erlab.analysis.fit.functions.general¶
Some general functions and utilities used in fitting.
Many functions are numba-compiled for speed.
Note
numba-compiled functions do not accept xarray objects. To broadcast over xarray
objects, use xarray.apply_ufunc().
Module Attributes
From |
Functions
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Interpolation formula for temperature dependent BCS-like gap magnitude. |
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Convolves |
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Dynes formula for superconducting density of states. |
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Fermi-dirac distribution. |
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Resolution-broadened Fermi edge. |
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Fermi-dirac edge with linear backgrounds above and below the fermi level. |
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Resolution-broadened Fermi edge with linear backgrounds above and below EF. |
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Gaussian parametrized with standard deviation and amplitude. |
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Gaussian parametrized with FWHM and peak height. |
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Lorentzian parametrized with HWHM and amplitude. |
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Lorentzian parametrized with FWHM and peak height. |
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Step function convolved with a Gaussian. |
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Resolution broadened step function with linear backgrounds. |
- erlab.analysis.fit.functions.general.TINY: float = 1e-15¶
From
lmfit.lineshapes, equal tonumpy.finfo(numpy.float64).resolution
- erlab.analysis.fit.functions.general.bcs_gap(x, a=1.76, b=1.74, tc=100.0)[source]¶
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=5, **kwargs)[source]¶
Convolves
funcwith gaussian of FWHMresolutioninx.- Parameters:
x (
ndarray[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:5) – Multiples of the standard deviation \(\sigma\) to pad with.**kwargs – Additional keyword arguments to
func.
- erlab.analysis.fit.functions.general.do_convolve_2d(x, y, func, resolution, pad=5, **kwargs)[source]¶
- erlab.analysis.fit.functions.general.dynes(x, n0=1.0, gamma=0.003, delta=0.01)[source]¶
Dynes formula for superconducting density of states.
The formula is given by [4]:
\[N_0 \text{Re}\left[\frac{|x| + i \Gamma}{\sqrt{(|x| + i \Gamma)^2 - \Delta^2}}\right]\]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)[source]¶
Fermi-dirac distribution.
\[\frac{1}{1 + e^{(x-x_0)/k_B T}}\]
- erlab.analysis.fit.functions.general.fermi_dirac_broad(x, center, temp, resolution)[source]¶
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[Any,dtype[float64]]) – The energy values at which to calculate the Fermi edge.center (
float) – The Fermi level.temp (
float) – The temperature in K.resolution (
float) – 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)[source]¶
Fermi-dirac edge with linear backgrounds above and below the fermi level.
- Parameters:
x (
ndarray[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
back0andback1corresponds to the linear background above and below EF (due to non-homogeneous detector efficiency or residual intensity on the phosphor screen during swept measurements), whiledos0anddos1corresponds 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)[source]¶
Resolution-broadened Fermi edge with linear backgrounds above and below EF.
- erlab.analysis.fit.functions.general.gaussian(x, center, sigma, amplitude)[source]¶
Gaussian parametrized with standard deviation and amplitude.
\[\frac{A}{\sqrt{2\pi\sigma^2}} \exp\left[-\frac{(x-x_0)^2}{2\sigma^2}\right]\]
- erlab.analysis.fit.functions.general.gaussian_wh(x, center, width, height)[source]¶
Gaussian parametrized with FWHM and peak height.
Note
\(\sigma=\frac{w}{2\sqrt{2\log{2}}}\)
- erlab.analysis.fit.functions.general.lorentzian(x, center, sigma, amplitude)[source]¶
Lorentzian parametrized with HWHM and amplitude.
\[\frac{A}{\pi\sigma\left[1 + \left(\frac{x-x_0}{\sigma}\right)^2\right]}\]
- erlab.analysis.fit.functions.general.lorentzian_wh(x, center, width, height)[source]¶
Lorentzian parametrized with FWHM and peak height.
Note
\(\sigma=w/2\)
- erlab.analysis.fit.functions.general.step_broad(x, center=0.0, sigma=1.0, amplitude=1.0)[source]¶
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.