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Pulse shaping and selective excitation. The effect of scalar coupling
JOURNAL
OF MAGNETIC
RESONANCE
79,568-576
(1988)
Pulse Shaping and Selective Excitation. The Effect of Scalar Coupling RENZO
BAZZO
AND
JONATHAN
Department of Biochemistry, University of Oxford, Received
South May
Parks
BOYD
Road,
Oxford
OXI3Qu,
England
4, 1988
Two-dimensional techniques have become invaluable tools for the elucidation of the structure of complex molecules in solution ( 1-4). Such experiments give a general map of the scalar or dipolar interactions among spins. However, the quality of the information can be limited. For example, if both high resolution in thefi dimension and high S/N ratios are required, then the demands on experimental time and instrumental stability might become prohibitive. Most 2D techniques have corresponding 1D or 2D analogs (5IO) which give access to particular regions of the complete 2D matrix. Both S/N and resolution may be improved as a result of restricting the frequency ranges in one or both dimensions. A similar approach for collecting only restricted portions of data will probably become the rule in 3D NMR ( 1 I ), where it is not usually feasible to record the complete spectrum ( 12). All these techniques rely upon the use of “soft” pulses, characterized by a large pulse width and a low RF power. The simplest way of applying a soft pulse is to use a rectangular pulse. Frequency selectivity, however, can be improved by shaping the pulse envelope (5, 13-17). If the average RF power of the pulse is much greater than the spin-spin coupling, then all multiplet components of a given spin are affected uniformly and the pulse is called semiselective (18). In this case the frequency-domain excitation profile and phase response for each pulse shape can be calculated using the Bloch equations. In many cases the frequency selectivity obtainable from semiselective pulses is insufficient. For example, soft pulses where the RF power is lower than the scalar coupling (hereafter called selective pulses) are used in coherence transfer experiments, to extract the multiplets of coupled spins from crowded spectra ( 17). The resulting spectra correspond to cross sections from a COSY experiment. A portion of a 2D COSY spectrum, in which thefi dimension is obtained directly in the frequency domain, can also be reconstructed by applying a selective excitation pulse whose frequency is changed by a fixed increment through a series of 1D spectra ( 19). The same idea has recently been extended to 3D spectroscopy (20). In all these experiments, where selective, rather than semiselective, pulses are used, the concept of frequency selectivity, traditionally referred to the tilting of the magnetization from the z direction to the x-y plane as a function of offset, is no longer sufficient. For selective pulses, the exchange among the different coherence compo0022-2364/88 $3.00 Copyright Q 1988 by Academic F’res, Inc. All rights of reproduction in any form reserved.
568
569
COMMUNICATIONS
nents belonging to a given spin or group of equivalent spins, as a function of offset, must be described. In this communication we show that an exact analytical solution can simply be obtained for arbitrary first-order spin systems, where the evolution due to the scalar coupling is explicitly considered. We also explore the properties of some common shaped selective pulses. Assuming for convenience a first-order two-spin system, one can describe the spin which is selectively irradiated-say spin- 1-by projecting the density operator of the spin system on the subspace spanned by the product operators ( 18) “belonging” to spin- 1. These are 1,x, II,, Ilr, 2IIxL 2IlyL 2IlrIZr. The part of the density operator which is ascribed to spin-2 is assumed to be unaffected by the selective pulse. Therefore, the state of spin-l is described by a sixdimensional vector M which we call the “coherence vector” of spin- 1. For uncoupled spins this vector reduces to the familiar three-dimensional magnetization vector whose motion can be followed using the Bloch equations. The evolution under the influence of a selective pulse can be represented by a 6 X 6 matrix U, which transforms the coherence vector according to M(t)
111
= UM(0).
