Chapter 4: The Structure of Crystalline Solids

Chapter 4 - 1

ISSUES TO ADDRESS...

• What are common crystal structures for

metals and ceramics?

• What features of a metal’s/ceramic’s atomic

structure determine its density?

• How do the crystal structures of ceramic

materials differ from those for metals?

Chapter 4: The Structure of Crystalline Solids고체결정질의구조

공통결정구조

비중

Chapter 4 - 2

Metallic Crystal Structures

• How can we stack metal atoms to minimize

empty space?

2-dimensions

vs.

Now stack these 2-D layers to make 3-D structures

빈공간을최소화하기위해원자가쌓이는방법은?

2차원 공간만 생각하면 좌측이 유리하나 3차원 공간이므로다른결과가나올수있다

Chapter 4 - 3

• Tend to be densely packed.

• Reasons for dense packing:

- Typically, only one element is present, so all atomic radii are

the same.

- Metallic bonding is not directional.

- Nearest neighbor distances tend to be small in order to lower

bond energy.

- Electron cloud shields cores from each other.

• Metals have the simplest crystal structures.

We will examine three such structures...

Metallic Crystal Structures

방향성없음

만일 전자가 없으면 양이온인코어(핵)이 서로 밀어내려고하는 힘이 작용하지만 전자구름이 이를 막아줘서(shield) 조밀하게쌓일수있다는의미

조밀하게쌓이는경향

Chapter 4 - 4

• Rare due to low packing density (only Po has this structure)

• Close-packed directions are cube edges.

• Coordination # = 6

(# nearest neighbors)

Simple Cubic Structure (SC)

Fig. 4.2, Callister & Rethwisch 9e.

단순입방결정구조

충진율이작으므로어떠한 금속도이러한 결정구조를갖지않는다. (반금속인폴로늄(Po; polonium)만존재)

• APF for a simple cubic structure = 0.52

최근접원자수는 6개

Atomic Packing Factor(원자충진률)

Chapter 4 - 5

• APF for a simple cubic structure = 0.52

APF =

a3

4

3π (0.5a) 31

atoms

unit cellatom

volume

unit cell

volume

Atomic Packing Factor (APF)

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

Adapted from Fig. 4.2 (a),

Callister & Rethwisch 9e.

close-packed directions

a

R = 0.5a

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

= There are eights of 1/8 atoms.

총단위정부피

단위정내원자부피

Chapter 4 - 6


Adapted from Fig. 4.1,


• Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded

differently only for ease of viewing.

Body Centered Cubic Structure (BCC)

ex: Cr, W, Fe (), Tantalum, Molybdenum

2 atoms/unit cell: 1 center + 8 corners x 1/8

체심입방격자

(# nearest neighbors)

Chapter 4 - 7

Atomic Packing Factor: BCC

APF =

4

3π ( 3a/4 )32

atoms

unit cell atom

volume

a3

unit cell

volume

length = 4R =

Close-packed directions:

3 a

• APF for a body-centered cubic structure = 0.68

aRAdapted from

Fig. 4.1(a), Callister &

Rethwisch 9e.

a

a2

a3청색삼각형

(밑면대각선길이)

Chapter 4 - 8


Adapted from Fig. 3.1, Callister & Rethwisch 9e.

• Atoms touch each other along face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded

differently only for ease of viewing.

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

면심입방격자

(최근접원자수)

Chapter 4 - 9

• APF for a face-centered cubic structure = 0.74

Atomic Packing Factor: FCC

maximum achievable APF

APF =

4

3π ( 2a/4 )34

atoms

unit cell atom

volume

a3

unit cell

volume

Close-packed directions:

length = 4R = 2 a

Unit cell contains:6 x 1/2 + 8 x 1/8

= 4 atoms/unit cella

2 a

Adapted from

Fig. 3.1(a),

Callister &

Rethwisch 9e.

