chapter 2 crystal structure

8/11/2019 Chapter 2 Crystal Structure

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Chapter 3

Crystal Structure andNoncrystalline Structure


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Learning objectivesDescribe what crystalline and noncrystalline materialsareDraw unit cells for face-centered cubic (FCC), body-centered cubic (BCC) and hexagonal close-packed(HCP) crystal structuresDerive the relationships between unit cell edge lengthand atomic radius for FCC and BCC crystal structuresCompute the densities for metals having FCC and BCCstructures

Write the designation for atom position, direction indicesand Miller indices for cubic crystalsClassify various types of crystalline imperfections (pointdefect, linear defect and planar defect)


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Crystalline and Amorphous

StructureMost of engineering materials arecrystalline atoms are arranged in a

regular and repeating mannerMetals are crystallineMinerals such as celestite (SrSo 4),

amethyst (SiO 2), alloys and some ceramicmaterials are also crystalline Amorphous without form, or non-crystalline such as polymers, glasses andsome metals


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Rare due to poor packing (only Po has this structure)

Simple cubic (SC)


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Face-centered cubic (FCC)

Isolated unit cellHard-sphere unit cell Atomic-site unit cell


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Face-centered cubic (FCC)

Relationship between thelattice constant, a, and

the atomic radius, R.

Typical metals: -Fe, Al, Ni,Cu, Ag, Pt, Au


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Body-centered cubic (BCC)



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Body-centered cubic (BCC)

Relationship between thelattice constant, a, and theatomic radius, R.

Typical metals: -Fe, V, Cr,Mo, W


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Hexagonal close-packed (HCP)

Relationship between edge length and atomic radius : a = 2RTypical metals: Be, Mg, -Ti, Zn, Zr



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R

Atoms/unit cell

Unit cell c ontains:6 x 1/2 + 8 x 1/8

= 4 atoms/unit cell

Unit cell c ontains:1 + 8 x 1/8

= 2 atoms/unit cell

SC BCC FCC

Unit cell contains:8 Corners x 1/8

= 1 atom/unit cell

a

R=0.5a

a a


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Metals density 1. Copper is an fcc metal with an atomic

radius of 0.128 nm. Calculate the densityof copper. Atomic mass of copper is

63.55 g/mol.2. Tungsten is a bcc metal with an atomic

radius of 0.137 nm. Calculate the densityof tungsten. Atomic mass of tungsten is183.85 g/mol


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Atomic packing factor (APF)fraction of unit-cell volume occupied byatoms

Calculate the APF for the BCC and FCC unit cell,assuming the atoms to be hard spheres.


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Theoretical Density, r

where n = number of atoms/unit cell A = atomic weight

V C = Volume of unit cell = a3

for cubic N A = Avogadros number = 6.022 x 10 23 atoms/mol

Density = r =

V C

N A

n Ar =

CellUnitofVolumeTotalCellUnitin AtomsofMass


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Space lattice and unit cell

Crystalline structure regularand repeating

Unit cell structural unit thatis repeated by translation informing a crystalline structure

Lattice constants length of aunit cell edge and/or anglebetween crystallographicaxes


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Seven Crystal

SystemsUnique unit cellshapes that can

be stackedtogether to fill 3-Dspace


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14 Bravais Lattices

Lattice points theoretical points arranged periodically in 3-D space


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Lattice positions

Atom positions in aBCC unit cell


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Directions in the unit cells: Miller indices is a notationsystem in crystallography for planes and directions incrystalShorthand notation1. determine the coordinates of two points2. subtract the coordinates of the tail from the head

3. Clear fraction and reduce the results to lowestintegers4. Enclose the number in a brackets [ ]. If negative signis produced, represent the negative sign with a bar overthe number


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Lattice direction


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Lattice planes Miller indices

1. Identify the points at which the plane intercepts2. Take receprocal of these inetercepts

3. Clear fractions4. Enclose the number in a brackets , no comma [ ]. If negativesign is produced, represent the negative sign with a bar over thenumber


