diff --git a/user_libs/DebyeFit/README.rst b/user_libs/DebyeFit/README.rst index 780dca2e..c7ad7da0 100644 --- a/user_libs/DebyeFit/README.rst +++ b/user_libs/DebyeFit/README.rst @@ -21,18 +21,18 @@ The generic form of dispersive media is .. math:: - \epsilon(\omega) = \epsilon^{'}(\omega) - j\epsilon^{''}(\omega), + \\epsilon(\\omega) = \\epsilon^{'}(\\omega) - j\\epsilon^{''}(\\omega), -where :math:`\omega` is the angular frequency, :math:`\epsilon^{'}` and :math:`\epsilon^{''}` are the real and imaginary parts of the permittivity respectively. +where :math:`\\omega` is the angular frequency, :math:`\\epsilon^{'}` and :math:`\\epsilon^{''}` are the real and imaginary parts of the permittivity respectively. This package provides scripts and tools which can be used to fit a multi-Debye expansion to dielectric data, defined as .. math:: - \epsilon(\omega) = \epsilon_{\infty} + \sum_{i=1}^{N}\frac{\Delta\epsilon_{i}}{1+j\omega t_{0,i}}, + \\epsilon(\\omega) = \\epsilon_{\\infty} + \\sum_{i=1}^{N}\\frac{\\Delta\\epsilon_{i}}{1+j\\omega t_{0,i}}, -where :math:`\epsilon(\omega)` is frequency dependent dielectric permittivity, :math:`\Delta\epsilon` - difference between the real permittivity at zero and infinite frequency. -:math:`\tau_{0}` is relaxation time (seconds), :math:`\epsilon_{\infty}` - real part of relative permittivity at infinite frequency, and :math:`N` is number of the Debye poles. +where :math:`\\epsilon(\\omega)` is frequency dependent dielectric permittivity, :math:`\\Delta\\epsilon` - difference between the real permittivity at zero and infinite frequency. +:math:`\\tau_{0}` is relaxation time (seconds), :math:`\\epsilon_{\\infty}` - real part of relative permittivity at infinite frequency, and :math:`N` is number of the Debye poles. To fit the data to a multi-Debye expansion, you can choose between Havriliak-Negami, Jonscher, or Complex Refractive Index Mixing (CRIM) models, as well as arbitrary dielectric data derived experimentally or calculated using a different function. @@ -65,7 +65,7 @@ The Havriliak-Negami relaxation is an empirical modification of the Debye relaxa .. math:: - \epsilon(\omega) = \epsilon_{\infty} + \frac{\Delta\epsilon}{\left(1+\left(j\omega t_{0}\right)^{a}\right)^{b}} + \\epsilon(\\omega) = \\epsilon_{\\infty} + \\frac{\\Delta\\epsilon}{\\left(1+\\left(j\\omega t_{0}\\right)^{a}\\right)^{b}} The ``HavriliakNegami`` class has the following structure: @@ -83,8 +83,8 @@ The ``HavriliakNegami`` class has the following structure: * ``f_min`` is first bound of the frequency range used to approximate the given function (Hz), * ``f_max`` is second bound of the frequency range used to approximate the given function (Hz), -* ``alpha`` is real positive float number which varies 0 < $\alpha$ < 1, -* ``beta`` is real positive float number which varies 0 < $\beta$ < 1, +* ``alpha`` is real positive float number which varies :math:`0 < \\alpha < 1`, +* ``beta`` is real positive float number which varies :math:`0 < \\beta < 1`, * ``e_inf`` is a real part of relative permittivity at infinite frequency, * ``de`` is a difference between the real permittivity at zero and infinite frequency, * ``tau_0`` is a relaxation time (seconds), @@ -106,7 +106,7 @@ Jonscher function is mainly used to describe the dielectric properties of concre .. math:: - \epsilon(\omega) = \epsilon_{\infty} + a_{p}*\left( -j*\frac{\omega}{\omega_{p}} \right)^{n} + \\epsilon(\\omega) = \\epsilon_{\\infty} + a_{p}*\\left( -j*\\frac{\\omega}{\\omega_{p}} \\right)^{n} The ``Jonscher`` class has the following structure: @@ -136,7 +136,7 @@ CRIM is the most mainstream approach for estimating the bulk permittivity of het .. math:: - \epsilon(\omega)^{d} = \sum_{i=1}^{m}f_{i}\epsilon_{m,i}(\omega)^{d} + \\epsilon(\\omega)^{d} = \\sum_{i=1}^{m}f_{i}\\epsilon_{m,i}(\\omega)^{d} The ``CRIM`` class has the following structure: