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Click to start. This is an effort to use the Microsoft PowerPoint animation features to illustrate the mathematical transformations for a Visualization. Click ….

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  1. Click to start This is an effort to use the Microsoft PowerPoint animation features to illustrate the mathematical transformations for a Visualization. Click ….. Only a two dimensional representations have been preferred since full three dimensional effort might complicate the matter in the beginning effort itself. Click ….. There is still room for improvements and inadequacies have been noted. But a preliminary view by many and their comments would indicate the required changes more effectively. Click ….. This PowerPoint file would be uploaded at the URL: http://ugc-inno-nehu.com/transform_tensors.ppt and which can be downloaded and viewers can try to implement the required modifications and indicate the desirable modifications to the author by email, comments and queries. Click to transit Dr.S. Aravamudhan

  2. a b a αaa 0 b 0 αbb y = Polarizability Tensor = = ^ ^ b b b b α α x a a a a F F F Inducing Field F Ellipse representing the polarizability tensor in the (system) material (molecule) along the principal axis for the interacting system Inducing field set with fixed Intensity along the y-axis μb=2 Resultant Induced Moment vector = μ μa=1 a b a 1 0 b 0 2 T μ = • • Click to start Click Refer to Slide #5 The moment ‘induced’ along the long b-axis is only for ‘inducing’field along the long axis= ‘b’ and no (response) ‘induced’ moment along the orthogonal short axis Similarly the moment ‘induced’ is along short a-axis for ‘inducing’field along short axis= ‘a’ and no (response) ‘induced’moment along the perpendicular long axis Click Assigned numerical values for ‘a’ can be 1 unit, and for ‘b’ it can be 2 units μa=1 μb=2 Click When the objects rotate, it would be a rotation of the associated ellipse Superscript‘T ’ stands for “Transpose of” Click Refer to Slide #3 Such an ellipse can be associated with a rectangular object (molecule) Click Refer to Slide #3 Elaboration of these follow in the next few slides CLICK to display next slide Dr.S. Aravamudhan

  3. Fx Fy Fz F Inducing field in the Laboratory frame is conventionally a column Vector A 3 x 1 matrix F F F = T ‘T’ stands for transpose F Fx Fy Fz Fx Fy Fz This transposed matrix is a row vector; a 1 x 3 matrix | |2 • Inducing field is applied in Laboratory (Fixed Frame) frame of reference This field would be resolvable in terms of the components along the in the Principal axis system of the susceptibility / polarizability Principal axes system; this is usually fixed reference frame within the molecule. Hence this system of axes would change in direction with respect to Laboratory fixed reference frame in the event of the molecule executing movements / undergoing translational and rotational motions ; Characteristic molecular fluctuations in fluids. Click mouse! Click to transit Mathematical description:- Click Cited in Slide#2 Dot product is the scalar product of the vectors and is represented below in matrix notation Row• Column 1x3 • 3x1 Click = Ellipse is two dimensional; a geometrical shape-only two components for the resolution of a vector Dr.S. Aravamudhan

  4. y b b b b b b Lab fixed axis Molecule fixed axis x a a a a a a Inducing Field Molecule fixed axis Molecule fixed axis Molecule fixed axis Molecule fixed axis Molecule fixed axis To view the tumbling molecules, Right Click the mouse and in the prop up menu click on “previous”, and then…. …click to view Tumbling Molecules When the Lab axes and the molecular system of axes have the coordinate axes one to one parallel, then the induced moment can be calculated without any transformations of coordinate systems. Click ….. Click Cited in Slide#3 Click When molecular fluctuations occur, then the molecular axes system tumbles with respect to the fixed laboratory axes. When the lab axes and molecular axes are rotated from one another, then the molecular physical quantity due to perturbations in the Laboratory axes system, can be related to the response (induced moments) only after appropriately transforming the physical quantities involved. Click to transit Photographic disposition at a particular instant during the fluctuations Dr.S. Aravamudhan

  5. = The L- matrix with the direction cosines of axes in Laboratory system and the corresponding (rotated) Molecular axes system. Components of [ ] inducing field along the principal axes a & bof polarizability tensor y y Polarizability Tensor in Molecular Principal axis system ^ ^ ^ ^ ^ ^ b = α α α α α α = = The applied inducing field vector in laboratory axes; a 3x1column vector F M x x a Inducing field in Lab axes Components of Response moment μM induced along the principal axes Lab fixed axis Is the Transpose of = T T T ^ ^ ^ L L L a b a -1 0 b 0 2 is the transpose of column vector which would be the 1x3 row vector T T ^ ^ ^ ^ F F F F F F F F F F F F L L L L a b a αaa 0 b 0 αbb In Molecular frame = = M M M = = = μ μ μ μ μ μ μ μM μM μM μM μM Transformed into Laboratory Frame = Energy of Interaction = • = Click ….Click If Molecular (a,b) PAS coincides with (x, y) Lab axes: Click As cited in Slide#2 Resultant induced moment Refer to slide#6 Click “Rotated” Molecule -fixed axis Click System Response: Along principal direction ‘a’, αaa response vector=(-1) * perturbing vector Along ‘b’, αbb response vector=( 2 )* perturbing vector Click Polarizability Tensor in Molecular Principal axis system Click to transit Polarizability Tensor transformed into Laboratory axis system Dr.S. Aravamudhan

  6. Inducing field in Lab axes = y y = ^ ^ ^ ^ ^ ^ b b α α α α α α = • = x x a a x y a laxlay b lbxlby Lab fixed axis x, y = = ^ Θ =45° c xCa= 45° yCa= -45° xCb= 135° yCb= 45° T T In terms of the angle of rotation ‘θ’ ^ ^ x y a 0.71 0.71 b -0.71 0.71 L L x y a cos(θ)sin(θ) b -sin(θ) cos(θ) T T T ^ ^ ^ ^ ^ ^ F F F F F F F F F ^ L L L L L L lax=cos (xca) Angle of rotation= ‘θ’=-90° x y a 0-1 b 1 0 = μ μ μ μ μ μ …..CLICK CLICK Energy of Interaction CLICK CLICK Cited in Slide#5 Angle of rotation=+45° and so on…. • • Direction in which the perturbing field (interacting with matter) is applied in the laboratory CLICK cos(45°) = sin(45°) = 0.707106 Induced moment ‘μ’in the Laboratory Polarizability Tensor in the Laboratory system of Axes ( transformed as above from molecular system of axes) CLICK CLICK to transit Dr.S. Aravamudhan

  7. F M System Response:- Induced moment Perturbing Field Sketch for a review Dr.S. Aravamudhan

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