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8.5: Zone Folding

  • Page ID
    7832
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    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

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    clipboard_e103fb5f286b46624a5b17dd531bb407b.png

    Example \(\PageIndex{1}\)

    2-D hexagonal Zone folding

    clipboard_e542cbfdaf5057e4524dc04be3a554577.png

    clipboard_eb31cbdc3bba83991dd9a3abb35c071c0.png

    clipboard_ee718e1fd0b5ee0b98d4eccbc05593915.png

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    clipboard_e4e107b1fa030aeed1d2bc8536042b459.png

    clipboard_e2b0ee378769cf8a522396991628a179e.png


    8.5: Zone Folding is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Dissemination of IT for the Promotion of Materials Science (DoITPoMS).

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