StretchedExponentialFTModel – Fourier transform of the stretched exponential¶

class qef.models.strexpft.StretchedExponentialFTModel(independent_vars=['x'], prefix='', missing=None, name=None, **kwargs)[source]¶

Fourier transform of the symmetrized stretched exponential

\[S(E) = A \int_{-\infty}^{\infty} dt/h e^{-i2\pi(E-E_0)t/h} e^{|\frac{x}{\tau}|^\beta}\]

Normalization and maximum at \(E=E_0\):

\[\int_{-\infty}^{\infty} dE S(E) = A max(S) = A \frac{\tau}{\beta} \Gamma(\beta^{-1})\]

Uses scipy.fftpack.fft for the Fourier transform

Fitting parameters:

If the time unit is picoseconds, then the reciprocal energy unit is mili-eV

guess(y, x=None, **kwargs)[source]¶

Guess starting values for the parameters of a model.

Parameters:	y (`ndarray`) – Intensities x (`ndarray`) – energy values kwargs (dict) – additional optional arguments, passed to model function.
Returns:	parameters with guessed values
Return type:	`Parameters`

qef.models.strexpft.strexpft(x, amplitude=1.0, center=0.0, tau=10.0, beta=1.0)[source]¶

Fourier transform of the symmetrized stretched exponential

\[S(E) = A \int_{-\infty}^{\infty} dt/h e^{-i2\pi(E-E_0)t/h} e^{|\frac{x}{\tau}|^\beta}\]

Normalization and maximum at \(E=E_0\):

\[\int_{-\infty}^{\infty} dE S(E) = A\]

\[max(S) = A \frac{\tau}{\beta} \Gamma(\beta^{-1})\]

Uses fft() for the Fourier transform

Parameters:	x (`ndarray`) – domain of the function, energy amplitude (float) – Integrated intensity of the curve center (float) – position of the peak tau (float) – relaxation time. beta (float) – stretching exponent If the time units are picoseconds, then the energy units are mili-eV.
Returns:	values – function over the domain
Return type:	`ndarray`