diff --git a/doc/src/Eqs/pair_srp1.jpg b/doc/src/Eqs/pair_srp1.jpg deleted file mode 100644 index bbbdc43e05..0000000000 Binary files a/doc/src/Eqs/pair_srp1.jpg and /dev/null differ diff --git a/doc/src/Eqs/pair_srp1.tex b/doc/src/Eqs/pair_srp1.tex deleted file mode 100644 index 45aa005d86..0000000000 --- a/doc/src/Eqs/pair_srp1.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -\begin{eqnarray*} - F^{SRP}_{ij} & = & C(1-r/r_c)\hat{r}_{ij} \qquad r < r_c \\ -\end{eqnarray*} - -\end{document} diff --git a/doc/src/Eqs/pair_srp2.jpg b/doc/src/Eqs/pair_srp2.jpg deleted file mode 100644 index c5d20dc212..0000000000 Binary files a/doc/src/Eqs/pair_srp2.jpg and /dev/null differ diff --git a/doc/src/Eqs/pair_srp2.tex b/doc/src/Eqs/pair_srp2.tex deleted file mode 100644 index 19cb19ae72..0000000000 --- a/doc/src/Eqs/pair_srp2.tex +++ /dev/null @@ -1,10 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -\begin{eqnarray*} - F_{i1}^{SRP} & = & F^{SRP}_{ij}(L) \\ - F_{i2}^{SRP} & = & F^{SRP}_{ij}(1-L) -\end{eqnarray*} - -\end{document} diff --git a/doc/src/Eqs/pair_sw.jpg b/doc/src/Eqs/pair_sw.jpg deleted file mode 100644 index f60f07fd27..0000000000 Binary files a/doc/src/Eqs/pair_sw.jpg and /dev/null differ diff --git a/doc/src/Eqs/pair_sw.tex b/doc/src/Eqs/pair_sw.tex deleted file mode 100644 index ebcc39d77b..0000000000 --- a/doc/src/Eqs/pair_sw.tex +++ /dev/null @@ -1,18 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -\begin{eqnarray*} - E & = & \sum_i \sum_{j > i} \phi_2 (r_{ij}) + - \sum_i \sum_{j \neq i} \sum_{k > j} - \phi_3 (r_{ij}, r_{ik}, \theta_{ijk}) \\ - \phi_2(r_{ij}) & = & A_{ij} \epsilon_{ij} \left[ B_{ij} (\frac{\sigma_{ij}}{r_{ij}})^{p_{ij}} - - (\frac{\sigma_{ij}}{r_{ij}})^{q_{ij}} \right] - \exp \left( \frac{\sigma_{ij}}{r_{ij} - a_{ij} \sigma_{ij}} \right) \\ - \phi_3(r_{ij},r_{ik},\theta_{ijk}) & = & \lambda_{ijk} \epsilon_{ijk} \left[ \cos \theta_{ijk} - - \cos \theta_{0ijk} \right]^2 - \exp \left( \frac{\gamma_{ij} \sigma_{ij}}{r_{ij} - a_{ij} \sigma_{ij}} \right) - \exp \left( \frac{\gamma_{ik} \sigma_{ik}}{r_{ik} - a_{ik} \sigma_{ik}} \right) -\end{eqnarray*} - -\end{document} diff --git a/doc/src/Eqs/pair_tersoff.jpg b/doc/src/Eqs/pair_tersoff.jpg deleted file mode 100644 index bd3e147192..0000000000 Binary files a/doc/src/Eqs/pair_tersoff.jpg and /dev/null differ diff --git a/doc/src/Eqs/pair_tersoff_1.jpg b/doc/src/Eqs/pair_tersoff_1.jpg deleted file mode 100644 index 79600b499c..0000000000 Binary files a/doc/src/Eqs/pair_tersoff_1.jpg and /dev/null differ diff --git a/doc/src/Eqs/pair_tersoff_1.tex b/doc/src/Eqs/pair_tersoff_1.tex deleted file mode 100644 index 7d34712491..0000000000 --- a/doc/src/Eqs/pair_tersoff_1.tex +++ /dev/null @@ -1,24 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -\begin{eqnarray*} - E & = & \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\ - V_{ij} & = & f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\ - f_C(r) & = & \left\{ \begin{array} {r@{\quad:\quad}l} - 1 & r < R - D \\ - \frac{1}{2} - \frac{1}{2} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) & - R-D < r < R + D \\ - 0 & r > R + D - \end{array} \right. \\ - f_R(r) & = & A \exp (-\lambda_1 r) \\ - f_A(r) & = & -B \exp (-\lambda_2 r) \\ - b_{ij} & = & \left( 1 + \beta^n {\zeta_{ij}}^n \right)^{-\frac{1}{2n}} \\ - \zeta_{ij} & = & \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk}) - \exp \left[ {\lambda_3}^m (r_{ij} - r_{ik})^m \right] \\ - g(\theta) & = & \gamma_{ijk} \left( 1 + \frac{c^2}{d^2} - - \frac{c^2}{\left[ d^2 + - (\cos \theta - \cos \theta_0)^2\right]} \right) -\end{eqnarray*} - -\end{document} diff --git a/doc/src/Eqs/pair_tersoff_2.jpg b/doc/src/Eqs/pair_tersoff_2.jpg