1. 7th March 2010Dr.S. Aravamudhan1 This is an effort to use the Microsoft PowerPoint animation features to illustrate the mathematical transformations for a Visualization. Only a two dimensional representations have been preferred since full three dimensional effort might complicate the matter in the beginning effort itself. 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. 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. http://ugc-inno-nehu.com/transform_tensors.ppt Click to start Click ….. Click to transit

2. 7th March 2010Dr.S. Aravamudhan2 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 Assigned numerical values for ‘a’ can be 1 unit, and for ‘b’ it can be 2 units μ a =1 μ b =2 b a b a μ a =1 μ b =2 Such an ellipse can be associated with a rectangular object (molecule) b a When the objects rotate, it would be a rotation of the associated ellipse CLICK to display next slide b a Ellipse representing the polarizability tensor in the (system) material (molecule) along the principal axis for the interacting system Polarizability Tensor = x y Inducing Field F Inducing field set with fixed Intensity along the y-axis F a b a 1 0 b 0 2 α ^ = a b a α aa 0 b 0 α bb = F F T α ^ Resultant Induced Moment vector = μ μ = Superscript ‘ T ’ stands for “Transpose of” Elaboration of these follow in the next few slides Click Refer to Slide #5 Refer to Slide #3 Click to start

3. 7th March 2010Dr.S. Aravamudhan3 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. Mathematical description:- Inducing field in the Laboratory frame is conventionally a column Vector F = A 3 x 1 matrix Dot product is the scalar product of the vectors and is represented below in matrix notation Row Column 1x3 3x1 = F F Ellipse is two dimensional; a geometrical shape-only two components for the resolution of a vector Click to transit This transposed matrix is a row vector; a 1 x 3 matrix F x F y F z Click mouse! Click FxFyFzFxFyFz F x F y F z F | | 2 F T ‘T’ stands for transpose Cited in Slide#2

4. 7th March 2010Dr.S. Aravamudhan4 x y Inducing Field Lab fixed axis b a Molecule fixed axis When molecular fluctuations occur, then the molecular axes system tumbles with respect to the fixed laboratory axes. b a Molecule fixed axis b a b a b a b a 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. 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 Click to transit Photographic disposition at a particular instant during the fluctuations Click Tumbling Molecules To view the tumbling molecules, Right Click the mouse and in the prop up menu click on “previous”, and then…. …click to view Cited in Slide#3 Click …..

5. 7th March 2010Dr.S. Aravamudhan5 b a Resultant induced moment a b a -1 0 b 0 2 α ^ = a b a α aa 0 b 0 α bb = Polarizability Tensor in Molecular Principal axis system α ^ = L ^ = The L- matrix with the direction cosines of axes in Laboratory system and the corresponding (rotated) Molecular axes system. F = The applied inducing field vector in laboratory axes; a 3x1 column vector L ^ T Is the Transpose of L ^ T F is the transpose of column vector which would be the 1x3 row vector F L ^ F = In Molecular frame F M Components of Response moment μ M induced along the principal axes Polarizability Tensor in Molecular Principal axis system Polarizability Tensor transformed into Laboratory axis system α ^ F μ = α ^ F M = μMμM μMμM L ^ T = μ μ = Transformed into Laboratory Frame μMμM F L ^ α ^ L ^ T = μ μMμM μMμM “Rotated” Molecule -fixed axis x y Inducing field in Lab axes F F M α ^ T F μ = Energy of Interaction F μ = System Response: Along principal direction ‘a’, α aa response vector=(-1) * perturbing vector Along ‘b’, α bb response vector=( 2 )* perturbing vector Click x y Lab fixed axis μ If Molecular (a,b) PAS coincides with (x, y) Lab axes: Click As cited in Slide#2 Click …. Click Components of [ ] inducing field along the principal axes a & b of polarizability tensor F M Click to transit Refer to slide#6

6. 7th March 2010Dr.S. Aravamudhan6 F L ^ α ^ L ^ T α ^ = μ T F μ = F μ = Energy of Interaction α ^ F μ = α ^ F L ^ L ^ T T F = α ^ F T F Direction in which the perturbing field (interacting with matter) is applied in the laboratory Polarizability Tensor in the Laboratory system of Axes ( transformed as above from molecular system of axes) μ α ^ Induced moment ‘μ’ in the Laboratory Inducing field in Lab axes F μ cos(45°) = sin(45°) = 0.707106 x y Lab fixed axis x, y b a Θ =45° c and so on…. x y a 0.71 0.71 b -0.71 0.71 L ^ x y a l ax l ay b l bx l by L ^ l ax =cos (x c a) ^ = In terms of the angle of rotation ‘θ’ x y a cos(θ) sin(θ) b -sin(θ) cos(θ) L ^ ^ xCa= 45° yCa= -45° xCb= 135° yCb= 45° Angle of rotation=+45° b a x y x y a 0 -1 b 1 0 L ^ = Angle of rotation= ‘θ’=-90° …..CLICK CLICK CLICK to transit Cited in Slide#5

7. 7th March 2010Dr.S. Aravamudhan7 Perturbing Field System Response:- Induced moment F M Sketch for a review