In evaluating Eq. [I], the convention used is that the inner products of rows of the matrix U with the “column vector” M are taken. A particular shaped pulse can be represented by a set of Nadjacent short rectangular pulses. In the general case the matrix U would then be the result of the product U = fi
Ui,
PI
i=l
where the matrix Ui describes the transformation
during the time At, = ti - ti - r
M( ti) = UiM( ti-1)
131
which is characterized by an RF field with power level Qi. The overall transformation can then be calculated, once the matrix Ui is known as a function of the frequency offset wl, the coupling constant J12, and the power Qi (in rad). The matrix Ui represents the unitary transformation [41
caused by the propagator exp [ -iXi zi
At,], with
= ~11lr + ~212z + Qi(IIx + IZX) + 2~J1211r~2r2
151
where w, and w2 are the frequency offsets and Qi is the RF power applied with phase +x. With the assumption that the selective pulse is not perturbing the coupled partner of the irradiated spin (C4i4 ~02)) the term QiIzx can be dropped from Eq. [ 51. In this approximation the diagonalization of the Hamiltonian can be worked out analytically (this conclusion would hold for any arbitrary first-order spin system). Then,
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COMMUNICATIONS
following a procedure recently used ( 2Z ) , the matrix Ui can be explicitly calculated and represented as
- f' -g+
;I
h: m
f-
gl-rn;
m;;
I+ -m+
m+ n+ _
;’
-“,’
;+
-g3;
-g-h-
-A;
mn-
-g+ h+
Ui =
h- -
(where the common subscript i has been omitted on the right) with + f(pr, D;) and analogously for g’, . . . , n’:
f * = f(P:, D))
/3: = l/2 arctg{ QJ(wr + aJlz)} ; PI = l/2 0:
=
[(WI
arCtg{ +
S?i/(Ul
-
7rJ1*)2+
a;]
aJl2)) 2
l/2.
; )
0: = [(w, - 7~4~)~ + Q:]1’2;
f(&
D) = 1/2[sin2(2P)
g(P, D) = -l/2
+ cos2(2~)cos(Dt)];
cos(2p)sin(Dt);
h(& D) = l/4 sin(4/3)[1
- cos(Dt)];
I(& D) = l/2 cos(Dt); m(& D) = -l/2
sin(2/3)sin(Dt);
n(& D) = 1/2[cos2(2p)
+ sin2(2/3)cos(Dt)]. WI While the calculations are more directly performed in the ordinary rotating frame using Eq. [ 5 1, it is perhaps instructive to consider the Hamiltonian in the tilted frame
0
500
HZ
FIG. 1. Selective coherence transfer experiment (I 7) (at 500 MHz) for the AX spin system of cytosine. A rectangular r/2 soft pulse ( 50 ms) on resonance was applied to spin- 1 (on the left), followed by a hard r/2 pulse. (a) The component IlyIzz is shown after transfer to spin-2 (on the right): I,&; (b) the component 1,,12, is shown after transfer to spin-2: I,&; (c)the 1D spectrum.
RECTANGULAR
I. 0
-1.0
1. 0
0. 5
-0. 5
0. 5
0. 0
0. 0
0. 0
0. 5
-0. 5
1.0
-1.0
-0. s m
b)
-1.0 11x 1.0
t
0. 5
r
1.0
-0.5
0. 5
0. 0
0. 0
-0. 5
0. 5
-0.
I.0
-1.0
-1.0
IlxI2z
,
‘.Or-----l
1 Ilz
-1.0
0. 0
“~“I-----
5
1.0 0. 5 0. 0 -0. 5
Ilz
-1.0
100
0
-100
100
0
-100
100
0
-100
OFFSET/HZ FIG. 2. The components of the coherence vector after a soft rectangular pulse, as a function of offset. The calculations used a nominal flip angle of r/2 on resonance (phase +x) and a pulse width of 50 ms (Q/~~=~Hz).(~)J,~=O;(~)J,~=~.~HZ;(C)J,~=~OHZ. 571
GAUSSIAN
a)
Q
1.0
-I. 0
0.5
-0. 5
0.0 m -0.5
0.5
-1.0 11x
1.0
112
1.0
-1.0
0.s
-0.s
(10
0.0
-0.5
0.5
+ -1.011x
1.0 -1.0
I.0 0.5
Tlb-
0.0 -a5
“1
0.0
-1.0 11x122
-0.5
-1
Y
“_1
I
-+d
1.0 IlyI2z
0.5 0.0 -0.5
1 -i:i 112
0.5 0.0 -0.5
0.0 0.5
‘1/-i
1.0
44
1 -1.0
112122 3
0.S
-‘.Or-----l -0.5
0.5
00
0.0
0.0
-0.5
0.5
0
-100
\/\/’ t!!i
‘.O
-0. 5
1.0 .I 1y -1.0
100
112
-1.0 I.0
-0.s
0.5
0.0
0.0
0.5
* -0.5
1.0 Ily122
. -1.0 112122
100
0
I
-100
100
1 I
0
-100
OFFS- T/HZ FIG. 3. The components of the coherence vector after a soft Gaussian pulse, as a function of offset. The calculations used a nominal hip angle of 7r/2 on resonance (phase +x) and a pulse width of 50 ms. The Gaussian function was truncated at the level corresponding to k3 standard deviations and was approximated by a histogram composed of 48 intervals. (a) J12 = 0; (b) Jr2 = 7.2 Hz; (c) J,r = 90 Hz. 572
HAL F-GAUSSIAN
I. 0
-1.0
1.0
0.5
-0. 5
0. 5
0. 0 -0.5
w
-1.0 1.0
0. 0
