erlab.analysis.fit.functions.general

Some general functions and utilities used in fitting.

Many functions are numba-compiled for performance.

Functions

active_shirley(x, peaks, k_steps, *[, ...])	Evaluate a Shirley-like background from the intensity of the peaks.
bcs_gap(x[, a, b, tc])	Interpolation formula for temperature dependent BCS-like gap magnitude.
do_convolve(x, func, resolution[, pad, ...])	Convolves func with gaussian of FWHM resolution in x.
do_convolve_2d(x, y, func, resolution[, pad])
dynes(x[, n0, gamma, delta])	Dynes formula for superconducting density of states.
fermi_dirac(x, center, temp)	Fermi-dirac distribution.
fermi_dirac_broad(x, center, temp, resolution)	Resolution-broadened Fermi edge.
fermi_dirac_linbkg(x, center, temp, back0, ...)	Fermi-dirac edge with linear backgrounds above and below the Fermi level.
fermi_dirac_linbkg_broad(x, center, temp, ...)	Resolution-broadened Fermi edge with linear backgrounds.
gaussian(x, center, sigma, amplitude)	Gaussian parametrized with standard deviation and amplitude.
gaussian_wh(x[, center, width, height])	Gaussian parametrized with FWHM and peak height.
lorentzian(x, center, gamma, amplitude)	Lorentzian parametrized with HWHM and amplitude.
lorentzian_wh(x[, center, width, height])	Lorentzian parametrized with FWHM and peak height.
sc_self_energy(x[, gamma1, gamma0, delta])	General phenomenological self-energy for superconductors.
sc_spectral_function(x[, amp, gamma1, ...])	Resolution-broadened superconducting spectral function.
step_broad(x[, center, sigma, amplitude])	Step function convolved with a Gaussian.
step_linbkg_broad(x, center, sigma, back0, ...)	Resolution broadened step function with linear backgrounds.
tll(x[, amp, center, alpha, temp, ...])	Resolution-broadened Tomonaga-Luttinger liquid (TLL) spectral function.
voigt(x[, center, sigma, gamma, amplitude])	Voigt profile.

erlab.analysis.fit.functions.general.active_shirley(x, peaks, k_steps, *, k_slope=0.0, lin_bkg=0.0, const_bkg=0.0)

Evaluate a Shirley-like background from the intensity of the peaks.

The model function is a Shirley-Végh-Salvi-Castle type background [Shirley, 1972, Végh, 1988, Salvi and Castle, 1998, Herrera-Gomez et al., 2014] together with a slope background [Herrera-Gomez et al., 2013] and a linear baseline. For more information about the model function, see Herrera-Gomez et al. [2014].

This function is intended for “active” fitting where the background is computed from individual peak components during the fitting process, so that the background updates as the peak parameters change.

The returned background is given by:

\[B(x) = \underbrace{c + m x}_{\text{baseline}} + \sum_i \underbrace{k_{\mathrm{step},\,i}\int_x^{x_{\mathrm{right}}} P_i(x')\,dx'}_{\text{step}} + \underbrace{k_{\mathrm{slope}}\int_x^{x_{\mathrm{right}}} \left(\int_{x'}^{x_{\mathrm{right}}} \sum_i P_i(t)\,dt\right)\,dx'}_{\text{slope}}\]

Parameters:

• x (array-like) – 1D coordinate array. Must be strictly monotonic (increasing or decreasing). Nonuniform spacing is allowed, but not tested extensively.

• peaks (list of array-like) – List of individual peak component arrays \(P_i(x)\), each 1D with the same shape as x.

• k_steps (Sequence[float]) – Per-peak scattering factors for each peak component. Must have the same length as peaks.

• k_slope (float, default: 0.0) – Scale factor for the slope-background.

• lin_bkg (float, default: 0.0) – Linear baseline slope \(m\).

• const_bkg (float, default: 0.0) – Constant baseline offset \(c\).

Returns:

background (dict) –

Dictionary with the following keys and values:

• "baseline": The linear baseline component.

• "shirley": The Shirley background component (if any k_steps are nonzero).