deleted file mode 100644 index 6cb8778a09..0000000000 Binary files a/doc/src/Eqs/pair_tersoff_2.jpg and /dev/null differ diff --git a/doc/src/Eqs/pair_tersoff_2.tex b/doc/src/Eqs/pair_tersoff_2.tex deleted file mode 100644 index 7b5beb8ef3..0000000000 --- a/doc/src/Eqs/pair_tersoff_2.tex +++ /dev/null @@ -1,14 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -\begin{eqnarray*} -\lambda_1^{i,j} &=& \frac{1}{2}(\lambda_1^i + \lambda_1^j)\\ -\lambda_2^{i,j} &=& \frac{1}{2}(\lambda_2^i + \lambda_2^j)\\ -A_{i,j} &=& (A_{i}A_{j})^{1/2}\\ -B_{i,j} &=& \chi_{ij}(B_{i}B_{j})^{1/2}\\ -R_{i,j} &=& (R_{i}R_{j})^{1/2}\\ -S_{i,j} &=& (S_{i}S_{j})^{1/2}\\ -\end{eqnarray*} - -\end{document} \ No newline at end of file diff --git a/doc/src/Eqs/pair_tersoff_mod.jpg b/doc/src/Eqs/pair_tersoff_mod.jpg deleted file mode 100644 index 2618943d85..0000000000 Binary files a/doc/src/Eqs/pair_tersoff_mod.jpg and /dev/null differ diff --git a/doc/src/Eqs/pair_tersoff_mod.tex b/doc/src/Eqs/pair_tersoff_mod.tex deleted file mode 100644 index eafc4fdeeb..0000000000 --- a/doc/src/Eqs/pair_tersoff_mod.tex +++ /dev/null @@ -1,24 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -\begin{eqnarray*} - E & = & \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\ - V_{ij} & = & f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\ - f_C(r) & = & \left\{ \begin{array} {r@{\quad:\quad}l} - 1 & r < R - D \\ - \frac{1}{2} - \frac{9}{16} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) - \frac{1}{16} \sin \left( \frac{3\pi}{2} \frac{r-R}{D} \right) & - R-D < r < R + D \\ - 0 & r > R + D - \end{array} \right. \\ - f_R(r) & = & A \exp (-\lambda_1 r) \\ - f_A(r) & = & -B \exp (-\lambda_2 r) \\ - b_{ij} & = & \left( 1 + {\zeta_{ij}}^\eta \right)^{-\frac{1}{2n}} \\ - \zeta_{ij} & = & \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk}) - \exp \left[ \alpha (r_{ij} - r_{ik})^\beta \right] \\ - g(\theta) & = & c_1 + g_o(\theta) g_a(\theta) \\ - g_o(\theta) & = & \frac{c_2 (h - \cos \theta)^2}{c_3 + (h - \cos \theta)^2} \\ - g_a(\theta) & = & 1 + c_4 \exp \left[ -c_5 (h - \cos \theta)^2 \right] \\ -\end{eqnarray*} - -\end{document} diff --git a/doc/src/Eqs/pair_tersoff_mod_c.jpg b/doc/src/Eqs/pair_tersoff_mod_c.jpg deleted file mode 100644 index 311ccc81eb..0000000000 Binary files a/doc/src/Eqs/pair_tersoff_mod_c.jpg and /dev/null differ diff --git a/doc/src/Eqs/pair_tersoff_mod_c.tex b/doc/src/Eqs/pair_tersoff_mod_c.tex deleted file mode 100644 index 8cea2d382c..0000000000 --- a/doc/src/Eqs/pair_tersoff_mod_c.tex +++ /dev/null @@ -1,10 +0,0 @@ -\documentclass[12pt]{article} -\pagestyle{empty} - -\begin{document} - -\begin{eqnarray*} - V_{ij} & = & f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) + c_0 \right] -\end{eqnarray*} - -\end{document} diff --git a/doc/src/Eqs/pair_tersoff_zbl.jpg b/doc/src/Eqs/pair_tersoff_zbl.jpg deleted file mode 100644 index 20d60d2256..0000000000 Binary files a/doc/src/Eqs/pair_tersoff_zbl.jpg and /dev/null differ diff --git a/doc/src/Eqs/pair_tersoff_zbl.tex b/doc/src/Eqs/pair_tersoff_zbl.tex deleted file mode 100644 index 902819aa1b..0000000000 --- a/doc/src/Eqs/pair_tersoff_zbl.tex +++ /dev/null @@ -1,33 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -\begin{eqnarray*} - E & = & \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\ - V_{ij} & = & (1 - f_F(r_{ij})) V^{ZBL}_{ij} + f_F(r_{ij}) V^{Tersoff}_{ij} \\ -f_F(r_{ij}) & = & \frac{1}{1 + e^{-A_F(r_{ij} - r_C)}}\\ - \\ - \\ - V^{ZBL}_{ij} & = & \frac{1}{4\pi\epsilon_0} \frac{Z_1 Z_2 \,e^2}{r_{ij}} \phi(r_{ij}/a) \\ - a & = & \frac{0.8854\,a_0}{Z_{1}^{0.23} + Z_{2}^{0.23}}\\ - \phi(x) & = & 0.1818e^{-3.2x} + 0.5099e^{-0.9423x} + 0.2802e^{-0.4029x} + 0.02817e^{-0.2016x}\\ - \\ - \\ - V^{Tersoff}_{ij} & = & f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\ - f_C(r) & = & \left\{ \begin{array} {r@{\quad:\quad}l} - 1 & r < R - D \\ - \frac{1}{2} - \frac{1}{2} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) & - R-D < r < R + D \\ - 0 & r > R + D - \end{array} \right. \\ - f_R(r) & = & A \exp (-\lambda_1 r) \\ - f_A(r) & = & -B \exp (-\lambda_2 r) \\ - b_{ij} & = & \left( 1 + \beta^n {\zeta_{ij}}^n \right)^{-\frac{1}{2n}} \\ - \zeta_{ij} & = & \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk}) - \exp \left[ {\lambda_3}^m (r_{ij} - r_{ik})^m \right] \\ - g(\theta) & = & \gamma_{ijk} \left( 1 + \frac{c^2}{d^2} - - \frac{c^2}{\left[ d^2 + - (\cos \theta - \cos \theta_0)^2\right]} \right) -\end{eqnarray*} - -\end{document} diff --git a/doc/src/pair_srp.rst b/doc/src/pair_srp.rst index a1138db382..822bd0dddb 100644 --- a/doc/src/pair_srp.rst +++ b/doc/src/pair_srp.rst @@ -56,19 +56,25 @@ Bonds of specified type *btype* interact with one another through a bond-pairwise potential, such that the force on bond *i* due to bond *j* is as follows -.. image:: Eqs/pair_srp1.jpg - :align: center +.. math:: -where *r* and *rij* are the distance and unit vector between the two -bonds. Note that *btype* can be specified as an asterisk "\*", which -case the interaction is applied to all bond types. The *mid* option -computes *r* and *rij* from the midpoint distance between bonds. The -*min* option computes *r* and *rij* from the minimum distance between -bonds. The force acting on a bond is mapped onto the two bond atoms -according to the lever rule, + F^{SRP}_{ij} & = C(1-r/r_c)\hat{r}_{ij} \qquad r < r_c + + +where *r* and :math:`\hat{r}_{ij}` are the distance and unit vector +between the two bonds. Note that *btype* can be specified as an +asterisk "\*", which case the interaction is applied to all bond types. +The *mid* option computes *r* and :math:`\hat{r}_{ij}` from the midpoint +distance between bonds. The *min* option computes *r* and +:math:`\hat{r}_{ij}` from the minimum distance between bonds. The force +acting on a bond is mapped onto the two bond atoms according to the +lever rule, + +.. math:: + + F_{i1}^{SRP} & = F^{SRP}_{ij}(L) \\ + F_{i2}^{SRP} & = F^{SRP}_{ij}(1-L) -.. image:: Eqs/pair_srp2.jpg - :align: center where *L* is the normalized distance from the atom to the point of closest approach of bond *i* and *j*\ . The *mid* option takes *L* as @@ -80,7 +86,7 @@ the data file or restart file read by the :doc:`read_data ` or :doc:`read_restart ` commands: * *C* (force units) -* *rc* (distance units) +* :math:`r_c` (distance units) The last coefficient is optional. If not specified, the global cutoff is used. @@ -114,7 +120,7 @@ Pair style *srp* turns off normalization of thermodynamic properties by particle number, as if the command :doc:`thermo_modify norm no ` had been issued. The pairwise energy associated with style *srp* is shifted to be zero -at the cutoff distance *rc*\ . +at the cutoff distance :math:`r_c`. ---------- @@ -127,7 +133,7 @@ This pair styles does not support mixing. This pair style does not support the :doc:`pair_modify ` shift option for the energy of the pair interaction. Note that as discussed above, the energy term is already shifted to be 0.0 at the -cutoff distance *rc*\ . +cutoff distance :math:`r_c`. The :doc:`pair_modify ` table option is not relevant for this pair style. diff --git a/doc/src/pair_sw.rst b/doc/src/pair_sw.rst index 425dcf1f16..2df9958ba1 100644 --- a/doc/src/pair_sw.rst +++ b/doc/src/pair_sw.rst @@ -39,12 +39,23 @@ Description The *sw* style computes a 3-body :ref:`Stillinger-Weber ` potential for the energy E of a system of atoms as -.. image:: Eqs/pair_sw.jpg - :align: center +.. math:: -where phi2 is a two-body term and phi3 is a three-body term. The -summations in the formula are over all neighbors J and K of atom I -within a cutoff distance = a\*sigma. + E & = \sum_i \sum_{j > i} \phi_2 (r_{ij}) + + \sum_i \sum_{j \neq i} \sum_{k > j} + \phi_3 (r_{ij}, r_{ik}, \theta_{ijk}) \\ + \phi_2(r_{ij}) & = A_{ij} \epsilon_{ij} \left[ B_{ij} (\frac{\sigma_{ij}}{r_{ij}})^{p_{ij}} - + (\frac{\sigma_{ij}}{r_{ij}})^{q_{ij}} \right] + \exp \left( \frac{\sigma_{ij}}{r_{ij} - a_{ij} \sigma_{ij}} \right) \\ + \phi_3(r_{ij},r_{ik},\theta_{ijk}) & = \lambda_{ijk} \epsilon_{ijk} \left[ \cos \theta_{ijk} - + \cos \theta_{0ijk} \right]^2 + \exp \left( \frac{\gamma_{ij} \sigma_{ij}}{r_{ij} - a_{ij} \sigma_{ij}} \right) + \exp \left( \frac{\gamma_{ik} \sigma_{ik}}{r_{ik} - a_{ik} \sigma_{ik}} \right) + + +where :math:`\phi_2` is a two-body term and :math:`\phi_3` is a +three-body term. The summations in the formula are over all neighbors J +and K of atom I within a cutoff distance :math:`a `\sigma`. Only a single pair\_coeff command is used with the *sw* style which specifies a Stillinger-Weber potential file with parameters for all @@ -86,24 +97,25 @@ and three-body coefficients in the formula above: * element 1 (the center atom in a 3-body interaction) * element 2 * element 3 -* epsilon (energy units) -* sigma (distance units) +* :math:`\epsilon` (energy units) +* :math:`\sigma` (distance units) * a -* lambda -* gamma -* costheta0 +* :math:`\lambda` +* :math:`\gamma` +* :math:`\cos\theta_0` * A * B * p * q * tol -The A, B, p, and q parameters are used only for two-body -interactions. The lambda and costheta0 parameters are used only for -three-body interactions. The epsilon, sigma and a parameters are used -for both two-body and three-body interactions. gamma is used only in the -three-body interactions, but is defined for pairs of atoms. -The non-annotated parameters are unitless. +The A, B, p, and q parameters are used only for two-body interactions. +The :math:`\lambda` and :math:`\cos\theta_0` parameters are used only +for three-body interactions. The :math:`\epsilon`, :math:`\sigma` and +*a* parameters are used for both two-body and three-body +interactions. :math:`\gamma` is used only in the three-body +interactions, but is defined for pairs of atoms. The non-annotated +parameters are unitless. LAMMPS introduces an additional performance-optimization parameter tol that is used for both two-body and three-body interactions. In the @@ -141,9 +153,9 @@ are usually defined by simple formulas involving two sets of pair-wise parameters, corresponding to the ij and ik pairs, where i is the center atom. The user must ensure that the correct combining rule is used to calculate the values of the three-body parameters for -alloys. Note also that the function phi3 contains two exponential +alloys. Note also that the function :math:`\phi_3` contains two exponential screening factors with parameter values from the ij pair and ik -pairs. So phi3 for a C atom bonded to a Si atom and a second C atom +pairs. So :math:`\phi_3` for a C atom bonded to a Si atom and a second C atom will depend on the three-body parameters for the CSiC entry, and also on the two-body parameters for the CCC and CSiSi entries. Since the order of the two neighbors is arbitrary, the three-body parameters for @@ -152,8 +164,8 @@ parameters for entries SiCC and CSiSi should also be the same. The parameters used only for two-body interactions (A, B, p, and q) in entries whose 2nd and 3rd element are different (e.g. SiCSi) are not used for anything