# Coordination Number Of Fcc

A crystalline material is one in which atoms are arranged in a regular pattern over large atomic distances. Thus, a long range order exists and upon solidification, atoms will positioned themselves in a regular 3-D pattern. Atoms are considered to be as solid spheres having well defined diameters. Lattice in a 3-D array of atoms arranged in space. Basic crystal structure governs by fracture strength of material.

**Unit Cells: **

The face-centered cubic (fcc) has a coordination number of 12 and contains 4 atoms per unit cell. The body-centered cubic (bcc) has a coordination numberof 8 and contains 2 atoms per unit cell. A sphere occupying a tetrahedral hole has a coordination number of 4. Tetrahedral holes are relatively small compared to cubic or octahedral holes. A fcc arrangement has both tetrahedral and octahedral holes. (Move your mouse over Figure 9B and see also the location of the octahedral holes in a cubic close-pack structure.).

These are the smallest unit of a structure which when repeated in all 3-dimensions produces the crystal structure.

Unit cells for most crystals are parallelepiped as parallelepiped having 3 sets of parallel planes. It represents the symmetry of crystal structure. It is the basic structural unit or building block of the crystal structure and defines the crystal structure by virtue of its geometry and the atoms position within.

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Space lattice + Basis = Unit cell

Way of filling-up of points in space lattice in atom is known as basis.

Space lattice- Infinite array of points in 3-D space in which each point located with respect to other.

**FCC Crystal Structure (Face Centered Cubic): **

The unit cell has a cubic geometry, with atom located at the corners and centers of all the cubic faces.

**i. Coordination Number****: **

It is equal to the number of nearest neighbour to an atom and number of atom touching it. For FCC the coordination number is 12.

**ii. Atoms Per Unit Cell****: **

As each corner atom is shared by eight unit cells thus, its contribution is 1/8^{th}, while each face atom is shared by two unit cells and its contribution is 1/2.

**Thus, total no. of atoms per unit cell are: **

= 1/8 × 8 + 1/2 × 6 = 4.

**iii. Atomic Packing Factor (APP)****: **

It is the ratio of volume of atoms per unit cell to total unit cell volume.

FCC structure has the maximum packing possible for spheres.

All sphere having the same diameter. Metals typically have large APP to maximize shielding provided by free electron cloud.

**Examples of Metals having FCC structure: **

a. Aluminum

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b. Nickel

c. Copper

d. Platinum

e. Gold

f. Silver

g. Lead

**BCC Structure (Body Centered Cubic): **

Atoms are locate at eight corners and a single atom at the centre of cube. There are two atoms per unit cell of a BCC structure. Coordination number of BCC crystal structure is 8, and its atomic packing factor is 0.68.

8 atoms at the corner × 1/8 = 1 atom

1 centre atom = 1 atom

Total = 2 atoms per unit cell.

Here, a^{2} + a^{2} + a^{2} = (diagonal of cube)^{2} = (4R)^{2 }

or 3a^{2} = 16R^{2 }

or a = 4R/√3

**Example of BCC structure are: **

a. Chromium

b. Tantalum

c. Iron (α)

d. Tungsten

e. Molybdenum

f. Potassium (k)

**Hexagonal Closed Packing (HCP)****: **

The third type of crystal is HCP. Here one dimension is greater than other two. The coordination number of HCP crystal is 12 and APF 0.74 same as FCC structure.

**Density Computations: **

Density = nA/VN

where, n = no. of atoms per unit cell.

A = At weight

V = Volume of unit cell

N = Avogadro’s number (6.023 x 10^{23} atoms/mol).

e.g.: graphite, Cd, Zn, Mg, Be, Ti, Mg, Zr

**Polymorphism****: **

Some metals as well as non-metals have more than one crystal structure, this phenomenon is called polymorphism and when found in elemental solids the condition is termed as allotropy.

e.g., graphite and diamond, α-iron, δ-iron, γ-iron.

These material have different structure at different temperature.

**Crystal Systems****: **

**The unit cell geometry is completely defined in terms of the:**

(i) Three edges are a, b and c.

(ii) Three inter axial angle α, β and γ.

Based on different possible combinations of (a, b & c) and (α, β and γ).

There are seven diff. possible crystal structures called crystal system.

**Lattice Parameter Relationships and Figures Showing Unit Cell Geometries for the Seven Crystal Systems****: **

**Crystallographic Directions****: **

A vector of convenient length is positioned such that it passes through the origin of the coordinate systems.The length of this vector projections on each of 3 axis is determined in term of unit cell dimensions a, b, c. The three number obtained are multiplied or divided by a common factor to reduce them to smallest integer values. The direction are represented as [uvw], i.e. not separated by commas.

