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Wigner-Seitz+reciprocal

# Wigner-Seitz+reciprocal - {it EH51 c Will 5 IN Will or ill...

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Unformatted text preview: {it}? EH51 c' Will 5' IN Will or ill l1! Elli} 1 ll: I‘ll: etﬂ tel Flame Lilli The mall‘ﬂCl-lﬂll ei'a reciprocal lattice: {a} the a—c section of the unit cell in a monoclinic (ml-i} direct lattice: tbl reclpmai lattice axes lie perpendi- cular la the end facet cf the direct cell: in} rcciptucal lattice points are spaced o' = ”de anti t:L = IfdmI: [all the lattice plane is ottrttpleted by extending the lattice: {e} the reciprocal lattice is mmpletctl by adding lay-ere above and below the ﬁrst plane. The b—asis is normal to 3 anti c. anti an the It“ attic is parallel to l: anti normal to the sec- tion containing 11* and c“. drawn. The length Lattice Direct lattice plane oblique mp: Wigner-Sch: cell Figure 2.1: cubic ﬁ-d F lattice: Wigner—Selle cell is a tit-entitle indecultedt‘ort. ﬁgure 19d. r lattice: Wigner-Sail: cell iv. a tmucatetl acrahrtlreu. Fugue 2.5“: of the. h'" axis is equal to iﬂJ. The reciprocal lattice layer containing the DH] point is then identical to Figure little. but is stacked a dis- tnttce of h* vertically below it1 and the layer containing the till} point is. 1t.'ei-tit::tli:-,r above it+ {Figure line}. Either layers then follow in the same way. For some purposes. it is convenient to multiplyI the length of the reciprocal axes h}! a constant. Thus. physics texts frequently use a reciprocal lattice spacing 2a- tintes that given aims-e. that it: ti:I = Iﬁfdjm, b" = 2Fflfplp, t5“ = Etrftlnul. Crystallographic-rs often are a reciprocal lattice scale multiplied h}! A, the wavelength of the radiation used to obtain a diffraction lttltttterrt+ so that: ﬂ' = Millet. b' = iNatttt f” = Ill-twill!- As with two-dimcncitntnl lattices. the pro- cedure required in construct the {primitive} Wigner-Selle cell in the direct lattice yields a cell called the ﬁrst Brilleuin zone when applied to the reciprocal lattice. The lattice that is reciprocal to a real space facocentred cubic F lattice is a hotly-centred cubic .i' lattice. The {primitive} Wigner-Sela cell of the cubic hotly-centred f lattice1 {Figure 3.9a}. a truncated octahedron. {Figure 2.9m. ia there-fete identical in shape to the {primitive} ﬁrst Brillouin zone of a face-centred cubic F lattice. in the same way. the lattice that is reciprocal to a real space hodyrccntrcd cubic ." lattice is a faccrccntrcti cubic F lattice. The Wignerrﬁeitz cell of the real space fawecrlted cubic F lattice. {Flaunt 2.9:). a regular titombic dooecaheriron. (figure can. is identical in shape to the ﬁrst Briilauin tone at" a hotly-centred centred cubic l lattice. ('l'ahle 2.3}. Reciprocal lattice oblique mp: 15' Briliouin zone Figure 13!: i lattice: l“ Brillauin acne is a truncated cctahedron. Figure 2.5”: F lattice: I“ Erillooin zone is a rhomhie dcdccahcdron. Figure 2.94 ...
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