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We know that two parallel planes have the same Miller indices, this implies that we can have an infinite number of parallel planes close to one another all of which have the same Miller indices. But it's claimed that

the interplanar distance between two adjacent planes is given as $$d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}}$$.

I can't understand what adjacent means here? to me the next parallel plane to any plane is infinitely close to it, hence the $$d_{h k l }=0$$

Can someone please explain what exactly is interplanar spacing? Thank you.

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    $\begingroup$ Take a simple cubic lattice, look along one cube-side direction, lets call it z, or (001), so h=0, k=0, and l=1. The planes of atoms are not infinitely close together. They are one cube side apart. $\endgroup$ – Jon Custer Nov 23 '20 at 14:46
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    $\begingroup$ But there aren’t planes of atoms there. $\endgroup$ – Jon Custer Nov 23 '20 at 15:02
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    $\begingroup$ Is it that the Miller indices of a plane are only defined only for planes passing via actual lattice points? I didn't find this addition of definition on the web or my text book. $\endgroup$ – Kashmiri Nov 23 '20 at 15:06
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    $\begingroup$ That is the point, yes. $\endgroup$ – Jon Custer Nov 23 '20 at 15:06
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    $\begingroup$ Oh Thank you. Could you copy paste your comment as an answer, I'd be glad to give a 10+ :). $\endgroup$ – Kashmiri Nov 23 '20 at 15:07
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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 $(\bar{10}\;20\;0);(10\;23\;20)$ etc. Atoms do not need to be on a lattice plane to add to the diffracted signal. The structural information is in the intensity of the diffracted spots not their position. (2 2 4) is parallel to (1 1 2) etc. as shown in the figure. In the general case it is usual to use vectors to find the perpendicular distance using formulas for finding a point at a given distance from a plane.

(Edit. I added (2 2 4) to indicate that it is a different plane but parallel to (1 1 2), in crystallography it is conventional to make all parallel planes have the same Miller indices, (h k l) (as they all scatter in the same way) by removing any common factor. )

planes

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  • $\begingroup$ Thank you, but why does the yellow plane have indices $(224)$ and not $(112)$ ? As far I've read I'd go on like this for the yellow plane : take reciprocal of the intercepts and I'd have $1/2,1/2,1$ and now reduce to lowest integers by multiplying by 2 I'd get $1 1 2$ which is incorrect as you've shown the indices to be $2 2 4$. $\endgroup$ – Kashmiri Nov 25 '20 at 3:50
  • $\begingroup$ I've used your image with credits to you, I hope you are okay with that, here physics.stackexchange.com/questions/596123/… $\endgroup$ – Kashmiri Nov 25 '20 at 4:34
  • $\begingroup$ yes, you are correct, these are conventionally given the same indices as the planes are parallel, I put 224 just to indicate the difference, then the indices are conventionally reduced to remove any common factor. $\endgroup$ – porphyrin Nov 25 '20 at 8:39
  • $\begingroup$ @porphyrin For looking at the geometry of the planes, removing a common factor from the Miller index makes sense. For indexing diffraction data and for Fourier transforms, you have to keep the higher-order indices intact. $\endgroup$ – Karsten Theis Nov 25 '20 at 15:05
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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 indices have no common divisor), the plane closest to a given origin will go through a/h, b/k and c/l. There will also be an plane through the origin, and the distance between the two will be the interplanar spacing.

enter image description here

In the figure, the plane through the origin is not shown. Instead, a second parallel plane (through 1, 1, 2) is shown. You might notice that the distance from the origin to the first plane is the same as the distance between the first and second plane.

For reflections of higher order, there are additional planes in between the planes of the corresponding first order reflection. The (4 4 2) reflections, for example, would have twice as many planes, with half the interplanar distance.

In a diffraction experiment, this corresponds to a larger Bragg angle. In using data from a diffraction experiment, these higher order reflections correspond to higher resolution.

[...]But it's claimed that [...]

The formula given by the OP is for cubic systems only. For other systems, you get more complicated formulas. For a simpler general treatment, you have to introduce the reciprocal lattice parameters $a^*, b^*, c^*$, and the relevant relation (so-called Laue condition) is:

$$ d^*_{hkl} = h a^* + k b^* + l c^*$$

To obtain $d_{hkl}$, take the reciprocal of the magnitude of $d^*_{hkl}$ .

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