Without mentioning the details of the experiments, it is to be
appreciated by this contribution that trying to interpret an experimental result set out a quest for retrieving a physical
picture, for a possible visualization, which seem to have got smeared out due to the necessities of the mathematical techniques
required in the specific context of bulksusceptibility demagnetization effects. In the presentation of this paper
the actual derivation of an equation would be projected and the effective simplification which resulted as a consequence of
the availability of such an equation would be elaborated. Ever since the breakthrough because of this derived equation, it
has been possible to find several contexts in Chemical research which benefit by the physical picture retained and retrieved.
The visualization is rendered much simpler now where till now it was not tractable. For the bulk susceptibility and the consequent
demagnetization in materials, an electron charge cloud is to be envisaged that can give rise to an induced dipole moment.
And, this dipole moment gives rise to field distribution around itself and the intensity of the field at a specified point
away from the dipole moment is to be estimated. The point dipole approximation is the way it becomes possible to tract this
influence of circulation of charges with a chargecloud description. There are obvious limitations to the point dipole approximation.
For the relative dispositions of the location of electron charge cloud and the location where its effect is to be interpreted,
the values become more often prohibitive in view of the validity criterion for the point dipole approximation. The well known
tables of demagnetization factor values available in literature (which required the setting up of elliptic integrals earlier)
can all be reproduced with at least comparable accuracy by the summation of contributions from closely packed spherical fragments
using the equation to be reported herein. All these could be accomplished only over a considerable length of time (Over 25
years) which in certain contexts finds a possibility to be a historically noteworthy biographical sketch of the experiences
of an author. As one of the presentations of this author was titled, the questions arising in this context have been enduring
questions^{1} ultimately aimed at trying to circumvent a practical constraint which seems to reduce the utility of an advanced
experimental technique. The referred technique is the multiple pulse linenarrowing experiments for the High Resolution Proton
NMR in single crystalline specimen of organic molecules. This technique achieves a selective averaging of the anisotropic
dipolar interaction in proton spin systems while retaining a scaled down chemical shift tensor information, which is also
an anisotropic interaction. For reliability criterion it becomes necessary to use spherical specimen and making spheres of
the organic molecular single crystals cannot be ensured as matter of routine. Thus by inquiring into what are the induced
field variations at nuclear (in this instance protons) sites within a molecule due to the characteristics of molecular electronic
structures (the nuclear shielding or chemical shifts) it has become possible to comment upon discreteness and continuum of
the material medium. This led to the setting up of criteria for evaluating the demagnetization factors by a summation procedure
which could provide conceptualizing constraints and a method of deriving equation to replace effectively the earlier evaluations
with a calculation which all through retains the perspectives of the original principles to be visualized at any stage of
the calculation procedure. The formalisms which were required to be set up earlier had to use mathematical physics to such
an extent that the visualization is obliterated and hence appealed less for the intuitive^{ 3} approaches. This abstract traces
the sequences in this context beginning with an introduction to the context in physics. Then a description of the criteria
set out for the derivation of an equation is given followed by the few steps derivation of the equation. As a conclusion the
effective simplification thus achieved are enumerated with indications of how the conceptual chemical foundations are strengthened.
A diamagnetic material, which is nonmagnetic, does acquire a magnetization when placed in a strong externally magnetic
field. This is explained on the basis of the circulating electrons in the constituent diamagnetic molecular species
making up the material which gives rise to the characteristic magnetic susceptibility ( a property which indicates as to what
extent the molecular electronic structure lets the electron motions to be affected by external magnetic fields present) because
of which the external magnetic field gives rise to induced magnetic moments within the molecules and thus the material medium.
Thus from the molecular specific mechanisms it becomes necessary to estimate the consequent results in the bulk of this material.
