Chapter 9 - The James Keeler Group

9 Coherence Selection:Phase Cycling and Gradient Pulses †9.1 IntroductionThe pulse sequence used in an NMR experiment is carefully designed toproduce a particular outcome. For example, we may wish to pass the spinsthrough a state of multiple quantum coherence at a particular point, or planfor the magnetization to be aligned along the z-axis during a mixing period.However, it is usually the case that the particular series of events wedesigned the pulse sequence to cause is only one out of many possibilities.For example, a pulse whose role is to generate double-quantum coherencefrom anti-phase magnetization may also generate zero-quantum coherenceor transfer the magnetization to another spin. Each time a radiofrequencypulse is applied there is this possibility of branching into many differentpathways. If no steps are taken to suppress these unwanted pathways theresulting spectrum will be hopelessly confused and uninterpretable.There are two general ways in which one pathway can be isolated fromthe many possible. The first is phase cycling. In this method the phases ofthe pulses and the receiver are varied in a systematic way so that the signalfrom the desired pathways adds and signal from all other pathways cancels.Phase cycling requires that the experiment is repeated several times,something which is probably required in any case in order to achieve therequired signal-to-noise ratio.The second method of selection is to use field gradient pulses. These areshort periods during which the applied magnetic field is madeinhomogeneous. As a result, any coherences present dephase and areapparently lost. However, this dephasing can be undone, and the coherencerestored, by application of a subsequent gradient. We shall see that thisdephasing and rephasing approach can be used to select particularcoherences. Unlike phase cycling, the use of field gradient pulses does notrequire repetition of the experiment.Both of these selection methods can be described in a unified frameworkwhich classifies the coherences present at any particular point according to acoherence order and then uses coherence transfer pathways to specify thedesired outcome of the experiment.9.2 Phase in NMRIn NMR we have control over both the phase of the pulses and the receiverphase. The effect of changing the phase of a pulse is easy to visualise in theusual rotating frame. So, for example, a 90° pulse about the x-axis rotatesmagnetization from z onto –y, whereas the same pulse applied about the y-axis rotates the magnetization onto the x-axis. The idea of the receiverphase is slightly more complex and will be explored in this section.†Chapter 9 "Coherence Selection: Phase Cycling and Gradient Pulses" © James Keeler2001, 2003 and 2004.9–1

The NMR signal – that is the free induction decay – which emerges fromthe probe is a radiofrequency signal oscillating at close to the Larmorfrequency (usually hundreds of MHz). Within the spectrometer this signalis shifted down to a much lower frequency in order that it can be digitizedand then stored in computer memory. The details of this down-shiftingprocess can be quite complex, but the overall result is simply that a fixedfrequency, called the receiver reference or carrier, is subtracted from thefrequency of the incoming NMR signal. Frequently, this receiver referencefrequency is the same as the transmitter frequency used to generate thepulses applied to the observed nucleus. We shall assume that this is the casefrom now on.The rotating frame which we use to visualise the effect of pulses is set atthe transmitter frequency, ω rf, so that the field due to the radiofrequencypulse is static. In this frame, a spin whose Larmor frequency is ω 0precessesat (ω 0– ω rf), called the offset Ω. In the spectrometer the incoming signal atω 0is down-shifted by subtracting the receiver reference which, as we havealready decided, will be equal to the frequency of the radiofrequency pulses.So, in effect, the frequencies of the signals which are digitized in thespectrometer are the offset frequencies at which the spins evolve in therotating frame. Often this whole process is summarised by saying that the"signal is detected in the rotating frame".9.1.1 Detector phaseThe quantity which is actually detected in an NMR experiment is thetransverse magnetization. Ultimately, this appears at the probe as anoscillating voltage, which we will write asS = cosω FID 0twhere ω 0is the Larmor frequency. The down-shifting process in thespectrometer is achieved by an electronic device called a mixer; thiseffectively multiplies the incoming signal by a locally generated referencesignal, S ref, which we will assume is oscillating at ω rf9–2Sref= cosωrfThe output of the mixer is the product S FIDS refS S = AcosωtcosωtFID ref rf 01=2A[ cos( ωrf+ ω0) t + cos( ωrf−ω0)t]The first term is an oscillation at a high frequency (of the order of twice theLarmor frequency as ω 0≈ ω rf) and is easily removed by a low-pass filter.The second term is an oscillation at the offset frequency, Ω. This is in linewith the previous comment that this down-shifting process is equivalent todetecting the precession in the rotating frame.We can go further with this interpretation and say that the second termrepresents the component of the magnetization along a particular axis(called the reference axis) in the rotating frame. Such a component variesas cos Ωt, assuming that at time zero the magnetization is aligned along thechosen axis; this is illustrated belowt

θt = 0time tAt time zero the magnetization is assumed to be aligned along the reference axis. After time t themagnetization has precessed through an angle θ = Ωt. The projection of the magnetization onto thereference axis is proportional to cos Ωt.Suppose now that the phase of the reference signal is shifted by φ,something which is easily achieved in the spectrometer. Effectively, thisshifts the reference axis by φ, as shown belowφθ–φt = 0time tShifting the phase of the receiver reference by φ is equivalent to detecting the component along an axisrotated by φ from its original position (the previous axis is shown in grey). Now the apparent angle ofprecession is θ = Ωt – φ. and the projection of the magnetization onto the reference axis is proportionalto cos (Ωt – φ).( )The component along the new reference axis is proportional to cos Ωt − φ .How this is put to good effect is described in the next section.9.1.2 Quadrature detectionWe normally want to place the transmitter frequency in the centre of theresonances of interest so as to minimise off-resonance effects. If thereceiver reference frequency is the same as the transmitter frequency, itimmediately follows that the offset frequencies, Ω, may be both positiveand negative. However, as we have seen in the previous section the effectof the down-shifting scheme used is to generate a signal of the form cos Ωt.Since cos(θ) = cos(–θ) such a signal does not discriminate between positiveand negative offset frequencies. The resulting spectrum, obtained byFourier transformation, would be confusing as each peak would appear atboth +Ω and –Ω.The way out of this problem is to detect the signal along twoperpendicular axes. As is illustrated opposite, the projection along one axisis proportional to cos(Ωt) and to sin(Ωt) along the other. Knowledge ofboth of these projections enables us to work out the sense of rotation of thevector i.e. the sign of the offset.The sin modulated component is detected by having a second mixer fedwith a reference whose phase is shifted by π/2. Following the abovediscussion the output of the mixer isyThe x and y projections of theblack vector are both positive.If the vector had precessed inthe opposite direction (shownshaded), and at the samefrequency, the projection alongx would be the same, but alongy it would be minus that of theblack vector.θx9–3

cos( Ωt−π 2)= cosΩtcosπ 2+sinΩtsinπ2= sinΩtThe output of these two mixers can be regarded as being the components ofthe magnetization along perpendicular axes in the rotating frame.The final step in this whole process is regard the outputs of the twomixers as being the real and imaginary parts of a complex number:cosΩt + isinΩt = exp( iΩt)The overall result is the generation of a complex signal whose phase variesaccording to the offset frequency Ω.9.1.3 Control of phaseIn the previous section we supposed that the signal coming from the probewas of the form cosω 0t but it is more realistic to write the signal ascos(ω 0t + φ sig) in recognition of the fact that in addition to a frequency thesignal has a phase, φ sig. This phase is a combination of factors that are notunder our control (such as phase shifts produced in the amplifiers and filtersthrough which the signal passes) and a phase due to the pulse sequence,which certainly is under our control.The overall result of this phase is simply to multiply the final complexsignal by a phase factor, exp(iφ sig):( ) ( iφsig )exp iΩtexpAs we saw in the previous section, we can also introduce another phase shiftby altering the phase of the reference signal fed to the mixer, and indeed wesaw that the cosine and sine modulated signals are generated by using twomixers fed with reference signals which differ in phase by π/2. If the phaseof each of these reference signals is advanced by φ rx, usually called thereceiver phase, the output of the two mixers becomes cos( Ωt − φ rx ) andsin( Ωt − φ rx ). In the complex notation, the overall signal thus acquiresanother phase factor( ) ( ) −exp iΩtexp iφexp iφsig( )Overall, then, the phase of the final signal depends on the differencebetween the phase introduced by the pulse sequence and the phaseintroduced by the receiver reference.9.1.4 LineshapesLet us suppose that the signal can be writtenSt ()= Bexp( iΩt) exp( iΦ) exp( −tT2)where Φ is the overall phase (= φsig−φrx ) and B is the amplitude. The term,exp(-t/T 2) has been added to impose a decay on the signal. Fouriertransformation of S(t) gives the spectrum S(ω):S( ω)= B A( ω)+ iD( ω)exp iΦ [1]rx[ ] ( )9–4

where A(ω) is an absorption mode lorentzian lineshape centred at ω = Ω andD(ω) is the corresponding dispersion mode lorentzian:( )2T2ω − Ω T2A( ω)= D( ω)=+ ( ω − )2 22 21 Ω T21+ ( ω −Ω)T2Normally we display just the real part of S(ω) which is, in this case,[ ( )]= ( )− ( )[ ]Re S ω B cosΦA ω sinΦD ωIn general this is a mixture of the absorption and dispersion lineshape. If wewant just the absorption lineshape we need to somehow set Φ to zero, whichis easily done by multiplying S(ω) by a phase factor exp(iΘ).S( ω) exp( iΘ)= B[ A( ω)+ iD( ω)] exp( iΦ) exp( iΘ)= BA( ω)+ iD( ω)exp i Φ Θ[ ] ( [ + ])As this is a numerical operation which can be carried out in the computerwe are free to choose Θ to be the required value (here –Φ) in order toremove the phase factor entirely and hence give an absorption modespectrum in the real part. This is what we do when we "phase thespectrum".9.1.5 Relative phase and lineshapeWe have seen that we can alter the phase of the spectrum by altering thephase of the pulse or of the receiver, but that what really counts is thedifference in these two phases.We will illustrate this with the simple vector diagrams shown below.Here, the vector shows the position of the magnetization at time zero and itsphase, φ sig, is measured anti-clockwise from the x-axis. The dot shows theaxis along which the receiver is aligned; this phase, φ rx, is also measuredanti-clockwise from the x-axis.If the vector and receiver are aligned along the same axis, Φ = 0, and thereal part of the spectrum shows the absorption mode lineshape. If thereceiver phase is advanced by π/2, Φ = 0 – π/2 and, from Eq. [1]S( ω)= B[ A( ω)+ iD( ω)] exp( −iπ2)= B − iA( ω)+ D( ω)[ ]This means that the real part of the spectrum shows a dispersion lineshape.On the other hand, if the magnetization is advanced by π/2, Φ = φ sig– φ rx= π/2 – 0 = π/2 and it can be shown from Eq. [1] that the real part of thespectrum shows a negative dispersion lineshape. Finally, if either phase isadvanced by π, the result is a negative absorption lineshape.Ωabsorptiondispersion9–5

yxφ rxφ9.1.6 CYCLOPSThe CYCLOPS phase cycling scheme is commonly used in even thesimplest pulse-acquire experiments. The sequence is designed to cancelsome imperfections associated with errors in the two phase detectorsmentioned above; a description of how this is achieved is beyond the scopeof this discussion. However, the cycle itself illustrates very well the pointsmade in the previous section.There are four steps in the cycle, the pulse phase goes x, y, –x, –y i.e. itadvances by 90° on each step; likewise the receiver advances by 90° oneach step. The figure below shows how the magnetization and receiverphases are related for the four steps of this cycley–xx–ypulse x y –x –yreceiver x y –x –yAlthough both the receiver and the magnetization shift phase on each step,the phase difference between them remains constant. Each step in the cyclethus gives the same lineshape and so the signal adds on all four steps, whichis just what is required.Suppose that we forget to advance the pulse phase; the outcome is quitedifferent9–6

pulse x x x xreceiver x y –x –yy–xx–yNow the phase difference between the receiver and the magnetization is nolonger constant. A different lineshape thus results from each step and it isclear that adding all four together will lead to complete cancellation (steps 2and 4 cancel, as do steps 1 and 3). For the signal to add up it is clearlyessential for the receiver to follow the magnetization.9.1.7 EXORCYLEEXORCYLE is perhaps the original phase cycle. It is a cycle used for 180°pulses when they form part of a spin echo sequence. The 180° pulse cyclesthrough the phases x, y, –x, –y and the receiver phase goes x, –x, x, –x. Thediagram below illustrates the outcome of this sequenceyy180°(±x)τ90°(x) – τ–xx–y180°(±y)τIf the phase of the 180° pulse is +x or –x the echo forms along the y-axis,whereas if the phase is ±y the echo forms on the –y axis. Therefore, as the180° pulse is advanced by 90° (e.g. from x to y) the receiver must beadvanced by 180° (e.g. from x to –x). Of course, we could just as well cyclethe receiver phases y, –y, y, –y; all that matters is that they advance in stepsof 180°. We will see later on how it is that this phase cycle cancels out theresults of imperfections in the 180° pulse.–y9–7

