$\sigma( \omega )d \omega = \sigma_E \delta( \omega - \omega_E ) d \omega$

s( w )* dw = s_E *delta( w - w_E )* dw

$\displaystyle\int_0^{\infty} \sigma_E( \omega ) d \omega =3 N$

@INT( sigma_E , omega , 0 , infty ) =3* N

$\sigma_E = 3 N$

s_E = 3* N

$\sigma_E( \omega ) d \omega = 3 N \delta( \omega - \omega_E ) d \omega$

s_E( w ) * dw = 3* N * d( w - w_E )* dw

$\ln Z = \beta N \eta -3 N \ln(1-e^{ -\beta \hbar \omega_E })$

ln Z = beta * N * eta -3* N *ln(1-exp(- beta * hbar * w_E ))

$\Theta_E =\displaystyle\frac{ \hbar \omega_E }{ k_B }$

Theta_E = hbar * w_E / k_B

$\ln Z_E =-\displaystyle\frac{3 N }{2}\displaystyle\frac{ \Theta_E }{ T }-\displaystyle\frac{ NV_0 }{ k_B T }-3 N \ln(1-e^{- \Theta_E / T })$

ln Z_E =-3* N * Theta_E/(2 * T )- N*V_0 /( k_B * T )-3* N *log(1-e^(- Theta_E / T ))

$\ln Z =-\displaystyle\frac{ V_0 }{ k_B T }+\displaystyle\frac{3 N }{2}\left(\displaystyle\frac{ \Theta_E }{ T }\right)^2-\displaystyle\frac{ N }{2}\left(\displaystyle\frac{ \Theta_E }{ T }\right)^3+\ldots$

ln Z =- V_0 /( k_B * T )+(3* N /2)*( Theta_E / T )^2-( N /2)*( Theta_E / T )^3+...

$\eta = - 3V_0 -\displaystyle\frac{3}{2}\hbar\omega_E$

eta = - 3* V_0 -(3/2)* hbar * omega_E