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Models

About implemented models

In principle, every model can be implemented in spectrafit by extending the module spectrafit.models by a new functions. It is important to know that the raise check have to be extend by the new function name in the solver_model and calculated_model.

__implemented_models__ = [
    "gaussian",
    "lorentzian",
    "voigt",
    "pseudovoigt",
    "exponential",
    "power",
    "linear",
    "constant",
    "erf",
    "atan",
    "log",
    "heaviside",
    "my_new_model",
]
...
for model in params:
    model = model.lower()
    if model.split("_")[0] not in __implemented_models__:
        raise KeyError(f"{model} is not supported")
    peak_kwargs[(model.split("_")[-1], model.split("_")[0])][
        model.split("_")[1]
    ] = params[model]

for key, _kwarg in peak_kwargs.items():
    if key[1] == "my_new_model":
        val += my_new_model(x, **_kwarg)
    ...


def my_new_model(
    x: np.array, amplitude: float = 1.0, center: float = 0.0, fwhmg: float = 1.0
) -> np.array:
    r"""Return a 1-dimensional `m`y_new_model` distribution."""


...
Further information about implemented own models in lmfit can be found in this example. So far, the built-in models of lmfit are not supported, yet.

Change in notation for the Full Maximum Half Widht (FWHM)

The notation for the Full Maximum Half Widht (FWHM) is adapted due to changes in the **kwargs-handling in the models; see also API and CHANGELOG. The notation becomes:

Method Old Notation New Notation
Gaussian-FWHM fwhm fwhmg
Lorentzian-FWHM fwhm fwhml
Pseudo-Voigt fwhm_g fwhmg
Pseudo-Voigt fwhm_l fwhml
Voigt fwhm fwhmv

Implemented models

Here is a list of implemented models of spectrafit:

Return a 1-dimensional Gaussian distribution.

\[ {\displaystyle g(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp ( -{\frac {1}{2}}{\frac {(x-\mu )^{2}}{\sigma ^{2}}} ) } \]

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the Gaussian distribution. Defaults to 1.0.

 1.0
center float

Center of the Gaussian distribution. Defaults to 0.0.

 0.0
fwhmg float

Full width at half maximum (FWHM) of the Gaussian distribution. Defaults to 1.0.

 1.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Gaussian distribution of x given.
Source code in spectrafit/models.py 
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@staticmethod
def gaussian(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    center: float = 0.0,
    fwhmg: float = 1.0,
) -> NDArray[np.float64]:
    r"""Return a 1-dimensional Gaussian distribution.

    $$
    {\displaystyle g(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp
    (  -{\frac {1}{2}}{\frac {(x-\mu )^{2}}{\sigma ^{2}}} ) }
    $$

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the Gaussian distribution.
             Defaults to 1.0.
        center (float, optional): Center of the Gaussian distribution.
             Defaults to 0.0.
        fwhmg (float, optional): Full width at half maximum (FWHM) of the Gaussian
            distribution. Defaults to 1.0.

    Returns:
        NDArray[np.float64]: Gaussian distribution of `x` given.
    """
    sigma = fwhmg * Constants.fwhmg2sig
    return np.array(amplitude / (Constants.sq2pi * sigma)) * np.exp(
        -((1.0 * x - center) ** 2) / (2 * sigma**2)
    )

Return a 1-dimensional Lorentzian distribution.

\[ f(x;x_{0},\gamma )={\frac {1}{\pi \gamma [ 1+ ( {\frac {x-x_{0}}{\gamma }})^{2} ] }} ={1 \over \pi \gamma } [ {\gamma ^{2} \over (x-x_{0})^{2}+\gamma ^{2}} ] \]

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the Lorentzian distribution. Defaults to 1.0.

 1.0
center float

Center of the Lorentzian distribution. Defaults to 0.0.

 0.0
fwhml float

Full width at half maximum (FWHM) of the Lorentzian distribution. Defaults to 1.0.

