G. Wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 (v2(n). In addition, p is proportional to the average of 3. Compensation by modulating the amplitude of the the field strength pulse train with a cosine function which results in dm=xv2(t)/△p the application of the disturbing field at +Av and Av, thereby cancelling the effects at the fre- p=m()p2()(△)2=n2()△ quency of 4. Applying the disturbing field twice, once before In these equations nr and p are given in units of nd once after a non-selective 180 pulse and radians. Eqs.(21)and(22)exhibit clear differen thereby refocussing the adverse effects between p and nr. The rotation p is always smaller than the phase shift pnr, and when the average strength of the applied field (va(n)) is zero, then the rotation p, 2.2.2. Amplitude-modulated pulses but not nr, vanishes completely, a situation encoun- Modulating the amplitude of a pulse permits the tered with composite pulse decoupling wherev2(n))= design of specific excitation profiles. Since the o due to the applied phase cycling schemes. A typical shape of a pulse and its excitation profile are not numerical example relates to the decoupling of alpl related by a Fourier transformation [16], more elabo- carbon resonances from carbonyl carbons which rate procedures must be used to find the optimal pulse separated by Av 18 000 Hz on a 600 MHz NMR shape based on a desired excitation profile. A large spectrometer. The application of an 8-ms-long, low- selection of pulse shapes have been developed and power WALTZ decoupling sequence to carbonyl haracterized [58]. From a given pulse shape the exci- carbons using a 1 kHz decoupling field strength on profile can be calculated by integrating the results in onr=1.4 rad(80%)and p=0. Using instead Bloch equations given in Appendix A. Software of waltz a rectangular 180 pulse at the carbonyl packages on commercial spectrometers include corre- frequency with its sixth null at 18000 Hz(Eq ( 20)to sponding routines(Bloch simulator) which help to refocus the effect of the carbonyl couplings, the choose the appropriate shape for a specific experiment values change to nr=0.13 rad(7.5.)and p=0.6. and to determine its parameters. In general, a good In the course of a multidimensional NMR experi- amplitude-modulated pulse should have an adequate ment selective decoupling may be applied during an frequency selectivity, uniform excitation, uniform volution time which is incremented In this situatic phase behaviour and a short duration. Some of these increases linearly and considering Eq. (21)it desired properties contradict each other. Improving becomes clear that the time-dependent phase nr the selectivity, for example, tends to increase the will manifest itself as a frequency shift v bulse duration which should be kept as short as pos- sible to counteract relaxation losses. During the appl vnr=(v2()2△y cation of a selective pulse, the magnetization components of interest accumulate an offset-depen ith the, numerical ex ample given above using dent phase error. If the phase error is approximately linear across the excitation bandwidth it can be refo- 27. 8 Hz. The numerical examples show that the non- cused by a non-selective 180 pulse after a suitable resonant effects described by or and p may cause delay within or after the selective pulse [59]. Far from severe signal loss and care has to be taken to corre for their influence. Four different procedures can be its irradiation frequency, a selective pulse may intro envisaged to compensate for non-resonant effects duce non-resonant phase and amplitude errors as described by Eqs. (21)and(22). These deficiencies [35 can be corrected using the same methods as described 1. Adjusting the phase and flip angle of the pulse for rectangular pulses in Section 2.2.1 directly following or preceding the occurrence of The performance of an amplitude-modulated pulse non-resonant effects depends on the initial state of the magnetization. A 2. A phase error occurring during an evolution time selective 180 pulse that provides good inversion can be corrected by applying a phase correction properties for longitudinal magnetization in general fter Fourier transformation does not perform well as a 180 refocusing pulse for
G Wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 09 transverse magnetization. For th s reason. some frequency shift vor during the application can be shapes of pulses, for example the BURP pulses [60 obtained by linearly increasing the pulse phase (t) are grouped into families with a member for excita- with time while keeping the carrier frequency fixed at tion, inversion and refocusing. The most frequently used shaped pulses are Gaussian, sinc with no one pair of side lobes, Gaussian cascades [61] which COs(2丌ot+中(t)=cos(2mu21+(重。