A good designed multiple-pulse NMR experiment is necessary to get a particular NMR spectrum. Because a given pulse sequence usually can affect the spins in some different ways and cause the final spectrum including the other unintended resonances that lead to ambiguities of interpretation.
There are two general methods can isolate these different possibilities. The first is phase cycling. In this method the phases of the pulses and the receiver are varied in a systematic way so that the signal from the desired pathways adds and signal from all other pathways cancels. Phase cycling experiment is required to repeat a number of times. The second method is to use field gradient pulses. These shorten the periods during which the applied magnetic field is made inhomogeneous. In a gradient pulse sequence any coherences present dephase are apparently lost. However, the aplication of a subsequent pulsed field gradient can undo this dephasing and cause some of the coherences to refocus. By a careful choice of the gradient pulses within a pulse sequence it is possible to ensure that only the coherences giving rise to the wanted signals are refocused. Unlike phase cycling, it is not require repeating the field gradient pulses in the experiment. Both methods can be described using the key concept of coherence order and by utilizing the idea of a coherence transfer pathway to specify the desired outcome of the experiment.
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A coherence of order p, represented by the density operator σ(p), evolves under a z-rotation of angle φ according to
Exp(-iφFz)σ(p)Exp(iφFz) = Exp(-ipφ) σ(p)
where Fz is the operator for the total z-component of the spin angular momentum. In words, a coherence of order p experiences a phase shift of -pφ. This equation is the definition of coherence order. The order or orders of any state can be determined by writing it in terms of raising and lowering operators and then simply inspecting the number of such operators in each term. A raising operator contributes +1 to the coherence order whereas a lowering operator contributes -1. A z-operator, Iiz, has coherence order 0 as it is invariant to z-rotations. I+1 and I-1 have coherence order +1 and -1, respectively. The overall coherence order of a product of operator can be found by adding together the coherence order of each operator in the product. For a system of N couple spins, the coherence order can take all integer values between -N to +N. Under free evolution, an operator simply acquires a phase factor exp(-iΩ(p1+p2+…)t), where the frequency Ω(p1+p2+…) is determined by the offset and coherence orders of the individual operators in the product Ω(p1+p2+…) = Ωp1+Ωp2+… . A pulse applied to equilibrium magnetization only generates equal amount of p = +1 and p = -1 coherence. An 180Ëš pulse simply reverses the sign of the coherence order. Only coherence order -1 is observable.
In designing a multiple-pulse NMR experiment, one way to specify the orders of coherence present at various points in the sequence is to use a coherence transfer pathway (CTP) diagram along with the timing diagram for the pulse sequence. An example of shown below, which gives the pulse sequence and CTP for the DQF COSY experiment.
In Figure 1 the solid lines under the sequence represent the coherence orders required during each part of the sequence; only the pulses cause a change in the coherence order. In addition, the values of âˆ†p are shown for each pulse. It is important to know that the CTP specified with the pulse sequence is just the desired pathway. We would need to establish separately that the pulse sequence can generate the coherences specified in the CTP and the spin system has the capability to support the coherences. For example, if there are no couplings, then no double quantum will be generated and thus selection of the above pathway will result in a null spectrum. The coherence transfer pathway must start with p = 0 as this is the coherence order to which equilibrium magnetization (z-magnetization) belongs. In addition, the pathway has to end with |p| = 1 as it is the only single quantum coherence that is observable. The usual convention is to assume that p = -1 is the detectable signal.
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Let us go back to the DQF COSY experiment. Note that the symmetrical pathways in t1 have been retained so that absorption mode lineshapes can be obtained. Also, both in generating the double quantum coherence and in reconverting it to observable magnetization, all possible pathways have been retained. If we do not do this, signal intensity is lost.
One way of viewing this sequence is to group the first two pulses together and view them as achieving the transformation 0 → ±2 i.e. âˆ†p = ±2. 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 unambiguously selects p = ±2 just before the last pulse, so phase cycling of the last pulse is not required.