 1.0

Returns:

Type Description
NDArray[np.float64]

Union[NDArray[np.float64], float]: Lorentzian distribution of x given.
Source code in spectrafit/models.py 
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@staticmethod
def lorentzian(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    center: float = 0.0,
    fwhml: float = 1.0,
) -> NDArray[np.float64]:
    r"""Return a 1-dimensional Lorentzian distribution.

    $$
    f(x;x_{0},\gamma )={\frac  {1}{\pi \gamma
    [ 1+ ( {\frac  {x-x_{0}}{\gamma }})^{2} ]
    }} ={1 \over \pi \gamma } [ {\gamma ^{2} \over (x-x_{0})^{2}+\gamma ^{2}} ]
    $$

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the Lorentzian distribution.
            Defaults to 1.0.
        center (float, optional): Center of the Lorentzian distribution. Defaults to
            0.0.
        fwhml (float, optional): Full width at half maximum (FWHM) of the Lorentzian
            distribution. Defaults to 1.0.

    Returns:
        Union[NDArray[np.float64], float]: Lorentzian distribution of `x` given.
    """
    sigma = fwhml * Constants.fwhml2sig
    return np.array(amplitude / (1 + ((1.0 * x - center) / sigma) ** 2)) / (
        pi * sigma
    )

Return a 1-dimensional Pseudo-Voigt distribution.

See also:

J. Appl. Cryst. (2000). 33, 1311-1316 https://doi.org/10.1107/S0021889800010219

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the Pseudo-Voigt distribution. Defaults to 1.0.

 1.0
center float

Center of the Pseudo-Voigt distribution. Defaults to 0.0.

 0.0
fwhmg float

Full width half maximum of the Gaussian distribution in the Pseudo-Voigt distribution. Defaults to 1.0.

 1.0
fwhml float

Full width half maximum of the Lorentzian distribution in the Pseudo-Voigt distribution. Defaults to 1.0.

 1.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Pseudo-Voigt distribution of x given.
Source code in spectrafit/models.py 
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@staticmethod
def pseudovoigt(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    center: float = 0.0,
    fwhmg: float = 1.0,
    fwhml: float = 1.0,
) -> NDArray[np.float64]:
    """Return a 1-dimensional Pseudo-Voigt distribution.

    !!! note "See also:"

        J. Appl. Cryst. (2000). 33, 1311-1316
        https://doi.org/10.1107/S0021889800010219

    Args:
        x (NDArray[np.float64]):  `x`-values of the data.
        amplitude (float, optional): Amplitude of the Pseudo-Voigt distribution.
            Defaults to 1.0.
        center (float, optional): Center of the Pseudo-Voigt distribution.
            Defaults to 0.0.
        fwhmg (float, optional): Full width half maximum of the Gaussian
            distribution in the Pseudo-Voigt distribution. Defaults to 1.0.
        fwhml (float, optional): Full width half maximum of the Lorentzian
            distribution in the Pseudo-Voigt distribution. Defaults to 1.0.

    Returns:
        NDArray[np.float64]: Pseudo-Voigt distribution of `x` given.
    """
    f = np.power(
        fwhmg**5
        + 2.69269 * fwhmg**4 * fwhml
        + 2.42843 * fwhmg**3 * fwhml**2
        + 4.47163 * fwhmg**2 * fwhml**3
        + 0.07842 * fwhmg * fwhml**4
        + fwhml**5,
        0.2,
    )
    n = (
        1.36603 * (fwhml / f)
        - 0.47719 * (fwhml / f) ** 2
        + 0.11116 * (fwhml / f) ** 3
    )
    return np.array(
        n
        * DistributionModels.lorentzian(
            x=x, amplitude=amplitude, center=center, fwhml=fwhml
        )
        + (1 - n)
        * DistributionModels.gaussian(
            x=x, amplitude=amplitude, center=center, fwhmg=fwhmg
        )
    )

Return a 1-dimensional Voigt distribution.

\[ {\displaystyle V(x;\sigma ,\gamma )\equiv \int_{-\infty }^{\infty }G(x';\sigma ) L(x-x';\gamma )\,dx'} \]

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
center float

Center of the Voigt distribution. Defaults to 0.0.

 0.0
fwhmv float

Full width at half maximum (FWHM) of the Lorentzian distribution. Defaults to 1.0.

 1.0
gamma float

Scaling factor of the complex part of the Faddeeva Function. Defaults to None.

