shaped radiofrequency pulses in high resolution nmr

ELSEVIER Journal of Progress in Nuclear Magnetic Resonance Spectroscopy 32 (1998) 59-106

Shaped radiofrequency pulses in high resolution NMR

Ray Freeman

Depar@nent of Chemistry, Cambridge Universiry, L.ensjeld Road, Cambridge CB.2 IEW, UK

Received 1 September 1997

1. Introduction ..................................... 60

2. Primitive pulses. ................................... 61

2.1. The linear approximation .............................. 61

2.2. Gaussian pulses .................................. 62

2.3. Sine pulses .................................... 63

2.4. Hermite pulses .................................. 63

3. Phase gradients .................................... 63

3.1. Half-Gaussian pulses ................................ 64

3.2. Spin pinging ................................... 65

4. Design strategies ................................... 67

4.1. Definition of the pulse shape ............................. 68

4.2. Gaussian cascades ................................. 68

4.3. Simulated annealing ................................ 68

4.4. Artificial neural networks .............................. 69

4.5. Genetic algorithms ................................. 7 1

4.6. Taboo search ................................... 72

4.7. Linearization of the Bloch equations. ......................... 72

5. Pure-phase pulses ................................... 73

5.1. BURP pulses ................................... 73

5.2. Relaxation effects ................................. 74

5.3. Initial conditions. ................................. 75

5.4. Close encounters. ................................. 77

6. Implementation .................................... 79

6.1. Soft pulses .................................... 79

6.2. The DANTE sequence ............................... 79

6.3. Simultaneous soft pulses .............................. 81

6.4. Pulses at two coupled sites. ............................. 82

6.5. Multisite excitation. ................................ 83

6.6. Binomial pulses .................................. 83

6.7. Polychromatic pulses ................................ 83

0022-2860/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved

PII SOO79-6565(97)00024- 1

60 R. Freeman/Progress in Nuclear Magnetic Resonance Spectroscopy 32 (1998) 59-106

6.8. Hadamard phase encoding ............. 6.9. Template excitation ............... 6.10. Excitation sculpting. ..............

7. Decoupling ....................

7.1. Adiabatic pulses ................. 7.2. Band-selective decoupling .............

8. Applications of soft pulses .............. 8.1. Band-selective excitation ............. 8.2. Multiplet-selective excitation. ........... 8.3. Line-selective excitation. ............. 8.4 Water suppression ................ 8.5. Resolution enhancement. ............. 8.6. Homonuclear equivalents of heteronuclear experiments 8.7. Selective correlation experiments .......... 8.8. Hartmann-Hahn coherence transfer ......... 8.9. Relaxation in Multispin systems .......... 8.10. Measurement of unresolved splittings. .......

9. Discussion .................... Appendix .................... References ....................

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Keywords: Radiofrequency pulses; Fourier transformation; High resolution NMR; Selective excitation

1. Introduction

Radiofrequency pulse excitation first came into widespread use in the early 1970s when Fourier transform methods began to revolutionize high resolution NMR methodology. At that time the aim was to excite the entire range of chemical shifts of the chosen spin species, so the pulses were intense and of short duration, usually tens of microseconds. For the sake of simplicity the intended pulse envelope was invariably rectangular, though instrumental shortcomings probably distorted this ‘ideal’ shape in practice. We would now call these ‘hard’ radiofrequency pulses, defined such that yB,/2~ > AF, where B, is the radiofrequency field intensity and AF is the entire range of chemical shifts.

When some degree of selectivity is required in the frequency domain, the radiofrequency pulses are reduced in intensity with a corresponding increase in duration (tr) to maintain the same flip angle, (Y = yB,t, radians. In general, a range of frequencies of the order of yB,/2?r (Hz) is excited by these ‘soft’

radiofrequency pulses. Outside this frequency range the effective field (the resultant of B, and the resonance offset AB) is tilted so that it lies close to the + Z axis of the rotating frame, and has very little effect on spins initially aligned along % Z. Depending on the intensity B,, soft pulses may be described as band-selective, covering a restricted range of chemical shifts, multiplet-selective, or line-selective. In the extreme case, where yB,/2r is small in comparison with the instrumental linewidth, it is possible to burn a hole in the experimental line profile by selective saturation [ 11.

Alexander [2] was probably the first to report a soft pulse experiment; it was designed to minimize the proton signal of water by executing a 360” rotation at that particular frequency offset while still exciting the solute resonance. Soft pulse experiments were employed in a variety of frequency-selective spin- spin and spin-lattice relaxation studies in liquids [3] but were soon overshadowed by hard-pulse Fourier transform methods [4] which had the important advantage that they monitored all the resonances in

R. Freeman/Progress in Nuclear Magnetic Resonance Spectroscopy 32 (1998) 59-106 61

the spectrum at the same time. Not until much later

was it appreciated that soft pulse experiments had much to offer in simplifying high resolution spectra,

restricting the size and duration of many two- dimensional experiments, or reducing the dimension- ality of multi-dimensional techniques [5,6].

More generally, the solvent suppression problem nearly proved to be the Achilles’ heel of Fourier transform NMR. The protons in water (at 110 moles dm-‘)

severely tax the dynamic range of available analog-to- digital converters when weak solute signals are under investigation, a common situation in biochemical studies. This results in digitization errors which transform into digitization noise in the spectrum and seriously limit the attainable sensitivity of the Fourier transform technique. Many present-day water suppression schemes [7] rely on some form of selective excitation or selective presaturation.

2. Primitive pulses

With the much wider utilization of soft pulse methods in the 1980s came the realization that a rectangular pulse envelope was not the ideal shape-it introduces sine-like sidelobes on each side of the main excitation [8]. Although these get progressively weaker with resonance offset, they extend to a surprisingly large distance and can easily

Fig. 1, Magnetization trajectories calculated for a 1 ms soft rectan-

gular pulse of nominal flip angle 90” and radiofrequency intensity

given by yB ,/2x = 250 Hz. The on-resonance trajectory is the arc

from + Z to + Y, giving a pure absorption signal. As the resonance

offset is increased from zero to 900 Hz in 100 Hz increments, the

effective field is progressively tilted towards the + Z axis. At the

hugest offset (900 Hz) the magnetization vector follows a circular

path, almost reaching the + Z axis; this condition is close to the

second zero-crossing on the excitation profile.

excite unwanted resonance responses. We can appreciate how this comes about by using the Bloch equations to calculate magnetization trajectories during a rectangular soft pulse for various offsets (A& from exact resonance (Fig. 1). At moderate offsets (U/B1 < l), the magnetization moves from the +.Z axis towards the XY plane, deviating from the ideal trajectory ( + Z- + Y) to give a small phase error in the detected signal. At larger offsets (dB/Br--4) the trajectory becomes an arc of a circle that touches the +Z axis. When the ratio ABIB, reaches ,115, J63, or

,‘143 the trajectory is a complete circle, corresponding to the second, fourth and sixth zero-crossing points of the frequency-domain excitation profile (Fig. 2). Where the magnetization path is an incomplete circle, a sidelobe response is excited. Only at very large resonance offsets (ABIB > 50) does the magnetization stay very close to the +Z axis so that we can

neglect the excitation altogether.

2.1. The linear approximation

To the extent that the NMR response is a linear function of flip angle (a! < 1 radian) the frequency- domain excitation profile is the Fourier transform of the radiofrequency pulse envelope. This is related to an important general rule known as the convolution theorem [9] which states that the Fourier transform of

Fig. 2. The frequency-domain excitation profile of a 1 ms soft

rectangular pulse calculated assuming a linear response (full

curve) and by means of the Blocb equations (dotted curve). The

deviations from linearity are greatest near resonance but negligible

at appreciable offsets.


the product of two functions g(t) and h(t) is equivalent to the convolution of their Fourier transforms G(f)*H(f). If an infinitely long radiofrequency wave g(t) is multiplied by a shaping function h(t), the frequency-domain spectrum is a delta function

G(f) convoluted with the Fourier transform H(f), essentially just the profile H(f).

For a rectangular 90” radiofrequency pulse, the response close to resonance is appreciably non-linear, as can be seen by comparing the excitation profile calculated through the Bloch equations with that

obtained by Fourier transformation (Fig. 2). Note however, that at larger resonance offsets the two

curves merge and become indistinguishable. The true test of linearity is that the magnetization trajectories never deviate too far from the +Z axis of the

rotating frame. There is an important consequence of the linear

approximation. It becomes evident in magnetic resonance imaging when a selective radiofrequency pulse is used in an applied magnetic field gradient, with a view to exciting a particular slice from an extended sample. In this situation, the nuclear spins represent a continuum of resonance frequencies defined by the field gradient. If the NMR response is strictly linear the frequency-domain excitation pattern is the exact Fourier transform of the pulse envelope; in turn the time-domain NMR signal is the exact Fourier transform of the excitation pattern. Consequently the time-

domain NMR signal follows the pulse shape exactly and is zero at the end of the pulse, and hence undetectable. Signals from different isochromats mutually interfere and give a zero resultant [lo- 121. This seems surprising, because we are accustomed to experiments where the response is not strictly linear, and of course many selective excitation experiments have been carried out in applied field gradients with 90” soft pulses. The solution is to form a spin echo

[ 131 so that the NMR signal is reconstituted at a time when the soft pulse was extinguished. This can be achieved by reversal of the sense of the applied field gradient.

2.2. Gaussian pulses

Under the linear approximation, because the transform of a rectangle is a sin& (sine) function, we see that the excitation sidelobes are a result of the abrupt

leading and trailing edges of the pulse envelope, suggesting that a ‘rounded-off’ shape would be more suitable. There is a Fourier transform theorem [9] that states that if a function S(t) can be differentiated k times before its derivative exhibits a discontinuity, then its Fourier transform S(f) falls off inversely as the kth power of frequency in the tails.

In this respect a Gaussian pulse envelope

S(t) = exp[ - a( t - to)12 (1)

should be ideal, for a Gaussian function can be

repeatedly differentiated without ever introducing a discontinuity. Soft Gaussian pulses [8,14,15] are therefore a good choice for selective excitation or selective spin inversion. The only slight complication

is that the pulse envelope must be truncated somewhere in the tails because an infinitely long pulse is

not a practical proposition. A step in amplitude of two or three percent does not distort the excitation pattern significantly (it introduces a very mild convolution with a sine function), and for the purposes of spin inversion (CY = ?r radians) even more severe truncation steps are readily accommodated.

The improvement afforded by Gaussian pulse shaping can be appreciated from Fig. 3, where absolute-value excitation profiles were calculated for

a one-millisecond rectangular pulse (full curve) and a one-millisecond Gaussian pulse (dotted curve). Although the central excitation peak from the

Fig. 3. Absolute-value mode excitation profiles calculated for a 1 ms

soft rectangular pulse (full curve with sidelobes) and a soft Gaussian

pulse of the same duration (dashed curve). The Gaussian pulse

envelope was truncated at the 2% level. Note the virtual absence

of sidelobes and much lower excitation level for the Gaussian pulse

at large offsets.


rectangular pulse is narrow, the sidelobes are very

prominent and extend to large offsets. At offsets

beyond the first sidelobes of the rectangular pulse, the Gaussian pulse gives far lower excitation levels. Absolute-value signals were considered here (rather than the pure absorption-mode) to give a more realis- tic measure of the danger of unwanted off-resonance excitation. Note the almost imperceptible oscillatory component in the tails of the Gaussian excitation profile; it arises because the Gaussian pulse envelope was truncated at the 2% points.

One might wonder how the Gaussian pulse achieves almost negligible excitation off-resonance

whereas the rectangular pulse of the same duration and flip angle gives an appreciable response. The magnetization trajectories give a clue to this apparent paradox. At appreciable offsets, the Gaussian pulse does cause quite large excursions, but the magnetization vector always returns to the + Z axis, so that the detected signal is vanishingly small. The movement starts very slowly, reaching a maximum speed at the middle of the pulse, and then decelerates again, tracing a trajectory in the form of a tear-drop [8]. In contrast, the rectangular pulse elicits a trajectory that is an arc of a circle, and in general this leaves an

appreciable transverse component of magnetization at the end of the pulse.

2.3. Sine pulses

In magnetic resonance imaging, soft-pulse excitation is mainly employed for ‘slice selection’-the excitation of a particular laminar section of the sample normal to an applied magnetic field gradient, with negligible perturbation of the rest of the sample.

This is usually the first step in obtaining a spin density map from nuclei contained between two close parallel planes. The ideal frequency-domain excitation profile would therefore be a rectangle (sometimes called the ‘top-hat’ shape) with abrupt transitions from full to negligible excitation at the edges and uniform excitation in between. In the linear approximation, where the Fourier transform relationship is justified, the required pulse shape would be a sine function. But because a true sine function has extensive (oscillatory) wings, and because the pulse duration must be minimized, it is usual to truncate the sine function at one of the zero-crossing points (often the

second or third). For this reason the frequency-domain

profile is not ideal. A more serious problem is that the assumption of a linear response is not justified; indeed

as described previously, we rely on the non-linearity if we are to observe any signal at all [lo- 121.

2.4. Hermite pulses

An interesting pulse shape related to the simple Gaussian is the Hermite pulse [16], the product of a Gaussian and an even-order polynomial. For example, if just the second-order term is included,

[ 1 - 0.8(t - to)*] exp[ - a(t - to)*] (2)

the absolute-value mode excitation near resonance is reduced in amplitude giving a ‘bull-nose’ shape to the excitation profile. This is useful for multiplet- selective experiments provided that the frequency- dependent phase shift can be corrected.

3. Phase gradients

Except when applied at exact resonance, soft radiofrequency pulses inevitably operate with the effective radiofrequency field tilted away from the +X axis towards the + 2 axis of the rotating frame. We define

the tilt angle 0 through the expression:

tan 8 = ABIB, (3)

Rotation about a tilted axis (and an increased effective field) implies a phase shift of the response with respect to the phase of a signal excited at exact resonance, as can be appreciated from Fig. 1. If we adjust for pure absorption at exact resonance, an increasing admixture of dispersion-mode appears as a function of resonance offset. In many situations this phase shift is

an approximately linear function of offset. Where this is strictly true, we can think of the phase gradient as arising through ‘free precession’ of magnetization vectors during the pulse, as though originating from a ‘focus’ located in time somewhere near the middle of the soft pulse. This is the case for the linear response regime of a time-symmetric, amplitude- modulated pulse such as a Gaussian, Hermite or sine

pulse. For simple applications, such as the direct

excitation of a restricted region of a high resolution


Fig. 4. Magnetization trajectories calculated for a full Gaussian pulse (a) and a half-Gaussian pulse (b) of the same duration. Note that after the

half-Gaussian pulse the magnetization always remains in the hemisphere where Y is positive, so there are no sidelobes in the frequency-domain excitation pattern.

spectrum, an approximately linear phase gradient poses no problem for it can be ‘corrected’ by the usual phase adjustment procedure, provided that the resonance lines are narrow enough that there is no significant phase shift across the linewidth. For more

complex pulse manipulations, where software ‘correction’ of the phase gradient is not feasible, a phase gradient can be a serious problem. It may of course be eliminated by a refocusing technique, using a 180” pulse and a suitable delay for echo formation, but this can introduce complications from echo modulation if there are homonuclear couplings, and loss of signal if spin-spin relaxation is appreciable.