An alternative view is to say that as only p = -1 is observable, selecting the transformation âˆ†p = +1 and -3 on the last pulse will be equivalent to selecting p = ±2 during the period just before the last pulse. Since the first pulse can only generate p = ±1 (present during t1), the selection of âˆ†p = +1 and -3 on the last pulse is sufficient to define the CTP completely.
A four step cycle to select âˆ†p = +1 involves the pulse going 0°, 90°, 180°, 270° and the receiver going 0°, 270°, 180°, 90°. As this cycle has four steps is automatically also selects âˆ†p = -3, just as required. The first of these cycles also selects âˆ†p = ±6 for the first two pulses i.e. filtration through six-quantum coherence; normally, we can safely ignore the possibility of such high-order coherences. The second of the cycles also selects âˆ†p = +5 and âˆ†p = -7 on the last pulse; again, these transfers involve such high orders of multiple quantum that they can be ignored.
Phase cycling has two major practical problems. The first is that a minimum time on the experiment is required to complete the cycle. This minimum time can become very long in two- and higher-dimensional experiments, much longer than would be needed to achieve the desired signal-to-noise ratio. In such cases the only way of reducing the experiment time is to record fewer increments which has the undesirable consequence of reducing the limiting resolution in the indirect dimensions.
The second problem is that phase cycling always relies on recording all possible contributions and then cancelling out the unwanted ones by combining subsequent signals. If the spectrum has high dynamic range, or if spectrometer stability is a problem, this cancellation is less than perfect. The result is unwanted peaks and t1-noise appearing in the spectrum. These problems become acute when dealing with proton detected heteronuclear experiments on natural abundance samples, or in trying to record spectra with intense solvent resonances.
Both of these problems are improved by moving to an alternative method of selection, the use of field gradient pulses. During a pulsed field gradient the applied magnetic field is made spatially inhomogeneous for a short time. As a result, transverse magnetization and other coherences dephase across the sample and are apparently lost. However, this loss can be reversed by the application of a subsequent gradient which undoes the dephasing process and thus restores the magnetization or coherence. The crucial property of the dephasing process is that it proceeds at a different rate for different coherences. For example, double-quantum coherence dephases twice as fast as single-quantum coherence. Thus, by applying gradient pulses of different strengths or durations it is possible to refocus coherences which have, for example, been changed from single- to double-quantum by a radiofrequency pulse.
Gradient pulses are introduced into the pulse sequence in such a way that only the wanted signals are observed in each experiment. In contrast to phase cycling, it does not rely on subtraction of unwanted signals, and is expected that the level of t1-noise will be much reduced. Again in contrast to phase cycling, no repetitions of the experiment are needed, enabling the overall duration of the experiment to be set strictly in accord with the required resolution and signal-to-noise ratio.
A field gradient pulse is a period during which the B0 field is made spatially inhomogeneous; for example an extra coil can be introduced into the sample probe and a current passed through the coil in order to produce a field which varies linearly in the z-direction. We can imagine the sample being divided into thin discs which, as a consequence of the gradient, all experience different magnetic fields and thus have different Larmor frequencies. At the beginning of the gradient pulse the vectors representing transverse magnetization in all these discs are aligned, but after some time each vector has precessed through a different angle because of the variation in Larmor frequency. After sufficient time the vectors are disposed in such a way that the net magnetization of the sample (obtained by adding together all the vectors) is zero. The gradient pulse is said to have dephased the magnetization.
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Some of the coherences can be refocused by a careful choice of the gradient pulses within a pulse sequence. The condition for refocusing is that the net phase acquired by the required pathway is zero, which can be written formally as
With more than two gradients in the sequence, there are many ways in which a given pathway can be selected. For example, the second gradient may be used to refocus the first part of the required pathway, leaving the third and fourth to refocus another part. Alternatively, the pathway may be consistently dephased and the magnetization only refocused by the final gradient, just before acquisition.
