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Since a comment does not allow the inclusion of a picture, perhaps this way is helpful: Have a look at the following picture, derived from an illustration about the Miller Indices on wikipedia. It ...

This answer uses vector algebra and the so-called reciprocal lattice to avoid trigonometry. There are probably other, more geometric ways to connect the Laue conditions to the lattice spacing. $d_{...

The lattice planes are a property of the lattice based on dividing up the unit cell. The cell can be divided up indefinitely to give all the Miller indices for example $(\bar{10}\;20\;0);(10\;23\;20)$ ...

Your adjacent plane isn't. Worth a thousand words, they say.

Because of translational symmetry, every plane has a set of symmetry-equivalent parallel planes (infinitely many). To get the Miller indexes, you have to find the plane closest to the origin that does ...

There are two equivalent ways to define the meaning of the Miller indices: via a point in the reciprocal lattice, or as the inverse intercepts along the lattice vectors. The reflecting plane are ...

I can't understand what adjacent means here? For first order reflections, you only consider parallel planes that contain crystal lattice points. If you have a first order reflection (i.e. the Miller ...

When Miller indices are different only in a common factor, the planes are in the same orientation. The higher order reflections share some planes with the lower order reflections. In the case ...

Let's say your structure contains a short piece of helical DNA, and it (the helix axis) starts at point P1 and goes to point P2. You want to know the Miller index associated with that direction. So ...

In answer to you first question, no if hkl differ, yes if they do not. I try to explain below. In a crystal the unit cell defines the repeating unit. Inside the unit cell the atoms are arranged as ...