| Larch: Data Analysis Tools for X-ray Spectroscopy

# 12.5. Some Builtin Line-shape Functions

Changed in version 0.9.34: The definitions of these line shapes are now taken from the lmfit package, and have a more uniform interface.

Larch provides a number of convenience functions for common line-shapes used in fitting of experimental data. This list is not exhaustive, but can be amended easily. All these functions return either a floating point scalar or array, depending on the input `x`.

gaussian(x, amplitude=1, center=0, sigma=1)

  a Gaussian or normal distribution function (see https://en.wikipedia.org/wiki/Normal_distribution), with three Parameters: `amplitude`, `center`, and `sigma`.

  \[f(x; A, \mu, \sigma) = \frac{A}{\sigma\sqrt{2\pi}} e^{[{-{(x-\mu)^2}/{{2\sigma}^2}}]}\]

  where the parameter `amplitude` corresponds to \(A\), `center` to \(\mu\), and `sigma` to \(\sigma\).

  The full width at half maximum is \(2\sigma\sqrt{2\ln{2}}\), approximately \(2.3548\sigma\). The maximum height is \(A/(\sigma\sqrt{2\pi})\).

lorentzian(x, amplitude=1, center=0, sigma=1)

  A Lorentzian or Cauchy-Lorentz distribution function (see https://en.wikipedia.org/wiki/Cauchy_distribution), with three Parameters: `amplitude`, `center`, and `sigma`.

  \[f(x; A, \mu, \sigma) = \frac{A}{\pi} \big[\frac{\sigma}{(x - \mu)^2 + \sigma^2}\big]\]

  where the parameter `amplitude` corresponds to \(A\), `center` to \(\mu\), and `sigma` to \(\sigma\). The full width at half maximum is \(2\sigma\). The maximum height is \(A/(\sigma\pi)\).

voigt(x, amplitude=1, center=0, sigma=1, gamma=None)

  A Voigt distribution function (see https://en.wikipedia.org/wiki/Voigt_profile), with four Parameters: `amplitude`, `center`, `sigma`, and `gamma` defined as:

  \[f(x; A, \mu, \sigma, \gamma) = \frac{A \textrm{Re}[w(z)]}{\sigma\sqrt{2\pi}}\]

  where

  \begin{eqnarray*} z &=& \frac{x-\mu +i\gamma}{\sigma\sqrt{2}} \\ w(z) &=& e^{-z^2}{\operatorname{erfc}}(-iz) \end{eqnarray*}

  and `erfc()` is the complimentary error function. As above, the parameter `amplitude` corresponds to \(A\), `center` to \(\mu\), and `sigma` to \(\sigma\). With the default value of `None`, `gamma` (\(\gamma\)) is constrained to have value equal to `sigma`, though it can be varied independently. For the case when \(\gamma = \sigma\), the full width at half maximum is approximately \(3.6013\sigma\).

pvoigt(x, amplitude=1, center=0, sigma=1, fraction=0.5)

  A pseudo-Voigt distribution function, which is a weighted sum of a Gaussian and Lorentzian distribution functions with the same values for `amplitude` (\(A\)) , `center` (\(\mu\)) and the same full width at half maximum (so constrained values of sigma`, \(\sigma\)). A paramater `fraction` (\(\alpha\)) controls controls the relative weight of the Gaussian and Lorentzian components, giving the full definition of

  \[f(x; A, \mu, \sigma, \alpha) = \frac{(1-\alpha)A}{\sigma_g\sqrt{2\pi}} e^{[{-{(x-\mu)^2}/{{2\sigma_g}^2}}]} + \frac{\alpha A}{\pi} \big[\frac{\sigma}{(x - \mu)^2 + \sigma^2}\big]\]

pearson7(x, amplitude=1, center=0, sigma=1, exponent=0.5)

  > A Pearson VII distribution function (see https://en.wikipedia.org/wiki/Pearson_distribution#The_Pearson_type_VII_distribution), with four parameers: `amplitude` (\(A\)), `center` (\(\mu\)), `sigma` (\(\sigma\)), and `exponent` (\(m\)) in

  > \[f(x; A, \mu, \sigma, m) = \frac{A}{\sigma{\beta(m-\frac{1}{2}, \frac{1}{2})}} \bigl[1 + \frac{(x-\mu)^2}{\sigma^2} \bigr]^{-m}\]

  where \(\beta\) is the beta function (see scipy.special.beta in `scipy.special`).

students_t(x, amplitude=1, center=0, sigma=1)

  A Student’s t distribution function (see https://en.wikipedia.org/wiki/Student%27s_t-distribution), with three Parameters: `amplitude` (\(A\)), `center` (\(\mu\)) and `sigma` (\(\sigma\)) in

  \[f(x; A, \mu, \sigma) = \frac{A \Gamma(\frac{\sigma+1}{2})} {\sqrt{\sigma\pi}\,\Gamma(\frac{\sigma}{2})} \Bigl[1+\frac{(x-\mu)^2}{\sigma}\Bigr]^{-\frac{\sigma+1}{2}}\]

  where \(\Gamma(x)\) is the gamma function.

breit_wigner(x, amplitude=1, center=0, sigma=1, q=1)

  A Breit-Wigner-Fano distribution function (see https://en.wikipedia.org/wiki/Fano_resonance>), with four Parameters: `amplitude` (\(A\)), `center` (\(\mu\)), `sigma` (\(\sigma\)), and `q` (\(q\)) in

  \[f(x; A, \mu, \sigma, q) = \frac{A (q\sigma/2 + x - \mu)^2}{(\sigma/2)^2 + (x - \mu)^2}\]

lognormal(x, cen=0, sigma=1)

  A Log-normal distribution function (see https://en.wikipedia.org/wiki/Lognormal), with three Parameters `amplitude` (\(A\)), `center` (\(\mu\)) and `sigma` (\(\sigma\)) in

  \[f(x; A, \mu, \sigma) = \frac{A e^{-(\ln(x) - \mu)/ 2\sigma^2}}{x}\]

Several builtin special functions can also be used to create lineshapes useful in fitting spectra and other x-ray data. Some of these are detailed in the Table of Useful Line shapes.

> Table of Useful Line shapes.

> function | description
> --- | ---
> gaussian(x, amplitude, center, sigma) | Gaussian, normal distribution
> lorentzian(x, amplitude, center, sigma) | Lorentzian distribution
> voigt(x, amplitude, center, sigma, gamma) | Voigt distribution
> pvoigt(x, amplitude, center, sigma, fraction) | pseudo-Voigt distribution
> pearson7(x, amplitude, center, sigma, exponent) | Pearson-7 distribution
> students_t(x, amplitude, center, sigma) | Student’s t distribution
> breit_wigner(x, amplitude, center, sigma, q) | Breit-Wigner-Fano distribution
> lognormal(x, amplitude, center, sigma) | Log-normal distribution
> arctan(x) | Arc-tangent function
> erf(x) | Error function
> erfc(x) | Complemented Error function (1-erf(x))
> gammaln(x) | log of absolute value of gamma(x)

Other standard special functions (Bessel functions, Legendre polynomials, etc) can be accessed from scipy.special:

    from scipy.special import j0 # Bessel function of order 0,
    from scipy.special import y1 # Bessel function of second kind of order 1

A host of functions to generate other distribution functions can be accessed from scipy.stats.