IS12Jy12JPulse sequence for INEPT.Filled rectangles represent 90°pulses and open rectanglesrepresent 180° pulses. Unlessotherwise indicated, all pulsesare of phase x.9.1.8 Difference spectroscopyOften a simple two step sequence suffices to cancel unwantedmagnetization; essentially this is a form of difference spectroscopy. Theidea is well illustrated by the INEPT sequence, shown opposite. The aim ofthe sequence is to transfer magnetization from spin I to a coupled spin S.With the phases and delays shown equilibrium magnetization of spin I, I z,is transferred to spin S, appearing as the operator S x. Equilibriummagnetization of S, S z, appears as S y. We wish to preserve only the signalthat has been transferred from I.The procedure to achieve this is very simple. If we change the phase ofthe second I spin 90° pulse from y to –y the magnetization arising fromtransfer of the I spin magnetization to S becomes –S xi.e. it changes sign.In contrast, the signal arising from equilibrium S spin magnetization isunaffected simply because the S zoperator is unaffected by the I spin pulses.By repeating the experiment twice, once with the phase of the second I spin90° pulse set to y and once with it set to –y, and then subtracting the tworesulting signals, the undesired signal is cancelled and the desired signaladds. It is easily confirmed that shifting the phase of the S spin 90° pulsedoes not achieve the desired separation of the two signals as both areaffected in the same way.In practice the subtraction would be carried out by shifting the receiverby 180°, so the I spin pulse would go y, –y and the receiver phase go x, –x.This is a two step phase cycle which is probably best viewed as differencespectroscopy.This simple two step cycle is the basic element used in constructing thephase cycling of many two- and three-dimensional heteronuclearexperiments.9.2 Coherence transfer pathwaysAlthough we can make some progress in writing simple phase cycles byconsidering the vector picture, a more general framework is needed in orderto cope with experiments which involve multiple-quantum coherence andrelated phenomena. We also need a theory which enables us to predict thedegree to which a phase cycle can discriminate against different classes ofunwanted signals. A convenient and powerful way of doing both thesethings is to use the coherence transfer pathway approach.9.2.1 Coherence orderCoherences, of which transverse magnetization is one example, can beclassified according to a coherence order, p, which is an integer takingvalues 0, ± 1, ± 2 ... Single quantum coherence has p = ± 1, double hasp = ± 2 and so on; z-magnetization, "zz" terms and zero-quantum coherencehave p = 0. This classification comes about by considering the phase whichdifferent coherences acquire is response to a rotation about the z-axis.A coherence of order p, represented by the density operator σ ( p), evolvesunder a z-rotation of angle φ according to9–8

( ) ( )= ( − )( ) ( )ppexp −iφFzσ exp iφFzexp ipφ σ[2]where F zis the operator for the total z-component of the spin angularmomentum. In words, a coherence of order p experiences a phase shift of–pφ. Equation [2] is the definition of coherence order.To see how this definition can be applied, consider the effect of a z-rotation on transverse magnetization aligned along the x-axis. Such arotation is identical in nature to that due to evolution under an offset, andusing product operators it can be written( ) ( )= +exp −iφI I exp iφI cosφ I sinφIz x z x y [3]The right hand sides of Eqs. [2] and [3] are not immediately comparable,but by writing the sine and cosine terms as complex exponentials thecomparison becomes clearer. Using[ ( )] =2i( )− ( − )[ ]11cosφ = exp( iφ)+ exp −i φ sinφ exp iφ exp iφEq. [3] becomes2( z) x ( z)( )exp −iφI I exp iφI[ ] + [ ( )− ( − )]11=2exp( iφ)+ exp −iφ Ix2iexp iφ exp iφI1 1 1 1=2 [ Ix +iIy] exp( iφ)+ 2 [ Ix −iIy] exp( −iφ)It is now clear that the first term corresponds to coherence order –1 and thesecond to +1; in other words, I xis an equal mixture of coherence orders ±1.The cartesian product operators do not correspond to a single coherenceorder so it is more convenient to rewrite them in terms of the raising andlowering operators, I +and I –, defined asI+ = Ix + i Iy I– = Ix – iIyfrom which it follows thatI = 1I +1x 2 [ +I– ] Iy =2i [ I+− I–][4]Under z-rotations the raising and lowering operators transform simply( ) ( )= ( )exp −iφIz I± exp iφIz exp miφI±which, by comparison with Eq. [2] shows that I +corresponds to coherenceorder +1 and I –to –1. So, from Eq. [4] we can see that I xand I ycorrespondto mixtures of coherence orders +1 and –1.As a second example consider the pure double quantum operator for twocoupled spins,2I I + 2I I1x 2y 1y 2xRewriting this in terms of the raising and lowering operators gives1 + + − −i ( I1 I2 − I1 I2)+ +The effect of a z-rotation on the term I1 I is found as follows:2exp −iφI exp iφI I I exp iφI exp iφI( 1z) ( −2z) 1+ 2+( 2z) ( 1z)( 1z) ( − ) 1+ 2+( 1z)( ) ( − ) = ( − )= exp −iφI exp iφ I I exp iφI= exp −iφ exp iφ I I exp 2iφI I1+ 2+ 1+ 2+y9–9

Thus, as the coherence experiences a phase shift of –2φ the coherence isclassified according to Eq. [2] as having p = 2. It is easy to confirm that theterm I1 I has p = –2. Thus the pure double quantum term,− 2− 2I1xI2y + 2I1yI, is an equal mixture of coherence orders +2 and –2.2xAs this example indicates, it is possible to determine the order or ordersof any state by writing it in terms of raising and lowering operators and thensimply inspecting the number of such operators in each term. A raisingoperator contributes +1 to the coherence order whereas a lowering operatorcontributes –1. A z-operator, I iz , has coherence order 0 as it is invariant to z-rotations.Coherences involving heteronuclei can be assigned both an overall orderand an order with respect to each nuclear species. For example the termI1 S has an overall order of 0, is order +1 for the I spins and –1 for the S+ 1– spins. The term I 1+ I 2+ S 1zis overall of order 2, is order 2 for the I spins andis order 0 for the S spins.9.2.2 Evolution under offsetsThe evolution under an offset, Ω, is simply a z-rotation, so the raising andlowering operators simply acquire a phase Ωtexp −iΩtIz I± exp iΩtIz exp miΩt I±For products of these operators, the overall phase is the sum of the phasesacquired by each term9–10( ) ( )= ( )( j jz ) ( −i iz ) i− j + ( i iz ) ( j jz )( i j)tIi− Ij+exp −iΩ tI exp iΩtI I I exp iΩtI exp iΩtI( )= exp i Ω −ΩIt also follows that coherences of opposite sign acquire phases of oppositesigns under free evolution. So the operator I 1+I 2+(with p = 2) acquires aphase –(Ω 1+ Ω 2)t i.e. it evolves at a frequency –(Ω 1+ Ω 2) whereas theoperator I 1–I 2–(with p = –2) acquires a phase (Ω 1+ Ω 2)t i.e. it evolves at afrequency (Ω 1+ Ω 2). We will see later on that this observation hasimportant consequences for the lineshapes in two-dimensional NMR.The observation that coherences of different orders respond differently toevolution under a z-rotation (e.g. an offset) lies at the heart of the way inwhich gradient pulses can be used to separate different coherence orders.9.2.3 Phase shifted pulsesIn general, a radiofrequency pulse causes coherences to be transferred fromone order to one or more different orders; it is this spreading out of thecoherence which makes it necessary to select one transfer among manypossibilities. An example of this spreading between coherence orders is theeffect of a non-selective pulse on antiphase magnetization, such as 2I 1xI 2z,which corresponds to coherence orders ±1. Some of the coherence may betransferred into double- and zero-quantum coherence, some may betransferred into two-spin order and some will remain unaffected. Theprecise outcome depends on the phase and flip angle of the pulse, but ingeneral we can see that there are many possibilities.

If we consider just one coherence, of order p, being transferred to acoherence of order p' by a radiofrequency pulse we can derive a verygeneral result for the way in which the phase of the pulse affects the phaseof the coherence. It is on this relationship that the phase cycling method isbased.We will write the initial state of order p as σ ( p), and the final state oforder p' as σ ( p' ) . The effect of the radiofrequency pulse causing the transferis represented by the (unitary) transformation U φwhere φ is the phase of thepulse. The initial and final states are related by the usual transformation( ) ( )Up – 1Up '0σ0= σ + terms of other orders [5]which has been written for phase 0; the other terms will be dropped as weare only interested in the transfer from p to p'. The transformation broughtabout by a radiofrequency pulse phase shifted by φ, U φ, is related to thatwith the phase set to zero, U 0, in the following way( ) ( )Uφ = exp −iφFzU0 exp iφFz[6]Using this, the effect of the phase shifted pulse on the initial state σ ( p)canbe writtenU σφ( p)– 1Uφ( ) ( )( )−( ) ( )p– 1= exp −iφFzU0exp iφFzσ exp iφFzU0exp iφFzThe central three terms can be simplified by application of Eq. [2]giving( ) ( − ) = ( )p– 1exp iφF σ exp iφF U exp ipφ σz( ) ( p)z 0( ) ( )( ) ( )ppUφσ U – 1φ= ( pφ) − φFzU σ U – 1exp i exp i0 0exp iφFzThe central three terms can, from Eq. [5], be replaced by σ ( p' ) to give( ) ( )φzppU σ U – 1= exp( ipφ) exp −iφF σ 'exp iφFφFinally, Eq. [5] is applied again to give( ) ( )( ) – ( p')φp 1U σ U exp i pφ exp –ip'φ σφ= ( ) ( )Defining ∆p = (p' – p) as the change is coherence order, this simplifies toz[7]p – 1p'U σ U exp – i∆ pφ σ[8]φ( ) ( )φ= ( )Equation [8] says that if the phase of a pulse which is causing a change incoherence order of ∆p is shifted by φ the coherence will acquire a phaselabel (–∆p φ). It is this property which enables us to separate differentchanges in coherence order from one another by altering the phase of thepulse.In the discussion so far it has been assumed that U φrepresents a singlepulse. However, any sequence of pulses and delays can be represented by asingle unitary transformation, so Eq. [8] applies equally well to the effect ofphase shifting all of the pulses in such a sequence. We will see that thisproperty is often of use in writing phase cycles.If a series of phase shifted pulses (or pulse sandwiches) are applied a9–11

phase (–∆p φ) is acquired from each. The total phase is found by adding upthese individual contributions. In an NMR experiment this total phaseaffects the signal which is recorded at the end of the sequence, even thoughthe phase shift may have been acquired earlier in the pulse sequence. Thesephase shifts are, so to speak, carried forward.9.2.4 Coherence transfer pathways diagramsIn designing a multiple-pulse NMR experiment the intention is to havespecific orders of coherence present at various points in the sequence. Oneway of indicating this is to use a coherence transfer pathway (CTP) diagramalong with the timing diagram for the pulse sequence. An example ofshown below, which gives the pulse sequence and CTP for the DQF COSYexperiment.t 1 t 2p210–1–2∆p=±1 ±1,±3+1,–3The solid lines under the sequence represent the coherence orders requiredduring each part of the sequence; note that it is only the pulses which causea change in the coherence order. In addition, the values of ∆p are shown foreach pulse. In this example, as is commonly the case, more than one orderof coherence is present at a particular time. Each pulse is required to causedifferent changes to the coherence order – for example the second pulse isrequired to bring about no less than four values of ∆p. Again, this is acommon feature of pulse sequences.It is important to realise that the CTP specified with the pulse sequence isjust the desired pathway. We would need to establish separately (forexample using a product operator calculation) that the pulse sequence isindeed capable of generating the coherences specified in the CTP. Also, thespin system which we apply the sequence to has to be capable of supportingthe coherences. For example, if there are no couplings, then no doublequantum will be generated and thus selection of the above pathway willresult in a null spectrum.The coherence transfer pathway must start with p = 0 as this is the orderto which equilibrium magnetization (z-magnetization) belongs. In addition,the pathway has to end with |p| = 1 as it is only single quantum coherencethat is observable. If we use quadrature detection (section 9.1.2) it turns outthat only one of p = ±1 is observable; we will follow the usual convention ofassuming that p = –1 is the detectable signal.9–12

9.3 Lineshapes and frequency discrimination9.3.1 Phase and amplitude modulationThe selection of a particular CTP has important consequences for lineshapesand frequency discrimination in two-dimensional NMR. These topics areillustrated using the NOESY experiment as an example; the pulse sequenceand CTP is illustrated opposite.If we imagine starting with I z, then at the end of t 1the operators presentare− cosΩtI+ sinΩtI1 y1 xThe term in I yis rotated onto the z-axis and we will assume that only thisterm survives. Finally, the z-magnetization is made observable by the lastpulse (for convenience set to phase –y) giving the observable term present att 2= 0 ascosΩt 1I xAs was noted in section 9.2.1, I xis in fact a mixture of coherence ordersp = ±1, something which is made evident by writing the operator in terms ofI +and I –1cosΩt I + I2 1( )+ −Of these operators, only I –leads to an observable signal, as this correspondsto p = –1. Allowing I –to evolve in t 2gives( ) −cosΩt exp iΩt I12 1 2The final detected signal can be written as1S t , t cosΩt exp iΩtC( )= ( )1 22 1 2This signal is said to be amplitude modulated in t 1; it is so called because theevolution during t 1gives rise, via the cosine term, to a modulation of theamplitude of the observed signal.The situation changes if we select a different pathway, as shownopposite. Here, only coherence order –1 is preserved during t 1. At the startof t 1the operator present is –I ywhich can be written( )1−2iI+ − I−Now, in accordance with the CTP, we select only the I –term. During t 1thisevolves to give( ) −exp iΩt I12i1Following through the rest of the pulse sequence as before gives thefollowing observable signal( )= ( ) ( )1SP t1, t2exp iΩt exp iΩt4 1 2This signal is said to be phase modulated in t 1; it is so called because theevolution during t 1gives rise, via exponential term, to a modulation of thephase of the observed signal. If we had chosen to select p = +1 during t 1thesignal would have been10–110–1–yt t 1 τ m 2–yt t 1 τ m 29–13