 None

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Voigt distribution of x given.
Source code in spectrafit/models.py 
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@staticmethod
def voigt(
    x: NDArray[np.float64],
    center: float = 0.0,
    fwhmv: float = 1.0,
    gamma: Optional[float] = None,
) -> NDArray[np.float64]:
    r"""Return a 1-dimensional Voigt distribution.

    $$
    {\displaystyle V(x;\sigma ,\gamma )\equiv
    \int_{-\infty }^{\infty }G(x';\sigma )
    L(x-x';\gamma )\,dx'}
    $$

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        center (float, optional): Center of the Voigt distribution. Defaults to 0.0.
        fwhmv (float, optional): Full width at half maximum (FWHM) of the Lorentzian
            distribution. Defaults to 1.0.
        gamma (float, optional): Scaling factor of the complex part of the
            [Faddeeva Function](https://en.wikipedia.org/wiki/Faddeeva_function).
            Defaults to None.

    Returns:
        NDArray[np.float64]: Voigt distribution of `x` given.
    """
    sigma = fwhmv * Constants.fwhmv2sig
    if gamma is None:
        gamma = sigma
    z = (x - center + 1j * gamma) / (sigma * Constants.sq2)
    return np.array(wofz(z).real / (sigma * Constants.sq2pi))

Return a 1-dimensional exponential decay.

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the exponential function. Defaults to 1.0.

 1.0
decay float

Decay of the exponential function. Defaults to 1.0.

 1.0
intercept float

Intercept of the exponential function. Defaults to 0.0.

 0.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Exponential decay of x given.
Source code in spectrafit/models.py 
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@staticmethod
def exponential(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    decay: float = 1.0,
    intercept: float = 0.0,
) -> NDArray[np.float64]:
    """Return a 1-dimensional exponential decay.

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the exponential function.
             Defaults to 1.0.
        decay (float, optional): Decay of the exponential function. Defaults to 1.0.
        intercept (float, optional): Intercept of the exponential function.
             Defaults to 0.0.

    Returns:
        NDArray[np.float64]: Exponential decay of `x` given.
    """
    return np.array(amplitude * np.exp(-x / decay) + intercept)

Return a 1-dimensional power function.

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the power function. Defaults to 1.0.

 1.0
exponent float

Exponent of the power function. Defaults to 1.0.

 1.0
intercept float

Intercept of the power function. Defaults to 0.0.

 0.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: power function of x given.
Source code in spectrafit/models.py 
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@staticmethod
def power(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    exponent: float = 1.0,
    intercept: float = 0.0,
) -> NDArray[np.float64]:
    """Return a 1-dimensional power function.

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the power function. Defaults to
            1.0.
        exponent (float, optional): Exponent of the power function. Defaults to 1.0.
        intercept (float, optional): Intercept of the power function. Defaults to
            0.0.

    Returns:
        NDArray[np.float64]: power function of `x` given.
    """
    return np.array(amplitude * np.power(x, exponent) + intercept)

Return a 1-dimensional linear function.

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
slope float

Slope of the linear function. Defaults to 1.0.

 1.0
intercept float

Intercept of the linear function. Defaults to 0.0.

 0.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Linear function of x given.
Source code in spectrafit/models.py 
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@staticmethod
def linear(
    x: NDArray[np.float64],
    slope: float = 1.0,
    intercept: float = 0.0,
) -> NDArray[np.float64]:
    """Return a 1-dimensional linear function.

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        slope (float, optional): Slope of the linear function. Defaults to 1.0.
        intercept (float, optional): Intercept of the linear function.
             Defaults to 0.0.

    Returns:
        NDArray[np.float64]: Linear function of `x` given.
    """
    return np.array(slope * x + intercept)

Return a 1-dimensional constant value.

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the constant. Defaults to 1.0.

 1.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Constant value of x given.
Source code in spectrafit/models.py 
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@staticmethod
def constant(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
) -> NDArray[np.float64]:
    """Return a 1-dimensional constant value.

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the constant. Defaults to 1.0.

    Returns:
        NDArray[np.float64]: Constant value of `x` given.
    """
    return np.array(np.linspace(amplitude, amplitude, len(x)))

Return a 1-dimensional error function.

\[ f(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt \]

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the error function. Defaults to 1.0.

 1.0
center float

Center of the error function. Defaults to 0.0.

 0.0
sigma float

Sigma of the error function. Defaults to 1.0.

 1.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Error function of x given.
Source code in spectrafit/models.py 
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@staticmethod
def erf(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    center: float = 0.0,
    sigma: float = 1.0,
) -> NDArray[np.float64]:
    r"""Return a 1-dimensional error function.

    $$
    f(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
    $$

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the error function.
                Defaults to 1.0.
        center (float, optional): Center of the error function. Defaults to 0.0.
        sigma (float, optional): Sigma of the error function. Defaults to 1.0.