3.1. Half-Gaussian pulses

The Fourier transform of a Gaussian time-domain function is another Gaussian in the frequency domain. In the linear approximation this applies to excitation by a Gaussian radiofrequency pulse provided that we monitor the absolute-value mode response (2 + v2)m, where u represents dispersion and v represents absorption. However, if we plot only the absorption-mode profile in the frequency domain and reject the dispersion contribution, we find that the sense of this signal oscillates, with a positive

lobe near exact resonance flanked by negative lobes further away, and weak positive lobes at even larger offsets. This is not a convenient profile for frequency- selective excitation.

A far more useful excitation profile is generated by half-Gaussian pulse [17], that is to say, a shape

defined by:

S(t) = exp[ - a(t - Q2] for t I to

S(t)=Ofort>tc (4)

Signal acquisition starts at to. When we compare the magnetization trajectories after a full Gaussian pulse (Fig. 4a) with those after a half-Gaussian pulse (Fig. 4b) we see that the latter never stray into the hemisphere where the signal is negative. The frequency-domain profile for the absorption mode

1 Fig. 5. The frequency domain excitation profile for a half-Gaussian

pulse, showing the absorption mode (full curve) and the much

broader dispersion mode (dashed curve). In some line-selective

coherence transfer experiments only the absorption mode is

important.


has only a single narrow positive peak (Fig. 5). How-

ever, the dispersion mode profile is much broader and has very extensive tails. We may rationalize this dif-

ference by imagining the half-Gaussian shape to be

made up of equal parts of a symmetrical full Gaussian curve and its antisymmetric counterpart:

S(t) = + exp[ - a(t - to)‘] for t % to

S(t) = - exp[ - a(t - t0)2] for t > to (3

The symmetrical full Gaussian has no discontinuity and excites the (symmetric) absorption mode response, while the antisymmetric full Gaussian has a large step at t = to and excites the very broad (antisymmetric) dispersion response.

Excitation by a half-Gaussian can be very useful in

soft-pulse coherence transfer experiments where the absorption-mode component is involved in the transfer, whereas the (broad) dispersion-mode response remains at the excitation site [17]. Line-selective excitation can be used as an alternative method of high-definition correlation spectroscopy where we can ‘zoom in’ on a particularly interesting region without having to examine the entire spectrum [17].

b

a

This technique has been called pseudo-correlation

spectroscopy ($-COSY) [18]. The undesirable broad dispersion-mode component

after a half-Gaussian pulse can be eliminated in a two-

scan difference-mode experiment if a ‘purge’ pulse is applied. This is a hard 90” pulse alternated in phase along the ?Y axes so that the X components of magnetization (dispersion) are returned to the +Z axis while the Y components (absorption) are conserved [18.19].

3.2. Spin pinging

One way to circumvent the problem of the phase gradient is to design a scheme that suppresses the dispersion-mode contributions. An ingenious technique introduced by Ping Xu [20] accomplishes this

and at the same time generates more favourable frequency-domain excitation profiles; this has been called ‘spin pinging’. Instead of applying the soft radiofrequency pulse to spins at Boltzmann equilibrium (magnetization aligned along +Z), spin

pinging starts with a hard 90” pulse that places the magnetization vector along +Y. This is followed

Y

Fig. 6. Effect of a soft pulse about the + X axis on magnetization initially aligned along + Y. (a) Rotation about the tilted effective field B,rr

through an angle CL (b) Definition of the tilted plane YZ’ in which the rotation occurs. (c) Rotation through an angle (Y in the YZ’ plane leaving a

projection Msin o on the - Z’ axis. (d) Viewed in the XZ plane. (e) Projection onto the + X axis. The dispersion component is Msin (Y sin 0.


immediately by a soft 180” pulse. Two scans are taken and the signals acquired in the difference mode. In the first scan, the soft 180” pulse is applied about the +X axis, in the second scan it is applied about the +Y axis.

We can appreciate how the dispersion component is suppressed by comparing the motion of the magnetization vectors in the two different scans. Consider

the motion of a typical magnetization vector M initially aligned along +Y and at a general resonance offset AB, which experiences a rectangular soft pulse of amplitude Bi and on-resonance flip angle oo. The

actual flip angle is:

Q! = (Y~B,~~IB,

where

(6)

B& = B; + AB* (7)

The effective field is tilted through an angle 0 defined by Eq. (3). Note that the tilt angle 8 and the effective

field B,ff are the same for the 180”(X) pulse as for the 180”( I’) pulse.

Consider first the effect of a soft pulse of nominal flip angle CY~, applied along an effective field in the XZ plane (Fig. 6). It is convenient to transform into a new co-ordinate system that is tilted through 0 in the XZ

plane so that Z - Z’ making Befl normal to the YZ

plane (Fig. 6b). The magnetization component M perpendicular to the effective field is rotated about Beff through an angle 1y, giving a projection onto the -Z’ axis equal to Msina! (Fig. 6~). When this is projected onto the XY plane (Fig. 6e) it leaves a detectable dispersion-mode component:

Disp = M sin f3 sin Q! (8)

In the second scan the effective field is in the ZY plane, tilted at an angle 8 with respect to the Y axis (Fig. 7a). After transformation into a new frame where

the XZ’ plane is normal to the effective field (Fig. 7b), we have a Z’ component - M sin 8 perpendicular to Beff. This is rotated through an angle (Y in the XZ’ plane (Fig. 7d), leaving a detectable dispersion component again given by Eq. (8). Consequently, the dispersion contribution vanishes in the difference mode, whatever the nominal flip angle of the soft

pulse. The absorption-mode response for a soft pulse

about the + X axis is given by:

(Abs), = M cos a (9)

while for a soft pulse applied about the +Y axis the

a

7 bz, - \ 1 \ I B eff

\ I \ \,: 8 M CT3 -__-__ Y

I\ I ’ ’ ‘\ I \ I \ I .

d 2

Fig. 7. Effect of a soft pulse about the + Y axis on magnetization initially aligned along + Y. (a) Rotation through an angle Q about an effective

field Bea tilted through 0 in the YZ plane. (b) Transformation to a new frame where the XZ’ plane is normal to B,R. (c) F’rojection onto Z’ axis.

(d)Rotation through an angle 01 in the XZ’ plane. (e) Projection onto the + X axis to give a dispersion component Msin OL sin 0.

R. Freeman/Progress in Nuclear Magnetic Resonance Spectroscopy 32 (I 998) 59-106 67

absorption response is

(Abs)y = M(cos2 0 + sin2 0 cos au) (10)

This expression is derived in the Appendix. It gives a difference-mode absorption signal:

(Ah), = - M cos2 c9( 1 - cos cr) (11)

This response (along the - Y axis) is clearly a maximum at exact resonance where 13 = o”, and for a soft pulse flip angle of 180”, when cos u = - 1. At large offsets the difference signal falls to zero as 8 - 90”.

The expression for the difference-mode frequency- domain excitation profile can be written:

(12)

This resembles a sine-squared function and it has the important property that the oscillatory sidelobes are far less prominent than those of a sine function (Fig. 8a). The general shape of the excitation profile is only mildly dependent on the nominal flip angle (Ye, rendering it rather tolerant of miscalibration of the flip angle, or of spatial inhomogeneity of the radiofrequency field B1. The excitation profile can be further improved by shaping the soft pulse according

a .

Fig. 8. Excitation profiles for a spin pinging experiment: hard 90”(x)

soft 180’(x) Acquire ( + ); hard 90”(X) soft 180”(Y) Acquire ( - ).

(a) with a one-second soft rectangular 180” pulse, (b) with a one-

second soft triangular 180” pulse. Note the attenuation of sidelobes

in (a) and their virtual elimination in (b).

to an isosceles triangle rather than a rectangle. Then the excitation profile is described by:

(Abs). = - ;IL&; (13)

The fourth power dependence virtually eliminates any sidelobe responses (Fig. 8b). Spin pinging is related to a method called ‘DANTE-Z’ proposed by Canet [21]. The latter technique starts with a soft 180” pulse and retrieves observable transverse magnetization by a subsequent hard 90” pulse. Both spin pinging and DANTE-Z exploit the favourable shaping properties of a soft 180” pulse compared with a soft 90” pulse.

4. Design strategies

In the limit that the NMR response is a linear function of flip angle, we may use the Fourier transform relationship to derive the pulse shape required to achieve a desired frequency-domain excitation profile. However, since most cases of practical interest involve appreciable non-linearities, the

general problem is more complicated. Indeed we rely on the non-linearity of the NMR response to achieve several useful new features of soft-pulse

excitation. This is an example of an inverse problem. Given the pulse shape, we can employ the Bloch equations to calculate magnetization trajectories for a suitable range of frequency offsets and hence predict the resultant excitation profile, but we cannot proceed directly from a desired excitation profile back to the pulse shape. The most general solution is to resort to iterative methods, where an initial pulse shape is continually refined. We define a target excitation profile and compare it with a trial profile generated by a trial pulse shape. The difference is an error functional

which eventually must be minimized, although it need not always proceed ‘downhill’ during the iteration. We seek the global minimum on this multidimensional hypersurface.

Much of the early work on pulse shaping was carried out by practitioners of magnetic resonance imaging [22-341 concerned with the design of slice- selective pulses and related problems. The need for ‘designer pulses’ in high-resolution spectroscopy was slower to emerge.


4.1. Definition of the pulse shape 4.2. Gaussian cascades

The first step is the specification of a ‘trial’ time- domain pulse shape with a suitable set of parameters that can then be varied during the iterative optimization. One very convenient scheme is to define the pulse shape as a finite Fourier series [35-391 using only the lower-order Fourier coefficients, This has the advantage of involving only a small number of vari-

able parameters, and it ensures that the pulse shape is always smooth with no sudden discontinuities. For example [39], if we define tp as the soft pulse duration,

and set w = 2alt,, then we can define a general pulse shape by:

The use of a Fourier series to define the pulse shape is dictated by the desire to avoid abrupt discontinuities and keep the envelope as smooth as possible. Of course there are other viable strategies, and it is inter-

esting to compare these different approaches. One of these is the ‘Gaussian cascade’ excitation pulse (G4) suggested by Emsley and Bodenhausen [40]. The

time-domain pulse envelope is defined as the super- position of four Gaussian curves of variable timing, intensity, and width. In addition, the senses of each

Gaussian component can be positive or negative. Gaussians were chosen because of their well-known rapid fall-off as a function of offset. As has been pointed out by Ngo and Morris [31], standard optimization routines are bedevilled by a multitude of local minima when used to design excitation patterns involving the Bloch equations. Emsley and Bodenhausen address this problem through two strategies. First they ensure a reasonably good initial trial shape by noting that the frequency domain pattern at appreciable offsets is predictable through the Fourier transform relationship [ 161. This helps to

guarantee that the starting point on the multidimensional error hypersurface is not too distant from the global minimum. Second, they readjust the form of the error function during the iteration. The optimization program is the well-known Simplex routine. That these methods lead to success is a tribute

to the practice of operator-controlled optimization. A satisfactory approximation to a pure-phase trapezoidal excitation pattern is achieved after only eleven iterations and twelve variable parameters. A related population inversion pulse (G3) is obtained

with only nine variable parameters.

yBi (t) = xw A0 + g A, cos(nwt) + B, sin(nwt) n=l

(14)

The zero-order Fourier coefficient A0 determines the pulse flip angle. In many applications, terms with n higher than ten can be ignored, giving 21 parameters, of which only 20 are in fact varied during the iteration. This restraint is an advantage because instrumental considerations dictate that very convoluted pulse shapes can be rather difficult to implement in practice. The Fourier series expansion in Eq. (14) is then used to calculate a histogram defining a pulse shape; usually 64 ordinates are sufficient to approximate the desired shape. Alternatively one could simply use the amplitudes of these ordinates as the basic shape parameters but then some other control over smoothness and continuity would have to be found

1381. The Bloch equations are used to calculate the trial

excitation profile which is then compared with a pre- defined ‘target’ profile. The differences constitute a multidimensional error surface which must be explored with the aim of finding the global minimum. There are several well-known ‘gradient descent’ methods to accomplish this, but they run into difficulties if there are many local minima into which the search routine can get trapped. We want the best solution rather than a merely good solution. Ideally the program should always reach the same solution irrespective of the initial trial pulse shape; if it does not, then we suspect that most, if not all, the solutions represent ‘false’ minima.

4.3. Simulated annealing

This scheme attacks the problem of local minima directly [41]. It is loosely based on the theory of annealing of a metal, where the adoption of a desired

solid-state structure is ‘encouraged’ by several cycles of slow cooling followed by reheating. As before, the program explores the multidimensional hypersurface defined by the sum of the squares of deviations between the trial and target excitation profiles. The simulated annealing algorithm normally follows a


downhill path on this error surface but occasionally

accepts an uphill jump that could allow it to escape being trapped in a local minimum. Uphill jumps are

only permitted with a low probability proportional to

exp( - AElkT), where AE represents the (positive) change in the error function, and T can be thought of as temperature. Note the analogy with the more familiar Boltzmann expression. The probability of a downward movement is always unity, but upward jumps are far less likely, and become less probable as AE increases or as T decreases. Iteration starts at a sufficiently high temperature T that virtually the entire

error surface is accessible. Then the temperature is lowered in a stepwise manner according to a suitable ‘annealing schedule’ so that ‘escape’ jumps are gradually reduced in probability. Increasing amounts of time are allowed as T decreases. The art is to find a good compromise between cooling too fast (‘quench- ing’) and cooling too slowly, which would involve an interminable computation. Once the search has settled into an acceptable global minimum, time can be saved by switching to conventional gradient descent methods to refine the solution.

An example of the application of the simulated annealing program is provided by a family of soft pulses designed to defeat the problem of phase gradient [39]. At first sight this appears to contradict the principle that soft pulses behave as if some free precession occurred during the pulse, leading to an inevitable dispersal of the phases of signal components at different offsets. In fact these new ‘pure- phase’ pulses exploit the non-linearity of the response so that magnetization vectors from signals at various offsets come to a focus along the + Y axis at the end of the pulse. Once the principal characteristics have been defined (band-selective, uniform response, pure- phase, or ‘BURP’ for short) the simulated annealing program converges towards a solution for the pulse shape that gives a good approximation to the desired excitation profile. These BURP pulses are examined in more detail later (Section 5.1).

