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If I have a crystal, say (001)-oriented ABO3, what does the 001 imply, from an experimental perspective? I understand that the 001 is referring to a plane as described by Miller indices, but what is it about this plane that is relevant? Is it that this plane is the same plane as the surface of the sample?

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When someone says (hkl)-oriented anything, they almost always mean the surface normal of the sample is parallel to (hkl). So Si-(111) would mean silicon oriented so the surface normal is along the (111) direction.

If you have a layered material, then having a (001) orientation means the c-axis is perpendicular to the surface of the sample. This is an extremely common orientation for layered crystals for example.

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It depends on context. In a surface science experiment, it would be the surface. In experiments on bulk properties, the surface could be irrelevant and rough, but one would measure some property along a cubic axis (or maybe paricularly along the c-axis), like velocity of sound or conductivity, etc.

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