PRESENTATION FILE: CMDAYS2011 Gauhati University: August th Aug S.Aravamudhan CMDAYS2011 CONDENSED MATTER DAYS 2011 PROGRAMME

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?

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

 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 corrective field is due to the molecular susceptibility  p and 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

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 link

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 are these significant 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 NMR 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

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

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

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

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: