1

CHAPTER 4CHAPTER 4

Crystal StructureCrystal Structure ：：Fundamental Crystallography, Crystal StructuFundamental Crystallography, Crystal Structu

re of Metals and X-Ray Diffractionre of Metals and X-Ray Diffraction

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4.1 Introduction

◎ In this chapter we expand our view of materials to

incorporate larger numbers of atoms

◎Crystallinity: Crystalline solids and Amorphous solids

． Amorphous solids ： show SRO (short range order )in

three dimensions , but no LRO (long range order)

． Crystalline solids ： exhibit both SRO and LRO in

three dimensions F3.22

structure

3

4.2 Crystal System to name the crystalline structure of a material,

to identify the crystalline structure of a material

to view the crystal to be composrd of lattice

point

◎ Crystal structure = Lattice + basis

． Lattice : arrangement of lattice points ． Lattice point≡ basis

4

The Basis

A crystal contains a structural unit, called the basis (or motif),that is repeated in three dimensions to generate the crystal structure.

The environment of each repeated unit is the same throughout the crystal (neglecting surface effects).

The basis may be a single atom or molecule, or it may be a small group of atoms, molecules, or ions.

Each repeated basis group has the same structure and the same spatial orientation as every other one in the crystal.

The basis must have the same stoichiometric composition as the crystal. f24.3

5

For NaCl, the basis consists of one Na+ ion and one Cl- ion. For Cu, the basis is a single Cu atom. For Zn, the basis consists of two Zn atoms. For diamond the basis is two C atoms; the two atoms of the basis are each surrounded tetrahedrally by four carbons, but the four bonds at one basis atom differ in orientation from those at the other atom. For CO2, the basis is four CO2 molecules. For benzene, the basis is four C6H6 molecule.

The Space Lattice.

If we place a single point at the same location in each repeated basis group, the set of points obtained forms the (space ) lattice of the crystal.

f13.2 f24.15 f3.7-2

f3.7-3 f3.7-4

f24.3

6

The Unit Cell.

The space lattice of a crystal can be divided into identical parallelepipeds by joining the lattice points with straight lines. (A parallelepiped Is a six-sided geometrical solid whose faces are all parallelograms.) each such parallelepiped is called a unit cell.

In crystallography, one chooses the unit cell so that it has the maximum symmetry and has the smallest volume consistent with the maximum symmetry; the maximum-symmetry requirement implies the maximum number of perpendicular unit-cell edges.

In two dimensions, a unit cell is a parallelogram with sides of length a and b and angle between these sides. In three dimensions, a unit cell is a parallelepiped with edges of length a、 b、 c and angles α 、 β 、 γ where α is the angle between edges b and c , etc.

f24.4

F3.4

7

Crystal System

In 1848 Bravais showed that there are 14 different kinds of lattices in three dimensions. The 14 Bravais lattices are grouped into seven crystal systems, based on unit-cell symmetry.

Unit cells that have lattice points only at their corners are called primitive (or simple) unit cells. Seven of the Bravais lattices have primitive (p or S) unit cells.

A body-centered lattice (denoted by the letter I or B, from the German Innenzen-trierte) has a lattice point within the unit cell as well as at each corner of the unit cell.

A face-centered (F) lattice has a lattice point on each of the six unit-cell faces as well as at the corners.

f24.5

f24.5 T3.2

8

The Letter c denotes an end-centered lattice with a lattice point

on each of the two faces bounded by edges of lengths a and b.

Number of lattice point per unit cell.

Each point at a unit-cell corner is shared among eight adjacent

unit cells in the lattice: four at the same level and four immediately

above or below. Therefore a primitive unit cell has 8/8 =1 lattice

point and 1 basis group per unit cell. Each point on a unit-cell face

is shared between two unit cells, so an F unit cell has 8/8+6/2 =4

lattice points and 4 basis groups per unit cell. F302 F301

9

4.3 CRYSTALS WITH ONE ATOM PER LATTICE SITE

AND HEXAGONAL CRYSTALS (usually metals)

Lattice Arrangement of Atoms

◎ Number of atoms per basis (lattice point)

◎ Number of atoms per lattice site

4.3.1 Body-centered Cubic Crystal

◎ ( BCC ) structure :e.g., tungsten

chromium , iron , molybdenum , and vanadium.