At any instant when the measured bulk magnetization is considered while inquiring into a specific point (a site which is within
the material, and hence, locatable as more closely related to a given molecule of the so many molecules within the medium)
value for the effective field value, the magnetic moment at any specific molecule has to be taken into account and the contribution
from the magnetic moments from near and far neighboring molecules have also to be taken into account). Qualitatively it seems
it is necessary inevitably to molecules as near and far neighbors to a specific site rather than categorically stating it
as contribution from all the other molecules in the medium with reference to the given molecule to which the site is locatable
as closest or the molecule containing the site within itself. Physicists use the term discrete region for the nearby
molecular regime, and the term continuum for the farther molecular regime. The characteristics of the contributions
from the continuum are not as simply inferred from any relatable individual molecular characteristics; not even as much as
what is possible for the contributions from discrete region and this is the necessity to demarcate the two regimes as pointed
out above. The quantitative measure for concrete demarcation of a boundary line of the two regions within the material has
been more or less vague. And, the real context where the discrete region can be significantly effecting the experimental measurements
was also not much evidenced except probably in hypothetical terms ^{2}
. For all contexts of the continuum bulk contributions at a point
within the medium, what matters seemingly is the geometrical shape that the material specimen possesses. Under certain suitable
hypothetical divisions of regions within the material for a spherical shape of the material specimen, the bulk contribution
totally vanishes irrespective of the molecular characteristics of the constituents of the material and this is true for any
point site within the material be at the centre or near the peripheral regions of the specimen. A similar characteristic is
attributable to ellipsoidal (ellipsoids of revolution) shapes for the specimen. More generally spherical shape is a special
case of the ellipsoidal criteria and this specialty is borne out conspicuously by the fact that the induced field contribution
from the bulk continuum is zero for spherical shape. For other ellipsoidal shape factors, the bulk continuum contribution
can be calculable as a fixed value over the entire specimen and this value can be related to the geometrical shape factor
characteristic of the ellipsoid. All this is evident from the statement that for ellipsoidal shapes the magnetization of the
specimen is homogeneous and for the special case of sphere the bulk susceptibility effect is absent uniformly through out
the sample. All this is valid as well for magnetic materials with strong, permanent internal fields. Clarifications
^{6} are now forthcoming on 1) the vagueness of demarcation of regions as stated above, 2) the elusive feature
of predominant dependence of the bulk effects on the geometrical factor for ellipsoids, 3) more specifically, the situation
of ‘no bulk effects’ as exclusively determined by the shape of specimen for spheres . For this improvement
on visualization, what was necessary was a mathematical formula for close packing of spheres with certain constraints that
would improve the applicability of the pointdipole approximation ^{7} for the calculation of induced field contributions;
and, to have the equations (for dipolar induced field) to be valid inherently even at distance ranges defined till now as
prohibitive for the validity. The constraints mentioned
above are required to stipulate that inherently the relevant relative distance ratios are in the valid range, while the distance
values are in the range of molecular dimensions or in the range of macroscopic sizes of specimen.
When the contribution of near neighbors are calculated, a discrete summation from individual neighbors becomes possible
for which the distance of the neighbor and the magnetic moment (related to the magnetic susceptibility at the neighbor, which
means, the susceptibility of the neighboring molecule) at that neighboring point are to be used. The “distance”
(R) of the molecule from the point where the contribution is calculated, should be large (of the order of 10 times larger)
compared to the molecular size (radius: r). This summation of the nearby molecules, in a single crystal lattice
of organic molecule can be calculated using a computer program which can be executed to completion within a reasonable computational
time^{ 4}. This is so because this near neighbor calculation need not be made beyond a distance of about 100 A° radius^{ 2} from the
point site. Extending the summation to all molecules beyond this radius does not add any more significantly to the total.