( )= ( ) ( )1SN t1, t2exp –iΩt exp iΩt4 1 2which is also phase modulated, except in the opposite sense. Note that ineither case the phase modulated signal is one half of the size of theamplitude modulated signal, because only one of the two pathways has beenselected.Although these results have been derived for the NOESY sequence, theyare in fact general for any two-dimensional experiment. Summarising, wefind• If a single coherence order is present during t 1the result is phasemodulation in t 1. The phase modulation can be of the form exp(iΩt 1) orexp(–iΩt 1) depending on the sign of the coherence order present.• If both coherence orders ±p are selected during t 1, the result isamplitude modulation in t 1; selecting both orders in this way is calledpreserving symmetrical pathways.9.3.2 Frequency discriminationThe amplitude modulated signal contains no information about the sign ofΩ, simply because cos(Ωt 1) = cos(–Ωt 1). As a consequence, Fouriertransformation of the time domain signal will result in each peak appearingtwice in the two-dimensional spectrum, once at F 1= +Ω and once at F 1=–Ω. As was commented on above, we usually place the transmitter in themiddle of the spectrum so that there are peaks with both positive andnegative offsets. If, as a result of recording an amplitude modulated signal,all of these appear twice, the spectrum will hopelessly confused. Aspectrum arising from an amplitude modulated signal is said to lackfrequency discrimination in F 1.On the other hand, the phase modulated signal is sensitive to the sign ofthe offset and so information about the sign of Ω in the F 1dimension iscontained in the signal. Fourier transformation of the signal S P(t 1,t 2) gives apeak at F 1= +Ω , F 2= Ω, whereas Fourier transformation of the signalS N(t 1,t 2) gives a peak at F 1= –Ω, F 2= Ω. Both spectra are said to befrequency discriminated as the sign of the modulation frequency in t 1isdetermined; in contrast to amplitude modulated spectra, each peak will onlyappear once.The spectrum from S P(t 1,t 2) is called the P-type (P for positive) or echospectrum; a diagonal peak appears with the same sign of offset in eachdimension. The spectrum from S N(t 1,t 2) is called the N-type (N for negative)or anti-echo spectrum; a diagonal peak appears with opposite signs in thetwo dimensions.It might appear that in order to achieve frequency discrimination weshould deliberately select a CTP which leads to a P– or an N-type spectrum.However, such spectra show a very unfavourable lineshape, as discussed inthe next section.9.3.3 LineshapesIn section 9.1.4 we saw that Fourier transformation of the signal9–14

St ()= exp( iΩt) exp( −tT2)gave a spectrum whose real part is an absorption lorentzian and whoseimaginary part is a dispersion lorentzian:S( ω)= A( ω)+ iD( ω)We will use the shorthand that A 2represents an absorption mode lineshapeat F 2= Ω and D 2represents a dispersion mode lineshape at the samefrequency. Likewise, A 1+represents an absorption mode lineshape at F 1=+Ω and D 1+represents the corresponding dispersion lineshape. A 1–and D 1–represent the corresponding lines at F 1= –Ω.The time domain signal for the P-type spectrum can be written as( )= ( ) ( ) ( − ) ( − )1SP t1, t2exp iΩt exp iΩt exp t T exp t T4 1 2 1 2 2 2where the damping factors have been included as before. Fouriertransformation with respect to t 2gives( )=4 ( 1) ( −1 2) [ 2+2]1SP t1, F2exp iΩt exp t T A iDand then further transformation with respect to t 1gives1SP ( F1,F2)= 4A1+ + iD1+A2 iD2[ ][ + ]The real part of this spectrum is1Re { SP( F1,F2)}= A+A − D+D[ ]4 1 2 1 2The quantity in the square brackets on the right represents a phase-twistlineshape at F 1= +Ω, F 2= ΩPerspective view and contour plot of the phase-twist lineshape. Negative contours are shown dashed.This lineshape is an inextricable mixture of absorption and dispersion, and itis very undesirable for high-resolution NMR. So, although a phasemodulated signal gives us frequency discrimination, which is desirable, italso results in a phase-twist lineshape, which is not.The time domain signal for the amplitude modulated data set can bewritten as1S t , t cos Ωt exp iΩt exp t T exp t TC( )= ( ) ( ) ( − ) ( − )1 22 1 2 1 2 2 2Fourier transformation with respect to t 2giveswhich can be rewritten as1S t , F cos Ωt exp t T A iDC( 1 2)= 2 ( 1) ( −1 2) [ 2+2]9–15

[ ] −( )= ( )+ ( − )( )[ + ]1SC t1, F2exp iΩt exp iΩt exp t T A iD4 1 1 1 2 2 2Fourier transformation with respect to t 1gives, in the real part of thespectrum{ ( )}= [ ]+4 [ 1 2 1 2]11Re SC F1, F24A1+ A2 – D1+D2A−A – D–DThis corresponds to two phase-twist lineshapes, one at F 1= +Ω, F 2= Ω andthe other at F 1= –Ω, F 2= Ω; the lack of frequency discrimination is evident.Further, the undesirable phase-twist lineshape is again present.The lineshape can be restored to the absorption mode by discarding theimaginary part of the time domain signal after the transformation withrespect to t 2, i.e. by taking the real part{ C( 1 2)}=2 ( 1) ( −1 2)21Re S t , F cos Ωt exp t T ASubsequent transformation with respect to t 1gives, in the real partA A+A A114 1+ 2 4 1−2which is two double absorption mode lineshapes. Frequency discriminationis lacking, but the lineshape is now much more desirable. The spectra withthe two phase-twist and two absorption mode lines are shown below on theleft and right, respectively.F 2F 1 F 1F 2009.3.4 Frequency discrimination with retention of absorption modelineshapesFor practical purposes it is essential to be able to achieve frequencydiscrimination and at the same time retain the absorption mode lineshape.There are a number of ways of doing this.9.3.4.1 States-Haberkorn-Ruben (SHR) methodThe key to this method is the ability to record a cosine modulated data setand a sine modulated data set. The latter can be achieved simply bychanging the phase of appropriate pulses. For example, in the case of theNOESY experiment, all that is required to generate the sine data set is toshift the phase of the first 90° pulse by 90° (in fact in the NOESY sequencethe pulse needs to shift from x to –y). The two data sets have to keptseparate.The cosine data set is transformed with respect to t 1and the imaginarypart discarded to give{ C( 1 2)}=2 ( 1) ( −1 2)21Re S t , F cos Ω t exp t T A[9]The same operation is performed on the sine modulated data set9–16

1S t , t sin Ωt exp iΩt exp t T exp t TS( 1 2)= 2 ( 1) ( 2) ( −1 2) ( −2 2)1Re { SS( t1, F2)}=2sin( t1) exp( −t1 T2)A2Ω [10]A new complex data set is now formed by using the signal from Eq. [9] asthe real part and that from Eq. [10] as the imaginary part( )= { ( )}+ ( )( ){ }S t , F Re S t , F i Re S t , FSHR 1 2 C 1 2 S 1 21=2exp( iΩt1) exp −t1 T2 A2Fourier transformation with respect to t 1gives, in the real part of thespectrum1Re { SSHR( F1 , F2)}= 2A1+A2This is the desired frequency discriminated spectrum with a pure absorptionlineshape.As commented on above, in NOESY all that is required to change fromcosine to sine modulation is to shift the phase of the first pulse by 90°. Thegeneral recipe is to shift the phase of all the pulses that precede t 1by90°/|p 1|, where p 1is the coherence order present during t 1. So, for a doublequantum spectrum, the phase shift needs to be 45°. The origin of this rule isthat, taken together, the pulses which precede t 1give rise to a pathway with∆p = p 1.In heteronuclear experiments it is not usually necessary to shift the phaseof all the pulses which precede t 1; an analysis of the sequence usually showsthat shifting the phase of the pulse which generates the transversemagnetization which evolves during t 1is sufficient.9.3.4.2 Echo anti-echo methodWe will see in later sections that when we use gradient pulses for coherenceselection the natural outcome is P- or N-type data sets. Individually, each ofthese gives a frequency discriminated spectrum, but with the phase-twistlineshape. We will show in this section how an absorption mode lineshapecan be obtained provided both the P- and the N-type data sets are available.As before, we write the two data sets as( )= ( ) ( ) ( − ) ( − )( )1SPt1, t24exp iΩt1 exp iΩt2 exp t1 T2 exp t2 T21SN( t1, t2)= exp( −iΩt ) exp( iΩt ) exp( −t T ) exp −t TWe then form the two combinations4 1 2 1 2 2 2( )= ( )+ ( )( )( )= [ ( )+ ( )]= sin( Ωt ) exp( iΩ t ) exp( −t T ) exp( −t T )S t , t S t , t S t , tC 1 2 P 1 2 N 1 21=2cos( Ωt1) exp( iΩt2) exp( −t1 T2) exp −t2 T21S t , t S t , t S t , tS 1 2 i P 1 2 N 1 212 1 2 1 22 2These cosine and sine modulated data sets can be used as inputs to the SHRmethod described in the previous section.An alternative is to Fourier transform the two data sets with respect to t 2to give9–17

9–181S t , F exp iΩt exp t T A iDP( 1 2)= 4 ( 1) ( −1 2) [ 2+2]( )= ( ) ( − )[ + ]1SNt1, F24exp – iΩt1 exp t1 T2 A2 iD2We then take the complex conjugate of S N(t 1,F 2) and add it to S P(t 1,F 2)1S t , F * exp iΩt exp t T A iDN( 1 2) =4 ( 1) ( −1 2) [ 2−2]( )= ( ) + ( )S t , F S t , F * S t , F+1 2 N 1 2 P 1 21=2exp( iΩt1) exp( −t1 T2)A2Transformation of this signal gives1S+ ( F1, F2)= 2 [ A1+ + iD1+] A2which is frequency discriminated and has, in the real part, the requireddouble absorption lineshape.9.3.4.3 Marion-Wüthrich or TPPI methodThe idea behind the TPPI (time proportional phase incrementation) orMarion–Wüthrich (MW) method is to arrange things so that all of the peakshave positive offsets. Then, frequency discrimination is not required asthere is no ambiguity.One simple way to make all offsets positive is to set the receiver carrierfrequency deliberately at the edge of the spectrum. Simple though this is, itis not really a very practical method as the resulting spectrum would be veryinefficient in its use of data space and in addition off-resonance effectsassociated with the pulses in the sequence will be accentuated.In the TPPI method the carrier can still be set in the middle of thespectrum, but it is made to appear that all the frequencies are positive byphase shifting some of the pulses in the sequence in concert with theincrementation of t 1.It was noted above that shifting the phase of the first pulse in the NOESYsequence from x to –y caused the modulation to change from cos(Ωt 1) tosin(Ωt 1). One way of expressing this is to say that shifting the pulse causesa phase shift φ in the signal modulation, which can be written cos(Ωt 1+ φ).Using the usual trigonometric expansions this can be writtencos( Ωt1 + φ)= cosΩt1cosφ −sinΩt1sinφIf the phase shift, φ, is –π/2 radians the result iscos( Ωt1 + π 2)= cosΩt1cos( −π 2)−sinΩt1sin( −π2)= sinΩt1This is exactly the result we found before.In the TPPI procedure, the phase φ is made proportional to t 1i.e. eachtime t 1is incremented, so is the phase. We will suppose thatφ( t1)= ω addt1The constant of proportion between the time dependent phase, φ(t 1), and t 1has been written ω add; ω addhas the dimensions of rad s –1 i.e. it is a frequency.Following the same approach as before, the time-domain function with theinclusion of this incrementing phase is thus

