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The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1. https://en.m.wikipedia.org/wiki/Miller_index

Does this mean, that parallel planes are generally equivalent, particularly does $(200)$ for example even exist or is this a wrong notation of $(100)$?

Also what is the correct notation for $(\bar{1}00)$? Since $(100)$ is parallel to $(\bar{1}00)$ I think $(100)$ is correct. Is there kind of a rule or a convention to invert the signs to obtain the maximum positive indices?

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I like Pister's fold-up crystal for this: http://www-bsac.eecs.berkeley.edu/~pister/crystal.pdf

One sees then that $(\bar{1}00)$ is on the other side of $(100)$ and that those are perpendicular to $(010)$. In a cubic crystal like silicon all six cubic planes $\{100\}$ are equivalent.

In non-centrosymmetric crystals opposite faces can be different. For example, in GaAs, one could say that the $(111)$ surface is gallium-terminated and $(\bar{1} \bar{1} \bar{1})$ is arsenic-terminated.

(200) etc is a notation for x-ray diffraction.

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