91影视

The Fermi energy${\mathit{E}}_{\mathbf{F}}$for a particle of mass$\mathit{m}$can be written as

${\mathit{E}}_{\mathbf{F}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{m}{\mathit{v}}_{\mathbf{F}}^{\mathbf{2}}\mathbf{\text{(1)}}$

Where,${\mathit{v}}_{\mathbf{F}}$is Fermi velocity,$\mathit{m}$is mass of electron, which is equal to

$\mathbf{9}\mathbf{.1}\mathbf{脳}{\mathbf{10}}^{\mathbf{鈭�}\mathbf{31}}\mathbf{}\mathbf{\text{kg}}$

The de Broglie wavelength$\mathit{位}\mathbf{\text{鈥�}}$of a particle of mass$\mathit{m}$and velocity$\mathit{v}$is given by

$\mathit{位}\mathbf{=}\frac{\mathbf{h}}{\mathbf{m}\mathbf{v}}\mathbf{\text{(2)}}$

$\mathit{位}\mathbf{=}\frac{\mathbf{h}}{\mathbf{m}\mathbf{v}}\mathbf{\text{(2)}}$

Where,$\mathbf{\text{h}}$is Planck's constant which is equal to$\mathbf{6}\mathbf{.64}\mathbf{脳}{\mathbf{10}}^{\mathbf{鈭�}\mathbf{34}}\mathbf{}\mathbf{\text{J}}\mathbf{鈰�}\mathbf{\text{s}}$

$\mathit{m}$is mass of electron, which is equal to$\mathbf{9}\mathbf{.1}\mathbf{脳}{\mathbf{10}}^{\mathbf{鈭�}\mathbf{31}}\mathbf{}\mathbf{\text{kg}}$

The condition for classical behaviour for a particle of wavelength$\mathit{位}\mathbf{\text{鈥�}}$is

${\mathbf{\left(}\frac{\mathbf{位}}{\mathbf{d}}\mathbf{\right)}}^{\mathbf{3}}\mathbf{<}\mathbf{<}\mathbf{1}\mathbf{\text{(3)}}$

Where,$\mathit{d}$is separation between the particles.

Fermi energy of conduction electrons in sodium is,

$\begin{array}{c}{E}_F=3.1\text{eV}\\ =\left(3.1\text{eV}\right)\left(\frac{1.6脳{10}^{鈭�19}\text{J}}{1\text{eV}}\right)\\ =4.9脳{10}^{鈭�19}\text{J}\end{array}$

Calculation:

Substitute the numerical values in equation (1)

$\begin{array}{c}{v}_F^2=\frac{2脳4.9脳{10}^{鈭�19}\text{kg}鈰�{\text{m}}^2/{\text{s}}^2}{(9.1脳{10}^{鈭�31}\text{kg})}\\ =1.076脳{10}^{12}{\text{m}}^2/{\text{s}}^2\\ v_F=1.03脳{10}^6\text{m}/\text{s}\end{array}$

Substitute the numerical values in equation (2)

$\begin{array}{c}位=\frac{6.64脳{10}^{鈭�34}{\text{kgm}}^2/\text{s}}{(9.1脳{10}^{鈭�31}\text{kg})(1.03脳{10}^6\text{m}/\text{s})}\\ =0.708脳{10}^{鈭�9}\text{m}\\ =0.708\text{nm}\end{array}$

Separation between the sodium atoms is,$d=$$0.37\text{nm}$

Substitute the numerical values in the left side of equation (3)

$\begin{array}{c}{\left(\frac{位}{d}\right)}^3=\left(\frac{0.708\text{nm}}{0.37\text{nm}}\right)\\ =1.9>1\end{array}$

The value of ${\left(\frac{位}{d}\right)}^3$is greater than 1. Therefore, it is necessary to treat the conduction electron gas as a quantum gas, if each sodium atom contributes one conduction electron to the electron gas and sodium atoms are spaced roughly$0.37\text{nm}$ .