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Double quantum coherence in pure nuclear quadrupole resonance spectroscopy
Journal of Molecular Structure, 192 (1989) 309-319 Elsevier Science Publishers B.V., Amsterdam - Printed
309
in The Netherlands
DOUBLE QUANTUM COHERENCE IN PURE NUCLEAR QUADRUPOLE RESONANCE SPECTROSCOPY
RAVINDER
REDDY and P.T. NARASIMHAN*
Department of Chemistry, Indian Institute of Technology, Kanpur 208 016 (India) (Received
16 March 1988)
ABSTRACT The results of a theoretical investigation on double quantum coherence in pure nuclear quadrupole resonance (NQR) for spin I = 5/2 with axial symmetry in a single crystal are presented. Using the tensor operator formalism it is shown that the creation and detection of double quantum coherence can be done by the application of a pulse scheme of the form (n/2), (2wo ) (n), ( CO,)r- (n) x(2wo). The effect of resonance off-set on the double quantum coherence is also considered.
INTRODUCTION
In pulsed nuclear magnetic resonance (NMR) or nuclear quadrupole resonance (NQR) spectroscopy magnetic dipole-allowed transitions between the eigenstates of the nuclear ensemble can be observed by perturbing the spin system with a radio frequency (rf) pulse which brings the eigenstates separated by ( Am 1 = 1 into coherent superposition. Such transitions are referred to as the single quantum transitions or single quantum coherences. The inherent time-dependence of each state, normally random in thermal equilibrium, acquires a phase coherence across the ensemble. This coherence is known as transverse magnetization and it gives rise to a free induction signal. However, if the magnetic dipole transition between particular states of the spin system is forbidden, i.e. )Am ) = 2, 3, ... . etc., then the direct excitation of such coherence by the resonant rf pulses is not possible. The first experimental NMR detection of coherence between forbidden levels separated by 1Am 1 = 2 was achieved by Hatanaka et al. [ 1,2] using a multistep process. They showed that in a three-level NMR system the creation and detection of the double quantum coherence ( 1Am 1 = 2 ) can be done by using a pulse scheme of the form (n/ a),(~,) (n),(w,)-T-(n),(o,) where CO,and wb are the frequencies of the two allowed transitions. Following the work of Hatanaka et al. [l, 21 multiple *Author to whom correspondence
0022-2860/89/$03.50
should be addressed.
0 1989 Elsevier Science Publishers
B.V.
310
quantum NMR spectroscopy has been extensively developed and applied to the solution of problems in molecular structure and dynamics [ 3-71. It is surprising to note that no study of multiple quantum coherence has so far been made in the area of pure NQR spectroscopy. This paper presents theoretical investigations of the creation and detection of double quantum coherence in pure NQR of physically equivalent nuclei with spin I = 5/2 in a single crystal. Our approach is based on the density matrix theory with the tensor operator formalism [8-l 11. DOUBLE QUANTUM COHERENCE FOR SPIN I = 5/2 CASE IN PURE NQR
We consider the case of a non-interacting nuclear spin ensemble with axially symmetric field gradient. The energy levels are depicted in Fig. 1. The pulse scheme considered is also shown there. In all the calculations in this paper we assume that the relaxation effects are not significant. We set h = 1, and express energy values in radians per second. The rf pulses are assumed to be applied along the x-axis of the electric field gradient. The total Hamiltonian can be written as X(t)
= =%o+&I(t)
+%b(t)
where Zo is the quadrupolar G
e2qQ = 41(21-l)
(1) Hamiltonian
and is given by [ 121
(2)
[31; -I21
and .Y&( t), X&(t) define the interaction between the nuclear spins and the respectively. The rf Hamiltoapplied rf pulses with frequencies and nians can be written as 2wQ,
@Q
x&(t)
= -2u,,,
(3)
cos(wQt)I,
and c&(t)
= -2016
(4)
cos(2~Qt)Ix
where ala = yHla and okib = yHlb; y is the gyromagnetic ratio of the nucleus and HIa and Hlb are the amplitudes of the applied rf pulses. Following Bowden et al. [8] the total Hamiltonian for spin I = 5/2 case can be written in terms of tensor operators X(t)
= -T;+2w,, (;,*
where
mQ
=
COS(C&T;
as
(U)+2C0i6
COS(%&$)T;
(a)
(5)
$e2qQ.
