Miller indices and FCC structure - non-existent planes

Below is a copy from Kittel's Introduction to Solid State Physics, 8th edition, page 42. It is a picture to illustrate that in FCC structures (loosely said and exactly cited) "no reflections can occur for which the indices are partly even and partly odd.". But how an FCC structure can have a peak named (400)? That translates to a plane that would be at $a/4$ where $a$ is the appropriate cubic cell constant and perpendicular to $x$ direction. There is no such plane in FCC structure.

I must admit that I am having huge trouble understanding the whole Miller indices business because if someone is referring to an $hkl$ reflection, it makes little sense because in order to see anything in the diffraction pattern, and the whole point of Bragg, is that there needs to be a constructive interference of two plane waves each scattered by different perpendicular planes. Another issue I have, if I look at plane that is perpendicular to $x$ axis and that intersects it at some point (other than 0), do I know that there is also a plane perpendicular to it at the origin (intersecting $x$ at 0)?

The answer to the question should answer the first paragraph, the second one is there only to illustrate items in crystallography that I do not understand.

1 Answer

The notation is for x-ray diffraction: (400) is the Bragg reflection from (200) planes in second order.

• In that case isn't the notation ambiguous? For example, in the BCC structure, the reflection from the "middle" plane made up by body centered atoms should be denoted (200) reflection but at the same time (200) would also be the second order reflection from the (100) plane. What am I missing? May 2, 2018 at 19:23
• The 100 reflections are forbidden in fcc and bcc (because reflection from the "middle" plane is 180 degrees out of phase with a cubic plane). The notation is not ambiguous, each peak can be assigned.
– user137289
May 2, 2018 at 19:28
• but there is the issue, (100) is the index of the cubic plane not of the middle plane from the atoms in the center of the cube (that plane should be indexed (200)). So while I agree that for hkl=100 the structure factor should disappear, at the same time (100) indexes the face of the cube not the middle plane... May 2, 2018 at 19:40
• But (200) denotes reflection from the set of planes with distances of half the cubic lattice parameter. Not just half of those planes.
– user137289
May 2, 2018 at 21:23