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?
제3장: 결정질 고체의 구조
학습목표...
• 원자/분자 구조에서 결정질과 비결정질 재료의 차이점 분석
• 면심입방격자, 체심입방격자, 육방조밀격자의 제도
• 면심입방격자와 체심입방격자 구조에서 단위정 모서리 길이와 원자 반지름 사이의 관계식 유도
• 주어진 단위정 차원에서 면심입방격자와 체심입방격자 구조를 갖는 금속의 밀도 계산
• 세 방향 지수가 주어질 때, 단위정 내에 이 지수와 일치하는 방향 제도
• 단위정 내에 그려진 면의 Miller 지수 표기
• 원자의 조밀 충진면으 적층에 의해 면심입방격자와 육방조밀격자가 생기는 과정을 설명
• 단결정과 다결정 재료의 구분
• 재료 성질에서 등방성과 이방성의 차이
• 치밀않음, 불규칙배열(random packing)
• 치밀, 규칙배열(ordered packing)
치밀하고 규칙적인 배열의 구조는 낮은 에너지상태를 갖는
Energy and Packing
Energy
r typical neighbor bond length
typical neighbor bond energy
Energy
r typical neighbor bond length typical neighbor
bond energy
• 장범위 규칙적 원자배열, 3차원적 패턴 결정질 재료(Crystalline materials...)
-모든 금속
-대부분의 세라믹 -일부의 폴리머
• 장범위 원자의 규칙성 없음
비결정질 재료(Noncrystalline materials...)
-복잡한 구조-급냉(rapid cooling)
crystalline SiO2
noncrystalline SiO2
“Amorphous" = Noncrystalline
Adapted from Fig. 3.23(b), Callister & Rethwisch 8e.
Adapted from Fig. 3.23(a), Callister & Rethwisch 8e.
Materials and Packing
Si Oxygen
• 예:
• 예:
금속의 결정 구조
• How can we stack metal atoms to minimize empty space?
2-dimensions
vs.
• 조밀한 원자 충진
• 이유:
- 일반적으로 오직 한 성분만이 존재, 같은 크기의 원자 - 금속결합: 방향성 무
- 작은 최인접 원자간 거리: 낮은 결합에너지
- 개개의 전자 구름은 중심부를 보호하고 있음(Electron cloud shields cores from each other)
• 가장 간단한 결정구조
We will examine three such structures...
금속의 결정 구조
• 낮은 충진밀도: 흔하지 않음 (only Polonium(Po) has this structure)
• 입방정의 모서리: 조밀방향 (Close-packed directions)
• 배위수(Coordination #) = 6
[최인접원자수(# nearest neighbors)]
단순입방정 (Simple Cubic Structure (SC))
(Courtesy P.M. Anderson)
• 단순입방정의 APF = 0.52
APF =
a 3 4
3 p (0.5a) 3 1
atoms unit cell
atom volume
unit cell volume
원자충진율 (Atomic Packing Factor (APF))
APF = Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
Adapted from Fig. 3.24,
close-packed directions
a
R=0.5a
contains 8 x 1/8 = 1 atom/unit cell
• 배위수(Coordination #) = 8
Adapted from Fig. 3.2, Callister & Rethwisch 8e.
• 체심과 모서리 원자는 입방의 대각선상에 서로 접촉
--Note: 모든 원자는 동일; 중심부의 다른 색깔의 원자는 쉽게 구별하기 위함
체심입방정 [Body Centered Cubic Structure (BCC)]
ex: Cr, W, Fe (), Tantalum, Molybdenum
원자충진율 (Atomic Packing Factor: BCC)
a
APF =
4
3 p ( 3 a/4 ) 3 2
atoms
unit cell atom
volume
a 3 volume
length = 4R =
Close-packed directions:
3 a
• APF for a body-centered cubic structure = 0.68
R a
Adapted from
Fig. 3.2(a), Callister &
Rethwisch 8e.
a 2 a
3
•
Coordination # = 12
Adapted from Fig. 3.1, Callister & Rethwisch 8e.
