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Crystals are solid material whose constituents, such as atoms, molecules or ions, are *arranged in a highly ordered microscopic pattern*, a crystal lattice that extends with regularity in all directions. Use for all crystallography and ordered structure topics.
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$c/a$ ratio for an ideal hexagonal close-packed (HCP) structure
The angle between $a_1$ and $a_2$ should be $2 \pi/3$, not $\pi/3$. Your equation is then
$$\left(\frac23 a\right)^2 + \left(\frac13 a\right)^2 + \left(\frac12 c\right)^2 + 2\left(\frac23 \frac13 a^2\ …