A Library of Lattices and Graphs

As similar lattices, finite lattices and graphs will appear in many models and simulations, it is of advantage to define commonly used lattices and graphs in a “library”, and just refer to them by name, e.g. as input parameter to a simulation. The first step is to divide the definition of the <LATTICEGRAPH> into four parts:

<verbatim>
<LATTICE name="square" dimension="2">
<BASIS>
    <VECTOR> 1 0 </VECTOR>
    <VECTOR> 0 1 </VECTOR>
</BASIS>
</LATTICE>

<FINITELATTICE name="rectangular periodic" dimension="2">
<LATTICE ref="square"/>
<PARAMETER name="L" />
<PARAMETER name="W" default="L" />
<EXTENT dimension="1" size="L"/>
<EXTENT dimension="2" size="W"/>
<BOUNDARY type="periodic"/>
</FINITELATTICE>

<UNITCELL name="complex example" dimension="2" vertices="2">
<VERTEX id="1" type="0"><COORDINATE> 0.3 0.7 </COORDINATE></VERTEX>
<VERTEX id="2" type="1"><COORDINATE> 0.6 0.3 </COORDINATE></VERTEX>
<EDGE><SOURCE vertex="1"/><TARGET vertex="1" offset="1 0"/></EDGE>
<EDGE><SOURCE vertex="1"/><TARGET vertex="1" offset="0 1"/></EDGE>
<EDGE><SOURCE vertex="1"/><TARGET vertex="2"/></EDGE>
</UNITCELL>
<LATTICEGRAPH>
<FINITELATTICE ref="rectangular periodic"/>
<UNITCELL ref="complex example"/>
</LATTICEGRAPH>

Here we made use of the predefinition of named lattices and unit cells (e.g. in a library), which we can then combine be referencing them in the <LATTICEGRAPH> element. Now we can define a <LATTICES> element, which can contain any number of <LATTICE>, <FINITELATTICE>, <UNITCELL>, <GRAPH> and <LATTICEGRAPH> elements. For example

<LATTICES>
<LATTICE name="chain lattice" dimension="1">
<BASIS><VECTOR>1</VECTOR></BASIS>
</LATTICE>
<LATTICE name="square lattice" dimension="2">
<PARAMETER name="a" default="1"/>
<BASIS><VECTOR>a 0</VECTOR><VECTOR>0 a</VECTOR></BASIS>
</LATTICE>
<LATTICE name="simple cubic lattice" dimension="3">
<PARAMETER name="a" default="1"/>
<BASIS>
    <VECTOR>a 0 0</VECTOR>
    <VECTOR>0 a 0</VECTOR>
    <VECTOR>0 0 a</VECTOR>
</BASIS>
</LATTICE>
<UNITCELL name="simple1d" dimension="1">
<VERTEX/>
<EDGE><SOURCE vertex="1" offset="0"/><TARGET vertex="1" offset="1"/></EDGE>
</UNITCELL>
<UNITCELL name="simple2d" dimension="2">
<VERTEX/>
<EDGE><SOURCE vertex="1" offset="0 0"/><TARGET vertex="1" offset="0 1"/></EDGE>
<EDGE><SOURCE vertex="1" offset="0 0"/><TARGET vertex="1" offset="1 0"/></EDGE>
</UNITCELL>
<UNITCELL name="simple3d" dimension="3" vertices="1">
<VERTEX/>
<EDGE><SOURCE vertex="1"/><TARGET vertex="1" offset="1 0 0"/></EDGE>
<EDGE><SOURCE vertex="1"/><TARGET vertex="1" offset="0 1 0"/></EDGE>
<EDGE><SOURCE vertex="1"/><TARGET vertex="1" offset="0 0 1"/></EDGE>
</UNITCELL>
<LATTICEGRAPH name = "square lattice 3x3">
<FINITELATTICE>
    <LATTICE ref="square lattice"/>
    <EXTENT dimension="1" size="3"/>
    <EXTENT dimension="2" size="3"/>
    <BOUNDARY dimension="1" type="periodic"/>
    <BOUNDARY dimension="2" type="open"/>
</FINITELATTICE>
<UNITCELL ref="simple2d"/>
</LATTICEGRAPH>
<LATTICEGRAPH name = "dimer">
<FINITELATTICE>
    <LATTICE ref="chain lattice"/>
    <EXTENT dimension="1" size="2"/>
    <BOUNDARY type="open"/>
</FINITELATTICE>
<UNITCELL ref="simple1d"/>
</LATTICEGRAPH>
<LATTICEGRAPH name = "simple cubic lattice">
<FINITELATTICE>
    <LATTICE ref="simple cubic lattice"/>
    <PARAMETER name="W" default="L"/>
    <PARAMETER name="H" default="W"/>
    <EXTENT dimension="1" size="L"/>
    <EXTENT dimension="2" size="W"/>
    <EXTENT dimension="3" size="H"/>
    <BOUNDARY type="periodic"/>
</FINITELATTICE>
<UNITCELL ref="simple3d"/>
</LATTICEGRAPH>
<LATTICEGRAPH name = "chain lattice">
<FINITELATTICE>
    <LATTICE ref="chain lattice"/>
    <EXTENT dimension="1" size ="L"/>
    <BOUNDARY type="periodic"/>
</FINITELATTICE>
<UNITCELL ref="simple1d"/>
</LATTICEGRAPH>
<LATTICEGRAPH name = "anisotropic square lattice">
<FINITELATTICE>
    <LATTICE ref="square lattice"/>
    <PARAMETER name="W" default="L"/>
    <EXTENT dimension="1" size="L"/>
    <EXTENT dimension="2" size="W"/>
    <BOUNDARY type="periodic"/>
</FINITELATTICE>
<UNITCELL ref="anisotropic2d"/>
</LATTICEGRAPH>
</LATTICES>

Lattices and Graphs Check Lattice Graph