◎ a0 /2=2r , or a0 (BCC)=4r/ (3.3)

(a0 a , r R)

◎ The total number of atoms per unit cell is two

{[8× (1/8)]+(1× 1)} ; CN(BCC)=8

3 3

F302 f3.3-1 f3.3-2

body-centered cubic BCC

basis

10

4.3.2 Face-centered Cubic Crystals

◎ Face-centered Cubic ( FCC ) structure : e.g. , aluminum ,

calcium , cooper , gold , lead , platinum , and silver.

( cubic close – packed structure )

◎ (FCC)=4r/ (3.1)

◎ There are four atoms per FCC cell {[8× (1/8)]+[(6×

1/2)]} , CN(FCC)=12

4.3.3 Hexagonal Close-packed structure

◎ Hexagonal close-packed ( HCP ) structure : e.g. ,

cadmium , cobalt , magnesium , titanium , yttrium ,

and zinc.

ao 2

F301

F303

f3.3-3

f3.3-4

Face-centered Cubic FCC

cubic close – packed

Hexagonal close-packed

HCP

Hexa

11

◎ The total number of atoms in the large HCP cell is six

{[12× (1/6)]+[2× (1/2)]+(3× 1)} , CN(HCP)=12.

◎ =2r

◎ c=(4/ ) =1.633 =3.266r (3.3-3)

◎ (large HCP)=( ) (3.3-4)

4.4 MILLER INDICES

4.4.1 Coordinates of Points

◎ Align the three coordinate axes with the edges of the

unit cell , with the origin at a corner of the cell.

E3.4 E3.5

ao

aoao

V uc 23 C2ao

F305

6

12

4.4.2 Indices of Directions

◎Miller indices for directions are obtained using the

following procedure:

※Method one

(1) Determine the coordinates of two points that lie in

the direction of interest ： h1, k1, l1, and h2, k2, l2.

(2) h = h2 － h1; k = k2 － k1 ; l = l2 － l1 .

(3) Clear fractions from the differences - h, k , and

l , to give indices in lowest integer values nh , nk , nl.

F306

• h ： k ： l determines the direction• h ： k ： l = nh ： nk ： nl

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(4) [nh’ nk’ nl’]

(5) For negative integers , as an example , if h<0 , we

write [h k l].

※Method two

(1) A vector of convenient length is positioned such that

it passes through the origin of the coordinate system.

(2) The length of the vector projection are measured in

terms of the unit cell dimensions a , b , c.

(3) These numbers are reduced to the smallest integer

values.

E3.4-2

E3.6 E3.7

14

◎ Families of directions : a cubic crystal as an example all the edges are equivalent, all the face diagonals are equivalent , all the body diagonals are equivalent , thus families of direction : <100> , <110> , <111>.

<100>: [100], [ī00], [010], [0 ī 0], [001] and [00 ī]

<110>: [110], [101], [011], [ ī10], [ī01], [0ī1],[1ī0],

[10ī], [01ī], [īī0], [ī0ī] and [0īī]

<111>:

(4) The three indices , are enclosed in square brackets , thus : [uvw].

A

15

◎ The angle between direction.

If A= ui + vj + wk and B= i+ j+ k , then

A·B= A B cos (3.4-1)

= {(u +v +w )/[( + + ( + + ]}

(3.4-2)

`u `v `w

1cos `u `v `w 2u 2v 2w 21

) 2`u 2`v 2`w 21

)

E3.4-4

• Directions in cubic crystals having the same indices without regard to order or sign (e.g., [123], [213], and [123]), are equivalent.

