Conference On Molecules to Materials: ICMM2006

Home | Sections 2 & 3 | Section 3 cntd & Section 4 | Section 4 cntd Figs6-9 | Section 5 ; References | EUROMAR2006 | An Abstract for the after-effects of an HR PMR study in Solids | Details of the particulars of particiaption in ICMM2006
Draft of Abstract for EPISTEME-2 at HBCSE,TIFR,Mumbai, Feb.2007

Early Research Studies on:"Explaining the trends of Nuclear Shielding caused by Magnetic moments related to Susceptibilties" have been detailed at the webpages at the URL: http://saravamudhan.tripod.com/ 

 A Simple Mathematical Formula For Close Packing Of Spheres With Constraints; And, The Inroad Into The Material From The  Molecule Within That Medium

S.Aravamudhan, Department of Chemistry, North Eastern Hill University, PO NEHU Campus Mawkynroh Umshing, Shillong 793022 Meghalaya

(Even though the author was not fully convinced that this topic would be relevant in the orgnizer's purview of the listed topics for the episteme-2, the author submitted this abstract to draw the attention of the reviewers for a scruitiny and a opinion.)

CLICK & VIEW ! MS Powerpoint Presentation File for the "Enduring Questions" of Refrence #1.

For an elaboration on the intramolecular aspects of induced fileds (measured as Chemical shifts) Click on Link: http://www.geocities.com/amudhan20012000/ismar_ca98.html

 

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 bulk-susceptibility 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 charge-cloud 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 questions1 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 line-narrowing 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 non-magnetic, 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 point-dipole 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 shape-dependent 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 fragmented-entities 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 Vsph= [4/3]•π•r3). Thus the magnetic dipole moment for the spherical fragment should be proportional to r3. 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 r3 (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 location-distance 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: 

  1. http://nehuacin.tripod.com/id3.html  - a web page documentation of the presentation at the 4th Alpine Conference on Solid State NMR: New Concepts and Applications , September 2005, Chamonix Mont-Blanc, France
  2. "Magnetized Materials: Contributions Inside Lorentz Ellipsoids", S.Aravamudhan, Indian J.Phys., Vol.79 (9), p985-989 (2005) This is Condensed Matter Days2004 proceedings’ special issue;

          "Bulk solid specimen shape dependences in the molecular, chemical-shift

          tensor determinations”

            S.Aravamudhan, Proceedings of the International Conference on   Molecule to Materials ICMM2006,

             SLIET, Longowal, Sangrur, Punjab, India; March 3-6, 2006.  The TEXT of this Full Paper of this presentation at

            ICMM2006 has   been posted by this author at the Webpage URL:  “Jump to EVENT-III” at 

            http://www.geocities.com/inboxnehu_sa/conference_events_2006.html

  1. http://www.geocities.com/saravamudhan1944/crsi_6nsc_iitk.html
  2. 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”.  

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

  2. (a) Local-field Effects and Effective-medium 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 619-622. http://saravamudhan.tripod.com/id1.html (c) Permanent and Induced Molecular Moments and Long Range Inter-molecular Forces: by AD Buckingham in advances in Chemical physics, vol XII intermolecular forces edited by Joseph O.Hierscfelder, Interscience publishers 1967, page 549.

  3. 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

 

 

 

      

                                                                                  

     

Click on the link below to display a webpage with links to contributions to events during July 2005 to February 2006. http://saravamudhan.tripod.com/id12.html#BOTTOM

The contents of pages: index, id1, id2, id3, & id4 of this ite contain the text of the
Full Paper appearing in the Proceedings of ICMM2006 March 3-6, 2006
Section on Materials for the Furure page 13-18; Article #2.3
 
The contents of page id5 is the Abstract for EPISTEME-2
 
The contents of page id6 is on EUROMAR2006