4.4. Artificial neural networks

Here is another scheme that mimics nature [42-481. Fundamental research aimed at establishing how the human brain functions has provided an offshoot in the field of numerical computation called the artificial

output

Fig. 9. Schematic diagram of an artificial neuron. Incoming signals

Sj are modified according to the synaptic weights W, and then

summed before passing to an amplifier with a biased sigmoid

response. The same output is fed to many neurons downstream.

neural network. This could be a hard-wired device, but for the present purposes it is implemented as a simulation in a digital computer. The key component is a simulated ‘neuron’ which receives several input signals Sj controlled in relative amplitudes by synaptic weights Wj. The input ‘voltages’ are weighted and summed according to

V=8+ 1 WjSj (15)

where 13 is a variable bias voltage. The result V is amplified according a sigmoid response function to provide a single output voltage that can be fed to several other neurons. In a sense, the neuron is either switched ‘on’ or ‘off’ by this process, depending on the input signals and their synaptic weights (Fig. 9).

Neurons are interconnected in layers. The first layer, sometimes called the retina, receives the basic information to be processed. Its output is fed to a second ‘hidden’ layer, which in turn feeds an output layer whose signals constitute the desired solution to

Retinal layer

Output layer

Fig. 10. A simulated neural network made up of three layers. Infor-

mation is fed to the retinal layer. All neurons in this layer are

connected to the hidden layer, and all neurons in the hidden layer

are connected to the output layer. Not shown (for simplicity) is a set of direct connections between the retinal layer and the output layer.

70 R. Freeman/Progress in Nuclear Magneric Resonance Spectroscopy 32 (1998) 59-106

the problem in hand (Fig. 10). Signals from the retinal layer may also be connected directly to the output layer. For the present purposes information is only fed in the forward direction; there is no feedback to earlier layers.

The aim is to solve the ‘inverse problem,’ in this case to determine the Fourier coefficients defining a

suitable pulse shape, given a set of parameters that define the desired frequency-domain excitation profile. The latter are signals fed to the retinal layer. Initially, of course, the output bears no relation to the required Fourier coefficients, why should it? The

network has first to be ‘trained’ by postulating a randomly-generated pulse shape Pk and by calculating the corresponding frequency-domain profile Fk through the Bloch equations.

These frequency-domain parameters Fk are fed to the retina, and the resulting set of output signals Ok is compared with the pulse shape parameters Pk. Since the initial synaptic weights and biases are quite random, the initial correlation between Ok and Pk

(measured as the inverse of the sum of the squares of the differences) is very poor, but can be improved by readjusting the synaptic weights Wj and bias 0. New trial sets Fk are fed to the retina one by one, and the Wj and 0 values are readjusted in an iterative cycle to minimize the differences between Ok and Pk.

The algorithm for altering the synaptic weights moves

backwards through the layers of the network, recursively adjusting Wj and 0 values, under the control of a parameter called the learning rate, which may decrease exponentially with time. The program approaches its goal asymptotically and may need to

be stopped after a predetermined number of presenta- tions. Success in training depends to some extent on providing a sufficient number and a suitable range of

trial profiles. All pupils need good teachers. Very gradually, as different trial excitation profiles are pre- sented to the retina, the neural network ‘learns’ to

recognize them and to suggest a reasonable approximation to the pulse shape. This acquired ‘knowledge’ resides in the Wj and 8 values associated with each neuron.

Naturally the training stage can be long and pro- tracted because every new presentation to the retina involves repeated calculations of the frequency- domain profile through the Bloch equations, once for every frequency offset in the range. However, the great advantage of the scheme is that the subsequent operating mode can be very fast, involving a single presentation of the desired profile and a prompt reading of the suggested Fourier coefficients (Fig. 11). The neural network is acting as a machine to invert the Bloch equations. The long training schedule has paid off with a rapid, repeatable ‘reflex’ response. It is

possible that human beings learn physical actions

Operating mode

change SynoPriC weights

Fig. 11. Schematic diagram showing how an artificial neural network is first trained by feeding with a series of frequency-domain patterns

generated from random pulse shapes, recursively adjusting the synaptic weights. Once the long training program is complete, the operating

mode (box) rapidly predicts a pulse shape suitable for any frequency-domain profile fed into the retinal layer.

R. FreenundProgress in Nuclear Magnetic Resonance Spectroscopy 32 (1998) 59-106 71

(say, catching a ball) by a similar mechanism. Speed could be of crucial importance in certain applications

of pulse design. One could imagine a situation in

magnetic resonance ‘in vivo’ spectroscopy where physicians would like to record the signal from some irregularly shaped organ within a patient, requir- ing the design of a new shaped localization pulse ‘on the spot’.

In a preliminary test of a neural network scheme,

the goal was a pulse shape suitable for generating an antiphase version of a J-doublet, the preparatory stage for many selective coherence transfer experiments [49]. This was called a ‘Janus pulse’ after the Roman god of gates, always depicted as facing in two opposite directions. The target excitation profile (Fig. 12) would be antisymmetric with respect to the center frequency, having two bands of uniform excitation and opposite phase, separated by a sharp central transition. In this application the retinal layer had 100 neurons, the hidden layer had 200 neurons (each connected to all the retinal neurons) and the output layer had only 16 neurons (each connected to all the earlier neurons). These 16 Fourier coefficients were considered sufficient to define the pulse shape. Experi- mental tests of the resulting Janus pulse on simple high resolution NMR spectra proved very promising

L491.

4.5. Genetic algorithms

When we contemplate the intricacy of living organisms and the enormous diversity of human bio- chemistry, Darwin’s ideas about natural selection

!4 J 3 max

Fig. 12. Target frequency-domain excitation profile for a Janus

pulse, designed for uniform excitation of an antiphase doublet

over a range from Jmi, to J,,,.

seem so revolutionary that they are still not universally accepted. Dawkins [50] has done much to dispel such scepticism in his book ‘The Blind

Watchmaker.’ He recalls the theological ideas of

William Paley, who argued that just as a complex mechanism such as a watch must have had a designer, a complicated organ such as the human eye must also have had a Maker. Dawkins demonstrates why this ‘argument from design’ is wrong, and shows how the organized complexity of the living world does indeed come about through a blind, apparently pur- poseless, evolutionary process. As part of his careful explanation, Dawkins demonstrates a simple pattern

generating algorithm that employs ‘genes’ to produce images on the screen, progressively altering the genes to produce further generations in a recursive process. The operator selects the most promising offspring at each stage. In many instances the images that are generated resemble life forms.

In fact there is a vast literature on genetic algorithms, distributed over many fields from cellular automata to engineering design [5 l-571. The variable parameters defining a given physical problem are represented numerically as ‘genes’ subject to random variations. Given a suitably ‘hostile’ environment (for example a set of engineering constraints), the design evolves with time and eventually the fittest design ‘survives.’ Once again science finds it profitable to imitate nature.

We could apply the same ideas to pulse shape design in several ways. For example two primitive pulse shapes could be ‘mated’ to produce new and interesting offspring, and by further ‘breeding,’ nudged towards some target with a desirable per- formance in the frequency domain [58]. Since there is no need to adhere to the reproductive rules of our

own world, three or more parents might interact at the same time, exchanging genes in any conceivable pattern. Alternatively, we might rely entirely on mutation of genes (asexual reproduction). As an example, we might define the pulse shape in terms of Fourier coefficients (Eq. (13)) and call the numerical values the genes (Fig. 13). By examining the resulting excitation profiles on a computer screen and by choosing the most promising offspring, the operator would guide the evolution in a profitable direction [59]. In this manner an amplitude-modulated pulse was designed that had properties similar to those of the


Trial Fourier

coefficients

4

Calculate trial

excitation profile

4

Increment Fourier iterate

coefficients i

4

Calculate new

excitation profiles

Fig. 13. Schematic diagram of a genetic algorithm used to design a

pulse shape that will excite a particular frequency-domain profile.

The operator must choose a promising profile at each stage and use

it as the parent for the next generation. When an acceptable profile

appears on tbe screen, the corresponding Fourier coefficients

(genes) are used to establish the pulse envelope.

BURP pulses, except that the focus point was delayed beyond the end of the pulse by a fixed interval (of the order of 0.1 ms) in order to allow time for pulse transients to decay or for field gradients to be switched

1601. Note that these applications of genetic algorithms

differ fundamentally from natural evolution in that they presuppose a final goal. There is another exciting possibility, the ‘discovery mode’. Assuming that the operator is also a skilled spectroscopist, we might hope that occasionally some surprising new excitation pattern might emerge during the evolution, quite different from the original target, and be recognized for its utility and originality.

4.6. Taboo search

There are many variations on the multidimensional optimization problem, stimulated by the fact that it offers solutions to a wide range of problems in physics, chemistry, mathematics and engineering.

We mention here the ‘taboo search’ [61,62] which offers an interesting new approach. In certain respects it resembles simulated annealing in that moves on the error hypersurface may be either downhill or uphill, and it has a distant resemblance to the artificial neural network because it contains an heuristic feature, as if the program were ‘learning’ from its ‘mistakes’. We retain the symbol AE for the change in error functional for a given step in the iteration. At each step,

the taboo search examines all feasible moves and selects the one with the lowest AE, not necessarily a negative value, so that in principle it is possible to climb out of a local minimum or merely make a neutral change in parameters. However, this choice is also constrained by certain taboo conditions.

A move is taboo for two possible reasons: (a) if AE exceeds some prespecified threshold or (b) if the new location is on the taboo list of length L, detailing the co-ordinates having the worst solutions (E) so far

experienced. The parameter L may be a constant or it may be changed for different stages of the search. The taboo list is continually updated by entering the most recent value and dropping the oldest. Conse- quently a given move only remains taboo for a specified number of iterations. A prospective move is then restricted to moves not on the taboo list, thus stimulating the search towards previously unexplored regions of hyperspace.

The second key feature is the construction of an elite

list which tabulates the most promising co-ordinates in the taboo list. If an unacceptable deterioration in AE occurs, possible taboo moves are re-examined and, if the elite list suggests that a taboo move

might be ‘interesting’, that move becomes admissible (its taboo status is revoked). This aspiration condition

is quantified in terms of an aspiration function, the best value of AE so far encountered for any of the taboo moves. This intensifies the search in regions of hyperspace already visited. The taboo and elite lists endow the program with a ‘learning’ feature based on past experience. Taboo search has been shown to compare favourably with simulated annealing when applied to certain standard test problems [62].

4.7. Linearization of the Bloch equations

Most of the difficulties outlined above stem from


the impossibility of simultaneously inverting the

Bloch equations for a large array of isochromats at different offsets from resonance. Consequently, the required soft radiofrequency pulse envelope cannot

be deduced directly from the desired frequency- domain excitation pattern. However, Ngo and Morris [31] have pointed out that the BIoch equations are

linear for small perturbations. That is to say, if two successive radiofrequency pulses induce two very small rotations 6wl and 8w2, the overall effect on the

nuclear magnetization vector is given by the vector sum of the effects of 6wl and 8w2 acting independently. The magnetization vector M, constrained so that its tip moves on the surface of a sphere, undergoes

such small displacements that they can be added like vectors on a flat surface. However, since the effect of a small rotation is a function of the time at which it is

applied, it is necessary to remove this time dependence by transforming the problem into a new time- varying co-ordinate frame where small perturbations

occurring at different times can be superposed. An initial guess is made for the pulse shape, but this

choice is not at all critical. The overall optimization problem is divided into a series of subproblems, each of which can be solved by minimizing the error functional. False minima are not involved because in each subproblem the search is directed towards a carefully chosen intermediate target. In this manner the error functional decreases monotonically as we move from the current situation to the intermediate target. The latter is modified as the iteration proceeds by con- sidering both the current excitation profile and the final target profile. We step carefully through the minefield of false minima by calculating the best

direction at each stage. The power of this method is demonstrated by its

ability to calculate a novel pulse shape very different

from the initial trial shape. For example, the problem of designing a slice-selective soft radiofrequency

pulse was solved starting from a simple rectangular

90” pulse, which evolved during the iteration into a symmetrical pulse envelope reminiscent of a truncated sine function. The final excitation pattern

had the required ‘top-hat’ shape [31].

5. Pure-phase pulses

As explained in Section 3, it is often inconvenient to have an induced phase gradient across a high reso-

lution spectrum, though most soft pulses inevitably generate such a gradient by delaying the point at which signal acquisition can be initiated. ‘Pure- phase’ pulses are designed so that magnetization vectors from a range of chemical sites are brought to a focus at the end of the soft pulse, so that they all

appear as pure absorption signals. We define an ‘effective bandwidth’ Af over which this compensation occurs. This is bounded by steep transition regions

where the response falls rapidly, and beyond these lie the suppression regions where both absorption and dispersion contributions are at a negligible level (Fig. 14). It is also very convenient to be able to vary the width of the excitation region at will, and to have a uniform response within this band. These pulses are therefore ‘band-selective’ and can be used to excite an entire spin multiplet or a range of different chemical shifts.

5.1. BURP pulses

One family of pure-phase soft pulses goes by the general name BURP (band-selective, uniform response, pure-phase). There are four main members of this family; E-BURP for excitation, U-BURP for

universal rotation, I-BURP for spin inversion, and RE-BURP for refocusing [39]. Band-selective pulses are used most commonly for excitation, starting with magnetization along the + Z axis, or as a ‘read’ pulse

- Effective bandwidth -

Fig. 14. Target frequency-domain absorption-mode profile for the pure-phase ‘BURP’ pulse. The dispersion-mode response is negligible

everywhere.


after some more complex preparation. One such pulse is E-BURP-2 [39] which has the pulse envelope (a 64-step histogram) and frequency-domain excitation profile illustrated in Fig. 15. In this case the effective excitation bandwidth (absorption-mode) is 400 Hz, flanked by steep transition regions and essentially negligible excitation elsewhere. Expansion of the bandwidth is achieved simply by scaling down the pulse duration, maintaining the same shape. So a

1 ms pulse would have an effective bandwidth of 4 kHz, whereas a 100 ms pulse would have a bandwidth of 40 Hz.

E-BURP pulses were designed by simulated annealing with the constraint A0 = 0.25. As a conse-

quence, E-BURP pulses only give the desired excitation profile if the flip angle is properly calibrated to the 90” condition. They achieve focusing along the +Y axis through an intricate interplay between diver- gence of magnetization vectors in the - Y hemisphere and convergence in the +Y hemisphere. This can be visualized by calculating a family of trajectories for

magnetization vectors covering a narrow range of offsets (Fig. 16). This ‘bunch’ of trajectories alternately expands and contracts, and finally achieves a focus near the +Y axis. Clearly the excursions of the magnetization vectors are so large that the non-linear

Fig. 15. The pulse envelope of E-BURP-Z shown as a 64-step histo-

gram (a) and the corresponding frequency-domain excitation profile

(b) showing absorption (full curve) and dispersion (dashed curve).

The effective bandwidth is 400 Hz. If the envelope is scaled down in

time, the excitation profile is correspondingly scaled up in fre-

quency (a 1 ms pulse gives an effective bandwidth of 4 kHz).

component of the motion is appreciable; indeed this is essential to achieve focusing over a significant band of frequencies.

Some insight into the relationship between pulse shape and excitation profile can be obtained by running the simulated annealing program with pulse

envelopes defined by a restricted number of Fourier coefficients [39]. With the A,, coefficient held constant at 0.25, the first search was initiated with only the first-order coefficients A, and BI. Then further pairs of A and B coefficients were introduced, up to nmax.

