How do atoms assemble into solid structures? How does the density of a material depend on its structure?

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1 제 3 장 : 결정질고체의구조 ISSUES TO ADDRESS... How do atoms assemble into solid structures? How does the density of a material depend on its structure? When do material properties vary with the sample (i.e., part) orientation? Chapter 3-1

2 학습목표... 원자 / 분자구조에서결정질과비결정질재료의차이점분석 면심입방격자, 체심입방격자, 육방조밀격자의제도 면심입방격자와체심입방격자구조에서단위정모서리길이와원자반지름사이의관계식유도 주어진단위정차원에서면심입방격자와체심입방격자구조를갖는금속의밀도계산 세방향지수가주어질때, 단위정내에이지수와일치하는방향제도 단위정내에그려진면의 Miller 지수표기 원자의조밀충진면으적층에의해면심입방격자와육방조밀격자가생기는과정을설명 단결정과다결정재료의구분 재료성질에서등방성과이방성의차이 Chapter 3-2

3 Energy and Packing 치밀않음, 불규칙배열 (random packing) Energy typical neighbor bond length typical neighbor bond energy r 치밀, 규칙배열 (ordered packing) Energy typical neighbor bond length typical neighbor bond energy r 치밀하고규칙적인배열의구조는낮은에너지상태를갖는경향이있다. Chapter 3-3

4 Materials and Packing 결정질재료 (Crystalline materials...) 장범위규칙적원자배열, 3차원적패턴 예 : -모든금속 -대부분의세라믹 -일부의폴리머 비결정질재료 (Noncrystalline materials...) 장범위원자의규칙성없음 예 : -복잡한구조 -급냉(rapid cooling) crystalline SiO2 Adapted from Fig. 3.23(a), Callister & Rethwisch 8e. Si Oxygen Amorphous" = Noncrystalline noncrystalline SiO2 Adapted from Fig. 3.23(b), Callister & Rethwisch 8e. Chapter 3-4

5 금속의결정구조 How can we stack metal atoms to minimize empty space? 2-dimensions vs. Now stack these 2-D layers to make 3-D structures Chapter 3-5

6 조밀한원자충진 금속의결정구조 이유 : - 일반적으로오직한성분만이존재, 같은크기의원자 - 금속결합 : 방향성무 - 작은최인접원자간거리 : 낮은결합에너지 - 개개의전자구름은중심부를보호하고있음 (Electron cloud shields cores from each other) 가장간단한결정구조 We will examine three such structures... Chapter 3-6

7 단순입방정 (Simple Cubic Structure (SC)) 낮은충진밀도 : 흔하지않음 (only Polonium(Po) has this structure) 입방정의모서리 : 조밀방향 (Close-packed directions) 배위수 (Coordination #) = 6 [ 최인접원자수 (# nearest neighbors)] (Courtesy P.M. Anderson) 3&resourceId=19134 Chapter 3-7

8 원자충진율 (Atomic Packing Factor (APF)) APF = Volume of atoms in unit cell* Volume of unit cell 단순입방정의 APF = 0.52 a *assume hard spheres close-packed directions contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.24, Callister & Rethwisch 8e. R=0.5a atoms unit cell APF = 1 volume 4 3 p (0.5a) 3 atom a 3 volume unit cell Chapter 3-8

9 체심입방정 [Body Centered Cubic Structure (BCC)] 체심과모서리원자는입방의대각선상에서로접촉 --Note: 모든원자는동일 ; 중심부의다른색깔의원자는쉽게구별하기위함 ex: Cr, W, Fe ( ), Tantalum, Molybdenum 배위수 (Coordination #) = 8 Adapted from Fig. 3.2, Callister & Rethwisch 8e. (Courtesy P.M. Anderson) 2 atoms/unit cell: 1 center + 8 corners x 1/8 Chapter 3-9

10 원자충진율 (Atomic Packing Factor: BCC) APF for a body-centered cubic structure = a a 2 a Adapted from Fig. 3.2(a), Callister & Rethwisch 8e. atoms unit cell APF = R 2 a 4 3 p ( 3 a/4 ) 3 a 3 Close-packed directions: length = 4R = 3 a volume unit cell volume atom Chapter 3-10