• 원자는 입방의 모서리와 면의 중심에 위치
--Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.
면심입방정
[Face Centered Cubic Structure (FCC)]ex: Al, Cu, Au, Pb, Ni, Pt, Ag
• APF for a face-centered cubic structure = 0.74
원자충진율 (Atomic Packing Factor: FCC)
최대의 APF
APF =
4
3 p ( 2 a/4 ) 3 4
atoms
unit cell atom
volume
a 3
unit cell volume Close-packed directions:
length = 4R = 2 a Unit cell contains:
6 x1/2 + 8 x1/8 = 4 atoms/unit cell a
2 a
Adapted from Fig. 3.1(a), Callister &
Rethwisch 8e.
A sites
B B B B B
B B
C sites
C C A C
B B sites
• ABCABC... Stacking Sequence
• 2D Projection
• FCC Unit Cell
FCC 적층 배열
B B B B B
B B B sites
C C A C
C C A C
A B C
• Coordination # = 12
• ABAB... Stacking Sequence
• APF = 0.74
• 3D Projection • 2D Projection
Adapted from Fig. 3.3(a), Callister & Rethwisch 8e.
조밀육방정
[Hexagonal Close-Packed Structure (HCP)]6 atoms/unit cell
ex: Cd, Mg, Ti, Zn
• c/a = 1.633
c
a
A sites
B sites
A sites Bottom layer
Middle layer Top layer
이론밀도(Theoretical Density, r)
where n = number of atoms/unit cell A = atomic weight
VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number
= 6.022 x 1023 atoms/mol Density = r =
VCNA r = n A
Cell
Unit
of Volume Total
Cell
Unit
in Atoms of
Mass
• Ex: Cr (BCC)
A = 52.00 g/mol R = 0.125 nm
n = 2 atoms/unit cell
rtheoretical
a = 4R/ 3 = 0.2887 nm
ractual R a
r
=a3
52.00 2
atoms unit cell
mol g
unit cell
volume atoms
mol 6.022x1023
Theoretical Density, r
= 7.18 g/cm3
= 7.19 g/cm3
Adapted from
Fig. 3.2(a), Callister &
Rethwisch 8e.
VCNA r = n A
재료의 밀도
r metals > r ceramics > r polymers 왜?
r(g/cm )3
Graphite/
Ceramics/
Semicond Metals/
Alloys
Composites/
fibers Polymers
1 2 2 0 30
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).
10
3 4 5
0.5
Magnesium Aluminum Steels
Titanium Cu,Ni Tin, Zinc Silver, Mo Tantalum Gold, W Platinum
G raphite Silicon Glass - soda Concrete Si nitride Diamond Al oxide Zirconia
H DPE, PS PP, LDPE PC
PTFE
PET PVC Silicone
Wood AFRE * CFRE * GFRE*
Glass fibers
Carbon fibers A ramid fibers
금속
(Metals have...)
• 조밀충진(close-packing) (metallic bonding)
• 대부분 큰 원자질량
Ceramics have...
• less dense packing • often lighter elements
Polymers have...
• low packing density (often amorphous)
• lighter elements (C,H,O)
Composites have...
일반적으로
• Some engineering applications require single crystals:
• Properties of crystalline materials often related to crystal structure.
(Courtesy P.M. Anderson)
-- Ex: Quartz fractures more easily along some crystal planes than others.
-- diamond single
crystals for abrasives
-- turbine blades
Fig. 8.33(c), Callister &
Rethwisch 8e. (Fig. 8.33(c) courtesy of Pratt and Whitney).
(Courtesy Martin Deakins, GE Superabrasives,
Worthington, OH. Used with permission.)
Crystals as Building Blocks
• Most engineering materials are polycrystals.
<Nb-Hf-W plate with an electron beam weld>
• Each "grain" is a single crystal.
• If grains are randomly oriented,
overall component properties are not directional.
Adapted from Fig. K, color inset pages of Callister 5e.
