Added equation for lj_cubic

git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@6858 f3b2605a-c512-4ea7-a41b-209d697bcdaa

doc/Eqs/pair_lj_cubic.tex

@@ -0,0 +1,12 @@
+\documentstyle[12pt]{article}
+
+\begin{document}
+
+\begin{eqnarray*}
+ E &=& u_{LJ}(r) \qquad r \leq r_s \\
+    &=&  u_{LJ}(r_s) + (r-r_s) u'_{LJ}(r) - \frac{1}{6} A_3 (r-r_s)^3 \qquad r_s < r \leq r_c \\
+    &=& 0 \qquad r > r_c 
+\end{eqnarray*}                           
+
+
+\end{document}

doc/pair_lj_cubic.html

@@ -23,19 +23,29 @@ pair_coeff * * 1.0 0.8908987
 <P><B>Description:</B>
 </P>
 <P>The <I>lj/cubic</I> style computes a truncated LJ interaction potential whose
-energy and force are continuous everywhere. This is
-achieved by replacing the LJ function outside the inflection point with
-a cubic function of distance, so that both the energy and force are
-continuous at the inflection point, and go to zero at the
-cutoff distance. The LJ potential inside the inflection point is
-unchanged. The location of the inflection point rs is defined
-by the LJ diameter, rs/sigma = (26/7)^1/6. The cutoff distance 
-is defined by rcut/rs = 67/48.
+energy and force are continuous everywhere. 
+Inside the inflection point the interaction is identical to the 
+standard 12/6 <A HREF = "pair_lj.html">Lennard-Jones</A> potential.
+The LJ function outside the inflection point is replaced 
+with a cubic function of distance. The energy, force and second
+derivative are continuous at the inflection point. 
+The cubic coefficient A3 is chosen so
+that both energy and force go to zero at the cutoff distance. 
+Outside the cutoff distance the energy and force are zero.
 </P>
-<P>This potential is commonly used to study the shock compression
+<CENTER><IMG SRC = "Eqs/pair_lj_cubic.jpg">
+</CENTER>
+<P>The location of the inflection point rs is defined
+by the LJ diameter, rs/sigma = (26/7)^1/6. The cutoff distance 
+is defined by rc/rs = 67/48. The analytic expression for the 
+the cubic coefficient 
+A3*rmin^3/epsilon = 27.93357 is given in the paper 
+Holian and Ravelo <A HREF = "#Holian">(Holian)</A>.
+</P>
+<P>This potential is commonly used to study the mechanical behavior
 of FCC solids, as in the paper by Holian and Ravelo <A HREF = "#Holian">(Holian)</A>.
 </P>
-<P>The following coefficients must be defined for each pair of atoms
+<P>The following coefficients must be defined for each pair of atom
 types via the <A HREF = "pair_coeff.html">pair_coeff</A> command as in the example
 above, or in the data file or restart files read by the
 <A HREF = "read_data.html">read_data</A> or <A HREF = "read_restart.html">read_restart</A>
@@ -46,8 +56,8 @@ commands, or by mixing as described below:
 </UL>
 <P>Note that sigma is defined in the LJ formula as the zero-crossing
 distance for the potential, not as the energy minimum, which
-is located at 2^(1/6)*sigma. In the above example, sigma = 0.8908987,
-so the energy minimum is located at r = 1.
+is located at rmin = 2^(1/6)*sigma. In the above example, sigma = 0.8908987,
+so rmin = 1.
 </P>
 <HR>
 

doc/pair_lj_cubic.txt

@@ -20,19 +20,29 @@ pair_coeff * * 1.0 0.8908987 :pre
 [Description:]
 
 The {lj/cubic} style computes a truncated LJ interaction potential whose
-energy and force are continuous everywhere. This is
-achieved by replacing the LJ function outside the inflection point with
-a cubic function of distance, so that both the energy and force are
-continuous at the inflection point, and go to zero at the
-cutoff distance. The LJ potential inside the inflection point is
-unchanged. The location of the inflection point rs is defined
-by the LJ diameter, rs/sigma = (26/7)^1/6. The cutoff distance 
-is defined by rcut/rs = 67/48.
+energy and force are continuous everywhere. 
+Inside the inflection point the interaction is identical to the 
+standard 12/6 "Lennard-Jones"_pair_lj.html potential.
+The LJ function outside the inflection point is replaced 
+with a cubic function of distance. The energy, force and second
+derivative are continuous at the inflection point. 
+The cubic coefficient A3 is chosen so
+that both energy and force go to zero at the cutoff distance. 
+Outside the cutoff distance the energy and force are zero.
 
-This potential is commonly used to study the shock compression
+:c,image(Eqs/pair_lj_cubic.jpg)
+ 
+The location of the inflection point rs is defined
+by the LJ diameter, rs/sigma = (26/7)^1/6. The cutoff distance 
+is defined by rc/rs = 67/48. The analytic expression for the 
+the cubic coefficient 
+A3*rmin^3/epsilon = 27.93357 is given in the paper 
+Holian and Ravelo "(Holian)"_#Holian.
+
+This potential is commonly used to study the mechanical behavior
 of FCC solids, as in the paper by Holian and Ravelo "(Holian)"_#Holian.
 
-The following coefficients must be defined for each pair of atoms
+The following coefficients must be defined for each pair of atom
 types via the "pair_coeff"_pair_coeff.html command as in the example
 above, or in the data file or restart files read by the
 "read_data"_read_data.html or "read_restart"_read_restart.html
@@ -43,8 +53,8 @@ sigma (distance units) :ul
 
 Note that sigma is defined in the LJ formula as the zero-crossing
 distance for the potential, not as the energy minimum, which
-is located at 2^(1/6)*sigma. In the above example, sigma = 0.8908987,
-so the energy minimum is located at r = 1.
+is located at rmin = 2^(1/6)*sigma. In the above example, sigma = 0.8908987,
+so rmin = 1.
 
 :line