The Outer Continuum in the Magnetized Material Specified Proton Site Lorentz Sphere The Outer Continuum in the Magnetized Material Lorentz Sphere of Lorentz Cavity Outer surface D out Inner Cavity surface D in D out = - D in Hence D out + D in =0 The various demarcations in an Organic Molecular Single Crystalline Spherical specimen required to Calculate the Contributions to the induced Fields at the specified site. D out/in values stand for the corresponding Demagnetization Factors 1. Contributions to Induced Fields at a POINT within the Magnetized Material.

A spherical sample is to have a homogeneously zero induced field within the specimen. Because of the convenience with which the summation procedure can be applied to find the value of induced field not only at the center but also at any point within the specimen, it has been possible to calculate the trend for the variation of induced field from the centre to the near-surface points. There is parabolic trend observable and this seems to be the possible trend in most of the cases of inhomogeneous field distributions as well except for the values of the parabola describing this trend. It seems possible to derive parabolic parameter values depending on the shape factors in the case of inhomogeneous field distributions in the non-ellipsoidal shapes. This would greatly reduce the necessity to do the summing over all the θ and φ values. This possiblility has been illustrate (in anticipation of the verification of the trends) in the remaining presentation in particular the last four slides.

The top or a Spindle Shaped object comes under the category of Shapes within which the Induced field distribution would be Inhomogeneous even if the Susceptibility is uniformly the same over the entire sample NMR Line for only Intra molecular Shielding Added intermolecular Contribultions causes a shift downfield or upfield homogeneous Inhomogeneous Magnetization can Cause Line shape alterations 4. The case of Shape dependence for homogeneously magnetized sample, and consideration of in-homogeneously magnetized material.

a b Outer a/b=1 outer a/b=0.25 Demagf=0.33 Demagf=0.708 inner a/b=1 Demagf=-0.33 Demagf= = =0.378 conventional combinations of shapes Fig.5[a] Conventional cases Current propositions of combinations Outer a/b=1 outer a/b=0.25 Demagf=0.33 Demagf=0.708 inner a/b=0.25 Demagf= = =0 Fig.5[b] The summed up contributions from within Lorentz sphere as a function of the radius of the sphere. The sum reaches a Limiting Value at around 50Aº. These are values reported in a M.Sc., Project (1990) submitted to N.E.H.University. T.C. stands for (shielding) Tensor Component 4. The case of Shape dependence for homogeneously magnetized sample, and consideration of in-homogeneously magnetized material.

YES a b Outer a/b=1 outer a/b=0.25 Demagf=0.33 Demagf=0.708 inner a/b=1 Demagf=-0.33 Demagf= = =0.378 conventional combinations of shapes Fig.5[a] Conventional cases Current propositions of combinations Outer a/b=1 outer a/b=0.25 Demagf=0.33 Demagf=0.708 inner a/b=0.25 Demagf= = =0 Fig.5[b] Would it be possible to Calculate such trends for summing within Lorentz Ellipsoids ? 3 rd Alpine Conference on SSNMR (Chamonix) poster contents Sept Till now the convergence characteristics were reported for Lorentz Spheres, that is the inner semi micro volume element was always spherical, within which the discrete summations were calculated. Even if the outer macro shape of the specimen were non-spherical (ellipsoidal) it has been conventional only to consider inner Lorentz sphere while calculating shape dependent demagnetization factors.

7/23/2015 1:55:50 AMMRSFall 2006/This slide 02m:01s for prev. 4 slides=05m:27s 7 RECAPITULATION on TOPICS in SOLID STATE Defining what is Conventionally known as Lorentz Sphere It becomes necessary to define an Inner Volume Element [I.V.E] in most of the contexts to distinguish the nearest neighbours (Discrete Region) of a specified site in solids, from the farther elements which can be clubbed in to be a continuum. The shape of the I.V.E. had always been preferentially (Lorentz) sphere. But, in the contexts to be addressed hence forth the I.V.E. need not be invariably a sphere. Even ellipsoidal I.V.E. or any general shape has to be considered and for the sake of continuity of terms used it may be referred to as Lorentz Ellipsoids / Lorentz Volume Elements. It has to be preferred to refer to hence forth as I.V.E. ( Volume element inside the solid material : small compared to macroscopic sizes and large enough compared to molecular sizes and intermolecular distances). Lorentz Spheres Spherical For outer shapes ellipsoidal cubical arbitrary Conventional Currently: The Discrete Region I.V.E. Shapes other than spherical I.V.E. need not be invariably a sphere general shape OR For any given shape