Fig. 1’7 illustrates how the excitation profiles (absorption and dispersion) gradually approach the ideal as more terms are introduced into the Fourier series expansion (Eq. (13)). When nmax reached 8 the excitation profile was considered adequate; this is E-BURP-2.

Later refinements concentrated on improving the uniformity of the excitation within the effective band-

width [63] and on further attenuation of the residual out-of-band excitation [64]. These improved pure- phase pulses are therefore to be preferred over the E-BURP-2 pulse. The Fourier coefficients defining the ‘quiet-SNEEZE’ pulse envelope [64] are set out in Table 1.

5.2. Relaxation effects

These pulse-shaping calculations have ignored relaxation effects during the pulse. In many applications with soft pulses in the millisecond

Fig. 16. Magnetization trajectories for a 1 ms E-BURP-2 pulse for a

family of resonance offsets near 600 Hz. Despite the large excur-

sions, these trajectories come to a focus near the +Y axis.

R. Freeman/Progress in Nuclear Magnetic Resonance Specrroscopy 32 (1998) 59-1W 75

n max = I

n max =2

n max = 3

n =4 max

n max =5

n max =6

n max = 7

n max = 8

Pulse shape Excitation profile I

Fig. 17. Pulse envelopes and the corresponding frequency-domain

excitation profiles obtained by means of a simulated annealing

program in which the number of Fourier coefficients (2nmax + 1)

defining the pulse shape was restricted. The final profile (n,,, = 8)

is the E-BURP-2 pulse.

range, this is a reasonable approximation, but for pulses of long duration, relaxation should be taken into account [65]. Apart from the expected reduction of signal intensity caused by relaxation losses, there is also an unfortunate distortion of the frequency- domain profile, most simply described as a rounding of the shoulders. Both spin-spin and spin-lattice relaxation are involved. Fortunately the E-BURP soft pulse can be reoptimized by allowing for

Table 1

Fourier coefficients” defining the time-domain envelope of the

‘Quiet-SNEEZE’ band-selective pulse [64]

n AlI

0 + 0.250

1 + 0.934

2 + 0.180

3 - 1.527 4 + 0.003 5 + 0.143

6 + 0.050 7 + 0.072

8 - 0.015

9 - 0.040

10 - 0.005

aTo be inserted into the expression:

B,

-

- 0.197

- 1.772

+ 0.204

+ 0.619

+ 0.076

+ 0.039

- 0.025

- 0.060

+ 0.005

+ 0.017

-@,(t)=w Aof

1 I? [A, cos(nut) + B, sin(nwt)] ll=l 1

relaxation during the calculation, making some simplifying assumptions, for example T, = T2, or T1 2s T2. If the process is initiated with only mild relaxation effects, it can be reasonably expected that the new solution lies close to the old, and therefore a standard optimization routine may be employed rather then a new simulated annealing run. In this manner satisfactory excitation profiles can be restored

[W. A set of excitation profiles was simulated in Fig. 18

for the case that Tl = T2 and for a range of different ratios of the relaxation time to the pulse duration t,. They show clearly that E-BURP-2 pulses suffer a severe distortion if T2 approaches t,; the edges of the profile ‘melt’ like an ice cube [66]. In contrast, SLURP pulses (silhouette largely unaffected by relaxation processes), designed to accommodate relaxation, showed a much better retention of the approximately trapezoidal shape for the same range of relaxation times. Note, however, that there is an inevitable loss of intensity through relaxation during the soft pulse.

5.3. Initial conditions

The E-BURP and SNEEZE pulses were both designed to operate on longitudinal magnetization,

76 R. Freeman/Progress in Nuclear Magnetic Resonance Spectmw~py 32 (1998) 59-106

-~-__-___.~

r___L.3-___ ___. b ______-___ ___.c

_____-_----- -d

r -5

-;5 -50 -Is :0 1

Fig. 18. Simulated excitation profiles for E-BURP-2 (top) and SLURP [bottom) calculated (assuming 7, = T1) for a range of values of T,lt,; (a) m,(b) 10.0, (c) 5.0, (d) 2.0, (e) 1 .O. (f) 0.75, (g) 0.5. The design of the E-BURP-2 soft pulse takes no account of relaxation duringthepuke, so the profiks are seriously distorted. The design of the SLURP soft poke incorporates a~ allowance for relaxation, so the profile retains the same general shape throughout, though intensity is lost through relaxation.

with the spin system either at BoItzmatm equilibrium retracing the path it would have taken from i-2 CO or after a population inversion. If we wish to return fY buf in the opposite sense. In the more genera1 pure Y-magnetization back to the i-Z axis, a phase- case where the magnetization has an arbitrary initial inverted and time-reversed E-BURP pulse will orientation, a universal 90” rotation is required; this is suffice, a typical magnetization vector simply provided by the U-BURP pulse 1391.

R. Freeman/Progress in Nuclear Magnetic Resonance Spectroscopy 32 (1998) 59-106 II

Population inversion is achieved with the time- symmetric soft 180” I-BURP pulse [39]. For spin-

echo experiments, where the magnetization is initially

in the XY plane of the rotating frame, the appropriate

soft 180” refocusing pulse is RE-BURP [39]. Pulses such as U-BURP and RE-BURP that act on magnetization with an arbitrary initial orientation, are sometimes called general rotation or plane

rotation pulses. They tend to be less robust than E-BURP and I-BURP with respect to instrumental shortcomings. None of the BURP pulses work well if not properly calibrated to the appropriate 90” or 180” condition.

5.4. Close encounters

In the excitation profile of soft pulses of the BURP family, the inner parts of the suppression region may be called penumbra to indicate that although the observed signal at these offsets is always low, the actual magnetization excursions during the pulse

[67] are in fact considerable (Fig. 19). This reflects the fact that a rather abrupt transition between focusing along +Y and focusing along +Z demands a

very non-linear response of the spin system. At much larger offsets (the umbra) the effective field is

tilted through a very large angle 6 and lies close to the

+Z axis, so the magnetization excursions are always very small.

Not surprisingly, if two band-selective soft pulses of the BURP type approach one another in frequency space so that their penumbra overlap, there is a significant interference effect and the excitation profiles

are distorted. We may express the degree of trespass in terms of AFlAf, where AF is the separation between centres and Af is the effective bandwidth of each pulse (assumed equal). For AF/Af > 3, the main features are wave-like disturbances in the previously flat suppression region, but closer encounters (AF/Af < 2)

cause gross distortions of the two excitation patterns (Fig. 20a) and the pulses are essentially unusable [68].

Fortunately this problem can be moderated. We see

from Fig. 19 that the magnetization excursions are fairly negligible during the first half of the soft pulse. By reversing the sense of the time evolution of one pulse of the pair, we can mitigate the interaction to a large extent and operate with a lower setting of AF (Fig. 20b). The only penalty is the

Fig. 19. Experimental time evolution of the absorption-mode proton signal in heavy water recorded during a 23 ms E-BURP-2, displayed as a

function of time and of resonance offset. At offsets greater than about 150 Hz the final observed signal is negligible but the signal excursions

during the pulse are considerable. These two regions are called the penumbra. Only at much larger offsets (the umbra) is the signal during the

pulse negligible.

78

a

R. Freeman/Progress in Nuclear Magnetic Resonance Spectroscopy 32 (1998) 59-106

offset

b

offset

Fig. 20. Close encounter between two simultaneous 100 ms quiet-SNEEZE pulses with effective bandwidths Af = 45 Hz, displayed as a

function of the separation between centres AF. (a) Note the severe wave-like disturbances where there would normally be a flat baseline (the

penumbra). At separations where AF is less than about 2Af, the desired excitation patterns are hopelessly distorted through interference effects

and the scheme is unusable. (b) When the time evolution of one pulse is reversed, the interference effects are minimized and the two excitation

profiles maintain an acceptable trapezoidal shape for separations AF greater than about 2AJ


introduction of a frequency-dependent phase shift for signals excited by the time-reversed pulse [68].

6. Implementation

6.1. Soft pulses

Shaped radiofrequency pulses are generated on modem high resolution spectrometers by means of a digitally-controlled waveform generator feeding a

linear radiofrequency power amplifier. The shape is defined by the spectrometer computer as a histogram made up of a time-sequence of k segments, where k is

usually in the range 64 to 256. Each segment represents a short radiofrequency pulse of constant intensity and fixed phase, but the intensity and phase vary from one segment to the next. This approximation of a smooth waveform as a discrete series of steps corresponds to the usual procedure for calculating the effect of a soft pulse through the Bloch equations or the Liouville-von Neumann equation. The Hamiltonian throughout each individual step is time- independent and the overall effect is a cascade of many such steps, or the resultant of a string of propagators.

For simplicity, many shaped pulses were designed with pure amplitude modulation, so the phase of each segment in the sequence should in principle remain constant. This extra degree of freedom proves to be very useful in practice, for by imposing a linear phase ramp we can displace the effective irradiation frequency away from the transmitter frequency. If the soft pulse duration is tp and there are k segments in the histogram with a phase increment A4 radians between adjacent segments, the frequency shift is Af = kA+l(27rtp) Hz. The increments A~#J should be small enough to avoid the problem of rotational aliasing (2na + A~#I = A~#J). We can visualize this process as a series of repeated jumps into new reference frames shifted in small steps A+. This jerky rotation of the reference frame is a close approxima-

tion to a smooth rotation at the frequency A$ The only difference is the introduction of some very weak sideband responses. In the case where phase modulation is already an integral feature of the shaped pulse, the effective frequency can still be shifted by super- imposing a phase ramp.

Normally the spectrometer computer stores a set of

data tables appropriate to the most popular pulse shapes, or calculates them from a set of Fourier coef-

ficients defining the shape. Calibration of the nominal

flip angle on resonance is an important step in setting up a new shaped-pulse experiment, particularly for pure-phase pulses, which only operate correctly when the flip angle is properly set. To be perfectly satisfied with this calibration the operator may need to record an array of NMR signals from a reference sample (for example HDO in heavy water) over a range of transmitter offsets and compare these with the theoretical excitation profile of the pulse in question. Fortunately there is usually control soft-

ware available which relates the soft pulse calibration to the much simpler calibration of a rectangular

pulse.

6.2. The DANTE sequence

Not all spectrometers are suitably equipped with waveform generators and linear amplifiers. The DANTE sequence [ 1,691 offers a simple alternative for the generation of shaped radiofrequency pulses, replacing each segment of the soft-pulse histogram with a short hard pulse followed by a short period of free precession. If the number of steps k is sufficiently

large, the zig-zag magnetization trajectory during the DANTE sequence gives a result very similar to that of the single soft pulse. This can be appreciated from Fig. 21, which shows the trajectory during a rectangular eleven-pulse DANTE sequence; the final

position is very similar to the fifth trajectory displayed in Fig. 1. This equivalence for sufficiently high values

of k was also established theoretically [70]. There is one major difference between excitation

with a single soft pulse and with a DANTE sequence. The individual hard pulses of a DANTE train have a

cumulative effect not only at exact resonance f. (when the magnetization vector is rotated directly from + Z to + Y) but also at a set of sideband condition f. 2 n/7

where II is an integer and r is the interval between hard pulses. Sidebands arise as a result of the rotational aliasing condition, because free precession through 2nn radians in the r interval is equivalent to no precession at all. This property is often put to good use in fine-tuning the excitation. By operating at the first sideband condition, we can make very fine


Fig. 21. The zig-zag magnetization trajectory followed during an

eleven-pulse DANTE sequence of duration 1 ms at a resonance

offset of 400 Hz. Rotations about the + X axis are the result of

short hard radiofrequency pulses, while rotations about the + 2 axis

arise from free precession. The final position is essentially the same

as that of the fifth trajectory in Fig. 1 for a rectangular soft pulse at

the same offset and a tilt of the effective field of about 1 radian.

adjustments to the effective irradiation frequency simply by varying 7. For sufficiently high pulse repetition rates the centreband and all other sideband responses are kept well away from the spectrum of interest. Sideband operation can be important in spectrometers where the transmitter frequency can only be changed in coarse steps.

Much weaker vestigial sidebands occur with soft pulse excitation as a result of the mild modulation imposed by the digitization steps; they only vanish completely if a completely smooth envelope is employed. The digitization rate is made high enough that these weak sidebands do not fall within the spectrum of interest.

Careful examination of the trajectory in Fig. 21 reveals an interesting detail-the first and last pulses were halved in intensity. Neglect of this precaution

results in an excitation profile that is slightly distorted with respect to that of the equivalent soft pulse; the sidelobes are accentuated and the baseline is slightly displaced (Fig. 22a). These effects are more serious the smaller the number of pulses k. If the last pulse is halved in amplitude (Fig. 22b) the baseline problem is corrected, and if the first pulse is halved, the sidelobes return to normal [7 11. The simplest explanation of this phenomenon is to reconsider the conversion of one segment of the soft pulse histogram into one step in

the DANTE sequence. If the segment has a width 7,

the correct procedure is to place the hard pulse at the centre of the r interval:

[ - r/2 - hard pulse - r/2 - lk

then the sidelobe and baseline distortions are corrected. If the hard pulse is placed at the beginning of the r interval and is the first pulse of the sequence, or if it is at the end of the r interval and is the last pulse of the sequence, it should be halved in intensity. This

correction is particularly important for shaped pulses such as the half-Gaussian, where the final intensity is very high. There is a well-known related problem in

the Fourier transformation of a digitized time-domain function.

Another useful feature of DANTE is that it allows

d

Fig. 22. Frequency-domain excitation patterns simulated for a ten-

pulse DANTE sequence showing the centreband and first sideband

response. (a) When all pulses have the same intensity there is a

baseline offset and increased oscillations in the region between

the main responses. (b) When the final pulse has half intensity the

baseline offset is corrected. (c) When the first pulse has half inten-

sity the oscillations are attenuated. (d) When both the first and last

pulses have half intensity both imperfections disappear.


observation of the NMR signal (one data point at a

time) in the free precession intervals between the hard

pulses [67,72]. An analogous operation is usually not feasible with a single soft pulse. This provides a useful ‘window’ for observing magnetization trajectories during soft pulse excitation with a DANTE train. For example an experimental observation of signal evolution during the E-BURP-2 pulse was compared with simulations based on the Bloch equations. Although only the real and imaginary components of the signal are monitored, the longitudinal com-

ponent can be deduced on the assumption that the relaxation losses are negligible on that time scale, and that radiation damping [73] can be neglected. Then we can write:

otherwise the excitation profile will be distorted in some manner.

The main limitation to pulse-width modulation occurs when the required hard-pulse flip angle is too short (for example, less than about 1 ps) because the phase glitch induced by the finite rise and fall times begins to present a problem. It can be circumvented by constructing a very short pulse as a combination of a forward and a backward pulse having a difference in flip angles equal to the desired value. If, as a result of the modulation, the pulse width is no longer negligible with respect to the free precession interval r, then this interval is readjusted to maintain a constant pulse repetition rate.

M;+M;+M;=M, (16) 6.3. Simultaneous soji pulses

Slight discrepancies between theory and experiment suggest that spatial inhomogeneity of the B1 field is a significant factor in the experimental case [67].

