The Solid State

Number of Particles in Unit Cells

A unit cell is a repeating structure whose specific arrangement makes up a cubic crystal system. However, a unit cell must not be confused with an atom or a particle. These atoms make up a unit cell. So let us understand how to calculate the number of particles in unit cells in a cubic crystal system.

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Number of Particles in Unit Cells

Let us remember what a unit cell is. It is the most basic unit of the crystal lattice. The arrangement of the unit cells will define the structure as well as the symmetry of the crystalline solids. Now a lattice point in the unit cell is generally where we would find a constituent particle. But it must be kept in mind that a lattice point can also be vacant.

Now, we must calculate the number of particles that are present in unit cells. For this let us first consider some general characteristics of unit cells and their structure. The contribution of atoms present at different lattice sites

  • A particle (atom, molecule or ion) which is located at the corners of the lattice, is shared by eight unit cells all in all. Hence the contribution of an atom at the corner to a particular unit cell is one-eighth
  • An atom on the face of the unit cell is shared by two unit cells. Hence the contribution of a particle at the face to a unit cell is 1/2
  • The constituent particle at the edge center is shared by four unit cells in the lattice. Hence its contribution to any one particular unit cell is one-fourth
  • An atom at the body center of a unit cell belongs entirely and only to that unit cell of the lattice. There is no sharing involved

So now let us calculate how many constituent particles are in all the different times of unit cells

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Primitive Cubic Unit Cells

As you know in a primitive unit cell the constituent particles, whether atoms, molecules or ions, are only located at the corners of the unit cell. As we saw above any particle at the corner of a unit cell contributes one-eighth of itself to one unit cell. Now in a cube, there are a total of eight corners.

And so in a primitive unit cell, there are eight particles at eight corners of the cubic structure. So the total contribution can be calculated as:

1/8 (contribution of corner atoms) × 8 (number of corners) = 1

Body-Centered Unit Cells

BCC Unit Cells

In a body centered unit cell, we have eight atoms at the corners and also one atom at the center of the unit cell. Now the particles at the center and those at the edges have different contributions. The one at the edges contributes one-eighth to a particular unit cell. The one at the center is not shared by any other unit cell in the lattice. So when we calculate the total number of atoms to a body centered unit cell, we get :

(1/8 × 8) + (1 × 1) = 2

Face Centered Unit Cells

Now in a face-centered unit cells particles are present at the edges and the faces of the cubic structure. The atoms at the eight corners have a one-eighth contribution to the unit cell. And the atoms at the face are shared equally between two unit cells in a lattice. So their contribution is half an atom. Note that in a cubic cell there are six faces. So the total number of atoms in a face-centered unit cell are:

(1/8 × 8) + (1/2 × 6) = 4

Solved Question for You

Q: A compound formed by elements A and B crystallizes in the cubic structure where A atoms are at the corners of a cube and B atoms are at the face centers. The formula of the compound is

  1. AB3
  2. AB
  3. A3B
  4. A3B3

Sol: The correct answer is “A”. The calculation will be as follows

Number of A atoms = 1/8 × 8 = 1

Number of B atoms = 1/2 × 6 = 3

Therefore the compound will be AB3

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