Homec4science
Diffusion lammps (lammm-master-deprecated) Browse

lammps/doc/pair_bop.html
311649293d08lammm-master-deprecated

Last Change
Raw File

pair_bop.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>pair_style bop 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>pair_style bop command</li>
<li class="wy-breadcrumbs-aside">
<a href="http://lammps.sandia.gov">Website</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="pair-style-bop-command">
<span id="index-0"></span><h1>pair_style bop command<a class="headerlink" href="#pair-style-bop-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>pair_style bop keyword ...
</pre></div>
</div>
<ul class="simple">
<li>zero or more keywords may be appended</li>
<li>keyword = <em>save</em></li>
</ul>
<div class="highlight-python"><div class="highlight"><pre>save = pre-compute and save some values
</pre></div>
</div>
</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>pair_style bop
pair_coeff * * ../potentials/CdTe_bop Cd Te
pair_style bop save
pair_coeff * * ../potentials/CdTe.bop.table Cd Te Te
comm_modify cutoff 14.70
</pre></div>
</div>
</div>
<div class="section" id="description">
<h2>Description<a class="headerlink" href="#description" title="Permalink to this headline"></a></h2>
<p>The <em>bop</em> pair style computes Bond-Order Potentials (BOP) based on
quantum mechanical theory incorporating both sigma and pi bondings.
By analytically deriving the BOP from quantum mechanical theory its
transferability to different phases can approach that of quantum
mechanical methods. This potential is similar to the original BOP
developed by Pettifor (<span class="xref std std-ref">Pettifor_1</span>,
<span class="xref std std-ref">Pettifor_2</span>, <span class="xref std std-ref">Pettifor_3</span>) and later updated
by Murdick, Zhou, and Ward (<a class="reference internal" href="#murdick"><span>Murdick</span></a>, <a class="reference internal" href="#ward"><span>Ward</span></a>).
Currently, BOP potential files for these systems are provided with
LAMMPS: AlCu, CCu, CdTe, CdTeSe, CdZnTe, CuH, GaAs. A sysstem with
only a subset of these elements, including a single element (e.g. C or
Cu or Al or Ga or Zn or CdZn), can also be modeled by using the
appropriate alloy file and assigning all atom types to the
singleelement or subset of elements via the pair_coeff command, as
discussed below.</p>
<p>The BOP potential consists of three terms:</p>
<img alt="_images/pair_bop.jpg" class="align-center" src="_images/pair_bop.jpg" />
<p>where phi_ij(r_ij) is a short-range two-body function representing the
repulsion between a pair of ion cores, beta_(sigma,ij)(r_ij) and
beta_(sigma,ij)(r_ij) are respectively sigma and pi bond ingtegrals,
THETA_(sigma,ij) and THETA_(pi,ij) are sigma and pi bond-orders, and
U_prom is the promotion energy for sp-valent systems.</p>
<p>The detailed formulas for this potential are given in Ward
(<a class="reference internal" href="#ward"><span>Ward</span></a>); here we provide only a brief description.</p>
<p>The repulsive energy phi_ij(r_ij) and the bond integrals
beta_(sigma,ij)(r_ij) and beta_(phi,ij)(r_ij) are functions of the
interatomic distance r_ij between atom i and j. Each of these
potentials has a smooth cutoff at a radius of r_(cut,ij). These
smooth cutoffs ensure stable behavior at situations with high sampling
near the cutoff such as melts and surfaces.</p>
<p>The bond-orders can be viewed as environment-dependent local variables
that are ij bond specific. The maximum value of the sigma bond-order
(THETA_sigma) is 1, while that of the pi bond-order (THETA_pi) is 2,
attributing to a maximum value of the total bond-order
(THETA_sigma+THETA_pi) of 3. The sigma and pi bond-orders reflect the
ubiquitous single-, double-, and triple- bond behavior of
chemistry. Their analytical expressions can be derived from tight-
binding theory by recursively expanding an inter-site Green&#8217;s function
as a continued fraction. To accurately represent the bonding with a
computationally efficient potential formulation suitable for MD
simulations, the derived BOP only takes (and retains) the first two
levels of the recursive representations for both the sigma and the pi
bond-orders. Bond-order terms can be understood in terms of molecular
orbital hopping paths based upon the Cyrot-Lackmann theorem
(<span class="xref std std-ref">Pettifor_1</span>). The sigma bond-order with a half-full
valence shell is used to interpolate the bond-order expressiont that
incorporated explicite valance band filling. This pi bond-order
expression also contains also contains a three-member ring term that
allows implementation of an asymmetric density of states, which helps
to either stabilize or destabilize close-packed structures. The pi
bond-order includes hopping paths of length 4. This enables the
incorporation of dihedral angles effects.</p>
<div class="admonition warning">
<p class="first admonition-title">Warning</p>
<p class="last">Note that unlike for other potentials, cutoffs for BOP
potentials are not set in the pair_style or pair_coeff command; they
are specified in the BOP potential files themselves. Likewise, the
BOP potential files list atomic masses; thus you do not need to use
the <a class="reference internal" href="mass.html"><em>mass</em></a> command to specify them. Note that for BOP
potentials with hydrogen, you will likely want to set the mass of H
atoms to be 10x or 20x larger to avoid having to use a tiny timestep.