The response of the nuclear spin system to the rf pulse excitations can be calculated using the Liouville-von Neumann equation. Before the application
311
Channel
Q
W,
NOR Energy Level Diagram for Spin I 512
Pulse Scheme
q
Fig. 1. Energy level diagram for spin I = 5/2 nuclear spin system with axially symmetric field gradient. Pulse scheme for double quantum coherence is shown on the left.
electric
of the rf pulses the spin system is assumed to be at thermal equilibrium under the influence of the pure quadrupolar Hamiltonian. In the high temperature approximation the thermal equilibrium density matrix is given by [ 131 a(tj0)
(Tr[ew( -$~)]]mlexp( -3%) - (Tr(l)l-1(l-(6~~kTT~ [Tr(l)l-’ [ 1-(~)p] C6) >
=
N-
the first term on the right hand side of the above equation, being a constant is not affected by any evolution of the spin system. Hence, we need to follow the evolution of the reduced density matrix (7) only. In the interaction
representation
defined by the transformation
operator
(8)
U = exp(+i ($,“Tgt) the total Hamiltonian &=
lJ2P(t)U-l=
is given by dig,+&
(9)
where &, = 2w,, cos(mot)UT:
(a)U-’
N wi, 2 -9(15)l’2
&jT: (a) 4 Ti (a)-3(219)i/2
T:(o)
1 (10)
312
and
28Ib =
2046 cos(2coQt)UT: (U)U-’ +
= Ll)lb;T: (a) 5 9(15)“2
T:(a)+
1 3(210)“2
T:(a)
1 (11)
In arriving at these equations we have made use of the evolution of the tensor operators under the influence of quadrupolar Hamiltonian wo/ (6) ‘12Ti (see Table 12 of ref. 9) and the high frequency terms have been dropped as nonsecular terms. In the interaction representation defined by the transformation operator given in eqn. (8)) the reduced density matrix satisfies the equation of motion
#(t)
=
dt
(12)
i[jl(t),%]
Since 2 is time independent p(t) is given by P(t)
= exp( -&t)fi(O)
exp(&)
(13)
In order to follow the evolution of density matrix, in the interaction representation, following the rf pulses we need to know the evolution of tensor operators under the action of the Hamiltonians &, and &,. The evolution of a few selected tensor operators under the influence of 21, and &,, calculated using the “nested commutation relationships” [8] are given in Tables 1 and 2, respectively. From the NQR energy level diagram (see Fig. 1) it is evident that double quantum coherence is possible between the states I+5/2>
-
(+3/2>
l-5/2>
-
l-3/2>
]+3/2>
-
I -l/2>
I -3/2>
-
]+1/2>
For spin I = 5/2 the operator set consists of 36 tensor operators [8]. Out of these, the operators that carry information about the double quantum coherence, i.e. I Am] = 2, are T2,(a,s),TZ (a,a),Ti (a,~) and TE (w) In what follows, we give general expressions for the density matrix of the spin system at various stages of the pulse sequence. The thermal equilibrium
313 TABLE 1 The evolution of a few selected tensor operators under the influence of the Hamiltonian $:,
UT;U-’
= &(25+3cos(<))+T;
5 P(cos(T) 4(105)“2
