91Ó°ÊÓ

• Copper is a monovalent metal.
• Molar mass of copper, A = 63.54 g/mol
• Density of copper, d = 8.96$\mathrm{g}/{\mathrm{cm}}^3$
• Fermi energy of copper,${\mathrm{E}}_{\mathrm{F}}=7.0\mathrm{eV}$

There are three main electrical properties of solids- resistivity , the temperature coefficient of resistivity $\mathbf{\left(}\mathbf{Î\pm }\mathbf{\right)}$and the number density of charge carriers (n) . Resistivity is described as the specific resistance of a substance. The temperature coefficient of resistivity is the reciprocal of resistivity when the rate of change of resistivity with absolute temperature is unity. Number density is the number of charge carriers present per unit volume.

Formulae:

The density of states associated with the conduction electrons of a metal (according to equation 41-5)

$N\left(E\right)=\frac{8\sqrt{2}\mathrm{Ï€}{m}^{3/2}}{h^3}{E}^{1/2}..........................\left(1\right)$

$\mathrm{where}\mathrm{h}=6.63×{10}^{-34}\mathrm{J}.\mathrm{s},\mathrm{m}=9.1×{10}^{-31}\mathrm{kg}$

The equation of Fermi energy according to Eq. 41-9,

$E_F={\left[\frac{3}{16\sqrt{2\mathrm{Ï€}}}\right]}^{2/3}\frac{h^2}{m}{n}^{2/3}..............................\left(2\right)$

The mass of an atom,

$M=A/N_A...................\left(3\right)$

$\mathrm{where}{\mathrm{N}}_{\mathrm{A}}=6.022×{10}^{23}{\mathrm{mol}}^{-1}$

The number density of conduction electrons,

$n=\frac{d}{M}..............................................\left(4\right)$

d= density of the atom, M = mass of a single atom

Substituting $E=E_F$in equation ,we get-

$N\left(E_F\right)=\frac{8\sqrt{2}\mathrm{Ï€}{m}^{3/2}}{h^3}{E_F}^{1/2}............................................\left(5\right)$

Comparing equation (5) and equation (2) , we can get the value of the density of states as follows:

$$\begin{gathered} \mathrm{N}\left({\mathrm{E}}_{\mathrm{F}}\right)=\frac{8\sqrt{2}{\mathrm{Ï€³¾}}^{3/2}}{{\mathrm{h}}^3}{\left[{\left(\frac{3}{16\sqrt{2}\mathrm{Ï€}}\right)}^{2/3}\frac{{\mathrm{h}}^2}{\mathrm{m}}{\mathrm{n}}^{2/3}\right]}^{1/2} \\ =\frac{8\sqrt{2}{\mathrm{Ï€³¾}}^{3/2}}{{\mathrm{h}}^3}{\left(\frac{3}{16\sqrt{2}\mathrm{Ï€}}\right)}^{2/3}\frac{{\mathrm{h}}^2}{\mathrm{m}}{\mathrm{n}}^{2/3} \\ =\frac{4\mathrm{m}}{{\mathrm{h}}^2}\sqrt[3]{3{\mathrm{Ï€}}^2\mathrm{n}} \\ =\frac{4{\mathrm{mc}}^2}{{\left(\mathrm{hc}\right)}^2}\sqrt[3]{3{\mathrm{Ï€}}^2\mathrm{n}} \\ \left({\mathrm{mc}}^2=5.11×{10}^5\mathrm{eV}\mathrm{and}\mathrm{hc}=1240\mathrm{eV}.\mathrm{nm}\right) \end{gathered}$$

Solving further,

$$\begin{gathered} \mathrm{N}\left({\mathrm{E}}_{\mathrm{F}}\right)=\frac{4\left(5.11×{10}^5\mathrm{eV}\right)}{{\left(1240\mathrm{eV}.\mathrm{nm}\right)}^2}\sqrt[3]{3{\mathrm{π}}^2\mathrm{n}} \\ =\left(4.11{\mathrm{nm}}^{-2}.{\mathrm{eV}}^{-1}\right){\mathrm{n}}^{1/3} \\ =\left(4.11×{10}^{18}{\mathrm{m}}^{-2}.{\mathrm{eV}}^{-1}\right){\mathrm{n}}^{1/3}....................................\left(\mathrm{a}\right) \end{gathered}$$

Hence, the value of density is $\left(4.11×{10}^{18}{\mathrm{m}}^{-2}.{\mathrm{eV}}^{-1}\right){\mathrm{n}}^{1/3}$.

Since, copper is a monovalent metal.

Now, the mass of the copper atom can be given using the data of molar mass in equation (iii) as follows:

$$\begin{gathered} \mathrm{M}=\frac{63.54\mathrm{g}/\mathrm{mol}}{6.022×{10}^{23}{\mathrm{mol}}^{-1}} \\ =1.055×{10}^{-22}\mathrm{g} \end{gathered}$$

Thus, the value of the number density of the conductions electrons of copper atom can be calculated using the given data in equation (iv) as follows:

$$\begin{gathered} \mathrm{n}=\frac{8.96\mathrm{g}/{\mathrm{cm}}^3}{1.055×{10}^{-22}\mathrm{g}} \\ =8.49×{10}^{28}{\mathrm{m}}^{-3} \\ =84.9{\mathrm{nm}}^{-3} \end{gathered}$$

Now, substituting this value in equation (a), we can get the density of states of the copper material as follows:

$$\begin{gathered} \mathrm{N}\left({\mathrm{E}}_{\mathrm{F}}=7\mathrm{eV}\right)=\left(4.11{\mathrm{nm}}^{-2}.{\mathrm{eV}}^{-1}\right){\left(84.9{\mathrm{nm}}^{-3}\right)}^{1/3} \\ =18{\mathrm{nm}}^{-3}.{\mathrm{eV}}^{-1} \\ =1.8×{10}^{28}{\mathrm{m}}^{-3}.{\mathrm{eV}}^{-1} \end{gathered}$$

Hence, the value of the density of states is $1.8×{10}^{28}{\mathrm{m}}^{-3}.{\mathrm{eV}}^{-1}$.

On comparing the calculated value of the density of states from the curve in Figure 41-6 , at${\mathrm{E}}_{\mathrm{F}}=7\mathrm{eV}$, both the values are approximately same.

Thus, the calculated value is correct.