2 物質の構造 単結晶 秩序 から 非晶質 乱れ まで 0) 凝縮系物質の形態 morphology polycrystal monocrystal single crystal 準結晶 quasicrystal 1) 結晶の構造 amorphous ー周期性と並進対称性ー 単結晶 でも 非晶質 でもない固体内秩序 h"p://shinbun.fan@miyagi.jp/arlcle/arlcle_20120223.php Daniel Shechtman Ho@Mg@ZnIalloy h"ps://ja.wikipedia.org/wiki/%e6%ba %96%E7%B5%90%E6%99%B6#/media/File:Ho@Mg@ZnQuasicrystal.jpg h"ps://news.engineering.iastate.edu/i 2011/10/05/iowa@state@ames@laboratory@ technion@scienlst@wins@nobel@prize@in@ chemistry/ 強靭 高抵抗
TranslaLon(al)Ivector T = n 1 a + n2 b ( n i = 0, ±1, ± 2,... ) primilveicell basisivector b θ a primilveivector { x 1a + x2b; 0 xi 1} (oblique)
単純(基本単位 胞の選び方 正方格子 (square) b 90 長方格子 (rectangular) b a 面心格子 (centered rectangular) b 基本格子ベクトルの選び方 90 a 90 a 六方格子 (hexagonal) 60
n C n θ = 2π n n = 1, 2, 3, 4, 6 IcrystalIla\ce IcrystalIfamily I Monoclinic I Oblique a b, θ 90 I Orthorhombic I Rectangular I CenteredI rectangular a b, θ = 90 I Hexagonal I Hexagonal a = b, θ = 120 (60 ) I ITetragonal I Square a = b, θ = 90
t n = n 1a + n2b + n3c ( n i = 0, ±1, ± 2,... )
http://www.iue.tuwien.ac.at/phd/karlowatz/node8.html π 6 0.52 3 8 π 0.68 2 6 π 0.74 2 6 π 0.74
基本格子ベクトルの取り方 a2 a3 a1 y z a a1 = ( x + y ) 2 a a2 = ( y + z ) 2 a a3 = ( z + x ) 2 a a1 = ( x + y z ) 2 a a2 = ( x + y + z ) 2 a a3 = ( x y + z ) 2 a1 = ax a2 = ay a3 = az a3 x a2 a1 h"p://www.askwillonline.com/2013/10/the@structure@of@materials.html 積層のちがいと積層欠陥 stacking fault 面心立方格子 (fcc) ABCABCABC A B C 六方細密充填 (hcp) ABABAB 1) 結晶の構造 ー周期性と並進対称性ー
R hkl R hkl = h a 1 + k a 2 + l a 3 [ hkl] [ 100] 100 R m1 00 010 010 R 0m2 0 a 1, a 2, a 3 ( h, k,l Z) [ ] [ 001] 001 R 00m3 h : k : l = 1 m 1 : 1 m 2 : 1 m 3 ( hkl) [ hkl] ( hkl) ( hkl) { hkl} h"ps://commons.wikimedia.org/wiki/file%3amiller_indices_felix_kling.svg
典型的な結晶構造 (1) ダイアモンド構造 せん亜鉛鉱構造 diamond structure zinc-blende structure 1 R = ( a1 + a2 + a3 ) 4 1 R = ( a1 + a2 + a3 ) 4 C, Si, Ge GaN, GaAs, ZnSe, CdTe 1) 結晶の構造 ー周期性と並進対称性ー 典型的な結晶構造 (2) CsCl構造 cesium chloride structure NaCl 構造 sodium chloride structure scc https://lampx.tugraz.at/~hadley/ss2/problems/ti/q1.php fcc
h"p://britneyspears.ac/physics/crystals/wcrystals.htm n( r ) n( r + t ) = n( r ) ( ) = n G G n r e i G r n( r + t ) = n( r ) e i G t t n = n 1 a + n2 b + n3 c G = 1
基本逆格子ベクトル b c A = 2π, B = 2π a b c c a, C = 2π a b c A a = 2π, A b = 0, B a = 0, B b = 2π, C a = 0 C b = 0 A c = 0 B c = 0 C c = 2π G = ha + kb + lc 逆格子ベクトル a b a b c ( h, k, l Z ) ２) 結晶構造の判定法 逆格子点の幾何学的意味 面ABC にOからおろした垂線の足 c C c ON = d b d R a N B ON OR = 2π となる ON 上の点 R! b 逆格子ベクトルの定義再考 O V = a b c = d ( a b sin θ ) C の大きさ 2π A a OR = 2π 2π a b 2π a b d C = = = d V d ( a b sin θ ) 2π a b a b C の向きまで考慮して C= = 2π V c a b ２) 結晶構造の判定法
k k exp i( k k ) r 2πrsinθ λ 2πrsin θ λ r O k k λ 0.1 nm = k r k r = ( k k ) r k r = kr cosϕ = krsinθ k r = k r cos ϕ = k rsin θ @ V = 10kV k = k = k = 2π λ CharacterisLcIradiaLonI X BremsstrahlungI X h"ps://miac.unibas.ch/pmi/01@ BasicsOfXray.html h"ps://physics.stackexchange.com/queslons/ 20385/peaks@on@top@of@bremsstrahlung Cu K α 1.54 Å! Mo K α 0.71 Å
回折条件 Bragg条件 強め合いの条件 k k R = 2π n ( n N ) R = n a + n b + n c は格子ベクトルで R a b c k k = nghkl Laue 条件 逆格子ベクトルによれば k k 4π sin θ k k = λ (hkl) 2π θ Ghkl k k = n d θ k d 2d sin θ = nλ (hkl) Bragg 条件 ( ) ( Ewald 球の考え方 [0 結晶方位を決定] k O Ghkl ) 1) 入射ベクトルを描く 2) 半径 k の球を描く k 3) 球上の逆格子点が散乱方向 C k P