-1) +i sin(r)
UT:(s)U-’
= Tf(s);(2cos(~)+5cos(~/2))-T:(s)~
UT:(s)U-’
= -T;(S)
(3)1/Z -T”(s)-7(2)1’2 1
2(2:)‘,zT:(S)
2 ( cos (O-cos(Y2)) 3(14)“2
3o ~(cos(~)-cos(~/2))+T:(s)~(5cos(~)+2cos(~/2)) 7( 14)“2 +;(s)+;T:(~)
%, =
w,
U = exp( -G&t,) <=
(8)1’201,tw
reduced density matrix in the interaction
B(O) =
representation
is given by
(;,2T;
Immediately following the application of an rf pulse with frequency width “twl” the density matrix is given by (see Table 2 ) +% (4+3 -l)+iqT’f(s)
2~0, and
cos(&) )+ (Io52)1,zT: (cos(&)
sin(CI,)+i(21i)1,zT:(a)
sin(&))
(14)
where & = (5) 1’2w&,l. Immediately after this, if we apply an rf pulse with a frequency mQ and width “t the density matrix just after the removal of the pulse is given by w2 ” then
314 TABLE 2 The evolution of a few selected tensor operators under the influence of the Hamiltonian %b
UT:(a)U-’
= T:(a)&(19+16cos(~/2))-T;(a)&(cos(
UT: (e)U-’
4 3(210)“*
= -T:
2
(cos(r/2)-l)+isin(t/2)
5( 14)“2
240 (a)---7(210)“’
(cos(U2)-l)+T:(u)----
10 (cos(5/2)-1) 9(14)“2 25 ,01/2
+TP(u)~(ll+lOcos(~/2))+isin(Q2)
UT:U-’
= $(4+3
cos(~))+~ (lo~)‘,~T1(cOh(C)
(a)
(15)“2 _-T:(s)+-
(21&2T:
(8)
= T:(s)~(2cos(~/2)+5cos(~))+T~(~)~(cos(~)-cos(~/2)) +isin(r)
UT;U-’
T3( 2 a )+L3(35)‘,“Tz
-1)
+;sin({)
UT::(s)U-’
2
72 --(l-c0~(~/2))+T;(a)&4+cos(~/2)) = T: (a)35(15)‘,2 +T:(a)p
UT: (a)U-’
2(5)“2 ~T’Z(a)+~T:(a)
(cos(U2)-l)-isin(U2)
(15PT2 + 2 7 0 -T4 3(7)“2
= T; -(cos(~)-l)+T,+(3+4cos(~)) go 7(105)“2
UT: (s)U-’
O
-i sin(
(5)“2 ?T:
+i sin(<)
LT2 7(7)“2
(s)
1
’
2(2)“2 (a) +--T:
(a)
90(cos(~)-cos(~/2))+T:(~)~(5cos(~/2)+2cos(~)) = T: W21t14J~,~
= T;(s)
cos(c/Z)+i
7(7$zT:
(s) +
sin(
(5)“2 3(14)‘/2T:(S)-(10)1/2
UT: (u)U-’
2
7
4(2)“’ - 7 UT; (s)U-’
1 (s)+ -T4 (35)“2
= Tg (a) cos(U2) +i sin(
LT4
3
(s)
-p 7(;02)‘,zT: (a) (5)“2
1
LT3 3(2)1/z 1(+3(,)1/2T: (a) +01/2T: (a) -@T:
2 (a)
315 TABLE 2 (continued)
UT$(s)U-’
= T:(s)
cos(
Ue(u)U-’
= T;(a)
-
cos(c/2)+isin(r/2)
T’:(s)+
(5)1’2 T4 (s) (14)‘,2 3
Tf (a) 25 2(5)“2T5 9(6)l” T33 (a)+3(6)1,*
(5)“* ---T5(u)++3(7y 1
fa) 3
U = exp( -G&t,) 5 = (5)1’*WIbtw
P(Gvl+4v2) =
$$JT&
+T% +‘-G(s)& +T’: (s)& +T: (s)& +T; (s)&]
(15)
In writing eqn. (15) we have made use of the results given in Table 1. The expressions for the B terms are given in Table 3. Now, if we apply an rf pulse with a frequency 2wo and width “tW3”, then the density matrix immediately following the pulse is given by D(L
+&.A?+GV3) = +[C,T;
+C,T:+C,TI
(a)
+C,T~(s)+C,T;(s)+C,T~(s)+C,T;(s)]
(16)
Equation (16) has been obtained in a manner similar to that of eqn. (14). The expressions for C terms are given in Table 4. The magnetization operators at WQ and 2mQ frequencies (90’ out of phase with the rf pulses applied) in the laboratory frame are given, respectively, by M,, = -sin(coQt)T; (a) Mzb =
-SiTl(&OQt)T;
(a)
In the interaction representation these are given by
1 the subscripts a and b refer to the
mQ
and
2mQ
frequencies.
