Bcc, Fcc, and Primitive Cubic Cell

Introduction

BCC stands for Body-Centered cubic unit cell, while FCC stands for Face-centered Cubic unit cell. These are types of atom arrangements. In BCC, the structure consists of various atoms arranged or packed in a cubic lattice where each or every single corner or edge of the cube/cubic shares a single atom with the atom in the centre. So, the atoms at the corners or edges are shared with eight different unit cells. And in FCC, atoms are present at every corner of the cube and also at the centre of every six faces of the cubic crystal. So, the atom present in/at the centres of the faces gets shared between two adjacent or one after another unit cells, and thus, only half of each atom contributes to an individual or unique cell.

What is a Unit Cell?

Unit Cells are defined as the building blocks of the crystals. These are the smallest repeating or similar units of a cube or crystal lattice. These unit cells are identical to each other such that they fill all the spaces without any overlapping. These unit cells are further defined based on the lattice points. In a unit cell, opposite faces are parallel to each other and the edge of the cell connects all the equivalent points. Mainly, there are three types of unit cells and they are −

• Simple or Primitive unit cell

• BCC(body-centred)

• FCC(face-centred)

Images Coming soon

Types of Unit Cell

Although there are varieties of types of unit cells. But here we will be discussing three major cubic unit cells and those are −

• Primitive or simple unit cell

• FCC(face-centred cell)

• BCC (body-centred cell

Now, briefly, we will explain each of them.

• Simple or Primitive unit cell - In a simple or primitive type unit cell, the constituent or component particles are found only in the corners of the crystal lattice. Here, each of the unit cells contains a one-eighth portion of all the eight corner atoms to form one complete atom in the unit cell.

Images Coming soon

• Face-centred unit cell - It is the most densely populated or dense unit cell. Here, along with the corners of the crystal, atoms are also present or found at the centre of the cube’s faces. So, the face-centred atoms are equally shared or divided between the two adjacent or one after another unit cells, and hence, only half of each atom is a part of the individual cell.

Images Coming soon

• Body-centred unit cell- It resembles similarities with the simple or primitive cubic unit cell, as both of them contain eight atoms at the corner of the cubic crystal. The difference is that BCC contains an atom found at/in the centre or middle of the cubic crystal and it has an open structure. And the atom present at the centre completely belongs or founds to the unit cell in which it resides.

Images Coming soon

Number of Atoms in the BCC Cell

As we know in the BCC unit cell there are eight atoms present at every corner of the cubic lattice and one atom is present at the centre/centre of the cubic lattice. And, each of the corner atoms contributes (1/8th)one-eighth of its portion to form a unit cell.

So, all the eight atoms at the corners will contribute = (8 × 1/8) = 1

And, in a body-centred cube, there is one/single atom present at/in the centre of the cubic lattice.

Therefore, the total number of atoms in total present in the bcc(body-centred) cell = (1 + 1)= 2.

Number of Atoms in FCC Cell

As we know in the FCC unit cell type there are eight atoms present at each or every corner of the cubic lattice and atoms are also present or found at the centre of the cube’s faces. Each corner atom contributes one-eighth of its portion to form a unit cell.

So, all the eight atoms at the corners will contribute = (8 × 1/8) = 1

And, there are six faces present in the cubic lattice and each particle present at/in the centre of the six faces gets shared with the neighbouring cube. So, all the six atoms at the faces will contribute =(6 × 1/2) = 3.

Therefore, the total number of atoms in total present in the fcc(face-centered) cell = (1 + 3) = 4.

The volume of HCP Unit Cell

Images Coming soon

As we know Volume = Area of base $\mathrm{\times }$ height. HCP stands for Hexagonal close-packed unit cell. It has a diamond-shaped structure with a hexagonal base and all sides are equal in length. Here, the atoms present at/in the corners of the base of the hexagonal unit cell are present in touch or contact with each other, thus the sides say a and b will form a relation with radius say r, that is a = b = 2r. The height(h) of the unit cell is, h = 2 dist between two adjacent layers $\mathrm{=\:2\:\times\sqrt{\frac{2}{3} a}}$ . Now, the base is a hexagon and as we know, a hexagon is made up of six equilateral triangles, and the area of the equilateral triangle is $\mathrm{\sqrt{3/4(side)^{2}}}$.