You can do this by using the <a class="reference internal" href="mass.html"><em>mass</em></a> command after using the
<code class="xref doc docutils literal"><span class="pre">pair_coeff</span></code> command to read the BOP potential
file.</p>
</div>
<p>One option can be specified as a keyword with the pair_style command.</p>
<p>The <em>save</em> keyword gives you the option to calculate in advance and
store a set of distances, angles, and derivatives of angles. The
default is to not do this, but to calculate them on-the-fly each time
they are needed. The former may be faster, but takes more memory.
The latter requires less memory, but may be slower. It is best to
test this option to optimize the speed of BOP for your particular
system configuration.</p>
<hr class="docutils" />
<p>Only a single pair_coeff command is used with the <em>bop</em> style which
specifies a BOP potential file, with parameters for all needed
elements. These are mapped to LAMMPS atom types by specifying
N additional arguments after the filename in the pair_coeff command,
where N is the number of LAMMPS atom types:</p>
<ul class="simple">
<li>filename</li>
<li>N element names = mapping of BOP elements to atom types</li>
</ul>
<p>As an example, imagine the CdTe.bop file has BOP values for Cd
and Te. If your LAMMPS simulation has 4 atoms types and you want the
1st 3 to be Cd, and the 4th to be Te, you would use the following
pair_coeff command:</p>
<div class="highlight-python"><div class="highlight"><pre>pair_coeff * * CdTe Cd Cd Cd Te
</pre></div>
</div>
<p>The 1st 2 arguments must be * * so as to span all LAMMPS atom types.
The first three Cd arguments map LAMMPS atom types 1,2,3 to the Cd
element in the BOP file. The final Te argument maps LAMMPS atom type
4 to the Te element in the BOP file.</p>
<p>BOP files in the <em>potentials</em> directory of the LAMMPS distribution
have a &#8221;.bop&#8221; suffix. The potentials are in tabulated form containing
pre-tabulated pair functions for phi_ij(r_ij), beta_(sigma,ij)(r_ij),
and beta_pi,ij)(r_ij).</p>
<p>The parameters/coefficients format for the different kinds of BOP
files are given below with variables matching the formulation of Ward
(<a class="reference internal" href="#ward"><span>Ward</span></a>) and Zhou (<a class="reference internal" href="pair_polymorphic.html#zhou"><span>Zhou</span></a>). Each header line containing a
&#8221;:&#8221; is preceded by a blank line.</p>
<hr class="docutils" />
<p><strong>No angular table file format</strong>:</p>
<p>The parameters/coefficients format for the BOP potentials input file
containing pre-tabulated functions of g is given below with variables
matching the formulation of Ward (<a class="reference internal" href="#ward"><span>Ward</span></a>). This format also
assumes the angular functions have the formulation of (<a class="reference internal" href="#ward"><span>Ward</span></a>).</p>
<ul class="simple">
<li>Line 1: # elements N</li>
</ul>
<p>The first line is followed by N lines containing the atomic
number, mass, and element symbol of each element.</p>
<p>Following the definition of the elements several global variables for
the tabulated functions are given.</p>
<ul class="simple">
<li>Line 1: nr, nBOt (nr is the number of divisions the radius is broken
into for function tables and MUST be a factor of 5; nBOt is the number
of divisions for the tabulated values of THETA_(S,ij)</li>
<li>Line 2: delta_1-delta_7 (if all are not used in the particular</li>
<li>formulation, set unused values to 0.0)</li>
</ul>
<p>Following this N lines for e_1-e_N containing p_pi.</p>
<ul class="simple">
<li>Line 3: p_pi (for e_1)</li>
<li>Line 4: p_pi (for e_2 and continues to e_N)</li>
</ul>
<p>The next section contains several pair constants for the number of
interaction types e_i-e_j, with i=1-&gt;N, j=i-&gt;N</p>