(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)
1 mm
Polycrystals
등방성(Isotropic) 이방성(Anisotropic)
• Single Crystals
-Properties vary with direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
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.)
• Polycrystals
-Properties may/may not vary with direction.
-If grains are randomly oriented: isotropic.
(Epoly iron = 210 GPa)
-If grains are textured, anisotropic.
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].)
단결정(Single) vs 다결정(Polycrystals)
E (diagonal) = 273 GPa
E (edge) = 125 GPa
동질이상(Polymorphism)
• 하나 이상의 결정구조를 갖는 재료[Two or more distinct crystal structures for the same material (allotropy/polymorphism)]
titanium , -Ti
carbon
diamond, graphite
BCC FCC BCC
1538ºC 1394ºC
912ºC
-Fe
-Fe
-Fe liquid
iron system
Fig. 3.4, Callister & Rethwisch 8e.
결정계(Crystal Systems)
7 crystal systems
14 crystal lattices
단위정(Unit cell):
결정의 완벽한 격자 형태를포함하는 가장 작은 반복 체적(smallest repetitive volume which contains the complete lattice pattern of a crystal)
a, b, and c are the lattice constants
입방
육방
정방
삼방
사방
단사
점 좌표(Point Coordinates)
단위정 중심에 대한 점 좌표(Point
coordinates for unit cell center are) a/2, b/2, c/2 ½ ½ ½
단위정 꼭지점에 대한 점 좌표(Point coordinates for unit cell corner are) 111
평행이동(Translation): 격자상수를
곱하여 정수로 표시(integer multiple of lattice constants) 결정격자 내에서는 동일한 위치(identical position in another unit cell)
z
x
y
a b
c
000
111
y z
2c
b b
결정 방향(Crystallographic Directions)
1. 적절한 크기의 벡터를 좌표축의 중심을 지나도록 한다.
2. 벡터가 각 축에 투영되는 길이를 결정하여 단위정의 크기인 a, b, c로 나타낸다.
3. 최소의 정수로 표기한다.
4. 콤마로 분리하지 않고, 모난 괄호 안에 표시한다.
[uvw]
ex:
1, 0, ½ => 2, 0, 1 => [ 201 ]
-1, 1, 1z
x
방법(Algorithm)
위의 바(overbar)는 음의 지수(음의 좌표)를 나타냄 [111]
=>
y
ex: linear density of Al in [110]
direction
a = 0.405 nm
선밀도(Linear Density)
• Linear Density of Atoms LD =
a
[110]
Unit length of direction vector Number of atoms
# atoms
length
3.5 nm 1
a 2
LD = 2 = -
Adapted from Fig. 3.1(a), Callister &
Rethwisch 8e.
육방 결정의 방향
(HCP Crystallographic Directions)1. Vector repositioned (if necessary) to pass through origin.
2. Read off projections in terms of unit cell dimensions a1, a2, a3, or c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas [uvtw]
[1120] ex: ½ , ½ , -1, 0 =>
Adapted from Fig. 3.8(a), Callister & Rethwisch 8e.
dashed red lines indicate
a2
a3
-a3 2
a 2
2 a 1 a3 -
a1 a2
z
방법
HCP Crystallographic Directions
• Hexagonal Crystals
– 4개의 축 Miller-Bravais 좌표계에서 방향은 다음과 같은 관계를 갖는다(i.e., u'v'w').
=
=
=
' w w
t v u
) v u
( + -
) ' u ' v 2 3 (
1 - ) ' v ' u 2 3 (
1 -
=
] uvtw [
] ' w ' v ' u
[
Fig. 3.8(a), Callister & Rethwisch 8e.
a3 -
a1 a2
z
결정면(Crystallographic Planes)
결정면(Crystallographic Planes)
• 밀리지수(Miller Indices): 면과 만나는
3좌표축의 역수, 최소의 정수값. 모든 평행한 면은 동일한 밀러지수(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)
결정면(Crystallographic Planes)
z
x
y
a b
c
4. Miller Indices (110)
example a b c z
y
a b
c
4. Miller Indices (100)
1. Intercepts 1 1 2. Reciprocals 1/1 1/1 1/
1 1 0 3. Reduction 1 1 0
1. Intercepts 1/2 2. Reciprocals 1/½ 1/ 1/
2 0 0 3. Reduction 2 0 0 example a b c
결정면(Crystallographic Planes)
z
x
y
a b
c
4. Miller Indices (634) example
1. Intercepts 1/2 1 3/4 a b c
2. Reciprocals 1/½ 1/1 1/¾ 2 1 4/3 3. Reduction 6 3 4
(001) (010),
Family of Planes {hkl}
(100), (010), (001),
Ex: {100} = (100),
결정면[Crystallographic Planes (HCP)]
• In hexagonal unit cells the same idea is used
example a1 a2 a3 c
4. Miller-Bravais Indices (1011)
1. Intercepts 1 -1 1 2. Reciprocals 1 1/
1 0
-1 -1
1 1 3. Reduction 1 0 -1 1
a2
a3
a1 z
Adapted from Fig. 3.8(b), Callister & Rethwisch 8e.
결정면(Crystallographic Planes)
• 결정면의 원자충진
• 철 포일도 촉매제로 사용이 가능. 노출면의 원자충진이 중요.
a) Fe의 (100)과 (111) 결정 면을 그리시오.
b) 각각의 결정 면에 대한 면밀도를 계산하시오.
Virtual Materials Science & Engineering (VMSE)
• VMSE is a tool to visualize materials science topics such as crystallography and polymer structures in three dimensions
VMSE: Metallic Crystal Structures &
Crystallography Module
• VMSE allows you to view crystal structures, directions, planes, etc. and manipulate them in three dimensions
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
VMSE: Crystallographic Planes Exercises
Additional practice on indexing crystallographic planes
면밀도(Planar Density) of (100) Iron
Solution: At T < 912ºC iron has the BCC structure.
(100)
Radius of iron R = 0.1241 nm
3 R 3 a = 4
Adapted from Fig. 3.2(c), Callister & Rethwisch 8e.
2D repeat unit
=
Planar Density =
a 2 1 atoms
2D repeat unit
=
nm
2atoms 12.1
m
2atoms
= 1.2 x 10
191
3 2
area 4
Planar Density of (111) Iron
Solution (cont): (111) plane 1 atom in plane/ unit surface cell
3
3 3 2
2
3 R R 16
3 4 a 2
3 ah
2
area =
=
=
=
atoms in plane
atoms above plane atoms below plane
a h 2
= 3
a 2
1
= =
nm
2atoms
7.0 m
2atoms 0.70 x 10
193 2 3 R
16
Planar Density =
atoms
2D repeat unit area
2D repeat unit
VMSE Planar Atomic Arrangements
• VMSE allows you to view planar arrangements and rotate them in 3 dimensions
BCC (110) Plane
http://bcs.wiley.com/he-bcs/Books?action=mininav&bcsId=5242&itemId=0470419970&assetId=199053&resourceId=19134
X-선 회절(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.
X-Rays to Determine Crystal Structure
X-ray intensity (from detector)
q d = n
2 sin q c
Measurement of
critical angle, q
c,
allows computation of planar spacing, d.
• 결정면에 대한 입사 X-선의
회절
(Incoming X-rays diffract from crystal planes).Adapted from Fig. 3.20, Callister & Rethwisch 8e.
reflections must be in phase for a detectable signal
spacing between planes d
q
q
extra distance travelled by wave “2”
회절의 조건;
여기서 면간 거리,
Bragg의 법칙
X-Ray Diffraction Pattern
Adapted from Fig. 3.22, Callister 8e.
(110)
(200)
(211)
z
x
a b y c
Diffraction angle 2q
Diffraction pattern for polycrystalline -iron (BCC)
Intensity (relative)
z
x
a b y c
z
x
a b y c