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Crystallographic Planes z

x

ya b

c

4. Miller Indices (110)

example a b cz

x

ya b

c


1. Intercepts 1 1 2. Reciprocals 1/1 1/1 1/

1 1 03. Reduction 1 1 0

1. Intercepts 1/2 2. Reciprocals 1/ 1/ 1/

2 0 03. Reduction 2 0 0

example a b c


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Crystallographic Planes z

x

ya b

c


example

1. Intercepts 1/2 1 3/4a b c

2. Reciprocals 1/ 1/1 1/

2 1 4/3

3. Reduction 6 3 4

(001)(010),

Family of Planes { hkl }

(100), (010),(001),Ex: {100} = (100),


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Single crystal : A material formed by the growth of a crystal nucleus withoutsecondary nucleation or impingement on other crystals; a regular three-dimensional structure extends throughout the material

Polycrystalline materials are solids that are composed of many crystallites ofvarying size and orientation. The variation in direction can be random (calledrandom texture) or directed, possibly due to growth and processing conditions.

Anisotropy: Is the property being directionally dependentModulus of Iron:[100]= 125 MPa[110]= 210 Mpa[111]= 272 MPa

Isotropic: Substances in which measured properties are independent ofdirection. Example; Tungsten , modulus is 384 in all direction [100], [110],[111].


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X-Rays to Determine Crystal Structure

X-ray

intensity(fromdetector)

q

q c

d = n

l2 sin q c

Measurement of

critical angle, qc,allows computation ofplanar spacing, d .

Incoming X -rays diffract from crystal planes.

Adapted from Fig. 3.37,Callister & Rethwisch 3e .

reflections mustbe in phase fora detectable signal

spacingbetweenplanes

d

q l

q extradistancetravelledby wave 2


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X-Ray Diffraction Pattern

Adapted from Fig. 3.20, Callister 5e.

(110)

(200)

(211)

z

x

ya b

c

Diffraction angle 2 q

Diffraction pattern for polycrystalline -iron (BCC)

I n t e

n s

i t y

( r e

l a t i

v e

)

z

x

ya b

cz

x

ya b

c


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Vacancy atoms Interstitial atoms Substitutional atoms

Point defects

Types of Imperfections

Dislocations Line defects

Grain Boundaries Area defects


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Vacancies :

-vacant atomic sites in a structure which is produced when anatom is missing from a normal sites.-Produced at high temperature or by radiation damage-At room temp few vacancies are present, but this numberincreases exponentially as we increase temp.

nv-=n exp (-Q/RT)nv is the number of vacancies per m 3

n is the number of lattice points per m 3

Q is the energy required to produce vacancyR is gas constant and T temp (K)

Point Defects in Metals

Vacancy distortionof planes


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Interstitial defects is produced when an extra atom isinserted into the lattice structure.

-Present as impurities-Once introduced, the number of interstitial atom in the

structure remains the same even the temperature ischanged.

Fig. 5.11, Callister & Rethwisch 3e.

self- interstitial

distortionof planes


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Point Defects in PolymersDefects due in part to chain packing errors and impurities suchas chain ends and side chains

Adapted from Fig. 5.7,Callister & Rethwisch 3e.

Adapted from Fig. 5.7,Callister & Rethwisch 3e.


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Line Defects ( Dislocations ) Are one-dimensional defects around which atoms aremisalignedProduced during solidification or deformation

Edge dislocation: extra half-plane of atoms inserted in a crystal structure

Screw dislocation: spiral planar ramp resulting from shear deformation


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Fig. 5.8, Callister & Rethwisch 3e.

Edge Dislocation


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Edge, Screw, and Mixed

Dislocations

Adapted from Fig. 5.10, Callister & Rethwisch 3e.

Edge

Screw

Mixed


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Significance of dislocations

Slip: The process by which a dislocationmoves and cause a material to deform is

called slipHigher the number of slip system easy tomaterial deform.

Dislocation move to the closed packdirection


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There are certain no of slip system for crystal:FCC=12

BCC=48HCP=3 or higher (depend on temperature),thats some materials shows DBTT (ductilebrittle transition temperature)


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Surface defects: Grain boundaries- boundary between two grain havingdifferent crystallographic orientation

-small grain gives higher strength


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Twin boundary produces during annealing

chapter 2 crystal structure

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