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Below is a Summary of the Results of Trying to Calculate
Demagnetization Factors of Magnetic-materials of Standard-shapes with Quanitative Geometrical Shape Parameters Describing
Their Shapes.
In the graphical plot the blue-line is a curve calculated
from the available analytical expression for the Demagnetization factors for
ellipsoidal solid shapes of homogeneously
magnetized specimen.The curve had to be stitched from two different equations one valid for values of "m">1 and the other
for "m"<1 and the curve appears as if it is obtained from a single analytical function valid allthrough.The green points
are obtained at discrete values of "m" by numerically substituting values at specified ratios of m=a/b or b/a and the red
points are obtained by the presnt method of calculation.It should be possible to improve the agreements even to a greater
extent.
All the above efforts
were calculation of induced fields at points (nuclear sites)inside a macroscopic specimen due to the magnetized material which
makes up that specimen and these are NOT calculations of induced fields OUTSIDE THE SPECIMEN when the specimen aquires a magnetization
when placed in an externally applied magnetic fields.
In the case of Calculating the Shielding
of proton in Benzene Molecule it seems it is all a case of being internal to the molecular specimen(INTRAmolecular).But actually
the effort will be to calculate the contribution due to the susceptibility tensors from the inside of the molecule at the
peripheral locations in the molecule (where the protons are) and hence this itself would be a case of calculating induced
fields at points outside the material very close to the megnetized material(here the molecular specimen is being refered to
as "specimen"(which had been usually used in the context of referring to a macroscopic extent and expanse of a magnetized
material containing several MOLES of the molecular substance)
IS THE CALCULATION OF INTRAMOLECULAR SHIELDING OF PROTONS
IN MOLECULE, A CASE OF CALCULATING FIELDS INSIDE THE SPECIMEN OR OUTSIDE THE SPECIMEN ?
What is indicated as a question above is explained further in
the following diagram. The delocalised six pi-electrons in the benzene molecule circulate at the regions found by joining
the 6 carbon atoms in a ring. It is reported that there are localised pi electron circulations at the carbon centres as well
as the more characteristic circulations of the delocalised cloud along the circumference of the aromatic ring. It is this
part of pi electron circulation around the six-fold molecular symmetry axis along the ring peripheri which is responsible
for the large deshielding effects characteristic of the aromatic ring protons. It seems possible, to apportion out this part
of circulation and associate and a diamagnetic susceptibility tensor for the changes in the circulation characteristics in
presence of an externally applied field. This can be thought of as giving rise to a magnetic dipole and using the dipole model
and a tensor form for the equaion to the induced field an estimate of the shielding tensor can be obtained.
As was pointed out in the explanation above in the following
picture an illustration depicts the delocalised electron circulation region in the frame work of the familiar six-member-ring.
Since the p-orbitals have lobes extending above and below the plane of the ring carbon atoms it is unrealistic to consider
a line for the flow of currents and hence the quantum chemical methods for the calculation of the current densities is resorted
to and this approach does yield values for suscptibility tensors associated with this circulation. In all these calculations
the main complication is to find appropriate origins for associating with the magnetic dipoles when the magnetic susceptibility
is explained to be a source of a magnetic dipole further resulting in seconadary fields induced. This aspect is considered
in the diagramatic feasibility study to obtain a ground to be convinced if after all these winding steps one still reaches
to a situation comparing well with the experimental values and other similar efforts earlier but much less straight-forward
from the point of view of the simple classical dipole model.
The Benzene molecular ring current thus
essentially gives a susceptibility tensor which can be thoght of giving rise to a magnetic dipole tensor and which when substituted
in the equation for the shielding tensor yields a numerical estimate for the shielding tensor at the site of the proton.And
just as much as the isotropic susceptibility value for the homogeneously magnetized specimen can be used and induced field
values can be obtained by a simple summing over the semimicro volume elements( the totality of the material having a istropic
volume susceptibility is considered to be made up of these volume elements - without any loss of the total material volume
- in such away that the individual volumes of these can be known definitely and a corresponding proportionate susceptibility
for these volumes to reproduce the total susceptibility of the material volume of the macroscopic specimen)the volume suscetibility
tensor can be subjected to a summation and would result in a shielding tensor value.The similarity of the role of the very
same induced fields in causing demagnetization effects and the shielding (shift) effects are well known to people working
in NMR of High Resolution NMR in liquids.
