.. _notation:

Symbols and Notation
====================

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**Symbol**                                              **Description**
:math:`\alpha, \beta, \gamma, \ldots`                   atom types
:math:`j, k, l, \ldots`                                 atom indices
:math:`b_{\mathrm{coh},\alpha}`                         the coherent scattering length of atom-types :math:`\alpha`
:math:`b_{\mathrm{coh},\alpha}^{\dagger}`               the complex conjugate of the coherent scattering length of atom-types :math:`\alpha`
:math:`b_{\mathrm{inc},\alpha}^{2}`                     the squared incoherent scattering length of atom-types :math:`\alpha`
:math:`c_{\alpha}`                                      concentration of atom-type :math:`\alpha` where :math:`c_{\alpha} = N_{\alpha} / N`
:math:`C_{\mathbf{vv}jj}(t)`                            the velocity autocorrelation function of particle :math:`j`
:math:`C_{\mathbf{vv}jj}(\omega)`                       the Fourier transform of :math:`C_{\mathbf{vv}jj}(t)`
:math:`D`                                               diffusion constant
:math:`\mathrm{DOS}(\omega)`                            density of states
:math:`\Delta_{j}^2(t)`                                 mean squared displacement of particle :math:`j`
:math:`E`                                               energy
:math:`\mathrm{EISF}(\mathbf{q})`                       elastic incoherent structure factor
:math:`F(\mathbf{q}, t)`                                intermediate scattering function
:math:`F_{\mathrm{coh}}(\mathbf{q}, t)`                 coherent intermediate scattering function
:math:`F_{\mathrm{inc}}(\mathbf{q}, t)`                 incoherent intermediate scattering function
:math:`F_{\mathrm{inc}}^{\text{G}}(\mathbf{q}, t)`      Gaussian approximation of the incoherent intermediate scattering function
:math:`J_{\mu\nu}(\mathbf{q}, t)`                       the current correlation function between the currents in the cartesian directions :math:`\mu` and :math:`\nu`
:math:`J_{\mu\nu}(\mathbf{q}, \omega)`                  the Fourier transform of :math:`J_{\mu\nu}(\mathbf{q}, t)`
:math:`J^{\mathrm{L}}(\mathbf{q}, t)`                   the longitudinal current correlation function
:math:`J^{\mathrm{L}}(\mathbf{q}, \omega)`              the Fourier transform of :math:`J^{\mathrm{L}}(\mathbf{q}, t)`
:math:`J^{\mathrm{T}}(\mathbf{q}, t)`                   the transverse current correlation function
:math:`J^{\mathrm{T}}(\mathbf{q}, \omega)`              the Fourier transform of :math:`J^{\mathrm{T}}(\mathbf{q}, t)`
:math:`k_{\mathrm{i}}`                                  the wavenumber of the incident neutron
:math:`k_{\mathrm{f}}`                                  the wavenumber of the scattered neutron
:math:`g(\mathbf{r})`                                   pair distribution function
:math:`G(\mathbf{r}, t)`                                the van Hove function
:math:`G^{\mathrm{d}}(\mathbf{r}, t)`                   distinct part of the van Hove function
:math:`G^{\mathrm{s}}(\mathbf{r}, t)`                   self part of the van Hove function
:math:`t`                                               time
:math:`t_{\mathrm{cor}}`                                the total time of the correlation function
:math:`t_{\mathrm{tot}}`                                the total time of the MD simulation
:math:`\Delta t`                                        the MD time step
:math:`S(\mathbf{q})`                                   static structure factor
:math:`S(\mathbf{q}, \omega)`                           dynamic structure factor
:math:`S_{\mathrm{coh}}(\mathbf{q}, \omega)`            dynamic coherent structure factor
:math:`S_{\mathrm{inc}}(\mathbf{q}, \omega)`            dynamic incoherent structure factor
:math:`S_{\mathrm{inc}}^{\text{G}}(\mathbf{q}, \omega)` Gaussian approximation of the dynamic incoherent structure factor
:math:`T`                                               temperature
:math:`m_{\mathrm{n}}`                                  neutron mass
:math:`n_{\mathrm{c}}`                                  total number of time steps calculated for the correlation function
:math:`n_{\mathrm{t}}`                                  total number of time steps in the MD simulation
:math:`N`                                               total number of atoms in the unit cell, simulation or volume :math:`V`
:math:`N_\alpha`                                        number of atom of type :math:`\alpha`
:math:`\rho`                                            density
:math:`r`                                               distance
:math:`\mathbf{r}(t)`                                   the position of an atom at time :math:`t`
:math:`q`                                               wavenumber
:math:`\mathbf{q}`                                      wavevector
:math:`\hat{\mathbf{q}}`                                a unit wavevector along :math:`\mathbf{q}`
:math:`\mathbf{v}(t)`                                   the velocity of an atom at time :math:`t`
:math:`W_\alpha`                                        weighting factor for atom-type :math:`\alpha`
:math:`W_{\alpha\beta}`                                 weighting factor for atom-type pair :math:`\alpha\beta`
:math:`w_\alpha`                                        individual weight for atom-type :math:`\alpha`
:math:`\omega`                                          angular frequency
:math:`\Delta \omega`                                   the frequency step
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