• "slope": The slope background component (if k_slope is nonzero).

Return type:

dict[str, ndarray[tuple[Any, …], dtype[_ScalarT]]]

Notes

• The integration endpoint, or right side, is always x[-1] (input order).

erlab.analysis.fit.functions.general.bcs_gap(x, a=1.76, b=1.74, tc=100.0)

Interpolation formula for temperature dependent BCS-like gap magnitude.

\[\Delta(T) \simeq a \cdot k_B T_c \cdot \tanh\left(b \sqrt{\frac{T_c}{T} - 1}\right)\]

Parameters:

• x (array-like) – The temperature values in kelvins at which to calculate the BCS gap.

• a (float, default: 1.76) – Proportionality constant. Default is 1.76.

• b (float, default: 1.74) – Proportionality constant. Default is 1.74.

• tc (float, default: 100.0) – The critical temperature in Kelvins. Default is 100.0.

erlab.analysis.fit.functions.general.do_convolve(x, func, resolution, pad=7, oversample=3, **kwargs)

Convolves func with gaussian of FWHM resolution in x.

Parameters:

• x (ndarray[tuple[Any, ...], dtype[float64]]) – An evenly spaced 1D array specifying where to evaluate the convolution.

• func (Callable) – Function to convolve.

• resolution (float) – FWHM of the gaussian kernel.

• pad (int, default: 7) – Multiples of the standard deviation \(\sigma\) to pad with.

• oversample (int, default: 3) – Factor by which to oversample x for convolution to reduce numerical artifacts.

• **kwargs – Additional keyword arguments to func.

erlab.analysis.fit.functions.general.do_convolve_2d(x, y, func, resolution, pad=5, **kwargs)

erlab.analysis.fit.functions.general.dynes(x, n0=1.0, gamma=0.003, delta=0.01)

Dynes formula for superconducting density of states.

The formula is given by [Dynes et al., 1978]:

\[N_0 \text{Re}\left[\frac{|x| + i \Gamma}{\sqrt{(|x| + i \Gamma)^2 - \Delta^2}}\right]\]

where \(x\) is the binding energy, \(N_0\) is the normal-state density of states at the Fermi level, \(\Gamma\) is the broadening term, and \(\Delta\) is the superconducting energy gap.

Parameters:

• x (array-like) – The input array of energy in eV.

• n0 (default: 1.0) – \(N_0\), by default 1.0.

• gamma (default: 0.003) – \(\Gamma\), by default 0.003.

• delta (default: 0.01) – The superconducting energy gap \(\Delta\), by default 0.01.

erlab.analysis.fit.functions.general.fermi_dirac(x, center, temp)

Fermi-dirac distribution.

\[f(x) = \frac{1}{1 + e^{(x-x_0)/k_B T}}\]

Parameters:

• x (ndarray[tuple[Any, ...], dtype[float64]]) – Energy values at which to calculate the Fermi edge.

• center (float) – The Fermi level.

• temp (float) – The temperature in K.

erlab.analysis.fit.functions.general.fermi_dirac_broad(x, center, temp, resolution)

Resolution-broadened Fermi edge.

The Fermi edge is calculated as:

\[\frac{1}{1 + e^{(x-x_0)/k_B T}} \otimes \text{g}(\sigma)\]

where \(\text{g}(\sigma)\) is a Gaussian kernel with standard deviation \(\sigma\). Note that the resolution is given in FWHM rather than the standard deviation.

Parameters:

• x (ndarray[tuple[Any, ...], dtype[float64]] | DataArray) – The energy values at which to calculate the Fermi edge.

• center (float | DataArray) – The Fermi level.

• temp (float | DataArray) – The temperature in K.

• resolution (float | DataArray) – The resolution of the Gaussian kernel in eV. Note that this is the FWHM of the Gaussian kernel, not the standard deviation.

erlab.analysis.fit.functions.general.fermi_dirac_linbkg(x, center, temp, back0, back1, dos0, dos1)

Fermi-dirac edge with linear backgrounds above and below the Fermi level.

\[I(x) = b_0 + b_1 x + \frac{d_0 - b_0 + (d_1 - b_1) x} {1 + e^{(x-x_0)/k_B T}}\]

Parameters:

• x (ndarray[tuple[Any, ...], dtype[float64]]) – The energy values at which to calculate the Fermi edge.