0.0
0. 5
-0.5
3 kdL.3rl 0. 5
-0.5
0.0
0. 0
-0.5
0.5
1.0
I. 0 0. 5 0. 0 -0.5
J
1.0
-0.5
0. 5
0. 0
0. 5
11x
IlzI2z
-1.0
4.0
0.0
0. 5
-1. 0 1.0
I. 0 -1. 0
-0.5
11
0. 5
0. 0
0. 0
0. 0
-0. 5
0. 5
-0. 5
m11x122 100
I. 0 0
-100
I12
-1.0 I. 0
-0. 5
-1.0
112
-1.0 1.0
1.0 -1.0
1
-I/-
I1 I22 El
-1. 0
0
100
-100
F-l112122 100
0
-100
OFFSET/HZ FIG. 4. The components of the coherence vector after a soft half-Gaussian The calculations used a nominal flip angle of ?r/2 on resonance (phase +x) The half-Gaussian function was truncated at the level corresponding to 3 approximated by a histogram composed of 24 intervals. (a) Jr2 = 0; (b) Jr2 =
573
pulse, as a function of offset. and a pulse width of 50 ms. standard deviations and was 7.2 Hz; (c) Jr* = 90 Hz.
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COMMUNICATIONS
8c I 0
0
8 c 8 I 0
0
0
t
f
a
575
COMMUNICATIONS
of the effective fields 8 ( see Eq. [ 8 ] of Ref. ( 21) : note that the last term should have a negative sign instead of the positive one, as reported). Assuming a first-order spin system and dropping the linear term 6&, we obtain APi = wleffIlr + 27rJL2[COS BIIJ~~ - sin dlI&r]
[71
with cd,&.= [UT + !2f]“2;
~91 = a&g{
Qi/col}.
When the pulse is selective (Oi < 2a JL2), it is the second part of the coupling term in Eq. [ 71 which is responsible for the excitation. Exactly at resonance ( w1 = 0; 8i = X/ 2), it commutes with the density operator expressed in the tilted frame (M-r cc Ii,) and, therefore, being the dominant term, no resulting excitation is predicted. On the other hand, further analysis shows that, when irradiation is exactly on a line (w, = fa J12), the same term, although small, is the one which provides the excitation. In this preliminary report we focus on the effect of the scalar coupling, neglecting factors such as relaxation, RF inhomogeneity, and B. inhomogeneity, which are expected to make a significant contribution when long irradiation times are used. In Fig. 1 we report some experimental data on the two-spin system of cytosine. For this system the theory predicts maximum and minimum excitation for the components 211xIZr and 211y12z, respectively, when using a a/2, 50 ms rectangular pulse (Q 2: 27rJ12) with phase +x, exactly at resonance (see Fig. 2b). In Figs. 2-4 we report the evolution of the different components of the coherence vector as a function of offset, assuming thermal equilibrium as the initial condition: M (0) = I ir. The effect of different coupling constants and different shaping functions is explored. Note that the high selectivity obtained for the y component of the magnetization using a waveform shaped as a half-Gaussian is retained for both the in-phase and the antiphase terms as reported earlier ( 17). It is clear from the diagram of Figs. 2-4~ that, for selective pulses, the same profile is repeated for values of the offset corresponding to each line of the multiplet, with sign alternation for the antiphase terms. Moreover, in-phase and antiphase terms which have the same phase (for example, I,, and 2IIxI2r) have the same excitation profile (ignoring the sign). When the average RF power is close to the value of the coupling constant (see Figs. 2-4b), the calculated excitation profile appears to be the result of the “overlap” between the portions relative to each line of the multiplet. These results can be rationalized by considering separately the molecules with IZZ = 1 / 2 and IZZ = - 1/ 2 (the selective pulse does not affect spin-2 ) . The Hamiltonian of Eq. [ 5 ] then becomes
FIG. 5. The components of the coherence vector after a soft hyperbolic secant pulse, as a function of offset. The calculations used Jr2 = 7.2 Hz. The hyperbolic secant function was truncated at the level in which the intensity is 0.75% of the maximum and was approximated by a histogram composed of 48 intervals. Different nominal flip angles 01on resonance are considered: (a) LY= 45” (25 ms); (b) (Y = 90 (50ms);(c)a= 135”(75ms);(d)cy= 180’(1OOms);(e)cY=270”(150ms);(f)~=360”(200ms).