    Returns:
        NDArray[np.float64]: Error function of `x` given.
    """
    return np.array(
        amplitude * 0.5 * (1 + erf(DistributionModels._norm(x, center, sigma)))
    )

Return a 1-dimensional Heaviside step function.

\[ f(x) = \begin{cases} 0 & x < 0 \\ 0.5 & x = 0 \\ 1 & x > 0 \end{cases} \]

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the Heaviside step function. Defaults to 1.0.

 1.0
center float

Center of the Heaviside step function. Defaults to 0.0.

 0.0
sigma float

Sigma of the Heaviside step function. Defaults to 1.0.

 1.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Heaviside step function of x given.
Source code in spectrafit/models.py 
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@staticmethod
def heaviside(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    center: float = 0.0,
    sigma: float = 1.0,
) -> NDArray[np.float64]:
    r"""Return a 1-dimensional Heaviside step function.

    $$
    f(x) = \begin{cases}
    0 & x < 0 \\
    0.5 & x = 0 \\
    1 & x > 0
    \end{cases}
    $$
    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the Heaviside step function.
                Defaults to 1.0.
        center (float, optional): Center of the Heaviside step function.
             Defaults to 0.0.
        sigma (float, optional): Sigma of the Heaviside step function.
             Defaults to 1.0.


    Returns:
        NDArray[np.float64]: Heaviside step function of `x` given.
    """
    return np.array(
        amplitude * 0.5 * (1 + np.sign(DistributionModels._norm(x, center, sigma)))
    )

Return a 1-dimensional arctan step function.

\[ f(x) = \frac{1}{\pi} \arctan(\frac{x - c}{s}) \]

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the arctan step function. Defaults to 1.0.

 1.0
center float

Center of the arctan step function. Defaults to 0.0.

 0.0
sigma float

Sigma of the arctan step function. Defaults to 1.0.

 1.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Arctan step function of x given.
Source code in spectrafit/models.py 
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@staticmethod
def atan(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    center: float = 0.0,
    sigma: float = 1.0,
) -> NDArray[np.float64]:
    r"""Return a 1-dimensional arctan step function.

    $$
    f(x) = \frac{1}{\pi} \arctan(\frac{x - c}{s})
    $$

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the arctan step function.
                Defaults to 1.0.
        center (float, optional): Center of the arctan step function.
             Defaults to 0.0.
        sigma (float, optional): Sigma of the arctan step function.
             Defaults to 1.0.

    Returns:
        NDArray[np.float64]: Arctan step function of `x` given.
    """
    return np.array(
        amplitude
        * 0.5
        * (1 + np.arctan(DistributionModels._norm(x, center, sigma)) / pi)
    )

Return a 1-dimensional logarithmic step function.

\[ f(x) = \frac{1}{1 + e^{-\frac{x - c}{s}}} \]

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the logarithmic step function. Defaults to 1.0.

 1.0
center float

Center of the logarithmic step function. Defaults to 0.0.

 0.0
sigma float

Sigma of the logarithmic step function. Defaults to 1.0.

 1.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Logarithmic step function of x given.
Source code in spectrafit/models.py 
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@staticmethod
def log(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    center: float = 0.0,
    sigma: float = 1.0,
) -> NDArray[np.float64]:
    r"""Return a 1-dimensional logarithmic step function.

    $$
    f(x) = \frac{1}{1 + e^{-\frac{x - c}{s}}}
    $$

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the logarithmic step function.
                Defaults to 1.0.
        center (float, optional): Center of the logarithmic step function.
             Defaults to 0.0.
        sigma (float, optional): Sigma of the logarithmic step function.
             Defaults to 1.0.

    Returns:
        NDArray[np.float64]: Logarithmic step function of `x` given.
    """
    return np.array(
        amplitude
        * 0.5
        * (1 + np.log(DistributionModels._norm(x, center, sigma)) / pi)
    )

Return a 1-dimensional cumulative Gaussian function.

\[ f(x) = \frac{1}{2} \left[1 + erf\left(\frac{x - c}{s \sqrt{2}}\right)\right] \]

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the Gaussian function. Defaults to 1.0.

 1.0
center float

Center of the Gaussian function. Defaults to 0.0.

