7th March 2010 Dr.S. Aravamudhan 1

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.

Click to start

Click …..

Click …..

Click …..

Click to transit

7th March 2010 Dr.S. Aravamudhan 2

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 axisSimilarly 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

μa=1 μb=2 b

a

b

aμa=1

μb=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 2α̂ =

a b

a αaa 0

b 0 αbb

=

F FT

α̂

Resultant Induced Moment

vector

= μμ= • •

Superscript ‘T ’

stands for “Transpose of”

Elaboration of these follow in the next few slides

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Click

Click

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Refer to Slide #5

Refer to Slide #3

Refer to Slide #3

Click to start

7th March 2010 Dr.S. Aravamudhan 3

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

=

FF •

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 Fx Fy Fz

Click mouse!

Click

ClickFx

Fy

Fz

Fx Fy Fz

F| |2

F T

‘T’ stands for transpose

Cited in Slide#2

7th March 2010 Dr.S. Aravamudhan 4

x

y

Inducing Field

Lab fixed axisb

a

Molecule fixed axis

When molecular fluctuations occur, then the molecular axes system tumbles with respect to the fixed laboratory axes.

b

aMolecule fixed axis

b

a

Molecule fixed axis

Molecule fixed axis

b

a

b

a

Molecule fixed axis

ba

Molecule fixed axis

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 …..

7th March 2010 Dr.S. Aravamudhan 5

b

aResultant induced moment

a b

a -1 0

b 0 2α̂ =

a b

a αaa 0

b 0 αbb

=Polarizability Tensor in Molecular Principal axis system

α̂ =

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

L̂T Is the Transpose

of L^

TF

is the transpose of column vector which would be the 1x3 row vector F

L^

F =In Molecular frameFM

Components of Response moment μM induced along the principal axes

Polarizability Tensor in Molecular Principal axis system

Polarizability Tensor transformed into Laboratory axis system

α̂ F μ=

α̂ FM= μM μML̂

T= μ

μ = Transformed into Laboratory FrameμM

FL^α̂L̂

T= μ

μM

μM

“Rotated” Molecule -fixed axis

x

y

Inducing field in Lab axes

F

FM

α̂T

F μ= Energy of InteractionF• μ =

System Response:

Along principal direction ‘a’, αaa response vector=(-1) * perturbing vector

Along ‘b’, αbb

response vector=( 2 )* perturbing vector

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Click

x

y

Lab fixed axis

μ

If Molecular (a,b) PAS coincides with (x, y) Lab axes:

Click

Click

As cited in Slide#2

Click

….ClickComponents of [ ] inducing field along the principal axes a & b of polarizability tensor

FM

Click to transit

Refer to slide#6

7th March 2010 Dr.S. Aravamudhan 6

FL^α̂L̂

T

α̂

= μT

F μ=F• μ = Energy of Interaction

α̂ F μ=

α̂ FL^

L̂TT

F =

α̂ FT

F • •Direction in which the perturbing field (interacting with matter) is applied in the laboratory

Polarizability Tensor in the Laboratory system ofAxes ( transformed as above from molecular system of axes)

μ

α̂

Induced moment ‘μ’ in the Laboratory

Inducing field in Lab axes

Fμ

cos(45°) = sin(45°) = 0.707106

x

y Lab fixed axis x, y

b a

Θ =45°cand so on….

x y

a 0.71 0.71

b -0.71 0.71L^

x y

a lax lay

b lbx lby

L^

lax =cos (xca) ^

=

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

ax

y

x y

a 0 -1

b 1 0 L^ =

Angle of rotation= ‘θ’=-90°

…..CLICK

CLICK

CLICK

CLICK

CLICK

CLICKCLICK to transit

Cited in Slide#5

7th March 2010 Dr.S. Aravamudhan 7

Perturbing Field

System Response:-

Induced moment

FM

Sketch for a review