cos( Ωt + φ( t ))= cos Ωt + ωaddt= cos Ω + ω t( )( )1 1 1 1In words, the effect of incrementing the phase in concert with t 1is to add afrequency ω addto all of the offsets in the spectrum. The TPPI methodutilizes this in the following way.In one-dimensional pulse-Fourier transform NMR the free inductionsignal is sampled at regular intervals ∆. After transformation the resultingspectrum displays correctly peaks with offsets in the range –(SW/2) to+(SW/2) where SW is the spectral width which is given by 1/∆ (this comesabout from the Nyquist theorem of data sampling). Frequencies outside thisrange are not represented correctly.Suppose that the required frequency range in the F 1dimension is from–(SW 1/2) to +(SW 1/2). To make it appear that all the peaks have a positiveoffset, it will be necessary to add (SW 1/2) to all the frequencies. Then thepeaks will be in the range 0 to (SW 1).As the maximum frequency is now (SW 1) rather than (SW 1/2) thesampling interval, ∆ 1, will have to be halved i.e. ∆ 1= 1/(2SW 1) in order thatthe range of frequencies present are represented properly.The phase increment is ω addt 1, but t 1can be written as n∆ 1for the nthincrement of t 1. The required value for ω addis 2π(SW 1/2) , where the 2π isto convert from frequency (the units of SW 1) to rad s –1 , the units of ω add.Putting all of this together ω addt 1can be expressed, for the nth increment asω t SW1additional 1= 2π⎛ ⎞ n1⎝ 2 ⎠ ( ∆ )SW11= 2π⎛ ⎞⎛⎞ n⎝ 2 ⎠⎜⎟⎝ 2 SW1 ⎠π= n2In words this means that each time t 1is incremented, the phase of the signalshould also be incremented by 90°, for example by incrementing the phaseof one of the pulses.A data set from an experiment to which TPPI has been applied is simplyamplitude modulated in t 1and so can be processed according to the methoddescribed above for cosine modulated data so as to obtain absorption modelineshapes. As the spectrum is symmetrical about F 1= 0, it is usual to use amodified Fourier transform routine which saves effort and space by onlycalculating the positive frequency part of the spectrum.add1t 1 = 0t 1 = ∆t 1 = 2∆t 1 = 3∆xt 1 = 4∆–y–xyxt 2t 2t 2t 2t 1t 2Illustration of the TPPI method.Each time that t 1 isincremented, so is the phase ofthe pulse preceding t 1 .9.3.4.4 States-TPPIWhen the SHR method is used, axial peaks (arising from magnetizationwhich has not evolved during t 1) appear at F 1= 0; such peaks can be anuisance as they may obscure other wanted peaks. We will see below(section 9.4.6) that axial peaks can be suppressed with the aid of phasecycling, all be it at the cost of doubling the length of the phase cycle.The States-TPPI method does not suppress these axial peaks, but moves9–19

them to the edge of the spectrum so that they are less likely to obscurewanted peaks. All that is involved is that, each time t 1is incremented, boththe phase of the pulse which precedes t 1and the receiver phase are advancedby 180° i.e. the pulse goes x, –x and the receiver goes x, –x.For non-axial peaks, the two phase shifts cancel one another out, and sohave no effect. However, magnetization which gives rise to axial peaksdoes not experience the first phase shift, but does experience the receiverphase shift. The sign alternation in concert with t 1incrementation adds afrequency of SW 1/2 to each peak, thus shifting it to the edge of the spectrum.Note that in States-TPPI the spectral range in the F 1dimension is –(SW 1/2)to +(SW 1/2) and the sampling interval is 1/2SW 1, just as in the SHR method.The nice feature of States-TPPI is that is moves the axial peaks out of theway without lengthening the phase cycle. It is therefore convenient to usein complex three- and four-dimensional spectra were phase cycling is at apremium.9.4 Phase cyclingIn this section we will start out by considering in detail how to write a phasecycle to select a particular value of ∆p and then use this discussion to leadon to the formulation of general principles for constructing phase cycles.These will then be used to construct appropriate cycles for a number ofcommon experiments.9.4.1 Selection of a single pathwayTo focus on the issue at hand let us consider the case of transferring fromcoherence order +2 to order –1. Such a transfer has ∆p = (–1 – (2) ) = –3.Let us imagine that the pulse causing this transformation is cycled aroundthe four cardinal phases (x, y, –x, –y, i.e. 0°, 90°, 180°, 270°) and draw up atable of the phase shift that will be experienced by the transferredcoherence. This is simply computed as – ∆p φ, in this case = – (–3)φ = 3φ.steppulse phasephase shift experienced bytransfer with ∆p = –3equivalent phase1 0 0 02 90 270 2703 180 540 1804 270 810 90The fourth column, labelled "equivalent phase", is just the phase shiftexperienced by the coherence, column three, reduced to be in the range 0 to360° by subtracting multiples of 360° (e.g. for step 3 we subtracted 360°and for step 4 we subtracted 720°).If we wished to select ∆p = –3 we would simply shift the phase of thereceiver in order to match the phase that the coherence has acquired; theseare the phases shown in the last column. If we did this, then each step of thecycle would give an observed signal of the same phase and so they fourcontributions would all add up. This is precisely the same thing as we did9–20

when considering the CYCLOPS sequence in section 9.1.6; in both casesthe receiver phase follows the phase of the desired magnetization orcoherence.We now need to see if this four step phase cycle eliminates the signalsfrom other pathways. As an example, let us consider a pathway with∆p = 2, which might arise from the transfer from coherence order –1 to +1.Again we draw up a table to show the phase experienced by a pathway with∆p = 2, that is computed as – (2)φsteppulsephasephase shift experienced bytransfer with ∆p = 2equivalentphaserx. phase toselect ∆p = –3difference1 0 0 0 0 02 90 –180 180 270 270 – 180 = 903 180 –360 0 180 180 – 0 = 1804 270 –540 180 90 90 – 180 = –90As before, the equivalent phase is simply the phase in column 3 reduced tothe range 0 to 360°. The fifth column shows the receiver (abbreviated torx.) phases determined above for selection of the transfer with ∆p = –3. Thequestion we have to ask is whether or not these phase shifts will lead tocancellation of the transfer with ∆p = 2. To do this we compute thedifference between the receiver phase, column 5, and the phase shiftexperienced by the transfer with ∆p = 2, column 4. The results are shown incolumn 6, labelled "difference". Looking at this difference column we cansee that step 1 will cancel with step 3 as the 180° phase shift between themmeans that the two signals have opposite sign. Likewise step 2 will cancelwith step 4 as there is a 180° phase shift between them. We conclude,therefore, that this four step cycle cancels the signal arising from a pathwaywith∆p = 2.An alternative way of viewing the cancellation is to represent the resultsof the "difference" column by vectors pointing at the indicated angles. Thisis shown below; it is clear that the opposed vectors cancel one another.0°step 1 2 3 4difference 0°90° 180° 270°Next we consider the coherence transfer with ∆p = +1. Again, we drawup the table and calculate the phase shifts experience by this transfer, whichare given by – (+1)φ = –φ.9–21

steppulsephasephase shift experiencedby transfer with ∆p = +1equivalentphaserx. phase toselect ∆p = –3difference1 0 0 0 0 02 90 –90 270 270 270 – 270 = 03 180 –180 180 180 180 – 180 = 04 270 –270 90 90 90 – 90 = 0Here we see quite different behaviour. The equivalent phases, that is thephase shifts experienced by the transfer with ∆p = 1, match exactly thereceiver phases determined for ∆p = –3, thus the phases in the "difference"column are all zero. We conclude that the four step cycle selects transfersboth with ∆p = –3 and +1.Some more work with tables such as these will reveal that this four stepcycle suppresses contributions from changes in coherence order of –2, –1and 0. It selects ∆p = –3 and 1. It also selects changes in coherence orderof 5, 9, 13 and so on. This latter sequence is easy to understand. A pathwaywith ∆p = 1 experiences a phase shift of –90° when the pulse is shifted inphase by 90°; the equivalent phase is thus 270°. A pathway with ∆p = 5would experience a phase shift of –5 × 90° = –450° which corresponds to anequivalent phase of 270°. Thus the phase shifts experienced for ∆p = 1 and5 are identical and it is clear that a cycle which selects one will select theother. The same goes for the series ∆p = 9, 13 ...The extension to negative values of ∆p is also easy to see. A pathwaywith ∆p = –3 experiences a phase shift of 270° when the pulse is shifted inphase by 90°. A transfer with ∆p = +1 experiences a phase of –90° whichcorresponds to an equivalent phase of 270°. Thus both pathways experiencethe same phase shifts and a cycle which selects one will select the other.The pattern is clear, this four step cycle will select a pathway with ∆p = –3,as it was designed to, and also it will select any pathway with ∆p = –3 + 4nwhere n = ±1, ±2, ±3 ...9.4.2 General RulesThe discussion in the previous section can be generalised in the followingway. Consider a phase cycle in which the phase of a pulse takes N evenlyspaced steps covering the range 0 to 2π radians. The phases, φ k, areφ k= 2πk/N where k = 0, 1, 2 ... (N – 1).To select a change in coherence order, ∆p, the receiver phase is set to–∆p × φ kfor each step and the resulting signals are summed. This cyclewill, in addition to selecting the specified change in coherence order, alsoselect pathways with changes in coherence order (∆p ± nN) wheren = ±1, ±2 ..The way in which phase cycling selects a series of values of ∆p whichare related by a harmonic condition is closely related to the phenomenon ofaliasing in Fourier transformation. Indeed, the whole process of phasecycling can be seen as the computation of a discrete Fourier transformationwith respect to the pulse phase; in this case the Fourier co-domains arephase and coherence order.9–22

The fact that a phase cycle inevitably selects more than one change incoherence order is not necessarily a problem. We may actually wish toselect more than one pathway, and examples of this will be given below inrelation to specific two-dimensional experiments. Even if we only requireone value of ∆p we may be able to discount the values selected at the sametime as being improbable or insignificant. In a system of m coupled spinsone-half, the maximum order of coherence that can be generated is m, thusin a two spin system we need not worry about whether or not a phase cyclewill discriminate between double quantum and six quantum coherences asthe latter simply cannot be present. Even in more extended spin systems thelikelihood of generating high-order coherences is rather small and so wemay be able to discount them for all practical purposes. If a high level ofdiscrimination between orders is needed, then the solution is simply to use aphase cycle which has more steps i.e. in which the phases move in smallerincrements. For example a six step cycle will discriminate between ∆p = +2and +6, whereas a four step cycle will not.9.4.3 Refocusing PulsesA 180° pulse simply changes the sign of the coherence order. This is easilydemonstrated by considering the effect of such a pulse on the operators I +and I –. For example:I ≡ I +πIx+ ( xiIy)⎯⎯ →( Ix– iIy)≡I–corresponds to p = +1 → p = –1. In a more complex product of operators,each raising and lowering operator is affected in this way so overall thecoherence order changes sign.We can now derive the EXORCYLE phase cycle using this property.Consider a 180° pulse acting on single quantum coherence, for which theCTP is shown opposite. For the pathway starting with p = 1 the effect of the180° pulse is to cause a change with ∆p = –2. The table shows a four-stepcycle to select this change10–1180°A 180° pulse simply changesthe sign of the coherence order.The EXORCYLE phase cyclingselects both of the pathwaysshown.Stepphase of180° pulsephase shift experienced bytransfer with ∆p = –2Equivalent phase= rx. phase1 0 0 02 90 180 1803 180 360 04 270 540 180The phase cycle is thus 0, 90°, 180°, 270° for the 180° pulse and0° 180° 0° 180° for the receiver; this is precisely the set of phases deducedbefore for EXORCYCLE in section 9.1.7.As the cycle has four steps, a pathway with ∆p = +2 is also selected; thisis the pathway which starts with p = –1 and is transferred to p = +1.Therefore, the four steps of EXORCYLE select both of the pathways shownin the diagram above.A two step cycle, consisting of 0°, 180° for the 180° pulse and 0°, 0° for9–23

the receiver, can easily be shown to select all even values of ∆p. Thisreduced form of EXORCYCLE is sometimes used when it is necessary tominimise the number of steps in a phase cycle. An eight step cycle, inwhich the 180° pulse is advanced in steps of 45°, can be used to select therefocusing of double-quantum coherence in which the transfer is fromp = +2 to –2 (i.e. ∆p = –4) or vice versa.10–19.4.4 Combining phase cyclesSuppose that we wish to select the pathway shown opposite; for the firstpulse ∆p is 1 and for the second it is –2. We can construct a four-step cyclefor each pulse, but to select the overall pathway shown these two cycleshave to be completed independently of one another. This means that therewill be a total of sixteen steps. The table shows how the appropriatereceiver cycling can be determinedStepphase of 1stpulsephase for∆p = 1phase of 2ndpulsephase for∆p = –2totalphaseequivalent phase =rx. phase1 0 0 0 0 0 02 90 –90 0 0 –90 2703 180 –180 0 0 –180 1804 270 –270 0 0 –270 905 0 0 90 180 180 1806 90 –90 90 180 90 907 180 –180 90 180 0 08 270 –270 90 180 –90 2709 0 0 180 360 360 010 90 –90 180 360 270 27011 180 –180 180 360 180 18012 270 –270 180 360 90 9013 0 0 270 540 540 18014 90 –90 270 540 450 9015 180 –180 270 540 360 016 270 –270 270 540 270 270In the first four steps the phase of the second pulse is held constant and thephase of the first pulse simply goes through the four steps 0° 90° 180° 270°.As we are selecting ∆p = 1 for this pulse, the receiver phases are simply 0°,270°, 180°, 90°.Steps 5 to 8 are a repeat of steps 1–4 except that the phase of the secondpulse has been moved by 90°. As ∆p for the second pulse is –2, therequired pathway experiences a phase shift of 180° and so the receiverphase must be advanced by this much. So, the receiver phases for steps 5–8are just 180° ahead of those for steps 1–4.In the same way for steps 9–12 the first pulse again goes through thesame four steps, and the phase of the second pulse is advanced to 180°.Therefore, compared to steps 1–4 the receiver phases in steps 9–12 need tobe advanced by – (–2) × 180° = 360° = 0°. Likewise, the receiver phases forsteps 13–16 are advanced by – (–2) × 270° = 540° = 180°.9–24

