Homec4science
Diffusion lammps (efficient_neuronet) Browse

lammps/doc/fix_box_relax.html
c6213f36d34cefficient_neuronet

Last Change
Raw File

fix_box_relax.html
View Options

<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>fix box/relax command &mdash; LAMMPS 15 May 2015 version documentation</title>
<link rel="stylesheet" href="_static/css/theme.css" type="text/css" />
<link rel="stylesheet" href="_static/sphinxcontrib-images/LightBox2/lightbox2/css/lightbox.css" type="text/css" />
<link rel="top" title="LAMMPS 15 May 2015 version documentation" href="index.html"/>
<script src="_static/js/modernizr.min.js"></script>
</head>
<body class="wy-body-for-nav" role="document">
<div class="wy-grid-for-nav">
<nav data-toggle="wy-nav-shift" class="wy-nav-side">
<div class="wy-side-nav-search">
<a href="Manual.html" class="icon icon-home"> LAMMPS
</a>
<div role="search">
<form id="rtd-search-form" class="wy-form" action="search.html" method="get">
<input type="text" name="q" placeholder="Search docs" />
<input type="hidden" name="check_keywords" value="yes" />
<input type="hidden" name="area" value="default" />
</form>
</div>
</div>
<div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="main navigation">
<ul>
<li class="toctree-l1"><a class="reference internal" href="Section_intro.html">1. Introduction</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_start.html">2. Getting Started</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_commands.html">3. Commands</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_packages.html">4. Packages</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_accelerate.html">5. Accelerating LAMMPS performance</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_howto.html">6. How-to discussions</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_example.html">7. Example problems</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_perf.html">8. Performance &amp; scalability</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_tools.html">9. Additional tools</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_modify.html">10. Modifying &amp; extending LAMMPS</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_python.html">11. Python interface to LAMMPS</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_errors.html">12. Errors</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_history.html">13. Future and history</a></li>
</ul>
</div>
&nbsp;
</nav>
<section data-toggle="wy-nav-shift" class="wy-nav-content-wrap">
<nav class="wy-nav-top" role="navigation" aria-label="top navigation">
<i data-toggle="wy-nav-top" class="fa fa-bars"></i>
<a href="Manual.html">LAMMPS</a>
</nav>
<div class="wy-nav-content">
<div class="rst-content">
<div role="navigation" aria-label="breadcrumbs navigation">
<ul class="wy-breadcrumbs">
<li><a href="Manual.html">Docs</a> &raquo;</li>
<li>fix box/relax command</li>
<li class="wy-breadcrumbs-aside">
<a href="http://lammps.sandia.gov">Homepage</a>
<a href="Section_commands.html#comm">Commands</a>
</li>
</ul>
<hr/>
</div>
<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
<div itemprop="articleBody">
<div class="section" id="fix-box-relax-command">
<span id="index-0"></span><h1>fix box/relax command<a class="headerlink" href="#fix-box-relax-command" title="Permalink to this headline"></a></h1>
<div class="section" id="syntax">
<h2>Syntax<a class="headerlink" href="#syntax" title="Permalink to this headline"></a></h2>
<div class="highlight-python"><div class="highlight"><pre>fix ID group-ID box/relax keyword value ...
