The Vector Model

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. 

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The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

After this short intermezzo, we turn back to introduce the last class of lattice models for amphiphiles, the vector models. Like the three-component model, they are based on the three state Ising model for ternary fluids however, they extend it in such a way that they account for the orientations of the amphiphiles explicitly amphiphiles (sites with 5 = 0) are given an additional degree of freedom a vector with length unity, which is sometimes constrained to point in one of the nearest neighbor directions, and sometimes completely free. It is set to zero on sites which are not occupied by amphiphiles. A possible interaction term which accounts for the peculiarity of the amphiphiles reads... 

JSji and j2 were vectors and j their resultant, the triangle whose sides are ju j2, and j3 would not dose unless j were limited by Eq. (7-43) hence, the name triangle condition, and hence, also, the analogy between quantum mechanics and the vector model. 

FIGURE 6.1 The vector model used to calculate dipole moments from bond moments. 

If polarized light passes through a medium that exhibits optical rotation, the motion along one of the circular vectors is slower than that of the other. The resultant vector is thus displaced from the original vector by some angle, . Figure 16.5 shows the vector model in which the phase difference is [Pg.588]

Holme, T. A., and Levine, R. D. (1988), An Algebraic Hamiltonian for Electronic Nuclear Degrees of Freedom Based on the Vector Model, Inti. J. Quant. Chem. 34, 457. 

Figure 3.5 Characterization of different polarizations of an axially symmetric system JMj for the special example J = 5/2. According to the vector model the projections M, = 5/2, 3/2, 1/2, —1/2, —3/2 and —5/2 are shown as tilted arrows precessing around the z-axis (in order to indicate this precession, the arrows have been drawn to both the left and right). The length of the arrows is used as a measure for the number of particles in the corresponding magnetic state, thus giving information about polarization. The polarizations shown are (a) an aligned, (b) an isotropic, and (c) an oriented system.

Clebsch-Gordan coefficients are chosen to be real and orthonormal.) The triangle condition from the vector model requires for the angular momenta... 

Starting from the vector model of a precessing spin, the polarization properties of an electron beam with electrons of spin (l/2)h are explained and then formulated... 

These quantum mechanical results are identical to the ones derived before based on the vector model (equs. (9.12)), which were used to interpret Fig. 9.2. 

The vector model cannot be interpreted in such a simple way in the case of a spin system with more than one nucleus. For weakly coupled spin systems, the single spin vector model may be applied for each nucleus, one after the other. Thus the coupling with the other nuclei can be incorporated into its precession frequency, since the definition of the weak coupling (J -C vM v,/1) means that the transitions of a nucleus only depend on the spin states of the other nuclei in the first order. The detected signal is the sum of the sine curves provided by the individual environment of the nuclei. 

The vector model cannot be applied to strongly coupled spin systems, since the precession of the individual nuclei cannot be separated from each other. 

There is a simple and well-known model for the description of DNMR phenomenon, provided no scalar coupling is involved.11 In Section 3.2.1, the macroscopic magnetisation vector of the vector model gives the detected signal (the FID). This vector rotates in the xy plane perpendicular to the direction of the external magnetic field during the detection its frequency is determined by the shielding effect of the chemical environment of the nuclei. 

The vector model of a single spin is the vector representation of the complex number in the individual density matrix of a single nucleus. This density matrix consists of only one complex number thus there is only one vector in the model. In the case of more than one nuclei, the density matrix is larger, there are more single quantum coherences and more vectors belong to one spin set in the model. Moreover, in case of a strongly coupled spin system, the density matrix has different numerical form for different basis sets of the vector space of the simulation (the basis can be one of the ) and

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The aim of this extension of the vector model is to show the connection between the two types of coherences present during precession and exchange processes based on the mathematical formulae in Section 3.5. 

According to our interpretation (see Section 3.3.1), there are two con-formers in this system A(l> was denoted as AB and A(-1 > was CD. The statistical (KMC) description of the exchange process does not influence the vector model. The only important thing is that there are some exchange points during detection. 

After that conversion precessions restart with the frequencies of the new conformer (E).This interpretation of the vector model shown in Figure 14 describes the evaluation of the elements of the density matrix only but not the detected signal. The FID is calculated from the actual elements of the eigenfunction representation of the density matrix (that is the (t) vector) as ... 

The fundamental problem in the case of a rotating molecule (as a rule, for simplicity s sake, we will consider a diatomic or a linear polyatomic one) is that of the interaction between the electronic motion and rotation of the nuclei. For better clarity and in conformity with the style of our further presentation, we will apply the vector model approach. 

Thus, accounting for non-zero value of A in the vector model of a molecule (Fig. 1.7) also leads to the appearance of a component d° of the dipole moment for a parallel transition. However, with increasing J, owing to the decrease in sin 6, the value of this component falls rapidly, which agrees with well-known experimental facts [194]. 

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. 

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... 

Using the vector model, when we want to describe the spin state of a particular nucleus, we can draw a vector in three-dimensional space, or we can describe the projection of that vector onto the three axes (the components of the vector). For example, a vector of length M0 on the — x axis could be described as... 

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