This may be thought of sort of convergence to a value by that radial distance when all the molecules within such a radial
sphere (a semimicro inner volume element IVE around the point site) has been included. But, this discrete sum so obtained
would not be zero in all cases and in general. It was also found that if the shape of IVE had been a general ellipsoidal shape,
not necessarily the special case of sphere, even then this convergent value is the same for the IVE near neighbor contribution
and the trend indicates this IVE can be of any arbitrary shape but the discrete summation effectively would be resulting in
the same convergent value as that of a sphere. Which means that the near neighbor summation results are independent of the
shape of the IVE described a boundary line but it should be sufficiently large to result in a convergent summed value for
the contributions from molecules within that IVE. But the bulk susceptibility effect which is effectively a summation of the
continuum region (similar to the discrete region) is well established to be shape dependent. The point now to be reconciled
is: the summation within IVE results in values which establish shape independence and beyond IVE the contribution from the
summation is not significant. On the other hand the bulk susceptibility shape dependent effects are experimentally established
and for this reason shapedependent demagnetization factors have been calculated and tabulated for convenience of applying
corrections for bulk susceptibility effects where ever appropriate and necessary. It is also to be realized that extending
the summation program adapting it for the summation in the continuum region seemed practically not viable for the reasons
of computational times required. Hence, for the continuum region still the calculation procedures have to be only the same
as the very early efforts by which the tabulation of demagnetization factors was achieved. To arrive at these tabulated values
the formalisms (based on the same dipole field criterion as for discrete summation) become much more mathematically involved
and diverting the attention to the complicated evaluation procedure rendering the physical descriptions elusive and visualizations
difficult. Because experimentally these tabulated values were found necessary and adequate, by practice it is known that bulk
susceptibility effects are shape dependent. Thus the result of summation within IVE with a clear visualization of the physical
reality stands in apparent contradiction to the inevitable shape dependences of demagnetization factors obtained as a result
of accomplishing a difficult task mathematically (setting up and evaluating elliptic integrals), which in practice is necessary
in experimental situations. Unless this traditional procedure of evaluating elliptic integrals is replaced by an equivalent
summation retaining the simple physical picture of a point dipole through out the evaluation, this conflict might not find
a resolvable solution. Then a via media could be stated as follows: since the inordinate increase in the computational time
is because of the increase in the number of contributing molecules with the increase in radius thus necessitating increased
number of summations, it would be possible to realize a summation only if there is only as many number of summations required
for the continuum as there are in the discrete region which is the above defined IVE. This would imply, the bulk of the sample
in the entire continuum region must be appropriately fragmented to result in much less number of fragmentedentities than
the actual number of total molecules in the continuum region. Then each fragment can be assigned a appropriate susceptibility
value similar to the way molecules have their respective characteristic susceptibility values to enable an induced magnetic
dipole moment to be placed at the centre of each of the fragmented entity. This would result in the ‘limited number
of summations’ required for implementing within short times. Consequently, the fragmenting should eventually account
for all the substance of the medium without leaving out any part of the material to be sure it is the contribution from the
entire continuum by the same principle, mechanism and equations for calculation. Thus close packing of spheres was a natural
fragmenting option. If a sphere of certain radius ‘r’ of the material is considered then from the value
of the Volume Susceptibility (susceptibility per unit volume) of the material and multiplying by the volume of the sphere
a magnetic susceptibility for that spherical fragment can be attributed to get a magnetic dipole moment in presence of external
magnetic field to be placed at the centre of such a sphere. Here the important relevant consequence is that the volume is
dependent on the cube of radius of the sphere ‘r’ (by the formula V_{sph}= [4/3]•π•r^{3}).
Thus the magnetic dipole moment for the spherical fragment should be proportional to r^{3}. For
the dipole model to be valid, the distance R of this spherical fragment should be large compared to r. If all
the fragmented spheres have the same radius r, then all the spheres have to be at a distance much larger than about
10 times r. Such a situation would again is not resulting any viable simplicity since the spheres have to closely packed
and the near and far spheres all would have same radius. Then a situation would arise that as in the discrete summation, beyond
a certain radius the fragmented spheres may not contribute significantly. This would result in possibly the same conflict
instead of resolving the conflict. It is to be noted that the dipole moment induced field at any point varies as reciprocal
of R^{ 3}. That is R^{ 3}. It is further to be noted that the magnetic moment varies directly as r^{3 }(the induced field would
increase, with the strength of magnetic moment) but the induced field varies as reciprocal of R ^{3} . Instead of fragmenting equal size spheres for the close packing, it is to be looked for
the possibility that as the distance R of the sphere increases the fragmented sphere also should increase in radius
r proportionately so that the ratio r/R is held constant while fragmenting. Thus it may be possible to close pack along
a radial vector line with fragmented spheres whose radii r is proportional to their locationdistance R from
the point site. Then each of the sphere must be contributing a same constant value to the induced field at the point site
even though the spheres are of different sizes but are placed at proportionately different distances. This type of fragmentation
would not result in the same kind of convergent situation while summing since at any value for the distance R the radius
of the sphere is correspondingly maintained at a proportionate value to yield the same contribution at a specific point site.