The major advantage of using DANTE for pulse shaping is that the required amplitude modulation can be achieved by varying the pulse width, keeping the hard pulse intensity constant. This neatly sidesteps the thorny problem of linearity,

because pulse width timing can be very accurate indeed. When using a single soft pulse we have to ensure that the entire sequence, from pulse-shape data table through the waveform generator, the power amplifier and the probe coil, is strictly linear,

The demands of modern spin gymnastics often

require soft pulse excitation (or spin inversion) at two or more chemical sites at the same time. This can be achieved by interleaving two DANTE sequences, one train having the hard pulses of constant phase, the other having a linear phase ramp [74]. To a reasonably good approximation we may assume that the two pulse trains do not interact, provided that their effective frequencies are not too close. To ensure that the signals excited at the two sites are both in pure absorption, the hard pulse phases are calculated to reach coincidence at the end of the combined sequence (Fig. 23). We may think of the two

Constant phase

Acquisition

Incrementedphase

Fig. 23. A combination of two interleaved DANTE trains, each with the same pulse repetition rate but with a linear phase incrementation of the

shaded pulses, so that this sequence acts at a different frequency.


DANTE sequences as acting in two different reference frames, one rotating faster than the other, but eventually coming into coincidence just as signal acquisition is started. The two trains need not be of

the same overall length but they must terminate at the same instant; they may therefore differ in frequency selectivity. They may, of course, have different shapes

and different overall flip angles.

6.4. Pulses at two coupled sites

So far it has been tacitly assumed so that we can neglect spin coupling effects during a soft pulse. While this is perfectly justifiable for hard pulses where the radiofrequency intensity yB1/2?r is large compared with Jis, it is not always the case for soft pulses. Fortunately it appears that if we use a pure- phase pulse of the BURP type to excite a single multiplet, the components of the multiplet behave independently and come to a focus at the end of the pulse in the same manner as resonances with different chemical shifts. Complications only begin to arise when two multiplets are excited simultaneously by two soft pulses. Even then, if there is no coupling between the two sites, the two multiplets are excited in the normal fashion with no disturbance of the phase or the relative intensities. This can be seen in the

400 MHz proton spectrum of strychnine (Fig. 24a) where spin multiplets at sites 12 and 15a are excited simultaneously with E-BURP-2 pulses [75]. There is no spin-spin coupling between these protons and the

two multiplets appear in pure absorption with no sign of any perturbation (Fig. 24b). In contrast, when the experiment is repeated at sites 15a and 15b which have a geminal coupling Jrs = 14.5 Hz, and if the

pulse duration t, is set to about 69 ms, both multiplets appear in the dispersion mode with an antiphase

doublet for the active splitting. This phase distortion is clearer if the receiver phase is shifted by 90” to give an absorption-mode display (Fig. 24~). The antiphase magnetization builds up as a function of t, and reaches a maximum at the condition .I&, = 1.53, corresponding to the pulse duration used in Fig. 24~. At longer pulse durations the antiphase

magnetization is converted into multiple-quantum coherence [76].

This phenomenon was called the double resonance two-spin effect (TSETSE) on the grounds that the pulses must ‘sting’ two sites to transmit the ‘disease’. One way to visualize the TSETSE effect is to imagine the refocusing of the S spin response to be perturbed by simultaneous irradiation of the I spins which ‘decouples’ the Jis interaction. Although the magnetization trajectories of the two S-spin lines

:I a

I L I I

PPm 4.0 3.0 2.0 I .o

Fig. 24. Experimental demonstration of the TSETSE effect. (a) Part of the conventional 400 MHz spectrum of strychnine. (b) Soft pulse

excitation of sites 12 and 15a which have no mutual coupling. (c) Soft pulse excitation of sites 15a and 15b which have a geminal couplingJts =

14.5 Hz. When the pulse duration tp is set to the condition Jrsrp = 1.5, the TSETSE effect generates dispersion-mode signals at both sites with antiphase active splittings. For clarity, the absorption mode was restored in trace (c) by shifting the receiver phase by 90”.


would normally come to a focus along the + Y axis at

the end of the BURP pulse, they are diverted from

their normal paths by the l-spin irradiation. This

phenomenon has been largely regarded as a nuisance

for soft pulse experiments [77-791, but it can be put to good use as a simple test for correlation through spin- spin coupling, exploiting the fact that only the two selected sites are involved.

6.5. Multisite excitation

Although the concept of interleaving two DANTE

sequences can be extended to encompass multisite excitation, if a suitable digitally-controlled waveform generator is available it is probably better to implement this in a different manner. Each excitation is represented as a soft-pulse histogram of k segments,

and each segment is associated with an amplitude (reflecting the desired modulation) and a phase (reflecting the required frequency shift) [80]. We may thus represent each segment by a vector. A number of soft pulses are combined by vector addition at the corresponding time segments (Fig. 25). The resultant is another histogram of k segments that behaves as a set of independent soft pulses. Very large numbers of simultaneous soft pulses can be generated in this manner, having arbitrary shapes, frequencies, intensities, selectivities and relative phases. Fortunately the dynamic range limitations of the waveform generator apply only to the resultant vector at each segment, not to each individual vector. Even higher dynamic ranges can be achieved by represent- ing a single vector in terms of two equal counter- rotating vectors, provided that suitable small phase increments (0.1”) are available [80].

6.6. Binomial pulses

The facility to combine several soft pulses at different frequencies and with different intensities

affords another method for suppression of frequency- domain sidelobes. A single soft rectangular pulse will of course produce a profile with sine-like oscillations in the tails (Fig. 26a). If we combine two such soft pulses of the same intensity with a frequency

separation Af given by

Af = 1/(2t,) (17)

d

time

Fig. 25. Combination of three soft pulse histograms by vector

addition at each time segment. For simplicity all the segments are

assumed to have the same amplitude, but the rates of phase ramping

differ, so that the three soft pulses have different effective frequen-

cies. The resultant has the same effect as the three individual soft

pulses acting independently.

where t, is the pulse duration, then the peaks of the sine-like oscillations in one profile fall in the troughs of the oscillations of its neighbour [81]. The registra- tion is almost exact at appreciable offsets but only approximate near resonance. This 1: 1 soft pulse combination gives a frequency-domain response with attenuated sidelobes (Fig. 26b). A combination of three soft pulses with intensities 1:2: 1 offers an even better excitation profile (Fig. 26~) and the 1:3:3: 1 arrangement has the oscillations essentially eliminated (Fig. 26d). Note, however, that the excitation band is broadened with respect to that of a single rectangular soft pulse.

6.7. Polychromatic pulses

Suppose we assemble a large array of soft pulses of equal intensity, maintaining a regular frequency separation Af= 1/(2t,) between adjacent pulses, and compensate any end-effects by gradually tapering off the intensities of the outermost pulses. This is called a polychromatic pulse [Sl]. If we also employ a small

84 R. Freeman/Progress in Nuclear Magnetic Resonance Spectroscopy 32 (1998) 59-l 06

b-.‘\_- c-A_ d L

Fig. 26. Frequency-domain excitation patterns for combinations of

rectangular soft pulses with frequency separations Af= l/(21,) and

binomial intensities. (a) A single soft pulse, showing sine-like oscil-

lations. (b) Two equal pulses (1: 1). (c) Three pulses with relative

intensities 1:2:1. (d) Four pulses with relative intensities 1:3:3:1.

Note the progressive attenuation of the sine-like oscillations.

flip angle (say 45”) we can neglect the interaction between the pulses, so they are linearly superposable. Under these conditions the overall frequency-domain absolute-magnitude response is very flat, and the phase gradient is approximately linear over the effective bandwidth. Polychromatic pulses provide

not only wideband uniform excitation, but also the possibility of ‘tailoring’ the excitation profile by adjusting the relative intensities of certain ‘elements’ within the array.

One very useful application is to generate a broad uniform excitation with a rejection notch at some suitable frequency, so as to suppress a strong solvent resonance. By adjusting the intensities of a small

cluster of elements, the width of the rejection notch can be adjusted. For example, the polychromatic pulse with relative intensities:

P= 1:4:7:8:8:8:8:8:4:0:4:8:8:8:8:8:7:4:1 (18)

has the excitation profile shown in Fig. 27, with a reasonably flat response over most of the frequency range but with a rejection notch at the centre. Related polychromatic pulses were used successfully for water suppression in high resolution NMR [82]. Polychromatic pulses are also well-suited to the generation of a Janus pulse, designed to excite antiphase doublets for a range of different values of the spin-spin coupling constant [82].

6.8. Hadamard phase encoding

Soft pulse experiments offer a viable alternative to two-dimensional spectroscopy if only a few correlations or Overhauser enhancements are required to solve a given chemical problem. One objection that could be raised is that we thereby sacrifice sensitivity. If we choose to perform N separate soft-pulse measurements, then the available spectrometer time must be divided into N tranches, whereas the two- dimensional alternative supplies all the information

Fig. 27. The experimental excitation profile (absolute magnitude) of a polychromatic pulse made up of 19 soft pulse elements separated by Af =

1/(2t,) and with relative intensities 1:4:7:8:8:8:8:8:4:0:4:8:8:8:8:8:7:4: 1. The profile was mapped by moving the HDO resonance in heavy water

over a 2000 Hz range in 25 Hz steps (unrelated to Aj). Note the rejection notch in the centre, designed for solvent suppression.


in a single experiment, and should therefore benefit from a sensitivity advantage of roughly JN.

Hadamard phase encoding [83-861 allows us to retrieve this sensitivity loss, thus making the soft-

pulse alternative much more attractive [87,88]. Consider the very simple case where we require four

correlations from four soft-pulse experiments. Instead of carrying out the four measurements one at a time, where each would have only one-quarter of the available time for signal accumulation, we apply all four soft pulses simultaneously. The trick is to repeat this multisite excitation experiment four times with different phase-encoding of the soft pulses. If a soft excitation pulse is shifted in phase by 1 80°, the corresponding correlation response is inverted. Suppose there are four correlations, represented by A - A’,

B - B’, C - C’ and D - D’ and that there is no interference between the four correlation experiments. Then we can establish a Hadamard matrix for the

phase-encoding:

A-A’ B-B’ C-C’ D-D’

Scan 1 + + + +

Scan 2 + + - -

Scan 3 + - + -

Scan 4 + - - +

The spectrum obtained from a single scan is quite complicated, containing results from all four correlations, but if we sum the results of all four scans we

find only the response A - A’, the others cancelling exactly. Similarly, by subtracting scans 3 and 4 from scans 1 and 2, we obtain only the response B - B’,

and so on. A complete separation is therefore feasible. The important point is that all the available spectrometer time has been used for each correlation experiment-there is a two-fold sensitivity gain over the experiment where the four correlations are examined separately.

Hadamard matrices are restricted to the order 4m,

where m is an integer, but if suitable spectrometer hardware is available it is possible to encode with

phase shifts through arbitrary angles 4 = 2ndk

where n and k are integers. For example, phase shifts of 2nd5 radians were used to encode Hartmann- Hahn coherence transfer experiments [89] in erythromycin [90].

Hadamard encoding was used successfully for the investigation of long-range carbon-proton couplings

in strychnine [88] and in melezitose [91]. Because of the low natural abundance of carbon-13, we can be sure that the individual polarization transfer experiments do not interfere with each other. For the strychnine experiment the soft carbon-13 pulses were implemented as half-Gaussians [17] and an INEPT sequence [92] was employed for polarization transfer to protons. Correlation through direct carbon-proton coupling was suppressed by the use of bilinear rotation decoupling [93] supplemented by phase cycling. Eight simultaneous correlation experiments were repeated eight times, coded according to a

Cl5 ---4-/-it--.-+

Cl4 -lr ,I

Cll- t -“V.++

II

Cl3 --II. IL

Fig. 28. Long-range carbon-proton correlations in strychnine,

carried out with eight simultaneous soft-pulse C - H coherence

transfer experiments, repeated eight times with phase encoding

defined by a Hadamard matrix. The individual correlation traces

were separated by decoding according to the same Hadamard

matrix. The source carbon sites are indicated on the left margin.

The conventional proton spectrum is shown at the bottom. Sensi-

tivity is improved by a factor of approximately J8.


Hadamard matrix of order eight. After decoding, the results are those set out in Fig. 28. No cross-talk can be discerned from interference between the different

correlation experiments.

6.9, Template excitation

The multisite excitation scheme described before (Section 6.5) defines a pulse-shape histogram in terms of a table of amplitudes and phases (Fig. 25) which drives a digitally-controlled waveform generator. Now an experimental free induction decay is stored in the computer as a table of real (Re) and

imaginary (Im) components, readily convertible into a table of amplitudes [(Re’ + Im2)‘“] and phases [arctan(Re/Im)]. What would happen if we fed the latter table into the waveform generator? First we should reverse the experimental data table in time so that all components come into phase at the end of the sequence rather than at the beginning. The waveform generator would then create a set of soft radiofrequency pulses at frequencies exactly matched to responses in the NMR spectrum. The relative intensities of these pulses would be proportional to the intensities of the corresponding NMR signals so they would excite the NMR responses in a non-linear fashion. This problem can be solved by first Fourier

transforming the free induction decay into the frequency-domain, setting all intensities to unity, and back-transforming to the time domain. Excitation with a set of soft pulses derived from this modified table would then be uniform at all sites. The NMR spectrum has created its own ‘template’ which excites a response only where there is an NMR signal [94]. A schematic illustration of this scheme is set out in Fig. 29.

Thanks to the stages of Fourier transformation and back transformation, the operator can intervene to create a modified template. For example, he might well wish to delete noise, artifacts or solvent peaks from the frequency-domain data before reconversion into the pulse generator table. Indeed, entire sections of the experimental spectrum could be nulled if necessary, so that these components are not excited in the soft-pulse version of the experiment. Various exotic shaping functions might be imposed on some of the pulse components by convoluting the appropriate frequency-domain responses with the Fourier

(solvent peaks, noise) amplitude & phase

Set intensities Real & imaginary

Inverse Fourier transformation

s(t) ~ Reverse time scale s(t) -_) S(4)

Fig. 29. Schematic diagram illustrating the generation of a set of soft

radiofrequency pulses from the experimental free induction decay.

This particular combination of soft pulses acts as a ‘template’ that

excites only those regions of the spectrum that contain NMR

signals.

transform of the desired pulse shape. The experimental spectrum might even be replaced by a completely artificial simulated spectrum.

Template excitation seems most likely to have practical utility in routine quantitative NMR spectroscopy where a selected component of a mixture needs to be monitored over time, or over many different batches of products from a repeated reaction. By

recording the NMR spectrum of a chosen pure component, a template is constructed which only excites at

the appropriate frequencies. Of course, in practice, some of these frequencies will be overlapped by signals from other components of the mixture, but the excitation can be nulled for these overlap regions without prejudice to the quantitative analysis, since the overall integral will only be reduced by a known factor. Alternatively the analysis can be calibrated by addition of a known amount of the chosen component.


