Magnetic Susceptibility: Further Insights into Macroscopic and Microscopic Fields and the Sphere of Lorentz

To make certain quantitative interpretations of spectra from NMR experiments carried out on heterogeneous samples, such as cells and tissues, we must be able to estimate the magnetic and electric fields experienced by the resonant nuclei of atoms in the sample. Here, we analyze the relationships between these fields and the fields obtained by solving the Maxwell equations that describe the bulk properties of the materials present. This analysis separates the contribution to these fields of the molecule in which the atom in question is bonded, the "host" fields, from the contribution of all the other molecules in the system, the "external" fields. We discuss the circumstances under which the latter can be found by determining the macroscopic fields in the sample and then removing the averaged contribution of the host molecule. We demonstrate that the results produced by the, so-called, "sphere of Lorentz" construction are of quite general validity in both static and time-varying cases. This analytic construct, however, is not "mystical" and its justification rests not on any sphericity in the system but on the local uniformity and isotropy, i.e., spherical symmetry, of the medium when averaged over random microscopic configurations. This local averaging is precisely that which defines the equations that describe the macroscopic fields. Hence, the external microscopic fields, in a suitably averaged sense, can be estimated from the macroscopic fields. We then discuss the calculation of the external fields and that of the resonant nucleus in NMR experiments