+2rvot) are based on individual Gaussian-shaped pulses, and ulses of the BURP family [60] The increment per unit time depends on the required The application of amplitude-modulated pulses in a offset frequency Poff from the carrier frequency vo. A sequence requires all stages of the transmitter positive phase increment will shift the frequency to pathway of the spectrometer to be linear, otherwise higher values, decrementing the phase lowers the the shape and hence the excitation profile deviate effective frequency of the pulse. For example, amide from the one selected. The direct determination of protons or carbonyl carbons resonate at a higher abso- the pulse length of an amplitude-modulated pulse te frequency than methyl resonances of protons or can be rather tedious. In this situation the use of a carbons, respectively. The time point 1=0 sets the Bloch simulator program which integrates the Bloch reference phase o for the off-resonance pulse. Any equation (A 1)seems more efficient for the further pulse on transverse magnetization of the same determination of the parameters of a shaped pulse nuclear species during the pulse sequence at a later by using the known parameters of a rectangular 90 time point T will have the phase (n)which in general pulse to determine the field strength of a given power deviates from o. The performance of pulses which setting. A prerequisite for such calculations is the invert z magnetization will usually be independent of inearity of the transmitter channel. In addition ampli- (T). However, off-resonance pulses that create or act tude-modulated pulses can be rather sensitive to RF on transverse magnetization require the phase (n)to inhomogeneity of the coil in the probe and therefore be adjusted properly, otherwise signal may be lost require good rF homogeneity for best performance q(T) can be calculated on the basis of Eq(24)and for example, the proper time chosen where更(门= 2.2.3. Amplitude- and phase- modulated pulses When the phase p(n)in Eq.(24) depends non- a pulse excites a spectral range around its irradia- linearly on the time f, the effective frequency changes tion frequency. However, some experimental tech during the pulse. An important group of pulses using niques require the pulse to excite at a frequency frequency sweeps during their application are the different from this position. For example, in the adiabatic pulses. These pulses excite, invert or refocus middle of a period of free precession a selective magnetization over a very wide frequency range at the 80 pulse may need to be applied to the amide cost of a longer pulse duration and a phase dispersion protons to decouple them from the alpha protons. If across the excitation bandwidth In applications the carrier is to remain on the water resonance. this such pul requires a homonuclear off-resonance selective pulse robust to RF inhomogeneities but not when al Off-resonance pulses have the advantage over switch for refocusing [62]. An additional feature ing the carrier frequency in that the latter method adiabatic pulses very attractive: doubling the RF generates a phase shift which must be taken into field strength of the pulse quadruples the bandwidth ccount and complicates the phase setting of subse covered[62]-a distinct advantage over conventional quent pulses applied to the precessing magnetization. pulses which excite a bandwidth proportional to the The centre of excitation of a pulse cannot only be strength of the pulse changed by changing its frequency but also by The concept of adiabatic pulses can be understood hanging its phase during the application. Two types with the help of a description in the rotating frame With the applied radiofrequency field B far above pulses shifted off-resonance by a fixed frequency resonance, the effective field Beff corresponds to and pulses where the frequency is swept during residual magnetic field Bz(Eq(A3)and Fig. 38 in their application. a pulse with a fixed off-resonance Appendix A)and the magnetization M stays aligned
210 G wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 along the positive z axis. Sweeping the frequency of conclusion that the amplitude of B, should be B, towards resonance will tip Berr away from the z axis smoothly reduced to zero towards the transverse plane. For a sufficiently slow As long as Eq. (26)stays fulfilled, the time-depe weep the magnetization M stays aligned with the dent phase (r)of the pulse can have any functional changing direction of the effective field. At resonance, form. For simplicity, very often linear sweeps are Benr and M lie in the transverse plane. As the frequency applied(Eq. (28). with the total sweep range Fand of the exciting field B, passes through resonance and the total duration of the sweep T, the change of Bper is subsequently swept to frequencies much lower than unit time(dB Jdn) becomes 2TFlYTs. Linear sweeps resonance, the magnetization M moves on towards the result in a quadratic time dependence for the phase negative z axis(Eqs. (A 3)and(A 4). For M to follow (n)of the constant frequency vo during the applica Berr the adiabatic passage condition must be fulfilled tion of an adiabatic pulse 129): cos(2πt+中()=cos(2πn!+(+(2xF/r)t2 de/dr< yBer with Befr=(B1(0)+B2(2) where B, represents the offset from the resonance Most applications of adiabatic pulses use single inver frequency and e describes the angle between the sion pulses or trains of inversion pulses in decoupling effective field and the z axis (Eq.(A4)). with sequences. When used to refocus transverse magneti Eqs.(25)and(A 4)one finds that the critical stage zation, their inherent long execution time may cause of any adiabatic sweep is the point where Bi sweeps problems with fast relaxing magnetization as well as with evolution due to scalar couplings which may erough resonance for a given magnetization compo- modulate the signals. On the other hand, depending Iting in t on the chemical shift range of scalar coupled nuclei. dB()dt≤yBi(t) (26) partial decoupling may be achieved during the adia batic pulse [63, 64]. Applications for refocusing are In addition to the condition described in Eq.