X線スペクトルの影響 特性X線 Ewald球ひとつ 反射点が少ない k O Ghkl 連続X線 Ewald球が沢山 反射点が沢山 C k P X線回折の種類 1) 単結晶X線回折 背面Laue法など SiIcrystalI(111)IbackwardIreflecLon h"p://mullwire.com/index.shtml h"p://minerva.union.edu/jonesc/photos %20ScienLfic.html
Photo51 超有名な単結晶回折の例 IAderIRaymondIGoslingI(RosalindIFranklin) h"ps://en.wikipedia.org/wiki/photo_51 h"ps://jp.pinterest.com/explore/dna@model/ ら線 単一ヘリックス 構造の回折パターン h"ps://www.aps.org/meelngs/march/vpr/2010/imagegallery/franklin.cfm
2) 粉末X線回折 θ-２θ法 (004)I ϕiscans (404)I (404)I h"ps://www.intechopen.com/books/ superconductors@new@developments/a@ fluorine@free@oxalate@route@for@the@ chemical@solulon@deposilon@of@ yba2cu3o7@films 3)Debye-Scherrer法 AI240 nmithickinico2o4ifilmigrowniati350 CI onimgal2o4i(001)isubstrate.i h"ps://www.researchgate.net/figure/ 257951207_fig2_Color@online@a@HR@XRD@2th@th@scan@ of@a@240nm@thick@nico2o4@film@grown@at@350c@on 粉末X線回折法 h"p://pd.chem.ucl.ac.uk/pdnn/inst2/linearea.htm h"p://www.chemgapedia.de/vsengine/vlu/vsc/de/ch/11/aac/vorlesung/ kap_5/vlu/kristallstrukturanalyse.vlu/page/vsc/de/ch/11/aac/vorlesung/ kap_5/kap5_7/kap57_2.vscml.html
散乱条件 全散乱波の振幅 n( r ) の空間積分 原子内の電子密度 A (t ) = dr n(r ) exp( iδk r ) e iω t 散乱波の強度 ここに Δk k k 2 2 I Δk A dr n(r ) exp( iδk r ) ( ) ig r n ( r ) = ng e を入れれば 2 I Δk A ( ) G 位相整合条件 構造因子 k Ghkl 2θ 2 i G Δk r ( n d r G e ) G V (G = Δk ) ) = 0 (G Δk ) i G Δk r ( d r e (structure factor) nhkl k 原子散乱因子 結晶構造因子 1 = dr n(r ) exp( ig r ) Vc 1 ig r ig r = e dr n(r )e Vc i i 結晶構造因子 原子散乱因子 fi = dr n(r ) exp( ig r ) sin ( 2kr sin θ ) = 4π dr n(r ) r 2 2kr sin θ Shkl = fi e i ighkl ri
S = f i e 2πi hn 1+kn 2 +ln 3 i πi ( h+k+l ) { = e = f i i r 1 = ( 0, 0, 0), r 2 = 1 2, 1 2, 1 2 ( ) 0 (h + k + l : odd) 2 f (h + k + l : even) (100) Cs +, I (200) S KCl = f (1+ e πih + e πik + e πil ) 0 (h, k,l : odd) = 4 f (otherwise) KCl (220) (420) (400) (222) S KBr = f i e 2πi ( hn 1+kn 2 +ln 3 ) i 0 (h, k, l : odd, even mixed) = 4 f (otherwise) KBr (200) Figure 17 Comparison of x-ray reflections from KCl and KBr powders. In KCl the numbers of electrons of K and Cl ions are equal. The scattering amplitudes f(k ) and f(cl ) are almost exactly equal, so that the crystal looks to x-rays as if it were a monatomic simple cubic lattice of lattice constant a/2. Only even integers occur in the reflection indices when these are based on a cubic lattice of lattice constant a. In KBr the form factor of Br is quite different to that of K, and all reflections of the fcc lattice are present. (Courtesy of R. van Nordstrand.) (420) (400) (331) (222) (311) (220) (111) 80 70 60 50 40 30 20 2u C.IKi"el,IIntroducLonItoISolidIStateIPhysicsI(Wiley)
点群 有限サイズ と空間群 無限サイズ 回転対称性 360 o 回転軸の周りに 回して重なる n C180 (2) C90 (4) ( x, y, z ) ( x, y, z ) ( x, y, z ) ( y, x, z ) C120 (3) C60 (6) ( x, y, z ) ( R120 (x, y), z ) 回反対称性 ( x, y, z ) ( R60 (x, y), z ) 回転操作の後 対称点に関して反転 1回回反対称 反転) ２回回反対称 =鏡映) (1) ( x, y, z ) ( x, y, z ) (2) ( x, y, z ) ( x, y, z ) 4回回反対称 (4) ( x, y, z ) ( y, x, z )
n m 360 o n 1 n x 4 1 ( x, y, z) ( x +1 2, y, z) z ( x, y, z) ( y, x, z +1 4) ( x, y, z) ( x + a, y, z) ( ) = ( a b) c a b c e a e = e a = a a a 1 = a 1 a = e a 1
C n (n = 2, 3, 4, 6) xy ( x, y, z) x, y, z ( ) ( x, y, z) x, y, z ( ) n