Unit-1
Crystallography:
is the experimental science of determining the arrangement of atoms in the crystalline solids
(see crystal structure). The word "crystallography" derives from
the Greek words crystallon "cold drop, frozen drop", with its meaning extending to all solids
with some degree of transparency, and grapho "I write". In July 2012, the United
Nations recognised the importance of the science of crystallography by proclaiming that 2014
would be the International Year of Crystallography.[1] X-ray crystallography is used to determine
the structure of large biomolecules such as proteins.
Before the development of X-ray diffraction crystallography (see below), the study
of crystals was based on their geometry. This involves measuring the angles of crystal faces
relative to theoretical reference axes (crystallographic axes), and establishing the symmetry of
the crystal in question. The former is carried out using a goniometer. The position in 3D space of
each crystal face is plotted on a stereographic net such as a Wulff net or Lambert net. The pole to
each face is plotted on the net. Each point is labelled with its Miller index. The final plot allows
the symmetry of the crystal to be established.
Crystallographic methods now depend on analysis of the diffraction patterns of a sample targeted
by a beam of some type. X-rays are most commonly used; other beams used
include electrons or neutrons. This is facilitated by the wave properties of the particles.
Crystallographers often explicitly state the type of beam used, as in the terms X-ray
diffraction, neutron diffraction and electron diffraction.
These three types of radiation interact with the specimen in different ways. X-rays interact with
the spatial distribution of electrons in the sample, while electrons are charged particles and
therefore feel the total charge distribution of both the atomic nuclei and electrons of the sample.
Neutrons are scattered by the atomic nuclei through the strong nuclear forces, but in addition,
the magnetic moment of neutrons is non-zero. They are therefore also scattered by magnetic
fields. When neutrons are scattered from hydrogen-containing materials, they produce diffraction
patterns with high noise levels. However, the material can sometimes be treated to
substitute deuterium for hydrogen. Because of these different forms of interaction, the three
types of radiation are suitable for different crystallographic studies
Crystallography is a useful method for materials scientists. In single crystals, the effects of the
crystalline arrangement of atoms is often easy to see macroscopically, because the natural shapes
of crystals reflect the atomic structure. In addition, physical properties are often controlled by
crystalline defects. The understanding of crystal structures is an important prerequisite for
understanding crystallographic defects. Mostly, materials do not occur as a single crystal, but in
poly-crystalline form (i.e., as an aggregate of small crystals with different orientations). Because
of this, the powder diffraction method, which uses diffraction patterns of polycrystalline samples
with a large number of crystals, plays an important role in structural determination.
Other physical properties are also linked to crystallography. For example, the minerals
in clay form small, flat, platelike structures. Clay can be easily deformed because the platelike
particles can slip along each other in the plane of the plates, yet remain strongly connected in the
direction perpendicular to the plates. Such mechanisms can be studied by
crystallographic texture measurements.
In another example, iron transforms from a body-centered cubic (bcc) structure to a facecentered cubic (fcc) structure called austenite when it is heated. The fcc structure is a closepacked structure unlike the bcc structure; thus the volume of the iron decreases when this
transformation occurs.
Crystallography is useful in phase identification. When performing any process on a material, it
may be desired to find out what compounds and what phases are present in the material. Each
phase has a characteristic arrangement of atoms. X-ray or neutron diffraction can be used to
identify which patterns are present in the material, and thus which compounds are present.
