heat treatment

1. Lecture 2
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1
Imperfections in the Atomic Arrangements
 It has been tacitly assumed that perfect order exists throughout crystalline materials
on an atomic scale.
 However, such an idealized solid does not exist; all contain large numbers of
various defects or imperfections.
 Therefore, the arrangement of the atoms in engineering materials contains
imperfections or defects.
 These defects often have a profound effect on the properties of materials.
 Many of the properties of materials are profoundly sensitive to deviations from
crystalline perfection; the influence is not always adverse.
 Often specific features are purposely created by the introduction of controlled
amounts or numbers of particular defects. In many applications, the presence of
such defects is useful.
 For example, defects known as dislocations are useful for increasing the strength of
metals and alloys; however, in single crystal silicon, used for manufacturing
computer chips, the presence of dislocations is undesirable.
 The imperfections are including; point defects (those associated with one or two
atomic positions); linear (or one-dimensional) defects; and interfacial defects, which
are two-dimensional.
Point defects
o Point defects are localized disruptions in otherwise perfect atomic arrangements in a
crystal structure.
o Even though we call them point defects, the disruption affects a region involving
several atoms (Figure 1).
o Point defects may be introduced by movement of the atoms when they gain energy by
heating, during processing of the material, by introduction of impurities, or alloying
elements.
o The distinction between an impurity and a dopant is as follows:
 Impurities are elements or compounds that are present from raw materials
or processing.

2. Lecture 2
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 Alloying elements are elements that are by design added, in known
concentrations with an intended beneficial effect on properties.
o In general, the effect of impurities is harmful, whereas the effect of alloying elements
on the properties of materials is useful.
Figure 1: Point defects: (a) vacancy, (b) interstitial atom, (c) small substitutional atom,
(d) large substitutional atom,
Vacancies
o The simplest of the point defects is a vacancy, or vacant lattice site.
o A vacancy is produced when an atom is missing from its normal site in the crystal
structure (Figure 1 (a) and Figure 2).
o All crystalline materials have vacancy defects and, in fact, it is not possible to create
such a material that is free of these defects.
o Vacancies are introduced into metals and alloys during solidification, at high
temperatures.
o Vacancies play an important role in determining the diffusion rate in a solid material.

3. Lecture 2
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Figure 2: Two-dimensional representations of a vacancy and a self-interstitial
o At room temperature (~298 K), the concentration of vacancies is small, but the
concentration of vacancies increases exponentially as the temperature increase, as
shown by Arrhenius equation:
where:
Nv is the number of vacancies per cm3
;
N is the number of atoms per cm3
;
T is the temperature in (K).
Qv is the energy required to produce one mole of vacancies, in cal/mol or
Joules/mol. (or Qv is the energy required for the formation of a vacancy, J/atom, or
eV/atom).
The value of k is depending on the units of Qv
k is the gas constant (Boltzmann’s constant), 1.987 cal/mol.K or 8.31 Joules/mol. K.
(or, 1.38 x 10-23
J/atom.K, or 8.62x 10-5
eV/atom.K).
o Due to the large thermal energy near the melting temperature, there may be as many as
one vacancy per 1000 atoms.
o Note that this equation provides for equilibrium concentration of vacancies at a given
temperature.
Vacancy
Self-interstitial

4. Lecture 2
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Self-interstitial
 A self-interstitial is an atom from the crystal that is crowded into an interstitial site, a
small void space that under ordinary circumstances is not occupied (Figure 2).
 In metals, a self-interstitial introduces relatively large distortions in the surrounding
lattice because the atom is substantially larger than the interstitial position in which it
is situated.
 Consequently, the formation of this defect is not highly probable, and it exists in very
small concentrations, which are significantly lower than for vacancies.
Example
Calculate the concentration of vacancies in copper at room temperature (25°C).
What temperature will be needed to heat treat copper such that the concentration of
vacancies produced will be 1000 times more than the equilibrium concentration of
vacancies at room temperature? Assume that 20,000 cal are required to produce a
mole of vacancies in copper. The lattice parameter of FCC copper is 0.36151 nm.
Solution
The lattice parameter of FCC copper is 0.36151 nm. There are four atoms per unit
cell; therefore, the number of copper atoms per cm3
is
At room temperature, T = 25 + 273 = 298 K:
Nv = 1.814 * 108
vacancies/cm3
.
We wish to find a temperature that will lead to a concentration of vacancies that is
1000 times higher than this number, or Nv = 1.814 * 1011
vacancies/cm3
.