<ul class="simple">
<li>Line 1: r_cut (for e_1-e_1 interactions)</li>
<li>Line 2: c_sigma, a_sigma, c_pi, a_pi</li>
<li>Line 3: delta_sigma, delta_pi</li>
<li>Line 4: f_sigma, k_sigma, delta_3 (This delta_3 is similar to that of
the previous section but is interaction type dependent)</li>
</ul>
<p>The next section contains a line for each three body interaction type
e_j-e_i-e_k with i=0-&gt;N, j=0-&gt;N, k=j-&gt;N</p>
<ul class="simple">
<li>Line 1: g_(sigma0), g_(sigma1), g_(sigma2) (These are coefficients for
g_(sigma,jik)(THETA_ijk) for e_1-e_1-e_1 interaction. <a class="reference internal" href="#ward"><span>Ward</span></a>
contains the full expressions for the constants as functions of
b_(sigma,ijk), p_(sigma,ijk), u_(sigma,ijk))</li>
<li>Line 2: g_(sigma0), g_(sigma1), g_(sigma2) (for e_1-e_1-e_2)</li>
</ul>
<p>The next section contains a block for each interaction type for the
phi_ij(r_ij). Each block has nr entries with 5 entries per line.</p>
<ul class="simple">
<li>Line 1: phi(r1), phi(r2), phi(r3), phi(r4), phi(r5) (for the e_1-e_1
interaction type)</li>
<li>Line 2: phi(r6), phi(r7), phi(r8), phi(r9), phi(r10) (this continues
until nr)</li>
<li>...</li>
<li>Line nr/5_1: phi(r1), phi(r2), phi(r3), phi(r4), phi(r5), (for the
e_1-e_1 interaction type)</li>
</ul>
<p>The next section contains a block for each interaction type for the
beta_(sigma,ij)(r_ij). Each block has nr entries with 5 entries per
line.</p>
<ul class="simple">
<li>Line 1: beta_sigma(r1), beta_sigma(r2), beta_sigma(r3), beta_sigma(r4),
beta_sigma(r5) (for the e_1-e_1 interaction type)</li>
<li>Line 2: beta_sigma(r6), beta_sigma(r7), beta_sigma(r8), beta_sigma(r9),
beta_sigma(r10) (this continues until nr)</li>
<li>...</li>
<li>Line nr/5+1: beta_sigma(r1), beta_sigma(r2), beta_sigma(r3),
beta_sigma(r4), beta_sigma(r5) (for the e_1-e_2 interaction type)</li>
</ul>
<p>The next section contains a block for each interaction type for
beta_(pi,ij)(r_ij). Each block has nr entries with 5 entries per line.</p>
<ul class="simple">
<li>Line 1: beta_pi(r1), beta_pi(r2), beta_pi(r3), beta_pi(r4), beta_pi(r5)
(for the e_1-e_1 interaction type)</li>
<li>Line 2: beta_pi(r6), beta_pi(r7), beta_pi(r8), beta_pi(r9),
beta_pi(r10) (this continues until nr)</li>
<li>...</li>
<li>Line nr/5+1: beta_pi(r1), beta_pi(r2), beta_pi(r3), beta_pi(r4),
beta_pi(r5) (for the e_1-e_2 interaction type)</li>
</ul>
<p>The next section contains a block for each interaction type for the
THETA_(S,ij)((THETA_(sigma,ij))^(1/2), f_(sigma,ij)). Each block has
nBOt entries with 5 entries per line.</p>
<ul class="simple">
<li>Line 1: THETA_(S,ij)(r1), THETA_(S,ij)(r2), THETA_(S,ij)(r3),
THETA_(S,ij)(r4), THETA_(S,ij)(r5) (for the e_1-e_2 interaction type)</li>
<li>Line 2: THETA_(S,ij)(r6), THETA_(S,ij)(r7), THETA_(S,ij)(r8),
THETA_(S,ij)(r9), THETA_(S,ij)(r10) (this continues until nBOt)</li>
<li>...</li>
<li>Line nBOt/5+1: THETA_(S,ij)(r1), THETA_(S,ij)(r2), THETA_(S,ij)(r3),
THETA_(S,ij)(r4), THETA_(S,ij)(r5) (for the e_1-e_2 interaction type)</li>
</ul>
<p>The next section contains a block of N lines for e_1-e_N</p>
<ul class="simple">
<li>Line 1: delta^mu (for e_1)</li>
<li>Line 2: delta^mu (for e_2 and repeats to e_N)</li>
</ul>
<p>The last section contains more constants for e_i-e_j interactions with
i=0-&gt;N, j=i-&gt;N</p>
<ul class="simple">
<li>Line 1: (A_ij)^(mu*nu) (for e1-e1)</li>
<li>Line 2: (A_ij)^(mu*nu) (for e1-e2 and repeats as above)</li>
</ul>
<hr class="docutils" />
<p><strong>Angular spline table file format</strong>:</p>
<p>The parameters/coefficients format for the BOP potentials input file
containing pre-tabulated functions of g is given below with variables
matching the formulation of Ward (<a class="reference internal" href="#ward"><span>Ward</span></a>). This format also