Crystallography covers the enumeration of the symmetry patterns which can be formed by atoms
in a crystal and for this reason has a relation to group theory and geometry
MILLER INDICES:
In Solid State Physics, it is important to be able to specify a plane or a set of planes in the
crystal. This is normally done by using the Miller indices. The use and definition of these Miller
indices are shown in fig. 9. Figure 9 This plane intercepts the a, b, c axes at 3a, 2b, 2c. The
reciprocals of these numbers are 2 1 , 2 1 , 3 1 . The smallest three integers having the same ratio
are 2, 3, 3, and thus the Miller indices of the plane are (233). The translation vectors of the
convenient unit cell, e.g. the cube edges of the fcc structure (which is not primitive) are used for
the coordinate axes. The plane shown intercepts at 3, 2, 2 on the axes. As, by definition, these
three points are not co-linear, therefore they can adequately define the plane. As we shall see
later, the reciprocal of these numbers are more useful. First, reciprocate these numbers thus
getting 1/3, 1/2, 1/2. Miller indices are the smallest integers having the same ratio as these. The
Miller indices of this plane is (hkl) = (2,3,3) By this definition, if the plane cuts an axis at
infinity, the corresponding index will be zero. By convention, if the intercept has a negative
value, the corresponding index is also negative. A minus sign is normally placed above that
index in the bracket. Miller indices are also used to denote a set of planes which are parallel. For
instance, the plane (200) is parallel to (100). The former cuts the x-axis at a v /2. Also by
symmetry, many sets of planes, e.g. all the faces of a cube, may be represented by a single set of
Miller indices (100). In this case the curly bracket is used, hence {100}. In other words the
{100} automatically includes the planes (100), (010) and (001). The direction of any general
vector can also be expressed in terms of a set of indices h, k and l. These indices are the smallest
integers that are proportional to the components of the vectors along the axes of the crystal (same
as the translation vectors of the unit cell). Unlike the Miller indices above, these integers used for
defining the direction of the general vector are not derived from the reciprocal. To distinguish
from the Miller indices for planes, i.e. (hkl) or {hkl}, the direction indices are enclosed in square
brackets, hence [hkl]. Also the indices [hkl] are used to denote many sets of equivalent directions
similar to the plane designations above. For a cubic system, it can be shown that the vector [hkl]
is normal to the planes defined by the Miller indices {hkl}. This fact happens to be extremely
useful in the analysis of x-ray diffraction. 1 Finally, the position of basis or atoms in the
conventional cell is often expressed in terms of the axes defining the cell. For instance, the
position of the body-centred atoms is 1/2, 1/2, 1/2 and the face-centred atom is 1/2, 1/2,
The orientation of planes is best represented by a vector normal to the plane. The direction of a
set of planes is indicated by a vector denoted by square brackets containing the Miller indices of
the set of planes. Miller indices are also used to describe crystal faces.
(hkl) denotes a set of planes [hkl] designates a vector (the direction of the planes) {hkl} set of
faces made equivalent by the symmetry of the system, thus: {100} for point group 1 this refers
only to the (100) face {100} for point group -1 this refers to (100) and (-100)
100) faces for mmm
{111} implies (111),(11-1),(1--11),(11-1),(-1-11),(-11-1),(1-1-1),(-1-1-1)
1) A
Packing Efficiency:
fv =
volume occupied by spheres in the unit cell
volume of the unit cell
The unit cell is characterized by three lengths (a, b, c) and three angles (a, b, g).
The quantities a and b are the lengths of the sides of the base of the cell and g is the
angle between these
se two sides.
The quantity c is the height of the unit cell.
The angles a and b describe the angles between the base and the vertical sides of the
unit cell.
In the hexagonal closest--packed structure, a = b = 2r and c = [4(2/3)1/2]r, where r is the
atomic radius of the atom.
a = b = 90o and g = 120o
The volume of the hexagonal unit cell:
V = 8(2)1/2r3
4
4
2 × πr 3 2 × πr 3
3
3
fv =
=
= 0.7405
V
8 2r 3
Fcc structure:
The ..ABCABC.. packing sequence of the fcc structure gives rise to a three-dimensional
three
structure with cubic symmetry.
Bcc structure:
The bcc structure has very little in common with the fcc structure - except the cubic
nature of the unit cell. Most importantly, it differs from the hcp and fcc structures
in that it is not a close-packed
close
structure.The
.The structure of the alkali metals are
characterized by a bcc unit cell.
