제3장 결정질 고체의 구조 - KOCWelearning.kocw.net › contents4 › document › lec › 2013 › Chosun › ... · 2013-06-27 · 결정면(Crystallographic Planes) •

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),


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,


close-packed directions

a

R=0.5a

contains 8 x 1/8 = 1 atom/unit cell

Chapter 3 - 9

• 배위수(Coordination #) = 8



• 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



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 =>



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)



Chapter 3 - 30


• 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


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


z

x

y a b

c


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),


Chapter 3 - 34


• 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).



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:

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제3장 결정질 고체의 구조 - KOCWelearning.kocw.net › contents4 › document › lec › 2013 › Chosun › ... · 2013-06-27 · 결정면(Crystallographic Planes) •