Concept explainers
Interpretation:
The expressions for the volume of a triclinic, monoclinic, orthorhombic and tetragonal unit cells have to be derived and also the volume of a given tetragonal unit cell has to be calculated.
Concept Introduction:
From a vector analysis, we know that volume of a geometric system can be alternatively written as
Answer to Problem 15A.4P
The volume of the tetragonal unit cell is found out to be
Explanation of Solution
First we have to derive the volume of a triclinic unit cell. Then we can take it as the reference to derive the volume of a hexagonal unit cell.
Derivation of the volume of a triclinic unit cell:
The cell dimensions of a triclinic unit cell is
Figure 1
There are many choices for the primitive vectors in the triclinic system. So we are taking
Then,
Therefore,
This is the expression for the volume of a triclinic unit cell.
Derivation of the volume of a monoclinic unit cell:
The cell dimensions of a monoclinic unit cell is
Now substituting these values in the expression of the volume of a triclinic unit cell, we can get the exact expression for the monoclinic unit cell.
The above expression is for the volume of a monoclinic unit cell.
Derivation of the volume of an orthorhombic unit cell:
The cell dimensions of a orthorhombic unit cell is
Now substituting these values in the expression of the volume of a triclinic unit cell, we can get the exact expression for the orthorhombic unit cell.
The above expression is for the volume of an orthorhombic unit cell.
Derivation of the volume of a tetragonal unit cell:
The cell dimensions of a tetragonal unit cell is
Now substituting these values in the expression of the volume of a triclinic unit cell, we can get the exact expression for the tetragonal unit cell.
The above expression is for the volume of a tetragonal unit cell.
Given data:
For a tetragonal unit cell,
The expression for the volume of a tetragonal unit cell is
Hence,
Therefore, the volume of the tetragonal unit cell is
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Chapter 15 Solutions
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