《1.4 倒易点阵与布里渊区(1).pdf》由会员分享，可在线阅读，更多相关《1.4 倒易点阵与布里渊区(1).pdf（38页珍藏版）》请在taowenge.com淘文阁网|工程机械CAD图纸|机械工程制图|CAD装配图下载|SolidWorks_CaTia_CAD_UG_PROE_设计图分享下载上搜索。
1、Chapter 1Formation of CrystalElectronAtomCrystalQuantum MechanicsCrystal DynamicsSemiconductorBindingCrystal StructureBorn-Oppenheimer ApproximationEnergy Band theoryFree electron theoryProfileTodays lectureReciprocal LatticeCrystal DiffractionBrillouin ZoneChapter 1 Formation of Crystal1.1 Quantum
2、Mechanics and atomic structure1.2 Interatomic bonding in solids1.3 Crystal structure and typical crystals1.4 Reciprocal Lattice and Brillouin Zone1.4.1 Reciprocal Lattice1.4.2 Crystal Diffraction1.4.3 Brillouin Zone Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)C
3、rystal diffractionReciprocal latticeBrillouin Zone Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)Crystal diffractionReciprocal latticeBrillouin ZoneImage of Reciprocal lattice1(21)2()12(21)kxkf xkxk (2)()f xf x01()(cossin)2nnnaf xanxbnx4444()sinsin3sin5sin7.357f
4、xxxxx()sinf xxFirst four Fourier approximations for a square waven=1234 Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)14sin(21)21nnxnCrystal diffractionReciprocal latticeBrillouin Zone(T)()nnrr1-1(2)()f xf x01()(cossin)2nnnaf xanxbnx01()daf x x1()cosdnaf xnx x1()
5、sindnbf xnx xiiii1cos(ee)2isin(ee)2nxnxnxnxnxnx i()enxnnf xc-i1()ed2nxncf xxiecosisinxxx0122()(cossin)pppppn xnCxSxaa()()n xan x/20/21()daann x xa/2/222()cosdapapCn xx xaa/2/222()sindapapSn xx xaa2i()epxappn xn2-i01()edpxaapnn xxa Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector
6、space(k-space)Crystal diffractionReciprocal latticeBrillouin Zoneie10 iecosisinxxx(2)()f xf x01()(cossin)2nnnaf xanxbnx01()daf x x1()cosdnaf xnx x1()sindnbf xnx xiiii1cos(ee)2isin(ee)2nxnxnxnxnxnx i()enxnnf xc-i1()ed2nxncf xxiecosisinxxx0122()(cossin)pppppn xnCxSxaa()()n xan x/20/21()daann x xa/2/22
7、2()cosdapapCn xx xaa/2/222()sindapapSn xx xaa2i()epxappn xnie10 iiii1cos(ee)2isin(ee)2nxnxnxnxnxnx iecosisinxxx Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)Crystal diffractionReciprocal latticeBrillouin Zonexix2piaGiiaaaa()e=eappGGn xnnG xG x()()n xan x2i()epxa
8、ppn xn-4/a-2/a 0 2/a 4/a-2a-a 0 a2aiaiamaimRa2iab2pipaGb1234倒易点阵倒易点阵倒易空间倒易空间波矢空间波矢空间 Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)Crystal diffractionReciprocal latticeBrillouin Zone()()nnrRri()eGGnnG rri)iii()ee()GGGGGGnnnenn eG(r RG rG RG rrRr2integersmmG R，1ie
9、G R122331vvvGbbb2ijijb aicosisinxexx123123uuuaaaR()eiGGnnG xx()()nnxax1D3D123123uuuaaaR0(,1,2,3)1ijiji jij 123123111231231 12233223 3()()()2uuuvvvu vu vu vmaaabbba bbbG RaaaiamaimRa2iab2pipaGb12345 Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)Crystal diffraction
10、Reciprocal latticeBrillouin Zone()()nnrRri()eGGnnG rr1(,)23ba a11()()2cv232323baabaaaa 111()22/ccvcv23a baaa23131222vvbaabaa122331vvvGbbb2ijijb a0(,1,2,3)1ijiji jij 32abvv（）1()()abvv23123aaabbb1 2 31 23(h h hhh hG）1 2 31 2 32h h hh h hdG1 2 31 2 31 2 31 2 31 2 311223311()2h h hh h hh h hh h hh h hhh
11、hdGhGGGbbbaOA123SC-SC FCC-BCC BCC-FCC0123 Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)Crystal diffractionReciprocal latticeBrillouin Zone Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)()eiGGnnG xx()()nnxax2iab2pipaGb1DCrysta