(17)
316 TABLE 3 Expressions for B terms
5 -((cos(&)-l)+A&(3+25cos(E;)) B2 = A14(105)“2 25 +A414(5)1/z W Ala-A 7(2)“’
B3 = isin
--% 27(70)“*
+A,+(2
cos(&)
i-5 cos(M2)
1 -A4
1 5(15)“ZA -Al----____ 2 -A 2(cos(r,) “3(14)“2 2(21)‘/2+ 14 (
B, = isin
-cos(&/2))
B6 = i sin(&/2)
where A, = ;(4+3
cos(&))
2 Az -- (105+cos(GbU Aa = isin(
(15)1’2 7 2
+A,+(5
cos(&) +2 cos(&/2))
317 TABLE 4 Expressions for C terms 1 C, = B,+4+3
90 -----_(cos(&)-l)+isin(&) cos(G)) +B’7(105)‘,2
C2 = 6 l-(cos(&)-1)+&+(3+4cos(&))+isin(&) (105)1/Z C, = isin
+8,;(2
cos(Q/2)+5cos(&))
+& A(cos(&)-cos(&/2))-isin(&/2) 21(14)“’ Cd = i sin(&)
C, = isin(&/2)
2 & ~(210)I,2+&
-
2 ?qiq=
+&
(
(cos(M-cos(tJ2))
cos(tJ2)
G = i sin(W) G =
isin
% = (5)1’2Wlbt,3
The signal in the interaction of the magnetization operator (A,i)OZTr[p*fi,i]
representation
is given by the expectation
value (18)
where i = a, b. RESULTS
The status of the spin system at various stages of the pulse sequence can be summarized as follows. The first pulse with frequency 2~0~ is a (x/2), pulse and hence we can set & = 7c/2 for this case. We can then obtain B ( twl) using eqn. (14). Immediately following the second pulse of width “&” and frequency WQ we have from eqns. (15)) (17) and (18) and using the orthogonality property of the tensor operators sin(t2)
(19)
318
(20) These results indicate that, when the flip angle of the second pulse of frequency mQ is different from (n) then we get free induction signals at both mQ and 2mQ, i.e. a and b channels, If we set & = n, then from eqns. (19) and (20) it is clear that there would be no signal in either the a or b channel. The density matrix eqn. (15) at this stage contains only the terms corresponding to quadrupolar and hexadecapole orders (Tg and Ti ) and Ti (s) , T$ (s) multipolar states which carry information about double quantum coherence. Let us now consider the situation following a (n/2) pulse of frequency 2aQ and a second pulse of mQ with a flip angle equal to (n). If we now apply a third (n) pulse of frequency 2mQ without any delay, i.e. tl = 0 (see Fig. 1) then immediately following the pulse we have from eqns. (16)- (18)
(21) and
(22)
= 0
Since the signal which is induced in channel a following the pulse in channel b, is arising entirely from Tz (s) and Ti (s) multipolar states it corresponds to the double quantum coherence signal. This means that the invisible double quantum coherence which has been created by the first two rf pulses in the sequence has been converted into an observable single quantum coherence, in channel a (detection frequency of mQ) by the second pulse in channel b. We have also investigated the influence of resonance off-set on the double quantum coherence, in the particular case where the off-set of rf pulses in channel b is exactly twice to that of rf pulses in channel a. In this case using Tables 1 and 2 and Table 12 of ref. 9 we have obtained for the signal, immediately following the last pulse in channel b (see Fig. 1)) the expressions cos (3A~07~) and (fiXLJ
= 0
(24)
where Am = (mQ--0). It may be noted here that the free precession frequency of the double quantum coherence is equal to three times the off-set, Ao, with respect to the rf
319
carrier frequency, CO,of channel a in this case. If the spin system is allowed to AW during the period r,, then we obtain evolve under -Ti, (6 ) l/2
@LX> =
-$$~[cos(3Awr,
+ Aor,)]
and (fix*)
= 6
(26)
In arriving at eqns. (25) and (26)) again we have made use of Tables 1 and 2 and Table 12 of ref. 9. CONCLUSIONS
Using the tensor should be possible NQR for the spin I by Hatanaka et al.
operator formalism, we have shown that in principle it to create and detect double quantum coherences in pure = 5/2 case using a pulse scheme similar to that employed [ 1,2] in NMR.
REFERENCES
7 8 9 10 11 12 13
H. Hatanaka, T. Terao and T. Hashi, J. Phys. Sot. Jpn., 39 (1975) 835. H. Hatanaka and T. Hashi, J. Phys. Sot. Jpn., 39 (1975) 1139. G. Bodenhausen, Prog. NMR Spectrosc., 14 (1981) 137. D.P. Weitekamp, Adv. Magn. Reson., 11 (1983) 111. G. Drobny, Annu. Rev. Phys. Chem., 26 (1985) 451. R.R. Ernst, G. Bodenhausen and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Clarendon Press, Oxford, 1987. M. Munowitz and A. Pines, Adv. Chem. Phys., 66 (1987) 1. G.J. Bowden and W.D. Hutchison, J. Magn. Reson., 67 (1986) 403. G.J. Bowden, W.D. Hutchison and J. Khachan, J. Magn. I&son., 67 (1986) 415. G.J. Bowden and W.D. Hutchison, J. Magn. Reson., 70 (1986) 361. G.J. Bowden and W.D. Hutchison, J. Magn. Reson., 71 (1987) 61. T.P. Das and E.L. Hahn, Nuclear Quadrupole Resonance Spectroscopy, Academic Press, New York and London, 1958. M. Goldman, Spin Temperature and NMR in Solids, Oxford University Press, Oxford, 1970.