∴R= 2𝑎/4

Chapter 4 - 10

A sites

B B

B

BB

B B

C sites

C C

CA

B

B sites

• ABCABC... Stacking Sequence

• 2D Projection

• FCC Unit Cell

FCC Stacking Sequence

B B

B

BB

B B

B sitesC C

CA

C C

CA

AB

C

A(맨아래 원자)와 C(맨위 원자)가어긋나게쌓여있음

Chapter 4 - 11


• ABAB... Stacking Sequence

• APF = 0.74

• 3D Projection • 2D Projection

Adapted from Fig. 4.3(a),


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

육방조밀결정

맨위 원자와 맨아래원자가동일위치임

예제 4.3에 HCP단위정부피구하는과정설명되어있음

Chapter 4 - 12

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

n Ar =

CellUnitofVolumeTotal

CellUnitinAtomsofMass

이론밀도

Chapter 4 - 13

• Ex: Cr (BCC)

A = 52.00 g/mol

R = 0.125 nm

n = 2 atoms/unit cell

ρtheoretical

ρactual

aR

r =

a3

52.002

atoms

unit cellmol

g

unit cell

volume atoms

mol

6.022 x 1023

Theoretical Density, ρ

= 7.18 g/cm3

= 7.19 g/cm3

Adapted from

Fig. 4.1(a), Callister &

Rethwisch 9e.

원자량

Chapter 4 - 14

금속의원자반지름과결정구조

Table_04_01

Chapter 4 - 15

Ceramic Crystal Structures

– 세라믹은보통금속과비금속원소간의화합물

– 세라믹은최소 2개이상의원소로구성되어있기때문에일반적으로금속의결정구조보다복잡

– 결합은이온결합또는공유결합

암염(NaCl)의결정구조 섬아연광(ZnS)의결정구조

Chapter 4 - 16

세라믹이온배열의기하학적형상(1)

– 금속원자는전자를공여해서양이온(cation)이됨

– 비금속원자는전자를받아음이온(anion)이됨

– 결정질세라믹재료의결정구조는 (1)각구성이온의전하량과(2)양이온대비음이온의크기에의해결정

(1) 전기적중성을이루기위해양이온전하량의합과음이온전하량

합이같은조건(예: CaF2, 여기서 Ca는 +2가이온, F는 -1가이온)

(2) 양이온과음이온반지름의비(rc/ra)에의해배위수(양이온주위음이온의수)가결정

안정한 또는 불안정한 양이온-음이온 배치(청색이 양이온,

적색이음이온)

Chapter 4 - 17

세라믹이온배열의기하학적형상(2)

세라믹 이온배열에서 양이온-

음이온 반지름비에 대한 배위수와배위형상

‒ 그러나 실제 이온은 딱딱한구가 아니므로 보여지는 배위수가절대관계는아님

‒ 이온반경은 배위수가 증가하면증가함

‒ 또한, 이온의 전하도 반경에영향을 준다. 일반적으로 전자를 잃은 양전하는 남아있는 원자가 전자를 더욱 견고하게 결속되어 이온반경이감소됨

Chapter 4 - 18

• Bonding:-- Can be ionic and/or covalent in character.

-- % ionic character increases with difference in electronegativity

of atoms.

• Degree of ionic character may be large or small:

Atomic Bonding in Ceramics

SiC: small

CaF2: large

(생략)

Chapter 4 - 19

Ceramic Crystal Structures

Oxide structures– oxygen anions larger than metal cations

– close packed oxygen in a lattice (usually FCC)

– cations fit into interstitial sites among oxygen ions

산화물

음이온 양이온

침입형

(생략)

Chapter 4 - 20

Factors that Determine Crystal Structure1. Relative sizes of ions – Formation of stable structures:

--maximize the # of oppositely charged ion neighbors.



- -

- -+

unstable

- -

- -+

stable

- -

- -+

stable

2. Maintenance of

Charge Neutrality :--Net charge in ceramic

should be zero.