• Only arrangement of atoms (or ions) are important for directions.

• Directions (with different indices) having similar atomic arrangement are equivalent.

16

What is projection?What is projection?

What is the projection of OC on X axis? On Y axis?What is the projection of OC on X axis? On Y axis?

What is the projection of AB on X axis? On Y axis?What is the projection of AB on X axis? On Y axis?

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For a cubic crystalFor a cubic crystal

<100>: [100], [ī00], [010], [0 ī 0], [001] and [00 ī]

How many edges does a cubic unit cell have?How many edges does a cubic unit cell have?

The same: Each set of the four parallel edges are the same and The same: Each set of the four parallel edges are the same and

can only be counted as one member of the family. But can only be counted as one member of the family. But

its opposite direction is counted as another family its opposite direction is counted as another family

member.member.

Equivalent:Equivalent: [100], [ī00], [010], [0 ī 0], [001] and [00 ī]

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4.4.3 Indices of Planes

◎ Miller indices for planes :

1. Identify the coordinates at which the plane intersects

the x , y , and z axes

2. Take the reciprocal of the intercepts

3. Clear fractions

4.Cite planes in parentheses : (h k l )

F3.9

E3.9 E3.10 E3.4-6 E3.4-7

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◎ Families of planes , {h k l} : all planes in a family are

equivalent in that they contain exactly the same

arrangement of atoms . In cube systems, for example ,

the members of {1 0 0} are (1 0 0) , (0 1 0) , (0 0 1) , and

their negatives (ī 0 0) , (0 ī 0) , and (0 0 ī);

and {111}: (ī ī ī), (ī 11), (1 ī ī), (11 ī), (ī ī1), (ī 1 ī),

(1 ī 1), and (111),

◎ Only arrangement of atoms are important for planes.

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※ Several important features and relationships :

1. Planes and their negatives are equivalent . The

negatives of directions are not equivalent but

rather point in opposite directions.

2. Planes are not necessarily equivalent to their

multiples . Directions are invariant to a multiplier.

3. In cubic crystals , a plane and a direction with the

same indices are orthogonal.

F 19.9

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4.4.4 Indices in the Hexagonal System

Miller-Bravais coordinate system :

f3.16

Utilizing a four-axis, or Miller-Bravais coordinate system

The three a1, a2, and a3 axes are within a single plane (called the basal plane). The z axis is perpendicular to this basal plane.

Four indices, as [uvtw]; the first three indices pertain to projections along the respective a1, a2, and a3 axes in the basal plane.

◎ Direction

F3.3 F3.7

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Example

aa11 : projections: 1, - : projections: 1, -½, -½, 0½, -½, 0 [2 [2ī ī0ī ī0]]

23

uvn

v 23

(3.6b)

vut

wnw

(3.6c)

(3.6d)

n is a factor that may be required to reduce u, v, t, and w to the smallest integers.

E3.8

Conversion from the three-index system to the four-index system,

uvtwwvu vun

u 23

(3.6a)

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◎ Planes four-index (hkil) scheme,

Basal planes: (0001)Basal planes: (0001)

intercepts: aintercepts: a11, , ；； aa22, , ；； aa33, , ；； C, 1.C, 1.

h = 0, k = 0, i = 0, l = 1h = 0, k = 0, i = 0, l = 1

ExampleExample

∞ ∞ ∞

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hkilhkl

khi

The three h, k, and l indices are identical for both indexing systems.

(3.7)E3.11E3.11

Prism plane(ABCD): (10ī0)Prism plane(ABCD): (10ī0)

intercepts: a1,+1; a2,intercepts: a1,+1; a2,; a3,-1; C,; a3,-1; C,..

h = 1, k = 0, i = -1, l = 0h = 1, k = 0, i = -1, l = 0

prism plane (DCGH): (01ī0)prism plane (DCGH): (01ī0)

family of prism planes: {10ī0}family of prism planes: {10ī0}

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4.5 DENSITIES AND PACKING FACTORS

OF CRYSTALLINE STRUCTURES

4.5.1 Linear Density

◎ linear density ( or LD) is the number of equivalent

lattice points per unit length along a direction.

