Problem set 34

1. The second neighbour distance in a simple cubic system having lattice constant $a$ is:
1. $\sqrt{2}a$
2. $a$
3. $\frac{\sqrt{3}}{2}a$
4. $\sqrt{3}a$
Second neighbours are present at the digonal position of the face. Hence, second neighbour distance is $\sqrt{2}a$.
Hence, answer is (A)
2. If the Debye characteristic frequency for copper is $6.55\times10^{12}Hz$, the Debye temperature for copper is :
1. $200K$
2. $314.5K$
3. $405K$
4. $150K$
\begin{align*} \theta_D&=\frac{h\nu_D}{k}\\ &={\scriptstyle\frac{6.62607004\times10^{-34}\times6.55\times10^{12}}{1.38\times10^{-23}}}\\ &=314.5K \end{align*}
3. If the concentration of Schotty defects in a fcc crystal is 1 in $10^{10}$ at $300^oK$, the energy of formation in eV of Schottky defects will be
1. 0.60
2. 0.50
3. 0.40
4. 1.0
$$n=Ne^{\left(-\frac{E_{schottky}}{k_BT}\right)}$$ Here, $n$ is number of defects defects, $N$ is number of atoms. $$\frac{n}{N}=e^{\left(-\frac{E_{schottky}}{k_BT}\right)}$$ $$E_{schottky}=-k_BT\ln{\frac{n}{N}}$$ $$\frac{n}{N}=\frac{1}{10^{10}}$$ \begin{align*} &E_{schottky}={\scriptstyle-1.38\times10^{-23}\times300\ln{\frac{1}{10^{10}}}J}\\ &{\scriptstyle=-\frac{1.38\times10^{-23}\times300}{1.6\times10^{-19}}(-10\ln{10}) eV}\\ &=0.6 eV \end{align*}
4. Van der Waals attractive interaction between inert gas atoms varies with interatomic separation $(R)$ as :
1. $-\frac{1}{R^2}$
2. $-\frac{1}{R^3}$
3. $-\frac{1}{R^6}$
4. $-\frac{1}{R^{12}}$
(C)$-\frac{1}{R^6}$
5. Metallic nickel crystallizes in fcc-type structure with lattice constant $a$. The interplanar distance between the diffracting planes (220) is this material is :
1. $\frac{a}{2\sqrt{2}}$
2. $\frac{a}{2}$
3. $\frac{\sqrt{3}}{2}a$
4. $\frac{a}{\sqrt{2}}$
\begin{align*} d_{hkl}&=\frac{a}{\sqrt{h^2+k^2+l^2}}\\ &=\frac{a}{\sqrt{2^2+2^2+0^2}}=\frac{a}{2\sqrt{2}} \end{align*} Hence, answer is (A)