--Reflected in chemical

formula:

CaF2 : Ca2+

cation

F-

F-

anions+

AmXp

m, p values to achieve

charge neutrality

(생략)

반대 극성의 최 근접이온수최대화

전기적중성도만족

Chapter 4 - 21

• Coordination Number increases with

Coordination Number and Ionic Radii

Adapted from Table 4.3,


2

rcationranion

Coord.

Number

< 0.155

0.155 - 0.225

0.225 - 0.414

0.414 - 0.732

0.732 - 1.0

3

4

6

8

linear

triangular

tetrahedral

octahedral

cubic

ZnS

(zinc blende)

NaCl(sodium

chloride)

CsCl(cesium chloride)

rcationranion

To form a stable structure, how many anions can

surround around a cation?

(생략)

배위수(양이온주위의음이온수)

음이온 반지름에 대한양이온의반지름비

Chapter 4 - 22

Computation of Minimum Cation-Anion

Radius Ratio

• Determine minimum rcation/ranion for an octahedral site (C.N. = 6)

a = 2ranion

2ranion + 2rcation = 2 2ranion

ranion + rcation = 2ranion

rcation = ( 2 -1)ranion

414.012anion

cation =-=r

r

(생략)

Chapter 4 - 23

Bond Hybridization

Bond Hybridization is possible when there is significant

covalent bonding

– hybrid electron orbitals form

– For example for SiC

• XSi = 1.8 and XC = 2.5

• ~ 89% covalent bonding

• Both Si and C prefer sp3 hybridization

• Therefore, for SiC, Si atoms occupy tetrahedral sites

(생략)

혼성화결합(전자 궤도가 중첩되어 최외각전자수가변화되는현상)

Chapter 4 - 24

• On the basis of ionic radii, what crystal structure

would you predict for FeO?

• Answer:

5500

1400

0770

anion

cation

.

.

.

r

r

=

=

based on this ratio,

-- coord # = 6 because

0.414 < 0.550 < 0.732

-- crystal structure is NaCl

Data from Table 4.4,


Example Problem: Predicting the Crystal

Structure of FeO

Ionic radius (nm)

0.053

0.077

0.069

0.100

0.140

0.181

0.133

Cation

Anion

Al3+

Fe2+

Fe3+

Ca2+

O2-

Cl-

F-

(생략)

(NaCl 다음페이지참조)

Chapter 4 - 25

Rock Salt Structure

Same concepts can be applied to ionic solids in general.

Example: NaCl (rock salt) structure

rNa = 0.102 nm

rNa/rCl = 0.564

cations (Na+) prefer octahedral sites



rCl = 0.181 nm

(생략)

6개의배위수

Chapter 4 - 26

MgO and FeO

O2- rO = 0.140 nm

Mg2+ rMg = 0.072 nm

rMg/rO = 0.514

cations prefer octahedral sites

So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms



MgO and FeO also have the NaCl structure

(생략)

(배위수는 6이나 주위 이온을 연결하는구조는 8면체자리임)

Chapter 4 - 27

AX Crystal Structures


CsCl(Cesium Chloride) structure:

Since 0.732 < 0.939 < 1.0,

cubic sites preferred

So each Cs+ has 8 neighbor Cl-

AX–Type Crystal Structures include NaCl, CsCl, and zinc blende

(생략)

Chapter 4 - 28

AX2 Crystal Structures

• Calcium Fluorite (CaF2)

• Cations in cubic sites

• UO2, ThO2, ZrO2, CeO2

• Antifluorite structure –

positions of cations and

anions reversed


Fluorite structure

(생략)

양이온과음이온의자리가뒤바뀐구조

Chapter 4 - 29

ABX3 Crystal Structures

Fig. 4.9, Callister &

Rethwisch 9e.

• Perovskite structure

Ex: complex oxide

BaTiO3

(생략)

Chapter 4 - 30

Density Computations for Ceramics

Number of formula units/unit cell

Volume of unit cell

Avogadro’s number

= sum of atomic weights of all anions in formula unit

= sum of atomic weights of all cations in formula unit

세라믹의밀도계산

온전한이온(일부가아닌..)