=

L

L ( )

( of the line contained

Length within one unit c )ell

along direction within one unit cellNumber of atoms

L LD( )

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As an example , consider the [1 1 0] direction in an FCC

crystal , the number of atoms is 2 [i .e . , 2×(1/2)+(1×1)] ,

the length of the line is 4r. for [1 1 0] in FCC is 1/(2r).

◎ The <110> family of directions has special significance

in the FCC structure , since these are the directions in

which atoms are in direct contact. As such , <110>

directions have the highest of any directions in the

FCC system . In any crystal system the directions with

the highest are termed the close-packed directions.

L

L

L

F312

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4.5.2 Planar Density

◎ Planar density ( or PD) is the number of atoms

per unit area on a plane of interest.

=

◎ Consider the planar density of the (1 1 1) plane in a FCC

crystal . The length of the side of each triangle is 4r . The

area is 4 ; the three atoms at the corners , 1/6

(i . e ., ) the three atoms along the edges , fraction

of ½ . The total number of atoms on the plane is two .

Planar density is 1/(2 ). f3.5-2

p

p ( on a plane within one unit cell)

( of the Plane contained

Number of atoms

Area within one unit cell)

2r 3

3 2r

o

o

36060

p PD( )

29

◎ This value of represents the highest possible planar

density for spherical atoms , any plane in any crystal

system that has a value of =1/(2 ) will be

referred to as a close-packed plane.

◎ For any specific crystal system , a family of planes

with a maximum value are referred to as highest

-density planes.

p

p 32r

p

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4.5.3 Volumetric Density

◎ Volumetric density ( or VD ) is the number of atoms

per unit volume.

◎ for FCC is 4/(16 )=1/(4 ) . This is the highest

volumetric density possible for spherical atoms . This

type of structure will be referred to as close-packed

structures . HCP crystals also have = 1/(4 )

f3.3-3

v

v

23r 2

3r

v 2

3r

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4.5.4 Atomic Packing Factors and Coordination

Numbers

◎ The ratio of the volume occupied by the atoms to the

total available volume is defined to be the atomic

packing factor (APF)

APF =

◎ APF(SC)=0.52 , APF(BCC)=0.68 , and APF(FCC)=

APF(HCP)=0.74.

◎ CN=12 for APF=0.74.

( of atoms in the unit cell)

( of the

Volume

V unit colume ell)(3.2)

F3.2

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4.5.5 Theoretical Density ,

A knowledge of the crystal structure of a metallic solid perA knowledge of the crystal structure of a metallic solid permits computation of its theoretical density through the relmits computation of its theoretical density through the relationshipationship

wherewhere

n = number of atoms associated with each unit celln = number of atoms associated with each unit cell

A = atomic weightA = atomic weight

VVcc = volume of the unit cell = volume of the unit cell

NNAA = Avogadro’s number (6.023 × 10 = Avogadro’s number (6.023 × 102323 atoms/mol) atoms/mol)

4.5.6. % Theoretical density 4.5.6. % Theoretical density (( = / 100% )= / 100% )

th

thc A

nA

V N (3.5)(3.5)

thmeasured

F 4.12

33

4.5.6 Close-Packed Structures

◎ Both FCC and HCP structures are characterized by an

APF 74% , a CN of 12 , the close-packed planes , the

close-packed directions and the difference is the

arrangement of their close-packed planes.