309
in The Netherlands
DOUBLE QUANTUM COHERENCE IN PURE NUCLEAR QUADRUPOLE RESONANCE SPECTROSCOPY
RAVINDER
REDDY and P.T. NARASIMHAN*
Department of Chemistry, Indian Institute of Technology, Kanpur 208 016 (India) (Received
16 March 1988)
ABSTRACT The results of a theoretical investigation on double quantum coherence in pure nuclear quadrupole resonance (NQR) for spin I = 5/2 with axial symmetry in a single crystal are presented. Using the tensor operator formalism it is shown that the creation and detection of double quantum coherence can be done by the application of a pulse scheme of the form (n/2), (2wo ) (n), ( CO,)r- (n) x(2wo). The effect of resonance off-set on the double quantum coherence is also considered.
INTRODUCTION
In pulsed nuclear magnetic resonance (NMR) or nuclear quadrupole resonance (NQR) spectroscopy magnetic dipole-allowed transitions between the eigenstates of the nuclear ensemble can be observed by perturbing the spin system with a radio frequency (rf) pulse which brings the eigenstates separated by ( Am 1 = 1 into coherent superposition. Such transitions are referred to as the single quantum transitions or single quantum coherences. The inherent time-dependence of each state, normally random in thermal equilibrium, acquires a phase coherence across the ensemble. This coherence is known as transverse magnetization and it gives rise to a free induction signal. However, if the magnetic dipole transition between particular states of the spin system is forbidden, i.e. )Am ) = 2, 3, ... . etc., then the direct excitation of such coherence by the resonant rf pulses is not possible. The first experimental NMR detection of coherence between forbidden levels separated by 1Am 1 = 2 was achieved by Hatanaka et al. [ 1,2] using a multistep process. They showed that in a three-level NMR system the creation and detection of the double quantum coherence ( 1Am 1 = 2 ) can be done by using a pulse scheme of the form (n/ a),(~,) (n),(w,)-T-(n),(o,) where CO,and wb are the frequencies of the two allowed transitions. Following the work of Hatanaka et al. [l, 21 multiple *Author to whom correspondence
0022-2860/89/$03.50
should be addressed.
0 1989 Elsevier Science Publishers
B.V.
310
quantum NMR spectroscopy has been extensively developed and applied to the solution of problems in molecular structure and dynamics [ 3-71. It is surprising to note that no study of multiple quantum coherence has so far been made in the area of pure NQR spectroscopy. This paper presents theoretical investigations of the creation and detection of double quantum coherence in pure NQR of physically equivalent nuclei with spin I = 5/2 in a single crystal. Our approach is based on the density matrix theory with the tensor operator formalism [8-l 11. DOUBLE QUANTUM COHERENCE FOR SPIN I = 5/2 CASE IN PURE NQR
We consider the case of a non-interacting nuclear spin ensemble with axially symmetric field gradient. The energy levels are depicted in Fig. 1. The pulse scheme considered is also shown there. In all the calculations in this paper we assume that the relaxation effects are not significant. We set h = 1, and express energy values in radians per second. The rf pulses are assumed to be applied along the x-axis of the electric field gradient. The total Hamiltonian can be written as X(t)
= =%o+&I(t)
+%b(t)
where Zo is the quadrupolar G
e2qQ = 41(21-l)
(1) Hamiltonian
and is given by [ 121
(2)
[31; -I21
and .Y&( t), X&(t) define the interaction between the nuclear spins and the respectively. The rf Hamiltoapplied rf pulses with frequencies and nians can be written as 2wQ,
@Q
x&(t)
= -2u,,,
(3)
cos(wQt)I,
and c&(t)
= -2016
(4)
cos(2~Qt)Ix
where ala = yHla and okib = yHlb; y is the gyromagnetic ratio of the nucleus and HIa and Hlb are the amplitudes of the applied rf pulses. Following Bowden et al. [8] the total Hamiltonian for spin I = 5/2 case can be written in terms of tensor operators X(t)
= -T;+2w,, (;,*
where
mQ
=
COS(C&T;
as
(U)+2C0i6
COS(%&$)T;
(a)
(5)
$e2qQ.
