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Chapter 3 - 1
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
Chapter 3 - 2
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
Chapter 3 - 3
• 치밀않음, 불규칙배열(random packing)
• 치밀, 규칙배열(ordered packing)
치밀하고 규칙적인 배열의 구조는 낮은 에너지상태를 갖는
경향이 있다.
Energy and Packing
Energy
r
typical neighbor bond length
typical neighbor bond energy
Energy
r
typical neighbor bond length
typical neighbor bond energy
Chapter 3 - 4
• 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:
Chapter 3 - 5
금속의 결정 구조
• 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 - 6
• 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
Chapter 3 - 7
• 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)
http://bcs.wiley.com/he-
bcs/Books?action=mininav&bcsId=5242&itemId=0470419970&assetId=19905
3&resourceId=19134
Chapter 3 - 8
• 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,
Callister & Rethwisch 8e.
close-packed directions
a
R=0.5a
contains 8 x 1/8 = 1 atom/unit cell
Chapter 3 - 9
• 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
2 atoms/unit cell: 1 center + 8 corners x 1/8 (Courtesy P.M. Anderson)
Chapter 3 - 10
원자충진율(Atomic Packing Factor: BCC)
a
APF =
4
3 p ( 3 a/4 ) 3 2
atoms
unit cell atom
volume
a 3
unit cell
volume
length = 4R =
Close-packed directions:
3 a
• APF for a body-centered cubic structure = 0.68
a R Adapted from
Fig. 3.2(a), Callister &
Rethwisch 8e.
a 2
a 3
Chapter 3 - 11
• 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
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 (Courtesy P.M. Anderson)
Chapter 3 - 12
• 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 x 1/2 + 8 x 1/8 = 4 atoms/unit cell
a
2 a
Adapted from
Fig. 3.1(a),
Callister &
Rethwisch 8e.
Chapter 3 - 13
A sites
B B
B
B B
B B
C sites
C C
C A
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
C A
C C
C A
A B
C
Chapter 3 - 14
• 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
Chapter 3 - 15
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 =
VC NA
n A r =
Cell Unit of Volume Total
Cell Unit in Atoms of Mass
Chapter 3 - 16
• 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
a R
r = a3
52.00 2
atoms
unit cell mol
g
unit cell
volume atoms
mol
6.022 x 1023
Theoretical Density, r
= 7.18 g/cm3
= 7.19 g/cm3
Adapted from
Fig. 3.2(a), Callister &
Rethwisch 8e.
VC NA
n A r =
Chapter 3 - 17
재료의 밀도
r metals > r ceramics > r polymers
왜?
Data from Table B.1, Callister & Rethwisch, 8e.
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
0.3
0.4
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... • intermediate values
일반적으로
Chapter 3 - 18
• 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
Chapter 3 - 19
• 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.
• Grain sizes typically range from 1 nm to 2 cm
(i.e., from a few to millions of atomic layers).
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)
Chapter 3 - 20
• 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
Chapter 3 - 21
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
Chapter 3 - 22
Fig. 3.4, Callister & Rethwisch 8e.
결정계(Crystal Systems)
7 crystal systems
14 crystal lattices
단위정(Unit cell): 결정의 완벽한 격자 형태를
포함하는 가장 작은 반복 체적(smallest repetitive volume
which contains the complete lattice pattern of a crystal)
a, b, and c are the lattice constants
Chapter 3 - 23
입방
육방
정방
삼방
사방
단사
삼사
Chapter 3 - 24
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
Chapter 3 - 25
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, 1
동등한 방향들을 묶어 족으로 표시(families of directions) <uvw>
z
x
방법(Algorithm)
위의 바(overbar)는 음의
지수(음의 좌표)를 나타냄
[ 111 ] =>
y
Chapter 3 - 26
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
1 3.5 nm
a 2
2 LD - = =
Adapted from
Fig. 3.1(a),
Callister &
Rethwisch 8e.
Chapter 3 - 27
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
projections onto a1 and a2 axes a1
a2
a3
-a3
2
a 2
2
a 1
- a3
a1
a2
z
방법
Chapter 3 - 28
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
Chapter 3 - 29
결정면(Crystallographic Planes)
Adapted from Fig. 3.10,
Callister & Rethwisch 8e.
Chapter 3 - 30
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)
Chapter 3 - 31
결정면(Crystallographic Planes) z
x
y a b
c
4. Miller Indices (110)
example a b c z
x
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
Chapter 3 - 32
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),
Chapter 3 - 33
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.
Chapter 3 - 34
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.
Chapter 3 -
Virtual Materials Science & Engineering (VMSE)
35
• VMSE is a tool to visualize materials science topics such as
crystallography and polymer structures in three dimensions
• Available in Student Companion Site at www.wiley.com/college/callister
and in WileyPLUS
Chapter 3 -
VMSE: Metallic Crystal Structures &
Crystallography Module
36
• VMSE allows you to view crystal structures, directions, planes,
etc. and manipulate them in three dimensions
Chapter 3 -
Unit Cells for Metals
37
• 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 -
VMSE: Crystallographic Planes Exercises
38
Additional practice on indexing crystallographic planes
Chapter 3 - 39
면밀도(Planar Density) of (100) Iron
Solution: At T < 912ºC iron has the BCC structure.
(100)
Radius of iron R = 0.1241 nm
R 3
3 4 a =
Adapted from Fig. 3.2(c), Callister & Rethwisch 8e.
2D repeat unit
= Planar Density = a 2
1
atoms
2D repeat unit
= nm2
atoms 12.1
m2
atoms = 1.2 x 1019
1
2
R 3
3 4 area
2D repeat unit
Chapter 3 - 40
Planar Density of (111) Iron Solution (cont): (111) plane 1 atom in plane/ unit surface cell
3 3 3
2
2
R 3
16 R
3
4
2 a 3 ah 2 area =
= = =
atoms in plane
atoms above plane
atoms below plane
a h 2
3 =
a 2
1
= = nm2
atoms 7.0
m2
atoms 0.70 x 1019
3 2 R 3
16 Planar Density =
atoms
2D repeat unit
area
2D repeat unit
Chapter 3 -
VMSE Planar Atomic Arrangements
41
• 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
Chapter 3 - 42
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.
Chapter 3 - 43
X-Rays to Determine Crystal Structure
X-ray intensity (from detector)
q
q c
d = n
2 sin q c
Measurement of
critical angle, qc,
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”
Chapter 3 - 44
회절의 조건;
여기서 면간 거리,
Bragg의 법칙
Chapter 3 - 45
Chapter 3 - 46
X-Ray Diffraction Pattern
Adapted from Fig. 3.22, Callister 8e.
(110)
(200)
(211)
z
x
y a b
c
Diffraction angle 2q
Diffraction pattern for polycrystalline -iron (BCC)
Inte
nsity (
rela
tive
)
z
x
y a b
c
z
x
y a b
c
Chapter 3 - 47
SUMMARY
• Atoms may assemble into crystalline or
amorphous 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).
• 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.
• 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 - 48
• Some materials can have more than one crystal
structure. This is referred to as polymorphism (or
allotropy).
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.
• X-ray diffraction is used for crystal structure and
interplanar spacing determinations.
Chapter 3 - 49
Core Problems:
Self-help Problems:
ANNOUNCEMENTS
Reading: