PRESENTATION FILE: CMDAYS2011 Gauhati University: August th Aug S.Aravamudhan CMDAYS2011 CONDENSED MATTER DAYS 2011 PROGRAMME A Slide show with automated animation timings. The show duration for this presentation file 33 mins.

16 Aug 2011S.Aravamudhan CMDAYS20112 "When a Magnetic Moment is Subdivided, do the Fragmented Moments Interact with Each Other?" S.ARAVAMUDHAN NORTH ESATERN HILL UNIVERSITY, SHILLONG Chemistry Department CMDAYS 2011 Gauhati University Aug 24-26, 2011 Additional materials for this presentation: The contents of this presentation file would be subjected to alterations as and when it is necessary. Download the updated file from this LINKLINK When A Magnetic Moment Is Subdivided, Do The Fragmented Moments Interact Among Themselves? WebPage for the participation in CMDAYS2011: ABSTRACT It has been possible to improve the validity of point dipole approximation by appropriately subdividing the magnetic moment of a magnetized material to calculate the induced fields within the material and calculate the demagnetization factors by a convenient summation procedure (1). Subdividing the magnetic moment in such a way that the vector addition of the fragmented moments yields the total magnetic moment value is followed by distributing the subdivided moments over the entire sample. In such a case will there be an interaction among the fragments? This would entail accounting for induced fields at one fragment due to the other fragments and the corresponding interaction energy. One way to explain this process is to uphold that the subdivision is only hypothetical and the boundaries between fragments are not real but only a mathematical convenience. On the other hand, the demagnetization factor by a simple procedure is possible only by this subdivision necessarily and the demagnetization obtained have physical significance. Hence it seems it has more basis to find out whether a subdivision of this kind which results in a physical quantity closer to true value, must be such that the interaction among the fragmented and distributed moments must be such as to cancel the effects of the interaction from all the others at the location of every fragment. A procedure to calculate such induced field values at a fragment would be reported and discussed for the above perspective. References: Proceedings of the 96 th Indian Science Congress, 2009;

C.Kittel, book on Solid State Physics Pages Lorentz Relation: E loc = E 0 +E 1 + E 2 +E 3 E 3 = intermolecular E 2 = N inner x P E 1 =N outer x P E 3 is the discrete sum at the center of the spherical cavity; does not depend upon macroscopic specimen shape. (Lorentz field) E 2 is usually for only a spherical Inner Cavity; with Demagnetization factor=0.33 ; E 2 = [N INNER or D INNER ] P E 1 is the contribution assuming the uniform bulk susceptibility and depend upon outer shape E 1 =[N OUTER or D OUTER ]P E 0 is the externally applied field 17 Aug 20113S.Aravamudhan CMDAYS2011 Additional materials for this presentation : Discrete summation

17 Aug 2011 These are Ellipsoids of Revolution and the three dimensional perspectives are imperative INDUCED FIELDS,DEMAGNETIZATION,SHIELDING Induced Field inside a hypothetical Lorentz’ cavity within a specimen = H`` Shielding Factor =  Demagnetization Factor = D a H`` = - . H 0 = - 4. . (D in - D out ) a. . H 0 out   = 4. . (D in - D out ) a.  When inner & outer shapes are spherical D in = D out Induced Field H`` = 0 polar axis ‘a’ equatorial axis ‘b’ m = a/b Induced Field / 4. . . H 0 = D ellipsoid polar axis ‘b’ equatorial axis ‘a’  = b/a Thus it can be seen that the the ‘D’-factor value depends only on that particular enclosing- surface shape ‘innner’ or ‘outer’ in References to ellipsoids are as per the Known conventions  >>>>  4S.Aravamudhan CMDAYS Evaluation requires solving integrals set up for appropriate shapes N + S - Field: Lines of Force Moment direction Discrete summation Only induced diamagnetic moment is considered The difference between induced moments and permanent moments: Induced moments arise when the specimen is exposed to magnetic field and the induction phenomena ensures the orientation of the moment is such a way the moment should be aligned (diamagnetism) for minimum potential energy. No further alignment and changes in orientation of the moment is necessary. The induced moments arise due to the polarization (P) of the electronic (static) charge cloud. Since it is the Currents (flowing charges) give rise to magnetic fields, time de[pendent changes in polarization are invoked and the Polarization currents ( ∂P / ∂t) appear in the expressions for current densities and charge densities in the context of Magnetism.. The permanent moments are present even in the absence of the magnetic field applied externally, and could be oriented in a direction that may require a realignment when the field is applied. In the context of the allowed discrete orientations of a spin, the ensemble of spins undergo a redistribution of the orientation of the spins and this phenomenon is referred to as the polarization of the spin-orientations. The vectorial direction of a Field happens to be opposite conventionally to the vector direction of the Moment. Links below are or getting clarifications on the conventions

 is the susceptibility which is inherent characteristic of the electronic structure of molecules/materials. Magnetization M p arising due to the interaction of the material at the Spot with the externally applied magnetic field of strength H 0 2-dimensional lattice Each point is occupied by a molecule. with susceptibility  p Magnetic moment μ p =  p H 0 Molecular to molar or volume susceptibility  requires summation over the appropriate number of molecules;  = n.  p. Till now no interaction due to the μ p at other sites has been considered (the native property). The native field at the molecular site, due to the molecular susceptibility  p should be equal to | μ p | =  p · H 0 = |H” p | (Only) shape dependent value (a gemetrical premultiplying factor)of the Demagnetization factor D arises due to the induced field from all the other molecules. When the induced field, due to such native moments, at a distant point is to be calculated, would it be of any consequence to know whether the native moments interact with each other? Would such an interaction if it is present alter the generic field (local filed) value at a site within the specimen? 17 Aug 20115S.Aravamudhan CMDAYS2011 The Magnetization M=Induced moment per unit volume; M= n. μ p ; where n is the number of molecules per unit volume Inquire for the appropriate units of Moment and Field Can the qualitative difference between the physical quantity “Moment” and the “Field” traced to account of r the necessity of the “H” field and the “B” field macroscopically ? Thus the demagnetization factor D ( a fraction, <1, ) can be pre-multiplying μ p to result in a reduced moment; D x μ p A possible way to answer the query stated as the title of this presentation is included in Slide #12Slide #12 The Electro-dynamic aspects of the micro- & macroscopic fields, and the generating currents and current densities for H and B fields find a reinterpretation in the review:

17 Aug 2011S.Aravamudhan CMDAYS E-08 Benzene Molecule & Its magnetic moment 2 A ˚ equal spacing Do these moments interact with each other? {χ M. (1-3.COS 2 θ)}/ (R M ) 3 Indfld vs distance χ || =-90 x [cgs units (molar)] Per molecule would be The above value divided by Avagadro number: x = x = x At a distance of 2 Angstrom from this moment (per unit field= χ || x 1G) The secondary field value would be x / (2 x ) 3 = x For χ || = x ; Secondary field would be x Further detailed consideration of the inter and intra molecular shielding contributions in terms of molecular, bond and atomic charge circulations (currents) are in the link and in the side #8 of this file linkside #8 That these moments should be interacting and accounting for this interaction in the discrete regime is all about the Lorentz sphere and the Field. Slide#3Slide#3

18 Aug 20117S.Aravamudhan CMDAYS2011 μpμp RpRp H p = μ p / R p 3 ∑ p H p μ x ∑ p 1 / R p 3 This is merely shape dependent It is precisely at this point of knowing the microscopic matter as it prevails, the Proton Magnetic Resonance seems to be answering to the details on the paradoxical situation: THE MICRO-MACRO PARADOX ! The shielding parameter measured by the proton magnetic resonance spectroscopy is a measure of the generic local field at the nuclear site within a molecule. This shielding has contributions from the neighboring moments present, and in favorable cases these can become significant contributions to alter the pattern of the spectra obtained. How these changes (to the spectral patterns) also become significant in the net macroscopic field distributions within the magnetized material: it is a matter of numerical values of induced fields at the neighboring moments compared to the native moment induced by the interaction with the externally applied (possibly) strong magnetic fields. It becomes necessary to correct for the bulk susceptibility effects depending on the shape.

These neighbor moment contributions are of the order of ppm of the strength the external fields as much as the native moments are. The (χ v susceptibility values are ppm cgs units) molecular susceptibility values (χ molecular = χ v / no. molecular units within volume ‘V’ ) are of order of CGS units to be multiplied by the external filed strength to arrive at the natively induced moments. Thus the native moments μ are themselves of the order of CGS units. If these moments are to induce secondary fields at a neighboring point, (~ μ /R 3 ) which is of the order of ppm, this induced secondary field would add to the native moment only insignificantly in comparison. Thus the interaction of the native moments would not be much consequence for the induced filed values at any location from all the other locations. THE MICRO-MACRO PARADOX IN INDUCED FIELD CALCULATIONS AND THE ROLE OF HR SOLID STATE NMRMICRO-MACRO PARADOX 18 Aug 20118S.Aravamudhan CMDAYS2011 H0H0 μ R Same These estimates have been possible since the validity of point dipole approximation has been assumed ppm Further detailed consideration of the inter and intra molecular shielding contributions in terms of molecular, bond and atomic charge circulations (currents) are in the link and in the side #8 of this file link side #8

18 Aug 2011S.Aravamudhan CMDAYS20119 Line defined by Polar angle θ / direction of radial vector When the magnetic moment is divided then, the divided elements are not molecular moments as considered in the previous slide. The magnetic moments of the divided elemental volumes are semi-micro magnitudes, each element comprising of several molecules. The magnitude of the moments is also larger and hence the induced fields at the neighboring element may also be more, depending on the element to element distance being larger in comparison to intermolecular distances. When the induced field is calculated at a farther point than the neighboring element, how much would be the inter-element contributions to alter the originally divided moment-magnitudes? It is to be pointed out at this juncture, the total induced field at a point values within the spheroidal specimen results only in a shape (and not the size) dependent pre-multiplying factor to the value of the induced moment (generated by the interaction with external field). It would be true that the individual elemental moments interact among each other but what matters is the effective total interaction. It is only in a spectroscopic analysis as for the Proton Magnetic Resonance Spectroscopy, it is possible to demarcate the intra molecular versus the intermolecular around a particular site and distinguish the strengths and significance of the long-range & short-range interaction scales. And, disentangle the microscopic and macroscopic consequences and observe the effects distinctly as if these are two different physical quantities even though it is all induced fields. Vector map: Slide#4 of 0_2_12Aug2011.ppt This demarcated typical region is considered in the next slide

16 Aug 2011S.Aravamudhan CMDAYS The numerical net value at the center amounts to nearly zero. This favorable numerical result indicates that the close packing criterion and the associated distances can be applied to know the actualities of contributions besides the afortiori reasons. Click for elaboration Refer to graph in next slide Appropriate radius and distances are noted and the calibration graph is used to estimate local field A calibration graph in next slide obtainable from the data below from the data Click here for an xls file Refer to previous slide Induced field at a point outside the magnetized specimen

03 Sept. 2011S.Aravamudhan CMDAYS Radius 2.5 mm Distance = 7.2 mm 7.2 / 2.5 = 2.88 ; 1 / (2.88) 3 = /2.5=1.08 ; 1/ (1.08) 3 = , 2.90E , 2.00E-07 Calculating the graphical plot: Click here for an xls file link A calibration graph is slide obtainable from the data in this URL from the data in this URL

16 Aug 2011S.Aravamudhan CMDAYS The following conclusive part is based on the contents of the materials in slide #5 of this ppt file.slide #5 As explained, the Magnetization is the induced moment per unit volume. And, in terms of the molecular moments, the native moments of the molecules are simply summed up over all the molecules in the unit volume to get this magnetization (or induced magnetic moment) and no interaction between the molecular moments is taken into account. And it is such a simple sum of moments which stands for the Total Magnetic Moment “M” of the specimen. It is this interaction (spelt out as unconsidered above) which is accounted for while calculating the induced secondary field (demagnetization factor) at distant point within the specimen; and, for this calculation, the total moment is divided and distributed. If the M is multiplied by the Demagnetization factor (shape-determined), then, dividing the resulting magnetic moment value cannot be subjected to the division and distribution in such a simple and straight forward manner. The way the magnetic moment is divided and distributed in the summation method is precisely to account for the interaction among the moments, hence the reply to the query (as in the title) could be that non interacting moments-sum is divided to consider the interaction among themselves. At this juncture it can be inferred that the subdivided fragments are being so considered as the consideration of molecular native moments within the Lorentz Sphere, the discrete regime.Lorentz Sphere Further considerations and details of summation within the discrete volume element (typically the Lorentz sphere) have been made available as internet resource and the same can be tractable from the webpage at: Click ……..

22 Aug 2011S.Aravamudhan CMDAYS Download the PowerPoint Files from the Web Directory and save these files in your resident disk in one and the same new directory created (may be named cmdays_gu) in your disk. More materials related to this symposium can be available from the Web subDirectory Also, you may display the URL: