QUB XRD Course. The crystalline state. The Crystalline State

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1 QUB XRD Course Introduction to Crystllogrphy 1 The crystlline stte Mtter Gseous Stte Solid stte Liquid Stte Amorphous (disordered) Crystlline (ordered) 2 The Crystlline Stte A crystl is constructed by the infinite repetition in spce of identicl building blocks. b Grid system + Building block Crystl 3

2 The Crystlline Stte Building block Grid system describes rrngement of groups of toms describes how building block repet in spce The lttice prmeters describe the infinite repetition unit. A volume element whose edges re successive grid lines. 4 The Crystlline Stte Lttice prmeters c b b c - sides αβγ- ngles 5 The 14 Brvis Lttices CubicP CubicI CubicF TetrgonlP TetrgonlI MonoclinicP MonoclinicC Triclinic TrigonlR Trigonl & Hexgonl P OrthorhombicP OrthorhombicC OrthorhombicI OrthorhombicF 6

3 The 14 Brvis Lttices All crystl structures must belong to one of the 14 spce, or Brvis, lttices: System Number of lttices Lttice symbols Restrictions on conventionl cell xes nd ngles Triclinic 1 P b c α β αβγ Monoclinic 2 P, C b c α = γ = 90º β Orthorhombic 4 P, C, I, F b c α = β = γ = 90º 7 The 14 Brvis Lttices System Number of lttices Lttice symbols Restrictions on conventionl cell xes nd ngles Tetrgonl 2 P, I = b c α = β = γ = 90º Cubic 3 P I or bcc F or fcc = b = c α = β = γ = 90º Trigonl 1 R = b = c α = β = γ < 120, 90º Hexgonl 1 P, C, I, F = b c α = β = 90º γ = Point Groups 2 fold rottion xis 9

4 Point Groups mirror plne 10 Point groups Ech rower is relted to the next by combintion of trnsltion nd reflection 11 Spce Groups 32 crystllogrphic point groups + 14 Brvis lttices (7 crystl clsses) 230 spce groups 12

5 Spce group Interntionl Tbles IUCR 13 Primitive cubic 14 Body Centred Cubic 15

6 Fce Centred Cubic Fce Centred Body Centred Primitive 17 A Simple Crystl Structure CsCl - Cesium Chloride Cs + 18

7 A Simple Crystl Structure Building block = 1 Cs ion + 1 Cl ion Grid system = primitive cube one Cs ion t ech corner site (0,0,0) one Cl ion in the center of the cube (½, ½, ½) Note: 8 corner sites - ech corner site shred by 8 cubes. One Cs ion + one Cl ion per cube. 19 A (not quite so) Simple Crystl Structure NCl - Sodium Chloride 20 A (not quite so) Simple Crystl Structure Lttice = FCC = Fce Centered Cubic Cl N FCC - toms t (0, 0, 0) (½, ½, ½) (½, ½,0) (0, 0, ½) (½,0, ½) (0, ½, 0) (0, ½, ½) (½, 0, 0) 21

8 Diffrction Diffrction is n interference phenomenon Wves interct with n object Simple exmple: opticl diffrction 22 Electromgnetic Wves/ Scttering k k o 23 Superposition of Wves A1 A1 A2 A2 A1+A2 A1+A2 24

9 Origin of Diffrction Phenomen 25 Diffrction Light of wvelength incident on two slits d prt: first mximum occurs when wves from ech slit re exctly in phse. i.e. when difference in pth-length is exctly = x d x φ 1st mximum φ = d sinφ x = d sinφ = 26 Diffrction If, insted of two slits, we hve lrge number of slits then the position of the first mximum remins the sme but the diffrction pttern becomes much shrper. Screen θ two slits mny slits 27

10 Diffrction In this experiment ech slit sctters the light nd becomes point source. Light 28 Diffrction If we replce the slit by n tom nd the light by X-rys, then the tom sctters the X-rys nd cts s point source. X-rys 29 Diffrction Plcing n tom on lttice (i.e. crystl) gives regulr rry of sctters. The (X-ry) wves scttered by these toms cn interfere in the sme wy s the (light) wves from the rry sctters in diffrction grting. 30

11 Diffrction A' A" A B C C' C" 2 3 θ B' B" d First order Second order Third order dsinθ dsinθ A D B A d θ C θ n = 2d sinθ B 31 Diffrction The condition for ll scttered wves to interfere constructively: = d sinθ + d sinθ = 2d sinθ (Brgg s lw) In 3-d crystl the toms re rrnged in plnes. The incident nd scttered bem directions must be coplnr with the norml to the plne (N). 32 Diffrction N N bisects incident nd reflected bems θ θ ngle of incidence = ngle of reflection (symmetricl) This is clled the Brgg Reflection. n= 2d sinθ is known (the wvelength of the X-ry bem) θ is mesured (the reflection ngle) d is clculted (the spcing between the lttice plnes) 33