◎ FCC ： cubic close-packed structure

◎ HCP ： hexagonal close-packed structure

f3.3-4 f3.5-3 f3.5-4

117 112 67 122

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4.8 CRYSTALLINE AND NONCRYSTALLINE

MATERIALS

Solids: F3.22

crystalline : with long rang order (of course , also with

short rang order)

semicrystalline : with some long range order

amorphous : with short range order only

166

35

Crystalline solid: f4-2.2 f4-2.3 f3-18

(1)sharp melting point

(2)well-developed faces and a characteristic shape

(3)Characteristic X-ray diffraction pattern (peaks)

shows a regular , ordered structure composed of

identical repeating units having the units having the

same orientation throughout the crystal (with short

and long-range order) motion

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Amorphous solid: f6-2.1

(1)When heated , it softens and melts over a wide

temperature range

(glass-transition temperature)

(2)does not have a characteristic crystal shape

(3)No characteristic X-ray diffraction peaks.

Semicrystalline solids:

(1)crystalline polymers

(2)Liquid crystal : fluids that show some degree of

long-range order. f24.8

F15-13 motion

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4.9 SINGLE CRYSTALS AND POLYCRYSTALLINE 4.9 SINGLE CRYSTALS AND POLYCRYSTALLINE

MATERIALSMATERIALS

◎Crystal:

• Single crystal

The whole piece (or body) of material has the same crystal structure and orientation.

• Polycrystalline

made up of a number of crystals (small single)with identical structures but different orientations.

※ grains : small (single)crystals 0.5 – 50 m. F3.17 f3.9-1

※Grain boundaries : internal surfaces of finite thickness where

crystals of different orientations meet.

f4.12

38

◎Many ceramic materials are in the form of polycrystalline

solids.

◎Noncrystalline structure (amorphous) can be formed by

cooling a material sufficiently quickly (quenching) that crystal

formation does not occur.

◎Polymers are usually either semicrystalline or amorphous.

◎Few materials are used in single-crystal form, single-crystal

materials have no grain boundaries, so they offer unique

mechanical, optical, and electrical properties.

◎Amorphous materials do not have grain boundaries either.

＊ Grain boundaries can scatter photons (light), phonons (heat)

and electrons, rendering the materials to be less transparent,

less thermally and electrically conductive.

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Examples :

1. Single-crystal quartz (SiO ) and perovskites are used as

transducers, such as : telephone receivers, and

phonograph cartridges.

2. Single-crystal germanium and silicon :

microelectronics industry.

3. Single-crystal nickel alloys : turbine blades.

4. Sapphire (A1 O ) and diamond (C) single crystals :

precious stones.

2

2 3

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4.10 POLYMORPHISM4.10 POLYMORPHISM

◎ Materials whose crystal structures change from one unit

cell to another at specific temperatures are termed polymorphic.

Iron, for example,

Fe (BCC, room temp) Fe ( FCC at 912 ) ℃

(BCC , 1394 )℃

Other examples : Silica carbon, (SiO2), alumina (A12O3), and

titania (TiO2)

◎ Property changes accompany structural changes : e.g., volume,

density, may increase or decrease.

912 ℃

1394 ℃

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◎ Many brittle materials cannot withstand the internal forces that develop as a result of these volume changes. These materials fail at the transformation temperature. An example is zirconia, ZrO2 :

Tetragonal monoclinic crack !!

◎ Another polymorphic materials is carbon.

1. The diamond structure has a 3-D tetrahedral network of covalent bonds with CN = 4, the highest melting temperature of any of the elements.

2. Graphite has a hexagonal two-dimensional layered structure. The bonds within each layer are covalent bonds, but the interlayer are weak secondary bonds which are easily broken giving graphite its excellent lubricating properties.

cooling, ~ 1000 ℃

f3.10-1

p.48

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4.11 ANISOTROPY4.11 ANISOTROPY

Isotropic : properties of a material are independent of direction :

Anisotropic : properties of a material depend on direction :

◎ properties, such as modulus of elasticity and coefficient of

thermal expansion can be estimated from the bond-energy

curve. But the curve is derived in the close-packed directions.