Clearly sparse NMR spectra (such as those of

carbon-13) are more amenable to this method than

crowded spectra. A good illustration is provided by

the 100 MHz carbon-13 spectrum of a freshly-

prepared solution of a-n-glucose in heavy water, which initially consists of the c+anomer alone. This slowly converts into a 3654 mixture of a and 0 forms (mutarotation). A template suitable for excitation of

a-glucose is easily obtained, and an analogous template for P-glucose can be constructed by subtracting the o responses from the equilibrium spectrum [94]. In this manner the slow interconversion can be monitored by recording separate spectra from the CY and /3 anomers, using cx and /3 templates for excitation

(Fig. 30).

6.10. Excitation sculpting

The recent implementation of pulsed field gradient equipment on modem high-resolution spectrometers has opened up several important new schemes for suppression of unwanted signal components, tending

a

to replace the more time-consuming phase cycling

methods [95-981. Shaka and coworkers [99-1011 have seized this opportunity to devise a new selective

excitation scheme called ‘excitation sculpting’.

Suppose we have a sequence S, made up of soft or hard pulses and free precession intervals, that can be represented as a pure rotation about some arbitrary axis (a unitary transformation). Normally S would be a 180” refocusing pulse in a spin echo sequence,

flanked by matched pulsed field gradients, G-S-G. We shall see that these applied gradients act to

dephase any magnetization components that do not receive a perfect 180” pulse, in a manner similar to

the WATERGATE sequence [ 1021. We are interested in how the sequence G-S-G

converts initial magnetization components mx, rn r and mZ into final magnetization components Mx, MY

and MZ. The sequence S is represented as a general rotation through an angle CY about an effective field Befi, tilted through 8 = arctan ABIB, in a vertical plane X’Z, which is in turn rotated through an angle @ with respect to the X2 plane (Fig. 3 1). Each of the angles (Y,

Fig. 30. Slow generation of the fl anomer Of D-ghCOSe (a) from the CY form (b) upon dissolution in heavy water. The carbon- 13 spectra of the two

forms were monitored separately by soft pulse excitation using templates designed for the cx and @ anomers. Spectra were recorded at 30 min.

intervals. There are resonances from the CY and fl forms near 70 ppm that are only 0.05 ppm apart, but they are well separated by this technique.


0, and 6 are some function of offset A$ We assume sequence is that refocused by S. Averaged over all

that the gradient pulse G is sufficiently long and values of the phase angle $J, the probability that initial

intense that transverse magnetization is dispersed transverse magnetization mXY is refocused by S

completely. involves this same factor P(Af):

The first point to note is that the sequence G-S-G has some remarkable properties, quite different from those of S alone. Consider, first of all, longitudinal magnetization. The first pulsed gradient G completely disperses all transverse magnetization before the pulse S, and the second pulsed gradient G disperses any new transverse magnetization created by S. We may therefore consider longitudinal magnetization separately;

the sequence does not mix longitudinal and transverse components 1991. The trigonometry is similar to that used to describe the spin pinging sequence [20]. See Eq. (10) and Eq. (11) in Section 3.2. For any particular offset Af we may write:

Mxr = $[cos*8( 1 - cos cr)]mxy (21)

This important result is derived by the vector model in the Appendix. It means that the frequency-domain profile for refocusing is exactly the same as the spin inversion profile. Expressed in a different manner, it implies that the pair of gradient pulses has suppressed

all contributions to the echo that have not experienced a perfect 180” pulse. The only complication is a frequency-dependent phase shift 2/3 of the final transverse signal arising from a phase shift /3 of the rotation axis of the refocusing pulse:

Mz = mz[cos cv cos*8 + sin*@ = mz[ 1 - 2P(Af)]

(19)

P(Af) is the probability that a given spin is inverted by S (it is a function of resonance offset AL and lies between zero and unity).

= i[cos*& 1 - cos cr)] (20)

Now consider transverse magnetization, The only transverse magnetization that survives the G-S-G

Fig. 3 1. Co-ordinate frame X’ Y’Z used to define a completely gen-

eral pulse of flip angle OL about an effective field (bold arrow) tilted

through 19 in the X’Z plane, which is rotated through an angle 0 with

respect to the XZ plane.

MX = !px[c0s28( 1 - cos ar)] sin 20 (22)

MY = $my[c0s28( 1 - cos a)] cos 2p (23)

Levitt has shown that this type of phase error in a spin echo experiment is reversed in sense if a second echo is generated by a second refocusing pulse identical to the first [103]. So even in the case of a completely general pulse S, the frequency-dependent phase shift 2/3 can be cancelled in a two-stage sequence:

G, -S-G,-G2-S-G2

where G, and G2 are two uncorrelated gradients (so that no components dephased by the G, gradients are refocused by the G2 gradients). In the two-stage sequence shown above, the attenuation functions are multiplicative:

Mz = [ 1 - 2P(Af)]*mz (24)

MX = VV..)l*mx (25)

MY = [Wf)l*my (26)

This two-stage sequence is therefore the one normally used.

If we neglect the phase error 2/3, the frequency- domain profile P(Af) given by (Eq. (20)) is identical to that (Eq. (11)) calculated for the spin pinging


technique [20]. Clearly both techniques are based on similar principles, though excitation sculpting enjoys

the practical advantage of not having to rely on

difference spectroscopy. Eq. (20) indicates that the

optimum flip angle is 180” (cos CY = -1) and that the amplitude profile is determined by the cos% factor (bearing in mind that 6 is a non-linear function of resonance offset AB).

The double-echo scheme can be concatenated by appending further two-stage sequences, each with a new pair of S pulses. This is what prompted the term ‘excitation sculpting’ to indicate that any desired frequency-domain excitation profile can be achieved

by progressively chipping away at some initial profile, relying on the specific spin inversion properties of a series of different pulses St, S2, Ss, etc.

Excitation sculpting can be thought of as a new method to generate ‘pure-phase’ spectra by first exciting transverse magnetization with a hard 90” pulse and then using the two-stage spin-echo sequence where S is a suitably shaped 180” pulse, for example, a hyper- bolic secant [ 104- 1061. It offers an alternative to the BURP pulses [39] and to spin pinging [20]. Provided that the overall sequence does not get too long, several stages may be concatenated.

Excitation sculpting was used for water suppression, with S implemented as a combination of a hard 180” pulse and a soft 180” pulse at the water frequency, similar to the WATERGATE scheme

[ 1021. With the two-stage sequence:

G, - soft 180” hard 180” - Gt - G2 - soft 180”

hard 180” - G2

a water suppression ratio greater than 30000 was achieved, in a spectrum where the solute resonances are all in pure absorption without any appreciable intensity distortion [99].

By incorporating the bilinear rotation decoupling (BIRD) sequence [93] excitation sculpting can be used as a very effective isotope filter, retaining the signals of protons directly bound .to carbon-13 but suppressing the remaining proton signals to a very high degree [ 1001. It was also applied to the transient nuclear Overhauser experiment to obtain very clean cross-relaxation spectra [loll, without the subtraction artifacts normally associated with the conventional steady-state difference mode [ 107,108].

7. Decoupling

7.1. Adiabatic pulses

So far we have considered amplitude-modulated soft pulses, using the phase parameter only in the form of a linear ramp to shift the effective irradiation frequency. Of course, more sophisticated phase modulation schemes are feasible. For example, a second-order (parabolic) phase modulation corresponds to a linear frequency sweep during the pulse. This is the basis of the adiabatic pulse, where the sweep rate is constrained to satisfy the adiabatic condition [ 109- 1111

(27)

where 6 is the inclination of the effective radiofrequency field B,g with respect to the +X axis of the rotating frame. Under this condition, the nuclear magnetization vector follows the direction of Befl as it traces a semicircular arc in the X2 plane, the angle 0 sweeping from near + ?r/2 radians through zero to near - 7r/2 radians. The degree to which the adiabatic condition is satisfied may be quantified in terms of the factor

(28)

Normally the Q factor is appreciably greater than unity [ 1061. Since an adiabatic pulse has a finite duration, the sweep must be truncated at the extremities, and this must be done smoothly to avoid violating the adiabatic condition [ill]. One useful shaping function is

B1 = Bt(max){ 1 - kin (@)I”} (29)

where the, index n controls the severity of the cutoff function [112].

Adiabatic pulses have the useful property that they can invert the nuclear spins over a wide frequency range with a rather modest radiofrequency level B,, and are insensitive to B1 provided it satisfies the adiabatic condition. This means that spatial inhomo- geneities in B, (and Bo) are unimportant in this con- text. Although the most common application is broadband heteronuclear decoupling [ 112- 1251


band-selective experiments are also feasible, and in this sense the adiabatic pulse qualifies as a shaped soft pulse.

7.2. Band-selective decoupling

By their very nature, proteins tend to have carbon- 13 chemical shifts clustered in specific regions of the spectrum, suggesting that band-selective carbon- 13 heteronuclear decoupling is to be preferred over the more usual broadband decoupling if radiofrequency heating threatens to be a problem [ 126,127]. No

power would then be wasted irradiating empty regions of the spectrum. There are also applications for band- selective proton-proton decoupling, where the requirements are more severe because outside the specified band the decoupling effect should be negligible. Composite-pulse cyclic decoupling schemes are therefore inappropriate because they were conceived with no thought to the degree of out-of-band excitation.

A direct attack on this problem is to analyse the Fourier spectrum of the composite pulses that make

up the decoupling sequence, reshaping them by eliminating the higher-order terms. These ‘rounded- off’ composite pulses avoid decoupling those regions

of the proton spectrum outside the specified decoupling bandwidth [128]. Another approach is to employ shaped band-selective spin inversion pulses of the I-BURP type [39] in a conventional decoupling sequence such as MLEV4 [129]. For effective decoupling the pulse repetition rate must be high compared with the J-splitting to be removed. Since the effective bandwidth Af for I-BURP pulses is approxi-

mately given by Af = 4(t&’ where (t&-l is now the pulse repetition rate, the minimum effective bandwidth is many tens of Hz, and can be further extended by increasing B2 and reducing t,.

An experimental illustration [130] of band- selective proton decoupling on a simple test sample (diphenylmethylsilane, J = 4 Hz) is shown in Fig. 32. With I-BURP-2 soft pulses applied at a repetition rate of approximately 40 Hz, and a decoupler field given by yBd2n = 20 Hz, there is efficient decoupling over a frequency range of more than 150 Hz, with essentially unperturbed doublets elsewhere, except

-150 Hz -

395 Hz

Fig. 32. Band-selective decoupling of the methyl protons from the SiH proton of diphenylmethylsilane (J = 4 Hz) using an MLEV-4 cycle of

soft I-BURP-2 pulses at a repetition rate of 40 Hz and a radiofrequency intensity yB*R?r = 20 Hz. The methyl doublet is effectively decoupled

over a 150 Hz range of decoupler offsets.


b

a

I

5.0 I

4.0

I

3.0 ppm

Fig. 33. Band-selective decoupling of the methyl resonances of erythromycin in the frequency range 0.9 ppm to 1.3 ppm by means of I-BURP-2

soft pulses in an MLEV-4 phase cycle. The six muhiplets indicated by arrows are simplified by removing the 1:3:3: 1 quartet structure while the

remaining multiplets are essentially unperturbed.

a

b

d

e h Fig. 34. Band-selective excitation of the 400 MHz proton spectrum of strychnine, illustrating how the effective excitation band can be increased

by decreasing the duration of the E-BURP-2 pulse. The pulse durations were (a) I5 ms, (b) 5 ms, (c) 4 ms, (d) 3 ms, and (e) 1.5 ms.


for transition regions roughly 50 Hz wide. A practical application [130] of this technique is the decoupling of the methyl protons of erythromycin resonating in the range 0.9 ppm to 1.3 ppm, showing how six proton multiplets are thereby simplified, while the remaining multiplets remain unperturbed (Fig. 33).

8. Applications of soft pulses

8.1. Band-selective excitation

Multidimensional NMR spectroscopy rapidly accumulates very large data tables, particularly where high resolution has to be maintained in all the frequency dimensions. The problem can be alleviated by exciting only a restricted band of frequencies, just the region of chemical interest, leaving the rest of the spins at equilibrium along the 2 axis, The ability of band-selective pure-phase pulses (such as E-BURP-2) to vary the effective excitation band is illustrated in Fig. 34 for a section of the 400 MHz high resolution proton spectrum of strychnine. This shows the progressive narrowing of the effective bandwidth as the E-BURP-2 pulses are increased in duration. Note that these changes are accompanied by very little distortion of the structure of the multiplets, and any out-of-band excitation is very weak.

ti.

I

Applied to two-dimensional spectroscopy, band selective excitation would be employed to reduce the number of tl increments required to cover the F, frequency band with the necessary resolution [131]. This curtails the overall length of the experiment, limits the data storage requirement, and shortens the processing and display stages. Note that it is not normally feasible to limit the F, spectral width by

violating the Nyquist condition, because low-pass filtration cannot be used in this dimension.

A 4I 3.2 ppm w

Fig. 35. (Top) A 3.2 ppm section of the 400 MHz proton spectrum

of strychnine. (Below) Soft-pulse excitation of each individual spin

multiplet by means of an E-BURP-2 pulse. Note the lack of distor-

tion with respect to the conventional spectrum, and the high level of

suppression of all other responses.

8.2. Multiplet-selective excitation

Ideally the multiplet should be recorded in pure absorption, with the relative intensities undistorted, as in a conventional hard-pulse excitation experiment. A band-selective pure-phase pulse is probably the best choice in this case. Fig. 35 shows a set of individual spin multiplets from the 400 MHz spectrum of strychnine, excited with a soft E-BURP-2 pulse centered on each multiplet in turn. Each multiplet has essentially the same form as in the conventional hard-pulse experiment.

There are several situations where it is useful to be Multiplet-selective excitation can be employed

able to excite a single spin multiple& leaving the rest in two-dimensional correlation spectroscopy (for

of the high resolution spectrum unaffected. This might example COSY) to obtain a cross peak with high be the first step in a selective coherence transfer resolution and high digital definition. The second experiment, or a means for obtaining a particular pulse of the COSY sequence would then be a hard

multiplet with very high digital definition prior to 90” pulse, exciting a long narrow strip of the two- extracting accurate values of the coupling constants. dimensional spectrum parallel to the F2 axis. This


“l--r--l “I 0 . 0 . 0 .

. 0 0 0 0 0

I I I I

Fig. 36. (a) Typical cross-peak pattern observed with the soft-pulse

exo-COSY sequence where all passive spins remain unperturbed

(assuming two passive couplings with the same sign). (b) Corre-

sponding pattern observed in a conventional COSY spectrum with a

90” mixing pulse.

exploits the fact that very fine digitization in the F2

dimension is easily achieved without any significant increase in the duration of the experiment.