and can be set to 0.0 if desired. -This is also true for the parameters in phi3 that are -taken from the ij and ik pairs (sigma, a, gamma) +This is also true for the parameters in :math:`\phi_3` that are +taken from the ij and ik pairs (:math:`\sigma`, *a*\ , :math:`\gamma`) ---------- diff --git a/doc/src/pair_tersoff.rst b/doc/src/pair_tersoff.rst index ab9d714499..f2e44d0362 100644 --- a/doc/src/pair_tersoff.rst +++ b/doc/src/pair_tersoff.rst @@ -50,12 +50,28 @@ Description The *tersoff* style computes a 3-body Tersoff potential :ref:`(Tersoff\_1) ` for the energy E of a system of atoms as -.. image:: Eqs/pair_tersoff_1.jpg - :align: center +.. math:: -where f\_R is a two-body term and f\_A includes three-body interactions. -The summations in the formula are over all neighbors J and K of atom I -within a cutoff distance = R + D. + E & = \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\ + V_{ij} & = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\ + f_C(r) & = \left\{ \begin{array} {r@{\quad:\quad}l} + 1 & r < R - D \\ + \frac{1}{2} - \frac{1}{2} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) & + R-D < r < R + D \\ + 0 & r > R + D + \end{array} \right. \\ + f_R(r) & = A \exp (-\lambda_1 r) \\ + f_A(r) & = -B \exp (-\lambda_2 r) \\ + b_{ij} & = \left( 1 + \beta^n {\zeta_{ij}}^n \right)^{-\frac{1}{2n}} \\ + \zeta_{ij} & = \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk}) + \exp \left[ {\lambda_3}^m (r_{ij} - r_{ik})^m \right] \\ + g(\theta) & = \gamma_{ijk} \left( 1 + \frac{c^2}{d^2} - + \frac{c^2}{\left[ d^2 + (\cos \theta - \cos \theta_0)^2\right]} \right) + + +where :math:`f_R` is a two-body term and :math:`f_A` includes three-body +interactions. The summations in the formula are over all neighbors +J and K of atom I within a cutoff distance = R + D. The *tersoff/table* style uses tabulated forms for the two-body, environment and angular functions. Linear interpolation is performed @@ -104,22 +120,24 @@ above: * element 2 (the atom bonded to the center atom) * element 3 (the atom influencing the 1-2 bond in a bond-order sense) * m -* gamma -* lambda3 (1/distance units) +* :math:`\gamma` +* :math:`\lambda_3` (1/distance units) * c * d -* costheta0 (can be a value < -1 or > 1) +* :math:`\cos\theta_0` (can be a value < -1 or > 1) * n -* beta -* lambda2 (1/distance units) +* :math:`\beta` +* :math:`\lambda_2` (1/distance units) * B (energy units) * R (distance units) * D (distance units) -* lambda1 (1/distance units) +* :math:`\lambda_1` (1/distance units) * A (energy units) -The n, beta, lambda2, B, lambda1, and A parameters are only used for -two-body interactions. The m, gamma, lambda3, c, d, and costheta0 +The n, :math:`\beta`, :math:`\lambda_2`, B, :math:`\lambda_1`, and A +parameters are only used for +two-body interactions. The m, :math:`\gamma`, :math:`\lambda_3`, c, d, +and :math:`\cos\theta_0` parameters are only used for three-body interactions. The R and D parameters are used for both two-body and three-body interactions. The non-annotated parameters are unitless. The value of m must be 3 or 1. @@ -149,7 +167,8 @@ SiCC entry. The parameters used for a particular three-body interaction come from the entry with the corresponding three elements. The parameters used only for two-body interactions -(n, beta, lambda2, B, lambda1, and A) in entries whose 2nd and 3rd +(n, :math:`\beta`, :math:`\lambda_2`, B, :math:`\lambda_1`, and A) +in entries whose 2nd and 3rd element are different (e.g. SiCSi) are not used for anything and can be set to 0.0 if desired. @@ -165,16 +184,24 @@ it reduces to the form of :ref:`Albe et al. ` when beta = 1 and m = 1. Note that in the current Tersoff implementation