1.Introduction
Often materials are subject to forces (loads) when they are used. Mechanical engineers calculate those forces and material scientists how materials deform (elongate, compress, twist) or break as a function of applied load, time, temperature, and other conditions.
Materials scientists learn about these mechanical properties by testing materials. Results from the tests depend on the size and shape of material to be tested (specimen), how it is held, and the way of performing the test. That is why we use common procedures, or standards, which are published by the ASTM.
Concepts of Stress and Strain
To compare specimens of different sizes, the load is calculated per unit area, also called normalization to the area. Force divided by area is called stress. In tension and compression tests, the relevant area is that perpendicular to the force. In shear or torsion tests, the area is perpendicular to the axis of rotation.
s = F/A0 tensile or compressive stress
t = F/A0 shear stress
The unit is the Megapascal = 106 Newtons/m2.
There is a change in dimensions, or deformation elongation, DL as a result of a tensile or compressive stress. To enable comparison with specimens of different length, the elongation is also normalized, this time to the length L. This is called strain, e.
e = DL/L
The change in dimensions is the reason we use A0 to indicate the initial area since it changes during deformation. One could divide force by the actual area, this is called true stress (see Sec. 6.7).
For torsional or shear stresses, the deformation is the angle of twist, q (Fig. 6.1) and the shear strain is given by:
g = tg q
Stress—Strain Behavior
Elastic deformation. When the stress is removed, the material returns to the dimension it had before the load was applied. Valid for small strains (except the case of rubbers).
Deformation is reversible, non permanent
Plastic deformation. When the stress is removed, the material does not return to its previous dimension but there is a permanent, irreversible deformation.
In tensile tests, if the deformation is elastic, the stress-strain relationship is called Hooke's law:
s = E e
That is, E is the slope of the stress-strain curve. E is Young's modulus or modulus of elasticity. In some cases, the relationship is not linear so that E can be defined alternatively as the local slope:
E = ds/de
Shear stresses produce strains according to:
t = G g
where G is the shear modulus.
Elastic moduli measure the stiffness of the material. They are related to the second derivative of the interatomic potential, or the first derivative of the force vs. internuclear distance (Fig. 6.6). By examining these curves we can tell which material has a higher modulus. Due to thermal vibrations the elastic modulus decreases with temperature. E is large for ceramics (stronger ionic bond) and small for polymers (weak covalent bond). Since the interatomic distances depend on direction in the crystal, E depends on direction (i.e., it is anisotropic) for single crystals. For randomly oriented policrystals, E is isotropic. .
Anelasticity
Here the behavior is elastic but not the stress-strain curve is not immediately reversible. It takes a while for the strain to return to zero. The effect is normally small for metals but can be significant for polymers.
Elastic Properties of Materials
Materials subject to tension shrink laterally. Those subject to compression, bulge. The ratio of lateral and axial strains is called the Poisson's ratio n.
n = elateral/eaxial
The elastic modulus, shear modulus and Poisson's ratio are related by E = 2G(1+n)
Tensile Properties
Yield point. If the stress is too large, the strain deviates from being proportional to the stress. The point at which this happens is the yield point because there the material yields, deforming permanently (plastically).
Yield stress. Hooke's law is not valid beyond the yield point. The stress at the yield point is called yield stress, and is an important measure of the mechanical properties of materials. In practice, the yield stress is chosen as that causing a permanent strain of 0.002 (strain offset, Fig. 6.9.)
The yield stress measures the resistance to plastic deformation.
The reason for plastic deformation, in normal materials, is not that the atomic bond is stretched beyond repair, but the motion of dislocations, which involves breaking and reforming bonds.
Plastic deformation is caused by the motion of dislocations.
Tensile strength. When stress continues in the plastic regime, the stress-strain passes through a maximum, called the tensile strength (sTS) , and then falls as the material starts to develop a neck and it finally breaks at the fracture point (Fig. 6.10).
Note that it is called strength, not stress, but the units are the same, MPa.
For structural applications, the yield stress is usually a more important property than the tensile strength, since once the it is passed, the structure has deformed beyond acceptable limits.
Ductility. The ability to deform before braking. It is the opposite of brittleness. Ductility can be given either as percent maximum elongation emax or maximum area reduction.
%EL = emax x 100 %
%AR = (A0 - Af)/A0
These are measured after fracture (repositioning the two pieces back together).
Resilience. Capacity to absorb energy elastically. The energy per unit volume is the
area under the strain-stress curve in the elastic region.
Toughness. Ability to absorb energy up to fracture. The energy per unit volume is the total area under the strain-stress curve. It is measured by an impact test (Ch. 8)
True Stress and Strain
When one applies a constant tensile force the material will break after reaching the tensile strength. The material starts necking (the transverse area decreases) but the stress cannot increase beyond sTS. The ratio of the force to the initial area, what we normally do, is called the engineering stress. If the ratio is to the actual area (that changes with stress) one obtains the true stress.
Elastic Recovery During Plastic Deformation
If a material is taken beyond the yield point (it is deformed plastically) and the stress is then released, the material ends up with a permanent strain. If the stress is reapplied, the material again responds elastically at the beginning up to a new yield point that is higher than the original yield point (strain hardening, Ch. 7.10). The amount of elastic strain that it will take before reaching the yield point is called elastic strain recovery (Fig. 6. 16).
Compressive, Shear, and Torsional Deformation
Compressive and shear stresses give similar behavior to tensile stresses, but in the case of compressive stresses there is no maximum in the s-e curve, since no necking occurs.
Hardness
Hardness is the resistance to plastic deformation (e.g., a local dent or scratch). Thus, it is a measure of plastic deformation, as is the tensile strength, so they are well correlated. Historically, it was measured on an empirically scale, determined by the ability of a material to scratch another, diamond being the hardest and talc the softer. Now we use standard tests, where a ball, or point is pressed into a material and the size of the dent is measured. There are a few different hardness tests: Rockwell, Brinell, Vickers, etc. They are popular because they are easy and non-destructive (except for the small dent).
Variability of Material Properties
Tests do not produce exactly the same result because of variations in the test equipment, procedures, operator bias, specimen fabrication, etc. But, even if all those parameters are controlled within strict limits, a variation remains in the materials, due to uncontrolled variations during fabrication, non homogenous composition and structure, etc. The measured mechanical properties will show scatter, which is often distributed in a Gaussian curve (bell-shaped), that is characterized by the mean value and the standard deviation (width).
Design/Safety Factors
To take into account variability of properties, designers use, instead of an average value of, say, the tensile strength, the probability that the yield strength is above the minimum value tolerable. This leads to the use of a safety factor N > 1 (typ. 1.2 - 4). Thus, a working value for the tensile strength would be sW = sTS / N.
Not tested: true stress-true stain relationships, details of the different types of hardness tests, but should know that hardness for a given material correlates with tensile strength. Varia
Tuesday, January 29, 2008
Chapter-5: DIFUSSION
5.1 Introduction
Many important reactions and processes in materials occur by the motion of atoms in the solid (transport), which happens by diffusion.
Inhomogeneous materials can become homogeneous by diffusion, if the temperature is high enough (temperature is needed to overcome energy barriers to atomic motion.
5.2 Diffusion Mechanisms
Atom diffusion can occur by the motion of vacancies (vacancy diffusion) or impurities (impurity diffusion). The energy barrier is that due to nearby atoms which need to move to let the atoms go by. This is more easily achieved when the atoms vibrate strongly, that is, at high temperatures.
There is a difference between diffusion and net diffusion. In a homogeneous material, atoms also diffuse but this motion is hard to detect. This is because atoms move randomly and there will be an equal number of atoms moving in one direction than in another. In inhomogeneous materials, the effect of diffusion is readily seen by a change in concentration with time. In this case there is a net diffusion. Net diffusion occurs because, although all atoms are moving randomly, there are more atoms moving in regions where their concentration is higher.
5.3 Steady-State Diffusion
The flux of diffusing atoms, J, is expressed either in number of atoms per unit area and per unit time (e.g., atoms/m2-second) or in terms of mass flux (e.g., kg/m2-second).
Steady state diffusion means that J does not depend on time. In this case, Fick’s first law holds that the flux along direction x is:
J = – D dC/dx
Where dC/dx is the gradient of the concentration C, and D is the diffusion constant. The concentration gradient is often called the driving force in diffusion (but it is not a force in the mechanistic sense). The minus sign in the equation means that diffusion is down the concentration gradient.
5.4 Nonsteady-State Diffusion
This is the case when the diffusion flux depends on time, which means that a type of atoms accumulates in a region or that it is depleted from a region (which may cause them to accumulate in another region).
5.5 Factors That Influence Diffusion
As stated above, there is a barrier to diffusion created by neighboring atoms that need to move to let the diffusing atom pass. Thus, atomic vibrations created by temperature assist diffusion. Also, smaller atoms diffuse more readily than big ones, and diffusion is faster in open lattices or in open directions. Similar to the case of vacancy formation, the effect of temperature in diffusion is given by a Boltzmann factor: D = D0 × exp(–Qd/kT).
5.6 Other Diffusion Paths
Diffusion occurs more easily along surfaces, and voids in the material (short circuits like dislocations and grain boundaries) because less atoms need to move to let the diffusing atom pass. Short circuits are often unimportant because they constitute a negligible part of the total area of the material normal to the diffusion flux. .
Many important reactions and processes in materials occur by the motion of atoms in the solid (transport), which happens by diffusion.
Inhomogeneous materials can become homogeneous by diffusion, if the temperature is high enough (temperature is needed to overcome energy barriers to atomic motion.
5.2 Diffusion Mechanisms
Atom diffusion can occur by the motion of vacancies (vacancy diffusion) or impurities (impurity diffusion). The energy barrier is that due to nearby atoms which need to move to let the atoms go by. This is more easily achieved when the atoms vibrate strongly, that is, at high temperatures.
There is a difference between diffusion and net diffusion. In a homogeneous material, atoms also diffuse but this motion is hard to detect. This is because atoms move randomly and there will be an equal number of atoms moving in one direction than in another. In inhomogeneous materials, the effect of diffusion is readily seen by a change in concentration with time. In this case there is a net diffusion. Net diffusion occurs because, although all atoms are moving randomly, there are more atoms moving in regions where their concentration is higher.
5.3 Steady-State Diffusion
The flux of diffusing atoms, J, is expressed either in number of atoms per unit area and per unit time (e.g., atoms/m2-second) or in terms of mass flux (e.g., kg/m2-second).
Steady state diffusion means that J does not depend on time. In this case, Fick’s first law holds that the flux along direction x is:
J = – D dC/dx
Where dC/dx is the gradient of the concentration C, and D is the diffusion constant. The concentration gradient is often called the driving force in diffusion (but it is not a force in the mechanistic sense). The minus sign in the equation means that diffusion is down the concentration gradient.
5.4 Nonsteady-State Diffusion
This is the case when the diffusion flux depends on time, which means that a type of atoms accumulates in a region or that it is depleted from a region (which may cause them to accumulate in another region).
5.5 Factors That Influence Diffusion
As stated above, there is a barrier to diffusion created by neighboring atoms that need to move to let the diffusing atom pass. Thus, atomic vibrations created by temperature assist diffusion. Also, smaller atoms diffuse more readily than big ones, and diffusion is faster in open lattices or in open directions. Similar to the case of vacancy formation, the effect of temperature in diffusion is given by a Boltzmann factor: D = D0 × exp(–Qd/kT).
5.6 Other Diffusion Paths
Diffusion occurs more easily along surfaces, and voids in the material (short circuits like dislocations and grain boundaries) because less atoms need to move to let the diffusing atom pass. Short circuits are often unimportant because they constitute a negligible part of the total area of the material normal to the diffusion flux. .
Chapter-4: IMPERFECTIONS
Imperfections in Solids
4.1 Introduction
Materials are often stronger when they have defects. The study of defects is divided according to their dimension:
0D (zero dimension) – point defects: vacancies and interstitials. Impurities.
1D – linear defects: dislocations (edge, screw, mixed)
2D – grain boundaries, surfaces.
3D – extended defects: pores, cracks.
Point Defects
4.2 Vacancies and Self-Interstitials
A vacancy is a lattice position that is vacant because the atom is missing. It is created when the solid is formed. There are other ways of making a vacancy, but they also occur naturally as a result of thermal vibrations.
An interstitial is an atom that occupies a place outside the normal lattice position. It may be the same type of atom as the others (self interstitial) or an impurity atom.
In the case of vacancies and interstitials, there is a change in the coordination of atoms around the defect. This means that the forces are not balanced in the same way as for other atoms in the solid, which results in lattice distortion around the defect.
The number of vacancies formed by thermal agitation follows the law:
NV = NA × exp(-QV/kT)
where NA is the total number of atoms in the solid, QV is the energy required to form a vacancy, k is Boltzmann constant, and T the temperature in Kelvin (note, not in oC or oF).
When QV is given in joules, k = 1.38 × 10-23 J/atom-K. When using eV as the unit of energy, k = 8.62 × 10-5 eV/atom-K.
Note that kT(300 K) = 0.025 eV (room temperature) is much smaller than typical vacancy formation energies. For instance, QV(Cu) = 0.9 eV/atom. This means that NV/NA at room temperature is exp(-36) = 2.3 × 10-16, an insignificant number. Thus, a high temperature is needed to have a high thermal concentration of vacancies. Even so, NV/NA is typically only about 0.0001 at the melting point.
stalline SiO2 (quartz) is still apparent in amorphous SiO2 (silica glass.)
4.3 Impurities in Solids
All real solids are impure. A very high purity material, say 99.9999% pure (called 6N – six nines) contains ~ 6 × 1016 impurities per cm3.
Impurities are often added to materials to improve the properties. For instance, carbon added in small amounts to iron makes steel, which is stronger than iron. Boron impurities added to silicon drastically change its electrical properties.
Solid solutions are made of a host, the solvent or matrix) which dissolves the solute (minor component). The ability to dissolve is called solubility. Solid solutions are:
homogeneous
maintain crystal structure
contain randomly dispersed impurities (substitutional or interstitial)
Factors for high solubility
Similar atomic size (to within 15%)
Similar crystal structure
Similar electronegativity (otherwise a compound is formed)
Similar valence
Composition can be expressed in weight percent, useful when making the solution, and in atomic percent, useful when trying to understand the material at the atomic level.
Miscellaneous Imperfections
4.4 Dislocations—Linear Defects
Dislocations are abrupt changes in the regular ordering of atoms, along a line (dislocation line) in the solid. They occur in high density and are very important in mechanical properties of material. They are characterized by the Burgers vector, found by doing a loop around the dislocation line and noticing the extra interatomic spacing needed to close the loop. The Burgers vector in metals points in a close packed direction.
Edge dislocations occur when an extra plane is inserted. The dislocation line is at the end of the plane. In an edge dislocation, the Burgers vector is perpendicular to the dislocation line.
Screw dislocations result when displacing planes relative to each other through shear. In this case, the Burgers vector is parallel to the dislocation line.
4.5 Interfacial Defects
The environment of an atom at a surface differs from that of an atom in the bulk, in that the number of neighbors (coordination) decreases. This introduces unbalanced forces which result in relaxation (the lattice spacing is decreased) or reconstruction (the crystal structure changes).
The density of atoms in the region including the grain boundary is smaller than the bulk value, since void space occurs in the interface.
Surfaces and interfaces are very reactive and it is usual that impurities segregate there. Since energy is required to form a surface, grains tend to grow in size at the expense of smaller grains to minimize energy. This occurs by diffusion, which is accelerated at high temperatures.
Twin boundaries: not covered
4.6 Bulk or Volume Defects
A typical volume defect is porosity, often introduced in the solid during processing. A common example is snow, which is highly porous ice.
4.7 Atomic Vibrations
Atomic vibrations occur, even at zero temperature (a quantum mechanical effect) and increase in amplitude with temperature. Vibrations displace transiently atoms from their regular lattice site, which destroys the perfect periodicity we discussed in Chapter 3.
4.1 Introduction
Materials are often stronger when they have defects. The study of defects is divided according to their dimension:
0D (zero dimension) – point defects: vacancies and interstitials. Impurities.
1D – linear defects: dislocations (edge, screw, mixed)
2D – grain boundaries, surfaces.
3D – extended defects: pores, cracks.
Point Defects
4.2 Vacancies and Self-Interstitials
A vacancy is a lattice position that is vacant because the atom is missing. It is created when the solid is formed. There are other ways of making a vacancy, but they also occur naturally as a result of thermal vibrations.
An interstitial is an atom that occupies a place outside the normal lattice position. It may be the same type of atom as the others (self interstitial) or an impurity atom.
In the case of vacancies and interstitials, there is a change in the coordination of atoms around the defect. This means that the forces are not balanced in the same way as for other atoms in the solid, which results in lattice distortion around the defect.
The number of vacancies formed by thermal agitation follows the law:
NV = NA × exp(-QV/kT)
where NA is the total number of atoms in the solid, QV is the energy required to form a vacancy, k is Boltzmann constant, and T the temperature in Kelvin (note, not in oC or oF).
When QV is given in joules, k = 1.38 × 10-23 J/atom-K. When using eV as the unit of energy, k = 8.62 × 10-5 eV/atom-K.
Note that kT(300 K) = 0.025 eV (room temperature) is much smaller than typical vacancy formation energies. For instance, QV(Cu) = 0.9 eV/atom. This means that NV/NA at room temperature is exp(-36) = 2.3 × 10-16, an insignificant number. Thus, a high temperature is needed to have a high thermal concentration of vacancies. Even so, NV/NA is typically only about 0.0001 at the melting point.
stalline SiO2 (quartz) is still apparent in amorphous SiO2 (silica glass.)
4.3 Impurities in Solids
All real solids are impure. A very high purity material, say 99.9999% pure (called 6N – six nines) contains ~ 6 × 1016 impurities per cm3.
Impurities are often added to materials to improve the properties. For instance, carbon added in small amounts to iron makes steel, which is stronger than iron. Boron impurities added to silicon drastically change its electrical properties.
Solid solutions are made of a host, the solvent or matrix) which dissolves the solute (minor component). The ability to dissolve is called solubility. Solid solutions are:
homogeneous
maintain crystal structure
contain randomly dispersed impurities (substitutional or interstitial)
Factors for high solubility
Similar atomic size (to within 15%)
Similar crystal structure
Similar electronegativity (otherwise a compound is formed)
Similar valence
Composition can be expressed in weight percent, useful when making the solution, and in atomic percent, useful when trying to understand the material at the atomic level.
Miscellaneous Imperfections
4.4 Dislocations—Linear Defects
Dislocations are abrupt changes in the regular ordering of atoms, along a line (dislocation line) in the solid. They occur in high density and are very important in mechanical properties of material. They are characterized by the Burgers vector, found by doing a loop around the dislocation line and noticing the extra interatomic spacing needed to close the loop. The Burgers vector in metals points in a close packed direction.
Edge dislocations occur when an extra plane is inserted. The dislocation line is at the end of the plane. In an edge dislocation, the Burgers vector is perpendicular to the dislocation line.
Screw dislocations result when displacing planes relative to each other through shear. In this case, the Burgers vector is parallel to the dislocation line.
4.5 Interfacial Defects
The environment of an atom at a surface differs from that of an atom in the bulk, in that the number of neighbors (coordination) decreases. This introduces unbalanced forces which result in relaxation (the lattice spacing is decreased) or reconstruction (the crystal structure changes).
The density of atoms in the region including the grain boundary is smaller than the bulk value, since void space occurs in the interface.
Surfaces and interfaces are very reactive and it is usual that impurities segregate there. Since energy is required to form a surface, grains tend to grow in size at the expense of smaller grains to minimize energy. This occurs by diffusion, which is accelerated at high temperatures.
Twin boundaries: not covered
4.6 Bulk or Volume Defects
A typical volume defect is porosity, often introduced in the solid during processing. A common example is snow, which is highly porous ice.
4.7 Atomic Vibrations
Atomic vibrations occur, even at zero temperature (a quantum mechanical effect) and increase in amplitude with temperature. Vibrations displace transiently atoms from their regular lattice site, which destroys the perfect periodicity we discussed in Chapter 3.
Chapter-3: STRUCTURE OF CRYSTALS
3.2 Fundamental Concepts
Atoms self-organize in crystals, most of the time. The crystalline lattice, is a periodic array of the atoms. When the solid is not crystalline, it is called amorphous. Examples of crystalline solids are metals, diamond and other precious stones, ice, graphite. Examples of amorphous solids are glass, amorphous carbon (a-C), amorphous Si, most plastics
To discuss crystalline structures it is useful to consider atoms as being hard spheres, with well-defined radii. In this scheme, the shortest distance between two like atoms is one diameter.
3.3 Unit Cells
The unit cell is the smallest structure that repeats itself by translation through the crystal. We construct these symmetrical units with the hard spheres. The most common types of unit cells are the faced-centered cubic (FCC), the body-centered cubic (FCC) and the hexagonal close-packed (HCP). Other types exist, particularly among minerals. The simple cube (SC) is often used for didactical purpose, no material has this structure.
3.4 Metallic Crystal Structures
Important properties of the unit cells are
· The type of atoms and their radii R.
· cell dimensions (side a in cubic cells, side of base a and height c in HCP) in terms of R.
· n, number of atoms per unit cell. For an atom that is shared with m adjacent unit cells, we only count a fraction of the atom, 1/m.
· CN, the coordination number, which is the number of closest neighbors to which an atom is bonded.
· APF, the atomic packing factor, which is the fraction of the volume of the cell actually occupied by the hard spheres. APF = Sum of atomic volumes/Volume of cell.
Unit Cell n CN a/R APF
SC 1 6 2 0.52
BCC 2 8 4Ö 3 0.68
FCC 4 12 2Ö 2 0.74
HCP 6 12 0.74
The closest packed direction in a BCC cell is along the diagonal of the cube; in a FCC cell is along the diagonal of a face of the cube.
3.5 Density Computations
The density of a solid is that of the unit cell, obtained by dividing the mass of the atoms (n atoms x Matom) and dividing by Vc the volume of the cell (a3 in the case of a cube). If the mass of the atom is given in amu (A), then we have to divide it by the Avogadro number to get Matom. Thus, the formula for the density is:
3.6 Polymorphism and Allotropy
Some materials may exist in more than one crystal structure, this is called polymorphism. If the material is an elemental solid, it is called allotropy. An example of allotropy is carbon, which can exist as diamond, graphite, and amorphous carbon.
3.11 Close-Packed Crystal Structures
The FCC and HCP are related, and have the same APF. They are built by packing spheres on top of each other, in the hollow sites (Fig. 3.12 of book). The packing is alternate between two types of sites, ABABAB.. in the HCP structure, and alternates between three types of positions, ABCABC… in the FCC crystals.
Crystalline and Non-Crystalline Materials
3.12 Single Crystals
Crystals can be single crystals where the whole solid is one crystal. Then it has a regular geometric structure with flat faces.
3.13 Polycrystalline Materials
A solid can be composed of many crystalline grains, not aligned with each other. It is called polycrystalline. The grains can be more or less aligned with respect to each other. Where they meet is called a grain boundary.
3.14 Anisotropy
Different directions in the crystal have a different packing. For instance, atoms along the edge FCC crystals are more separated than along the face diagonal. This causes anisotropy in the properties of crystals; for instance, the deformation depends on the direction in which a stress is applied.
3.15 X-Ray Diffraction Determination of Crystalline Structure – not covered
3.16 Non-Crystalline Solids
In amorphous solids, there is no long-range order. But amorphous does not mean random, since the distance between atoms cannot be smaller than the size of the hard spheres. Also, in many cases there is some form of short-range order. For instance, the tetragonal order of cry
Atoms self-organize in crystals, most of the time. The crystalline lattice, is a periodic array of the atoms. When the solid is not crystalline, it is called amorphous. Examples of crystalline solids are metals, diamond and other precious stones, ice, graphite. Examples of amorphous solids are glass, amorphous carbon (a-C), amorphous Si, most plastics
To discuss crystalline structures it is useful to consider atoms as being hard spheres, with well-defined radii. In this scheme, the shortest distance between two like atoms is one diameter.
3.3 Unit Cells
The unit cell is the smallest structure that repeats itself by translation through the crystal. We construct these symmetrical units with the hard spheres. The most common types of unit cells are the faced-centered cubic (FCC), the body-centered cubic (FCC) and the hexagonal close-packed (HCP). Other types exist, particularly among minerals. The simple cube (SC) is often used for didactical purpose, no material has this structure.
3.4 Metallic Crystal Structures
Important properties of the unit cells are
· The type of atoms and their radii R.
· cell dimensions (side a in cubic cells, side of base a and height c in HCP) in terms of R.
· n, number of atoms per unit cell. For an atom that is shared with m adjacent unit cells, we only count a fraction of the atom, 1/m.
· CN, the coordination number, which is the number of closest neighbors to which an atom is bonded.
· APF, the atomic packing factor, which is the fraction of the volume of the cell actually occupied by the hard spheres. APF = Sum of atomic volumes/Volume of cell.
Unit Cell n CN a/R APF
SC 1 6 2 0.52
BCC 2 8 4Ö 3 0.68
FCC 4 12 2Ö 2 0.74
HCP 6 12 0.74
The closest packed direction in a BCC cell is along the diagonal of the cube; in a FCC cell is along the diagonal of a face of the cube.
3.5 Density Computations
The density of a solid is that of the unit cell, obtained by dividing the mass of the atoms (n atoms x Matom) and dividing by Vc the volume of the cell (a3 in the case of a cube). If the mass of the atom is given in amu (A), then we have to divide it by the Avogadro number to get Matom. Thus, the formula for the density is:
3.6 Polymorphism and Allotropy
Some materials may exist in more than one crystal structure, this is called polymorphism. If the material is an elemental solid, it is called allotropy. An example of allotropy is carbon, which can exist as diamond, graphite, and amorphous carbon.
3.11 Close-Packed Crystal Structures
The FCC and HCP are related, and have the same APF. They are built by packing spheres on top of each other, in the hollow sites (Fig. 3.12 of book). The packing is alternate between two types of sites, ABABAB.. in the HCP structure, and alternates between three types of positions, ABCABC… in the FCC crystals.
Crystalline and Non-Crystalline Materials
3.12 Single Crystals
Crystals can be single crystals where the whole solid is one crystal. Then it has a regular geometric structure with flat faces.
3.13 Polycrystalline Materials
A solid can be composed of many crystalline grains, not aligned with each other. It is called polycrystalline. The grains can be more or less aligned with respect to each other. Where they meet is called a grain boundary.
3.14 Anisotropy
Different directions in the crystal have a different packing. For instance, atoms along the edge FCC crystals are more separated than along the face diagonal. This causes anisotropy in the properties of crystals; for instance, the deformation depends on the direction in which a stress is applied.
3.15 X-Ray Diffraction Determination of Crystalline Structure – not covered
3.16 Non-Crystalline Solids
In amorphous solids, there is no long-range order. But amorphous does not mean random, since the distance between atoms cannot be smaller than the size of the hard spheres. Also, in many cases there is some form of short-range order. For instance, the tetragonal order of cry
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