</pre></div>
</div>
<ul class="simple">
<li>ID, group-ID are documented in <a class="reference internal" href="fix.html"><em>fix</em></a> command</li>
<li>box/relax = style name of this fix command</li>
</ul>
<pre class="literal-block">
one or more keyword value pairs may be appended
keyword = <em>iso</em> or <em>aniso</em> or <em>tri</em> or <em>x</em> or <em>y</em> or <em>z</em> or <em>xy</em> or <em>yz</em> or <em>xz</em> or <em>couple</em> or <em>nreset</em> or <em>vmax</em> or <em>dilate</em> or <em>scaleyz</em> or <em>scalexz</em> or <em>scalexy</em> or <em>fixedpoint</em>
<em>iso</em> or <em>aniso</em> or <em>tri</em> value = Ptarget = desired pressure (pressure units)
<em>x</em> or <em>y</em> or <em>z</em> or <em>xy</em> or <em>yz</em> or <em>xz</em> value = Ptarget = desired pressure (pressure units)
<em>couple</em> = <em>none</em> or <em>xyz</em> or <em>xy</em> or <em>yz</em> or <em>xz</em>
<em>nreset</em> value = reset reference cell every this many minimizer iterations
<em>vmax</em> value = fraction = max allowed volume change in one iteration
<em>dilate</em> value = <em>all</em> or <em>partial</em>
<em>scaleyz</em> value = <em>yes</em> or <em>no</em> = scale yz with lz
<em>scalexz</em> value = <em>yes</em> or <em>no</em> = scale xz with lz
<em>scalexy</em> value = <em>yes</em> or <em>no</em> = scale xy with ly
<em>fixedpoint</em> values = x y z
x,y,z = perform relaxation dilation/contraction around this point (distance units)
</pre>
</div>
<div class="section" id="examples">
<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline"></a></h2>
<div class="highlight-python"><div class="highlight"><pre>fix 1 all box/relax iso 0.0 vmax 0.001
fix 2 water box/relax aniso 0.0 dilate partial
fix 2 ice box/relax tri 0.0 couple xy nreset 100
</pre></div>
</div>
</div>
<div class="section" id="description">
<h2>Description<a class="headerlink" href="#description" title="Permalink to this headline"></a></h2>
<p>Apply an external pressure or stress tensor to the simulation box
during an <a class="reference internal" href="minimize.html"><em>energy minimization</em></a>. This allows the box
size and shape to vary during the iterations of the minimizer so that
the final configuration will be both an energy minimum for the
potential energy of the atoms, and the system pressure tensor will be
close to the specified external tensor. Conceptually, specifying a
positive pressure is like squeezing on the simulation box; a negative
pressure typically allows the box to expand.</p>
<hr class="docutils" />
<p>The external pressure tensor is specified using one or more of the
<em>iso</em>, <em>aniso</em>, <em>tri</em>, <em>x</em>, <em>y</em>, <em>z</em>, <em>xy</em>, <em>xz</em>, <em>yz</em>, and <em>couple</em>
keywords. These keywords give you the ability to specify all 6
components of an external stress tensor, and to couple various of
these components together so that the dimensions they represent are
varied together during the mimimization.</p>
<p>Orthogonal simulation boxes have 3 adjustable dimensions (x,y,z).
Triclinic (non-orthogonal) simulation boxes have 6 adjustable
dimensions (x,y,z,xy,xz,yz). The <a class="reference internal" href="create_box.html"><em>create_box</em></a>, <a class="reference internal" href="read_data.html"><em>read data</em></a>, and <a class="reference internal" href="read_restart.html"><em>read_restart</em></a> commands
specify whether the simulation box is orthogonal or non-orthogonal
(triclinic) and explain the meaning of the xy,xz,yz tilt factors.</p>
<p>The target pressures <em>Ptarget</em> for each of the 6 components of the
stress tensor can be specified independently via the <em>x</em>, <em>y</em>, <em>z</em>,
<em>xy</em>, <em>xz</em>, <em>yz</em> keywords, which correspond to the 6 simulation box
dimensions. For example, if the <em>y</em> keyword is used, the y-box length
will change during the minimization. If the <em>xy</em> keyword is used, the
xy tilt factor will change. A box dimension will not change if that
component is not specified.</p>
<p>Note that in order to use the <em>xy</em>, <em>xz</em>, or <em>yz</em> keywords, the
simulation box must be triclinic, even if its initial tilt factors are
0.0.</p>
<p>When the size of the simulation box changes, all atoms are re-scaled
to new positions, unless the keyword <em>dilate</em> is specified with a
value of <em>partial</em>, in which case only the atoms in the fix group are
re-scaled. This can be useful for leaving the coordinates of atoms in
a solid substrate unchanged and controlling the pressure of a
surrounding fluid.</p>
<p>The <em>scaleyz</em>, <em>scalexz</em>, and <em>scalexy</em> keywords control whether or
not the corresponding tilt factors are scaled with the associated box
dimensions when relaxing triclinic periodic cells. The default
values <em>yes</em> will turn on scaling, which corresponds to adjusting the
linear dimensions of the cell while preserving its shape. Choosing
<em>no</em> ensures that the tilt factors are not scaled with the box
dimensions. See below for restrictions and default values in different
situations. In older versions of LAMMPS, scaling of tilt factors was
not performed. The old behavior can be recovered by setting all three
scale keywords to <em>no</em>.</p>
<p>The <em>fixedpoint</em> keyword specifies the fixed point for cell relaxation.
