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1
CHAPTER 1
CRYSTAL STRUCTURES AND BCC CRYSTAL SYSTEM
1.1-THE IMPORTANCE OF NANOMECHANICS
If one likes to have the shortest and most complete definition of
nanotechnology one should refer to the statement by the US National
Science and Technology Council which states: "The essence of
nanotechnology is the ability to work at molecular level, atom by atom,
to create large structures with fundamentally new molecular
organization. The aim is to exploit these properties by gaining control
of structures and devices at atomic, molecular, and supramolecular
levels and to learn to efficiently manufacture and use these devices". In
short, nanotechnology is the ability to build micro and macro material
with atomic precision.
Nanomechanics is the branch of nanotechnology that studies the
mechanical behavior of nanomaterials, nanostructures and nanosystems.
As in the aeronautics and aerospace industry the research for high
performance materials with size and weight reduction is very strong, it
is easy to understand the importance of nanomechanics in this industry.
Just for this reason, aeronautics and aerospace industry is one of the
main driving forces of research in nanomechanics.
The main aspects of nanomechanics is to investigate elastic and
inelastic behavior in continuum and uses atomistic/molecular approach
at the nanometer scale. Using experimentally collected data and
multiscale modeling, nanomechanics establish mechanisms of
deformation and failure of nanostructured materials and nanoscale
structures. This purpose is obtained studying the properties of the
atomic/molecular structures with which many materials are built.
Many engineering materials, like Aluminium, Copper, Iron, Cobalt etc,
are made up with different types of crystal structures like BCC, FCC
and HCP. Each structure has its own unique arrangement and based on
atomic radius and atomic distance, different crystal structure has
different density and so on for other material properties.
It is clear that a very important role in understanding materials
properties is played by the crystalline structures, and this work is
focused on the crystal structure BCC (body-centered cubic).
1.2-LATTICES AND CRYSTAL SYSTEMS
Metals and many important classes of non-metallic solids are
crystalline. It means that the constituent atoms are arranged in a pattern
that repeats itself periodically in three dimensions. The actual
arrangement of the atoms is described by the crystal structure.
1.2.1-Lattice and basis
The starting point to understand the crystal structure is the ideal crystal.
The ideal crystal is an infinite structure formed regularly repeating an
atom or group of atoms, called basis, on a space filling lattice.
In this definition, there are two important terms: lattice and basis.
3
The lattice + basis scheme is only a convenient tool to systematically
describe the crystal structure.
A lattice is an infinite arrangements of points in a regular pattern. To be
a lattice, the arrangements and orientation of all points viewed relative
to any one point must be the same, no matter which vantage point is
chosen, In other words, the arrangement must have translational
symmetry. In the following figures it's possible to see an example of a
two-dimensional lattice and an arrangement of points that is not a
lattice:
Figure 1.1 - Two dimensional arrangement of points that satisfy the
definition of a lattice
Figure 1.2 - Two dimensional arrangement of points that do not satisfy
the definition of a lattice
In figure 1.2 there is a honeycomb pattern, where the arrangement
around each point is the same, but the orientation changes.
The basis is a motif unit of atoms that is translationally invariant from
one lattice site to the next. In other words, a basis can be represented by
a molecule attached to each lattice site, but with no difference between
the bonding of the atoms within the motif and the bonding of the atoms
with the neighboring motifs.
5
1.2.2-Primitive lattice vectors and primitive unit cell, conventional
unit cell
To generate a set of points that satisfy the definition of lattice given
above, it's possible to define the lattice point R using the following
equation:
R[l] = li∙Ai , li ∈
Z , (1)
where Ai are three linearly independent vectors and Z is the set of all
integers. The three vectors A1, A2, and A3 are called primitive lattice
vectors. They are, in the most general case, not orthogonal to each
other, but in all cases they do not lie in the same plane. The lattice is
generated by taking all possible integer combinations of the primitive
lattice vectors.
Furthermore, the primitive lattice vectors define a unit cell, called
primitive unit cell, that, when repeated through the space, generates the
lattice. At the very end, we can think of the lattice as being composed of
an infinite number of primitive unit cells packed together in a space-
filling pattern.