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COMMUNICATIONS
and the Bloch equations can be used to calculate the evolution of the operators describing the two separate lines ( 17). The alternative approach followed in this paper, where the coupling interaction is considered explicitly, provides further insight into the generation and the evolution of the different terms which describe the selectively irradiated spin. In Fig. 5 are the results obtained with the hyperbolic secant for different values of the nominal flip angle, assuming an average RF power Q,/27r = 5 Hz, J,2 = 7.2 Hz, and M(0) = I,. Due to the striking properties of this function (25), all the intensity, which is spread among the different components at intermediate steps, is refocused along the longitudinal axis when the flip angle is adjusted to be 27r rad. Selective pulses can be used not only to achieve selective excitation but also as mixing pulses. General procedures such as phase cycling can be used to select particular coherence transfer pathways in pulse sequences containing selective pulses. Therefore, a detailed examination of the transfer functions, assuming any initial condition, should help in designing the most convenient shaping function for a particular purpose. ACKNOWLEDGMENTS We thank
SERC
and ICI for financial
support.
We are grateful
to Iain D. Campbell
for helpful
comments.
REFERENCES 1. J. JEENER, “Ampere International Summer School, Basko Polje, Yugoslavia, 197 1.” 2. W. P. AUE, E. BARTHOLDI, AND R. R. ERNST, J. Chem. Phys. 64,2229 (1976). 3. R. R. ERNST, G. BODENHAUSEN, AND A. WOKAUN, “Principles of Nuclear Magnetic One and Two Dimensions,” Oxford Univ. Press, London, 1987. 4. 5. 6. 7. 8.
K. C. H. H. S.
Resonance
in
WOTHRICH, “NMR of Proteins and Nucleic Acids,” Wiley, New York, 1986. BAUER, R. FREEMAN, T. FRENKIEL, J. KEELER, AND. A. J. SHAKA, J. Mugn. Reson. 58,442 (1984). OXZHIUNAT AND R. FREEMAN, J. Mugn. Reson. 60,164 (1984). KESSLER, H. OSCHKINAT, C. GRIESINGER, AND W. BERMEL, J. Mugn. Reson. 70,106 (1986). SUBRAMANIAN AND A. BAX, J Magn. Reson. 71,325 (1987). J. C. MADSEN, C. GRIESINGER, 0. W. SORENSEN, AND R. R. ERNST, J. Magn. (1987). J. P. WALTHO, AND J. KEELER, J. Magn. Reson. 74,386 (1987). 0. W. SORENSEN, AND R. R. ERNST, J. Magn. Reson. 73,574 (1987). C. GRIESINGER, P. J. KRAULIS, 0. W. SORENSEN, R. R. ERNST, A. M. GRONENBORN,
9. R. BRUSCHWEILER,
Reson. 73,380 10. J. CAVANAGH, Il. C. GRIESINGER, 12.
H. OSCHKINAT,
13.
AND G. M. CLORE, Nature (London) 332,374 L. E. CROOKS, J. HOENNINGER, M. AFVXAWA,
(1988). L. KAUFMAN,
R. MCREE,
J. WATTS,
AND
J. R.
SINGER, SPIE, Recent Future Dev. Med. Imaging II 206, 120 (1979). 14. J. BAUM, R. TYCKO, AND A. PINES, J. Chem. Phys. 79,4643 (1983). 15. S. L. MCCALLAND E. L. HAHN, Phys. Rev. 183,457 (1969). 16. M. S. SILVER, R. I. JOSEPH, AND D. I. HOULT, Phys. Rev. A 31, 2753 (1985). 17. J. FRIEDRICH, S. DAVIES, AND R. FREEMAN, J. Magn. Reson. 75,390 (1987). 18.
0.
SORENSEN,
trosc. 16,
G. W. EICH, M. H. LEVI=, 163 (1983). J. FRIEDRICH, AND R. FREEMAN, J. FRIEDRICH, AND R. FREEMAN,
19. 20.
S. DAVIES, S. DAVIES,
21.