5. Lecture 2
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Example
Calculate the equilibrium number of vacancies per cubic meter for copper at
1000o
C. The energy for vacancy formation is 0.9 eV/atom; the atomic weight and
density (at 1000 o
C) for copper are 63.5 g/mol and 8.4 g/cm3
, respectively.
Solution
It is first necessary, however, to determine the value of N, the number of atomic
sites per cubic meter for copper, from its atomic weight ACu, its density ρ, and
Avogadro’s number
NA, according to
Thus, the number of vacancies at 1000o
C (1273 K) is equal to

6. Lecture 2
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Interstitial Sites
 In all crystal structures, there are small holes between the usual atoms into which
smaller atoms may be placed. These locations are called interstitial sites.
 An atom, when placed into an interstitial site, touches two or more atoms in the lattice.
This interstitial atom has a coordination number equal to the number of atoms it
touches.
 Figure 3 shows interstitial locations in the SC, BCC, and FCC structures.
Figure 3: The location of the interstitial sites in cubic unit cells.
 The cubic site, with a coordination number of eight, occurs in the SC structure at the
body-centered position (
1
2
1
2
1
2
).
 Octahedral sites give a coordination number of six (not eight). They are known as
octahedral sites because the atoms contacting the interstitial atom form an octahedron.
 Tetrahedral sites give a coordination number of four.
 Example, the octahedral sites in BCC unit cells are located at the faces of the cube; a
small atom placed in the octahedral site touches the four atoms at the corners of the
face, the atom at the center of the unit cell, plus another atom at the center of the
adjacent unit cell, giving a coordination number of six.
 In FCC unit cells, octahedral sites occur at the center of each edge of the cube, as well
as at the body center of the unit cell.
 Interstitial atoms whose radii are slightly larger than the radius of the interstitial site
may enter that site, pushing the surrounding atoms slightly apart.
 If the interstitial atom is too large, it prefers to enter a site having a larger coordination
number (Table 3-6).
 Therefore, an atom with a radius ratio between 0.225 and 0.414 enters a tetrahedral
site; if its radius is somewhat larger than 0.414, it enters an octahedral site instead.

7. Lecture 2
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Table 1: The coordination number and the radius ratio
Example
Calculate the number of octahedral sites that uniquely belong to one FCC unit cell.
Solution
The octahedral sites include the twelve edges of the unit cell, with the coordinates
plus the center position, (
1
2
1
2
1
2
).
Each of the sites on the edge of the unit cell is shared between four unit cells, so only 1/4 of
each site belongs uniquely to each unit cell. Therefore, the number of sites belonging
uniquely to each cell is

8. Lecture 2
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Substitutional Defects
 A substitutional defect is introduced when one atom is replaced by a different type
of atom as in Figure 1(c) and (d).
 The substitutional atoms occupy the normal lattice site.
 Substitutional atoms may either be larger than the normal atoms in the crystal
structure, in which case the surrounding interatomic spacings are reduced, or
smaller causing the surrounding atoms to have larger interatomic spacings.
 In either case, the substitutional defects disturb the surrounding crystal.
 The substitutional defect can be introduced either as an impurity or as a deliberate
alloying addition (Alloying element), and, once introduced, the number of defects is
relatively independent of temperature.
 Examples of substitutional defects include incorporation of dopants such as
phosphorus (P) or boron (B) into Si.
 Similarly, adding copper (Cu) to nickel (Ni), copper atoms will occupy
crystallographic sites where nickel atoms would normally be present.
 The substitutional atoms will often increase the strength of the metallic material.