物性物理学I_2.pptx

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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

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

php Daniel Shechtman Ho@Mg@ZnIalloy h"ps://ja.wikipedia.

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1 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 Daniel Shechtman h"ps://ja.wikipedia.org/wiki/%e6%ba h"ps://news.engineering.iastate.edu/i chemistry/ 強靭高抵抗

2 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)

3 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

$n C n θ = 2π n n = 1, 2, 3, 4, 6 IcrystalIla\ce IcrystalIfamily I Monoclinic I Oblique a b, θ 90 I Orthorhombic I$

4 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

5 t n = n 1a + n2b + n3c ( n i = 0, ±1, ± 2,... )

6 π π π π 0.74

a3 x a2 a1 h"p://www.askwillonline.com/2013/10/the@structure@of@materials.

7 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:// 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

8 R hkl R hkl = h a 1 + k a 2 + l a 3 [ hkl] [ 100] 100 R m 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

9 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 fcc

10 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

11 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 = 2π λ CharacterisLcIradiaLonI X BremsstrahlungI X h"ps://miac.unibas.

12 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 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

13 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

14 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

15 Photo51 超有名な単結晶回折の例 IAderIRaymondIGoslingI(RosalindIFranklin) h"ps://en.wikipedia.org/wiki/photo_51 ら線単一ヘリックス構造の回折パターン h"ps://

2) 粉末X線回折 θ-２θ法 (004)I ϕiscans (404)I (404)I h"ps://www.intechopen.

ﬂuorine@free@oxalate@route@for@the@ chemical@solulon@deposilon@of@

onimgal2o4i(001)isubstrate.i h"ps://www.researchgate.

of@a@240nm@thick@nico2o4@ﬁlm@grown@at@350c@on 粉末X線回折法 h"p://pd.chem.ucl.ac.

16 2) 粉末X線回折 θ-２θ法 (004)I ϕiscans (404)I (404)I h"ps:// 3)Debye-Scherrer法 AI240 nmithickinico2o4iﬁlmigrowniati350 CI onimgal2o4i(001)isubstrate.i h"ps:// 粉末X線回折法 h"p://pd.chem.ucl.ac.uk/pdnn/inst2/linearea.htm h"p:// kap_5/vlu/kristallstrukturanalyse.vlu/page/vsc/de/ch/11/aac/vorlesung/ kap_5/kap5_7/kap57_2.vscml.html

17 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

18 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) u C.IKi"el,IIntroducLonItoISolidIStateIPhysicsI(Wiley)

19 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 )

20 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

21 C n (n = 2, 3, 4, 6) xy ( x, y, z) x, y, z ( ) ( x, y, z) x, y, z ( ) n

<4D F736F F F696E74202D208DDE97BF955D89BF8A778AEE F8BF38AD48C512E B93C782DD8EE682E890EA97705D>

<4D F736F F F696E74202D208DDE97BF955D89BF8A778AEE F8BF38AD48C512E B93C782DD8EE682E890EA97705D> 材料科学の枠組み基礎編材料評価学基礎格子晶系空間群 ( 対称性 ) いろいろな結晶の構造結晶と逆格子 ( 回折結晶学 ) X 線と結晶応用編電子顕微鏡放射光中性子線 1 格子 (Lattice) 3 次元の周期的な点の配列点のまわりの環境が同一である, 空間の点の無限の配列 c R n1a n2b n3c 格子定数 (Lattice parameters a or Lattice