• center (float) – The Fermi level.

• temp (float) – The temperature in K.

• back0 (float) – The constant background above the Fermi level.

• back1 (float) – The slope of the background above the Fermi level.

• dos0 (float) – The constant background below the Fermi level.

• dos1 (float) – The slope of the background below the Fermi level.

Note

back0 and back1 corresponds to the linear background above and below EF (due to non-homogeneous detector efficiency or residual intensity on the phosphor screen during swept measurements), while dos0 and dos1 corresponds to the linear density of states below EF including the linear background.

erlab.analysis.fit.functions.general.fermi_dirac_linbkg_broad(x, center, temp, resolution, back0, back1, dos0, dos1)

Resolution-broadened Fermi edge with linear backgrounds.

\[I(x) = \left[ b_0 + b_1 x + \frac{d_0 - b_0 + (d_1 - b_1) x} {1 + e^{(x-x_0)/k_B T}} \right] \otimes \text{g}(\sigma)\]

erlab.analysis.fit.functions.general.gaussian(x, center, sigma, amplitude)

Gaussian parametrized with standard deviation and amplitude.

\[G(x) = \frac{A}{\sqrt{2\pi\sigma^2}} \exp\left[-\frac{(x-x_0)^2}{2\sigma^2}\right]\]

erlab.analysis.fit.functions.general.gaussian_wh(x, center=0.0, width=1.0, height=1.0)

Gaussian parametrized with FWHM and peak height.

\[G(x) = h \exp\left[-\frac{16 \log{2} (x-x_0)^2}{w^2}\right]\]

Note

\(\sigma=\frac{w}{2\sqrt{2\log{2}}}\)

erlab.analysis.fit.functions.general.lorentzian(x, center, gamma, amplitude)

Lorentzian parametrized with HWHM and amplitude.

\[L(x) = \frac{A}{\pi\gamma\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\]

erlab.analysis.fit.functions.general.lorentzian_wh(x, center=0.0, width=1.0, height=1.0)

Lorentzian parametrized with FWHM and peak height.

\[L(x) = \frac{h}{1 + 4\left(\frac{x-x_0}{w}\right)^2}\]

Note

\(\gamma=w/2\)

erlab.analysis.fit.functions.general.sc_self_energy(x, gamma1=1e-5, gamma0=0.0, delta=0.0)

General phenomenological self-energy for superconductors.

The function is given by [Norman et al., 1998]:

\[\Sigma(x) = -i \Gamma_1 + \frac{\Delta^2}{x + i \Gamma_0}\]

where \(\Gamma_1\) is the single-particle scattering rate, \(\Gamma_0\) is the pair-breaking rate, and \(\Delta\) is the energy gap.

Parameters:

• x (array-like) – The input array of energy in eV.

• gamma1 (default: 1e-5) – \(\Gamma_1\), the single-particle scattering rate.

• gamma0 (default: 0.0) – \(\Gamma_0\), the pair-breaking scattering rate.

• delta (default: 0.0) – The energy gap \(\Delta\).

erlab.analysis.fit.functions.general.sc_spectral_function(x, amp=1.0, gamma1=1e-5, gamma0=0.0, delta=0.0, lin_bkg=0.0, const_bkg=0.0, resolution=0.01)

Resolution-broadened superconducting spectral function.

The superconducting spectral function with a linear background is calculated as:

\[I(x) = \left[ A \cdot A_{sc}(x, \Sigma) + \left(m \cdot |x| + b \right) \right] \otimes \text{g}(\sigma)\]

where \(A_{sc}(x, \Sigma)\) is the superconducting spectral function calculated using the self-energy \(\Sigma(x)\) given by sc_self_energy(). \(\text{g}(\sigma)\) is a Gaussian kernel with standard deviation \(\sigma\). Note that the resolution parameter is the FWHM of the Gaussian kernel, not the standard deviation.