Another way of looking at this is to consider each step individually. Forexample, compared to step 1, in step 14 the first pulse has been advanced by90° so the phase from the first pulse is – (1) × 90° = –90°. The second pulsehas been advanced by 270° so the phase from this is – (–2) × 270° = 540°.The total phase shift of the required pathway is thus –90 + 540 = 450°which is an equivalent phase of 90°. This is the receiver phase shown in thefinal column.The key to devising these sequences is to simply work out the two fourstepcycles independently and then merge them together rather than tryingto work on the whole cycle. One writes down the first four steps, and thenduplicates this four times as the second pulse is shifted. We would find thesame steps, in a different sequence, if the phase of the second pulse isshifted in the first four steps.We can see that the total size of a phase cycle grows at an alarming rate.With four phases for each pulse the number of steps grows as 4 l where l isthe number of pulses in the sequence. A three-pulse sequence such asNOESY or DQF COSY would therefore involve a 64 step cycle. Such longcycles put a lower limit on the total time of an experiment and we may endup having to run an experiment for a long time not to achieve the desiredsignal-to-noise ratio but simply to complete the phase cycle.Fortunately, there are several "tricks" which we can use in order toshorten the length of a phase cycle. To appreciate whether or not one ofthese tricks can be used in a particular sequence we need to understand insome detail what the sequence is actually doing and what the likelyproblems are going to be.9.4.5 Tricks9.4.5.1 The first pulseAll pulse sequences start with equilibrium magnetization, which hascoherence order 0. It can easily be shown that when a pulse is applied toequilibrium magnetization the only coherence orders that can be generatedare ±1. If retaining both of these orders is acceptable (which it often is), it istherefore not necessary to phase cycle the first pulse in a sequence.There are two additional points to make here. If the spins have notrelaxed completely by the start of the sequence the initial magnetization willnot be at equilibrium. Then, the above simplification does not apply.Secondly, the first pulse of a sequence is often cycled in order to suppressaxial peaks in two-dimensional spectra. This is considered in more detail insection 9.4.6.9–25

210–1–2Pulse sequence for generatingdouble-quantum coherence.Note that the 180° pulse simplycauses a change in the sign ofthe coherence order.9.4.5.2 Grouping pulses togetherThe sequence shown opposite can be used to generate multiple quantumcoherence from equilibrium magnetization; during the spin echo anti-phasemagnetization develops and the final pulse transfers this into multiplequantum coherence. Let us suppose that we wish to generate doublequantum, with p = ±2, as show by the CTP opposite.As has already been noted, the first pulse can only generate p = ±1 andthe 180° pulse only causes a change in the sign of the coherence order. Theonly pulse we need to be concerned with is the final one which we want togenerate only double quantum. We could try to devise a phase cycle for thelast pulse alone or we could simply group all three pulses together andimagine that, as a group, they achieve the transformation p = 0 to p = ±2 i.e.∆p = ±2. The phase cycle would simply be for the three pulses together togo 0°, 90°, 180°, 270°, with the receiver going 0°, 180°, 0°, 180°.It has to be recognised that by cycling a group of pulses together we areonly selecting an overall transformation; the coherence orders presentwithin the group of pulses are not being selected. It is up to the designer ofthe experiment to decide whether or not this degree of selection is sufficient.The four step cycle mentioned above also selects ∆p = ±6; again, wewould have to decide whether or not such high orders of coherence werelikely to be present in the spin system. Finally, we note that the ∆p valuesfor the final pulse are ±1, ±3; it would not be possible to devise a four stepcycle which selects all of these pathways.9.4.5.3 The last pulseWe noted above that only coherence order –1 is observable. So, althoughthe final pulse of a sequence may cause transfer to many different orders ofcoherence, only transfers to p = –1 will result in observable signals. Thus,if we have already selected, in an unambiguous way, a particular set ofcoherence orders present just before the last pulse, no further cycling of thispulse is needed.9.4.5.4 Example – DQF COSYA good example of the applications of these ideas is in devising a phasecycle for DQF COSY, whose pulse sequence and CTP is shown below.t 1 t 2p210–1–2∆p=±1 ±1,±3+1,–3Note that we have retained symmetrical pathways in t 1so that absorptionmode lineshapes can be obtained. Also, both in generating the doublequantum coherence, and in reconverting it to observable magnetization, allpossible pathways have been retained. If we do not do this, signal intensityis lost.One way of viewing this sequence is to group the first two pulses9–26

together and view them as achieving the transformation 0 → ±2 i.e.∆p = ±2. This is exactly the problem considered in section 9.4.5.2, wherewe saw that a suitable four step cycle is for the first two pulses to go 0°, 90°,180°, 270° and the receiver to go 0°, 180°, 0°, 180°. This unambiguouslyselects p = ±2 just before the last pulse, so phase cycling of the last pulse isnot required (see section 9.4.5.3).An alternative view is to say that as only p = –1 is observable, selectingthe transformation ∆p = +1 and –3 on the last pulse will be equivalent toselecting p = ±2 during the period just before the last pulse. Since the firstpulse can only generate p = ±1 (present during t 1), the selection of ∆p = +1and –3 on the last pulse is sufficient to define the CTP completely.A four step cycle to select ∆p = +1 involves the pulse going 0°, 90°,180°, 270° and the receiver going 0°, 270°, 180°, 90°. As this cycle hasfour steps is automatically also selects ∆p = –3, just as required.The first of these cycles also selects ∆p = ±6 for the first two pulses i.e.filtration through six-quantum coherence; normally, we can safely ignorethe possibility of such high-order coherences. The second of the cycles alsoselects ∆p = +5 and ∆p = –7 on the last pulse; again, these transfers involvesuch high orders of multiple quantum that they can be ignored.9.4.6 Axial peak suppressionPeaks are sometimes seen in two-dimensional spectra at co-ordinates F 1= 0and F 2= frequencies corresponding to the usual peaks in the spectrum. Theinterpretation of the appearance of these peaks is that they arise frommagnetization which has not evolved during t 1and so has not acquired afrequency label.A common source of axial peaks is magnetization which recovers due tolongitudinal relaxation during t 1. Subsequent pulses make thismagnetization observable, but it has no frequency label and so appears atF 1= 0. Another source of axial peaks is when, due to pulse imperfections,not all of the original equilibrium magnetization is moved into thetransverse plane by the first pulse. The residual longitudinal magnetizationcan be made observable by subsequent pulses and hence give rise to axialpeaks.A simple way of suppressing axial peaks is to select the pathway ∆p = ±1on the first pulse; this ensures that all signals arise from the first pulse. Atwo-step cycle in which the first pulse goes 0°, 180° and the receiver goes0°, 180° selects ∆p = ±1. It may be that the other phase cycling used in thesequence will also reject axial peaks so that it is not necessary to add anexplicit axial peak suppression steps. Adding a two-step cycle for axialpeak suppression doubles the length of the phase cycle.9.4.7 Shifting the whole sequence – CYCLOPSIf we group all of the pulses in the sequence together and regard them as aunit they simply achieve the transformation from equilibriummagnetization, p = 0, to observable magnetization, p = –1. They could becycled as a group to select this pathway with ∆p = –1, that is the pulses9–27

going 0°, 90°, 180°, 270° and the receiver going 0°, 90°, 180°, 270°. This issimple the CYCLOPS phase cycle described in section 9.1.6.If time permits we sometimes add CYCLOPS-style cycling to all of thepulses in the sequence so as to suppress some artefacts associated withimperfections in the receiver. Adding such cycling does, of course, extendthe phase cycle by a factor of four.This view of the whole sequence as causing the transformation ∆p = –1also enables us to interchange receiver and pulse phase shifts. For example,suppose that a particular step in a phase cycle requires a receiver phase shiftθ. The same effect can be achieved by shifting all of the pulses by –θ andleaving the receiver phase unaltered. The reason this works is that all of thepulses taken together achieve the transformation ∆p = –1, so shifting theirphases by –θ shift the signal by – (–θ) = θ, which is exactly the effect ofshifting the receiver by θ. This kind of approach is sometimes helpful ifhardware limitations mean that small angle phase-shifts are only availablefor the pulses.9.4.8 Equivalent cyclesFor even a relatively simple sequence such as DQF COSY there are anumber of different ways of writing the phase cycle. Superficially these canlook very different, but it may be possible to show that they really are thesame.For example, consider the DQF COSY phase cycle proposed in section9.4.5.4 where we cycle just the last pulsestep 1st pulse 2nd pulse 3rd pulse receiver1 0 0 0 02 0 0 90 2703 0 0 180 1804 0 0 270 90Suppose we decide that we do not want to shift the receiver phase, but wantto keep it fixed at phase zero. As described above, this means that we needto subtract the receiver phase from all of the pulses. So, for example, in step2 we subtract 270° from the pulse phases to give –270°, –270° and –180°for the phases of the first three pulses, respectively; reducing these to theusual range gives phases 90°, 90° and 180°. Doing the same for the othersteps gives a rather strange looking phase cycle, but one which works in justthe same way.step 1st pulse 2nd pulse 3rd pulse receiver1 0 0 0 02 90 90 180 03 180 180 0 04 270 270 180 0We can play one more trick with this phase cycle. As the third pulse isrequired to achieve the transformation ∆p = –3 or +1 we can alter its phaseby 180° and compensate for this by shifting the receiver by 180° also.9–28

Doing this for steps 2 and 4 only givesstep 1st pulse 2nd pulse 3rd pulse receiver1 0 0 0 02 90 90 0 1803 180 180 0 04 270 270 0 180This is exactly the cycle proposed in section 9.4.5.4.9.4.9 Further examplesIn this section we will use a shorthand to indicate the phases of the pulsesand the receiver. Rather than specifying the phase in degrees, the phases areexpressed as multiples of 90°. So, EXORCYCLE becomes 0 1 2 3 for the180° pulse and 0 2 0 2 for the receiver.9.4.9.1 Double quantum spectroscopyA simple sequence for double quantum spectroscopy is shown belowττt 1t 2210–1–2Note that both pathways with p = ±1 during the spin echo and with p = ±2during t 1are retained. There are a number of possible phase cycles for thisexperiment and, not surprisingly, they are essentially the same as those forDQF COSY. If we regard the first three pulses as a unit, then they arerequired to achieve the overall transformation ∆p = ±2, which is the same asthat for the first two pulses in the DQF COSY sequence. Thus the samecycle can be used with these three pulses going 0 1 2 3 and the receivergoing 0 2 0 2. Alternatively the final pulse can be cycled 0 1 2 3 withthe receiver going 0 3 2 1, as in section 9.4.5.4.Both of these phase cycles can be extended by EXORCYCLE phasecycling of the 180° pulse, resulting in a total of 16 steps.9.4.9.2 NOESYThe pulse sequence for NOESY (with retention of absorption modelineshapes) is shown belowt 1τ mix t 210–1If we group the first two pulses together they are required to achieve thetransformation ∆p = 0 and this leads to a four step cycle in which the pulsesgo 0 1 2 3 and the receiver remains fixed as 0 0 0 0. In this experimentaxial peaks arise due to z-magnetization recovering during the mixing time,and this cycle will not suppress these contributions. Thus we need to addaxial peak suppression, which is conveniently done by adding the simple9–29

cycle 0 2 on the first pulse and the receiver. The final 8 step cycle is 1stpulse: 0 1 2 3 2 3 0 1, 2nd pulse: 0 1 2 3 0 1 2 3, 3rd pulse fixed,receiver: 0 0 0 0 2 2 2 2.An alternative is to cycle the last pulse to select the pathway ∆p = –1,giving the cycle 0 1 2 3 for the pulse and 0 1 2 3 for the receiver. Onceagain, this does not discriminate against z-magnetization which recoversduring the mixing time, so a two step phase cycle to select axial peaks needsto be added.9.4.9.3 Heteronuclear ExperimentsThe phase cycling for most heteronuclear experiments tends to be rathertrivial in that the usual requirement is simply to select that component whichhas been transferred from one nucleus to another. We have already seen insection 9.1.8 that this is achieved by a 0 2 phase cycle on one of the pulsescausing the transfer accompanied by the same on the receiver i.e. adifference experiment. The choice of which pulse to cycle depends more onpractical considerations than with any fundamental theoreticalconsiderations.The pulse sequence for HMQC, along with the CTP, is shown belowI∆∆t 2S1p I 0–11p S 0–1t 1Note that separate coherence orders are assigned to the I and S spins.Observable signals on the I spin must have p I= –1 and p S= 0 (any othervalue of p Swould correspond to a heteronuclear multiple quantumcoherence). Given this constraint, and the fact that the I spin 180° pulsesimply inverts the sign of p I, the only possible pathway on the I spins is thatshown.The S spin coherence order only changes when pulses are applied tothose spins. The first 90° S spin pulse generates p S= ±1, just as before. Asby this point p I= +1, the resulting coherences have p S= +1, p I= –1(heteronuclear zero-quantum) and p S= +1, p I= +1 (heteronuclear doublequantum).The I spin 180° pulse interconverts these midway during t 1, andfinally the last S spin pulse returns both pathways to p S= 0. A detailedanalysis of the sequence shows that retention of both of these pathwaysresults in amplitude modulation in t 1(provided that homonuclear couplingsbetween I spins are not resolved in the F 1dimension).Usually, the I spins are protons and the S spins some low-abundanceheteronucleus, such as 13 C. The key thing that we need to achieve is tosuppress the signals arising from vast majority of I spins which are notcoupled to S spins. This is achieved by cycling a pulse which affects thephase of the required coherence but which does not affect that of the9–30