**Crystallographic Planes: **

The layers of atoms or the planes along which atoms are arranged are known as crystallographic planes. The relation of a set of planes to the axes of the unit cell is designated by Miller indices. One corner of the unit cell is assumed to be the origin of the space coordinates and any set of planes is identified by the reciprocals of its intersections with these coordinates. The unit of the coordinates is the lattice parameter of the crystal. If a plane is parallel to an axis, it intersects it at infinity.

In figure (b), or the cubic system, the crosshatched plane BCHG intersects the Y axis at one unit from the origin and is parallel to the X and Z axes or intersects them at infinity. Therefore,

The shown plane has Miller indices of (010). If a plane cuts any axis on the negative side of the origin. The index will be negative and is indicated by placing a minus sign above the index, as (h k̅ l). For example, the Miller indices of the plane ADEF which goes through the origin (point A) cannot be determined without changing the location of the origin. Any point in the cube may be selected as the origin.

For convenience, take point B. The plane ADEF is parallel to the X-axis (BC) and the Z-axis (BG) but intersects the Y-axis at -1. The plane has Miller indices of (0 1̅ 0). As another example, the Miller indices of the plane BDJ (figure) may be determined as-

Hence this plane has Miller indices of (112). If the Miller indices of a plane result in fractions, these fractions must be cleared.

**Example: **

Consider a plane that intersects the x-axis at 1, the y-axis at 3, and the z-axis at 1.

Taking reciprocals gives indices of 1, 1/3, and 1. Multiplying through by 3 to clear fractions results in Miller indices of (313) for the plane. All parallel planes have the same indices. Parentheses (hkl) around Miller indices signify a specific plane or set of parallel planes. Braces signify a family of planes of the same “form” (which are equivalent in the crystal), such as the cube faces of a cubic crystal: {100} = (100) + (010) + (001) + (1̅00) + (01̅0) + (001̅). Reciprocals are not used to determine the indices of a direction.

In order to arrive at a point on a given direction, consider that starting at the origin. It is necessary to move a distance u times the unit distance along the x-axis, v times the unit distance b along the y-axis, and w times the unit distance c along the z-axis. If u, v and w are the smallest integers to meet the desired motion, they are the indices of the direction and are enclosed in square brackets (uvw). A group of similar directions are enclosed in an angular brackets <uvw>.

**Example:**

In figure to determine the direction AC starting at the origin (point A), it is necessary to move one unit along x-axis to point D and one unit in the direction of y- axis to reach the point C. The direction AC would have indices of [110].

In a cubic crystal the direction has the same indices as the plane to which it is perpendicular. An approximate idea of the packing of atoms on a particular plane may be obtained by visualizing a single unit cell of the b.c.c. and f.c.c. structure.

**Considering the atoms as the lattice points, the number of atoms on a particular plane would be:**

**Linear and Planer Densities****: **

For a particular material, equivalent directions have equal linear density.

Linear density (LD) = No. of atoms centered on unit length of direction vector/Length of the direction vector.

Consider an e.g., of FCC unit cell and the [110] direction, therein.

No of atoms centered on direction vector are 1/2 + 1 + 1/2 = 2 atoms

Length of Direction vectors R + 2R + R = 4R

**(ii) Planer Density: **

Equivalent crystallographic planer have equal planer density.

PD = No. of atoms centered on a plane/Area of the plane

for the same case of (110) plane with an FCC unit cell.

No. of atoms centered on plane ABCD = 1/2+1/2 + 1/2+1/2 = 2

Area of plane ABCD = 4R x a = 4R x 2 √2 R = 8 √2 R^{2}.

L.D and P.D are important considerations as slip taken place on most dense packed crystallographic planes, an in those planes, along directions having greatest atomic packing.

**Close-Packed Crystal Structures****: **

HCP – AB AB AB AB…

FCC – ABC ABC ABC…

For HCP let A be first plane of closed packed atoms. The second plane atoms (B plane) are placed is the voids made by atoms of first plane, while atoms of next A plane are directly above the (A) atoms of first plane. Thus forming AB AB AB… structure.

For FCC structure B plane atoms also rest in triangular voids of A plane atoms while C plane atoms rest in triangular voids of B plane atoms. This sequence is repeated.

## Coordination Number Of Fcc Bcc Hcp

**Single Crystals****: **

When the periodic and repeated arrangement of atoms is perfect and extends throughout entire specimens without interruption. The result is single crystal. All unit cells have the same orientation. Single crystals exist as nuclei grows at the expense of surrounding liquid. Grains formed have different orientations and boundaries separating these different orientations are called grain boundaries.

**Anisotropy:**

The physical properties of some substances depend upon the crystallographic direction in which measurements are taken for e.g.: The elastic modules, the electrical conductivity and refraction index may have different values in [100] & [111] directions. This directionality of properties is termed as ‘Anisotropy’ and it is associated with variance of atomic and ionic spacing with crystallographic direction. Substances for which physical properties are independent of direction of measurement are isotropic materials.