12、l diffractionReciprocal latticeBrillouin Zone Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)()()nnrRri()eGGnnG rr112233vvvGbbb2ijijb a0(,1,2,3)1ijiji jij 3D()eiGGnnG xx()()nnxax2iab2pipaGb1DCrystal diffractionReciprocal latticeBrillouin ZoneReal lattice Reciproca
13、l latticeVolumez ay ax a321aaa112122132ayzazxaxyaaa112122132 a-xy+z ax-yz ax+y-zaaaz/2by/2bx/2b321aaay x bz x bz y b232221aaa212223bxyzbxyzbxyzaaa 3/2a3/22a3/24aSCFCCBCCSCBCCFCC Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)Crystal diffractionReciprocal latticeBr
14、illouin Zone2ijijb a0(,1,2,3)1ijiji jij Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)Crystal diffractionReciprocal latticeBrillouin Zone()()nnrRri()eGGnnG rr112233vvvGbbb2ijijb a0(,1,2,3)1ijiji jij 3D()eiGGnnG xx()()nnxax2iab2pipaGb1D32bavv（）1 2 31 2 3v v vv v v
15、G1 2 31 2 32v v vv v vdG12v23baa23131222vvbaabaa32bavv（）1 2 31 2 3v v vv v vG1 2 31 2 32v v vv v vdG()()nnrRri()eGGnnG rr112233vvvGbbb2ijijb a0(,1,2,3)1ijiji jij 1 12233uuuRaaaa1Ra2a3b1b2b3G Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)Crystal diffractionRecipro
16、cal latticeBrillouin Zone0(0,)cos()u xtAt00cos()cos()(),xAtAtcu x t0002cos(2cos()cos()xAtAtxTAtkxc Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)coscosoxc,Crystal diffractionReciprocal latticeBrillouin Zone12222vTccTvTTcTk2 nk00(,)cos()cos()2u x tAtAAAAxAtxk2kkk2
17、iab2ipaG2 nki()(,)etxku x tA Fourier series Reciprocal lattice(space)Reciprocal space&wave-vector space(k-space)Crystal diffractionReciprocal latticeBrillouin Zonea1Ra2a3b1b2b3Gkpk2()2kkEmHow to learn?Basic Concept1Periodic functionFourier series Reciprocal vectorReciprocal latticeWave-vector spaceT
18、he Bragg lawLaue equationEwald structureTrain of Thought2Crystal diffractionReciprocal latticeBrillouin ZoneBrillouin ZoneChapter 1 Formation of Crystal1.1 Quantum Mechanics and atomic structure1.2 Interatomic bonding in solids1.3 Crystal structure and typical crystals1.4 Reciprocal Lattice and Bril
19、louin Zone1.4.1 Reciprocal Lattice1.4.2 Crystal Diffraction1.4.3 Brillouin Zone The Bragg law Laue equation Ewald structureElectron beamCrystalCrystal diffractionReciprocal latticeBrillouin Zone The Bragg law Laue equation Ewald structureWilliam Lawrence Bragg(1890-1971)Sir William Henry Bragg(1862-
20、1942)Crystal diffractionReciprocal latticeBrillouin ZoneNobel Prize 19152dsin=nCrystal diffractionReciprocal latticeBrillouin Zone The Bragg law Laue equation Ewald structureA.Piccard,E.Henriot,P.Ehrenfest,E.Herzen,Th.De Donder,E.Schrdinger,J.E.Verschaffelt,W.Pauli,W.Heisenberg,R.H.Fowler,L.Brilloui
21、n;A.P.Debye,M.Knudsen,W.L.Bragg,H.A.Kramers,P.A.M.Dirac,A.H.Compton,L.de Broglie,M.Born,N.Bohr;I.Langmuir,M.Planck,M.Skodowska-Curie,H.A.Lorentz,A.Einstein,P.Langevin,Ch.E.Guye,C.T.R.Wilson,O.W.Richardson The Bragg law Laue equation Ewald structure2dsin=nElectron beamCrystalCrystal diffractionRecipr
22、ocal latticeBrillouin Zone2 nk2nkcoscos()ddmdnn()2 md kk2 nd GkkGkGOA The Bragg law Laue equation Ewald structureCrystal diffractionReciprocal latticeBrillouin ZoneElectron-beamkOkkGRLRd=Lk-k=G The Bragg law Laue equation Ewald structureCrystal diffractionReciprocal latticeBrillouin ZoneElectron-bea
