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1 PROTON I II Molecular + Region I + Region II Lorentz Sphere Contribution Total Experimental = Bulk Susceptibility Effects SHIELDING H igh R esolution P roton M agnetic R esonance 1. Contributions to Induced Fields at a POINT within the Magnetized Material.

2 The Outer Continuum in the Magnetized Material Specified Proton Site Lorentz Sphere The Outer Continuum in the Magnetized Material Lorentz Sphere of Lorentz Cavity Outer surface D out Inner Cavity surface D in D out = - D in Hence D out + D in =0 The various demarcations in an Organic Molecular Single Crystalline Spherical specimen required to Calculate the Contributions to the induced Fields at the specified site. D out/in values stand for the corresponding Demagnetization Factors 1. Contributions to Induced Fields at a POINT within the Magnetized Material.

3 Equation 1 Equation 2 In equation 1 above E Loc on the Left Hand Side of the equation, obviously depends on the value of P for estimation. And, in equation 2, the value for to is to be estimated from the value of E Loc. If P is related to E mac (instead of to E Loc in equation-2) which is the applied field, then the paradoxical situation would not be posed. The paradox is that E Loc is a field which is a value including the effect of P, and hence to know E Loc for equation 2 the value of P must have been known already.

4 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

5 Magnetic Field {χ M. (1-3.COS 2 θ)}/ (R M ) 3 6.0E-08 Benzene Molecule & Its magnetic moment Each moment contributes to induced field 2 A ˚ equal spacing

6 Isotropic Susceptibility Tensor = Induced field Calculations using these equations and the magnetic dipole model have been simple enough when the summation procedures were applied as would be described in this presentation. 2. Calculation of induced field with the Magnetic Dipole Model using point dipole approximations. σ1σ1 +σ2+σ2 +σ 3.. σ inter = At this site Magnified

7 7/23/2015 1:57:34 AMMRSFall 2006/This slide 01m:40s for prev.6 slides=08m:03s 7 First and Foremost it is to be pointed out that trying to sum the induced field contributions in the discrete region by taking the molecule by molecule contributions from the neighborhood, results in a convergence of the summed value to a total sum, which can be termed as Intermolecular contributions. This convergence is depicted below in the actual single crystalline case of an Organic molecular single crystalline system. The summed up contributions from within Lorentz sphere as a function of the radius of the sphere. The sum reaches a Limiting Value at around 50Aº. These are values reported in a M.Sc., Project (1990) submitted to N.E.H.University. T.C. stands for (shielding) Tensor Component.Convergence occurred in this particular case at 50 Aº radius.

8 7/23/2015 1:57:34 AMMRSFall 2006/This slide 01m:15s for prev. 7 slides=09m:43s 8 Magnetic Field {χ M. (1-3.COS 2 θ)}/ (R M ) 3 Benzene Molecule & Its magnetic moment Each moment contributes to induced field 2 A ˚ equal spacing Induced field varies as R -3 Number of molecules in successive shells increase as R 2 Product of above two vary as R -1 The above distance dependences can be depicted graphically Lattice summation Shell by Shell Dashed lines: convergence limit When the R becomes large, the R-1 term contribution becomes smaller and smaller to become insignificant Magnetic field direction -ve zone +ve (1-3.cos 2 θ) term causes +ve & -ve contributing zones See drawing below

9 7/23/2015 1:57:34 AMMRSFall 2006/This slide 01m:32s for prev. slide=01m:10s 9 Bulk Susceptibility Effects In HR PMR SOLIDS Single crystal arbitray shape Single Crystal Spherical Shape Induced Fields at the Molecular Site Lorentz Sphere cavity DEFINING: LORENTZ SPHERE & CAVITY Which is essentially a spherical Inner Volume Element [I.V.E] Necessity for considering the shape of I.V.E to be (Lorentz) E llipsoids is being pointed out in this authors results The Shielding Tensor at the Proton site in the Molecule is determined Proton site

10 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

11 1. Experimental determination of Shielding tensors by HR PMR techniques in single crystalline solid state, require Spherically Shaped Specimen. The bulk susceptibility contributions to induced fields is zero inside spherically shaped specimen. 2. The above criterion requires that a semi micro spherical volume element is carved out around the site within the specimen and around the specified site this carved out region is a cavity which is called the Lorentz Cavity. Provided the Lorentz cavity is spherical and the outer specimen shape is also spherical, then the criterion 1 is valid. 3. In actuality the carving out of a cavity is only hypothetical and the carved out portion contains the atoms/molecules at the lattice sites in this region as well. The distinction made by this hypothetical boundary is that all the materials outside the boundary is treated as a continuum. For matters of induced field contributions the materials inside the Lorentz sphere must be considered as making discrete contributions. Illustration in next slide depicts pictorially the above sequence

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Combination. In mathematics a combination is a way of selecting several things out of a larger group, where (unlike permutations) order does not matter.

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