Coordination Numbers (CN)=8
Net number of spheres in unit cell =(8×1/8)+(1×1)=2
Density calculations:
p = M/V
Where:
p = density
M = mass
V = volume
The Density Calculator uses the formula that density (p) is equal to mass (M) divided by
volume (V) or p=M/V. The calculator can use any two of the values to calculate the third.
Density is defined as mass per unit volume. Along with values, enter the known units of
measure for each and this calculator will convert among units.
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Calculate p Given M and V
Calculate density given mass and volume.
o p = M/V
Calculate M Given p and V
Calculate
ulate mass given density and volume.
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o M = pV
Calculate V Given p and M
Calculate volume given density and mass.
o V = p/M
Cube root of Volume
Cube root of volume, 3√ V, is shown to help in the mental visualization of the actual
volume.
Grain and grain boundries:
Grain boundary in a solid crystalline material is a region separating two crystals (grains) of
the same phase. These two grains differ in mutual orientations and the grain boundary thus
represents a transition region, where the atoms are shifted from their regular positions as
compared to the crystal interior [12, 19, 20]. Grain boundaries represent the simplest
interface: If the adjoining grains differ in chemical composition and/or in parameters of the
crystal lattice, the interface between them is called phase boundary (interphase boundary,
heterophase boundary). The grain boundary is also called homophase boundary in this
classification. In general, interfaces represent a crystallographic and/or chemical discontinuity
with an average width less than two atomic diameters although they may be sometimes more
diffuse spreading over appreciable number of interplanar spacings [12,23]. In this context,
free surface is the interface between solid and vacuum [20]. Only those aspects of grain
boundaries that are necessary for further reading will be mentioned in this chapter. For more
thorough information other sources are recommended to the reader, especially the
comprehensive book of Sutton and Bluffing [12] and selected parts of the book edited by
Wolf and Yip [24].
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There is another class of defects called interfacial or planar defects:
– They occupy an area or surface and are therefore bidimensional.
– They are of great importance in mechanical metallurgy.
Examples of these form of defects include:
– grain boundaries
– twin boundaries
– anti-phase boundaries
– free surface of materials
Of all these, the grain boundaries are the most important from the mechanical
properties point of view.
Crystalline solids (most materials) generally consist of millions of individual grains
separated by boundaries.
Each grain (or subgrain) is a single crystal.
Within each individual grain there is a systematic packing of atoms. Therefore each
grain has different orientation (see Figure 16-1) and is separated from the neighboring
grain by grain boundary.
When the misorientation between two grains is small, the grain boundary can be
described by a relatively simple configuration of dislocations (e.g., an edge dislocation
wall) and is, fittingly, called a low-angle boundary.
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When the misorientation is large (high-angle
(high angle grain boundary), more complicated
structures are involved (as in a configuration of soap bubbles simulating the atomic
planes in crystal lattices).
The grain boundaries are therefore:
– where grains meet in a solid.
– transition regions between the neighboring crystals.
– Where there is a disturbance in the atomic ppacking,
These transition regions (grain boundaries) may consist of various kinds of dislocation
arrangements.
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In general, a grain boundary has five degrees of freedom. We need three degrees to
specify the orientation of one grain with respect to the other, and We need the other
two degrees to specify the orientation of the boundary with respect to one of the
grains.
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Grain structure is usually specified by giving the average diameter or using a
procedure due to ASTM according to which grains size is specified by a number n in
n
the expression N = 2n-1
, where N is the number of grains per square inch when the
sample is examined at 100x.
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The misorientation at the boundary is related to spacing between dislocations, D, by
the following relation:
b
b
D=
≈
θ  θ
2 sin  
2
where b is the Burgers vector.
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As the misorientation q increases, the spacing between dislocations is reduced, until,
at large angles, the description of the boundary in terms of simple dislocation
arrangements does not make sense.