Chapter 4 - 31

Densities of Material Classes

ρmetals > ρceramics > ρpolymers

Why?

Data from Table B.1, Callister & Rethwisch, 9e.

ρ(g

/cm

)

3

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

1

2

20

30Based on data in Table B1, Callister

*GFRE, CFRE, & AFRE are Glass,Carbon, & Aramid Fiber-ReinforcedEpoxy composites (values based on60% 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

TantalumGold, WPlatinum

Graphite

Silicon

Glass -sodaConcrete

Si nitrideDiamondAl oxide

Zirconia

HDPE, PSPP, LDPE

PC

PTFE

PETPVCSilicone

Wood

AFRE*

CFRE*

GFRE*

Glass fibers

Carbon fibers

Aramid fibers

Metals have...• close-packing

(metallic bonding)

• often large atomic masses

Ceramics have...• less dense packing

• often lighter elements

Polymers have...• low packing density

(often amorphous)

• lighter elements (C,H,O)

Composites have...• intermediate values

In general

Chapter 4 - 32

Silicate CeramicsMost common elements on earth are Si & O

• SiO2 (silica) polymorphic forms are quartz,

crystobalite, & tridymite

• The strong Si-O bonds lead to a high melting

temperature (1710ºC) for this material

Si4+

O2-

Figs. 4.10 & 4.11, Callister & Rethwisch

9e crystobalite

규산염

(생략)

Chapter 4 - 33

Bonding of adjacent SiO44- accomplished by the

sharing of common corners, edges, or faces

Silicates

Mg2SiO4 Ca2MgSi2O7

Adapted from Fig.

4.13, Callister &

Rethwisch 9e.

Presence of cations such as Ca2+, Mg2+, & Al3+

1. maintain charge neutrality, and

2. ionically bond SiO44- to one another

(생략)

Chapter 4 - 34

• Quartz is crystalline

SiO2:

• Basic Unit: Glass is noncrystalline (amorphous)

• Fused silica is SiO2 to which no

impurities have been added

• Other common glasses contain

impurity ions such as Na+, Ca2+,

Al3+, and B3+

(soda glass)



Glass Structure

SiO4 tetrahedron4-

Si4+

O2-

Si4+

Na+

O2-

(생략)

Chapter 4 - 35

Layered Silicates

• Layered silicates (e.g., clays, mica, talc)

– SiO4 tetrahedra connected together to form 2-D plane

• A net negative charge is associated with each (Si2O5)

2-

unit

• Negative charge balanced by adjacent plane rich in positively charged cations


Rethwisch 9e.

(생략)

Chapter 4 - 36

• Kaolinite clay alternates (Si2O5)2- layer with Al2(OH)4

2+

layer

Layered Silicates (cont)

Note: Adjacent sheets of this type are loosely bound to one another by van der Waal’s forces.


Rethwisch 9e.

(생략)

Chapter 4 - 37

Polymorphic Forms of Carbon

Diamond– tetrahedral bonding of

carbon• hardest material known

• very high thermal

conductivity

– large single crystals –gem stones

– small crystals – used to grind/cut other materials

– diamond thin films• hard surface coatings –

used for cutting tools, medical devices, etc.


Rethwisch 9e.

보석

여러 형태의, 다양한.

정사면체

탄소원자하나가주위 4개의 탄소원자와결합된형태의공유결합

탄소는 다이아몬드와 흑연의 2가지 동소체로 존재하며, 일반적으로 금속, 세라믹, 폴리머재료의분류방법에속하지않으나흑연은세라믹으로분류됨

Chapter 4 - 38

Polymorphic Forms of Carbon (cont)

Graphite– layered structure – parallel hexagonal arrays of

carbon atoms

– weak van der Waal’s forces between layers

– planes slide easily over one another -- good lubricant


Rethwisch 9e.

