Packing in fcc, bcc and hcp lattices

JEE Chemistry Solid State, packing in fcc, bcc and hcp lattices, , , , , , ,

The three most common crystal structures are face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close-packed (HCP) lattices. The packing in each of these structures is as follows:

  1. FCC lattice: In an FCC lattice, each atom is surrounded by 12 nearest neighbors arranged in the form of an octahedron. The atoms are located at the corners and the center of each face of the cube. The coordination number is 12 and the packing efficiency is 74%.
  2. BCC lattice: In a BCC lattice, each atom is surrounded by 8 nearest neighbors arranged in the form of a cube. The atoms are located at the corners and the center of the cube. The coordination number is 8 and the packing efficiency is 68%.
  3. HCP lattice: In an HCP lattice, each atom is surrounded by 12 nearest neighbors arranged in the form of a hexagon. The atoms are located at the corners and the center of each hexagon. The coordination number is 12 and the packing efficiency is 74%.

In terms of packing efficiency, FCC and HCP lattices are more efficient than the BCC lattice. This is because in the FCC and HCP lattices, the atoms are arranged in a more closely packed manner than in the BCC lattice. However, the BCC lattice is more stable than the FCC and HCP lattices at high temperatures and pressures.

What is Required Packing in fcc, bcc and hcp lattices

The Required Packing in FCC, BCC, and HCP lattices is the minimum number of atoms needed to fill a unit cell of the respective lattice type.

  1. FCC lattice: In an FCC lattice, the required packing is four atoms per unit cell. Each corner atom contributes 1/8th of an atom, and each face-centered atom contributes 1/2 of an atom to the unit cell.
  2. BCC lattice: In a BCC lattice, the required packing is two atoms per unit cell. Each corner atom contributes 1/8th of an atom, and the center atom contributes 1 whole atom to the unit cell.
  3. HCP lattice: In an HCP lattice, the required packing is six atoms per unit cell. Each corner atom contributes 1/6th of an atom, and each hexagon-centered atom contributes 2/3 of an atom to the unit cell.

It’s worth noting that while the packing fraction (or efficiency) of FCC and HCP lattices is higher than that of the BCC lattice, the BCC lattice has a more closely packed structure, which makes it more stable under certain conditions.

Who is Required Packing in fcc, bcc and hcp lattices

Instead, the term typically used in crystallography is “Primitive Unit Cell”, which refers to the smallest unit cell that can be used to build up the crystal lattice.

For FCC, BCC, and HCP lattices, the primitive unit cell is as follows:

  1. FCC lattice: The primitive unit cell of an FCC lattice is a cube with atoms located at each of the eight corners, and at the center of each of the six faces. The unit cell contains four atoms.
  2. BCC lattice: The primitive unit cell of a BCC lattice is a cube with atoms located at each of the eight corners, and one atom located at the center of the cube. The unit cell contains two atoms.
  3. HCP lattice: The primitive unit cell of an HCP lattice is a hexagonal prism with atoms located at each of the two bases and at the center of each of the hexagonal faces. The unit cell contains six atoms.

Atomic packing factor

In crystallography, nuclear pressing element (APF), pressing productivity, or pressing portion is the small portion of volume in a precious stone construction that is involved by constituent particles. It is a dimensionless amount and in every case not as much as solidarity. In nuclear frameworks, by show, the still up in the air by expecting that molecules are unbending circles. The range of the circles is taken to be the most extreme worth with the end goal that the particles don’t cover. For one-part gems (those that contain just a single kind of molecule), the pressing division is addressed numerically by{\displaystyle \mathrm {APF} ={\frac {N_{\mathrm {particle} }V_{\mathrm {particle} }}{V_{\text{unit cell}}}}}

where Nparticle is the quantity of particles in the unit cell, Vparticle is the volume of every molecule, and Vunit cell is the volume involved by the unit cell. It tends to be demonstrated numerically that for one-part structures, the most thick game plan of iotas has an APF of around 0.74 (see Kepler guess), got by the nearby pressed structures. For different part structures, (for example, with interstitial amalgams), the APF can surpass 0.74.

The nuclear pressing variable of a unit cell is pertinent to the investigation of materials science, where it makes sense of numerous properties of materials. For instance, metals with a high nuclear pressing variable will have a higher “functionality” (flexibility or pliability), like how a street is smoother when the stones are nearer together, permitting metal particles to effectively slide past each other more.

FCC and HCP lattices

There are two straightforward standard grids that accomplish this most noteworthy typical thickness. They are called face-focused cubic (FCC) (additionally called cubic close pressed) and hexagonal close-stuffed (HCP), in view of their evenness. Both depend on sheets of circles organized at the vertices of a three-sided tiling; they vary in how the sheets are stacked upon each other. The FCC grid is additionally referred to mathematicians as that produced by the A3 root foundation.

