8.5: Zone Folding

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    From LibreTexts, under CC BY-NC-SA 2.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    An important property of the Brillouin Zones is that, because the reciprocal lattice is periodic, there exists for any point outside the first zone a unique reciprocal lattice vector that will translate that point back inside the first zone. Each point in reciprocal space is only unique up to a reciprocal lattice vector. Each Zone contains every single physically distinguishable point, and so they all have the same area (in 2-D) or volume (in 3-D).

    This is easiest to see by example. The illustrations below will show how the first six zones for the 2-D square and hexagonal lattices can be translated or 'folded' back on top of the first zone.

    Example \(\PageIndex{1}\)

    2-D square Zone folding

    clipboard_e4f7563a18fe07ed6a4cd7bd64a518ca3.png

    clipboard_ea978b6cc8caba9fe23f42b4f7c0e358a.png

    clipboard_e611f1dc9157ee2ce2068a12ce6206ac9.png

    clipboard_e9f958b1b112e82b15e36c82c55fee70a.png

    clipboard_e103fb5f286b46624a5b17dd531bb407b.png

    Example \(\PageIndex{1}\)

    2-D hexagonal Zone folding

    clipboard_e542cbfdaf5057e4524dc04be3a554577.png

    clipboard_eb31cbdc3bba83991dd9a3abb35c071c0.png

    clipboard_ee718e1fd0b5ee0b98d4eccbc05593915.png

    clipboard_e4471ee399c23f5cafc081634f0427859.png

    clipboard_e4e107b1fa030aeed1d2bc8536042b459.png

    clipboard_e2b0ee378769cf8a522396991628a179e.png

    This page titled 8.5: Zone Folding was last modified on Thu, 26 Nov 2020 20:32:09 GMT and is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Dissemination of IT for the Promotion of Materials Science (DoITPoMS) via source content that was edited to the style and standards of the LibreTexts platform.