The response of the nuclear spin system to the rf pulse excitations can be calculated using the Liouville-von Neumann equation. Before the application
311
Channel
Q
W,
NOR Energy Level Diagram for Spin I 512
Pulse Scheme
q
Fig. 1. Energy level diagram for spin I = 5/2 nuclear spin system with axially symmetric field gradient. Pulse scheme for double quantum coherence is shown on the left.
electric
of the rf pulses the spin system is assumed to be at thermal equilibrium under the influence of the pure quadrupolar Hamiltonian. In the high temperature approximation the thermal equilibrium density matrix is given by [ 131 a(tj0)
(Tr[ew( -$~)]]mlexp( -3%) - (Tr(l)l-1(l-(6~~kTT~ [Tr(l)l-’ [ 1-(~)p] C6) >
=
N-
the first term on the right hand side of the above equation, being a constant is not affected by any evolution of the spin system. Hence, we need to follow the evolution of the reduced density matrix (7) only. In the interaction
representation
defined by the transformation
operator
(8)
U = exp(+i ($,“Tgt) the total Hamiltonian &=
lJ2P(t)U-l=
is given by dig,+&
(9)
where &, = 2w,, cos(mot)UT:
(a)U-’
N wi, 2 -9(15)l’2
&jT: (a) 4 Ti (a)-3(219)i/2
T:(o)
1 (10)
312
and
28Ib =
2046 cos(2coQt)UT: (U)U-’ +
= Ll)lb;T: (a) 5 9(15)“2
T:(a)+
1 3(210)“2
T:(a)
1 (11)
In arriving at these equations we have made use of the evolution of the tensor operators under the influence of quadrupolar Hamiltonian wo/ (6) ‘12Ti (see Table 12 of ref. 9) and the high frequency terms have been dropped as nonsecular terms. In the interaction representation defined by the transformation operator given in eqn. (8)) the reduced density matrix satisfies the equation of motion
#(t)
=
dt
(12)
i[jl(t),%]
Since 2 is time independent p(t) is given by P(t)
= exp( -&t)fi(O)
exp(&)
(13)
In order to follow the evolution of density matrix, in the interaction representation, following the rf pulses we need to know the evolution of tensor operators under the action of the Hamiltonians &, and &,. The evolution of a few selected tensor operators under the influence of 21, and &,, calculated using the “nested commutation relationships” [8] are given in Tables 1 and 2, respectively. From the NQR energy level diagram (see Fig. 1) it is evident that double quantum coherence is possible between the states I+5/2>
-
(+3/2>
l-5/2>
-
l-3/2>
]+3/2>
-
I -l/2>
I -3/2>
-
]+1/2>
For spin I = 5/2 the operator set consists of 36 tensor operators [8]. Out of these, the operators that carry information about the double quantum coherence, i.e. I Am] = 2, are T2,(a,s),TZ (a,a),Ti (a,~) and TE (w) In what follows, we give general expressions for the density matrix of the spin system at various stages of the pulse sequence. The thermal equilibrium
313 TABLE 1 The evolution of a few selected tensor operators under the influence of the Hamiltonian $:,
UT;U-’
= &(25+3cos(<))+T;
5 P(cos(T) 4(105)“2
