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When I hear someone talking about a (100) plane or a (111) plane or an (hkl) in general, my first thought is, is the system cubic. The reason I think this is because I tend NOT to think of the planes associated with their miller indices, but the vectors that are normal to planes. In cubic systems, [hkl] vectors are normal to their corresponding (hkl) planes. Hence, direct space (hkl) planes are parallel to their reciprocal space counterparts. In cubic systems, these relationships are true.

I think this is a bad way to think of the relationship between Direct and Reciprocal space, as not all systems are cubic and I fumble when dealing with other sorts of lattices. Does anyone have some advice of a more fundamental way to view the relationship between the two spaces? Preferably, one that is quite intuitive. I realize they are Fourier Transforms of one another. I am sorry if this is a silly question.

For example, in an HCP system, how should I envision the (hkl) plane? It is not normal to the [hkl] vector? is there no simple intuitive way to see such things? I can certainly do the math I suppose.

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  • $\begingroup$ My suggestion: Get an image of the direct lattice for cubic, hexagonal etc. and get the reciprocal image from x-ray diffraction and try and see if you can find an intuitive visual relationship between the two.. I have tried but I did not find any visual correspondence between the two.. So I just accept the fact that one is the Fourier inversion of the other.. I find it difficult to intuitively visualize even 1-D Fourier inverses let alone 2-D $\endgroup$
    – mcodesmart
    Commented Sep 29, 2013 at 22:05
  • $\begingroup$ There is the notion of dual basis. The dual basis $\vec e^j$ of a basis $\vec e_i$ is such that $\vec e^j.\vec e_i = \delta^j_i$ $\endgroup$
    – Trimok
    Commented Sep 30, 2013 at 12:36

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The reciprocal vector [hkl] is perpendicular to the planes described by the Miller indices (hkl). This is a general relationship, not specific to cubic systems. So yes, it works for hexagonal too. Same is true for the relationship that gives the spacing of the planes as the inverse of the magnitude of the corresponding reciprocal vector (up to a factor of $ 2 \pi $). If you look up the proofs of these relationships you will see that no assumption about the crystal symmetry is made.

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