흑연

Chapter 4 - 39

Crystallinity in Polymers

• Ordered atomic

arrangements involving

molecular chains

• Crystal structures in terms

of unit cells

• Example shown

– polyethylene unit cell


Rethwisch 9e.

결정성

폴리머 재료도 결정질이 존재할 수 있으나 금속이나 세라믹 재료와 달리 분자가 포함되어있기때문에원자배열이더복잡

폴리머 결정성(polymer crystallinity)은 분자사슬이 규칙적으로 적층되어 원자배열이 규칙성을갖도록된상태를의미

결정화도(degree of crystallinity)는 순수 비정질에서 95%의결정질까지가능

금속은거의전체가결정질세라믹은전체가결정질혹은비정질가능

Chapter 4 - 40

• 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

(Courtesy Martin Deakins,

GE Superabrasives, Worthington,

OH. Used with permission.)

Crystals as Building Blocks단결정

기본구조(예: 분자)

Chapter 4 - 41

• 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).

Fig. K, color inset pages

of Callister 5e. (Courtesy of Paul E.

Danielson, Teledyne Wah

Chang Albany)

1 mm

Polycrystals

Isotropic

Anisotropic다결정

하프늄(Hf), 니오븀(Nb), 텅스텐(W)

이방성(비등방성)

grain: 결정립 등방성

Chapter 4 - 42

• Single Crystals

- Properties vary with

direction: anisotropic.

- Example: the modulus

of elasticity (E) in BCC iron:

Data from Table 3.3,


• Polycrystals

- Properties may/may not

vary with direction.

- If grains are randomly

oriented: isotropic.

(Epoly iron = 210 GPa)

- If grains are textured,

anisotropic.

200 μm Adapted from Fig.

6.19(b), Callister &

Rethwisch 9e.

Single vs PolycrystalsE (diagonal) = 273 GPa

E (edge) = 125 GPa

질감이 있게배치됨

Chapter 4 - 43

Polymorphism

• Two or more distinct crystal structures for the same

material (allotropy/polymorphism)

titanium

α, β -Ti

carbon

diamond, graphite

BCC

FCC

BCC

1538° C

1394° C

912° C

δ-Fe

γ-Fe

α-Fe

liquid

iron system

동질이상

동소체/동질이상

동소체에서 결정구조는 외부의 온도와압력에의해결정된다

Chapter 4 - 44

선밀도와면밀도

선밀도와 면밀도는 슬립현상을 이해하는 데 중요하며, 슬립은 면밀도가 가장 높은결정면에서선밀도가가장높은방향으로일어난다.

• 선밀도(linear density, LD): 특정한결정방향의단위벡터상에중심이놓여있는원자들의단위길이에대한개수(예; [110]방향에서의선밀도)

• 면밀도(planar density, PD): 특정한결정면에중심을둔원자들의단위면적당개수(예; FCC와 BCC에서 (110)면의면밀도)

FCC BCC

Chapter 4 - 45

ex: linear density of Al in [110]

direction

a = 0.405 nm

Linear Density

• Linear Density of Atoms LD =

a

[110]

Unit length of direction vector

Number of atoms

# atoms

length

13.5 nm

a2

2LD -==

방향벡터의단위길이

방향벡터상에중심을둔원자수

선밀도

Chapter 4 - 46

Planar Density of (100) Iron

Solution: At T < 912° C iron has the BCC structure.

(100)

Radius of iron R = 0.1241 nm

R3

34a =

2D repeat unit

= Planar Density =a 2

1

atoms

2D repeat unit

= nm2

atoms12.1

m2

atoms= 1.2 x 1019

1

2

R3

34area

2D repeat unit

Fig. 4.2(c), Callister & Rethwisch 9e [from W. G. Moffatt, G. W.

Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. I,

Structure, p. 51. Copyright © 1964 by John Wiley & Sons, New York.

Reprinted by permission of John Wiley & Sons, Inc.]

