, Y, 2) with respect to the corresponding QPAS of the unit which undergoes torsional oscillation. ( 0: ) is the thermal average of 0;. The last term J& in eqn. (3) is the Hamiltonian for heteronuclear dipoledipole interaction and is given by
Fi being the ith component of the field due to the non-resonant nuclei (S) at the site of the resonant nuclei (I). In order to simplify the evaluation of the expressions involving exponential operators we express 2 in a representation where &!o, the main term in the Hamiltonian, is diagonal. This representation is denoted throughout by a prime ( ’ ) as superscript and is realized by the unitary transformation
R=R-‘=
’ [ cos sin 0t/2 r/2
- cos r/2
0
sin t/2
sin 05/2
-cos sin 0512 512
cos 0t/2
where tan 5 = q/ (3 ) *12. Using eqn. (9) in eqn. (3 ) we obtain 2’
= %b+&+A~%!;,+xI,+%‘;s
1
(9)
(10)
where for example, %‘& represents R%$R-‘. We give here the form of the various terms in the right hand side of eqn. (10) in terms of the single transition operators [ 15, 161 J?& = K[I;-3-I;-4)
(11)
325
where K = T
(1+ $/3) ‘j2.
X& = -2~0, cos c&[ ((3)“”
cos {+sin 5) (I$“+I&“)
+ ((3)“” AS;,
sin <-cos
<) (I$“-I$“)+
(1$4+1$3)]
(12)
= AK(I&-3-I;-4)
(13)
J+o (1+$/3)l’2
(14)
where AK= 8$
= A(I;-2-I;-4)
+B(I$3-I$4)
(15)
where A = 9
(4(3)‘/2~8g
sin c- (3)1’2(3+q)62y(
(3)‘j2 cos r
-sin 0 - (3)“2(3-q)0$(
(3)“” cos t+sin r)}.
and
B=
-2
{4(3)“2@i COS<+ (3)1'2(3+Tj)@$(
(3)‘12 sin 5
+COs O+ (3)“2(3-?j)0:( Z’;s = F,{ ((3)“”
sin <-cos
(I$;“+Ik”,
0 (Ik”-1%“)
- ((3)“”
(3)“”
sin<-cos
0)
cos c+sin <)
+ <1k”+1$“,}+
FY{ ( (3)“2 sin r+cos <) (Ig4+Ig3)
(16)
- ( (3)‘12 cos <-sin 5) (I&“I”;“) - (I&“-I”,-“)}+ FZ{2 cos <(I&4+Ig3)
+2 sin &I$“-Ig-“)
+ (I_&4-Ig3)}
We next take eqn. (10) to quadrupole interaction frame (QIF) by using the transformation ( 17 ) U = exp( -Mht)
(17)
In &IF X’ becomes j&f = U+X’U %f = U+Z’:,U+U+AZ’~U+U+~~U+U+~;sU ,&=
““‘+A&b+&+&
(18)
326
Note that we express all the terms in the Hamiltonian which have been transformed by the R matrix and subsequently by the U operator with a prime superscript plus a tilde. The non-zero commutators used to evaluate eqn. (18) are given in the Appendix. We have the following expressions for various terms on the right hand side of eqn. (18). 5Pcf = -2w,
cos wt[ ((3)“’
+ (I’,-2+I‘$3)sinKt}+ A*b
= AX;,
((3)“”
sin c-cos
(19)
5) (I_$“-I%-“) + (Ik4+Ig3)]
(29)
+B{ (I_$“+I$“)
= Fx{ (I$“+Ik”)
+ ((3)“”
cos Kt-
sin <-cos
(I’,-“-I”,-“)
+Fz{2 cos <(I;-4+I;-3) ((3)“”
+Fv(sin
sin Kt}
(21)
5) (I$“-1%“))
+FY{ ( (3)*12 sin r+cos 4) (Ik4+Ig3)
-FX
cos Kt
= AK(I;-3-1$-4)
G%‘$= A(I~2-I~4) 2is
cos l+sin LJ{ (I$“+I$“)
(22)
- (I&“-I”;“)}
+ (I;-4-I;-3)}
cos <+sin LJ{ (I$2+I$4)cos
Kt- (Ig2-I$4)sin
Kt}
5- (3)“” cos {>{ (Ik2+13;4)cos Kt+ (Ik2-Ig4)sin
Kt}
+ Fz2 sin <( (I$” - 1%“) cos Kt - (I$&+
Ik3) sin Kt}
In eqns. (19), (21) and (22) we truncate the time dependent part in the firstorder approximation [ 1 ] to obtain &,
= -L(I$2+1$4)
(23)
where L = a,( (3)“” cos r+sin <). j&
= A(I;-2-I;-4)
(24)
and 2is
= FX { (I_$“+I$“) +Fy{ ((3)“”
+ ( (3)‘12 sin <-cos
5) (I_$“-1%“))
sin <+cos <) (.Ik4+Ig3) - (I$“-I”;“)}
+FZ{2 cos &I;-4+I;-3)
(25)
+ (I;-“-I;-“)}
When an rf pulse is “on” we have the total Hamiltonian of the system as 2’
= &+A&+&+~;s~%‘;f
(18)
and when the pulse is “off” eqn. (18) reduces to 2’
= 2fnt = A*;,+&+%‘;s
(26)
since during the “off” period only the internal interactions are in operation.
321 THE CYCLICITY OF THE PHASE-ALTERNATED
This pulse sequence
is represented
PULSE SEQUENCE (PAPS)
schematically
tzl - (z-t~~-2Z-t~~-r)jv
as (271
Here the superscript represents the phase of the pulses and subscript the pulse pulse. After a width. The window is 22. The first pulse tzi is the preparatory time t, = 42 the events within the bracket repeat. In order that the above pulse sequence be cyclic, it must satisfy the condition [ 11 Ntc U,(Nt,)
= T exp[ -i
s
&&‘)dt’]
= 1
(28)
0
for N integer. T is the Dyson time ordering operator and %‘kfis given in eqn. (23). Urf is called the propagator. Using the Magnus expansion to express the solution of eqn. (28)) we see that a sufficient condition for the pulse sequence to be cyclic is - r(i) _ Xrf -0
(29)
for all i. Clearly if 2; is phase alternated such that for each pulse with rotation angle cy about the direction X corresponding to the single transition operators I$’ and I$* constituting the *kf, there is a subsequent pulse about -X with rotation angle ly so as to compensate for the first pulse, then 2:j”’ = 0 and to zeroth order Urf(Ntc) = 1. It is also easy to see that since %& contains only Ik” and 1%” operators and since I;* and 1%” commute 9kji) = 0. PAPS being cyclic we can use average Hamiltonian theory to study the response. EVALUATION OF THE INITIAL DENSITY MATRIX b(O) AT THE END OF FIRST PULSE, i.e. PREPARATORY PULSE
Before the application of the preparatory pulse the system is assumed to be at equilibrium under the influence of the Hamiltonian 2 = ~?b + interactions. JY;,t Z &Yb where Zint is the sum of the internal line-broadening Using the high temperature approximation [ 171 we obtain ~~~;~~~[exp(-a)l)-‘exp(-a);[Tirl)l’(~-~)
(30)
The first term is unaffected by any evolution of the system, so we follow the evolution of the second term only which is proportional to the reduced density matrix p defined by setting Q=
[D(l)]-'
[
1
-(+I
328
Thus, pes = %;a = K(I&3-1;-4)
(32)
In QIF, it is Pep = K(Ii-3--1;-4)
(33)
when we apply a pulse of duration twl along the x axis; the density immediately after the pulse is given by p(O) = exp( -&&,)p,,
matrix
exp(i%&,)
(34)
Substituting eqn. (23) for *& and using the commutators pendix, we get the initial density matrix
given in the Ap-
cos cyi - (12;1+13;4) sin vi}
P(O) = K{ @“-I;-:“,
(35)
where w1 = ry, ( (3) “’ cos <+ sin <) &. ZEROTH
ORDER
AVERAGE
HAMILTONIANS
FOR
THE
LINE-BROADENING
INTERACTIONS
The zeroth order average Hamiltonians for the above mentioned line-broadening interactions during PAPS can now be calculated. We give here only the final results. The commutators needed to evaluate them are given in the Appendix. Firstly, A2’$n
= AK
(I’-2z-13-4z)cos
@+ (12-1y+13-1y)sin 2
[
1
v/z cos fi 2 2
where y/, = o1 ( (3 ) ‘I2 cos lj+ sin 5) tw2
(36)
secondly, j@“)
= A
(I’-2z-
13-4z)cos F+
(12-1y+13-4y)sin
5
1
cos 5
(37)
and thirdly,
2 &” = kl(Ik4+Ik3)
(I$“-I$“)cos
+k2
$-
(Ig4+13;l)sin
+k3 1 (I&4-12J3)cos
+k4(Ik4+I$3)
F-
(I$3-I$4)sin
(I~4-I~3)C0s
+k6 i
+ lz, (I;-” + Ii-“)
5
cos F 1
( F 1 cos 7
F+
(Ik2+Ig4)sin
F
cos y 1 (33)
Here k, = F,; k, = Fx( (3)“” sin {-cos r+cosl); k5 = F,; and k, = BF+os r. COMMUTATION
c); k, = -F,,; k4 = FY( (3)‘j2 sin
OF j?(O) WITH THE AVERAGE HAMILTONIANS
OF THE LINE-
BROADENING INTERACTIONS
Using eqn. (35) along with eqns. (36)- (38) and the necessary in the Appendix, it can easily be shown that [p(0),A.%‘ti”)] [p(o),.R$O’]
= 0 = 0
[p(o),G?yp] =
commutators
(39) (40)
0
(41)
= 0
(42)
when v2 = 2~~. Therefore, [b(O),$,!,“,‘] where .g;;i’
= Aj$9’$) + j@“’ +2;&0’
(43)
CONCLUSION
Since [p(O),~~~~‘] = 0, p(O) is a constant of motion during PAPS with respect to the three line-broadening interactions, namely, efg inhomogeneity, torsional oscillations of the units constituting the solid and heteronuclear dipolar interaction. Therefore, these interactions do not contribute to the linewidth during PAPS. We therefore conclude that PAPS will narrow the spectral line of a pure NQR signal from physically equivalent spin I = 3/2 nuclei in a single crystal sample when the rf pulses are applied along the x-axis with the pulse width of tw2 = 2t,,. A similar condition on pulse width was obtained earlier by Osokin [ 51 for spin I = 1 nuclei. This condition on the pulse widths therefore appears to be a general one. APPENDIX
Wokaun and Ernst [ 151 and Vega [ 161 have developed the single transition operator formalism or generalized fictitious spin-l/2 operator formalism which has been used in this paper. They give the form and properties of these operators in detail. We give here the non-zero commutators used in this paper to evaluate expressions involving exponential operators at various places mentioned.
