Question

Section 10.6 notes that as causes of resistance, ionic vibrations give way to lattice imperfections at around 10 K. A typical spring constant between atoms in a solid is or order of magnitude${\mathbf{10}}^3\frac{N}{m}$ and typical spacing is nominally${\mathbf{10}}^{-10}\mathbf{m}$ . Estimate how much the vibrating atoms locations might deviate, as a function of their normal separations, at 10 K.

Short answer

The vibrational distance is only $0.5\%$of the nominal separation.

Step-by-step solution

 Step 1: Determine the formulas

Consider the formulato determine the thermal energy as:

$\mathbf{E}\mathbf{=}{\mathbf{k}}_n\mathbf{T}$ ….. (1)

Here, $k_n$is Boltzmann constant and the temperature is T.

Consider the expression for the vibrational distance as:

$x=\sqrt{\frac{2\left(\text{spring\hspace{0.17em}potential\hspace{0.17em}energy}\right)}{\kappa }}$

Determine the number of vibrating atoms locations that might deviate.

From equation (1) solve for the thermal energy as:

$\begin{array}{c}{E}_T=(1.38\times {10}^{-23}\frac{\text{J}}{\text{K}})\left(10\text{\hspace{0.17em}K}\right)\\ =1.38\times {10}^{-22}\text{\hspace{0.17em}J}\end{array}$

Solve for the expression for the vibrational energy as:

$\begin{array}{c}x=\sqrt{\frac{2(1.38\times {10}^3)}{{10}^3\frac{\text{N}}{\text{m}}}}\\ =5\times {10}^{-23}\text{\hspace{0.17em}m}\end{array}$

Solve for the ratio of the distance as:

$\begin{array}{c}\frac{x}{a}=\frac{5\times {10}^{-23}}{{10}^{-10}}\\ =0.5\%\end{array}$

Therefore, the vibrational distance is$0.5\%$ only of the nominal separation.