ICMM2006

The Sections 2 & 3 of the reference 2 contain the relevant discussion to explain well the preliminaries required to proceed towards the induced fields at points within in bulk of the material medium

4. THE ELLIPSOIDAL SEMI MICRO VOLUME ELEMENTS INSTEAD OF SPHERES FOR THE DISCRETE SUMMATION

The sections 4 & 5 of reference 2 contain the discussions for this aspect. It has been argued out that as much as there is reason for induced field to be zero within the magnetized spherical specimen, there is equally a valid argument as to why the induced fields within an ellipsoidal specimen also can be zero. However, this point of view has till now not been brought forward and it is in the context of intermolecular contributions in HR PMR in solids the ground could be laid for these reasons to be brought to stay. To gain better insight into the arguments and reasons for prevailing practices it became necessary to investigate the trends and final limiting values for convergence of the sum of contributions within the semi micro ellipsoidal volume elements. Such calculations indicate that even if during the summation the summed values are different for spheres and ellipsoids after a critical value for the spherical radius and the major axis length value for the ellipsoids the limiting values are the same. Which means for the sphere the value being zero for cubic lattice for the ellipsoidal element also the sum goes to zero after a certain size of the ellipsoid and for several values of ellipticities indicating that ellipsoidal elements for all constants and types of lattices, can be replaced by a an effectively spherical volume element. That shape dependent factor which is supposed to be the value of such a summation over the entire set of lattice points in the macroscopic specimen seems inexplicable from this result that the discrete summation results in the same sum total value for induced field contribution within sphere and ellipsoids of any shape factor (ellipticity). Thus why does the demagnetization factor different? Only because outer shape and inner shapes are different? Even this can be argued that the inner contribution not being dependent upon shape can be replaced with any shape that the macroscopic specimen has and the dimensions being of semi micro ranges.
In such an event a certain trends indicated that the contributions from inside the semi micro volume element seem to dominate in value over the entire contribution from the remaining bulk of the specimen by several orders of magnitude that, the importance of macroscopic shape is relative and the demagnetization factor is only a multiplicative factor for the shape and when the multiplied value is less in magnitude than the semi micro volume element contribution, the dependences on shape factors may not be important for practical purposes for the induced field values at any particular site within the macroscopic specimen. Since this is all the arguments for diamagnetic specimen, similar calculations for paramagnetic and other magnetic materials seem warranted and in the presentation this point of view would be emphasized and highlighted. Thus from the graphical plot (in Fig.6) it seems it is obvious that the sum of contributions of induced field at the centre of a sphere is zero for typical variety of cubic lattice constants. It is possible to compare the situation for inner sphere and the inner ellipsoid the remaining factors remaining the same. This is shown in Fig.7. The graph on top of Fig.7is the same as in Fig.6.The lower graphical plot is for the same cubic lattice parameters but the sphere radius is scaled with required ellipticity and each color represents an ellipticity.

This is the text of the Full Paper for the presentation made at the Sant Longowal Institute of Technology, Longowal, Sangrur, Punjab, INDIA on the occasion of the "International Conference on Molecules to Materials" - The ICMM2006 held during March 3-6, 2006.