How is Required Packing in fcc, bcc and hcp lattices

Instead, the correct term is “Primitive Unit Cell”, which is the smallest repeating unit that contains all the symmetry elements of the crystal structure. The shape and size of the primitive unit cell is determined by the crystal structure.

For the FCC, BCC, and HCP lattices, the primitive unit cells have specific shapes and sizes, as follows:

  1. FCC lattice: The primitive unit cell of an FCC lattice is a cube with atoms located at each of the eight corners and at the center of each of the six faces. The cube has edge length “a” and the distance between opposite faces is also “a”.
  2. BCC lattice: The primitive unit cell of a BCC lattice is a cube with atoms located at each of the eight corners and one atom located at the center of the cube. The cube has edge length “a” and the distance between opposite faces is sqrt(3) times “a”.
  3. HCP lattice: The primitive unit cell of an HCP lattice is a hexagonal prism with atoms located at each of the two bases and at the center of each of the hexagonal faces. The prism has a height of “c” and a base with edge length “a”. The angle between the base and the sides of the prism is 120 degrees.

The primitive unit cell determines the crystal structure and its properties. The arrangement of atoms in the primitive unit cell, as well as the size and shape of the cell, contribute to the crystal’s overall symmetry, density, and other physical properties.

Case Study on Packing in fcc, bcc and hcp lattices

One example of the importance of packing in these lattices is the study of metallic alloys. Metallic alloys are materials made up of two or more metallic elements, and their properties are highly dependent on the crystal structure and packing of atoms.

For instance, consider the case of the Cu-Au alloy system, where copper (Cu) and gold (Au) are mixed together. Depending on the relative concentrations of Cu and Au atoms, different crystal structures can form. If the concentration of Cu is low, the alloy forms an FCC structure, while if the concentration of Cu is high, the alloy forms a BCC structure.

The packing of atoms in the FCC and BCC structures is different, which leads to differences in the mechanical, thermal, and electrical properties of the alloy. For example, the BCC structure is generally stronger and less ductile than the FCC structure due to its more compact packing of atoms.

Another example is the study of HCP metals, which have a hexagonal close-packed structure. HCP metals are often used in high-temperature applications because of their high strength, good creep resistance, and low thermal expansion coefficient. The packing of atoms in the HCP lattice, which involves stacking of hexagonal layers, is responsible for these properties.

Overall, understanding the packing of atoms in FCC, BCC, and HCP lattices is crucial for the study and design of materials with specific properties.

White paper on Packing in fcc, bcc and hcp lattices

Introduction:

The packing of atoms in a crystal lattice is a crucial factor that determines the crystal structure and properties of a material. FCC, BCC, and HCP lattices are among the most common crystal structures in metallic materials and play a critical role in determining their mechanical, thermal, and electrical properties.

FCC Lattice:

The FCC (Face-Centered Cubic) lattice is the most common crystal structure in metals, including aluminum, copper, and gold. In the FCC lattice, atoms are located at each of the eight corners of a cube and at the center of each of the six faces. This arrangement results in a densely packed structure with a coordination number of 12, which means that each atom has 12 nearest neighbors. The FCC lattice has a high symmetry and exhibits isotropic behavior, making it ideal for materials that require good ductility, toughness, and formability.

BCC Lattice:

The BCC (Body-Centered Cubic) lattice is another common crystal structure in metals, including iron, chromium, and tungsten. In the BCC lattice, atoms are located at each of the eight corners of a cube and at the center of the cube. This arrangement results in a less densely packed structure with a coordination number of 8, which means that each atom has 8 nearest neighbors. The BCC lattice has a lower symmetry and exhibits anisotropic behavior, making it ideal for materials that require high strength, hardness, and wear resistance.

HCP Lattice:

The HCP (Hexagonal Close-Packed) lattice is a common crystal structure in metals, including titanium, zinc, and magnesium. In the HCP lattice, atoms are arranged in a hexagonal pattern with each layer stacked on top of the other in an ABCABC… sequence. This arrangement results in a densely packed structure with a coordination number of 12, which means that each atom has 12 nearest neighbors. The HCP lattice has a high symmetry and exhibits anisotropic behavior, making it ideal for materials that require high strength, creep resistance, and low thermal expansion coefficient.

Conclusion:

In conclusion, the packing of atoms in FCC, BCC, and HCP lattices plays a crucial role in determining the crystal structure and properties of metallic materials. Understanding the differences in packing and coordination numbers between these lattices is essential for the design and engineering of materials with specific properties.