So, the area of the base = $\mathrm{6 \times \sqrt{3/4\cdot a^{2}}}$

Therefore, the Volume of HCP = $\mathrm{(6\times \sqrt{3/4a^{2}}\times 2\times \sqrt{2/3}a )}$

Now, putting a = 2r, we will get, Volume = $\mathrm{24\sqrt{2}r^{3}}$

Classification of Crystal Structure by Symmetry

Crystal symmetry is defined as a reflection of internal atomic arrangement.

And the crystals are classified or distributed into seven major crystallographic systems based on their symmetry. And they are - Tetragonal, Cubic, Orthorhombic, Monoclinic, Hexagonal, triclinic and Rhombohedral or trigonal.

Crystal System	Possible variations	Edge lengths or Axial distance	Axial angles	Examples
Cubic	Primitive, BCC, FCC	a = b = c	$\mathrm{\alpha =\beta =\gamma =90^{\circ}}$	Cu, NaCl
Tetragonal	BCC, Primitive	a = b $\mathrm{ eq }$ c	$\mathrm{\alpha =\beta =\gamma =90^{\circ}}$	$\mathrm{TiO_{2},CaSO_{4}}$
Orthorhombic	Primitive, BCC, FCC	a or A $\mathrm{ eq}$ b or B $\mathrm{ eq}$ c or C	$\mathrm{\alpha =\beta =\gamma =90^{\circ}}$	$\mathrm{KNO_{3},BaSO_{4}}$
Monoclinic	Primitive, Endcentred	A or A $\mathrm{ eq}$ b or B $\mathrm{ eq}$ c or C	$\mathrm{\alpha =\gamma =90 ^{\circ}\:\beta eq 90^{\circ}}$	$\mathrm{Na_{2}SO_{4}.10H_{2}O}$ Monoclinic sulphur
Rhombohedral	Primitive	a or A= b or B = c or C	$\mathrm{\alpha =\beta =\gamma eq 90^{\circ}}$	Calcite, HgS
Hexagonal	Primitive	a or A= b or B $\mathrm{ eq}$ c or C	$\mathrm{\alpha =\beta= 90^{\circ}\:\gamma =120^{\circ}}$	ZnO, CdS, Graphite
Triclinic	Primitive	a or A $\mathrm{ eq}$ b or B $\mathrm{ eq}$ c or C	$\mathrm{\alpha eq \beta eq \:\gamma eq 90^{\circ}}$	$\mathrm{K_{2}Cr_{2}O_{7}}$

Conclusion

In this article, we discussed BCC, FCC primitive unit cells, their structures, their atom arrangements, the number of atoms for BCC and FCC unit cells, the definition of the unit cell, and types of the unit cell that are majorly three named as a primitive cubic unit cell, then body-centred(bcc) and fcc(face-centred) unit cell. Then, we briefly come to know about each one of them. The volume of HCP (Hexanol closed-packed) is found to be equal to $\mathrm{24\sqrt{2}r^{3}}$. And, finally, the classification of crystal structure by symmetry includes seven systems of atom arrangements and they are - Cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, and triclinic.

FAQs

Q1. What are primitive unit cells?

Ans. Unit cells where the component or constituent or component particles that are present or found only at the corner positions are termed primitive unit cells.

Q2. Define Centred type Unit cells?

Ans. Centred type unit cells are those unit cells that contain atoms at other positions also along with the atoms found in the corners of the cube or cubic lattice.

Q 3. The number of atoms in the BCC and FCC unit cells?

Ans. In the BCC(body-centred) unit cell, there are in total of 2 atoms present while in the FCC unit cell, in total there are 4 atoms present there.

Q 4. What is the volume of the HCP unit cell?

Ans. The volume of HCP unit cell = $\mathrm{24\sqrt{2}r^{3}}$, where r is the radius.

Q5. Name all the crystal systems formed based on symmetry.

Ans. Following are all seven crystal systems or crystals based on symmetry-

• Cubic unit cell

• Tetragonal cell

• Orthorhombic unit cell

• Hexagonal cell

• Rhombohedral cell

• Monoclinic cell

• Triclinic cell.