in LAMMPS, m must be specified as either 3 or 1. Tersoff used a slightly different but equivalent form for alloys, which we will refer to as Tersoff\_2 -potential :ref:`(Tersoff\_2) `. The *tersoff/table* style implements +potential :ref:`(Tersoff\_2) `. +The *tersoff/table* style implements Tersoff\_2 parameterization only. LAMMPS parameter values for Tersoff\_2 can be obtained as follows: -gamma\_ijk = omega\_ik, lambda3 = 0 and the value of +:math:`\gamma_{ijk} = \omega_{ik}`, :math:`\lambda_3 = 0` and the value of m has no effect. The parameters for species i and j can be calculated using the Tersoff\_2 mixing rules: -.. image:: Eqs/pair_tersoff_2.jpg - :align: center +.. math:: + + \lambda_1^{i,j} & = \frac{1}{2}(\lambda_1^i + \lambda_1^j)\\ + \lambda_2^{i,j} & = \frac{1}{2}(\lambda_2^i + \lambda_2^j)\\ + A_{i,j} & = (A_{i}A_{j})^{1/2}\\ + B_{i,j} & = \chi_{ij}(B_{i}B_{j})^{1/2}\\ + R_{i,j} & = (R_{i}R_{j})^{1/2}\\ + S_{i,j} & = (S_{i}S_{j})^{1/2} + Tersoff\_2 parameters R and S must be converted to the LAMMPS parameters R and D (R is different in both forms), using the following diff --git a/doc/src/pair_tersoff_mod.rst b/doc/src/pair_tersoff_mod.rst index dbe0b4d95b..94cc9300ed 100644 --- a/doc/src/pair_tersoff_mod.rst +++ b/doc/src/pair_tersoff_mod.rst @@ -49,21 +49,40 @@ potential :ref:`(Tersoff\_1) `, :ref:`(Tersoff\_2) ` wit modified cutoff function and angular-dependent term, giving the energy E of a system of atoms as -.. image:: Eqs/pair_tersoff_mod.jpg - :align: center +.. math:: -where f\_R is a two-body term and f\_A includes three-body interactions. + E & = \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\ + V_{ij} & = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\ + f_C(r) & = \left\{ \begin{array} {r@{\quad:\quad}l} + 1 & r < R - D \\ + \frac{1}{2} - \frac{9}{16} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) - \frac{1}{16} \sin \left( \frac{3\pi}{2} \frac{r-R}{D} \right) & + R-D < r < R + D \\ + 0 & r > R + D + \end{array} \right. \\ + f_R(r) & = A \exp (-\lambda_1 r) \\ + f_A(r) & = -B \exp (-\lambda_2 r) \\ + b_{ij} & = \left( 1 + {\zeta_{ij}}^\eta \right)^{-\frac{1}{2n}} \\ + \zeta_{ij} & = \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk}) + \exp \left[ \alpha (r_{ij} - r_{ik})^\beta \right] \\ + g(\theta) & = c_1 + g_o(\theta) g_a(\theta) \\ + g_o(\theta) & = \frac{c_2 (h - \cos \theta)^2}{c_3 + (h - \cos \theta)^2} \\ + g_a(\theta) & = 1 + c_4 \exp \left[ -c_5 (h - \cos \theta)^2 \right] \\ + + +where :math:`f_R` is a two-body term and :math:`f_A` includes three-body interactions. The summations in the formula are over all neighbors J and K of atom I within a cutoff distance = R + D. The *tersoff/mod/c* style differs from *tersoff/mod* only in the formulation of the V\_ij term, where it contains an additional c0 term. -.. image:: Eqs/pair_tersoff_mod_c.jpg - :align: center +.. math:: -The modified cutoff function f\_C proposed by :ref:`(Murty) ` and + V_{ij} & = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) + c_0 \right] + + +The modified cutoff function :math:`f_C` proposed by :ref:`(Murty) ` and having a continuous second-order differential is employed. The -angular-dependent term g(theta) was modified to increase the +angular-dependent term :math:`g(\theta)` was modified to increase the flexibility of the potential. The *tersoff/mod* potential is fitted to both the elastic constants @@ -105,30 +124,30 @@ not blank or comments (starting with #) define parameters for a triplet of elements. The parameters in a single entry correspond to coefficients in the formulae above: -element 1 (the center atom in a 3-body interaction) -element 2 (the atom