Alternatively, the second pulse of the COSY sequence may be implemented as two simultaneous soft pulses, one applied to site I and one applied to site S (if the cross peak of interest arises from I - S coherence transfer). Then all other regions of the two-dimensional spectrum are suppressed. This ‘exo-COSY’ technique [64] serves to emphasize the

deceptive simplicity of the conventional hard-pulse COSY experiment. Note that because the two simultaneous soft 90” pulses affect only the active spins I and 8, leaving all passive spins unperturbed, the structure of the final cross peak [132] resembles that observed in COSY-45 [133], E-COSY [134] or

the Z-COSY [ 1351 experiment (Fig. 36a), rather than the conventional COSY pattern (Fig. 36b).

8.3. Line-selective excitation

If the duration of the soft pulse is increased to the

order of one second, and if relaxation losses during the pulse still remain acceptable, then line-selective excitation is feasible. The applications lie in the field of multidimensional spectroscopy. For example, half- Gaussian soft pulses [17] can be used to excite one transition at a time in an alternative version of correlation spectroscopy where the F, dimension is explored directly, by incrementing the irradiation frequency in very small steps (say 0.5 Hz) with very

high selectivity. This ‘pseudo-correlation spectroscopy’ [ 1361 sacrifices the multiplex advantage of the conventional two-dimensional experiment, but

has the advantage that selected regions may be examined without having to excite the entire spectral width. It also permits the examination of the fine

structure of cross peaks in three-dimensional correla-

tion spectra [ 137,138] where the conventional hard pulse technique would yield only poor resolution because of the practical restrictions on the size of the data table.

If the spin-pinging technique [20] is employed for line-selective excitation in pseudo-correlation spec-

troscopy, then the diagonal peaks (which would normally appear in the dispersion-mode) are almost completely suppressed, because spin pinging cancels the dispersion components. Fig. 37 shows the pseudo-

COSY spectrum of the four coupled protons in meta-

bromonitrobenzene [ 1391. There are twelve well-defined cross peaks but essentially no diagonal peaks in this spectrum. This could be very useful when it is necessary to detect cross peaks close to the principal diagonal. The cross peaks can be displayed with high resolution and high digital definition with this technique.

8.4. Water suppression

The dynamic range problem created by the presence of a very intense water peak in high resolution spectra of aqueous solutions has spawned an entire field of water suppression techniques, several of which employ soft radiofrequency pulses. For

example, the ‘water-eliminated Fourier transform’ (WEFT) technique [ 1401 aims to catch the water resonance at the null-point of its inversion-recovery curve,

while the solute spins are already approaching Boltzmann equilibrium, relying on the appreciably longer spin-lattice relaxation time of water. Naturally this method is more efficient if the initial spin inversion pulse is frequency selective [ 1411.

The composite selective pulse scheme of Redfield [ 1421 was specifically designed to have a broad null at the water frequency. A soft-pulse scheme that uses a polychromatic pulse to provide uniform excitation except for a notch at the water frequency was already mentioned in Section 6.7.

One of the most effective water suppression sequences is WATERGATE [102] which employs two matched field gradients and a combination of a hard 180” pulse with a soft 180” pulse tuned to the


I . . I . .

. .

. .

ES

F 1

Hz 300 200 100 0

F 2

Fig. 37. Pseudo-correlation spectrum of merabromonitrobenzene recorded by stepping the frequency of the soft pulse (duration 1.3 s) in 0.3 Hz

increments across the F, frequency dimension. The spin-pinging sequence was used, thus suppressing the diagonal peaks. The conventional

high resolution spectrum is shown along the top margin.

frequency of the water resonance: field Ba, because spin-spin relaxation is usually

gradient - hard 180” soft 180” - gradient

The hard 180“ pulse refocuses all the signals dephased by the first gradient, except for the water signal which receives an effective 360” pulse. Shaka [99] has improved on this scheme as outlined in Section 6.10.

8.5. Resolution enhancement

In high resolution liquid-phase NMR of small-to- medium sized molecules, the limiting factor on resolution is the spatial inhomogeneity of the static

quite slow. One might therefore imagine that the resolution could be significantly improved by reducing the volume of the sample, using, for example, a small spherical microcell. In practice this turns out not to be the case, because small samples are disproportionately affected by the discontinuities of diamagnetic susceptibility at the sample walls. The effect can be mitigated by winding the radiofrequency coil directly onto a small cylindrical sample tube, surrounding them both with a perfluorinated organic liquid carefully matched in susceptibility. However, this strategy [143] was only used to improve the


sensitivity of extremely small (non-spinning)

samples, with no thought to resolution enhancement.

Here one can take a leaf out of the magnetic reso-

nance imager’s book and improve the resolution by reducing the eflective volume of the sample. This is achieved by soft-pulse excitation in an applied field gradient, usually a Z-gradient because this is the direction in which the natural gradients are most serious, being unaffected by sample spinning. Soft-pulse excitation is then confined to a flat, roughly disc-shaped volume. After excitation, the applied field gradient is extinguished, leaving the spins free to precess in the residual gradients of the magnet, except that the effect of natural Z gradients was markedly reduced. Fig. 38 shows how the resolution of the multiplet of the low-

field ring proton of furan-2-aldehyde is improved from 0.6 Hz to 0.08 Hz by selective excitation in an applied field gradient [144]. The more intense the imposed gradient, the smaller the effective sample volume. In principle, the effective volume could be restricted in all three dimensions.

8.6. Homonuclear equivalents of heteronuclear

experiments

An intrinsic property of the soft radiofrequency pulse is its ability to perform the equivalent of a heteronuclear experiment on a homonuclear system. We take as illustrative example the well-known INEPT polarization transfer sequence [92] which now forms part of many heteronuclear multidimensional experiments. Sensitivity is improved by starting with protons (which have a favourable Boltzmann distribution of populations), transferring polarization to carbon- 13 or nitrogen-15, and then returning the polarization to protons to benefit from their inherently

high detection efficiency [ 145,146]. The INEPT polarization transfer scheme exploits

spin echo modulation to generate antiphase magnetization of the I spins, transferring this polarization to the coupled S spins by means of a hard 90” pulse:

I : 90”(X) - 7 - 1 SO”(X) - 7 - 90”(Y)

s: 180” - 7 - 90” Acq.

Note the 90’ phase shift of the final Z-spin 90” pulse. The r interval is adjusted to the condition rJ,s = l/4 so as to prepare antiphase I-spin magnetization at the

end of the second r interval. The 90”(Y) pulse

converts this into a population disturbance which is

then ‘read’ by the 90” pulse on the S spins. The observed signal is therefore an antiphase multiplet, but it can be converted into an in-phase signal by appending a second stage of spin echo refocusing [147].

The homonuclear version of the INEPT experiment [148] may be illustrated by reference to a particular application. An E-BURP-2 soft pulse is used for selective excitation of proton 15a of strychnine, with simultaneous Gaussian 180” pulses to refocus proton 15a, and invert proton 14. Polarization transfer and detection of the signal from proton 14 (the S spin) is achieved by a hard 90” pulse.

I : soft 90”(X) - 7 - soji 180”(X) - 7 - 90”(Y)

s: soft 180” - 7 - 90”(Y) Acq.

Note that the two 180” soft pulses define the two sites

under investigation; all other signals are suppressed by phase cycling. The result is a broad antiphase doublet for proton 14 since there are many small unresolved passive couplings at this site. When the soft-pulse experiment is repeated with a second refocusing period, proton 14 is converted into an (unresolved) in-phase doublet,

There is a useful trick that can be used when we have both the in-phase and antiphase versions of the same doublet [ 148,149]. Individually, neither version of the doublet can be used to get an accurate measure of the coupling constant, because the antiphase case overestimates the splitting, and the in-phase case underestimates it, leaving the doublet unresolved. But by reprocessing the data in the time domain we

can extract an accurate value of the coupling constant. The homonuclear INEPT experiment gives a detected time-domain signal that may be written S = So

sin(xJrst), while the refocused version gives a signal S = St, cos(n;llst). We multiply these by cos( nJot) and sin(rJat) respectively, where Jo is a large, accurately known ‘dummy’ splitting, then add the results:

Sa sin(?r&t) cos(7rJot) + SO cos(TJIst) sin(7rJat)

= Sa sin[n(Jts + J,)t] (30)

Fourier transformation gives an antiphase doublet


.,A b

Hz

Fig. 38. Resolution enhancement by restricting the effective sample

volume. The conventional 300 MHz spectrum of the low-field ring

proton of furan-2-aldehyde (a) shows a linewidth of 0.6 Hz, but

when excited by a DANTE sequence in an applied Z gradient and

subsequently detected without the gradient, the linewidth is reduced

to 0.08 Hz (b).

with a large resolved splitting ]Jts + Jo] Hz, from which an accurate value of Jis can be extracted since Jo is known. An alternative treatment generates an in-phase doublet with the same splitting, exploiting

the identity:

So cos(?r.&) cos(7rJat) - Sa sin(?rJ&) sin(7rJst)

= S, cos[*(Jts + J&l (31)

The technique is called ‘J-extension’ [ 1491.

Fig. 39 shows the in-phase and antiphase versions of the response from proton 14 of strychnine obtained by these soft-pulse INEPT experiments [148]. The splitting of interest cannot be resolved in the in- phase response (Fig. 39a) and is grossly overestimated (7.1 Hz) in the antiphase response (Fig. 39b). When the splitting is ‘extended’ by adding a dummy splitting of 40.0 Hz the new splitting is large enough to be

measured by conventional methods (Fig. 39c and d), yielding a coupling constant Jts = 4.8 Hz.

8.7. Selective correlation experiments

Two-dimensional spectroscopy has proved so suc-

cessful for correlation experiments that the possibility of obtaining the same results by soft-pulse methods is often overlooked. Yet many chemical applications are based on only a small number of selected correlations, rendering the rest of the information in the two- dimensional correlation spectrum redundant. In the soft-pulse alternative, the large volume of information in a two-dimensional spectrum is traded for a wealth of fine detail in the one-dimensional analogue.

When performed in the same total time, the two- dimensional experiment and its one-dimensional counterpart have essentially the same sensitivity, because the signal loss through relaxation during the evolution time tl is just about matched by relaxation

at

b -------7 V

A

._\A ,dL

4 200 Hz *

Fig. 39. The in-phase (a) and antiphase (b) signals from proton 14 in strychnine obtained by polarization transfer from proton 15a by the

homonuclear INEPT technique. In (a) the splitting of interest is not resolved, while in (b) it is grossly overestimated (7.1 Hz) owing to

destructive interference in the overlap region. (c) and (d) The method of J-extension adds a known dummy splitting of 40.0 Hz, allowing

the unknown coupling constant (Jts = 4.8 Hz) to be evaluated.


losses during the soft pulse and any subsequent fixed

delay. However, if several soft-pulse experiments

need to be carried out in the same total spectrometer time, then Hadamard phase encoding should be

employed, in order to retain the highest possible sensitivity (Section 6.8.).

The simplest one-dimensional correlation experiments are designed by replacing the first hard excitation pulse with a soft pulse, keeping the remainder of the sequence unchanged. For example the one-dimensional COSY scheme would be written:

soft 90”(X) - r - hard 90”(X) Acquire

where the interval 7 is now a fixed period related to the reciprocal of the active coupling Jis. The most efficient transfer of antiphase coherence occurs when r = 1/(2.Zis). Some allowance may be necessary for losses caused by spin-spin relaxation during the soft pulse and the fixed delay. The usual embellish- ments of refocused COSY [150] and relayed COSY [ 1501 are readily implemented. Similarly, the ubiquitous NOESY experiment is easily converted into its one-dimensional analogue [151]. In this case there is an advantage in initiating the sequence with a soft half-Gaussian pulse followed by a hard 290” purge pulse so that, with the appropriate phase cycling, any cross-relaxation effects during the soft pulse may be neglected [ 1501. Similar considerations are applicable to the one-dimensional version of the ROESY [152] experiment.

8.8. Hartmann-Hahn coherence transfer

Hartmann and Hahn [153] demonstrated how

coherence could be transferred from one chemical

site I to another site S by the application of spin lock-

ing fields B 1 at site I and B2 at site S, and by satisfying

the Hartmann-Hahn matching condition:

YI& = ~sB2 (32)

Most Hartmann-Hahn experiments are carried out in the solid phase with the I spins in high abundance coupled together by strong dipolar interactions, and the S spins in low abundance.

In liquid-phase high resolution spectroscopy the situation is quite different, and the key interaction is the spin-spin coupling, but the same matching condition is applicable. If an IS spin system is perturbed from equilibrium, the Hartmann-Hahn effect transfers coherence back and forth from one site to the other in a cyclic manner at a frequency equal to i.Zrs Hz, and this oscillation continues until damped by spin-spin relaxation. In contrast to other coherence transfer techniques, the observed signal is an in-phase

multiplet. An illustrative example is shown in Fig. 40 for the

IS protons in uracil (.Z,s = 14 Hz). The system is prepared by means of a soft 90”( - X) pulse to generate an inverted signal at site Z, and by a simultaneous soft

90”( + X) pulse to give a positive-going signal at site S [80]. E-BURP-2 multiplet-selective pulses are well suited for this purpose. Spin-lock fields (B1 = B2) are then applied simultaneously at sites Z and S for a variable period 7. We see that as T is increased, an increasing amount of negative coherence is transferred to site S while an equal amount of positive coherence is transferred to site I. When 7 reaches

l/.ZIs (70 ms), the response at site S is completely

II I spin S spin

Fig. 40. Cyclic interchange of coherence between two proton sites in uracil (J ts = 14 Hz). The system was first prepared with the I-spin

magnetization along - Y and the S-spin magnetization along + Y. Then equal spin lock fields were applied at the two sites for a duration r,

where r was incremented from 10 to 140 ms in 10 ms steps. When r = 70 ms, the inverted signal appears at site S and the upright signal at site I. A complete cycle is completed for r = 140 ms.


inverted, and the response at site I is upright. As the spin-lock duration is further increased, the coherence swings back again, reaching a full cycle of interchange when r reaches 2/Jis (140 ms) [ 1541.

The maximum coherence transfer from site I to site S occurs when the spin lock duration r = l/Jis. We may use this as a simplified version of the TOCSY [155] or HOHAHA [ 1561 experiment, where, instead of dissipating the available coherence indiscrimi- nately throughout the entire coupled spin system, we

direct it all to a single designated site. The use of two simultaneous (selective) spin-lock fields defines the two chemical sites under investigation and excludes all others. It does not matter if the resonances of the two active sites are overlapped by other responses; difficulties only arise if any extraneous responses at the two sites are themselves coupled. The experiment can therefore be used to pick out particular responses from a congested region, for example the proton resonances in sugars, where the anomeric proton is usually in the clear. It may even be used to probe the crowded region directly.