Then is it possible to find how many such sphere would be needed for packing along a line and is it a closely packing number?
It was in fact a surprise that a formula resulted for realizing such a close packing. Once the criteria for the constraints
were set thus, it did not require much effort to derive an equation to get this number of close packed spheres along a given
radial vector length.
In the actual presentation, the derivation and the derived equations ^{5} would be projected with the brief mention
of the actual technicalities of the specialized subject of investigation, which in this context could be revealing as to how
while solving a set out problem at hand has a concomitant lesson on the methodologies of research and how learning process
evolves at each step of approach to a solution.
REFERENCES:
 https://nehuacin.tripod.com/id3.html  a web page documentation of the presentation at the 4^{th} Alpine Conference on Solid State NMR: New Concepts
and Applications , September 2005, Chamonix MontBlanc, France

"Magnetized Materials: Contributions Inside Lorentz Ellipsoids", S.Aravamudhan, Indian J.Phys.,
Vol.79 (9), p985989 (2005) This is Condensed Matter Days2004 proceedings’ special issue;
"Bulk solid specimen shape dependences in the molecular,
chemicalshift
tensor determinations”
S.Aravamudhan, Proceedings of the International Conference on Molecule to Materials ICMM2006,
SLIET, Longowal, Sangrur, Punjab, India; March 36, 2006. The TEXT
of this Full Paper of this presentation at
ICMM2006 has been posted by this
author at the Webpage URL: “Jump to EVENTIII” at
http://www.geocities.com/inboxnehu_sa/conference_events_2006.html
 http://www.geocities.com/saravamudhan1944/crsi_6nsc_iitk.html

There has been a report in
literature of an effort to evolve a “rapid computational technique” for estimating geomagnetic anomalies”
under the subject matter of GEOMAGNETISM. See the Poster Sheets (in particular SHEET_1_5) at http://geocities.com/inboxnehu_sa/Poster_Sheets_Ampere.html "Rapid Computation of Magnetic Anomalies and Demagnetization
Effects Caused by Bodies of Arbitrary Shape", P.Vallabh Sharma Pure and Applied Geophysics Vol.64, page 89 (1966). On page 102 sec 4.5 of this paper the
Effect of Susceptibility and on Inhomogeneous
Magnetization” is discussed and on page 103 Sec.5 “Average
Demagnetization Factor for Rock Specimens is discussed
with the elaboration on the “CONCEPT OF
AVERAGE DEMAGNETIZATION FACTOR”.

http://geocities.com/inboxnehu_sa/conference_events_2005.html ‘EventVII’ link for full paper for the presentation at RCMA2005, Department
of Physics, University of Manipur

(a)
Localfield Effects and Effectivemedium theory:
A Microscopic perspective D.E.Aspnes, American Journal of
Physics, 1982, 50(8), pages 704 to 708 (b)
Local Fields in Solids: microscopic aspects of
dielectrics S.E.Schnatterly and C.Tario, Reviews of
Modern Physics, 1992, 64, pages 619622. http://saravamudhan.tripod.com/id1.html (c) Permanent and Induced
Molecular Moments and Long Range Intermolecular Forces: by AD Buckingham in advances in Chemical physics, vol XII intermolecular
forces edited by Joseph O.Hierscfelder, Interscience publishers 1967, page 549.

Some
of the explanations of basics of magnetic resonance phenomenon has been included as Links at the Webpage URL:
http://www.geocities.com/saravamudhan1944/inno_course_contents.html
8. "Bulk
Susceptibility Effects and NMR Chemical Shifts:The Perspective for
Molecules to Materials" Abstract NMRS2007