assumes the angular functions have the formulation of (<a class="reference internal" href="pair_polymorphic.html#zhou"><span>Zhou</span></a>).</p>
<ul class="simple">
<li>Line 1: # elements N</li>
</ul>
<p>The first line is followed by N lines containing the atomic
number, mass, and element symbol of each element.</p>
<p>Following the definition of the elements several global variables for
the tabulated functions are given.</p>
<ul class="simple">
<li>Line 1: nr, ntheta, nBOt (nr is the number of divisions the radius is broken
into for function tables and MUST be a factor of 5; ntheta is the power of the
power of the spline used to fit the angular function; nBOt is the number
of divisions for the tabulated values of THETA_(S,ij)</li>
<li>Line 2: delta_1-delta_7 (if all are not used in the particular</li>
<li>formulation, set unused values to 0.0)</li>
</ul>
<p>Following this N lines for e_1-e_N containing p_pi.</p>
<ul class="simple">
<li>Line 3: p_pi (for e_1)</li>
<li>Line 4: p_pi (for e_2 and continues to e_N)</li>
</ul>
<p>The next section contains several pair constants for the number of
interaction types e_i-e_j, with i=1-&gt;N, j=i-&gt;N</p>
<ul class="simple">
<li>Line 1: r_cut (for e_1-e_1 interactions)</li>
<li>Line 2: c_sigma, a_sigma, c_pi, a_pi</li>
<li>Line 3: delta_sigma, delta_pi</li>
<li>Line 4: f_sigma, k_sigma, delta_3 (This delta_3 is similar to that of
the previous section but is interaction type dependent)</li>
</ul>
<p>The next section contains a line for each three body interaction type
e_j-e_i-e_k with i=0-&gt;N, j=0-&gt;N, k=j-&gt;N</p>
<ul class="simple">
<li>Line 1: g0, g1, g2... (These are coefficients for the angular spline
of the g_(sigma,jik)(THETA_ijk) for e_1-e_1-e_1 interaction. The
function can contain up to 10 term thus 10 constants. The first line
can contain up to five constants. If the spline has more than five
terms the second line will contain the remaining constants The
following lines will then contain the constants for the remainaing g0,
g1, g2... (for e_1-e_1-e_2) and the other three body
interactions</li>
</ul>
<p>The rest of the table has the same structure as the previous section
(see above).</p>
<hr class="docutils" />
<p><strong>Angular no-spline table file format</strong>:</p>
<p>The parameters/coefficients format for the BOP potentials input file
containing pre-tabulated functions of g is given below with variables
matching the formulation of Ward (<a class="reference internal" href="#ward"><span>Ward</span></a>). This format also
assumes the angular functions have the formulation of (<a class="reference internal" href="pair_polymorphic.html#zhou"><span>Zhou</span></a>).</p>
<ul class="simple">
<li>Line 1: # elements N</li>
</ul>
<p>The first two lines are followed by N lines containing the atomic
number, mass, and element symbol of each element.</p>
<p>Following the definition of the elements several global variables for
the tabulated functions are given.</p>
<ul class="simple">
<li>Line 1: nr, ntheta, nBOt (nr is the number of divisions the radius is broken
into for function tables and MUST be a factor of 5; ntheta is the number of
divisions for the tabulated values of the g angular function; nBOt is the number
of divisions for the tabulated values of THETA_(S,ij)</li>
<li>Line 2: delta_1-delta_7 (if all are not used in the particular</li>
<li>formulation, set unused values to 0.0)</li>
</ul>
<p>Following this N lines for e_1-e_N containing p_pi.</p>
<ul class="simple">
<li>Line 3: p_pi (for e_1)</li>
<li>Line 4: p_pi (for e_2 and continues to e_N)</li>
</ul>
<p>The next section contains several pair constants for the number of
interaction types e_i-e_j, with i=1-&gt;N, j=i-&gt;N</p>
<ul class="simple">
<li>Line 1: r_cut (for e_1-e_1 interactions)</li>
<li>Line 2: c_sigma, a_sigma, c_pi, a_pi</li>