23、mRd=Lk-k=G The Bragg law Laue equation Ewald structureCrystal diffractionReciprocal latticeBrillouin Zone The Bragg law Laue equation Ewald structureElectron beamCrystalCrystal diffractionReciprocal latticeBrillouin ZoneHow to learn?Basic Concept1Periodic functionFourier series Reciprocal vectorReci
24、procal latticeWave-vector spaceThe Bragg lawLaue equationEwald structureTrain of Thought2Crystal diffractionReciprocal latticeBrillouin ZoneBrillouin ZoneChapter 1 Formation of Crystal1.1 Quantum Mechanics and atomic structure1.2 Interatomic bonding in solids1.3 Crystal structure and typical crystal
25、s1.4 Reciprocal Lattice and Brillouin Zone1.4.1 Reciprocal Lattice1.4.2 Crystal Diffraction1.4.3 Brillouin Zone Brillouin Zone 2D Brillouin Zone 3D Brillouin Zone Interface&Crystal diffraction12233The reciprocal lattice point is in the middle of its first BZ.All BZs have the same volume.Every BZ jus
26、t contains one lattice point.Crystal diffractionReciprocal latticeBrillouin Zone Brillouin Zone 2D Brillouin Zone 3D Brillouin Zone Interface&Crystal diffractionCrystal diffractionReciprocal latticeBrillouin ZoneA.Piccard,E.Henriot,P.Ehrenfest,E.Herzen,Th.De Donder,E.Schrdinger,J.E.Verschaffelt,W.Pa
27、uli,W.Heisenberg,R.H.Fowler,L.Brillouin;B.P.Debye,M.Knudsen,W.L.Bragg,H.A.Kramers,P.A.M.Dirac,A.H.Compton,L.de Broglie,M.Born,N.Bohr;I.Langmuir,M.Planck,M.Skodowska-Curie,H.A.Lorentz,A.Einstein,P.Langevin,Ch.E.Guye,C.T.R.Wilson,O.W.Richardson Brillouin Zone 2D Brillouin Zone 3D Brillouin Zone Interf
28、ace&Crystal diffractionCrystal diffractionReciprocal latticeBrillouin ZonePhotograph of the first conference in 1911 at the Hotel Metropole.Seated(L-R):W.Nernst,M.Brillouin,E.Solvay,H.Lorentz,E.Warburg,J.Perrin,W.Wien,M.Skodowska-Curie,and H.Poincar.Standing(L-R):R.Goldschmidt,M.Planck,H.Rubens,A.So
29、mmerfeld,F.Lindemann,M.de Broglie,M.Knudsen,F.Hasen hrl,G.Hostelet,E.Herzen,J.H.Jeans,E.Rutherford,H.Kamerlingh Onnes,A.Einstein and P.Langevin.Brillouin Zone 2D Brillouin Zone 3D Brillouin Zone Interface&Crystal diffractionCrystal diffractionReciprocal latticeBrillouin ZoneGdk221222sin222sin2 sinGG
30、GkGGddGkk GBragg Plane Brillouin Zone 2D Brillouin Zone 3D Brillouin Zone Interface&Crystal diffractionCrystal diffractionReciprocal latticeBrillouin ZoneAll wave vectors k which tips are on the boundary of BZ can be Bragg reflected.Only waves whose wave vector drawn from the origin of k-space termi
31、nates on the boundary of BZ will satisfy the condition for diffraction.Basic Concept3Train of Thought1Conclusions2Math and physical Details4How to learn?Crystal diffractionReciprocal latticeBrillouin ZoneSummary&Basic ConceptsBasic Concept1Periodic functionFourier series Reciprocal vectorReciprocal latticeWave-vector spaceThe Bragg lawLaue equationEwald structureTrain of Thought2Conclusions3Crystal diffractionReciprocal latticeBrillouin ZoneBrillouin ZoneWe knowandwe see.Fourier series SummaryThank you for your attention!Crystal diffractionReciprocal latticeBrillouin Zone