3 rd Alpine Conference On SSNMR : results from Poster

σ exp – σ inter = σ intra Discrete Summation Converges in Lorentz Sphere to σ inter Bulk Susceptibility Contribution = 0 Bulk Susceptibility Contribution = 0 Similar to the spherical case. And, for the inner ellipsoid convergent σ inter is the same as above σ exp (ellipsoid) should be = σ exp (sphere) HR PMR Results independent of shape for the above two shapes !!

The questions which arise at this stage 1. How and Why the inner ellipsoidal element has the same convergent value as for a spherical inner element? 2. If the result is the same for a ellipsoidal sample and a spherical sample, can this lead to the further possibility for any other regular macroscopic shape, the HR PMR results can become shape independent ? This requires the considerations on: The Criteria for Uniform Magnetization depending on the shape regularities. If the resulting magnetization is Inhomogeneous, how to set a criterian for zero induced field at a point within on the basis of the Outer specimen shape and the comparative inner cavity shape?

In fact, the effort towards this step wise inquiry began with the realization of the simple summation procedure for calculating demagnetization factor values. Thus if one has to proceed further to inquire into the field distributions inside regular shapes for which the magnetization is not homogeneous, then there must be simpler procedure for calculating induced fields within the specimen, at any given point within the specimen since the field varies from point to point, there would be no possibility to calculate at one representative point and use this value for all the points in the sample. A rapid and simple calculation procedure could be evolved and as a testing ground, it was found to reproduce the demagnetization factor values with good accuracy which compared well with the tabulated values available in the literature. Results presented at the 2 nd Alpine Conference on SSNMR, Sept. 2001

Using the Summation Procedure induced fields within specimen of TOP (Spindle) shape and Cylindrical shape could be calculated at various points and the trends of the inhomogeneous distribution of induced fields could be ascertained. Zero ind. Field Points Poster Contribution at the 17thEENC/32ndAmpere, Lille, France, Sept Graphical plot of the Results of Such Calculation would be on display

1.Reason for the conevergence value of the Lorentz sphere and ellipsoids being the same. 2.Calculation of induced fields within magnetized specimen of regular shapes. (includes other-than sphere and ellipsoid cases as well) 3. Induced field calculations indicate that the point within the specimen should be specified with relative coordinate values. The independent of the actual macroscopic measurements, the specified point has the same induced field value provided for that shape the point is located relative to the standardized dimension of the specimen. Which means it is only the ratios are important and not the actual magnitudes of distances. These two points would have the same induced field values (both midpoints) These two points would have the same induced field values (both at ¼) Lorentz cavity The two coinciding points of macroscopic specimen and the cavity are in the respective same relative coordinates. Hence the net induced field at this point can be zero Further illustrations in next slide Added Results to be discussed at 4 th Alpine Conference

Symbols for Located points Inside the cavity Points in the macroscopic specimen Relative coordinate of the cavity point and the Bulk specimen point are the same. Hence net induced field can be zero In the cavity the cavity point is relatively at the midpoint of cavity. The point in bulk specimen is relatively at the relative ¼ length. Hence the induced field contributions cannot be equal and of opposite sign Applying the criterion of equal magnitude demagnetization factor and opposite sign specimen length cavity length This type of situation as depicted in these figures for the location of site within the cavity at an off-symmetry position, raises certain questions for the discrete summation and the sum values. This is considered in the next slide

In all the above inner cavities, the field was calculated at a point which is centrally placed in the inner cavity. Hence the discrete summation could be carried out about this point of symmetry. For a spherical and ellipsoidal inner cavity, the induced field calculations were carried out at a point which is a center of the cavity. If the point is not the point of symmetry, then around this off-symmetry position the discrete summation has to be calculated. The consequence of such discrete summation may not be the same as what was reported in 3 rd Alpine conference for ellipsoidal cavities, but centrally placed points. This is the aspect which will have to be investigated from this juncture onwards after the presentation at the 4 th Alpine Conference. The case of anisotropic bulk susceptibility can be figured out without doing much calculations further afterwards.