R. BAZZOAND
J. BOYD,
G. BODENHAUSEN,
AND
J. Magn. Reson. 75,540 J. Mugn. Reson. 76,555 J. Magn. Reson. 75,452 (1987).
R. R. ERNST, (1987). (1988).
Prog. NMR Spec-
OF MAGNETIC
RESONANCE
79,568-576
(1988)
Pulse Shaping and Selective Excitation. The Effect of Scalar Coupling RENZO
BAZZO
AND
JONATHAN
Department of Biochemistry, University of Oxford, Received
South May
Parks
BOYD
Road,
Oxford
OXI3Qu,
England
4, 1988
Two-dimensional techniques have become invaluable tools for the elucidation of the structure of complex molecules in solution ( 1-4). Such experiments give a general map of the scalar or dipolar interactions among spins. However, the quality of the information can be limited. For example, if both high resolution in thefi dimension and high S/N ratios are required, then the demands on experimental time and instrumental stability might become prohibitive. Most 2D techniques have corresponding 1D or 2D analogs (5IO) which give access to particular regions of the complete 2D matrix. Both S/N and resolution may be improved as a result of restricting the frequency ranges in one or both dimensions. A similar approach for collecting only restricted portions of data will probably become the rule in 3D NMR ( 1 I ), where it is not usually feasible to record the complete spectrum ( 12). All these techniques rely upon the use of “soft” pulses, characterized by a large pulse width and a low RF power. The simplest way of applying a soft pulse is to use a rectangular pulse. Frequency selectivity, however, can be improved by shaping the pulse envelope (5, 13-17). If the average RF power of the pulse is much greater than the spin-spin coupling, then all multiplet components of a given spin are affected uniformly and the pulse is called semiselective (18). In this case the frequency-domain excitation profile and phase response for each pulse shape can be calculated using the Bloch equations. In many cases the frequency selectivity obtainable from semiselective pulses is insufficient. For example, soft pulses where the RF power is lower than the scalar coupling (hereafter called selective pulses) are used in coherence transfer experiments, to extract the multiplets of coupled spins from crowded spectra ( 17). The resulting spectra correspond to cross sections from a COSY experiment. A portion of a 2D COSY spectrum, in which thefi dimension is obtained directly in the frequency domain, can also be reconstructed by applying a selective excitation pulse whose frequency is changed by a fixed increment through a series of 1D spectra ( 19). The same idea has recently been extended to 3D spectroscopy (20). In all these experiments, where selective, rather than semiselective, pulses are used, the concept of frequency selectivity, traditionally referred to the tilting of the magnetization from the z direction to the x-y plane as a function of offset, is no longer sufficient. For selective pulses, the exchange among the different coherence compo0022-2364/88 $3.00 Copyright Q 1988 by Academic F’res, Inc. All rights of reproduction in any form reserved.
568
569
COMMUNICATIONS
nents belonging to a given spin or group of equivalent spins, as a function of offset, must be described. In this communication we show that an exact analytical solution can simply be obtained for arbitrary first-order spin systems, where the evolution due to the scalar coupling is explicitly considered. We also explore the properties of some common shaped selective pulses. Assuming for convenience a first-order two-spin system, one can describe the spin which is selectively irradiated-say spin- 1-by projecting the density operator of the spin system on the subspace spanned by the product operators ( 18) “belonging” to spin- 1. These are 1,x, II,, Ilr, 2IIxL 2IlyL 2IlrIZr. The part of the density operator which is ascribed to spin-2 is assumed to be unaffected by the selective pulse. Therefore, the state of spin-l is described by a sixdimensional vector M which we call the “coherence vector” of spin- 1. For uncoupled spins this vector reduces to the familiar three-dimensional magnetization vector whose motion can be followed using the Bloch equations. The evolution under the influence of a selective pulse can be represented by a 6 X 6 matrix U, which transforms the coherence vector according to M(t)
111
= UM(0).