11 면심입방정 [Face Centered Cubic Structure (FCC)] 원자는입방의모서리와면의중심에위치 --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. ex: Al, Cu, Au, Pb, Ni, Pt, Ag Coordination # = 12 (Courtesy P.M. Anderson) Adapted from Fig. 3.1, Callister & Rethwisch 8e. 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 Chapter 3-11

12 원자충진율 (Atomic Packing Factor: FCC) APF for a face-centered cubic structure = 0.74 최대의 APF 2 a a Adapted from Fig. 3.1(a), Callister & Rethwisch 8e. atoms unit cell APF = Close-packed directions: length = 4R = 2 a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell p ( 2 a/4 ) 3 a 3 volume atom volume unit cell Chapter 3-12

13 ABCABC... Stacking Sequence 2D Projection A sites B sites C sites A FCC 적층배열 B B C B C B B C B B FCC Unit Cell A B C Chapter 3-13

14 조밀육방정 [Hexagonal Close-Packed Structure (HCP)] ABAB... Stacking Sequence 3D Projection A sites 2D Projection Top layer c B sites Middle layer a Coordination # = 12 APF = 0.74 c/a = A sites Adapted from Fig. 3.3(a), Callister & Rethwisch 8e. Bottom layer 6 atoms/unit cell ex: Cd, Mg, Ti, Zn Chapter 3-14

15 이론밀도 (Theoretical Density, r) Density = r = r = Mass of Atoms in Unit Cell Total Volume of Unit Cell n A V C N A where n = number of atoms/unit cell A = atomic weight V C = Volume of unit cell = a 3 for cubic N A = Avogadro s number = x atoms/mol Chapter 3-15

16 Theoretical Density, r r = Ex: Cr (BCC) A = g/mol R = nm n = 2 atoms/unit cell n A V C N A Adapted from Fig. 3.2(a), Callister & Rethwisch 8e. volume unit cell atoms unit cell r = R a 3 a x a = 4R/ 3 = nm g mol r theoretical r actual atoms mol = 7.18 g/cm 3 = 7.19 g/cm 3 Chapter 3-16

17 일반적으로 r metals > r ceramics > r polymers 왜? 금속 (Metals have...) 조밀충진 (close-packing) (metallic bonding) 대부분큰원자질량 Ceramics have... less dense packing often lighter elements Polymers have... low packing density (often amorphous) lighter elements (C,H,O) Composites have... intermediate values 재료의밀도 r (g/cm 3 ) Metals/ Alloys Platinum Gold, W Tantalum Silver, Mo Cu,Ni Steels Tin, Zinc Titanium Aluminum Magnesium Graphite/ Ceramics/ Semicond Polymers Data from Table B.1, Callister & Rethwisch, 8e. Composites/ fibers B ased on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers in an epoxy matrix). Zirconia Al oxide Diamond Si nitride Glass - soda Concrete Silicon G raphite PTFE Silicone PVC PET PC H DPE, PS PP, LDPE Glass fibers GFRE* Carbon fibers CFRE * A ramid fibers AFRE * Wood Chapter 3-17

18 Crystals as Building Blocks Some engineering applications require single crystals: -- diamond single crystals for abrasives (Courtesy Martin Deakins, GE Superabrasives, Worthington, OH. Used with permission.) Properties of crystalline materials often related to crystal structure. -- Ex: Quartz fractures more easily along some crystal planes than others. -- turbine blades Fig. 8.33(c), Callister & Rethwisch 8e. (Fig. 8.33(c) courtesy of Pratt and Whitney). (Courtesy P.M. Anderson) Chapter 3-18

19 Polycrystals 이방성 (Anisotropic) Most engineering materials are polycrystals. 1 mm Adapted from Fig. K, color inset pages of Callister 5e. (Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany) <Nb-Hf-W plate with an electron beam weld> 등방성 (Isotropic) Each "grain" is a single crystal. If grains are randomly oriented, overall component properties are not directional. Grain sizes typically range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers). Chapter 3-19

20 단결정 (Single) vs 다결정 (Polycrystals) Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (E poly iron = 210 GPa) -If grains are textured, anisotropic. E (diagonal) = 273 GPa E (edge) = 125 GPa Data from Table 3.3, Callister & Rethwisch 8e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.) 200 mm Adapted from Fig. 4.14(b), Callister & Rethwisch 8e. (Fig. 4.14(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].) Chapter 3-20