By default, it is the center of the box. Whatever point is
chosen will not move during the simulation. For example, if the lower
periodic boundaries pass through (0,0,0), and this point is provided
to <em>fixedpoint</em>, then the lower periodic boundaries will remain at
(0,0,0), while the upper periodic boundaries will move twice as
far. In all cases, the particle positions at each iteration are
unaffected by the chosen value, except that all particles are
displaced by the same amount, different on each iteration.</p>
<div class="admonition warning">
<p class="first admonition-title">Warning</p>
<p class="last">Appling an external pressure to tilt dimensions <em>xy</em>,
<em>xz</em>, <em>yz</em> can sometimes result in arbitrarily large values of the
tilt factors, i.e. a dramatically deformed simulation box. This
typically indicates that there is something badly wrong with how the
simulation was constructed. The two most common sources of this error
are applying a shear stress to a liquid system or specifying an
external shear stress tensor that exceeds the yield stress of the
solid. In either case the minimization may converge to a bogus
conformation or not converge at all. Also note that if the box shape
tilts to an extreme shape, LAMMPS will run less efficiently, due to
the large volume of communication needed to acquire ghost atoms around
a processor&#8217;s irregular-shaped sub-domain. For extreme values of
tilt, LAMMPS may also lose atoms and generate an error.</p>
</div>
<div class="admonition warning">
<p class="first admonition-title">Warning</p>
<p class="last">Performing a minimization with this fix is not a
mathematically well-defined minimization problem. This is because the
objective function being minimized changes if the box size/shape
changes. In practice this means the minimizer can get &#8220;stuck&#8221; before
you have reached the desired tolerance. The solution to this is to
restart the minmizer from the new adjusted box size/shape, since that
creates a new objective function valid for the new box size/shape.
Repeat as necessary until the box size/shape has reached its new
equilibrium.</p>
</div>
<hr class="docutils" />
<p>The <em>couple</em> keyword allows two or three of the diagonal components of
the pressure tensor to be &#8220;coupled&#8221; together. The value specified
with the keyword determines which are coupled. For example, <em>xz</em>
means the <em>Pxx</em> and <em>Pzz</em> components of the stress tensor are coupled.
<em>Xyz</em> means all 3 diagonal components are coupled. Coupling means two
things: the instantaneous stress will be computed as an average of the
corresponding diagonal components, and the coupled box dimensions will
be changed together in lockstep, meaning coupled dimensions will be
dilated or contracted by the same percentage every timestep. The
<em>Ptarget</em> values for any coupled dimensions must be identical.