Alloy:
a
lloy
Low-carbon steels:
Not very responsive to heat treatments
soft, weak, tough and ductile
Machinable, weldable, not expensive
YS~275 MPa, TS~415
TS~415-550MPa, 25% el.
Contain less than 0.25%C
Medium-carbon steels:
Contain 0.25-0.60 wt.% carbon
Can be heat-treated but only in thin sections
Stronger than low-C steels but less ductile and less tough
Good wear resistance
Railway wheels & tracks, gears
High carbon steels:
0.60 -1.4 wt.% C
Hardest, strongest, least ductile of all steels
Almost always used in tempered condition
Especially wear resistant
Form hard and wear resistant carbides with alloying elements
Used in cutting tools, dies, knives, razors, springs and high strength wires
Nonferrous alloys:
Aluminum alloys
Copper alloys
Magesnium alloys
Nickel alloys
Titanium alloys
Refractory metals
Superalloys
Age hardening:
Cast irons:
Theoretically contains > 2.14 wt.% carbon
Usually contains between 3.0-4.5 wt.% C, hence very brittle
Also 1-3 wt.% silicon
Since they become liquid easily between 1150 C and 1300 C, they can be easily cast
Inexpensive
Machinable, wear resistant
4 types: gray cast iron, nodular cast iron, white cast iron, malleable cast iron
Copper alloys:
Soft, ductile, difficult to machine
Highly resistant to corrosion
Excellent electrical & thermal conductivity
Can be alloyed to improve hardness
Cold worked to get the maximum hardness
Cu-Zn = brass; Cu-X = bronzes
Magnesium alloys:
Lowest density of all structural metals= 1.7 gm/cc
Relatively soft and low elastic modulus (45 GPa)
Have to be heated to be deformation processed
Burns easily in the molten and powder states
Susceptible to corrosion in marine environments
Competing with plastics
Nickel alloys:
Quite ductile and formable
Highly corrosion resistant, especially at high temperature
Essential part of austenitic stainless steels
Used in pumps, valves in seawater and petroleum environments
Titanium alloys:
Low density, high melting point
High specific strength and elastic modulus
superior corrosion resistance in many environments
Absorb interstitials at high temperatures
Highly reactive with other materials and hence non-conventional processing techniques
have been developed
Highly used in aerospace applications
Solid solution:
A solid solution is a solid-state solution of one or more solutes in a solvent. Such a mixture is
considered a solution rather than a compoundwhen the crystal structure of the solvent remains
unchanged by addition of the solutes, and when the mixture remains in a single homogeneous
phase. This often happens when the two elements (generally metals) involved are close together
on the periodic table;; conversely, a chemical compound generally results when two metals
involved are not near each other on the periodic table.[4]
The solid solution needs to be distinguished from a mechanical mixture of powdered solids like
two salts, sugar and salt, etc. The mechanical mixtures have total or partial miscibility gap in
solid state. Examples of solid solutions include crystallized salts from their liquid mixture,
metalalloys,, moist solids. In the case of metal alloys intermetallic compounds occur frequently.
The binary phase diagram in Fig. 2 shows the phases of a mixture of two substances in varying
concentrations,
and . The region labeled " " is a solid solution, with
acting as the
solute in a matrix of . On the other end of the concentration scale, the region labeled " " is
also a solid solution, with
acting as the solute in a matrix of . The large solid region in
between the and solid solutions, labeled " + ", is not a solid solution. Instead, an
examination of the microstructure of a mixture in this range would reveal two phases — solid
solution -in- and solid solution -in- would form separate phases,
perhaps lamella or grains
Hume rothery rules:
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Hume-Rothery (1899-1968)
1968) was a metallurgist who studied the alloying of metals. His
research was conducted at Oxford University where in 1958, he was appointed to the first
chair in metallurgy.
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His research led to some simple and useful rules on the extent to which an element might
dissolve in a metal . The rules that he derived are paraphrased here. The rules are still