330
For taking A?& to QIF and to evaluate p”(0)
[ (I&3-Ig4),
(Ik;2+I$4)]
= i(I$‘+Ig3)
(I;“-I$“),
(1$2+1$3)]
= -i(I$2+I$4)
[
To evaluate pes and A2a”)
[ (I$“+I$“), [
(I$“+I$“),
(Ik3-Ig4)]
= i(12;‘+Ig4)
(12;‘+13;4)] = -i(I&3-Iim4)
For taking Xk, i.e. quadrupolar Hamiltonian due to torsional oscillations of units in the solid, to &IF
[ (Ik”-I%“), (I’-3x+I$4)] [(I&I;-;“,,
(I:_“- Iy)]
= i(I&3-12;4) = -i(Iy+Iy)
For taking Y&, i.e. heteronuclear dipole-dipole interaction Hamiltonian, to QIF
[ (Ig,-“-I%;“), (Ik2+Ig4)]
= i(I$2-I$4)
[
(I~z-I~4),
(Ik2-I3;4)]
= -i(Ijf2+I$f4)
[
(Ig2-Iff4),
(I$‘+13J4)] = -;(I_$“-1%“)
[
(Ig2-Ig4),
(I$“-I$“)]
[
(I~m2-I~4), (I&“-1$“)]
[
(Ik2-Ig4),
= i(Ik2+IG4) = i(Ig4+Ik3)
(12;4+Ik3)] = -i(I$3-I$4)
To evaluate &Lo’
[ (I$“+IG”), (I$“-I$“)]
= i(Ig4+Ig1)
[
(I$‘+I$“),
(12;“+13Jl)] = -i(I$‘-1%‘)
[
(I_t+j”+I$-“),(Ig4-12;3)]
= ;(I_$“-1%“)
[
(Ik”+I$“),
= -i(Ik4-12J3)
(I$“-I:“)]
331
To evaluate [p(O), 2&O) ]
REFERENCES 1 2
7 8 9 10 11 12 13 14 15 16 17
(a) U. Haeberlen, High Resolution NMR in Solids, Academic Press, New York, 1976. (b) M. Mehring, Principles of High Resolution NMR in Solids, Springer Verlag, Berlin, 1983. T.P. Das and E.L. Hahn, Nuclear Quadrupole Resonance Spectroscopy, Academic Press, New York, 1958. R.A. Marino and SM. Klainer, J. Chem. Phys., 67 (1977) 3388. E.D. Ostroff and J.S. Waugh, Phys. Rev. Lett., 16 (1966) 1097. D.Ya. Osokin, J. Mol. Struct., 83 (1982) 243 and references cited therein. (a) N.E. Ainbinder, G.B. Furman, G.E. Kibrik, A.Ya. Poljakov and I.G. Shaposhnikov, Z. Naturforsch., Teil A, 41 (1986) 366. (b) N. Reddy, Ph.D. Thesis, Indian Institute of Technology, Kanpur, India, 1987 (unpublished). (c) N. Reddy and P.T. Narasimhan, J. Mol. Struct., 192 (1989) 345. R. Reddy and P.T. Narasimhan, Ninth Int. Symp. Nuclear Quadrupole Resonance, Kanpur, India, Jan. 1988, to be published. R.S. Cantor and J.S. Waugh, J. Chem. Phys., 73 (1980) 1054. A.K. Hitrin, G.E. Karnaukh and B.N. Provotorov, J. Mol. Struct., 83 (1982) 269. M. Matti Maricq, Phys. Rev. B, 33 (1986) 4501. V.L. Ermakov and D.Ya. Osokin, Mol. Phys., 53 (1984) 1335. J.S. Waugh, C.H. Wang, L.M. Huber and R.L. Vold, J. Chem. Phys., 48 ( 1968) 662. N.E. Ainbinder and G.B. Furman, Sov. Phys-JETP, 58(3) (1983) 575. N.N. Bogolyubov and Yu.A. Mitropol’skii, Analytic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1964. A. Wokaun and R.R. Ernst, J. Chem. Phys., 67 (1977) 1752. S. Vega, J. Chem. Phys., 68 (1978) 5518. M. Goldman, Spin Temperature and Nuclear Magnetic Resonance in Solids, Oxford University Press, London, 1970.