.. _models: Models ====== .. _czjzek_model: Czjzek distribution ------------------- A Czjzek distribution model [#f1]_ is a random distribution of the second-rank traceless symmetric tensors about a zero tensor. An explicit form of a traceless symmetric second-rank tensor, :math:`{\bf S}`, in Cartesian basis, follows, .. math:: {\bf S} = \left[ \begin{array}{l l l} S_{xx} & S_{xy} & S_{xz} \\ S_{xy} & S_{yy} & S_{yz} \\ S_{xz} & S_{yz} & S_{zz} \end{array} \right], where :math:`S_{xx} + S_{yy} + S_{zz} = 0`. The elements of the above Cartesian tensor, :math:`S_{ij}`, can be decomposed into second-rank irreducible spherical tensor components [#f3]_, :math:`R_{2,k}`, following .. math:: S_{xx} &= \frac{1}{2} (R_{2,2} + R_{2,-2}) - \frac{1}{\sqrt{6}} R_{2,0}, \\ S_{xy} &= S_{yx} = -\frac{i}{2} (R_{2,2} - R_{2,-2}), \\ S_{yy} &= -\frac{1}{2} (R_{2,2} + R_{2,-2}) - \frac{1}{\sqrt{6}} R_{2,0}, \\ S_{xz} &= S_{zx} = -\frac{1}{2} (R_{2,1} - R_{2,-1}), \\ S_{zz} &= \sqrt{\frac{2}{3}} R_{20}, \\ S_{yz} &= S_{zy} = \frac{i}{2} (R_{2,1} + R_{2,-1}). In the Czjzek model, the distribution of the second-rank traceless symmetric tensor is based on the assumption of a random distribution of the five irreducible spherical tensor components, :math:`R_{2,k}`, drawn from an uncorrelated five-dimensional multivariate normal distribution. Since :math:`R_{2,k}` components are complex, random sampling is performed on the equivalent real tensor components, which are a linear combination of :math:`R_{2,k}`, and are given as .. math:: U_1 &= \frac{1}{\sqrt{6}} R_{2,0}, \\ U_2 &= -\frac{1}{\sqrt{12}} (R_{2,1} - R_{2,-1}), \\ U_3 &= \frac{i}{\sqrt{12}} (R_{2,1} + R_{2,-1}), \\ U_4 &= -\frac{i}{\sqrt{12}} (R_{2,2} - R_{2,-2}), \\ U_5 &= \frac{1}{\sqrt{12}} (R_{2,2} + R_{2,-2}), where :math:`U_i` forms an ortho-normal basis. The components, :math:`U_i`, are drawn from a five-dimensional uncorrelated multivariate normal distribution with zero mean and covariance matrix, :math:`\Lambda=\sigma^2 {\bf I}_5`, where :math:`{\bf I}_5` is a :math:`5 \times 5` identity matrix and :math:`\sigma` is the standard deviation. In terms of :math:`U_i`, the traceless second-rank symmetric Cartesian tensor elements, :math:`S_{ij}`, follows .. math:: S_{xx} &= \sqrt{3} U_5 - U_1, \\ S_{xy} &= S_{yx} = \sqrt{3} U_4, \\ S_{yy} &= -\sqrt{3} U_5 - U_1, \\ S_{xz} &= S_{zx} = \sqrt{3} U_2, \\ S_{zz} &= 2 U_1, \\ S_{yz} &= S_{zy} = \sqrt{3} U_3, and the explicit matrix form of :math:`{\bf S}` is .. math:: {\bf S} = \left[ \begin{array}{l l l} \sqrt{3} U_5 - U_1 & \sqrt{3} U_4 & \sqrt{3} U_2 \\ \sqrt{3} U_4 & -\sqrt{3} U_5 - U_1 & \sqrt{3} U_3 \\ \sqrt{3} U_2 & \sqrt{3} U_3 & 2 U_1 \end{array} \right]. In a shorthand notation, we denote a Czjzek distribution of second-rank traceless symmetric tensor as :math:`S_C(\sigma)`. .. _ext_czjzek_model: Extended Czjzek distribution ---------------------------- An Extended Czjzek distribution model [#f2]_ is a random perturbation of the second-rank traceless symmetric tensors about a non-zero tensor, which is given as .. math:: S_T = S(0) + \rho S_C(\sigma=1), where :math:`S_T` is the total tensor, :math:`S(0)` is the non-zero dominant second-rank tensor, :math:`S_C(\sigma=1)` is the Czjzek random model attributing to the random perturbation of the tensor about the dominant tensor, :math:`S(0)`, and :math:`\rho` is the size of the perturbation. In the above equation, the :math:`\sigma` parameter from the Czjzek random model, :math:`S_C`, has no meaning and is set to one. The factor, :math:`\rho`, is defined as .. math:: \rho = \frac{||S(0)|| \epsilon}{\sqrt{30}}, where :math:`\|S(0)\|` is the 2-norm of the dominant tensor, and :math:`\epsilon` is a fraction. ---- .. [#f1] Czjzek, G., Fink, J., Götz, F., Schmidt, H., Coey, J. M. D., Atomic coordination and the distribution of electric field gradients in amorphous solids Phys. Rev. B (1981) **23** 2513-30. `DOI: 10.1103/PhysRevB.23.2513 `_ .. [#f2] Caër, G.L., Bureau, B., Massiot, D., An extension of the Czjzek model for the distributions of electric field gradients in disordered solids and an application to NMR spectra of 71Ga in chalcogenide glasses. Journal of Physics: Condensed Matter, (2010), **22**. `DOI: 10.1088/0953-8984/22/6/065402 `_ .. [#f3] Grandinetti, P. J., Ash, J. T., Trease, N. M. Symmetry pathways in solid-state NMR, PNMRS 2011 **59**, *2*, 121-196. `DOI: 10.1016/j.pnmrs.2010.11.003 `_