Chapter 1Crystal Structure and

Noncrystalline Structure

Learning objectives

Describe what crystalline and noncrystalline materials are

Draw unit cells for face-centered cubic (FCC), body-centered cubic (BCC) and hexagonal close-packed (HCP) crystal structures

Derive the relationships between unit cell edge length and atomic radius for FCC and BCC crystal structures

Compute the densities for metals having FCC and BCC structures

Write the designation for atom position, direction indices and Miller indices for cubic crystals

Crystalline and Amorphous Structure Most of engineering materials are

crystalline – atoms are arranged in a regular and repeating manner

Metals are crystalline Minerals such as celestite (SrSo4),

amethyst (SiO2), alloys and some ceramic materials are also crystalline

Amorphous – without form, or non-crystalline such as polymers, glasses and some metals

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

Simple cubic (SC)

Face-centered cubic (FCC)

Isolated unit cellHard-sphere unit cellAtomic-site unit cell

Face-centered cubic (FCC)

Relationship between the lattice constant, a, and the atomic radius, R.

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

Body-centered cubic (BCC)

Isolated unit cellHard-sphere unit cellAtomic-site unit cell

Body-centered cubic (BCC)

Relationship between the lattice constant, a, and the atomic radius, R.

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

Hexagonal close-packed (HCP)

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

Isolated unit cellHard-sphere unit cellAtomic-site unit cell

R

Atoms/unit cell

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell

Unit cell contains: 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

Metals’ density

1. Copper is an fcc metal with an atomic radius of 0.128 nm. Calculate the density of 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 density of tungsten. Atomic mass of tungsten is 183.85 g/mol

5

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

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

atoms

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

13

Theoretical Density,

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 = =

VC NA

n A =

Cell Unit of VolumeTotal

Cell Unit in Atomsof Mass

Space lattice and unit cell

Crystalline structure – regular and repeating

Unit cell – structural unit that is repeated by translation in forming a crystalline structure

Lattice constants – length of a unit cell edge and/or angle between crystallographic axes

Seven Crystal Systems Unique unit cell

shapes that can be stacked together to fill 3-D space

14 Bravais Lattices

Lattice points – theoretical points arranged periodically in 3-D space

Lattice positions

Atom positions in a BCC unit cell

Directions in the unit cells: Miller indices is a notation system in crystallography for planes and directions in crystal

Shorthand notation 1. determine the coordinates of two points 2. subtract the coordinates of the tail from the head 3. Clear fraction and reduce the results to lowest

integers 4. Enclose the number in a brackets [ ]. If negative sign

is produced, represent the negative sign with a bar over the number

Lattice direction

Lattice planes – Miller indices

1. Identify the points at which the plane intercepts 2. Take receprocal of these inetercepts 3. Clear fractions 4. Enclose the number in a brackets , no comma [ ]. If negative sign

is produced, represent the negative sign with a bar over the number

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Crystallographic Planes z

x

ya b

c

4. Miller Indices (110)

example a b cz

x

ya b

c

4. Miller Indices (100)

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

1 1 03. Reduction 1 1 0

1. Intercepts 1/2 2. Reciprocals 1/½ 1/ 1/

2 0 03. Reduction 2 0 0

example a b c

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Crystallographic Planesz

x

ya 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),

HOME WORK FOR VLNV-MECH

What are the Miller indices of the cubic crystallographic planes shown?

Single crystal : A material formed by the growth of a crystal nucleus without secondary 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 of varying size and orientation. The variation in direction can be random (called random texture) or directed, possibly due to growth and processing conditions.

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

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

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X-Rays to Determine Crystal Structure

X-ray intensity (from detector)

c

d n

2 sinc

Measurement of critical angle, c, allows computation of planar spacing, d.

• Incoming X-rays diffract from crystal planes.

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

reflections must be in phase for a detectable signal

spacing between planes

d

incoming

X-rays

outg

oing

X-ra

ys

detector

extra distance travelled by wave “2”

“1”

“2”

“1”

“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

Diffraction pattern for polycrystalline -iron (BCC)

Inte

nsity

(re

lativ

e)

z

x

ya b

cz

x

ya b

c

Assignment for VLNV Mech

1. Draw direction vector in unit cubes for the following cubic directions

2. Draw in unit cubes the crystal planes that have the following Miller indices

(a) [111], (b) [110], (c)[121], (d)[113]

(a) (111), (b) (321), (c)(121), (d)(331)

(e) (302), (f) (014), (g)(342)

3. Determine the expected diffraction angle for the first-order reflection from the (310) set of planes for BCC chromium when monochromatic radiation of wavelength 0.0757 nm is used.

4. Using the data for α-iron in Table 3.1, compute the interplanar spacings for the (111) and (211) sets of planes.

5. The metal niobium has a BCC crystal structure. If the angle of diffraction for the (211) set of planes occurs at 75.99° (first-order reflection) when monochromatic x-radiation having a wavelength of 0.1659 nm isused, compute (a) the interplanar spacing for this set of planes, and (b) the atomic radius for the niobium atom.