21 동질이상 (Polymorphism) 하나이상의결정구조를갖는재료 [Two or more distinct crystal structures for the same material (allotropy/polymorphism)] titanium, -Ti carbon diamond, graphite iron system liquid 1538ºC BCC -Fe 1394ºC FCC -Fe BCC 912ºC -Fe Chapter 3-21

22 결정계 (Crystal Systems) 단위정 (Unit cell): 결정의완벽한격자형태를포함하는가장작은반복체적 (smallest repetitive volume which contains the complete lattice pattern of a crystal) 7 crystal systems 14 crystal lattices a, b, and c are the lattice constants Fig. 3.4, Callister & Rethwisch 8e. Chapter 3-22

23 입방 육방 정방 삼방 사방 단사 삼사 Chapter 3-23

24 c 점좌표 (Point Coordinates) z 111 단위정중심에대한점좌표 (Point coordinates for unit cell center are) a/2, b/2, c/2 ½ ½ ½ x a z 000 b 2c y 단위정꼭지점에대한점좌표 (Point coordinates for unit cell corner are) 111 b b y 평행이동 (Translation): 격자상수를곱하여정수로표시 (integer multiple of lattice constants) 결정격자내에서는동일한위치 (identical position in another unit cell) Chapter 3-24

25 결정방향 (Crystallographic Directions) x z y 방법 (Algorithm) 1. 적절한크기의벡터를좌표축의중심을지나도록한다. 2. 벡터가각축에투영되는길이를결정하여단위정의크기인 a, b, c 로나타낸다. 3. 최소의정수로표기한다. 4. 콤마로분리하지않고, 모난괄호안에표시한다. [uvw] ex: 1, 0, ½ => 2, 0, 1 => [ 201 ] -1, 1, 1 => [ 111 ] 위의바 (overbar) 는음의지수 ( 음의좌표 ) 를나타냄 동등한방향들을묶어족으로표시 (families of directions) <uvw> Chapter 3-25

26 선밀도 (Linear Density) Linear Density of Atoms LD = Number of atoms Unit length of direction vector [110] ex: linear density of Al in [110] direction a = nm a Adapted from Fig. 3.1(a), Callister & Rethwisch 8e. # atoms length LD = 2 2 a = 3.5 nm - 1 Chapter 3-26

27 육방결정의방향 (HCP Crystallographic Directions) a 3 z a 2 Adapted from Fig. 3.8(a), Callister & Rethwisch 8e. a 1 ex: ½, ½, -1, 0 => - 방법 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a 1, a 2, a 3, or c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvtw] [ 1120 ] a 3 a 2 2 a 1 2 dashed red lines indicate projections onto a 1 and a 2 axes a 1 a 2 -a 3 Chapter 3-27

28 HCP Crystallographic Directions Hexagonal Crystals 4 개의축 Miller-Bravais 좌표계에서방향은다음과같은관계를갖는다 (i.e., u'v'w'). a 3 z a 2 a 1 Fig. 3.8(a), Callister & Rethwisch 8e. - [ u ' v ' w '] u v t w = = = = 1 ( 2 u '- v ') 3 1 ( 2 v '- u ') 3 -( u + v ) w ' [ uvtw ] Chapter 3-28

29 결정면 (Crystallographic Planes) Adapted from Fig. 3.10, Callister & Rethwisch 8e. Chapter 3-29

30 결정면 (Crystallographic Planes) 밀리지수 (Miller Indices): 면과만나는 3 좌표축의역수, 최소의정수값. 모든평행한면은동일한밀러지수 (reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.) Algorithm 1. Read off intercepts of plane with axes in terms of a, b, c 2. Take reciprocals of intercepts 3. Reduce to smallest integer values 4. Enclose in parentheses, no commas i.e., (hkl) Chapter 3-30

31 결정면 (Crystallographic Planes) z example a b c 1. Intercepts Reciprocals 1/1 1/1 1/ Reduction Miller Indices (110) example a b c 1. Intercepts 1/2 2. Reciprocals 1/½ 1/ 1/ Reduction Miller Indices (100) x x a a c c z b b y y Chapter 3-31