<em>Couple xyz</em> can be used for a 2d simulation; the <em>z</em> dimension is
simply ignored.</p>
<hr class="docutils" />
<p>The <em>iso</em>, <em>aniso</em>, and <em>tri</em> keywords are simply shortcuts that are
equivalent to specifying several other keywords together.</p>
<p>The keyword <em>iso</em> means couple all 3 diagonal components together when
pressure is computed (hydrostatic pressure), and dilate/contract the
dimensions together. Using &#8220;iso Ptarget&#8221; is the same as specifying
these 4 keywords:</p>
<div class="highlight-python"><div class="highlight"><pre>x Ptarget
y Ptarget
z Ptarget
couple xyz
</pre></div>
</div>
<p>The keyword <em>aniso</em> means <em>x</em>, <em>y</em>, and <em>z</em> dimensions are controlled
independently using the <em>Pxx</em>, <em>Pyy</em>, and <em>Pzz</em> components of the
stress tensor as the driving forces, and the specified scalar external
pressure. Using &#8220;aniso Ptarget&#8221; is the same as specifying these 4
keywords:</p>
<div class="highlight-python"><div class="highlight"><pre>x Ptarget
y Ptarget
z Ptarget
couple none
</pre></div>
</div>
<p>The keyword <em>tri</em> means <em>x</em>, <em>y</em>, <em>z</em>, <em>xy</em>, <em>xz</em>, and <em>yz</em> dimensions
are controlled independently using their individual stress components
as the driving forces, and the specified scalar pressure as the
external normal stress. Using &#8220;tri Ptarget&#8221; is the same as specifying
these 7 keywords:</p>
<div class="highlight-python"><div class="highlight"><pre>x Ptarget
y Ptarget
z Ptarget
xy 0.0
yz 0.0
xz 0.0
couple none
</pre></div>
</div>
<hr class="docutils" />
<p>The <em>vmax</em> keyword can be used to limit the fractional change in the
volume of the simulation box that can occur in one iteration of the
minimizer. If the pressure is not settling down during the
minimization this can be because the volume is fluctuating too much.
The specified fraction must be greater than 0.0 and should be &lt;&lt; 1.0.
A value of 0.001 means the volume cannot change by more than 1/10 of a
percent in one iteration when <em>couple xyz</em> has been specified. For
any other case it means no linear dimension of the simulation box can
change by more than 1/10 of a percent.</p>
<hr class="docutils" />
<p>With this fix, the potential energy used by the minimizer is augmented
by an additional energy provided by the fix. The overall objective
function then is:</p>
<img alt="_images/fix_box_relax1.jpg" class="align-center" src="_images/fix_box_relax1.jpg" />
<p>where <em>U</em> is the system potential energy, <em>P*_t is the desired
hydrostatic pressure, *V</em> and <em>V*_0 are the system and reference
volumes, respectively. *E*_*strain</em> is the strain energy expression
proposed by Parrinello and Rahman <a class="reference internal" href="#parrinello1981"><span>(Parrinello1981)</span></a>.
Taking derivatives of <em>E</em> w.r.t. the box dimensions, and setting these
to zero, we find that at the minimum of the objective function, the
global system stress tensor <strong>P</strong> will satisfy the relation:</p>
<img alt="_images/fix_box_relax2.jpg" class="align-center" src="_images/fix_box_relax2.jpg" />
<p>where <strong>I</strong> is the identity matrix, <strong>h**_0 is the box dimension tensor of
the reference cell, and **h**_0*d* is the diagonal part of
**h**_0. **S**_*t* is a symmetric stress tensor that is chosen by LAMMPS
so that the upper-triangular components of **P</strong> equal the stress tensor
specified by the user.</p>
<p>This equation only applies when the box dimensions are equal to those
of the reference dimensions. If this is not the case, then the
converged stress tensor will not equal that specified by the user. We
can resolve this problem by periodically resetting the reference
dimensions. The keyword <em>nreset_ref</em> controls how often this is done.