12 Diffrction For crystl the bem is reflected only when the crystl is correctly oriented. N 2θ No reflection N 2θ Reflection 34 Lttice Plnes nd Miller Indices The lttice is described by 3 xes:, b, c. Ech plne must intercept these xes. The plne intercepts the xes t ¼, ½b, c. c/l c (hkl) b/k /h () b 3Å c (???) 8Å 2Å b 1Å 0 1Å 2Å 3Å 4Å (b) 35 Lttice Plnes To find the Miller Indices: Find intercepts on, b, c xes ¼ ½ 1 Tke reciprocls (hkl) = (421) All lttice plnes cn be indexed in the sme wy. 36

13 Miller Indices The plne is prllel to the xis it crosses t b c The plne is prllel to the b xis it crosses b t The plne crosses c t 1 The Miller Indices of this plne is (0 0 1) 37 Miller Indices The plne crosses t 1 b c The plne crosses b t 1 The plne crosses c t 1 The Miller Indices of this plne is (1 1 1) 38 Miller Indices The plne crosses t b c The plne crosses b t ½ The plne crosses c t 1 The Miller Indices of this plne is (0 2 1) 39

14 Lttice Plnes c d100 b d200 (100) (200) (110) (110) (111) (102) 40 Lttice Plnes Rel crystl structure CsCl = 4.11Å, =1.54 Clculte: d (hkl) nd θ hkl for the following (hkl) hkl d θ 2θ Lttice Plnes Rel crystl structure CsCl = 4.11Å, = 1.54 Clculte: d (hkl) nd θ hkl for the following (hkl) hkl d θ 2θ

15 Intensities Wht influences the intensities of Brgg reflections? Exmple: CsCl Cs + Cl - 43 Wvefronts A θ d (100) Cs + Cl - B 44 Wvefronts The digrm shows the (100) plnes scttering in phse. f Z The reflecting power of toms (normlly clled the tomic scttering fctor) is relted to the number of electrons in the tom. Cs + = 54 electrons Cl - = 18 electrons the reflected bem from Cs + toms hs n mplitude 3x lrger thn the bem from Cl - toms sin θ/ Difference in phse X-ry bem Atom 45

16 Wvefronts A θ d (100) Cs + Cl - B 46 Wvefronts Look t the wve front A - B of the reflected bem Bems from Cl - toms (on plnes d 100 prt) re in phse. Bems from Cs + toms (lso on plnes d 100 prt) re in phse. But, since Cs + plnes re exctly hlf-wy between Cl - plnes, bems from Cs + nd Cl - plnes re exctly out of phse. 47 Wvefronts Amplitude of diffrcted bem A(54-18) = A(36) Intensity = I 100 A2 (36) 2 = 1296A 2 (A is some constnt) Wek reflection 48

17 Reflection The (200) plnes 49 Reflection This time ll toms sctter in-phse. Amplitude of diffrcted bem A ( ) = A72 Intensity of diffrcted bem I (200) A 2 x 72 2 Strong reflection = 5184A 2 50 Reflection The (110) plnes Cs + nd Cl - ions ll lie in the (110) plnes Cs + nd Cl - sctter in phse 51

18 Reflection 4.11 d = = = 2.91Å (110) 2 2 o θ 110 = I A ( ) = 5184A (110) 2 2 Strong reflection 52 Reflection Now look the (111) plnes Cl - ions lie in (111) plnes nd re d(111) prt Cl - ions sctter in phse Cs + ions lie mid-wy between Cl - plnes Cs + ions sctter out of phse 53 Reflection 4.11 d = = = 2.373Å (111) 3 3 o θ (111) = I A (54 18) = 1296A (111) 2 2 Wek reflection 54

19 Reflection The (222) reflection must be strong: I(222) = A 2 (54+18) 2 = 5184A 2 d(222) = 1.187Å θ(222) = Reflection To summrize: hkl d 2θ I Å 21.6 wek Å strong Å wek Å strong from lttice from building block Crystl structure 56 CsCl Counts 1) CsCl º2Thet 57

20 Form Fctor The form fctor reduces intensities of higher ngle Brgg reflections: Temperture fctor Lorentz-polriztion fctor Instrumentl fctors Smple fctors 58 A (not quite so) Simple Crystl Structure NCl - Sodium Chloride 59 Reflection NCl (FCC) Cl - N + Top view d (100) d (200) d(220) d (110) 60

21 Reflection (100) bsent completely (200) strong (110) bsent completely (220) strong 61 Reflection The (111) plnes 62 Reflection Cl - toms lie in (111) plnes N + toms lie in between I A 2 (18-8) 2 = 100A 2 = (111) is quite wek Sctter out of phse Cl - toms lie in (222) plnes N + toms lie in (222) plnes I A 2 (18+8) 2 = 262A 2 = (222) is quite strong Sctter in phse 63

22 Summry The diffrction pttern is like finger print of the crystl structure: d vlues reflect the unit cell prmeters ( grid ) intensities reflect the toms/molecules ( building blocks ) 64