In evaluating Eq. [I], the convention used is that the inner products of rows of the matrix U with the “column vector” M are taken. A particular shaped pulse can be represented by a set of Nadjacent short rectangular pulses. In the general case the matrix U would then be the result of the product U = fi
Ui,
PI
i=l
where the matrix Ui describes the transformation
during the time At, = ti - ti - r
M( ti) = UiM( ti-1)
131
which is characterized by an RF field with power level Qi. The overall transformation can then be calculated, once the matrix Ui is known as a function of the frequency offset wl, the coupling constant J12, and the power Qi (in rad). The matrix Ui represents the unitary transformation [41
caused by the propagator exp [ -iXi zi
At,], with
= ~11lr + ~212z + Qi(IIx + IZX) + 2~J1211r~2r2
151
where w, and w2 are the frequency offsets and Qi is the RF power applied with phase +x. With the assumption that the selective pulse is not perturbing the coupled partner of the irradiated spin (C4i4 ~02)) the term QiIzx can be dropped from Eq. [ 51. In this approximation the diagonalization of the Hamiltonian can be worked out analytically (this conclusion would hold for any arbitrary first-order spin system). Then,
570
COMMUNICATIONS
following a procedure recently used ( 2Z ) , the matrix Ui can be explicitly calculated and represented as
- f' -g+
;I
h: m
f-
gl-rn;
m;;
I+ -m+
m+ n+ _
;’
-“,’
;+
-g3;
-g-h-
-A;
mn-
-g+ h+
Ui =
h- -
(where the common subscript i has been omitted on the right) with + f(pr, D;) and analogously for g’, . . . , n’:
f * = f(P:, D))
/3: = l/2 arctg{ QJ(wr + aJlz)} ; PI = l/2 0:
=
[(WI
arCtg{ +
S?i/(Ul
-
7rJ1*)2+
a;]
aJl2)) 2
l/2.
; )
0: = [(w, - 7~4~)~ + Q:]1’2;
f(&
D) = 1/2[sin2(2P)
g(P, D) = -l/2
+ cos2(2~)cos(Dt)];
cos(2p)sin(Dt);
h(& D) = l/4 sin(4/3)[1
- cos(Dt)];
I(& D) = l/2 cos(Dt); m(& D) = -l/2
sin(2/3)sin(Dt);
n(& D) = 1/2[cos2(2p)
+ sin2(2/3)cos(Dt)]. WI While the calculations are more directly performed in the ordinary rotating frame using Eq. [ 5 1, it is perhaps instructive to consider the Hamiltonian in the tilted frame
0
500
HZ
FIG. 1. Selective coherence transfer experiment (I 7) (at 500 MHz) for the AX spin system of cytosine. A rectangular r/2 soft pulse ( 50 ms) on resonance was applied to spin- 1 (on the left), followed by a hard r/2 pulse. (a) The component IlyIzz is shown after transfer to spin-2 (on the right): I,&; (b) the component 1,,12, is shown after transfer to spin-2: I,&; (c)the 1D spectrum.
RECTANGULAR
I. 0
-1.0
1. 0
0. 5
-0. 5
0. 5
0. 0
0. 0
0. 0
0. 5
-0. 5
1.0
-1.0
-0. s m
b)
-1.0 11x 1.0
t
0. 5
r
1.0
-0.5
0. 5
0. 0
0. 0
-0. 5
0. 5
-0.
I.0
-1.0
-1.0
IlxI2z
,
‘.Or-----l
1 Ilz
-1.0
0. 0
“~“I-----
5
1.0 0. 5 0. 0 -0. 5
Ilz
-1.0
100
0
-100
100
0
-100
100
0
-100
OFFSET/HZ FIG. 2. The components of the coherence vector after a soft rectangular pulse, as a function of offset. The calculations used a nominal flip angle of r/2 on resonance (phase +x) and a pulse width of 50 ms (Q/~~=~Hz).(~)J,~=O;(~)J,~=~.~HZ;(C)J,~=~OHZ. 571
GAUSSIAN
a)
Q
1.0
-I. 0
0.5
-0. 5
0.0 m -0.5
0.5
-1.0 11x
1.0
112
1.0
-1.0
0.s
-0.s
(10
0.0
-0.5
0.5
+ -1.011x
1.0 -1.0
I.0 0.5
Tlb-
0.0 -a5
“1
0.0
-1.0 11x122
-0.5
-1
Y
“_1
I
-+d
1.0 IlyI2z
0.5 0.0 -0.5
1 -i:i 112
0.5 0.0 -0.5
0.0 0.5
‘1/-i
1.0
44
1 -1.0
112122 3
0.S
-‘.Or-----l -0.5
0.5
00
0.0
0.0
-0.5
0.5
0
-100
\/\/’ t!!i
‘.O
-0. 5
1.0 .I 1y -1.0
100
112
-1.0 I.0
-0.s
0.5
0.0
0.0
0.5
* -0.5
1.0 Ily122
. -1.0 112122
100
0
I
-100
100
1 I
0
-100
OFFS- T/HZ FIG. 3. The components of the coherence vector after a soft Gaussian pulse, as a function of offset. The calculations used a nominal hip angle of 7r/2 on resonance (phase +x) and a pulse width of 50 ms. The Gaussian function was truncated at the level corresponding to k3 standard deviations and was approximated by a histogram composed of 48 intervals. (a) J12 = 0; (b) Jr2 = 7.2 Hz; (c) J,r = 90 Hz. 572
HAL F-GAUSSIAN
I. 0
-1.0
1.0
0.5
-0. 5
0. 5
0. 0 -0.5
w
-1.0 1.0
0. 0
0.0
0. 5
-0.5
3 kdL.3rl 0. 5
-0.5
