Question

The ratio of the volume of a tetragonal lattice unit cell to that of a hexagonal lattice unit cell is:

(both having same respective lengths)

• A. $\frac{\sqrt{3}}{2}abc$
• B. $\frac{2}{3\sqrt{3}}$
• C. $\frac{2}{\sqrt{3}}\frac{a^{2}c}{b}$
• D. $1.33$

Answer 1

Answer: (D)

$1.33$

The volume of the unit cell is given by the expression

$V=abc[1-co{s}^2\alpha -co{s}^2\beta -co{s}^2\gamma +2cos\alpha cos\beta cos\gamma ]$

For tetragonal lattice $a=b\ne c$ and $\alpha =\beta =\gamma ={90}^0$ and

$cos{90}^0=0$

The volume of tetragonal unit cell is given by the expression

$V=abc[1-co{s}^2{90}^0-co{s}^2{90}^0-co{s}^2{90}^0+2cos{90}^{o}cos{90}^{o}cos{90}^o]$

$V=abc[1-0-0-0+2\times 0\times 0times0]$

$V=abc$

For hexagonal unit cell $a=b\ne c$ and $\alpha =\beta ={90}^o$ and $\gamma ={120}^0$ and $cos{90}^o=0$, $cos{120}^o=-0.5$

The volume of the hexagonal unit cell is

$V=abc[1-co{s}^2{90}^o-co{s}^2{90}^o-co{s}^2{120}^o+2cos{90}^{o}cos{90}^{o}cos{120}^o]$

$V=abc[1-0-0-(-0.5{)}^2+0\times 0\times (-0.5)]$

$V=abc[1-0.25]$

$V=0.75abc$

The ratio of the volume of a tetragonal lattice unit cell to that of a hexagonal lattice unit cell is (both having same respective lengths)

$\frac{abc}{0.75abc}=1.333$