An interesting extension of this principle is to con- catenate several Hartmann-Hahn transfers [ 1571. After coherence transfer from site I to site 5, a second stage can be initiated by moving the frequencies of the spin lock fields to site S and site R, still satisfying the homonuclear Hartmann-Hahn condition (B, = B2) and adjusting the spin-lock period to the new condition r = l/Jsk. Since the transfer always involves in-phase magnetization, the process can be extended

at will, provided that the relaxation losses during the aggregate spin-lock time are acceptable. Fig. 41 is a

schematic diagram illustrating the sequence for three successive stages of Hartmann-Hahn coherence

transfer. As a practical illustration, six consecutive coher-

ence transfers were implemented in the proton spectrum of sucrose octa-acetate [ 1541 starting at Hl and terminating at H7, one of two non-equivalent

methylene protons (Fig. 42). In this experiment only a single spin-lock field was employed, centred at the mean chemical shift (6t + 6,)/2, rather than two simultaneous spin-lock fields at 6t and As. The Hartmann-Hahn condition then requires that effective fields at the two sites be equal. This is rather less frequency-selective, but it achieves faster coherence transfer, the maximum interchange occurring for the condition r = 1/(2Jis). This is a distinct advantage when many stages are used, because it helps reduce losses through spin-spin relaxation during the spin- lock intervals. Note that the six-stage transfer requires a knowledge of the seven proton chemical shifts and the six coupling constants, though these values need not be particularly accurate. The experiment serves as a structural tool, indicating that these seven protons are coupled in a chain in the order demonstrated.

Multiple Hartmann-Hahn coherence transfer has many ramifications. It can be used to measure spin- lattice relaxation times by prefixing the sequence illustrated in Fig. 41 with a hard 180” pulse followed by a variable relaxation delay tR. The signal from the

Spin-lock fi

- Spin-lock & - Spin-lock f2 _

P Spin-lock f3 - Spin-lock A

Fig. 41. Schematic representation of a sequence for the implementation of three consecutive stages of coherence transfer by means of the

homonuclear Hartmann-Hahn experiment. Coherence is passed from site 1 to site 4 in three stages.


h

I I I I I 6.0 5.5 5.0 4.5 4.0 ppm

Fig. 42. Six consecutive stages of coherence transfer along a chain of seven coupled protons in sucrose octa-acetate, implemented by successive

Hartmann-Hahn coherence transfers. At each stage, a single spin-lock field E, is placed at the mid-point of the two relevant proton shifts, with

a duration determined by U(2.Q where J is the appropriate proton-proton coupling. (a) The conventional spectrum. (b) Selective excitation of

Hl. (c) Coherence transfer Hl + H2. (d) Hl -t H2 + H3. (e) Hl -t H2 -t H3 + H4. (f) Hl - H2 -+ H3 + H4 + H5. (g) HI + H2 + H3 -+

H4-+H5*H6.(h)Hl+H2+H3+H4+H5+H6+H7.

proton at the end of the chain then recovers at a rate determined by the T1 value of the first proton in the chain. By thus effectively ‘transferring’ the relaxation behaviour of the first proton to the last proton, it is possible to study relaxation of a site that is not directly

observable owing to severe overlap [158]. A similar principle can be applied to the problem of

determining the connectivities of carbon atoms. Although the INADEQUATE experiment [ 1591 gives an unequivocal answer to this question, it suffers from poor intrinsic sensitivity, since it relies on find-

ing two coupled carbon-13 spins in the same molecule, a relatively rare event. If a chain of concatenated proton-proton coherence transfers is terminated with a non-selective stage of polarization transfer to carbon-13, the carbon sites are identified one-by- one, and high sensitivity is retained [ 1601.

Fig. 43 is a schematic diagram illustrating three such Hartmann-Hahn transfers in the glucose ring of sucrose octa-acetate, identifying sites C2, C3 and C4. In practice it is necessary to decouple the protons from carbon- 13 during the proton-proton transfers, to

refocus the antiphase carbon-13 signals, and to decouple carbon-13 from protons during the final acquisition stage. Fig. 44 shows the experimental results used to identify sites Cl through C6 of the glucose ring. The determination can be carried out

with an increasingly long chain of Hartmann-Hahn transfers, or by several overlapping short chains. For example, once C5 was identified, the assignment of C6 requires only the short transfer chain H.5 - H6 - C6. Note the reassuring absence of artifacts in the

carbon- 13 spectra.

8.9. Relaxation in multispin systems

We take it for granted that individual spin-lattice relaxation times can be measured in systems of many coupled spins by using hard-pulse methods to invert all the spins in the same manner. This is the classic ‘inversion-recovery’ experiment [4]. Yet we know that in many situations in proton spectroscopy the

relaxation of one proton is affected by dipolar interaction with its near neighbours in the same


a C6

g-

f

e

b

C

Fig. 43. Identification of carbon sites through Hartmann-Hahn

coherence transfer along a chain of coupled protons (decoupled

from carbon- 13) followed by non-selective polarization transfer

to carbon-13.

molecule-the origin of the nuclear Overhauser

effect. These interactions come into play if one proton is selectively inverted at the beginning of the inversion-recovery experiment, all the neighbour protons remaining at Boltzmann equilibrium. The recovery curve for this chosen proton is no longer strictly exponential, as the initial population disturbance starts to spill over onto the other protons through cross relaxation [161-1631. However, if the measurements are restricted to the initial section of the recovery curve these slower evolutions can be neglected. If the relaxation mechanism is purely dipolar, then this initial rate is 3/2 times slower than in the conventional hard-pulse inversion recovery technique, a manifestation of the famous ‘3/2 effect’ [ 1 lo]. If the relaxation of the chosen proton is only partly determined by dipolar interaction with its neighbours in the same molecule, then the ratio of the non-selective relaxation rate to the selective

b Cl

PPm 100 80 60

Fig. 44. (a) The 100 MHz carbon-13 spectrum of sucrose octa-

acetate in acetone-de,. The responses marked F are from the fructose

ring and are not involved in this Hartmann-Hahn experiment.

(b) Coherence transfer HI - Cl in the glucose ring. (c) HI -

H2 - C2. (d) Hl + H2 - H3 - C3. (e) Hl - H2 4 H3 - H4

+C4.(f)Hl-H2+H3+H4+H5+CS.(g)H5dH6+C6.

Non-selective excitation

a

HI Selective excitation

Fig. 45. Spin-lattice relaxation in a glucopyranose derivative.

(a) Recovery after hard-pulse spin inversion of all the protons.

(b) Recovery after soft-pulse spin inversion of HI, with all other

protons at Boltzmann equilibrium. The ratio of the initial rates of

recovery observed in (a) and (b) is 1.5, indicating that HI is relaxed

exclusively by intramolecular dipolar interactions.


relaxation rate is equal to the nuclear Overhauser which can then be used to analyse the modulated enhancement: decay observed in a second experiment:

R -==1+?J R

(33) sel

S spins : soft 90”(X) - r - soft 180”(Y) - r - Acquire

An experimental demonstration of this phenomenon is shown in Fig. 45 for spin-lattice relaxation of the

protons in 3,4,6-tri-O-acetyl-l-0-benzoyl-2-chloro- 2-deoxy-@-D-glucopyranose [ 1641. We concentrate on the spin-lattice relaxation of Hl. In the hard- pulse inversion recovery experiment (Fig. 45a) the relaxation time is measured as 1.97 -’ 0.05 s. In the alternative mode, where Hl is selectively inverted by a 38 ms soft pulse, the reciprocal of the initial relaxation rate gives a time constant of 2.99 + 0.05 s (Fig. 45b). Consequently the ratio RnorjRsel = 312,

indicating purely dipolar relaxation in this case. These experiments were in fact carried out with a rectangular soft pulse; with modem technology there would be no difficulty in performing such experiments

with suitably shaped soft pulses, or in extending them to schemes where any number of protons are inverted in any desired combination.

I spins soft 180“

Thus the coupling constant Jts and the spin-spin

relaxation time can be measured [ 1661.

8.10, Measurement of unresolved splittings

It is sometimes important to be able to measure spin-spin splittings that are just beyond the limit of the instrumental resolving power. One approach to this problem was already outlined in Section 8.5. Another method relies on the fact that for small-to- medium sized molecules, the natural line widths are often appreciably narrower that the instrumental line width. So a spin-echo experiment is in order. Hahn and Maxwell [165] showed that a Carr-Purcell spin echo from a site S is modulated by spin-spin coupling to an adjacent site I, if both the I and S

spins experience 180” pulses:

An illustration is shown in Fig. 46 for the small para coupling in metabromonitrobenzene. A soft E- BURP-2 pulse was used to excite the S spins in a spin

echo sequence in which the refocusing pulse was RE- BURP [39]. This generated an exponential decay (Fig. 46a) indicating a spin-spin relaxation time of 1.27 2 0.04 s. When the experiment was repeated with a simultaneous I-BURP inversion pulse on the coupled I spins, a modulated decay was recorded (Fig. 46b). By dividing the ordinates in the modulated

decay by the corresponding ordinates of the monotonic decay, almost full cycle of the modulating cosine wave is obtained before the quotient became unreliable because of low signal amplitudes (Fig. 46~). This provides an accurate value for the small para

coupling constant, Jis = 0.358 t 0.009 Hz. By comparing the corresponding monotonic and modulated decays of spin echoes in furan-2-aldehyde it proved possible to measure an even smaller (unresolved) coupling of 0.11 Hz.

9. Discussion

A4 = Ma cos(?r&t) exp( - tlT2) (34)

However, if the l-spin 180” pulse is omitted, the echoes from the S spins decay monotonically with a time constant equal to the spin-spin relaxation

time TZ. Consequently by performing the soft-pulse experiment:

S spins : soft 90”(X) - r - soft 180”(Y) - r - Acquire

we obtain the spin-spin relaxation time of the S spins,

Soft-pulse experiments offer an attractive alternative to the ubiquitous multi-dimensional spectroscopy, particularly when a given chemical or structural problem can be solved with one or two pieces of spectroscopic information. The selective experiment is usually faster, uses less data storage and is simpler to display. Then there is another range of high resolution experiments where selective excitation is an essential feature. In the majority of these applications there is a decided advantage in deploying shaped pulses, principally to avoid unwanted excitation of adjacent resonances, but also to design an appropriate excitation profile. This

review has sought to demonstrate the increasing scope of soft-pulse experiments, and to indicate the technical improvements in pulse shaping that are now


1.0 -

0.8 -

0.6 a

1

0.4 1

0.2 -

0.0 1

1 1.0: ,

0.8 :

0.6 : I b I

0.4 7

0.2 : 0.0 : J!J()

-0.2 : i ,““,“‘.,~“‘,““,~~‘~,‘~~‘,~~“1’~~’,””,””

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

C

-0.8

-1.2 1”“,“.‘,.“‘~““,‘.“,.‘.‘,““,““,“..,”’.,’.’. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.

seconds

5

Fig. 46. (a) Exponential decay of spin echoes from the low-field proton of metabromonitrobenzene excited by a soft E-BURP-2 pulse and

refocused by a soft RE-BURP pulse. (b) Modulated spin-echo decay curve of the same proton obtained by applying a soft I-BURP inversion

pulse to the high-field proton at the same time as the RE-BURP pulse. (c) Separation of the modulation term cos(r_Ftst) by dividing the echo

amplitudes in (b) by the corresponding amplitudes in (a). This yields a coupling constant of 0.358 2 0.009 Hz. Diagram courtesy of Hsi-Wei Jia.

available on most modem spectrometers. The cover- age of applications of soft-pulse experiments is not

comprehensive, but seeks to highlight some of the more interesting innovations.

Appendix

The theoretical treatments of spin pinging [20] and

of excitation sculpting [99] are very similar. Hwang and Shaka [99] have used rotation operators to demonstrate that the sequence G-S-G exhibits a frequency-domain profile for spin echo refocusing that is identical to the spin inversion profile of the pulse S. The same result can be derived in a more pictorial manner using the vector model of nuclear magnetization.

We follow the evolution of a typical isochromat m


b

Fig. 47. (a) Tilted reference frame in which the X’Y plane is normal to the effective field BeE. (b) Rotation through an angle 01 in the tilted plane.

from a small volume element of the sample. All the isochromats start in phase aligned along the +Y axis

of the rotating frame. The first gradient pulse G causes m to precess through an angle 4, leaving components my = mcos r#~ along the +Y axis and mx = msin C$ along the fX axis. (Individual isochromats are dispersed essentially uniformly around the XY plane.) Next a pulse S is applied about the +X axis, creating an effective field B,a that is tilted through an

angle 6 = arctan (ABIBi). In order to calculate the effect of S, we transform to

a new tilted co-ordinate frame X’YZ’ where 2 is the effective field axis (Fig. 47a). Tilting has no effect on Y magnetization. We consider first the effect on my =

mcos 4. After rotation through an angle CY (Fig. 47b) and transformation back into the XYZ frame, we have

a new Y component:

rnly = mcos C$ cos a (35)

and a new X component:

rnlx = mcos 9 sin (Y sin 8 (36)

The evolution of mx = msin 4 involves a component along Beff which is not changed by rotation, and a component perpendicular to B,H which is rotated through an angle (Y. Together they create a new X component:

mttx = msin C$ sin2 6 cos (Y + msin $J cos2 8 (37)

and a new (negative) Y component:

mNY = - msin C/J sin e sin Q! (38)

The total magnetization along the + Y axis is therefore:

mty + rn’ly = mcos 4 cos Q - msin 4 sin 8 sin (Y

(39)

and the total magnetization along the + X axis is:

rntx + rn’lx = mcos 4 sin (Y sin 8 + msin 4 sin2 6 cos o!

+ msin 4 cos2 6 (40)

Next these magnetization components precess through the same angle r#~ through the action of the second gradient pulse G: This generates an X component:

+ +msin2r$[cos (Y + sin2 8 cos a! + cos2 01

+ m(cos2 4 - sin2 C#J) sin ~9 sin o (41)

When averaged over all possible 4 values, all these terms vanish because (sin24) = 0, (cos’ C#I) = (sin2 4) = $. The corresponding Y component is given

by:

+ mcos2 + cos a - msin2+ sin 0 sin (Y - msin2 $J

(sin2 8 cos (Y + cos2 0) (42)

When averaged over all possible C#I values this reduces

to:

+ $m[cos a - sin2 8 cos (Y - cos2 81

= + irn cos2 B(cos a - 1) (43)

This is the derivation of Eq. (21) and demonstrates that the spin echo has the same frequency-domain profile as the spin inversion profile. It is also the expression for the absorption-mode signal in a spin

pinging experiment [20]. So far we have assumed that the radiofrequency

pulse is applied along the + X axis. If the pulse S is phase shifted through an angle /3 with respect to + X, the spin echo is focused along an axis shifted through


2/3 with respect to the - Y axis, as is well known from the analysis of Levitt [ 1031. In particular if /3 is 90”, we have focusing along + Y rather than - Y, according to the Meiboom-Gill modification. If a second echo is formed by appending a further sequence G2-S-G2 (where G, f G2), the phase shift of the echo is cancelled.

References

[ 1] G.A. Morris, R. Freeman, J. Magn. Reson. 29 (1978) 433.

[2] S. Alexander, Rev. Sci. Instr. 32 (1961) 1066.

[3] R. Freeman, S. Wittekoek, J. Magn. Reson. 1 (1969) 238.

[4] R.L. Vold, J.S. Waugh, M.P. Klein, D.E. Phelps, J. Chem.

Phys. 43 (1968) 3831.

[5] H. Kessler, S. Mronga, G. Gemmecker, Magn. Reson. Chem.

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