As noted in Section \(3.15\), for single crystals of some substances, the physical properties are anisotropic that is, they depend on crystallographic direction. One such property is the modulus of elasticity. For cubic single crystals, the modulus of elasticity in a general \([u v w]\) direction, \(E_{\text {uww }}\) is described by the relationship $$ \begin{aligned} \frac{1}{E_{u v w}}=& \frac{1}{E_{(100)}}-3\left(\frac{1}{E_{(100)}}-\frac{1}{E_{\\{111)}}\right) \\\ &\left(\alpha^{2} \beta^{2}+\beta^{2} \gamma^{2}+\gamma^{2} \alpha^{2}\right) \end{aligned} $$ where \(E_{\\{100)}\) and \(E_{\langle 111\rangle}\) are the moduli of elasticity in the \([100]\) and \([111]\) directions, respectively; \(\alpha, \beta\), and \(\gamma\) are the cosines of the angles between \([u v w]\) and the respective [100], [010], and [001] directions. Verify that the \(E_{\\{110\rangle}\) values for aluminum, copper, and iron in Table \(3.4\) are correct.

Understanding anisotropic physical properties is crucial in many fields of science and engineering, especially when dealing with materials like crystals. Anisotropy refers to the directional dependence of a material's physical properties, meaning these properties change depending on the orientation of measurement. This is in contrast to isotropic materials, where properties like elasticity, thermal conductivity, and strength are the same in all directions. Anisotropy arises from the molecular or atomic structure of a material, and it's particularly noticeable in single crystals. In single crystals, the orientation of atoms in the lattice structure can greatly influence mechanical properties such as the modulus of elasticity—an indicator of the material's ability to resist deformation when a force is applied.

In practical terms, engineers and material scientists must consider anisotropic properties when designing components to ensure they can withstand forces from various directions. When measuring such properties in a material, the direction along which the measurement is taken can greatly affect the result, thus requiring careful analysis and understanding of the material's internal structure.

The crystallographic direction is a directional vector that is related to the crystal lattice. It's a key concept in crystallography, the study of the arrangement of atoms in crystalline solids. These directions are not random but are defined based on the crystal's lattice structure and are identified by Miller Indices—a symbolic vector notation that uses integers to represent the orientation of a crystal plane or direction. For instance, the directions [100], [010], and [001] represent the fundamental axes in a cubic crystal system, which are perpendicular to each other.

Knowing the crystallographic direction is essential when discussing properties such as modulus of elasticity in crystals. Depending on the direction in which the force is applied, the response of the material may vary dramatically. That's why scientists and engineers need to understand and calculate how different directions within the crystal structure influence its mechanical and physical properties. This concept is also what relates anisotropy to crystallographic directions, as the directional dependence of properties is inherently linked to the material's crystallographic structure.

The single crystal cubic modulus is a term that refers to the modulus of elasticity specific to single-crystal materials with a cubic lattice structure. The modulus of elasticity, also known as Young's modulus, is a measure of a material's stiffness and is represented by 'E'. In isotropic materials, 'E' is uniform in all directions, but in cubic single crystals, it varies with crystallographic direction due to anisotropy. The single crystal cubic modulus along any [uvw] direction, denoted as \(E_{uvw}\), is determined not by a single fixed value but by a relationship that involves the moduli of elasticity in the principal directions, combined with the crystallographic orientation of the measured direction in terms of the directional cosines \(\alpha, \beta, \gamma\).

To give learners a clear understanding, consider how modulus of elasticity in a given direction is determined by both the inherent properties along the primary crystal orientations ([100], [110], [111], etc.) and the specific angle that the direction forms with these axes. The formula provided in the exercise implies that even within a single crystal, the mechanical response is not straightforward but is instead a nuanced outcome dependent on the intricate interplay between directionality and atomic arrangement. Thus, when engineers and materials scientists deal with single crystal components, calculations like the ones illustrated in the problem ensure that their designs are well-suited to the material's unique directional properties.