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Cell["3.029 Spring 2022\[LineSeparator]Lecture 08 - 02/23/2022", "Subtitle",
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Cell["Physical Properties of Crystals", "Chapter",
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While certain physical properties, like the mass density of a material, are \
isotropic, preferred directions in crystals imply that others, such as the \
stiffness tensor of elastic materials, depend on direction.\
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Since physical properties represent measurable quantities, they are by \
definition measured in a particular reference frame.\
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It stands to reason, that if a set of measurements in a different reference \
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Cell["Coordinate Transformations (Review)", "Section",
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We can define a transformation between two sets of reference frames\
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Cell["\<\
In-fact, you\[CloseCurlyQuote]ve seen this before in the notebook \
`3.019_L12=Change_of_Basis_Stiffness_Outline.nb` from last semester (which I\
\[CloseCurlyQuote]ve uploaded again on canvas)!\
\>", "Item",
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Cell["\<\
We will quickly review some concepts here, and extend the analysis to \
Thermodynamic stability\
\>", "Item",
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Cell[CellGroupData[{

Cell["Einstein Summation Convention", "Subsection",
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Cell["\<\
How do we express the same physical property tensor, but in the blue \
coordinate system?\
\>", "Subitem",
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In isotropic materials, we can express the intensity of magnetization to the \
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This is an \[OpenCurlyDoubleQuote]intrinsic\[CloseCurlyDoubleQuote] tensor \
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Fortunately, there's a very convenient way to express these in the Wolfram \
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A list of rules of how to specify each component\[LineSeparator]here we\
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A list of symmetries, e.g. using \[OpenCurlyDoubleQuote]Generators\
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In 3.013 last semester, you talked about elastic properties of solids which \
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Cell["\<\
In the linear elastic regime, the stress increases linearly with increasing \
strain. \
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When the stress (or strain) is large enough, the material softens (becomes \
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A good mental picture is a Hooke's law spring's force versus displacement \
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Cell["We will investigate the nature of stresses and strains. ", "Item",
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An important take-away is that stresses and strains are not so simple as the \
plot above\
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they are not scalars that can be represented by a single axis  \
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Cell["Instead, stresses and strains are rank-2 tensors", "Subsubitem",
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Cell["\<\
It is instructive to think of stress as representing the ratio between two \
vector quantities\
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Cell[TextData[{
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Armed with our definitions for stress and strain as rank-2 tensors\
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An equivalent definition to the compliance would be to introduce the \
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which has the same \[OpenCurlyDoubleQuote]intrinsic\[CloseCurlyDoubleQuote] \
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 A strained crystal does work, in-fact we\[CloseCurlyQuote]ve already seen \
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So far, we\[CloseCurlyQuote]ve seen how physical properties are tensors which \
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And we\[CloseCurlyQuote]ve derived the most general symbolic tensors for some \
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The next step is to see how crystal symmetry constrains these independent \
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This is summarized nicely by Neumann's principle, which states that:\
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\"The symmetry elements of any physical property of a crystal must include \
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This therefore enforces \[OpenCurlyDoubleQuote]crystal symmetry\
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As an example, let\[CloseCurlyQuote]s suppose we are looking at a material \
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And rotate the rank-2 permeability tensor we derived above using our \
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