bonded to the center atom) -element 3 (the atom influencing the 1-2 bond in a bond-order sense) -beta -alpha -h -eta -beta\_ters = 1 (dummy parameter) -lambda2 (1/distance units) -B (energy units) -R (distance units) -D (distance units) -lambda1 (1/distance units) -A (energy units) -n -c1 -c2 -c3 -c4 -c5 -c0 (energy units, tersoff/mod/c only):ul +* element 1 (the center atom in a 3-body interaction) +* element 2 (the atom bonded to the center atom) +* element 3 (the atom influencing the 1-2 bond in a bond-order sense) +* :math:`\beta` +* :math:`\alpha` +* h +* :math:`\eta` +* :math:`\beta_{ters}` = 1 (dummy parameter) +* :math:`\lambda_2` (1/distance units) +* B (energy units) +* R (distance units) +* D (distance units) +* :math:`\lambda_1` (1/distance units) +* A (energy units) +* n +* c1 +* c2 +* c3 +* c4 +* c5 +* c0 (energy units, tersoff/mod/c only):ul -The n, eta, lambda2, B, lambda1, and A parameters are only used for -two-body interactions. The beta, alpha, c1, c2, c3, c4, c5, h +The n, :math:`\eta`, :math:`\lambda_2`, B, :math:`\lambda_1`, and A parameters are only used for +two-body interactions. The :math:`\beta`, :math:`\alpha`, c1, c2, c3, c4, c5, h parameters are only used for three-body interactions. The R and D parameters are used for both two-body and three-body interactions. The c0 term applies to *tersoff/mod/c* only. The non-annotated diff --git a/doc/src/pair_tersoff_zbl.rst b/doc/src/pair_tersoff_zbl.rst index 2ad3be9d7d..e60de64f28 100644 --- a/doc/src/pair_tersoff_zbl.rst +++ b/doc/src/pair_tersoff_zbl.rst @@ -38,26 +38,53 @@ based on a Coulomb potential and the Ziegler-Biersack-Littmark universal screening function :ref:`(ZBL) `, giving the energy E of a system of atoms as -.. image:: Eqs/pair_tersoff_zbl.jpg - :align: center +.. math:: -The f\_F term is a fermi-like function used to smoothly connect the ZBL + E & = \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\ + V_{ij} & = (1 - f_F(r_{ij})) V^{ZBL}_{ij} + f_F(r_{ij}) V^{Tersoff}_{ij} \\ + f_F(r_{ij}) & = \frac{1}{1 + e^{-A_F(r_{ij} - r_C)}}\\ + \\ + \\ + V^{ZBL}_{ij} & = \frac{1}{4\pi\epsilon_0} \frac{Z_1 Z_2 \,e^2}{r_{ij}} \phi(r_{ij}/a) \\ + a & = \frac{0.8854\,a_0}{Z_{1}^{0.23} + Z_{2}^{0.23}}\\ + \phi(x) & = 0.1818e^{-3.2x} + 0.5099e^{-0.9423x} + 0.2802e^{-0.4029x} + 0.02817e^{-0.2016x}\\ + \\ + \\ + V^{Tersoff}_{ij} & = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\ + f_C(r) & = \left\{ \begin{array} {r@{\quad:\quad}l} + 1 & r < R - D \\ + \frac{1}{2} - \frac{1}{2} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) & + R-D < r < R + D \\ + 0 & r > R + D + \end{array} \right. \\ + f_R(r) & = A \exp (-\lambda_1 r) \\ + f_A(r) & = -B \exp (-\lambda_2 r) \\ + b_{ij} & = \left( 1 + \beta^n {\zeta_{ij}}^n \right)^{-\frac{1}{2n}} \\ + \zeta_{ij} & = \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk}) + \exp \left[ {\lambda_3}^m (r_{ij} - r_{ik})^m \right] \\ + g(\theta) & = \gamma_{ijk} \left( 1 + \frac{c^2}{d^2} - + \frac{c^2}{\left[ d^2 + (\cos \theta - \cos \theta_0)^2\right]} \right) + + +The :math:`f_F` term is a fermi-like function used to smoothly connect the ZBL repulsive potential with the Tersoff potential. There are 2 -parameters used to adjust it: A\_F and r\_C. A\_F controls how "sharp" -the transition is between the two, and r\_C is essentially the cutoff +parameters used to adjust it: :math:`A_F` and :math:`r_C`. :math:`A_F` +controls how "sharp" +the transition is between the two, and :math:`r_C` is essentially the cutoff for the ZBL potential. For the ZBL portion, there are two terms. The first is the Coulomb repulsive term, with Z1, Z2 as the number of protons in each nucleus, -e as the electron charge (1 for metal and real units) and epsilon0 as -the permittivity of vacuum. The second part is the ZBL universal +e as the electron charge (1 for metal and real units) and :math:`\epsilon_0` +as the permittivity of vacuum. The second part is the ZBL universal screening function, with a0 being the Bohr radius (typically 0.529 Angstroms), and the remainder of the coefficients provided by the original paper. This screening function should be applicable to most systems. However, it is only accurate for small separations (i.e. less than 1 Angstrom). -For the Tersoff portion, f\_R is a two-body term and f\_A includes +For the Tersoff portion, :math:`f_R` is a two-body term and :math:`f_A` +includes three-body interactions. The summations in the formula are over all neighbors J and K of atom I within a cutoff distance = R + D. @@ -102,29 +129,32 @@ in the formula above: * element 2 (the atom bonded to the center atom) * element 3 (the atom influencing the 1-2 bond in a bond-order sense) * m -* gamma -* lambda3 (1/distance units) +* :math:`\gamma` +* :math:`\lambda_3` (1/distance units) * c * d -* costheta0 (can be a value < -1 or > 1) +* :math:`\cos\theta_0` (can be a value < -1 or > 1) * n -* beta -* lambda2 (1/distance units) +* :math:`\beta` +* :math:`\lambda_2` (1/distance units) * B (energy units) * R (distance units) * D (distance units) -* lambda1 (1/distance units) +* :math:`\lambda_1` (1/distance units) * A (energy units) -* Z\_i -* Z\_j +* :math:`Z_i` +* :math:`Z_j` * ZBLcut (distance units) * ZBLexpscale (1/distance units) -The n, beta, lambda2, B, lambda1, and A parameters are only used for -two-body interactions. The m, gamma, lambda3, c, d, and costheta0 +The n, :math:`\beta`, :math:`\lambda_2`, B, :math:`\lambda_1`, and A +parameters are only used for +two-body interactions. The m, :math:`\gamma`, :math:`\lambda_3`, c, d, +and :math:`\cos\theta_0` parameters are only used for three-body interactions. The R and D parameters are used for both two-body and three-body interactions. The -Z\_i,Z\_j, ZBLcut, ZBLexpscale parameters are used in the ZBL repulsive +:math:`Z_i`, :math:`Z_j`, ZBLcut, ZBLexpscale parameters are used in the +ZBL repulsive portion of the potential and in the Fermi-like function. The non-annotated parameters are unitless. The value of m must be 3 or 1. @@ -153,7 +183,8 @@ SiCC entry. The parameters used for a particular three-body interaction come from the entry with the corresponding three elements. The parameters used only for two-body interactions -(n, beta, lambda2, B, lambda1, and A) in entries whose 2nd and 3rd +(n, :math:`\beta`, :math:`\lambda_2`, B, :math:`\lambda_1`, and A) +in entries whose 2nd and 3rd element are different (e.g. SiCSi) are not used for anything and can be set to 0.0 if desired. @@ -172,12 +203,19 @@ different but equivalent form for alloys, which we will refer to as Tersoff\_2 potential :ref:`(Tersoff\_2) `. LAMMPS parameter values for Tersoff\_2 can be obtained as follows: -gamma = omega\_ijk, lambda3 = 0 and the value of +:math:`\gamma = \omega_{ijk}`, :math:`\lambda_3 = 0` and the value of m has no effect. The parameters for species i and j can be calculated using the Tersoff\_2 mixing rules: -.. image:: Eqs/pair_tersoff_2.jpg - :align: center +.. math:: + + \lambda_1^{i,j} & = \frac{1}{2}(\lambda_1^i + \lambda_1^j)\\ + \lambda_2^{i,j} & = \frac{1}{2}(\lambda_2^i + \lambda_2^j)\\ + A_{i,j} & = (A_{i}A_{j})^{1/2}\\ + B_{i,j} & = \chi_{ij}(B_{i}B_{j})^{1/2}\\ + R_{i,j} & = (R_{i}R_{j})^{1/2}\\ + S_{i,j} & = (S_{i}S_{j})^{1/2}\\ + Tersoff\_2 parameters R and S must be converted to the LAMMPS parameters R and D (R is different in both forms), using the following