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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.

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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.

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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.

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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:

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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:

Figure 1.8 - The 14 Bravais lattices

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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.

The third neighbor directions are the 110

, the distance between the

reference atom and the third-neighbor is √2∙a.

The basic reference frame is given by the vectors:

a1=a/2[-1 1 1]T

a2=a/2[1 -1 1]T

a3=a/2[1 1 -1]T

The volume of the basic cell is:

VBCC = a3/2