Parameters:

• x (array-like) – The input array of energy in eV.

• amp (float | DataArray, default: 1.0) – The overall scale factor \(A\).

• gamma1 (float | DataArray, default: 1e-5) – \(\Gamma_1\), the single-particle scattering rate.

• gamma0 (float | DataArray, default: 0.0) – \(\Gamma_0\), the pair-breaking scattering rate.

• delta (float | DataArray, default: 0.0) – The energy gap \(\Delta\).

• lin_bkg (float | DataArray, default: 0.0) – The slope of the linear background \(m\).

• const_bkg (float | DataArray, default: 0.0) – The constant background \(b\).

• resolution (float | DataArray, default: 0.01) – The broadening in eV. Note that this is the FWHM of the Gaussian kernel, not the standard deviation.

erlab.analysis.fit.functions.general.step_broad(x, center=0.0, sigma=1.0, amplitude=1.0)

Step function convolved with a Gaussian.

The broadened step function is calculated as:

\[\frac{A}{2}\cdot\text{erfc}\left(\frac{x - x_0}{\sqrt{2\sigma^2}}\right)\]

where \(\text{erfc}\) is the complementary error function.

Parameters:

• x (ndarray[tuple[Any, ...], dtype[float64]]) – The input array of x values.

• center (float, default: 0.0) – The center of the step function.

• sigma (float, default: 1.0) – The standard deviation of the Gaussian.

• amplitude (float, default: 1.0) – The amplitude of the step.

erlab.analysis.fit.functions.general.step_linbkg_broad(x, center, sigma, back0, back1, dos0, dos1)

Resolution broadened step function with linear backgrounds.

erlab.analysis.fit.functions.general.tll(x, amp=1.0, center=0.0, alpha=0.1, temp=10.0, resolution=0.01, const_bkg=0.0)

Resolution-broadened Tomonaga-Luttinger liquid (TLL) spectral function.

The TLL spectral function is calculated as [Ohtsubo et al., 2015]:

\[I(x,T) = \left[A T^\alpha \cosh\left(\frac{\epsilon}{2}\right) \left|\Gamma\left(\frac{1 + \alpha}{2} + i \frac{\epsilon}{2\pi}\right)\right|^2 f(\epsilon,T)\right] \otimes \text{g}(\sigma) + B\]

where \(\epsilon=(x - x_0)/k_B T\) is the temperature-normalized energy, \(\Gamma\) is the gamma function, \(f(\epsilon,T) = 1/(e^{\epsilon}+1)\) is the Fermi-Dirac distribution, \(\text{g}(\sigma)\) is a Gaussian kernel with standard deviation \(\sigma\), and \(B\) is a constant background.

Note that the resolution parameter is the FWHM of the Gaussian kernel, not the standard deviation.

Parameters:

• x (ndarray[tuple[Any, ...], dtype[float64]] | DataArray) – The energy values at which to calculate the TLL spectral function.

• amp (float | DataArray, default: 1.0) – The amplitude.

• center (float | DataArray, default: 0.0) – The center.

• alpha (float | DataArray, default: 0.1) – The power law exponent.

• temp (float | DataArray, default: 10.0) – The temperature in K.

• resolution (float | DataArray, default: 0.01) – The resolution of the Gaussian kernel in eV. Note that this is the FWHM of the Gaussian kernel, not the standard deviation.

• const_bkg (float | DataArray, default: 0.0) – A constant background to add to the broadened TLL function.

erlab.analysis.fit.functions.general.voigt(x, center=0.0, sigma=0.0, gamma=0.5, amplitude=1.0)

Voigt profile.

The Voigt profile can be expressed as:

\[V(x) = A \frac{\text{Re}[w(z)]}{\sigma \sqrt{2\pi}}, \quad z = \frac{x - x_0 + i \gamma}{\sigma \sqrt{2}}\]

where \(w(z)\) is the Faddeeva function. This implementation uses scipy.special.voigt_profile to compute the Voigt profile.