Maxwell coils

Fig. 2.1.3 A pair of saddle coils creates an x or y gradient (left), a pair of Maxwell coils creates a z gradient (right). The arrows on the coils indicate the current direction, the circles with...

In related techniques, the homogeneous magnetic field of standard NMR magnets is degraded, for instance, by a Maxwell coil pair [Tanl], or pulsed Bq field gradients are... 

FIGURE JA2 A filamentary, single-turn Maxwell coil pair z gradient c< and a typical unshielded z gradient coil. 

Equation (24.9) predicts that, for Maxwell coils, the gradient deviates 10 percent from the ideal linearity at z/a = 0.66. For n = 0.3 m, this deviation takes place at z = 0.2 m (e.g., the field of view would be 40 cm if 10 percent deviation from linearity was the maximum desired). 

Maxwell coils, quadratic shape coils, etc. However, Helmholtz coils, given in Figure 4.49, proved very suitable for the generation of the homc eneous magnetic field. 

The velocity gradient leads to an altered distribution of configuration. This distortion is in opposition to the thermal motions of the segments, which cause the configuration of the coil to drift towards the most probable distribution, i.e. the equilibrium s configurational distribution. Rouse derivations confirm that the motions of the macromolecule can be divided into (N-l) different modes, each associated with a characteristic relaxation time, iR p. In this case, a generalised Maxwell model is obtained with a discrete relaxation time distribution. 

The following observations seem to lead to a key for the understanding of the quasistatic treatment of the stress-optical behaviour (777). With the aid of third eq. (5.4) and eq. (5.8), the ratio of Maxwell constant and intrinsic viscosity, as valid for Gaussian coil molecules in a matching solvent, can be formulated as follows ... 

Thus, the observations made at the beginning of this section [see eqs. (5.10) and (5.11a)] with respect to coil molecules and rigid rods, are confirmed for the behaviour of the dumb-bell models. In particular, a comparison of eqs. (5.18) and (5.22) shows that, other than for the intrinsic viscosity, the Maxwell constant can only be calculated when, besides (hi) also (hi) is known. This remark will be of importance for the next section, where theories for short chain molecules will be discussed. 

In the center of a Maxwell pair the radial component of the magnetic field is half the longitudinal component. With the main field B pointing parallel to the coil axis, the total magnetic field is... 

Fig. 16. The spin-echo signal as a function of the magnetic field gradient for isotropic self-diffusion in a Maxwell pair coil system. Note the deviation from ideality as the polarizing field, B , decreases.

The use of the fringe field of superconducting magnets for the measurement of very slow diffusion processes is now well established [Kimmich and Fisher, 1994). In considering the violation of the small inhomogeneous field approximation, we note that the fringe field may not be as well defined as that produced by quadrupolar coils or Maxwell pair coils. In order to provide a simple model, we will consider the magnetic field distribution created by a simple coil of radius... 

In order to investigate the effect caused by finite dimensions of a transmitter coil let us first derive formulae for the vector potential of the electrical type corresponding to a current element. As is well known, complex amplitudes of the field are described by Maxwell s equations ... 

So far we have only considered the movement in the z-direction. The anti-Helmholtz coils produce a magnetic quadrupole field with three components. From Maxwell s equation divB = 0 and the condition dB /dx = dBy/dy, which follows from the rotational symmetry of the arrangement, we obtain the relations... 

Let / , be the resistance per unit length of the coil. The total resistance of a Maxwell pair then is 4iraRi. The power P dissipated in the gradient coil depends on the product of the total resistance and the square of the current ... 

Axial gradient coil sets of commercial MR scanners typically use more than the two coils that make up a Maxwell pair. Normally, gradient field series expansions are made and coil location, radius, and current are selected to obtain high linearity or low inductance. One such axial coil ... 

As a consequence of the Maxwell equations, linear field gradients are accompanied by additional spatially dependent field components. Then, an asymmetric coil generates concomitant field terms of zeroth and first order in space. The formalism was used experimentally to compensate for artifacts observed in three different imaging methods an image shift in standard echo planar imaging (EPI), an echo shift in diffusion-weighted EPI, and a phase shift in a flow quantification technique based on phase contrast images. 

For polyacrylamide there are two rheological effects which can be explained in terms of its random coil structure. Firstly, it was discussed above that polyacrylamide is much more sensitive than xanthan to solution salinity and hardness. This is explained by the fact that the salinity causes the molecular chain to collapse, which results in a much smaller molecule and hence in a lower viscosity solution. The second effect which can be explained in terms of the polyacrylamide random coil structure is the viscoelastic behaviour of this polymer. This is shown both in the dynamic oscillatory measurements and in the flow through the stepped capillaries (Chauveteau, 1981). When simple models of random chains are constructed, such as the Rouse model (Rouse, 1953 Bird et al, 1987), the internal structure of these bead and spring models gives rise to a spectrum of relaxation times, Analysis of this situation shows that these relaxation times define response times for the molecule, as indicated in the simple Maxwell model for a viscoelastic fluid discussed above. Thus, because of the internal structure of a flexible coil molecule, one would expect to observe some viscoelastic behaviour. This phenomenon is discussed in much more detail by Bird et al (1987b), in which a range of possible molecular models are discussed and the significance of these to the constitutive relationship between stress and deformation rate and deformation history is elaborated. 

SO that for Marshall and Mentzner s definition a" = 0.5. Marshall and Men tzner (1964), using the contra variant convected Maxwell model to obtain a relaxation time for their polymer solutions, noticed the onset of viscoelastic behaviour at a Deborah number between 0.1 and 1.0 for flow through sintered bronze. Some of their results for various flexible coil polymers are shown in Figure 6.10. 

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