It is important to note that the choice of primitive lattice vectors for a
given lattice is not unique. There are in fact an infinite number of
possibilities, but the choice is not arbitrary, and must satisfy some
important requirements. Primitive lattice vectors must connect lattice
points and the primitive unit cell they define must contain only one
lattice point. When calculating the number of points contained in a
primitive unit cell, the lattice points at the corners of the call are shared
equally amongst all cells in contact with that point.
It is shown an example of a two-dimensional lattice, primitive lattice
vectors and primitive unit cell they define (figure 1.3) :
Figure 1.3 - Primitive lattice vectors that define a primitive unit cell
Also, it is represented the same two-dimensional lattice, but with
different possible primitive lattice vectors and consequently primitive
unit cell (figure 1.4) :
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Figure 1.4 - Alternative choices for primitive lattice vectors
An important property of the primitive unit cell, that follows directly
from the requirements discussed above, is that its volume, given by
Ω = ׀ A1 ∙ A2 ×
A3 2׀ ( )
remains the same for any choice of primitive lattice vectors.
However, it is often more convenient to work with larger unit cell that
more obviously reveal the symmetries of the lattice they generate. This
set of vectors is called non-primitive lattice vectors, and the cell they
generate is the non-primitive unit cell. This cell is larger than the
minimal primitive cell, but sill generates the lattice when repeated
through the space. As the volume is larger, a non-primitive unit cell
contains more than one lattice points. Similarly to primitive unit cells,
there is an infinite number of possible non-primitive unit cells for a
given lattice. Although their shapes can be complex, the convention is
to select the minimal parallelepiped that shares the symmetry properties
of the lattice. This kind of cell is called conventional unit cell of the
lattice.
Crystallographer refer to the non-primitive lattice vectors of the
conventional unit cell as the crystal axes, and denote them a, b, c. The
magnitude of the crystal axis a = |a|, b = |b|, c = |c|, are called the lattice
constants or lattice parameters, and the angles between them are α, β, γ.
An example of conventional unit cell is shown (figure 1.5) :
Figure 1.5 - The conventional unit cell
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1.2.3-Miller indices
Within the lattice, it is possible to define point sites, directions and
planes. The common way to define these entities in a lattice is to use the
Miller indices.
For an easier representation, the conventional unit cell used to show the
Miller indices is a cubic one, with lattice parameters having the same
magnitude equal to "a", and all the three angles are 90° (figure 1.6) :
Figure 1.6 - Cubic cell illustrating method of describing the orientation
of planes
Any plane A'B'C' in figure 1.4 can be defined by the intercepts OA',
OB', OC' with three principal axis x, y, and z. The Miller indices is to
take the reciprocals of the ratios of the intercepts to the corresponding
unit cell dimensions (lattice parameters). Thus the plane A'B'C' is given
by
and the numbers are then reduced to the three smallest integers in these
ratios.
Planes parallel to the plane of lattice points nearer to the origin O have
the same Miller indices. When a plane intercepts one of the principal
axes on the negative side of the origin, it is placed a minus sign above
the index relative to that axe.
Any direction LM in Fig 1.5 is described by the line parallel to LM
through the origin O, in this case OE. The direction is given by the three
smallest integers in the ratios of the lengths of the projections of OE
resolved along the three principal axes, namely in this figure OA, OB,
OC, to the corresponding lattice parameters of the unit cell.
11
Figure 1.7 - Cubic cell illustrating the method of describing directions
and point sites
Thus, if the cubic unit cell is given by OA, OB, and OC, the direction
LM is
By convention, brackets [ ] and ( ) imply specific directions and planes
respectively, and and { } refer respectively to directions and planes
of the same type.
An important property is that in cubic crystals, the Miller indices of a
plane are the same as the indices of the direction normal to that plane.
Regarding point sites, the coordinates of any point in a crystal relative
to a chosen origin site are described by the fractional displacements of
the point along the three principal axes divided by the corresponding
lattice parameters of the unit cell.
1.2.4-Point symmetry operation
There are some operation that can be applied on a given lattice. In
particular, if a given lattice posses a particular symmetry, it is possible
to apply on it a point symmetry operation, that is a transformation of the
lattice specified with respect to a single point that remains unchanged
during the process.
The point symmetry operations consist of three basic types, and
combinations thereof: rotation, reflection and inversion.
rotation:
A rotation operator rotates the lattice by some angle about an axis
passing through a lattice point. A lattice is said to possess an n-fold
rotational symmetry about a given axis if the lattice remains unchanged
after a rotation of 2π/n about it. For an isolated molecule or other finite
structure, n can be any integer value. However, for infinite lattices with
translational symmetry, n can only take on the values 1,2,3,4 and 6. The
rotation axis generally coincides with some convenient crystal
direction.
reflection across a plane:
13
A reflection or "mirror" operation, denoted with m, corresponds to what
the name intuitively suggest. We define a plane passing through at least
one lattice point. Than for every point in the lattice we draw the
perpendicular line from the point to the plane, and determine the
distance, d, from point to plane along this line. We then move the lattice
point perpendicularly to the other side of the mirror plane at distance d.
inversion:
The inversion operator, called 1, has the straightforward effect of
transforming any lattice point (R1,R2,R3) to (-R1,-R2,-R3). The origin is
left unchanged, and is referred to as "the center of inversion" or
sometimes "center of symmetry". All lattices must possess at least this
symmetry.
1.3-BRAVAIS LATTICES
The possible crystal systems are classified by asking what conditions
are imposed on a lattice if it is to have any of these operations as point
symmetry operation. These conditions will came in the form of
restrictions on the relative lengths of a, b, and c and on the angles
between them. For each of them there can be more than one unique
arrangement of lattice points. The end result is the 14 unique "Bravais
lattices", in the name of the French physicist August Bravais which, in
1848 discovered that there are 14 unique lattices in three dimensional
crystalline systems. In figure 1.8 are represented all the 14 Bravais
lattices:
15
The seven crystal systems are classified for their symmetry property.
Triclinic
Is the least symmetric lattice that is possible. Other than the trivial
identity operator, 1, the triclinic system has only one symmetry, the
inversion 1. For this crystal system there is only the primitive (triclinic-
P) Bravais lattice.
Monoclinic
It possesses only a two-fold rotation axes 2 or equivalently a reflection
plane of symmetry, m. The axis of symmetry lies along one of the lattice
vectors, conventionally chosen to be c. There are two unique
monoclinic Bravais lattice, the primitive (monoclinic-P) and the base-
centered (monoclinic-C).
Orthorhombic
It has two two-fold axes, which automatically implies a third. There are
four unique orthorhombic Bravais lattices: the primitive (orthorhombic-
P), base-centered (orthorhombic-C), body-centered (orthorhombic-I),
and face-centered (orthorhombic-F).
Tetragonal
It has a single four-fold axes of symmetry, 4. There are only two unique
Bravais lattices: the primitive (tetragonal-P) and the body-centered
(tetragonal-I).
Trigonal and hexagonal
The hexagonal system possesses a single six-fold axes of symmetry, 6,
while the trigonal system possesses a single three-fold axes of
symmetry, 3. These two symmetry conditions lead to the identical set of
restrictions on the lattice vectors. Although the two lattices have the
same conditions imposed on them by their respective symmetries,
crystals in the two systems are not the same, because the symmetry of
the basis atoms within the cell determines the final crystal system.
Cubic
It is the most symmetric system, requiring four three-fold axes. There
are three unique Bravais lattices: the primitive (cubic-P), the body
centered (cubic-I or BCC) and the face-centered (cubic-F or FCC).
1.4-BCC CRYSTAL SYSTEM
The present work is focused on the body-centered cubic (BCC) Bravais
lattice. Before to enter in the deep of the work, it’s important to talk
about some characteristics of the BCC crystal system. In the figure 1.9
it is showed the BCC lattice structure, where it is represented the
reference atom and the non-primitive basic cell, either in blue, the first-
neighbor atoms in green, and the second-neighbor atoms in red.
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Figure 1.9 - BCC Bravais lattice with first and second neighbors in
evidence
The close-packed directions are the111
, and the distance between the
reference atom and the first neighbor is √3∙a/2, where “a” is the lattice
parameter of the conventional unit cell represented in the figures.
The second-neighbor directions are the 100
, the distance between the
reference atom and the second-neighbor is a.