If this keyword is not used, or is given a value of zero, then the
reference dimensions are set to those of the initial simulation domain
and are never changed. A value of <em>nstep</em> means that every <em>nstep</em>
minimization steps, the reference dimensions are set to those of the
current simulation domain. Note that resetting the reference
dimensions changes the objective function and gradients, which
sometimes causes the minimization to fail. This can be resolved by
changing the value of <em>nreset</em>, or simply continuing the minimization
from a restart file.</p>
<div class="admonition warning">
<p class="first admonition-title">Warning</p>
<p class="last">As normally computed, pressure includes a kinetic-
energy or temperature-dependent component; see the <a class="reference internal" href="compute_pressure.html"><em>compute pressure</em></a> command. However, atom velocities are
ignored during a minimization, and the applied pressure(s) specified
with this command are assumed to only be the virial component of the
pressure (the non-kinetic portion). Thus if atoms have a non-zero
temperature and you print the usual thermodynamic pressure, it may not
appear the system is converging to your specified pressure. The
solution for this is to either (a) zero the velocities of all atoms
before performing the minimization, or (b) make sure you are
monitoring the pressure without its kinetic component. The latter can
be done by outputting the pressure from the fix this command creates
(see below) or a pressure fix you define yourself.</p>
</div>
<div class="admonition warning">
<p class="first admonition-title">Warning</p>
<p class="last">Because pressure is often a very sensitive function of
volume, it can be difficult for the minimizer to equilibrate the
system the desired pressure with high precision, particularly for
solids. Some techniques that seem to help are (a) use the
&#8220;min_modify line quadratic&#8221; option when minimizing with box
relaxations, (b) minimize several times in succession if need be,
to drive the pressure closer to the target pressure, (c) relax
the atom positions before relaxing the box, and (d) relax the box
to the target hydrostatic pressure before relaxing to a target
shear stress state. Also note that
some systems (e.g. liquids) will not sustain a non-hydrostatic applied
pressure, which means the minimizer will not converge.</p>
</div>
<hr class="docutils" />
<p>This fix computes a temperature and pressure each timestep. The
temperature is used to compute the kinetic contribution to the
pressure, even though this is subsequently ignored by default. To do
this, the fix creates its own computes of style &#8220;temp&#8221; and &#8220;pressure&#8221;,
as if these commands had been issued:</p>
<div class="highlight-python"><div class="highlight"><pre>compute fix-ID_temp group-ID temp
compute fix-ID_press group-ID pressure fix-ID_temp virial
</pre></div>
</div>
<p>See the <a class="reference internal" href="compute_temp.html"><em>compute temp</em></a> and <a class="reference internal" href="compute_pressure.html"><em>compute pressure</em></a> commands for details. Note that the
IDs of the new computes are the fix-ID + underscore + &#8220;temp&#8221; or fix_ID
+ underscore + &#8220;press&#8221;, and the group for the new computes is the same
as the fix group. Also note that the pressure compute does not
include a kinetic component.</p>
<p>Note that these are NOT the computes used by thermodynamic output (see
the <a class="reference internal" href="thermo_style.html"><em>thermo_style</em></a> command) with ID = <em>thermo_temp</em>
and <em>thermo_press</em>. This means you can change the attributes of this
fix&#8217;s temperature or pressure via the
<a class="reference internal" href="compute_modify.html"><em>compute_modify</em></a> command or print this temperature
or pressure during thermodynamic output via the <a class="reference internal" href="thermo_style.html"><em>thermo_style custom</em></a> command using the appropriate compute-ID.
It also means that changing attributes of <em>thermo_temp</em> or
<em>thermo_press</em> will have no effect on this fix.</p>
</div>
<hr class="docutils" />
<div class="section" id="restart-fix-modify-output-run-start-stop-minimize-info">
<h2>Restart, fix_modify, output, run start/stop, minimize info<a class="headerlink" href="#restart-fix-modify-output-run-start-stop-minimize-info" title="Permalink to this headline"></a></h2>
<p>No information about this fix is written to <a class="reference internal" href="restart.html"><em>binary restart files</em></a>.</p>
<p>The <a class="reference internal" href="fix_modify.html"><em>fix_modify</em></a> <em>temp</em> and <em>press</em> options are
supported by this fix. You can use them to assign a
<a class="reference internal" href="compute.html"><em>compute</em></a> you have defined to this fix which will be used
in its temperature and pressure calculation, as described above. Note
that as described above, if you assign a pressure compute to this fix
that includes a kinetic energy component it will affect the
minimization, most likely in an undesirable way.</p>
<div class="admonition warning">
<p class="first admonition-title">Warning</p>
<p class="last">If both the <em>temp</em> and <em>press</em> keywords are used in a
single thermo_modify command (or in two separate commands), then the
order in which the keywords are specified is important. Note that a
<a class="reference internal" href="compute_pressure.html"><em>pressure compute</em></a> defines its own temperature
compute as an argument when it is specified. The <em>temp</em> keyword will
override this (for the pressure compute being used by fix npt), but
only if the <em>temp</em> keyword comes after the <em>press</em> keyword. If the
<em>temp</em> keyword comes before the <em>press</em> keyword, then the new pressure
compute specified by the <em>press</em> keyword will be unaffected by the
<em>temp</em> setting.</p>
</div>
<p>This fix computes a global scalar which can be accessed by various
<a class="reference internal" href="Section_howto.html#howto-15"><span>output commands</span></a>. The scalar is the
pressure-volume energy, plus the strain energy, if it exists.</p>
<p>No parameter of this fix can be used with the <em>start/stop</em> keywords of
the <a class="reference internal" href="run.html"><em>run</em></a> command.</p>
<p>This fix is invoked during <a class="reference internal" href="minimize.html"><em>energy minimization</em></a>, but
not for the purpose of adding a contribution to the energy or forces
being minimized. Instead it alters the simulation box geometry as
described above.</p>
</div>
<div class="section" id="restrictions">
<h2>Restrictions<a class="headerlink" href="#restrictions" title="Permalink to this headline"></a></h2>
<p>Only dimensions that are available can be adjusted by this fix.
Non-periodic dimensions are not available. <em>z</em>, <em>xz</em>, and <em>yz</em>, are
not available for 2D simulations. <em>xy</em>, <em>xz</em>, and <em>yz</em> are only
available if the simulation domain is non-orthogonal. The
<a class="reference internal" href="create_box.html"><em>create_box</em></a>, <a class="reference internal" href="read_data.html"><em>read data</em></a>, and
<a class="reference internal" href="read_restart.html"><em>read_restart</em></a> commands specify whether the
simulation box is orthogonal or non-orthogonal (triclinic) and explain
the meaning of the xy,xz,yz tilt factors.</p>
<p>The <em>scaleyz yes</em> and <em>scalexz yes</em> keyword/value pairs can not be used
for 2D simulations. <em>scaleyz yes</em>, <em>scalexz yes</em>, and <em>scalexy yes</em> options
can only be used if the 2nd dimension in the keyword is periodic,
and if the tilt factor is not coupled to the barostat via keywords
<em>tri</em>, <em>yz</em>, <em>xz</em>, and <em>xy</em>.</p>
</div>
<div class="section" id="related-commands">
<h2>Related commands<a class="headerlink" href="#related-commands" title="Permalink to this headline"></a></h2>
<p><a class="reference internal" href="fix_nh.html"><em>fix npt</em></a>, <a class="reference internal" href="minimize.html"><em>minimize</em></a></p>
</div>
<div class="section" id="default">
<h2>Default<a class="headerlink" href="#default" title="Permalink to this headline"></a></h2>
<p>The keyword defaults are dilate = all, vmax = 0.0001, nreset = 0.</p>
<hr class="docutils" />
<p id="parrinello1981"><strong>(Parrinello1981)</strong> Parrinello and Rahman, J Appl Phys, 52, 7182 (1981).</p>
</div>
</div>
</div>
</div>
<footer>
<hr/>
<div role="contentinfo">
<p>
&copy; Copyright .
</p>
</div>
Built with <a href="http://sphinx-doc.org/">Sphinx</a> using a <a href="https://github.com/snide/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>.
</footer>
</div>
</div>
</section>
</div>
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT:'./',
VERSION:'15 May 2015 version',
COLLAPSE_INDEX:false,
FILE_SUFFIX:'.html',
HAS_SOURCE: true
};
</script>
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
<script type="text/javascript" src="_static/sphinxcontrib-images/LightBox2/lightbox2/js/jquery-1.11.0.min.js"></script>
<script type="text/javascript" src="_static/sphinxcontrib-images/LightBox2/lightbox2/js/lightbox.min.js"></script>
<script type="text/javascript" src="_static/sphinxcontrib-images/LightBox2/lightbox2-customize/jquery-noconflict.js"></script>
<script type="text/javascript" src="_static/js/theme.js"></script>
<script type="text/javascript">
jQuery(function () {
SphinxRtdTheme.StickyNav.enable();
});
</script>
</body>
</html>
c4science · Help