Added equation for lj_cubic

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\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}

<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

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>

<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 rcut/rs = 67/48.

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 shock compression

<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>

</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>

[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

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.

: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 rcut/rs = 67/48.

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 shock compression

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

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