-1) +i sin(r)
UT:(s)U-’
= Tf(s);(2cos(~)+5cos(~/2))-T:(s)~
UT:(s)U-’
= -T;(S)
(3)1/Z -T”(s)-7(2)1’2 1
2(2:)‘,zT:(S)
2 ( cos (O-cos(Y2)) 3(14)“2
3o ~(cos(~)-cos(~/2))+T:(s)~(5cos(~)+2cos(~/2)) 7( 14)“2 +;(s)+;T:(~)
%, =
w,
U = exp( -G&t,) <=
(8)1’201,tw
reduced density matrix in the interaction
B(O) =
representation
is given by
(;,2T;
Immediately following the application of an rf pulse with frequency width “twl” the density matrix is given by (see Table 2 ) +% (4+3 -l)+iqT’f(s)
2~0, and
cos(&) )+ (Io52)1,zT: (cos(&)
sin(CI,)+i(21i)1,zT:(a)
sin(&))
(14)
where & = (5) 1’2w&,l. Immediately after this, if we apply an rf pulse with a frequency mQ and width “t the density matrix just after the removal of the pulse is given by w2 ” then
314 TABLE 2 The evolution of a few selected tensor operators under the influence of the Hamiltonian %b
UT:(a)U-’
= T:(a)&(19+16cos(~/2))-T;(a)&(cos(
UT: (e)U-’
4 3(210)“*
= -T:
2
(cos(r/2)-l)+isin(t/2)
5( 14)“2
240 (a)---7(210)“’
(cos(U2)-l)+T:(u)----
10 (cos(5/2)-1) 9(14)“2 25 ,01/2
+TP(u)~(ll+lOcos(~/2))+isin(Q2)
UT:U-’
= $(4+3
cos(~))+~ (lo~)‘,~T1(cOh(C)
(a)
(15)“2 _-T:(s)+-
(21&2T:
(8)
= T:(s)~(2cos(~/2)+5cos(~))+T~(~)~(cos(~)-cos(~/2)) +isin(r)
UT;U-’
T3( 2 a )+L3(35)‘,“Tz
-1)
+;sin({)
UT::(s)U-’
2
72 --(l-c0~(~/2))+T;(a)&4+cos(~/2)) = T: (a)35(15)‘,2 +T:(a)p
UT: (a)U-’
2(5)“2 ~T’Z(a)+~T:(a)
(cos(U2)-l)-isin(U2)
(15PT2 + 2 7 0 -T4 3(7)“2
= T; -(cos(~)-l)+T,+(3+4cos(~)) go 7(105)“2
UT: (s)U-’
O
-i sin(
(5)“2 ?T:
+i sin(<)
LT2 7(7)“2
(s)
1
’
2(2)“2 (a) +--T:
(a)
90(cos(~)-cos(~/2))+T:(~)~(5cos(~/2)+2cos(~)) = T: W21t14J~,~
= T;(s)
cos(c/Z)+i
7(7$zT:
(s) +
sin(
(5)“2 3(14)‘/2T:(S)-(10)1/2
UT: (u)U-’
2
7
4(2)“’ - 7 UT; (s)U-’
1 (s)+ -T4 (35)“2
= Tg (a) cos(U2) +i sin(
LT4
3
(s)
-p 7(;02)‘,zT: (a) (5)“2
1
LT3 3(2)1/z 1(+3(,)1/2T: (a) +01/2T: (a) -@T:
2 (a)
315 TABLE 2 (continued)
UT$(s)U-’
= T:(s)
cos(
Ue(u)U-’
= T;(a)
-
cos(c/2)+isin(r/2)
T’:(s)+
(5)1’2 T4 (s) (14)‘,2 3
Tf (a) 25 2(5)“2T5 9(6)l” T33 (a)+3(6)1,*
(5)“* ---T5(u)++3(7y 1
fa) 3
U = exp( -G&t,) 5 = (5)1’*WIbtw
P(Gvl+4v2) =
$$JT&
+T% +‘-G(s)& +T’: (s)& +T: (s)& +T; (s)&]
(15)
In writing eqn. (15) we have made use of the results given in Table 1. The expressions for the B terms are given in Table 3. Now, if we apply an rf pulse with a frequency 2wo and width “tW3”, then the density matrix immediately following the pulse is given by D(L
+&.A?+GV3) = +[C,T;
+C,T:+C,TI
(a)
+C,T~(s)+C,T;(s)+C,T~(s)+C,T;(s)]
(16)
Equation (16) has been obtained in a manner similar to that of eqn. (14). The expressions for C terms are given in Table 4. The magnetization operators at WQ and 2mQ frequencies (90’ out of phase with the rf pulses applied) in the laboratory frame are given, respectively, by M,, = -sin(coQt)T; (a) Mzb =
-SiTl(&OQt)T;
(a)
In the interaction representation these are given by
1 the subscripts a and b refer to the
mQ
and
2mQ
frequencies.