(26), not yet very common, but their applicability has adiabatic passage requires that no relaxation occur been demonstrated [63, 65, 66] during the sweep period. In practical implementations the field cannot be swept starting at an infinitely large 2.3. Magnetic field gradients offset; however, when choosing a finite starting value the effective field and the magnetization M are not Pulsed magnetic field gradients(PFGs) have been aligned. Consequently, M precesses around B which degrades the performance of adiabatic pulses used in NMR spectroscopy for more than thirty years to study the diffusion of molecules in solution [67, 681 To reduce this initial precession the starting angle e must be as small as possible. For small values of e resolution NMR in solution (69-73] only since the one finds with Eq(A 4) introduction of actively shielded gradient coils ⊙=B1B2 (27) hich offer short recovery times(Section 3.5.2)for the re-establishment of the Based on Eq (27)small values of e require large magnetic field after the application of a PFG.The initial offsets for the start of the sweep For an ang. resonance frequency of nuclei placed in a magnetic e corresponding to 1(0.0175 rad) and with field gradient becomes dependent on their spatial yB=2 kHz, an offset of at least 1 15 kHz is neces location. A linear PFG with the strength G per unit sary. Eq.(27)suggests an alternative approach to length along the z direction produces the following reduce the initial offset substantially. If the sweep pendence of the resonance frequency w(z)on the z tarts with an amplitude of B i at zero and is smoothly coordinate increased to its nominal value, then the offset where the sweep must start can be reduced, making adiabatic sweeps more efficient [62 The same consideration where wo is the resonance frequency without the for the end of the adiabatic sweep leads to the application of a PFG. The evolution of the operators
G. Wider/Progress in Nuclear Magnetic Resonance spectroscopy 32(1998)193-275 used to describe an NMR experiment (Section 2.1.2) high-resolution NMR spectroscopy only in recent becomes dependent on the vertical position in the years replacing or supplementing phase cycling sample. Using the shift operator basis the following chemes for the selection of a specific coherence transformations are obtained due to the action of a transfer pathway(Fig. 1). Eg.(30)illustrates the ent pulse applied for a duration T basic principle used Every application of a PFG intro- 30) duces a phase factor of the form y GzT. For the desired coherence pathway the sum of all these phase factors must be zero. All other pathways do not result in detectable signal if sufficiently strong gradients are sed(Eq.(31). For multiple quantum coherence In these transformation rules the evolution due to every operator in the product will evolve according to Eq (30). Consequently, the sensitivity of the pre chemical shift and J coupling are not included.The cession frequency hase factor y Gz describing the spatial dispersion of of a coherence to magne gradients will be proportional to its order. Thus homo- the resonance frequencies depends on the gyro- nuclear double quantum coherence will be twice as magnetic ratio y which makes the action of a gradient sensitive to magnetic field gradients as single quan less efficient for nuclei with a small applying an identical gradient on transverse proton coherence will be unaffected. For heteronuclear mul and nitrogen magnetization will result in a spread of tiple quantum coherences, the different gyromagnetic esonance frequencies across the sample which is 10 ratios have to be taken nt(Eq.(30) times larger for proton than for nitrogen nucle application of gradients has the potential to select the Eq ( 30) describes how the evolution of magnetization desired signal in one scan in contrast to phase cycling components depends on the z coordinate. However, which requires the repetition of the experiment and the magnitude of the macroscopic magnetization the subtraction of unwanted signals. unfortunately will be the integral over the sample length L which only half the signal defocused by a gradient results in the following relationship refocused if a 90 RF pulse is applie d between the two sin(y GTL/2) gradients. This drawback can be verified using M= yGrL/2 Sinc(y GTL/2) (31) Eqs. (A9)and(30). Such a signal loss constitutes a common feature when gradients are used for pathway The sinc function in Eq (31)has a damped oscillatory selection. As discussed in Section 4.5.3, experimental behaviour and is zero only at specific values. In techniques exist for some cases that prevent this signal between the zero crossings the detectable signal can loss. Further signal losses can occur due to diffusion h appreciable values and can interfere with the losses. In between the defocusing and refocusing performance of the experiment requiring the value gradients, the molecules with the nuclei diffuse to a for y GrL to be optimized. For example, a moderate different loo different location preventing a complete refocusing radient with a duration of 1 ms and a strength of (Eq. (51). Diffusion losses are largest for small 0. I T m applied to proton magnetization using a molecules, particularly the solvent magnetization typical commercial probe will show a maximum Magnetic field gradients can also be applied in the nal recovery between the first and second zero cross form of spatially inhomogeneous RF pulses [74,75] ing of about O5% of