면밀도 면(100)의뜻

Chapter 4 - 47

Planar Density of (111) IronSolution (cont): (111) plane 1 atom in plane/ unit surface cell

atoms in plane

atoms above plane

atoms below plane

ah2

3=

a2

1

= = nm2

atoms7.0

m2

atoms0.70 x 1019

3 2R3

16Planar Density =

atoms

2D repeat unit

area

2D repeat unit

333

2

2

R3

16R

3

42a3ah2area ====

앞장의 (100)면에비해면밀도가작음

Chapter 4 - 48

결정구조에따른원자적층방법

• FCC와 HCP 원자적층위치

HCP HCPFCC

Chapter 4 - 49

결정구조에따른원자적층방법

• 세라믹원자적층방법

사면체자리(tetrahedral

position): 4개원자틈

팔면체자리(octahedral

position): 6개원자틈

암염(NaCl)의 적층 예:

6개의 음이온 사이에하나의양이온배치

양이온과음이온의비율은 1:1임

Chapter 4 - 50

X선회절: 결정구조의파악• 고체내원자나분자의배열파악은 X-선회절분석으로가능하다

파장이 같은 두 파동이 같은 위상차(phase)를 갖고산란되었다가 경로차가 파장의 정수배를 갖고 간섭을 일으키면 진폭이 합해져 보강간섭이 일어난다.

회절(diffraction)현상의원인이된다.

산란된 두 파동이 같은 위상차(phase)를 갖고 산란되었다가 경로차가파장의절반의정수배가될경우소멸간섭이일어난다.

X-선은고체의원자간격정도의극히짧은파장과높은 에너지를 갖는 전자파의 일종으로 고체 재료에 투사될 때, 빔의 진행 경로에 놓여있는 원자나이온에의해모든방향으로산란된다.

회절보강간섭의결과

Chapter 4 - 51

X-Ray Diffraction

• Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.

• Can’t resolve spacings λ

• Spacing is the distance between parallel planes of atoms.

회절격자

회절방사선

파장보다작게될수없다

Chapter 4 - 52

X선회절과 Bragg의법칙

n 은반사지수로정수이다.

Bragg의법칙

면간거리 dhkl 을측정하여원자간거리를알아낸다.

결정면사이의거리정보는원자의종류를추정하는데도움이됨

Chapter 4 - 53

X선회절과 Bragg의법칙

X-선 Diffractometer(디프렉토미터): 분말 시편의 회절각도를측정하는기기, 결과로부터결정구조를파악

납분말의회절패턴

시편 S를회전시켜가면서회절이생기는각도를측정

X선광원

검출부

Chapter 4 - 54

X-Rays to Determine Crystal Structure

X-ray intensity (from detector)

θ

θc

d =nλ

2 sinθc

• Incoming X-rays diffract from crystal planes.



reflections must be in phase for a detectable signal

spacing between planes

d

θλ

θ

extra distance travelled

by wave “2”

Measurement of

critical angle, , allows

computation of planar

spacing, d.

θ

Chapter 4 - 55

X-Ray Diffraction Pattern

Adapted from Fig. 3.22, Callister 8e.

(110)

(200)

(211)

z

x

ya b

c

Diffraction angle 2θ

Diffraction pattern for polycrystalline α-iron (BCC)

Inte

nsity (

rela

tive)

z

x

ya b

c

z

x

ya b

c

Chapter 4 - 56

Summary

• X-ray diffraction is used for crystal structure and interplanar

spacing determinations.

• 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.

• Some materials can have more than one crystal structure.

This is referred to as polymorphism (or allotropy).

• Ceramic crystal structures are based on:

-- maintaining charge neutrality

-- cation-anion radii ratios.

• Interatomic bonding in ceramics is ionic and/or covalent.

동소체

Chapter 4 - 57

Core Problems:

Self-help Problems:

ANNOUNCEMENTS

Reading:

Chapter 4: The Structure of Crystalline Solids

Documents