PRESENTATION FILE: CMDAYS2011 Gauhati University: August 24-26th 2011

16 Aug 2011

http://www.ugc-inno-nehu.com/cmdays_abstract_accepted.htmlhttp://www.ugc-inno-nehu.com/cmdays2011_gu.html

1S.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 2011 S.Aravamudhan CMDAYS2011 2

"When a Magnetic Moment is Subdivided, do the Fragmented Moments Interact with Each Other?"

S.ARAVAMUDHANNORTH ESATERN HILL UNIVERSITY, SHILLONG

Chemistry Department

CMDAYS 2011Gauhati UniversityAug 24-26, 2011

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When A Magnetic Moment Is Subdivided, Do The Fragmented

Moments Interact Among Themselves?

WebPage for the participation in CMDAYS2011:

http://www.ugc-inno-nehu.com/cmdays2011_gu.html

ABSTRACTIt 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 96th Indian Science Congress, 2009; http://www.ugc-inno-nehu.com/isc2009nehu.html http://saravamudhan.tripod.com/id6.html

7/23/2008 7:41:01 AM MRSFall 2006/This slide 01m:15s for prev. 2 slides=02m:42s

3

The Outer Continuum in the Magnetized Material

Specified Proton Site

LorentzSphere

The Outer Continuum in the Magnetized Material

LorentzSphere of

LorentzCavity

Outer surface D outInner Cavity surface Din

D out = - Din HenceD out + Din =0

The various demarcations in an Organic MolecularSingle Crystalline Spherical specimen required to Calculatethe Contributions to the induced Fields at the specified site.

D out/in values stand for the corresponding Demagnetization Factors

In the NEXT Slides: INDUCED

FIELDS,DEMAGNETIZATION,

SHIELDING:: Current lessons on I .V.E.

i=ii /R3i [1-(3.RRi /R5

i)]

1. Contributions to Induced Fields at a POINT within the Magnetized Material.

Inner Volume Element

Equation for Discrete summation

Discrete summation: an animated Illustration in Slides # 8 & 9 !!!

C.Kittel, book onSolid State PhysicsPages 405-409Lorentz Relation: Eloc = E0 +E1 + E2 +E3

E3= intermolecular

E2 = Ninner x P

E1=Nouter x P

E3 is the discrete sum at the center of the spherical cavity; does not depend upon macroscopic specimen shape. (Lorentz field) E2 is usually for only a spherical Inner Cavity; with Demagnetization factor=0.33 ; E2 = [NINNER or DINNER] P E1 is the contribution assuming the uniform bulk susceptibility and depend upon outer shape E1 =[NOUTER or DOUTER]P E0 is the externally applied field

17 Aug 2011 3S.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 = Da 

H`` = - . H0 = - 4 . . (Din- Dout)a . . H0

out

= 4 . . (Din- Dout)a .

When inner & outer shapes are spherical

Din = Dout Induced Field H`` = 0

polar axis ‘a’

equatorial axis ‘b’

m = a/b

Induced Field / 4 . . . H0 = 0.333 - Dellipsoid

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 CMDAYS2011

0.333

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

http://www.youtube.com/user/aram1121944/#p/u/8/LctfaPFRjJ8 http://www.av8n.com/physics/electric-dipole.htmhttp://en.wikipedia.org/wiki/Dipole http://hyperphysics.phy-astr.gsu.edu/hbase/electric/dipole.htmlhttp://www.youtube.com/watch?v=513aIefo6QA

is the susceptibility which is inherent characteristic of the electronic structure of molecules/materials. Magnetization Mp arising due to the interaction of the material at the Spot with the externally applied magnetic field of strength H0

2-dimensional lattice

Each point is

occupied by a molecule. with susceptibility p

Magnetic moment μp = p H0

Molecular to molar or volume susceptibility requires summation over the appropriate number of molecules; = n. p.

Till now no interaction due to the μp 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 |μp| = p · H0 = |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 2011 5S.Aravamudhan CMDAYS2011

The Magnetization M=Induced moment per

unit volume; M= n. μp ; 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 μp

to result in a reduced moment; D x μp

A possible way to answer the query stated as the title of this presentation is included in Slide #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:http://www.ugc-inno-nehu.com/cmdays2011/micros_magntzn_llhirst_revmodphys69_607_1997.pdf

17 Aug 2011 S.Aravamudhan CMDAYS2011 6

Indfld vs recpcubedist

-2.5E-05

-2.0E-05

-1.5E-05

-1.0E-05

-5.0E-06

0.0E+00

2.0E-08 1.0E-07 1.8E-07 2.6E-07 3.4E-07 4.2E-07

Indfld vs recpcubedist

indfld vs recpcubedist

-2.5E-05

-2.0E-05

-1.5E-05

-1.0E-05

-5.0E-06

0.0E+00

2.0E-08 2.8E-08 3.6E-08 4.4E-08 5.2E-08 6.0E-08

indfld vs recpcubedist

6.0E-08

Benzene Molecule &Its magneticmoment

2 A˚ equal spacing

Do these moments interact

with each other?

{χM . (1-3.COS2θ)}/(RM)3

i=ii /R3i [1-(3.RRi /R5

i)]

Indfld vs distance

Indfld vs distance

Indfld vs distance

Indfld vs distance

χ|| =-90 x 10-6 [cgs units (molar)]Per molecule would be The above value divided by Avagadro number: 6.023 x 1023 =14.9427 x 10-29 = 1.49427 x 10-30 At a distance of 2 Angstrom from this moment (per

unit field= χ|| x 1G)

The secondary field value would be -1.49427 x 10-30 / (2 x 10-8 )3

= -1.8678 x 10-5

For χ|| = -100 x 10-6 ; Secondary field would be -2.07533 x 10-5

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

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#3

18 Aug 2011 7S.Aravamudhan CMDAYS2011

μp

Rp

Hp = μp/Rp3

∑p Hp

μ x ∑p 1/Rp

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.

http://nehuacin.tripod.com/id5.html

These neighbor moment contributions are of the order of ppm of the strength the external fields as much as the native moments are.

The (χv susceptibility values are ppm cgs units) molecular susceptibility values (χmolecular = χv / no. molecular units within volume ‘V’ ) are of order of 10-30 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 10-30 CGS units. If these moments are to induce secondary fields at a neighboring point, (~ μ /R3) 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

http://nehuacin.tripod.com/id5.html

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H0

μ

R

Same10-30

10-36 10-37

10-42

These estimates have been possible since the validity of point dipole approximation has been assumed

10-36

10-36

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

18 Aug 2011 S.Aravamudhan CMDAYS2011 9

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 2011 S.Aravamudhan CMDAYS2011 10

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 fieldA calibration graph in

next slide obtainable from the data below

Click here for an xls file

Refer to previous slide

Induced field at a point outside the magnetized specimen

03 Sept. 2011 S.Aravamudhan CMDAYS2011 11Radius 2.5 mm

Distance = 7.2 mm

7.2 / 2.5 = 2.88 ; 1 / (2.88)3 = 0.0419

2.7/2.5=1.08 ; 1/ (1.08)3 = 0.793832

0.793832, 2.90E-06

0.0419, 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

16 Aug 2011 S.Aravamudhan CMDAYS2011 12

The following conclusive part is based on the contents of the materials in slide #5 of this ppt file.

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.

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:

http://www.ugc-inno-nehu.com/cmdays2011_gu.html

Click……..

22 Aug 2011 S.Aravamudhan CMDAYS2011 13

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