Spectral Weight of a Gaussian oscillator

Given a Gaussian oscillator of the form:

$$\epsilon_2 = A\left[e^{-4\ln(2)\left(\frac{\omega-\omega_0}{\sigma}\right)^2} - e^{-4\ln(2)\left(\frac{\omega+\omega_0}{\sigma}\right)^2}\right]$$

as defined in [1] so that $\epsilon(\omega)$ is Kramers-Kronig consistent by calculating $\epsilon_1$ with a KK analysis, we want to obtain the spectral weight of such an oscillator. The spectral weight is given by

$$SW_{G} = \int\limits_0^{\infty}\sigma_1(\omega) d\omega = \int\limits_0^{\infty}\omega\epsilon_0\epsilon_2(\omega) d\omega $$

where $\sigma_1$ is the real part of the optical conductivity and $\epsilon_2$ is the imaginary part of the dielectric function.

Preliminaries

Computational Preliminaries



In [1]:

    
from sympy import *
init_printing()
omega = symbols('omega')
A, sigma, omega_0 = symbols('A sigma omega_0', real = True, positive = True)

Calculus Preliminaries

First, to get familiar with the integrals, the strategy will be to calculate $\int\limits_0^{\infty}Ae^{-C\left(\frac{\omega\mp\omega_0}{\sigma}\right)^2}$ with $C = 4\ln(2)$.



In [2]:

    
C = symbols('C', real = True, positive = True)
I_m = Integral(A*exp(-C*((omega-omega_0)/sigma)**2), (omega, 0, oo))
I_p = Integral(A*exp(-C*((omega+omega_0)/sigma)**2), (omega, 0, oo))
I_m-I_p









    Out[2]:





$$\int_{0}^{\infty} A e^{- \frac{C}{\sigma^{2}} \left(\omega - \omega_{0}\right)^{2}}\, d\omega - \int_{0}^{\infty} A e^{- \frac{C}{\sigma^{2}} \left(\omega + \omega_{0}\right)^{2}}\, d\omega$$

Performing a change of variables $x = \frac{\omega\mp\omega_0}{\sigma}$



In [3]:

    
x = symbols('x')
I_m = I_m.transform((omega-omega_0)/sigma, x)
I_p = I_p.transform((omega+omega_0)/sigma, x)
I_m-I_p









    Out[3]:





$$\int_{- \frac{\omega_{0}}{\sigma}}^{\frac{1}{\sigma} \left(- \omega_{0} + \infty\right)} A \sigma e^{- C x^{2}}\, dx - \int_{\frac{\omega_{0}}{\sigma}}^{\infty} A \sigma e^{- C x^{2}}\, dx$$

The area of integration gets reduced to the interval $[-\frac{\omega_0}{\sigma}, \frac{\omega_0}{\sigma}]$. And exploiting the even integrand, it ends like



In [4]:

    
2*A*sigma*Integral(exp(-C*x**2), (x, 0, omega_0/sigma))









    Out[4]:





$$2 A \sigma \int_{0}^{\frac{\omega_{0}}{\sigma}} e^{- C x^{2}}\, dx$$

Performing the integral



In [5]:

    
_.doit()









    Out[5]:





$$\frac{A \sigma}{\sqrt{C}} \sqrt{\pi} \operatorname{erf}{\left (\frac{\sqrt{C} \omega_{0}}{\sigma} \right )}$$

Obtaining the Spectral Weight

Now, performing the real integral for the Gaussian spectral weight. Again splitting the integral in two pieces with $C = 4\ln(2)$.



