Dielectric function tools ------------------------- The :py:class:~._signals.dielectric_function.DielectricFunction class inherits from :py:class:~._signals.complex_signal.ComplexSignal and can thus access complex properties. To convert a :py:class:~._signals.complex_signal.ComplexSignal to a :py:class:~._signals.dielectric_function.DielectricFunction, make sure that the signal dimension and signal type are properly set: .. code-block:: python >>> s.set_signal_type('DielectricFunction') Note that :py:class:~._signals.dielectric_function.DielectricFunction is complex and therefore is a subclass of :py:class:~._signals.complex_signal1d.ComplexSignal1D. Number of effective electrons ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The Bethe f-sum rule gives rise to two definitions of the effective number (see :ref:[Egerton2011] ): .. math:: n_{\mathrm{eff1}}\left(-\Im\left(\epsilon^{-1}\right)\right)=\frac{2\epsilon_{0}m_{0}}{\pi\hbar^{2}e^{2}n_{a}}\int_{0}^{E}E'\Im\left(\frac{-1}{\epsilon}\right)dE' n_{\mathrm{eff2}}\left(\epsilon_{2}\right)=\frac{2\epsilon_{0}m_{0}}{\pi\hbar^{2}e^{2}n_{a}}\int_{0}^{E}E'\epsilon_{2}\left(E'\right)dE' where :math:n_a is the number of atoms (or molecules) per unit volume of the sample, :math:\epsilon_0 is the vacuum permittivity, :math:m_0 is the electron mass and :math:e is the electron charge. The :py:meth:~._signals.dielectric_function.DielectricFunction.get_number_of_effective_electrons method computes both. Compute the electron energy-loss signal ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The :py:meth:~._signals.dielectric_function.DielectricFunction.get_electron_energy_loss_spectrum "naively" computes the single-scattering electron-energy loss spectrum from the dielectric function given the zero-loss peak (or its integral) and the sample thickness using: .. math:: S\left(E\right)=\frac{2I_{0}t}{\pi a_{0}m_{0}v^{2}}\ln\left[1+\left(\frac{\beta}{\theta(E)}\right)^{2}\right]\Im\left[\frac{-1}{\epsilon\left(E\right)}\right] where :math:I_0 is the zero-loss peak integral, :math:t the sample thickness, :math:\beta the collection semi-angle and :math:\theta(E) the characteristic scattering angle.