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<center><b> Intrinsic Local Modes </b></center><br>
<center>
<applet code="MonatILMApplet.class" width="600" height="440">
<param name="minwidthpanel" value="300">
<center>
<i>Java</i> support is required for this interface. <br>
In order to see <i>Intrinsic Local Modes</i> applet<br>
you need a <i>Java</i> compatible browser.
</center>
</applet>
</center>
<hr>
<center><b> Intrinsic Local Modes </b></center>
<hr width="75%">
Consider a chain of particles of mass <i><b>m</b></i> where the
nearest-neighbors are connected by the anharmonic springs. The
anharmonic interparticle potential has the following form <br>
<center>
<img src="images/potential.gif" WIDTH="138" HEIGHT="34"> ,
</center>
where
<b><i>K</i><sub>2</sub>>0</b>
and
<b><i>K</i><sub>4</sub>>0</b>
are the harmonic and quartic anharmonic terms, respectively, and
<i><b>x</b></i> is the deviation of the spring's length from its
equilibrium value.
<p>
Such lattice supports intrinsic local modes (ILMs) with their
frequencies above the phonon band characterized by the maximal
harmonic plane waves frequency
<img align="middle" src="images/wmeql.gif" WIDTH="115" HEIGHT="31">.
<br>
The eigenvector of the intrinsic local mode can be found within the
rotating-wave apporximation (RWA) where the displacement of the
<i>n</i>th particle from its equilibrium position
<b><i>u<sub>n</sub></i></b>
is described by the following ansatz <br>
<center><nobr><img src="images/unrwa.gif" WIDTH="201" HEIGHT="26"></nobr></center> <br>
where
<img align="middle" src="images/alpha.gif" WIDTH="18" HEIGHT="16">
is the amplitude of the mode, and
<img align="middle" src="images/phin.gif" WIDTH="25" HEIGHT="25">
characterizes its ac displacement pattern. Substitution of the above
ansatz into the classical equations of motion <br>
<center><nobr><img src="images/motioneq.gif" WIDTH="399" HEIGHT="36"></nobr></center> <br>
allows one to find the mode eigenvector.
The ILM's eigenvector is a wave package which transfers to a lattice
envelope soliton in a limit of a weak anharmonicicty.
<p>
A similar ansatz can give the eigenvector of a moving ILM.
<p>
A more complete description of the intrinsic local modes you can find in
a review article: <br>
S. A. Kiselev, S. R. Bickham, and A. J. Sievers,
"Properties of Intrinsic Localized Modes in One-Dimensional Lattices",
<i>Comments Cond. Mat. Phys</i>, <b>17</b>, 135-173 (1995).
<hr width="75%">
The above <font color="red"><b><i>applet</i></b></font> allows you to
watch vibrating ILMs in the lattice of 15 particles with periodic
boundaries. The evolution of the chain is calculated by the
molecular-dynamics technique. The parameters of the lattice are the
following:
<nobr>
<b><i>m</i>=1,
<i>K</i><sub>2</sub>=1,
<i>K</i><sub>4</sub>=10</b>.
</nobr>
<br>
You can launch either an <i>Odd-Parity ILM</i>
(when a central particle has the highest amplitude)
or an <i>Even-Parity ILM</i>
(when two central particles have the highest and opposite amplitudes).
You can also launch a <i>Moving ILM</i>.
<p>
The time is shown in units of the shortest period of
small amplitude plane wave vibrations,
<img align="middle" src="images/tmeql.gif" WIDTH="115" HEIGHT="31">.
<p>
Energy is shown in arbitrary units. The <font color="red"><b>kinetic</b></font>
energy of the particle and the <font color="ccaa00"><b>potential</b></font>
energy of the bond are shown as the <font color="red"><b>red</b></font> and the
<font color="#ccaa00"><b>yellow</b></font> bars, respectively.
<p>
If you wait for a while you will see a spectrum of the particles'
vibrations. It will be shown in the left panel.
As the time of the evolution goes the spectrum resolution improves.
The frequency unit is the maximal plane wave frequency,
<img align="middle" src="images/wmeql.gif" WIDTH="115" HEIGHT="31">.
<hr>
<font size="-2">
Last modified: December 1, 1996<br>
<a HREF="http://www.lightlink.com/sergey">Sergey Kiselev</a>,
<i><a href="mailto:sergey@lightlink.com">sergey@lightlink.com</a></i><br>
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