In [6]:

    
epsilon_0 = symbols('epsilon_0', real = True, positive = True)
NumCoeff = A*epsilon_0*sigma
SW_m = Integral(NumCoeff*exp(-C*((omega-omega_0)/sigma)**2), (omega, 0, oo))
SW_p = Integral(NumCoeff*exp(-C*((omega+omega_0)/sigma)**2), (omega, 0, oo))
SW_m-SW_p









    Out[6]:





$$\int_{0}^{\infty} A \epsilon_{0} \sigma e^{- \frac{C}{\sigma^{2}} \left(\omega - \omega_{0}\right)^{2}}\, d\omega - \int_{0}^{\infty} A \epsilon_{0} \sigma e^{- \frac{C}{\sigma^{2}} \left(\omega + \omega_{0}\right)^{2}}\, d\omega$$

Performing the same change of variables as before $x = \frac{\omega\mp\omega_0}{\sigma}$



In [7]:

    
SW_m = SW_m.transform((omega-omega_0)/sigma, x)
SW_p = SW_p.transform((omega+omega_0)/sigma, x)
SW_m-SW_p









    Out[7]:





$$\int_{- \frac{\omega_{0}}{\sigma}}^{\frac{1}{\sigma} \left(- \omega_{0} + \infty\right)} A \epsilon_{0} \sigma^{2} e^{- C x^{2}}\, dx - \int_{\frac{\omega_{0}}{\sigma}}^{\infty} A \epsilon_{0} \sigma^{2} e^{- C x^{2}}\, dx$$

For the term linear in $x$ (odd), the interval of integration is again $[-\frac{\omega_0}{\sigma}, \frac{\omega_0}{\sigma}]$, which renders the integral to $0$. Reordering the rest of the integrals and keeping the leading $A\epsilon_0\sigma$ coefficient out.



In [8]:

    
SW_1 = Integral(omega_0*exp(-C*x**2), (x, -omega_0/sigma, oo))
SW_2 = Integral(omega_0*exp(-C*x**2), (x, omega_0/sigma, oo))
SW = SW_1 + SW_2
SW









    Out[8]:





$$\int_{- \frac{\omega_{0}}{\sigma}}^{\infty} \omega_{0} e^{- C x^{2}}\, dx + \int_{\frac{\omega_{0}}{\sigma}}^{\infty} \omega_{0} e^{- C x^{2}}\, dx$$

This is the same integral with overlapping intervals. Once in the range $[-\frac{\omega_0}{\sigma}, \frac{\omega_0}{\sigma}]$ and twice in the range $[\frac{\omega_0}{\sigma}, \infty]$. Absorbing $\omega_0$ to the numerical prefactor and leaving it out for clarity.



In [9]:

    
NumCoeff *= omega_0
SW_1 = Integral(exp(-C*x**2), (x, -omega_0/sigma, omega_0/sigma))
SW_2 = Integral(exp(-C*x**2), (x, omega_0/sigma, oo))
SW = SW_1 + 2*SW_2
SW









    Out[9]:





$$\int_{- \frac{\omega_{0}}{\sigma}}^{\frac{\omega_{0}}{\sigma}} e^{- C x^{2}}\, dx + 2 \int_{\frac{\omega_{0}}{\sigma}}^{\infty} e^{- C x^{2}}\, dx$$

The first integral is already known from the preparatory calculations, but we recalculate it again here. This give us a closed form for the spectral weight of a Gaussian oscillator.



In [10]:

    
SW = SW.subs(C, 4*ln(2))
result = SW.doit().simplify()
result *= NumCoeff 
result









    Out[10]:





$$\frac{\sqrt{\pi} A \epsilon_{0} \omega_{0} \sigma}{2 \sqrt{\log{\left (2 \right )}}}$$

Reference

[1] De Sousa Meneses, D., Malki, M., & Echegut, P. (2006). Structure and lattice dynamics of binary lead silicate glasses investigated by infrared spectroscopy. Journal of Non-Crystalline Solids, 352(8), 769–776. doi:10.1016/j.jnoncrysol.2006.02.004

Attribution and License

Produced by Ignacio Vergara Kausel vergara at ph2.uni-koeln.de

Available at https://github.com/ivergara/science_notebooks

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