unwanted coherence. The obvious targets are the two S spin 90° pulses,each of which is required to give the transformation ∆p S= ±1. A two stepcycle with either of these pulses going 0 2 and the receiver doing the samewill select this pathway and, by difference, suppress any I spinmagnetization which has not been passed into multiple quantum coherence.It is also common to add EXORCYCLE phase cycling to the I spin 180°pulse, giving a cycle with eight steps overall.9.4.10 General points about phase cyclingPhase cycling as a method suffers from two major practical problems. Thefirst is that the need to complete the cycle imposes a minimum time on theexperiment. In two- and higher-dimensional experiments this minimumtime can become excessively long, far longer than would be needed toachieve the desired signal-to-noise ratio. In such cases the only way ofreducing the experiment time is to record fewer increments which has theundesirable consequence of reducing the limiting resolution in the indirectdimensions.The second problem is that phase cycling always relies on recording allpossible contributions and then cancelling out the unwanted ones bycombining subsequent signals. If the spectrum has high dynamic range, orif spectrometer stability is a problem, this cancellation is less than perfect.The result is unwanted peaks and t 1-noise appearing in the spectrum. Theseproblems become acute when dealing with proton detected heteronuclearexperiments on natural abundance samples, or in trying to record spectrawith intense solvent resonances.Both of these problems are alleviated to a large extent by moving to analternative method of selection, the use of field gradient pulses, which is thesubject of the next section. However, as we shall see, this alternativemethod is not without its own difficulties.9.5 Selection with field gradient pulses9.5.1 IntroductionLike phase cycling, field gradient pulses can be used to select particularcoherence transfer pathways. During a pulsed field gradient the appliedmagnetic field is made spatially inhomogeneous for a short time. As aresult, transverse magnetization and other coherences dephase across thesample and are apparently lost. However, this loss can be reversed by theapplication of a subsequent gradient which undoes the dephasing processand thus restores the magnetization or coherence. The crucial property ofthe dephasing process is that it proceeds at a different rate for differentcoherences. For example, double-quantum coherence dephases twice as fastas single-quantum coherence. Thus, by applying gradient pulses of differentstrengths or durations it is possible to refocus coherences which have, forexample, been changed from single- to double-quantum by a radiofrequencypulse.Gradient pulses are introduced into the pulse sequence in such a way that9–31

only the wanted signals are observed in each experiment. Thus, in contrastto phase cycling, there is no reliance on subtraction of unwanted signals,and it can thus be expected that the level of t 1-noise will be much reduced.Again in contrast to phase cycling, no repetitions of the experiment areneeded, enabling the overall duration of the experiment to be set strictly inaccord with the required resolution and signal-to-noise ratio.The properties of gradient pulses and the way in which they can be usedto select coherence transfer pathways have been known since the earliestdays of multiple-pulse NMR. However, in the past their wide applicationhas been limited by technical problems which made it difficult to use suchpulses in high-resolution NMR. The problem is that switching on thegradient pulse induces currents in any nearby conductors, such as the probehousing and magnet bore tube. These induced currents, called eddycurrents, themselves generate magnetic fields which perturb the NMRspectrum. Typically, the eddy currents are large enough to disrupt severelythe spectrum and can last many hundreds of milliseconds. It is thusimpossible to observe a high-resolution spectrum immediately after theapplication of a gradient pulse. Similar problems have beset NMR imagingexperiments and have led to the development of shielded gradient coilswhich do not produce significant magnetic fields outside the sample volumeand thus minimise the generation of eddy currents. The use of thistechnology in high-resolution NMR probes has made it possible to observespectra within tens of microseconds of applying a gradient pulse. With suchapparatus, the use of field gradient pulses in high resolution NMR is quitestraightforward, a fact first realised and demonstrated by Hurd whose workhas pioneered this whole area.9.5.2 Dephasing caused by gradientsA field gradient pulse is a period during which the B 0field is made spatiallyinhomogeneous; for example an extra coil can be introduced into the sampleprobe and a current passed through the coil in order to produce a field whichvaries linearly in the z-direction. We can imagine the sample being dividedinto thin discs which, as a consequence of the gradient, all experiencedifferent magnetic fields and thus have different Larmor frequencies. At thebeginning of the gradient pulse the vectors representing transversemagnetization in all these discs are aligned, but after some time each vectorhas precessed through a different angle because of the variation in Larmorfrequency. After sufficient time the vectors are disposed in such a way thatthe net magnetization of the sample (obtained by adding together all thevectors) is zero. The gradient pulse is said to have dephased themagnetization.It is most convenient to view this dephasing process as being due to thegeneration by the gradient pulse of a spatially dependent phase. Supposethat the magnetic field produced by the gradient pulse, B g, varies linearlyalong the z-axis according toB Gz9–32g =where G is the gradient strength expressed in, for example, T m –1 or G cm –1 ;

the origin of the z-axis is taken to be in the centre of the sample. At anyparticular position in the sample the Larmor frequency, ω L(z), depends onthe applied magnetic field, B 0, and B gωL = γ( B0 + Bg)= γ( B0+ Gz ) ,where γ is the gyromagnetic ratio. After the gradient has been applied fortime t, the phase at any position in the sample, Φ(z), is given byΦ()= z γ ( B0 + Gz)t . The first part of this phase is just that due to the usualLarmor precession in the absence of a field gradient. Since this is constantacross the sample it will be ignored from now on (which is formally thesame result as viewing the magnetization in a frame of reference rotating atγB 0). The remaining term γGzt is the spatially dependent phase induced bythe gradient pulse.If a gradient pulse is applied to pure x-magnetization, the followingevolution takes place at a particular position in the sampleγGztIzI ⎯ →cos γGzt I sin γGzt I .x⎯⎯ ( ) + ( )The total x-magnetization in the sample, M x, is found by adding up themagnetization from each of the thin discs, which is equivalent to the integralM x ()= t1r maxx12 r max∫– 1 2 r maxcos( γGzt)dzwhere it has been assumed that the sample extends over a region ± 1 2r max.Evaluating the integral gives an expression for the decay of x-magnetizationduring a gradient pulse( )sin Gr tMx( 1 2γmaxt)=12γGrmaxtThe plot below shows M x (t) as a function of time1.0y[11]M x0.50.010 20 30 40 50γGr max t-0.5The black line shows the decay of magnetization due to the action of a gradient pulse. The grey line isan approximation, valid at long times, for the envelope of the decay.Note that the oscillations in the decaying magnetization are imposed on anoverall decay which, for long times, is given by 2/(γGtr max). Equation [11]embodies the obvious points that the stronger the gradient (the larger G) thefaster the magnetization decays and that magnetization from nuclei withhigher gyromagnetic ratios decays faster. It also allows a quantitative9–33

assessment of the gradient strengths required: the magnetization will havedecayed to a fraction α of its initial value after a time of the order of2 ( γGαr max ) (the relation is strictly valid for α

9.5.3 Selection by refocusingThe method by which a particular coherence transfer pathway is selectedusing field gradients is illustrated opposite. The first gradient pulse encodesa spatially dependent phase, Φ 1and the second a phase Φ 2whereΦ1 = sp1 1γBg,1τ1 and Φ2 = sp2 2γBg,2τ2.After the second gradient the net phase is (Φ 1+ Φ 2). To select the pathwayinvolving transfer from coherence order p 1to coherence order p 2, this netphase should be zero; in other words the dephasing induced by the firstgradient pulse is undone by the second. The condition (Φ 1+ Φ 2) = 0 can berearranged tosB1 g,1τ1 p2sB2 g,2τ = – 2p1. [13]For example, if p 1= +2 and p 2= –1, refocusing can be achieved by makingthe second gradient either twice as long (τ 2= 2τ 1), or twice as strong(B g,2= 2B g,1) as the first; this assumes that the two gradients have identicalshape factors. Other pathways remain dephased; for example, assuming thatwe have chosen to make the second gradient twice as strong and the sameduration as the first, a pathway with p 1= +3 to p 2= –1 experiences a netphaseΦ1 + Φ2 = 3sBg,1τ1 – sBg,2τ1 = sBg,1τ1.Provided that this spatially dependent phase is sufficiently large, accordingthe criteria set out in the previous section, the coherence arising from thispathway remains dephased and is not observed. To refocus a pathway inwhich there is no sign change in the coherence orders, for example, p 1= –2to p 2= –1, the second gradient needs to be applied in the opposite sense tothe first; in terms of Eqn. [13] this is expressed by having s 2= –s 1.The procedure can easily be extended to select a more complexcoherence transfer pathway by applying further gradient pulses as thecoherence is transferred by further pulses, as illustrated opposite. Thecondition for refocusing is again that the net phase acquired by the requiredpathway be zero, which can be written formally as∑ spi iγiBg,iτi= 0 .iWith more than two gradients in the sequence, there are many ways inwhich a given pathway can be selected. For example, the second gradientmay be used to refocus the first part of the required pathway, leaving thethird and fourth to refocus another part. Alternatively, the pathway may beconsistently dephased and the magnetization only refocused by the finalgradient, just before acquisition.At this point it is useful to contrast the selection achieved using gradientpulses with that achieved using phase cycling. From Eqn. [13] it is clearthat a particular pair of gradient pulses selects a particular ratio ofcoherence orders; in the above example any two coherence orders in theratio –2 : 1 or 2 : –1 will be refocused. This selection according to ratio ofRFgτ1τ 2p 1p 2Illustration of the use of a pairof gradients to select a singlepathway. The radiofrequencypulses are on the line marked"RF" and the field gradientpulses are denoted by shadedrectangles on the line marked"g".RFg τ 1τ τ 3τ249–35

coherence orders is in contrast to the case of phase cycling in which a phasecycle consisting of N steps of 2π/N radians selects a particular change incoherence order ∆p = p 2– p 1, and further pathways which have ∆p = (p 2–p 1) ± mN, where m = 0, 1, 2 ...It is straightforward to devise a series of gradient pulses which will selecta single coherence transfer pathway. It cannot be assumed, however, thatsuch a sequence of gradient pulses will reject all other pathways i.e. leavecoherence from all other pathways dephased at the end of the sequence.Such assurance can only be given be analysing the fate of all other possiblecoherence transfer pathways under the particular gradient sequenceproposed. In complex pulse sequences there may also be several differentways in which gradient pulses can be included in order to achieve selectionof the desired pathway. Assessing which of these alternatives is the best, inthe light of the requirement of suppression of unwanted pathways and theeffects of pulse imperfections may be a complex task.ppp(a)RF210–1–2(b)RFg210–1–2(c)RFg210–1–29.5.3.1 Selection of multiple pathwaysAs we have seen earlier, it is not unusual to want to select two or morepathways simultaneously, for example either to maximise the signalintensity or to retain absorption-mode lineshapes. A good example of this isthe double-quantum filter pulse sequence element, shown opposite.The ideal pathway, shown in (a), preserves coherence orders p = ± 2 duringthe inter-pulse delay. Gradients can be used to select the pathway –2 to –1or +2 to –1, shown in (b) and (c) respectively. However, no combination ofgradients can be found which will select simultaneously both of thesepathways. In contrast, it is easy to devise a phase cycle which selects bothof these pathways (section 9.4.5.4). Thus, selection with gradients will inthis case result in a loss of half of the available signal when compared to anexperiment of equal length which uses selection by phase cycling. Such aloss in signal is, unfortunately, a very common feature when gradients areused for pathway selection.9.5.3.2 Selection versus suppressionCoherence order zero, comprising z-magnetization, zz-terms andhomonuclear zero-quantum coherence, does not accrue any phase during agradient pulse. Thus, it can be separated from all other orders simply byapplying a single gradient. In a sense, however, this is not a gradientselection process; rather it is a suppression of all other coherences. Agradient used in this way is often called a purge gradient. In contrast toexperiments where selection is achieved, there is no inherent sensitivity losswhen gradients are used for suppression. We will see examples below ofcases where this suppression approach is very useful.9.5.3.3 Gradients on other axesThe simplest experimental arrangement generates a gradient in which themagnetic field varies in the z direction, however it is also possible togenerate gradients in which the field varies along x or y. Clearly, thespatially dependent phase generated by a gradient applied in one direction9–36