<li>Line 3: delta_sigma, delta_pi</li>
<li>Line 4: f_sigma, k_sigma, delta_3 (This delta_3 is similar to that of
the previous section but is interaction type dependent)</li>
</ul>
<p>The next section contains a line for each three body interaction type
e_j-e_i-e_k with i=0-&gt;N, j=0-&gt;N, k=j-&gt;N</p>
<ul class="simple">
<li>Line 1: g(theta1), g(theta2), g(theta3), g(theta4), g(theta5) (for the e_1-e_1-e_1
interaction type)</li>
<li>Line 2: g(theta6), g(theta7), g(theta8), g(theta9), g(theta10) (this continues
until ntheta)</li>
<li>...</li>
<li>Line ntheta/5+1: g(theta1), g(theta2), g(theta3), g(theta4), g(theta5), (for the
e_1-e_1-e_2 interaction type)</li>
</ul>
<p>The rest of the table has the same structure as the previous section (see above).</p>
<hr class="docutils" />
<p><strong>Mixing, shift, table tail correction, restart</strong>:</p>
<p>This pair style does not support the <a class="reference internal" href="pair_modify.html"><em>pair_modify</em></a>
mix, shift, table, and tail options.</p>
<p>This pair style does not write its information to <a class="reference internal" href="restart.html"><em>binary restart files</em></a>, since it is stored in potential files. Thus, you
need to re-specify the pair_style and pair_coeff commands in an input
script that reads a restart file.</p>
<p>This pair style can only be used via the <em>pair</em> keyword of the
<a class="reference internal" href="run_style.html"><em>run_style respa</em></a> command. It does not support the
<em>inner</em>, <em>middle</em>, <em>outer</em> keywords.</p>
</div>
<hr class="docutils" />
<div class="section" id="restrictions">
<h2>Restrictions<a class="headerlink" href="#restrictions" title="Permalink to this headline"></a></h2>
<p>These pair styles are part of the MANYBODY package. They are only
enabled if LAMMPS was built with that package (which it is by default).
See the <a class="reference internal" href="Section_start.html#start-3"><span>Making LAMMPS</span></a> section for more
info.</p>
<p>These pair potentials require the <a class="reference internal" href="newton.html"><em>newtion</em></a> setting to be
&#8220;on&#8221; for pair interactions.</p>
<p>The CdTe.bop and GaAs.bop potential files provided with LAMMPS (see the
potentials directory) are parameterized for metal <a class="reference internal" href="units.html"><em>units</em></a>.
You can use the BOP potential with any LAMMPS units, but you would need
to create your own BOP potential file with coefficients listed in the
appropriate units if your simulation does not use &#8220;metal&#8221; units.</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="pair_coeff.html"><em>pair_coeff</em></a></p>
</div>
<div class="section" id="default">
<h2>Default<a class="headerlink" href="#default" title="Permalink to this headline"></a></h2>
<p>non-tabulated potential file, a_0 is non-zero.</p>
<hr class="docutils" />
<p id="pettofor-1"><strong>(Pettifor_1)</strong> D.G. Pettifor and I.I. Oleinik, Phys. Rev. B, 59, 8487
(1999).</p>
<p id="pettofor-2"><strong>(Pettifor_2)</strong> D.G. Pettifor and I.I. Oleinik, Phys. Rev. Lett., 84,
4124 (2000).</p>
<p id="pettofor-3"><strong>(Pettifor_3)</strong> D.G. Pettifor and I.I. Oleinik, Phys. Rev. B, 65, 172103
(2002).</p>
<p id="murdick"><strong>(Murdick)</strong> D.A. Murdick, X.W. Zhou, H.N.G. Wadley, D. Nguyen-Manh, R.
Drautz, and D.G. Pettifor, Phys. Rev. B, 73, 45206 (2006).</p>
<p id="ward"><strong>(Ward)</strong> D.K. Ward, X.W. Zhou, B.M. Wong, F.P. Doty, and J.A.
Zimmerman, Phys. Rev. B, 85,115206 (2012).</p>
<p id="zhou"><strong>(Zhou)</strong> X.W. Zhou, D.K. Ward, M. Foster (TBP).</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="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></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