(17)
316 TABLE 3 Expressions for B terms
5 -((cos(&)-l)+A&(3+25cos(E;)) B2 = A14(105)“2 25 +A414(5)1/z W Ala-A 7(2)“’
B3 = isin
--% 27(70)“*
+A,+(2
cos(&)
i-5 cos(M2)
1 -A4
1 5(15)“ZA -Al----____ 2 -A 2(cos(r,) “3(14)“2 2(21)‘/2+ 14 (
B, = isin
-cos(&/2))
B6 = i sin(&/2)
where A, = ;(4+3
cos(&))
2 Az -- (105+cos(GbU Aa = isin(
(15)1’2 7 2
+A,+(5
cos(&) +2 cos(&/2))
317 TABLE 4 Expressions for C terms 1 C, = B,+4+3
90 -----_(cos(&)-l)+isin(&) cos(G)) +B’7(105)‘,2
C2 = 6 l-(cos(&)-1)+&+(3+4cos(&))+isin(&) (105)1/Z C, = isin
+8,;(2
cos(Q/2)+5cos(&))
+& A(cos(&)-cos(&/2))-isin(&/2) 21(14)“’ Cd = i sin(&)
C, = isin(&/2)
2 & ~(210)I,2+&
-
2 ?qiq=
+&
(
(cos(M-cos(tJ2))
cos(tJ2)
G = i sin(W) G =
isin
% = (5)1’2Wlbt,3
The signal in the interaction of the magnetization operator (A,i)OZTr[p*fi,i]
representation
is given by the expectation
value (18)
where i = a, b. RESULTS
The status of the spin system at various stages of the pulse sequence can be summarized as follows. The first pulse with frequency 2~0~ is a (x/2), pulse and hence we can set & = 7c/2 for this case. We can then obtain B ( twl) using eqn. (14). Immediately following the second pulse of width “&” and frequency WQ we have from eqns. (15)) (17) and (18) and using the orthogonality property of the tensor operators sin(t2)
(19)
318
(20) These results indicate that, when the flip angle of the second pulse of frequency mQ is different from (n) then we get free induction signals at both mQ and 2mQ, i.e. a and b channels, If we set & = n, then from eqns. (19) and (20) it is clear that there would be no signal in either the a or b channel. The density matrix eqn. (15) at this stage contains only the terms corresponding to quadrupolar and hexadecapole orders (Tg and Ti ) and Ti (s) , T$ (s) multipolar states which carry information about double quantum coherence. Let us now consider the situation following a (n/2) pulse of frequency 2aQ and a second pulse of mQ with a flip angle equal to (n). If we now apply a third (n) pulse of frequency 2mQ without any delay, i.e. tl = 0 (see Fig. 1) then immediately following the pulse we have from eqns. (16)- (18)
(21) and
(22)
= 0
Since the signal which is induced in channel a following the pulse in channel b, is arising entirely from Tz (s) and Ti (s) multipolar states it corresponds to the double quantum coherence signal. This means that the invisible double quantum coherence which has been created by the first two rf pulses in the sequence has been converted into an observable single quantum coherence, in channel a (detection frequency of mQ) by the second pulse in channel b. We have also investigated the influence of resonance off-set on the double quantum coherence, in the particular case where the off-set of rf pulses in channel b is exactly twice to that of rf pulses in channel a. In this case using Tables 1 and 2 and Table 12 of ref. 9 we have obtained for the signal, immediately following the last pulse in channel b (see Fig. 1)) the expressions cos (3A~07~) and (fiXLJ
= 0
(24)
where Am = (mQ--0). It may be noted here that the free precession frequency of the double quantum coherence is equal to three times the off-set, Ao, with respect to the rf
319
carrier frequency, CO,of channel a in this case. If the spin system is allowed to AW during the period r,, then we obtain evolve under -Ti, (6 ) l/2
@LX> =
-$$~[cos(3Awr,
+ Aor,)]
and (fix*)
= 6
(26)
In arriving at eqns. (25) and (26)) again we have made use of Tables 1 and 2 and Table 12 of ref. 9. CONCLUSIONS
Using the tensor should be possible NQR for the spin I by Hatanaka et al.
operator formalism, we have shown that in principle it to create and detect double quantum coherences in pure = 5/2 case using a pulse scheme similar to that employed [ 1,2] in NMR.
REFERENCES
7 8 9 10 11 12 13
H. Hatanaka, T. Terao and T. Hashi, J. Phys. Sot. Jpn., 39 (1975) 835. H. Hatanaka and T. Hashi, J. Phys. Sot. Jpn., 39 (1975) 1139. G. Bodenhausen, Prog. NMR Spectrosc., 14 (1981) 137. D.P. Weitekamp, Adv. Magn. Reson., 11 (1983) 111. G. Drobny, Annu. Rev. Phys. Chem., 26 (1985) 451. R.R. Ernst, G. Bodenhausen and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Clarendon Press, Oxford, 1987. M. Munowitz and A. Pines, Adv. Chem. Phys., 66 (1987) 1. G.J. Bowden and W.D. Hutchison, J. Magn. Reson., 67 (1986) 403. G.J. Bowden, W.D. Hutchison and J. Khachan, J. Magn. I&son., 67 (1986) 415. G.J. Bowden and W.D. Hutchison, J. Magn. Reson., 70 (1986) 361. G.J. Bowden and W.D. Hutchison, J. Magn. Reson., 71 (1987) 61. T.P. Das and E.L. Hahn, Nuclear Quadrupole Resonance Spectroscopy, Academic Press, New York and London, 1958. M. Goldman, Spin Temperature and NMR in Solids, Oxford University Press, Oxford, 1970.