 0.0
fwhmg float

Full width at half maximum of the Gaussian function. Defaults to 1.0.

 1.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Cumulative Gaussian function of x given.
Source code in spectrafit/models.py 
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@staticmethod
def cgaussian(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    center: float = 0.0,
    fwhmg: float = 1.0,
) -> NDArray[np.float64]:
    r"""Return a 1-dimensional cumulative Gaussian function.

    $$
    f(x) = \frac{1}{2} \left[1 + erf\left(\frac{x - c}{s \sqrt{2}}\right)\right]
    $$

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the Gaussian function. Defaults to
            1.0.
        center (float, optional): Center of the Gaussian function. Defaults to 0.0.
        fwhmg (float, optional): Full width at half maximum of the Gaussian
             function. Defaults to 1.0.


    Returns:
        NDArray[np.float64]: Cumulative Gaussian function of `x` given.
    """
    sigma = fwhmg * Constants.fwhmg2sig
    return np.array(
        amplitude * 0.5 * (1 + erf((x - center) / (sigma * np.sqrt(2.0))))
    )

Return a 1-dimensional cumulative Lorentzian function.

\[ f(x) = \frac{1}{\pi} \arctan\left(\frac{x - c}{s}\right) + \frac{1}{2} \]

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the Lorentzian function. Defaults to 1.0.

 1.0
center float

Center of the Lorentzian function. Defaults to 0.0.

 0.0
fwhml float

Full width at half maximum of the Lorentzian function. Defaults to 1.0.

 1.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Cumulative Lorentzian function of x given.
Source code in spectrafit/models.py 
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@staticmethod
def clorentzian(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    center: float = 0.0,
    fwhml: float = 1.0,
) -> NDArray[np.float64]:
    r"""Return a 1-dimensional cumulative Lorentzian function.

    $$
    f(x) = \frac{1}{\pi} \arctan\left(\frac{x - c}{s}\right) + \frac{1}{2}
    $$

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the Lorentzian function.
                Defaults to 1.0.
        center (float, optional): Center of the Lorentzian function.
             Defaults to 0.0.
        fwhml (float, optional): Full width at half maximum of the Lorentzian
            function. Defaults to 1.0.

    Returns:
        NDArray[np.float64]: Cumulative Lorentzian function of `x` given.
    """
    sigma = fwhml * Constants.fwhml2sig
    return np.array(amplitude * (np.arctan((x - center) / sigma) / pi) + 0.5)

Return a 1-dimensional cumulative Voigt function.

\[ f(x) = \frac{1}{2} \left[1 + erf\left(\frac{x - c}{s \sqrt{2}}\right)\right] \]

Parameters:

Name Type Description Default
x NDArray[np.float64]

x-values of the data.

 required
amplitude float

Amplitude of the Voigt function. Defaults to 1.0.

 1.0
center float

Center of the Voigt function. Defaults to 0.0.

 0.0
fwhmv float

Full width at half maximum of the Voigt function. Defaults to 1.0.

 1.0
gamma float

Gamma of the Voigt function. Defaults to 1.0.

 1.0

Returns:

Type Description
NDArray[np.float64]

NDArray[np.float64]: Cumulative Voigt function of x given.
Source code in spectrafit/models.py 
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@staticmethod
def cvoigt(
    x: NDArray[np.float64],
    amplitude: float = 1.0,
    center: float = 0.0,
    fwhmv: float = 1.0,
    gamma: float = 1.0,
) -> NDArray[np.float64]:
    r"""Return a 1-dimensional cumulative Voigt function.

    $$
    f(x) = \frac{1}{2} \left[1 + erf\left(\frac{x - c}{s \sqrt{2}}\right)\right]
    $$

    Args:
        x (NDArray[np.float64]): `x`-values of the data.
        amplitude (float, optional): Amplitude of the Voigt function. Defaults to
            1.0.
        center (float, optional): Center of the Voigt function. Defaults to 0.0.
        fwhmv (float, optional): Full width at half maximum of the Voigt function.
            Defaults to 1.0.
        gamma (float, optional): Gamma of the Voigt function. Defaults to 1.0.

    Returns:
        NDArray[np.float64]: Cumulative Voigt function of `x` given.
    """
    sigma = fwhmv * Constants.fwhmv2sig
    return np.array(
        amplitude
        * 0.5
        * (1 + erf((x - center) / (sigma * np.sqrt(2.0))))
        * np.exp(-(((x - center) / gamma) ** 2))
    )

Important constants for the models

For calculating the models a few math constants are needed, which are implemented in the constants module.