cannot be refocused by a gradient applied in a different direction. Insequences where more than one pair of gradients are used, it may beconvenient to apply further gradients in different directions to the first pair,so as to avoid the possibility of accidentally refocusing unwanted coherencetransfer pathways. Likewise, a gradient which is used to destroy allcoherences can be applied in a different direction to gradients used forpathway selection.9.5.4 Refocusing and inversion pulsesRefocusing and inversion pulses play an important role in multiple-pulseNMR experiments and so the interaction between such pulses and fieldgradient pulses will be explored in some detail. As has been noted above insection 9.4.3, a perfect refocusing pulse simply changes the sign of the orderof any coherences present, p → –p. If the pulse is imperfect, there will betransfer to coherence orders other than –p.A perfect inversion pulse simply inverts z-magnetization or, moregenerally, all z-operators: I z→ –I z. If the pulse is imperfect, it will generatetransverse magnetization or other coherences. Inversion pulses are usedextensively in heteronuclear experiments to control the evolution ofheteronuclear couplings.We start out the discussion by considering the refocusing of coherences,illustrated opposite. The net phase, Φ, at the end of such a sequence is( p) ( p')Φ = Ω δ + spγBgτ + Ω δ + sp'γBgτwhere Ω ( p)is the frequency with which coherence of order p evolves in the( − p) ( )absence of a gradient; note that Ω =−Ωp . The net phase is zero if, andonly if, p' = –p. With sufficiently strong gradients all other pathwaysremain dephased and the gradient sequence thus selects the refocusedcomponent. As is expected for a spin echo, the underlying evolution of thecoherence (as would occur in the absence of a gradient) is also refocused bythe selection of the pathway shown. Any transverse magnetization which animperfect refocusing pulse might create is also dephased.Placing equal gradients either side of a refocusing pulse therefore selectsthe coherence transfer pathway associated with a perfect refocusing pulse.This selection works for all coherence orders so, in contrast to thediscussion in section 9.5.3.1, there is no loss of signal. Such a pair ofgradients are often described as being used to "clean up" a refocusing pulse,referring to their role in eliminating unwanted pathways.We cannot use gradients to select the pathway associated with aninversion pulse as p = 0 both before and after the pulse. However, we canapply a gradient after the pulse to dephase any magnetization which mightbe created by an imperfect pulse. Taking the process a step further, we canapply a gradient both before and after the pulse, with the two gradients inopposite directions. The argument here is that this results in the maximumdephasing of unwanted coherences – both those present before the pulse andthose that might be generated by the pulse. Again, this sequence is oftendescribed as being used to "clean up" an inversion pulse.RFδ180°g τ τpp'Gradient sequence used to"clean up" a refocusing pulse.Note that the two gradients areof equal area. The refocusedpathway has p' = –p.RFgτ180°Gradient sequence used to"clean up" an inversion pulse.τδ9–37

ISgδτ180°In a heteronuclear experiment(separate pulses shown on theI and S spins) a 180° pulse onthe S spin refocuses the IScoupling over the period 2δ.The gradient pulses shown areused to "clean up" the inversionpulse.RFg210–1–2τδt 1t 2τ1τ 2Simple pulse sequence forDQF COSY with gradientselection and retention ofsymmetrical pathways in t 1 .The second gradient is twicethe area of the first.In heteronuclear experiments an inversion pulse applied to one nucleus isused to refocus the evolution of a coupling to another nucleus. Forexample, in the sequence shown opposite the centrally placed S spin 180°pulse refocuses the IS coupling over the period 2δ. The pair of gradientsshown have no net effect on I spin coherences as the dephasing due to thefirst gradient is exactly reversed by the second. The gradient sequence canbe thought of as "cleaning up" the S spin inversion pulse.9.5.4.1 Phase errors due to gradient pulsesFor the desired pathway, the spatially dependent phase created by a gradientpulse is refocused by a subsequent gradient pulse. However, the underlyingevolution of offsets (chemical shifts) and couplings is not refocused, andphase errors will accumulate due to the evolution of these terms. Sincegradient pulses are typically of a few milliseconds duration, these phaseerrors are substantial.In multi-dimensional NMR the uncompensated evolution of offsetsduring gradient pulses has disastrous effects on the spectra. This isillustrated here for the DQF COSY experiment, for which a pulse sequenceusing the gradient pulses is shown opposite. The problem with thissequence is that the double quantum coherence will evolve during delay τ 1.Normally, the delay between the last two pulses would be a fewmicroseconds at most, during which the evolution is negligible. However,once we need to apply a gradient the delay will need to be of the order ofmilliseconds and significant evolution takes place. The same considerationsapply to the second gradient.We can investigate the effect of this evolution by analysing the result fora two-spin system. In the calculation, it will be assumed that only theindicated pathway survives and that the spatially dependent part of theevolution due to the gradients can be ignored as ultimately it is refocused.The coherence with order of + 2 present during τ 1evolves according to( ( ) )Ω1τ1I1z+ Ω2τ1I2zI I ⎯⎯⎯⎯ ⎯ →I I exp – i Ω + Ω τ ,1+ 2+1+ 2+1 2 1where Ω 1and Ω 2are the offsets of spins 1 and 2, respectively. After thefinal 90° pulse and the second gradient the observable terms on spin 1 are[ ]( ( ) ) +exp – i Ω + Ω τ cosΩτ 2I I sinΩτ 2I I [14]i2 1 2 1 1 2 1x 2z 1 2 1y 2zwhere it has been assumed that τ 2is sufficiently short that evolution of thecoupling can be ignored. It is clearly seen from Eqn. [14] that, due to theevolution during τ 2, the multiplet observed in the F 2dimension will be amixture of dispersion and absorption anti-phase contributions. In addition,there is an overall phase shift due to the evolution during τ 1. The phasecorrection needed to restore this multiplet to absorption depends on both thefrequency in F 2and the double-quantum frequency during the first gradient.Thus, no single linear frequency dependent phase correction could phasecorrect a spectrum containing many multiplets. The need to control thesephase errors is plain.9–38

The general way to minimise these problems is to associate each gradientwith a refocusing pulse, as shown opposite. In sequence (a) the gradient isplaced in one of the delays of a spin echo; the evolution of the offset duringthe first delay τ is refocused during the second delay τ. So, overall there areno phase errors due to the evolution of the offset.An alternative is the sequence (b). Here, as in (a), the offset is refocusedover the whole sequence. The first gradient results in the usually spatiallydependent phase and then the 180° pulse changes the sign of the coherenceorder. As the second gradient is opposite to the first, it causes furtherdephasing; effectively, it is as if a gradient of length 2τ is applied.Sequence (b) will give the same dephasing effect as (a) if each gradient in(b) is of duration τ/2; the overall sequence will then be of duration τ. Iflosses due to relaxation are a problem, then clearly sequence (b) is to bepreferred as it takes less time than (a).The sequence below shows the gradient-selected DQF COSY pulsesequence modified by the inclusion of extra 180° pulses to remove phaseerrors. Note that although the extra 180° pulses are effective at refocusingoffsets, they do not refocus the evolution of homonuclear couplings. It isessential, therefore, to keep the gradient pulses as short as is feasible.(a)RFg(b)RFg180°τ ττ180°τ τττt 11RFτ11τ2τ 2τ 2gττIn many pulse sequences there are already periods during which theevolution of offsets is refocused. The evolution of offsets during a gradientpulse placed within such a period will therefore also be refocused, making itunnecessary to include extra refocusing pulses. Likewise, a gradient may beplaced during a "constant time" evolution period of a multi-dimensionalpulse sequence without introducing phase errors in the correspondingdimension; the gradient simply becomes part of the constant time period.This approach is especially useful in three- and four-dimensionalexperiments used to record spectra of 15 N, 13 C labelled proteins.9.5.5 SensitivityThe use of gradients for coherence selection has consequences for thesignal-to-noise ratio of the spectrum when it is compared to a similarspectrum recorded using phase cycling. If a gradient is used to suppress allcoherences other than p = 0, i.e. it is used simply to remove all coherences,leaving just z-magnetization or zz terms, there is no inherent loss ofsensitivity when compared to a corresponding phase cycled experiment. If,however, the gradient is used to select a particular order of coherence thesignal which is subsequently refocused will almost always be half theintensity of that which can be observed in a phase cycled experiment. Thisfactor comes about simply because it is likely that the phase cycledexperiment will be able to retain two symmetrical pathways, whereas thegradient selection method will only be able to refocus one of these.9–39

The foregoing discussion applies to the case of a selection gradientplaced in a fixed delay of a pulse sequence. The matter is different if thegradient is placed within the incrementable time of a multi-dimensionalexperiment, e.g. in t 1of a two-dimensional experiment. To understand theeffect that such a gradient has on the sensitivity of the experiment it isnecessary to be rather careful in making the comparison between thegradient selected and phase cycled experiments. In the case of the latterexperiments we need to include the SHR or TPPI method in order toachieve frequency discrimination with absorption mode lineshapes. If agradient is used in t 1we will need to record separate P- and N-type spectraso that they can be recombined to give an absorption mode spectrum. Wemust also ensure that the two spectra we are comparing have the samelimiting resolution in the t 1dimension, that is they achieve the samemaximum value of t 1and, of course, the total experiment time must be thesame.The detailed argument which is needed to analyse this problem is beyondthe scope of this lecture; it is given in detail in J. Magn. Reson Ser. A, 111,70-76 (1994) † . The conclusion is that the signal-to-noise ratio of anabsorption mode spectrum generated by recombining P- and N-typegradient selected spectra is lower, by a factor 1 2 , than the correspondingphase cycled spectrum with SHR or TPPI data processing.The potential reduction in sensitivity which results from selection withgradients may be more than compensated for by an improvement in thequality of the spectra obtained in this way. Often, the factor which limitswhether or not a cross peak can be seen is not the thermal noise level by thepresence of other kinds of "noise" associated with imperfect cancellationetc.9.5.6 DiffusionThe process of refocusing a coherence which has been dephased by agradient pulse is inhibited if the spins move either during or between thedefocusing and refocusing gradients. Such movement alters the magneticfield experienced by the spins so that the phase acquired during therefocusing gradient is not exactly opposite to that acquired during thedefocusing gradient.In liquids there is a translational diffusion of both solute and solventwhich causes such movement at a rate which is fast enough to causesignificant effects on NMR experiments using gradient pulses. As diffusionis a random process we expect to see a smooth attenuation of the intensity ofthe refocused signal as the diffusion contribution increases. These effectshave been known and exploited to measure diffusion constants since thevery earliest days of NMR.†There is an error in this paper: in Fig. 1(b) the penultimate S spin 90° pulse should bephase y and the final S spin 90° pulse is not required.9–40

The effect of diffusion on the signal intensity from the simple echosequence shown opposite is relatively simple to analyse and captures all ofthe essential points. Note that the two gradient pulses can be placedanywhere in the intervals δ either side of the 180° pulse. For a singleuncoupled resonance, the intensity of the observed signal, S, expressed as afraction of the signal intensity in the absence of a gradient, S 0is given byS ⎛ 2 2 2= − G⎛ τ−⎞ ⎞exp⎜γ τ ∆ DS ⎝ ⎝03⎠⎟[15]⎠where D is the diffusion constant, ∆ is the time between the start of the twogradient pulses and τ is the duration of the gradient pulses; relaxation hasbeen ignored. For a given pair of gradient pulses it is diffusion during theinterval between the two pulses, ∆, which determines the attenuation of theecho. The stronger the gradient the more rapidly the phase varies across thesample and thus the more rapidly the echo will be attenuated. This is the2 2 2physical interpretation of the term γ G τ in Eqn. [15].Diffusion constants generally decrease as the molecular mass increases.A small molecule, such as water, will diffuse up to twenty times faster thana protein with molecular weight 20,000. The table shows the loss inintensity due to diffusion for typical gradient pulse pair of 2 ms duration andof strength 10 G cm –1 for a small, medium and large sized molecule; data isgiven for ∆ = 2 ms and ∆ = 100 ms. It is seen that even for the most rapidlydiffusing molecules the loss of intensity is rather small for ∆ = 2 ms, butbecomes significant for longer delays. For large molecules, the effect issmall in all cases.180°δ δRF∆g τ τFraction of transverse magnetization refocused after a spin echo with gradient refocusing a∆/ms small molecule b medium sized molecule c macro molecule d2 0.99 1.00 1.00100 0.55 0.88 0.97aCalculated for the pulse sequence shown above for two gradients of strength 10 G cm –1and duration, τ, 2 ms; relaxation is ignored. b Diffusion constant, D, taken as that for water,which is 2.1 × 10 –9 m 2 s –1 at ambient temperatures. c Diffusion constant taken as 0.46 × 10 –9m 2 s –1 d. Diffusion constant taken as 0.12 × 10 –9 m 2 s –1 .9.5.6.1 Minimisation of Diffusion LossesThe foregoing discussion makes it clear that in order to minimise intensitylosses due to diffusion the product of the strength and durations of thegradient pulses, G 2 τ 2 , should be kept as small as is consistent withachieving the required level of suppression. In addition, a gradient pulsepair should be separated by the shortest time, ∆, within the limits imposedby the pulse sequence. This condition applies to gradient pairs the first ofwhich is responsible for dephasing, and the second for rephasing. Once thecoherence is rephased the time that elapses before further gradient pairs isirrelevant from the point of view of diffusion losses.9–41