The separation distance between atoms in any direction other

than the close packed directions will be greater than .

◎ Single crystals exhibit some degree of anisotropy.

(because the atomic packing depends on direction.)

f0103F2.8

43

◎ An example of a highly anisotrophic crystalline material is

graphite. Its coefficient of thermal expansion along the c axis

is more than 25 times greater than that in any direction

parallel to the basal planes.

◎ While single crystals are anisotrophic, most polycrystals are

nearly isotropic on the macroscopic scale.

◎ Anisotropic can also occur on a larger structural scale

examples : wood, steel-reinforced concrete, carbon-fiber-

reinforced epoxy, oriented polymers.

f3.10-1

F3.17 F4.12

44

4.12 DETERMINATION OF CRYSTAL STRUCTURES4.12 DETERMINATION OF CRYSTAL STRUCTURES

X-Ray Diffraction f3-17 f24.17 f3.15

X-Ray Diffraction is generated by interaction between X-ray photons and electrons in the material.

Figure 24.17b shows the x-ray beam incident at angle to one of the (210) planes. Most of the x-ray photons will pass through this plane with no change in direction, but a small fraction will collide with electrons in the atoms of this plane and will be scattered, that is, will undergo a change in direction. The x-ray photons are scattered in all directions.

The zero-path-difference condition gives 0 = pq (cos - cos ), so = . Thus, waves scattered from a plane of lattice points at an angle equal to the angle of incidence are in phase with one another. Waves scattered at other angles will generally be out phase with one another and will give destructive interference.

F3.18 F3.19

f3.18

45

The single plane of lattice points acts as a “mirror” and “reflects” a small fraction of the incident x-rays.

The x-ray beam will penetrate the crystal to a depth of millions of layers of planes. For constructive interference between x-rays scattered by the entire set of planes, the waves constructively scattered (“reflected”) by two adjacent planes must have a path difference of an integral number of x-ray wavelengths.

2d sin = n n = 1,2,3,…….hkl

The Bragg equation is the fundamental equation of x-ray crystallography.

f3.12-2

n=1 1st order diffraction

n=2 2nd order diffraction

46

For a set of planes to give a diffracted beam of sufficient intensity to be observed, each plane of the set must have a high density of electrons;

This requires a high density of atoms, so each plane must have a high density of lattice points. (i.e., Planes with low indices). Because the number of sets of such planes in limited, the number of values of d is limited and the Bragg condition will be met only for certain values of

X-ray crystallographers prefer to write in the form

2dnh, nk, nl sin =

Lattice Geometry

The distance L between the points x1, y1, z1 and x2, y2 and z2, is L=[(x1-x2)2a2 + (y1-y2)2b2 + (z1-z2)2c2 + 2(x1+x2) (y1-y2) ab cos + 2 (y1-y2) (z1-z2) bc cos + 2 (z1-z2) (x1-x2) ca cos ]1/2

hkl

F19.6 F 19.9F21-3

47

For a cubic crystal this equation simplifies to

L = a [ (x2-x1)2 + (y2-y1)2 + (z2-z1)2 ]1/2

The volume of a unit cell is given by

V = abc (1-cos2 - cos2 - cos2 + 2 cos cos cos )1/2

If the unit cell is cubic, orthorhombic, or tetragonal, this equation reduces to v = abc

The perpendicular distance d between adjacent planes of a set is given by

d = v [h2b2c2sin2 + k2a2c2sin2 + 12a2b2sin2 + 2hlab2c

(cos cos – cos ) + 2hkabc2 (cos cos – cos ) + 2kla2bc

(cos cos – cos )] –1/2

where v is the unit cell volume

48

For the special case that the unit cell axes are mutually perpendicular (i.e., for orthorhombic, tetragonal, and cubic unit cells), i.e., α=β=γ=90°

dnh,nk,nl =

2d sin = n

2/12

2

2

2

2

2

,,)

a

h(

1

c

l

b

k

d lkh

n

d lkh ,,

2/12

22

2

22

2

22

,,)(

1

c

ln

b

kn

a

hn

d nlnknh

sin,,2sin)(2,,

nlnknhlkn

dn

d

f19.9

hkl

49

For cubic unit cells: a = b = c

(for n = 1)

Interplanar Spacing (dh,k,l) of Cubic Lattice

(1) Simple cubic

For primitive cubic crystals dhkl may have the following values:

(100) (110) (111) (200) (120) (112) (220)

θ1 θ2 θ3 θ4 θ5 θ6 θ7

sin2 ,, lkhd

2 2 2hkl

ad

h k l

, / 2, / 3, / 4, / 5, / 6, / 8, .,a a a a a a a etc

2 2 2

2sin

a

h k l

50

Hence, only the reflections 111, 200, 311, 222, 400, 331, 420, etc., are observed. Consequently, the spacing corresponding to the hkl reflections that may appear on a powder diagram of a face-centered cubic crystal are:

.,,20/,19/,16/,12/,11/,8/,4/,3/ etcaaaaaaaa

f19.10

(2) Face-centered cubic (FCC) The diffraction patterns of face-centered crystals show an absence of all reflections for which the indices hkl are not all even or all odd.

51

(3) Body-centered cubic (BCC) An examination of the diffraction data for body ecentered crystals shows that hkl reflections for which the sum h+k+l is odd are not observed. Accordingly, the interplanar spacing found for a body-centered cubic lattice are which are the distances between the (110), (200), (211), and (220) planes.

Other XRD Phenomena and Techniques

.,,8/,6/,4/,2/ etcaaaa

f19.8f19.8 f19.11f19.11 f3.12-3f3.12-3 F3.20F3.20 F3.21F3.21

52

The locations of the spots in the diffraction pattern depend only on the nature of the spae lattice and the lengths and angles of the unit cell and that the intensities of the spots depend on the kinds of atoms and the locations of atoms (interactomics distances) within the basis of the crystal structure.

f3.5 f3.6 f3.11 f13.12 f3.13

Analysis of the locations of the diffraction spots gives the space lattice and the lengths and angles of the unit cell. The unit-cell content z is calculated using the know crystal density. Analysis of the electron probability density (x, y, z) as a function of position in the unit cell. From contours of , the locations of the nuclei are obvious, and bond lengths and angles can be found.

53

The Use of x-Ray DiffractionThe Use of x-Ray Diffraction

(1) Identify the crystalline structure of a sample with known chemical composition.

Arrangement and the spacing of atoms in crystalline materials

(2) Identify the compound (chemical composition) and crystalline structure of an unknown sample.

x-Ray diffraction also provides compounds, an x-ray diffraction pattern is unique for each crystalline substance. Thus, if an exact match can be found between the pattern of unknown and an standard sample, chemical identity can be assumed.

(3) Estimate the relative amounts of different phases

Quantitative information concerning a crystalline compound in a mixture, for example, the percentage of graphite in a graphite charcoal mixture.

165

Types of motion and energy

(sensitive to T : motion and energy T ) (unit ofmotion orparticles underconsideration)

gas

liquid solid 

single atoms

(di- or many atomic) molecules

Translation (atoms or molecules)

○ ○

○ ×

Rotation(atoms)

× ○

○

×

Vibration(atoms or bondings)

× ○

○

○

Electronic(electrons)

○ ○

○

○

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167

III. Structure of MaterialsIII. Structure of Materialsraw materials synthesis

shape forming sintering fabrication Material

products

Processing techniques and operation conditions Structure

( 結構 )

Properties

( 性質 )

Structure ：Atomic scale structure (~0.1nm)

types of atoms, types of bondings, coordination number

Crystalline structure (>~10nm)

Microscopic (scale) structure (~0.1µm) ： SEM

Macroscopic structure (~0.1cm)

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