Figure 2 is the pulse sequence for recording absorption mode HMQC spectra. The centrally placed I spin 180° pulse results in no net dephasing of the I spin part of the heteronuclear multiple quantum coherence by the two gradients G1 i.e. the dephasing of the I spin coherence caused by the first is undone by the second. However, the S spin coherence experiences a net dephasing due to these two gradients and this coherence is subsequently refocused by G2. Two 180° S spin pulses together with the delays τ1 refocus shift evolution during the two gradients G1. The centrally placed 180° I spin pulse refocuses chemical shift evolution of the I spins during the delays âˆ† and all of the gradient pulses (the last gradient is contained within the final delay, âˆ†). The refocusing condition is
where the + and - signs refer to the P- and N-type spectra respectively. The switch between recording these two types of spectra is made simply by reversing the sense of G2. The P- and N-type spectra are recorded separately and then combined to give a frequency discriminated absorption mode spectrum.
In the case that I and S are proton and carbon-13 respectively, the gradients G1 and G2 are in the ratio 2 : ± 1. Proton magnetization not involved in heteronuclear multiple quantum coherence, i.e. magnetization from protons not coupled to carbon-13, is refocused after the second gradient G1 but is then dephased by the final gradient G2. Provided that the gradient is strong enough these unwanted signals, and the t1-noise associated with them, will be suppressed.
The basic pulse sequence for the HSQC experiment is shown in Figure 3(a). For a coupled two spin system the transfer can be described as proceeding via the spin ordered state 2IzSz which exists at point a in the sequence. In the absence of significant relaxation magnetization from uncoupled I spins is present at this point as Iy. Thus, a field gradient applied at point a will dephase the unwanted magnetization and leave the wanted term unaffected. The main practical difficulty with this approach is that the uncoupled magnetization is only along y at point a provided all of the pulses are perfect; if the pulses are imperfect there will be some z magnetization present which will not be eliminated by the gradient. In the case of observing proton - carbon-13 or proton - nitrogen-15 HSQC spectra from natural abundance samples, the magnetization from uncoupled protons is very much larger than the wanted magnetization, so even very small imperfections in the pulses can give rise to unacceptably large residual signals. However, for globally labeled samples the degree of suppression has been shown to be sufficient, especially if some minimal phase cycling or other procedures are used in addition. Indeed, such an approach has been used successfully as part of a number of three- and four-dimensional experiments applied to globally carbon-13 and nitrogen-15 labeled proteins.
The key to obtaining the best suppression of the uncoupled magnetization is to apply a gradient when transverse magnetization is present on the S spin. An example of the HSQC experiment utilizing such a principle is given in Figure 3(b). Here, G1 dephases the S spin magnetization present at the end of t1, and after transfer to the I spins, refocusing is effected by G2. An extra 180° pulse to S in conjunction with the extra delay τ1 ensures that phase errors which accumulate during G1 are refocused; G2 is contained within an existing spin echo. The refocusing condition is
where the - and + signs refer to the N- and P-type spectra respectively. As before, an absorption mode spectrum is obtained by combining the N- and P-type spectra, which can be selected simply by reversing the sense of G2.
The basic HMQC and HSQC sequences can be extended to give two- and three-dimensional experiments such as HMQC-NOESY and HMQC-TOCSY. The HSQC experiment is often used as a basic element in other two-dimensional experiments. For example, in proteins the proton - nitrogen-15 NOE is usually measured by recording a two-dimensional spectrum using a pulse sequence in which native nitrogen-15 magnetization is transferred to proton for observation. The difference between two such spectra recorded with and without pre-saturation of the entire proton spectrum reveals the NOE. Suppression of the water resonance in the control spectrum causes considerable difficulties, which are conveniently overcome by use of gradient pulses for selection.
Pulsed-field gradients appear to offer a solution to many of the difficulties associated with phase cycling, in particular they promise higher quality spectra and the freedom to choose the experiment time solely on the basis of the required resolution and sensitivity are attractive features. However, these improvements are conditional. When gradient selection is used, attention has to be paid to their effect on sensitivity and lineshapes, and dealing with these issues usually results in a more complex pulse sequence. Indeed it seems that the potential loss in sensitivity when using field gradient pulses is the most serious drawback of such experiments. Nevertheless, in a significant number of cases the potential gains, seen in the broadest sense, seem to outweigh the losses.