(a)RFg(b)RFgt 1t 2t 1t 2In two-dimensional NMR, diffusion can lead to line broadening in the F 1dimension if t 1intervenes between a gradient pair. Consider the twoalternative pulse sequences for recording a simple COSY spectrum shownopposite. In (a) the gradient pair are separated by the very short time of thefinal pulse, thus keeping the diffusion induced losses to an absoluteminimum. In (b) the two gradients are separated by the incrementable timet 1; as this increases the losses due to diffusion will also increase, resulting inan extra decay of the signal in t 1. The extra line broadening due to this2 2 2decay can be estimated from Eqn. [15], with ∆ = t 1, as γ G τ D π Hz. Fora pair of 2 ms gradients of strength 10 G cm –1 this amounts ≈ 2 Hz in thecase of a small molecule.This effect by which diffusion causes an extra line broadening in the F 1dimension is usually described as diffusion weighting. Generally it ispossible to avoid it by careful placing of the gradients. For example, thesequences (a) and (b) are in every other respect equivalent, thus there is noreason not to chose (a). It should be emphasised that diffusion weightingoccurs only when t 1intervenes between the dephasing and refocusinggradients.9.5.7 Some examples of gradient selection9.5.7.1 IntroductionReference has already been made to the two general advantages of usinggradient pulses for coherence selection, namely the possibility of a generalimprovement in the quality of spectra and the removal of the requirement ofcompleting a phase cycle for each increment of a multi-dimensionalexperiment. This latter point is particularly significant when dealing withthree- and four-dimensional experiments.The use of gradients results in very significant improvement in thequality of proton-detected heteronuclear experiments, especially whenunlabelled samples are used. In such experiments, gradient selection resultsin much lower dynamic range in the free induction decay as compared tophase cycled experiments.As has been discussed above, special care needs to be taken inexperiments which use gradient selection in order to retain absorption modelineshapes.In the following sections the use of gradient selection in several differentexperiments will be described. The gradient pulses used in these sequenceswill be denoted G 1, G 2etc. where G iimplies a gradient of duration τ i,strength B g,iand shape factor s i. There is always the choice of altering theduration, strength or, conceivably, shape factor in order to establishrefocusing. Thus, for brevity we shall from now on write the spatiallydependent phase produced by gradient G iacting on coherence of order p asγpG iin the homonuclear case or9–42∑γ jpG j ijin the heteronuclear case; the sum is over all types of nucleus.

9.5.7.2 Double-quantum Filtered COSY(a)(b)RFt 1G 1 G 2τ1τ 2RFt 1τ 1G 1 G 2gg210–1–2210–1–2This experiment has already been discussed in detail in previous sections;sequence (a) is essentially that described already and is suitable forrecording absorption mode spectra. The refocusing condition is G 2= 2G 1;frequency discrimination in the F 1dimension is achieved by the SHR orTPPI procedures. Multiple quantum filters through higher orders can beimplemented in the same manner.In sequence (b) the final spin echo is not required as data acquisition isstarted immediately after the final radiofrequency pulse; phase errors whichwould accumulate during the second gradient pulse are thus avoided. Ofcourse, the signal only rephases towards the end of the final gradient, sothere is little signal to be observed. However, the crucial point is that, as themagnetization is all in antiphase at the start of t 2, the signal grows from zeroat a rate determined by the size the couplings on the spectrum. Providedthat the gradient pulse is much shorter that 1/J, where J is a typical protonprotoncoupling constant, the part of the signal missed during the gradientpulse is not significant and the spectrum is not perturbed greatly. Analternative procedure is to start to acquire the data after the final gradient,and then to right shift the free induction decay, bringing in zeroes from theleft, by a time equal to the duration of the gradient.9.5.7.3 HMQCI∆∆t 2Sτ 1t 1τ 1g1p I 0–11p S 0–1G 1 G 1 G 2There are several ways of implementing gradient selection into the HMQCexperiment, one of which, which leads to absorption mode spectra, is shownabove. The centrally placed I spin 180° pulse results in no net dephasing ofthe I spin part of the heteronuclear multiple quantum coherence by the twogradients G 1i.e. the dephasing of the I spin coherence caused by the first isundone by the second. However, the S spin coherence experiences a netdephasing due to these two gradients and this coherence is subsequently9–43

efocused by G 2. Two 180° S spin pulses together with the delays τ 1 refocusshift evolution during the two gradients G 1. The centrally placed 180° I spinpulse refocuses chemical shift evolution of the I spins during the delays ∆and all of the gradient pulses (the last gradient is contained within the finaldelay, ∆). The refocusing condition ism2γsG1 − γIG2= 0where the + and – signs refer to the P- and N-type spectra respectively. Theswitch between recording these two types of spectra is made simply byreversing the sense of G 2. The P- and N-type spectra are recordedseparately and then combined in the manner described in section 9.3.4.2 togive a frequency discriminated absorption mode spectrum.In the case that I and S are proton and carbon-13 respectively, thegradients G 1and G 2are in the ratio 2:±1. Proton magnetization not involvedin heteronuclear multiple quantum coherence, i.e. magnetization fromprotons not coupled to carbon-13, is refocused after the second gradient G 1but is then dephased by the final gradient G 2. Provided that the gradient isstrong enough these unwanted signals, and the t 1-noise associated withthem, will be suppressed.9.5.7.4 HSQCI∆2∆2ya∆2∆2t 2St 1bg1p I 0–11p S 0–1GThe sequence above shows the simplest way of implementing gradients intothe HSQC experiment. An analysis using product operators shows that atpoint a the required signal is present as the operator 2I zS zwhereas theundesired signal (from I spins not coupled to S spins) is present as I y. Thus,a field gradient applied at point a will dephase the unwanted magnetizationand leave the wanted term unaffected. This is an example of using gradientsnot for selection, but for suppression of unwanted magnetization (seesection 9.5.3.2).The main practical difficulty with this approach is that the unwantedmagnetization is only along y at point a provided all of the pulses areperfect; if the pulses are imperfect there will be some z-magnetizationpresent which will not be eliminated by the gradient. In the case ofobserving proton- 13 C or proton- 15 N HSQC spectra from natural abundancesamples, the magnetization from uncoupled protons is very much largerthan the wanted magnetization, so even very small imperfections in the9–44

pulses can give rise to unacceptably large residual signals. However, forglobally labelled samples the degree of suppression is often sufficient andsuch an approach is used successfully in many three- and four-dimensionalexperiments applied to globally 13 C and 15 N labelled proteins.The key to obtaining the best suppression of the uncoupled magnetizationis to apply a gradient when transverse magnetization is present on the Sspin. An example of the HSQC experiment utilising such a principle isshown belowI∆2∆2∆2∆2t 2St 1τ 1g1p I 0–11p S 0–1G 1 G 2HSQC pulse sequence with gradient selection. The CTP for an N-type spectrum is shown by the fullline and for the P-type spectrum by the dashed line.Here, G 1dephases the S spin magnetization present at the end of t 1and, aftertransfer to the I spins, G 2refocuses the signal. An extra 180° pulse to S inconjunction with the extra delay τ 1ensures that phase errors whichaccumulate during G 1are refocused; G 2is contained within the final spinecho which is part of the usual HSQC sequence. The refocusing conditionismγSG1 − γIG2 = 0where the – and + signs refer to the N- and P-type spectra respectively. Asbefore, an absorption mode spectrum is obtained by combining the N- andP-type spectra, which can be selected simply by reversing the sense of G 2.9.6 Zero-quantum dephasing and purge pulsesBoth z-magnetization and homonuclear zero-quantum coherence havecoherence order 0, and thus neither are dephased by the application of agradient pulse. Selection of coherence order zero is achieved simply byapplying a gradient pulse which is long enough to dephase all othercoherences; no refocusing is used. In the vast majority of experiments it isthe z-magnetization which is required and the zero-quantum coherence thatis selected at the same time is something of a nuisance.A number of methods have been developed to suppress contributions tothe spectrum from zero-quantum coherence. Most of these utilise theproperty that zero-quantum coherence evolves in time, whereas z-magnetization does not. Thus, if several experiments in which the zeroquantumhas been allowed to evolve for different times are co-added,cancellation of zero-quantum contributions to the spectrum will occur. Likephase cycling, such a method is time consuming and relies on a difference9–45

procedure. However, it has been shown that if a field gradient is combinedwith a period of spin-locking the coherences which give rise to these zeroquantumcoherences can be dephased. Such a process is convenientlyconsidered as a modified purging pulse.9.6.1 Purging pulsesA purging pulse consists of a relatively long period of spin-locking, takenhere to be applied along the x-axis. Magnetization not aligned along x willprecess about the spin-locking field and, because this field is inevitablyinhomogeneous, such magnetization will dephase. The effect is thus topurge all magnetization except that aligned along x. However, in a coupledspin system certain anti-phase states aligned perpendicular to the spin-lockaxis are also preserved. For a two spin system (with spins k and l), theoperators preserved under spin-locking are I kx, I lxand the anti-phase state2IkyIlz− 2IkzIly. Thus, in a coupled spin system, the purging effect of thespin-locking pulse is less than perfect.The reason why these anti-phase terms are preserved can best be seen bytransforming to a tilted co-ordinate system whose z-axis is aligned with theeffective field seen by each spin. For the case of a strong B 1field placedclose to resonance the effective field seen by each spin is along x, and so theoperators are transformed to the tilted frame simply by rotating them by–90° about y9–46−π2I−π2Iky Tly TIkx⎯⎯⎯→IkzIlx⎯⎯⎯→Ilz− π 2( Iky+ Ily)2I I −2I I ⎯⎯⎯⎯⎯→2I I −2I Iky lzkz lyT T T Tky lx kx lyOperators in the tilted frame are denoted with a superscript T. In this framethe x-magnetization has become z, and as this is parallel with the effectivefield, it clearly does not dephase. The anti-phase magnetization along y hasbecome2I I − 2I IT T T Tky lx kx lywhich is recognised as zero-quantum coherence in the tilted frame. Likezero-quantum coherence in the normal frame, this coherence does notdephase in a strong spin-locking field. There is thus a connection betweenthe inability of a field gradient to dephase zero-quantum coherence and thepreservation of certain anti-phase terms during a purging pulse.Zero-quantum coherence in the tilted frame evolves with time at afrequency, Ω TZQ, given byΩ T 2 2 2= Ω + ω ΩZQ( k 1 ) − ( l+ ω21 )where Ω iis the offset from the transmitter of spin i and ω 1is the B 1fieldstrength. If a field gradient is applied during the spin-locking period thezero quantum frequency is modified toTΩ ()= r Ω + γ B () r ω 2Ω γ B r ω 2ZQ2({ k g } + ) − + ()1l g2({ } +1 )This frequency can, under certain circumstances, become spatially

dependent and thus the zero-quantum coherence in the tilted frame willdephase. This is in contrast to the case of zero-quantum coherence in thelaboratory frame which is not dephased by a gradient pulse.The principles of this dephasing procedure are discussed in detailelsewhere (J. Magn. Reson. Ser. A 105, 167-183 (1993) ). Here, we note thefollowing features. (a) The optimum dephasing is obtained when the extraoffset induced by the gradient at the edges of the sample, γB g(r max), is of theorder of ω 1. (b) The rate of dephasing is proportional to the zero-quantumfrequency in the absence of a gradient, (Ω k– Ω l). (c) The gradient must beswitched on and off adiabatically. (d) The zero-quantum coherences mayalso be dephased using the inherent inhomogeneity of the radio-frequencyfield produced by typical NMR probes, but in such a case the optimumdephasing rate is obtained by spin locking off-resonance so thattan – 1( ω1 Ω kl , )≈ 54°.(e) Dephasing in an inhomogeneous B 1field can be accelerated by the useof special composite pulse sequences.The combination of spin-locking with a gradient pulse allows theimplementation of essentially perfect purging pulses. Such a pulse could beused in a two-dimensional TOCSY experiment whose pulse sequence isshown below as (a).(a)(b)yDIPSIt t 1 2t 1τ mt 2gG A G A G A G 2Pulse sequences using purging pulses which comprise a period of spin locking with a magnetic fieldgradient. The field gradient must be switched on and off in an adiabatic manner.In this experiment, the period of isotropic mixing transfers in-phasemagnetization (say along x) between coupled spins, giving rise to crosspeakswhich are absorptive and in-phase in both dimensions. However, themixing sequence also both transfers and generates anti-phase magnetizationalong y, which gives rise to undesirable dispersive anti-phase contributionsin the spectrum. In sequence (a) these anti-phase contributions areeliminated by the use of a purging pulse as described here. Of course, at thesame time all magnetization other than x is also eliminated, giving a nearperfect TOCSY spectrum without the need for phase cycling or otherdifference measures.These purging pulses can be used to generate pure z-magnetizationwithout contamination from zero-quantum coherence by following themwith a 90°(y) pulse, as is shown in the NOESY sequence (b). Zero-quantumcoherences present during the mixing time of a NOESY experiment giverise to troublesome dispersive contributions in the spectra, which can beeliminated by the use of this sequence.9–47