Mathematical constants for the curve models.

Constants

  1. Natural logarithm of 2

    \[ ln2 = \log{2} \]

  2. Square root of 2 times pi

    \[ sq2pi = \sqrt{2 \pi} \]

  3. Square root of pi

    \[ sqpi = \sqrt{ \pi} \]

  4. Square root of 2

    \[ sq2 = \sqrt{2} \]

  5. Full width at half maximum to sigma for Gaussian

    \[ fwhmg2sig = \frac{1}{ 2 \sqrt{2\log{2}}} \]

  6. Full width at half maximum to sigma for Lorentzian

    \[ fwhml2sig = \frac{1}{2} \]

  7. Full width at half maximum to sigma for Voigt according to the article by Olivero and Longbothum1, check also XPSLibary website.

    $$ fwhm_{\text{Voigt}} \approx 0.5346 \cdot fwhm_{\text{Gaussian}} + \sqrt{ 0.2166 fwhm_{\text{Lorentzian}}^2 + fwhm_{\text{Gaussian}}^2 }

    $$

    In case of equal FWHM for Gaussian and Lorentzian, the Voigt FWHM can be defined as:

    \[ fwhm_{\text{Voigt}} \approx 1.0692 + 2 \sqrt{0.2166 + 2 \ln{2}} \cdot \sigma \]
    \[ fwhmv2sig = \frac{1}{fwhm_{\text{Voigt}}} \]

  1. J.J. Olivero, R.L. Longbothum, Empirical fits to the Voigt line width: A brief review, Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 17, Issue 2, 1977, Pages 233-236, ISSN 0022-4073, https://doi.org/10.1016/0022-4073(77)90161-3. ↩

Source code in spectrafit/models.py 
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@dataclass(frozen=True)
class Constants:
    r"""Mathematical constants for the curve models.

    !!! info "Constants"

        1. Natural logarithm of 2

            $$
            ln2 = \log{2}
            $$

        2. Square root of 2 times pi

            $$
            sq2pi = \sqrt{2 \pi}
            $$

        3. Square root of pi

            $$
            sqpi = \sqrt{ \pi}
            $$

        4. Square root of 2

            $$
            sq2 = \sqrt{2}
            $$

        5. Full width at half maximum to sigma for Gaussian

            $$
            fwhmg2sig = \frac{1}{ 2 \sqrt{2\log{2}}}
            $$

        6. Full width at half maximum to sigma for Lorentzian

            $$
            fwhml2sig = \frac{1}{2}
            $$

        7. Full width at half maximum to sigma for Voigt according to the article by
            Olivero and Longbothum[^1], check also
            [XPSLibary website](https://xpslibrary.com/voigt-peak-shape/).

            $$
            fwhm_{\text{Voigt}} \approx 0.5346 \cdot fwhm_{\text{Gaussian}} +
              \sqrt{ 0.2166 fwhm_{\text{Lorentzian}}^2  + fwhm_{\text{Gaussian}}^2 }

            $$

            In case of equal FWHM for Gaussian and Lorentzian, the Voigt FWHM can be
            defined as:

            $$
            fwhm_{\text{Voigt}} \approx 1.0692 + 2 \sqrt{0.2166 + 2 \ln{2}} \cdot \sigma
            $$

            $$
            fwhmv2sig = \frac{1}{fwhm_{\text{Voigt}}}
            $$

        [^1]:
            J.J. Olivero, R.L. Longbothum,
            _Empirical fits to the Voigt line width: A brief review_,
            **Journal of Quantitative Spectroscopy and Radiative Transfer**,
            Volume 17, Issue 2, 1977, Pages 233-236, ISSN 0022-4073,
            https://doi.org/10.1016/0022-4073(77)90161-3.
    """

    ln2 = log(2.0)
    sq2pi = sqrt(2.0 * pi)
    sqpi = sqrt(pi)
    sq2 = sqrt(2.0)
    fwhmg2sig = 1 / (2.0 * sqrt(2.0 * log(2.0)))
    fwhml2sig = 1 / 2.0
    fwhmv2sig = 1 / (2 * 0.5346 + 2 * sqrt(0.2166 + log(2) * 2))