qspec.normal_chi2_convolved_f_pdf  (  f ,  f_lab ,  alpha ,  m ,  t ,  scale_e ,  e0 = 0 ,  relativistic = True  )[source]

The probability density at the frequency $f$ in the rest frame of an atom with kinetic energy $E_x$, determined through the Doppler shift of $f_\text{lab}$, and distributed according to a convolution of a normal and a $\chi^2_1$ distribution $$\begin{aligned} \rho^{\prime\prime}(f) = \left|\frac{\partial v_x}{\partial f}(f, f_\text{lab})\right| \rho^\prime(v_x(f, f_\text{lab})), \qquad\int\limits_0^\infty \rho^{\prime\prime}(f)\mathrm{d}f = 1, \end{aligned}$$ where $\rho^\prime$ is the probability density function normal_chi2_convolved_vx_pdf.

Parameters:

farray_like

The frequency quantiles $f$ (arb. units).

f_labarray_like

The laser frequency $f_\text{lab}$ in the laboratory frame ([f]).

alphaarray_like

The angle $\alpha$ between the laser and the velocity of the atom (rad).

marray_like

The mass $m$ of the atom (u).

tarray_like

The temperature $T$ of the environment (K).

scale_earray_like

The standard deviation $\sigma_{E_x}$ of the normal distribution (eV).

e0array_like

The mean energy $E_0$ of the normal distribution (eV).

relativisticbool

Kinetic energies are calculated either relativistically (True) or classically (False).

Returns:

rho_fndarray

The probability density in thermal equilibrium at the frequency f ([1/f]).