the original intensity which, for either using the inherent RF inhomogeneity of the example, results in a substantial residual signal for the transmitter coil or using a separate coil designed to solvent resonance deliver inhomogeneous B fields [76. The first ulsed magnetic field gradients find three main method is frequently used in the form of spin-lock applications: (i)spatial encoding of coherences, (ii) purge pulses which destroy magnetization compo elimination of unwanted coherences and (iii selection nents that are not aligned along the axis defined by of coherences. Whereas point (i) has already been the phase of the RF pulse [77]. The second more sed in diffusion measurements for a long time, points fficient method requires special hardware and ha (ii)and (iii)have become increasingly important in not yet found many applications. Radiofrequency
212 G wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 gradients possess the inherent advantage over static digitized at equidistant time points and stored on a pulsed magnetic field gradients that they can be computer applied frequency selective [78]. On the other hand, The sampling theorem [79] specifies that for the limitations arise due to rF heating effects a d the representation of the analog signal in digital form rather modest maximal strength for RF gradients the sampling rate of the digitizer must be at least which typically is limited to 0. I T m twice the highest frequency in the signal. With a quadrature detection system the highest frequency in 2. 4. Data acquisition the spectrum is half the spectral range covered by the signals. In other words, the time increment, the 2. 4.1. Digitizing the signal dwell time 4, between two digitized points must be decay(FID)of the m quence the free induction equal to or smaller than the inverse of the sweep width At the end of a pulse tization of one nuclear 2vN which covers the frequency range from -VN to species is measured. The sensitivity of the detection +rN with the carrier frequency being at zero fre- is proportional to y (50)), and whenever quency. Slower sampling results in folding of the sig feasible the magnetization should be transferred to nals with absolute frequencies larger than vN into the and detected on the nuclear species with the largest spectral range of interest. The folded signals will be gyromagnetic ratio. This procedure requires an represented by an incorrect frequency and in efficient coupling between the different species of will have a phase that differs from that of unfolded nuclei to retain the sensitivity advantage, despite the signals. One-dimensional H NMR spectra of proteins inevitable signal losses during the transfer. When contain a very large number of resonances and folding orking with macromolecules in solution, the proton of additional signals into the spectral range is not constitutes the preferred nucleus for detection sired, Thi However, irrespective of the nuclear species detected, experiments where folding quite frequently can be the resonance frequency is tens or hundreds of mega- tolerated and helps to reduce the spectral range hertz, whereas the difference of resonance frequencies additional dimensions of one particular nuclear species in different environ- The data can be digitized with two different ments, the chemical shift range, is very small, often sampling techniques. The first method digitizes the only a few kilohertz. Technically speaking, a high two quadrature channels simultaneously and delivers modulated by data points that can be low frequency signals which represent the spectrum This simultaneous digitization method simply f interest. By subtracting the carrier frequency from measures the orthogonal components of the preces- the signal, one obtains a modified signal that contains ing magnetization which correspond to the two quad only frequencies between zero and a few kilohertz. rature channels. The minimal sampling rate 1/4 Spectrometers use frequency mixing schemes to becomes equal to the sweep width 2PN and a complex transform the high resonance frequencies to a lower Fourier transform of the FId produces the spectrum frequency range since the necessary high dynamic The second method digitizes the two quadrature ange digitizers exist only up to frequencies of a few channels sequentially with a sampling rate of 4vN hundred kilohertz. In addition, a quadrature detection Consequently, two time-shifted signals are measured scheme(Section 3.3. 1) delivers two signals which are and correct processing requires that every second data 90 out of phase, which allows discrimination point pair must be inverted in sign before the signal is between positive and negative frequencies with submitted to a real Fourier transformation, resulting in espect to the carrier frequency. The carrier frequency the spectrum [80]. The sequential digitization can be and the frequency of the rF pulses on the observe understood using the concept of the rotating frame of channel are usually identical and consequently reference at the carrier frequency (Fig. 38 in quadrature detection enables the user to place the Appendix A). Including the effect of the sign inver- carrier in the middle of the spectrum resulting in a sion, the sampling method corresponds to more efficient excitation of the resonances(Fig. 7). sampling of the magnetization components along the The low frequency analog signal thus obtained x, y, - x,-y, x, y, -x,... axis in the rotating frame