The vector model of NMR

Now consider a collection of similar spin-1 nuclei in the applied static field. As stated, the orientation parallel to the applied field, a, has slightly lower energy than the antiparallel orientation, P, so at equihhrium there will he an excess of nuclei in the a state as defined hy the Boltzmann distrihution  

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Figure 2.4. In the vector model of NMR many like spins are represented by a bulk magnetisation vector. At equilibrium the excess of spins in the a state places this parallel to the +z-axis.

Chapter 3 introduces the vector model of NMR. This model has its limitations, but it is very useful for understanding how pulses excite NMR signals. We can also use the vector model to understand the basic, but very important, NMR experiments such as pulse-acquire, inversion recovery and most importantly the spin echo. 

This section summarizes primarily the classical description of NMR based on the vector model of the Bloch equations. Important concepts like the rotating frame, the effect of rf pulses, and the free precession of transverse magnetization are introduced. More detailed accounts, still on an elementary level, are provided in textbooks [Deri, Farl, Fukl]. 

The classical vector model of NMR and the basic one-pulse Fourier transform experiment... 

Now let s look in detail at each process, so that we can understand the NMR acquisition parameters needed to set up the experiment. After the pulse, we will follow through the hardware devices in a block diagram and try to understand a little about the NMR hardware. It turns out that processing of the NMR signal in the hardware is strictly analogous to the theoretical steps we will use in viewing the NMR experiment with the vector model, so it is essential to understand it in general. 

Before we can understand any experiment more complicated than a simple spectrum, we need to develop some theoretical tools to help us describe a large population of spins and how they respond to RF pulses and delays. The vector model uses a magnetic vector to represent one peak (one NMR line) in the spectrum. The vector model is easy to understand but because it represents a quantum phenomenon in terms of classical physics, it can describe only the simpler NMR experiments. It is important to realize that the vector model is just a convenient way of picturing the NMR phenomenon in our minds and is not really an accurate description of what is going on. As human beings, however, we need a physical picture in our minds and the vector model provides it by analogy to macroscopic objects. 

The vector model is a way of visualizing the NMR phenomenon that includes some of the requirements of quantum mechanics while retaining a simple visual model. We will jump back and forth between a classical spinning top model and a quantum energy diagram with populations (filled and open circles) whenever it is convenient. The vector model explains many simple NMR experiments, but to understand more complex phenomena one must use the product operator (Chapter 7) or density matrix (Chapter 10) formalism. We will see how these more abstract and mathematical models grow naturally from a solid understanding of the vector model. 

It is very difficult to describe coherence transfer using the vector model. To understand it we will need to expand our theoretical picture to include product operators. Product operators are a shorthand notation that describes the spin state of a population of spins by dividing it into symbolic components called operators. You might wonder why you would trade in a nice pictorial system for a bunch of equations and symbols. The best reason I can give is that the vector model is useless for describing most of the interesting NMR experiments, and product operators offer a bridge between the familiar vectors and the more formal and... 

These operators and the rules that govern chemical shift and 7-coupling evolution in time can be used to describe any combination of RF pulses and delays, giving a prediction of the observable magnetization (and thus the spectrum) at the end of the sequence. This gives us the means of understanding all of the ID and 2D NMR experiments. By comparison, the vector model can explain only a few of the ID experiments. 

The INEPT experiment [26] (Insensitive Nuclei Enhanced by Polarisation Transfer) was one of the forerunners of many of the pulse NMR experiments developed over subsequent years and still constitutes a feature of some of the most widely used multidimensional experiments in modem pulse NMR. Its purpose is to enable non-selective polarisation transfer between spins, and its operation may be readily understood with reference to the vector model. Most often it is the proton that is used as the source nucleus and these discussion will relate to XH spin systems throughout, although it should be remembered that any high-y spin- /2 nucleus constitutes a suitable source. 

The BIRD pulse [14] is in fact a cluster of pulses (Fig. 6.12) used as a tool in NMR to differentiate spins that possess a heteronuclear coupling from those that do not. The effect of the pulse can vary depending on the phases of the pulses within the cluster, so we concentrate here on the selective inversion described above. For illustrative purposes, proton pulse phases of x, y, X will be considered as this provides a clearer picture with the vector model, although equivalent results are achieved with phases x, x, —x, as in the original publication. The scheme (Fig. 6.14) begins with a proton excitation pulse followed by a spin-echo. Since carbon-12 bound protons have no one-bond... 

Chapter 6 introduces the product operator formalism for analysing NMR experiments. This approach is quantum mechanical, in contrast to the semi-classical approach taken by the vector model. We will see that the formalism is well adapted to describing pulsed NMR experiments, and that despite its quantum mechanical rigour it retains a relatively intuitive approach. Using product operators we can describe important phenomena such as the evolution of couplings during spin echoes, coherence transfer and the generation of multiple quantum coherences. 

For most kinds of spectroscopy it is sufficient to think about energy levels and selection rules this is not true for NMR. For example, using this energy level approach we cannot even describe how the most basic pulsed NMR experiment works, let alone the large number of subtle two-dimensional experiments which have been developed. To make any progress in understanding NMR experiments we need some more tools, and the first of these we are going to explore is the vector model. 

The vector model, introduced in Chapter 3, is very useful for describing basic NMR experiments but unfortunately is not applicable to coupled spin systems. When it comes to two-dimensional NMR many of the experiments are only of interest in coupled spin systems, so we really must have some way of describing the behaviour of such systems under multiple-pulse experiments. 

The product operator formalism is a complete and rigorous quantum mechanical description of NMR experiments and is well suited to calculating the outcome of modem multiple-pulse experiments. One particularly appealing feature is the fact that the operators have a clear physical meaning and that the effects of pulses and delays can be thought of as geometrical rotations, much in the same way as we did for the vector model in Chapter 3. 

Figure 6 Explanation of a CIDNP multiplet effect with vector models (projections). From left to right, starting state influence of the first nuclear spin (A) only influence of the second nuclear spin population differences resulting spectrum. The labels on the vector models denote the electron spin of radical 1 or 2, the labels in the NMR spectrum the nuclear transitions. Further explanation, see text.

Figure 5. Explanation of an S- T0-type CIDNP net effect with vector models (left), resulting schematic population diagram (center), and NMR spectrum (right). The example describes a radical pair with one proton in radical 1, triplet precursor, product of the singlet exit channel, gq > g2, and positive hyperfine coupling constant. For the vector models, a clockwise sense of precession has been chosen, the labels 1 and 2 designate the radical and a> and /i) the nuclear spin state, and the dotted vertical lines in the projections give the amount of singlet character. For further details, see the text.

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