0.0
0. 0
-0.5
0.5
1.0
I. 0 0. 5 0. 0 -0.5
J
1.0
-0.5
0. 5
0. 0
0. 5
11x
IlzI2z
-1.0
4.0
0.0
0. 5
-1. 0 1.0
I. 0 -1. 0
-0.5
11
0. 5
0. 0
0. 0
0. 0
-0. 5
0. 5
-0. 5
m11x122 100
I. 0 0
-100
I12
-1.0 I. 0
-0. 5
-1.0
112
-1.0 1.0
1.0 -1.0
1
-I/-
I1 I22 El
-1. 0
0
100
-100
F-l112122 100
0
-100
OFFSET/HZ FIG. 4. The components of the coherence vector after a soft half-Gaussian The calculations used a nominal flip angle of ?r/2 on resonance (phase +x) The half-Gaussian function was truncated at the level corresponding to 3 approximated by a histogram composed of 24 intervals. (a) Jr2 = 0; (b) Jr2 =
573
pulse, as a function of offset. and a pulse width of 50 ms. standard deviations and was 7.2 Hz; (c) Jr* = 90 Hz.
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8c I 0
0
8 c 8 I 0
0
0
t
f
a
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of the effective fields 8 ( see Eq. [ 8 ] of Ref. ( 21) : note that the last term should have a negative sign instead of the positive one, as reported). Assuming a first-order spin system and dropping the linear term 6&, we obtain APi = wleffIlr + 27rJL2[COS BIIJ~~ - sin dlI&r]
[71
with cd,&.= [UT + !2f]“2;
~91 = a&g{
Qi/col}.
When the pulse is selective (Oi < 2a JL2), it is the second part of the coupling term in Eq. [ 71 which is responsible for the excitation. Exactly at resonance ( w1 = 0; 8i = X/ 2), it commutes with the density operator expressed in the tilted frame (M-r cc Ii,) and, therefore, being the dominant term, no resulting excitation is predicted. On the other hand, further analysis shows that, when irradiation is exactly on a line (w, = fa J12), the same term, although small, is the one which provides the excitation. In this preliminary report we focus on the effect of the scalar coupling, neglecting factors such as relaxation, RF inhomogeneity, and B. inhomogeneity, which are expected to make a significant contribution when long irradiation times are used. In Fig. 1 we report some experimental data on the two-spin system of cytosine. For this system the theory predicts maximum and minimum excitation for the components 211xIZr and 211y12z, respectively, when using a a/2, 50 ms rectangular pulse (Q 2: 27rJ12) with phase +x, exactly at resonance (see Fig. 2b). In Figs. 2-4 we report the evolution of the different components of the coherence vector as a function of offset, assuming thermal equilibrium as the initial condition: M (0) = I ir. The effect of different coupling constants and different shaping functions is explored. Note that the high selectivity obtained for the y component of the magnetization using a waveform shaped as a half-Gaussian is retained for both the in-phase and the antiphase terms as reported earlier ( 17). It is clear from the diagram of Figs. 2-4~ that, for selective pulses, the same profile is repeated for values of the offset corresponding to each line of the multiplet, with sign alternation for the antiphase terms. Moreover, in-phase and antiphase terms which have the same phase (for example, I,, and 2IIxI2r) have the same excitation profile (ignoring the sign). When the average RF power is close to the value of the coupling constant (see Figs. 2-4b), the calculated excitation profile appears to be the result of the “overlap” between the portions relative to each line of the multiplet. These results can be rationalized by considering separately the molecules with IZZ = 1 / 2 and IZZ = - 1/ 2 (the selective pulse does not affect spin-2 ) . The Hamiltonian of Eq. [ 5 ] then becomes
FIG. 5. The components of the coherence vector after a soft hyperbolic secant pulse, as a function of offset. The calculations used Jr2 = 7.2 Hz. The hyperbolic secant function was truncated at the level in which the intensity is 0.75% of the maximum and was approximated by a histogram composed of 48 intervals. Different nominal flip angles 01on resonance are considered: (a) LY= 45” (25 ms); (b) (Y = 90 (50ms);(c)a= 135”(75ms);(d)cy= 180’(1OOms);(e)cY=270”(150ms);(f)~=360”(200ms).