32 결정면 (Crystallographic Planes) z example a b c 1. Intercepts 1/2 1 3/4 2. Reciprocals 1/½ 1/1 1/¾ 2 1 4/3 3. Reduction Miller Indices (634) x a c b y Family of Planes {hkl} Ex: {100} = (100), (010), (001), (100), (010), (001) Chapter 3-32

33 결정면 [Crystallographic Planes (HCP)] In hexagonal unit cells the same idea is used z example a 1 a 2 a 3 c 1. Intercepts Reciprocals 1 1/ Reduction a 2 a 3 4. Miller-Bravais Indices (1011) a 1 Adapted from Fig. 3.8(b), Callister & Rethwisch 8e. Chapter 3-33

34 결정면 (Crystallographic Planes) 결정면의원자충진 철포일도촉매제로사용이가능. 노출면의원자충진이중요. a) Fe 의 (100) 과 (111) 결정면을그리시오. b) 각각의결정면에대한면밀도를계산하시오. Chapter 3-34

35 Virtual Materials Science & Engineering (VMSE) VMSE is a tool to visualize materials science topics such as crystallography and polymer structures in three dimensions Available in Student Companion Site at and in WileyPLUS Chapter 3-35

36 VMSE: Metallic Crystal Structures & Crystallography Module VMSE allows you to view crystal structures, directions, planes, etc. and manipulate them in three dimensions Chapter 3-36

37 Unit Cells for Metals VMSE allows you to view the unit cells and manipulate them in three dimensions Below are examples of actual VMSE screen shots FCC Structure HCP Structure Chapter 3-37

38 VMSE: Crystallographic Planes Exercises Additional practice on indexing crystallographic planes Chapter 3-38

39 면밀도 (Planar Density) of (100) Iron Solution: At T < 912ºC iron has the BCC structure. 2D repeat unit (100) a = R Adapted from Fig. 3.2(c), Callister & Rethwisch 8e. atoms 2D repeat unit 1 Planar Density = a 2 area 2D repeat unit = Radius of iron R = nm R 2 = atoms 12.1 nm 2 = 1.2 x Chapter 3-39 atoms m 2

40 Planar Density of (111) Iron Solution (cont): (111) plane 1 atom in plane/ unit surface cell 2 a atoms in plane atoms above plane atoms below plane atoms 2D repeat unit Planar Density = area 2D repeat unit area = 2 ah = 3 a = 3 R = R atoms nm 2 = 7.0 = h 2 = 0.70 x a Chapter 3-40 R atoms m 2 2

41 VMSE Planar Atomic Arrangements VMSE allows you to view planar arrangements and rotate them in 3 dimensions BCC (110) Plane Chapter 3-41

42 X- 선회절 (X-Ray Diffraction) Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation. Can t resolve spacings Spacing is the distance between parallel planes of atoms. Chapter 3-42

43 X-Rays to Determine Crystal Structure 결정면에대한입사 X- 선의회절 (Incoming X-rays diffract from crystal planes). extra distance travelled by wave 2 q q d reflections must be in phase for a detectable signal Adapted from Fig. 3.20, Callister & Rethwisch 8e. spacing between planes Measurement of critical angle, q c, allows computation of planar spacing, d. X-ray intensity (from detector) d = n 2 sin q c q q c Chapter 3-43

44 회절의조건 ; Bragg 의법칙 여기서면간거리, Chapter 3-44

45 Chapter 3-45

46 Intensity (relative) z c X-Ray Diffraction Pattern z c z c a x b y (110) a x b y a x (211) b y (200) Adapted from Fig. 3.22, Callister 8e. Diffraction angle 2q Diffraction pattern for polycrystalline -iron (BCC) Chapter 3-46

47 SUMMARY Atoms may assemble into crystalline or amorphous structures. Common metallic crystal structures are FCC, BCC, and HCP. Coordination number and atomic packing factor are the same for both FCC and HCP crystal structures. We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP). Crystallographic points, directions and planes are specified in terms of indexing schemes. Crystallographic directions and planes are related to atomic linear densities and planar densities. Chapter 3-47

48 SUMMARY Materials can be single crystals or polycrystalline. Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains. Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy). X-ray diffraction is used for crystal structure and interplanar spacing determinations. Chapter 3-48

49 Reading: ANNOUNCEMENTS Core Problems: Self-help Problems: Chapter 3-49