The extent of anisotropic effects is a function of symmetry of the crystal structure i.e. degree of anisotropy increases with decreasing structure symmetry. Thus monoclinic, triclinic structure are highly anisotropic. For many polycrystalline materials, the crystallographic orientations of individual grains are random.

Each grain is anisotropic in nature but the specimen composed of aggregates of grains behaves isotropically. Also the magnitude of measured value represents some average of the directional values. Sometimes the grain in polycrystalline materials have preferential direction of orientation or growth, in which case the material is said to have a “Texture”.

**Measurement of Grain Size****:**

Straight lines of same length are drawn on micro pictograph that show the grain structures. The grains intersected by each line segment are counted, and the length of line is divided by an average of the number of grains intersected over all the line segments. The average grain is then found by divided this result by linear magnification of the photomicrographs.

Another way is through ASTM size number charts. Which has grain size number from 1-10 (Higher number indicates fines grain). Grain structure is photographed at a magnification of 100. The grain size is expressed as grain size number of the chart that most nearly matches the grain in the micrograph.

Average number of grains per inch at magnification of 100, [N = 2^{n-1}]

Where, n = given size number.

**Plastic Deformation of Polycrystalline Materials****: **

In poly crystal line material there are many preferred planes and direction for different grains due to their random orientation. Thus, in case of poly-crystalline materials “gross plastic deformation” take place, where in individual grains undergo comparable distortion and the grain boundaries do not come apart or open up. Before deformation the grains are equiaxed but after deformation they get elongated in the direction of applied forces.

Polycrystalline materials are stronger than single crystal material because they require more stresses to initiate slip and yielding. These higher stresses are consequences of constraints that are imposed on the grains during deformation.

**Bauschinger Effect****: **

When the specimen unload the strength of material got increase due to work hardening and if reloading it in opposite direction then back stress now support the movement of dislocation. As a result of that yield points in opposite direction appears prematurely. This phenomenon called Bauschinger effect.

At temperature = 0.4 melting temperature, upon cold working the number of dislocation in the material increases. The amount of dislocation multiplication takes place, is the property of material and it is expressed by work hardening exponent (n) in the flow curve equations σ_{f} = Kϵ^{n}

where, k = strength coefficient, σ_{f} = true stress and ϵ = true strain.

Higher the value of ‘n’, creation of dislocation is increased by cold working. Finally, these dislocation gather at the grain boundary and created dislocation forest.

These dislocation are repulsive in nature, which results in creation of back stress in the movement of dislocation. This is finally results in increase in strength of material upon unloading the specimen from the region of work hardening and reloading in opposite direction.

**Recovery****: **

During recovery, some of the stored internal strain energy is relieved by virtue of dislocation motion in the absence of an externally applied stress, as a result of enhanced atomic diffusion at an elevated temperature. There is some reduction in the number of dislocation also. Physical properties such as electrical and thermal conductivities are recovered to their pre cold worked states.

**Recrystallizatio****n:**

Even after recovery the grains are still in a high strain energy state. Recrystallization is the formation of new set of strain free grains that are required. Dislocation densities are reduced and properties are restored to pre cold work conditions. The driving force is the difference in the internal energy of strained and unstrained material.

The new grains forms initially as a small nuclei and grow at the expense of parent material. It also refines the grain structure. Extent of recrystallization depends upon both time and temperature. The degree of recrystallization increases with time. Recrystallization temperature is the temperature at which recrystallization just reaches completion in 1 hour. It typically ranges between 0.3Tm- 0.5Tm for most materials and alloy, where T_{m} = melting point temperature.

As the amount of cold work increases, the degree of recrystallization is enhanced and thus the recrystallization temperature is lowered and approaches a constant value at high deformation. There is some critical degree of cold work below which recrystallization will not take place. The recrystallization rate increases exponentially with temperature.

A larger the initial grain size, the greater degree of deformation is required to produce an equivalent recrystallization temperature. A minimum amount of deformation is needed to cause recrystallization temperature. Smaller the degree of deformation, higher will be recrystallization temperature. Greater the degree of deformation and lower the annealing temperature, the smaller will be the recrystallization grain size. Alloying raises the recrystallization upto 0.7 T_{m}.

**Grain Growth****: **

After recrystallization is complete, the grains will continue to grow if the metal specimen is left at elevated temperature this phenomenon is called grain growth. Grains boundaries are associated with higher energies, and as the grain size increases the total grain boundary area decrease resulting in reduction of total energy. This is the driving force for grain growth.

Here large grains grown at the expense of smaller grains which further shrink in size. Average grain size increases. This is done by short range diffusion of atoms from one side of boundary to the other. The direction of boundary movement and atomic motion are opposite to each other.

## Sc Bcc Fcc

**Time dependences of grain growth is: **

d^{n} – do^{n} = kt

where k, n ar constants and ‘do’ is initial grain size.

Grain growth proceeds more rapidly at higher temperature as diffusion rate increases with temperature.