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and the Bloch equations can be used to calculate the evolution of the operators describing the two separate lines ( 17). The alternative approach followed in this paper, where the coupling interaction is considered explicitly, provides further insight into the generation and the evolution of the different terms which describe the selectively irradiated spin. In Fig. 5 are the results obtained with the hyperbolic secant for different values of the nominal flip angle, assuming an average RF power Q,/27r = 5 Hz, J,2 = 7.2 Hz, and M(0) = I,. Due to the striking properties of this function (25), all the intensity, which is spread among the different components at intermediate steps, is refocused along the longitudinal axis when the flip angle is adjusted to be 27r rad. Selective pulses can be used not only to achieve selective excitation but also as mixing pulses. General procedures such as phase cycling can be used to select particular coherence transfer pathways in pulse sequences containing selective pulses. Therefore, a detailed examination of the transfer functions, assuming any initial condition, should help in designing the most convenient shaping function for a particular purpose. ACKNOWLEDGMENTS We thank
SERC
and ICI for financial
support.
We are grateful
to Iain D. Campbell
for helpful
comments.
REFERENCES 1. J. JEENER, “Ampere International Summer School, Basko Polje, Yugoslavia, 197 1.” 2. W. P. AUE, E. BARTHOLDI, AND R. R. ERNST, J. Chem. Phys. 64,2229 (1976). 3. R. R. ERNST, G. BODENHAUSEN, AND A. WOKAUN, “Principles of Nuclear Magnetic One and Two Dimensions,” Oxford Univ. Press, London, 1987. 4. 5. 6. 7. 8.
K. C. H. H. S.
Resonance
in
WOTHRICH, “NMR of Proteins and Nucleic Acids,” Wiley, New York, 1986. BAUER, R. FREEMAN, T. FRENKIEL, J. KEELER, AND. A. J. SHAKA, J. Mugn. Reson. 58,442 (1984). OXZHIUNAT AND R. FREEMAN, J. Mugn. Reson. 60,164 (1984). KESSLER, H. OSCHKINAT, C. GRIESINGER, AND W. BERMEL, J. Mugn. Reson. 70,106 (1986). SUBRAMANIAN AND A. BAX, J Magn. Reson. 71,325 (1987). J. C. MADSEN, C. GRIESINGER, 0. W. SORENSEN, AND R. R. ERNST, J. Magn. (1987). J. P. WALTHO, AND J. KEELER, J. Magn. Reson. 74,386 (1987). 0. W. SORENSEN, AND R. R. ERNST, J. Magn. Reson. 73,574 (1987). C. GRIESINGER, P. J. KRAULIS, 0. W. SORENSEN, R. R. ERNST, A. M. GRONENBORN,
9. R. BRUSCHWEILER,
Reson. 73,380 10. J. CAVANAGH, Il. C. GRIESINGER, 12.
H. OSCHKINAT,
13.
AND G. M. CLORE, Nature (London) 332,374 L. E. CROOKS, J. HOENNINGER, M. AFVXAWA,
(1988). L. KAUFMAN,
R. MCREE,
J. WATTS,
AND
J. R.
SINGER, SPIE, Recent Future Dev. Med. Imaging II 206, 120 (1979). 14. J. BAUM, R. TYCKO, AND A. PINES, J. Chem. Phys. 79,4643 (1983). 15. S. L. MCCALLAND E. L. HAHN, Phys. Rev. 183,457 (1969). 16. M. S. SILVER, R. I. JOSEPH, AND D. I. HOULT, Phys. Rev. A 31, 2753 (1985). 17. J. FRIEDRICH, S. DAVIES, AND R. FREEMAN, J. Magn. Reson. 75,390 (1987). 18.
0.
SORENSEN,
trosc. 16,
G. W. EICH, M. H. LEVI=, 163 (1983). J. FRIEDRICH, AND R. FREEMAN, J. FRIEDRICH, AND R. FREEMAN,
19. 20.
S. DAVIES, S. DAVIES,
21.
R. BAZZOAND
J. BOYD,
G. BODENHAUSEN,
AND
J. Magn. Reson. 75,540 J. Mugn. Reson. 76,555 J. Magn. Reson. 75,452 (1987).
R. R. ERNST, (1987). (1988).
Prog. NMR Spec-