Problem 2. Structure-Property Relations: Tensor and Matrix Operations in Describing Materials Properties For a BaTiO; crystal at room temperature, its x₁ [100], x2 [010], and x3 [001] directions constitute a set of orthogonal right-handed axes. To transform this set of axes system to a new right-handed axes system with direction x₁= [011], x₂= [011], (2.a) find the transformation matrix (a) - the set of direction cosines relating the new axes to the old.

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Problem 2. Structure-Property Relations: Tensor and Matrix Operations in Describing
Materials Properties
For a BaTiO; crystal at room temperature, its x1 [100], x2 [010], and x3 [001] directions
constitute a set of orthogonal right-handed axes. To transform this set of axes system to a
new right-handed axes system with direction x1'= [011], x2'= [0 ī1].
(2.a) find the transfomation matrix (a) - the set of direction cosines relating the new axes
to the old.
[30
(2.b) If the dielectric property in the original system is given as le, /ɛ]=|0 30 0
0 0 50
find out the dielectric constant ɛ11' along the direction x1'/[011].
(2.c) Provide a sketch of the directional
dependence of its dielectric property (i.e., e'
as a function of e and 4) for 0s 0 < 180
degrees. A two-dimensional drawing is fine
when proper labels are given.
Transcribed Image Text:Problem 2. Structure-Property Relations: Tensor and Matrix Operations in Describing Materials Properties For a BaTiO; crystal at room temperature, its x1 [100], x2 [010], and x3 [001] directions constitute a set of orthogonal right-handed axes. To transform this set of axes system to a new right-handed axes system with direction x1'= [011], x2'= [0 ī1]. (2.a) find the transfomation matrix (a) - the set of direction cosines relating the new axes to the old. [30 (2.b) If the dielectric property in the original system is given as le, /ɛ]=|0 30 0 0 0 50 find out the dielectric constant ɛ11' along the direction x1'/[011]. (2.c) Provide a sketch of the directional dependence of its dielectric property (i.e., e' as a function of e and 4) for 0s 0 < 180 degrees. A two-dimensional drawing is fine when proper labels are given.
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