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<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
|
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</font><font size="6">
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</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
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<div style="width:220px;float:left;text-align:center;font-weight:900;font-size:12px;">
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<a href="detail1.html">1. Analysis of cluster dynamics</a>
|
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</div>
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<div style="width:220px;float:left;text-align:center;">
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<a href="detail2.html"> 2. Clusters in external fields</a>
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</div>
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<div style="width:220px;float:left;text-align:center;">
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<a href="../tddft-md/formal.html"> 3. Theoretical developments </a>
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</a>
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<div id="WideContent">
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<div id="contentBoxWide">
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<div id="contentBoxHeader">
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<p> Analysis of cluster dynamics</p>
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</div>
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<div id="contentBoxContent">
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<!-- START CONTENT HERE -->
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<p>
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<img src="figs/na8p_mie.gif" width="250" align="right">
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<br><br>
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The basic dynamical property of a metal cluster is the optical
|
||||
absorption spectrum which has a pronounced collection of strength in
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the region of the Mie plasmon. TDLDA driven with small amplitude
|
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excitations allows to explore the optical response [<a href="../literatur.html#own1155">9</a>].
|
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The figure beneath shows results for Na<sub>8</sub><sup>+</sup> as
|
||||
example (taken from [<a href="../literatur.html#own1315"><font color="red">???</font></a>]) in comparison
|
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to experiment (upper panel) and CI calculations (<font color="red"><b>???</b></font>)(second from above).
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The overall position of the peak strength is nicely reproduced by all
|
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methods, even by the semiclassical approach. CI produces the most
|
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detailed spectrum. The green bars show the discrete spectrum as it
|
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emerges from the CI calculation, and the red curve results from Lorentzian
|
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smoothing which simulates to some extend the finite experimental
|
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resolution and thermal fluctuations. The enormous number of spectral
|
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lines (green) is due to electronic correlations which are absent in
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TDLDA. Nonetheless, the unavoidable smoothing overrules these details
|
||||
and makes TDLDA spectra competitive. It is noteworthy that also the
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semiclassical approximation (Vlasov-LDA) performs surprisingly well.
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This provides a good starting point for the subsequent applications
|
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in more energetic situations.
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<br>
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</p>
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<p>
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<br><br><br><br><br>
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<img src="figs/na_vgl_small.gif" width="400" align="left">
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<br><br>
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Laser induced direct photo-emission of electrons allows to conclude on
|
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the clusters single-electron states by measuring the photo-electron
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spectra (PES). TDLDA with appropriate self-interaction correction
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(SIC) [<a href="../literatur.html#own1252">277</a>] allows to simulate that
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process in detail [<a href="../literatur.html#own1227">251</a>] . The figure to
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the left shows two examples for two clusters which are nearly
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spherical (taken from [<a href="../literatur.html#own1285">304</a>]). The arrows
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indicate the level classification according to principal quantum
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number and angular momentum. The PES depend, of course, on the
|
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direction of emission (checked here are the case where the cluster
|
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axis is ``perpendicular'' or ``parallel'' to the laser
|
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polarization). Experiments take an average over all direction.
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The summed theoretical PES agree fairly well with the data.
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</p>
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<p>
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<br><br><br><br>
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<img src="figs/na41p+3_comb.gif" width="350" align="right">
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<br><br><br><br><br>
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Pump and probe (P&P) techniques are an extremely powerful tool for
|
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time-resolved analysis. The complexity of clusters allows an
|
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enormous manifold of P&P scenarios. The figure to the right sketches
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a simple and robust scenario for a nearly spherical cluster, actually
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Na<sub>41</sub><sup>+</sup> [<a href="../literatur.html#own1246">290</a>]. The idea is to map the
|
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radius vibrations of the cluster by an off-resonant laser pulse. The
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pump pulse ionizes the Na<sub>41</sub><sup>+</sup> within 50 fs by three more charge
|
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units, see second panel from top for dipole response (black line) and first
|
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panel for ionization. The generated Coulomb pressure drives
|
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oscillations of the radius <i>R<sub>ion</sub></i>, shown in the lowest panel.
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</br></br>
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The Mie plasmon
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frequency depends on the cluster extension as w<sub>Mie</sub>~
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R<sup>-3/2</sup> and oscillates with opposite phase, see third panel. Thus
|
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the changing distance to the off-resonant laser frequency (green
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horizontal line) modulates the dipole response to probe pulses
|
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accordingly (second panel) which, in turn, yields changing ionization
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through the probe pulse as function of delay time. The final
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ionization (upper panel) becomes then a direct map of the underlying
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breathing oscillations of the cluster.
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</p>
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<center>
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<table width="70%">
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<tr>
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<td align="right">
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<li style="margin-top:1px;border-top:1px solid #B0C4DE; "><a href="../index.html">Home</a></li>
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<li><a href="../staff.html">Staff</a></li>
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<li><a href="../publications.html">Publications/Talks</a></li>
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</div>
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</div>
|
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<div id="image">
|
||||
<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
|
||||
</font><font size="6">
|
||||
</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
|
||||
|
||||
</div>
|
||||
|
||||
<a name="oben">
|
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<div id="content">
|
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<div style="margin:15px;width:770px;border:1px solid gray;float:left;font-size:10px;">
|
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<div style="width:220px;float:left;text-align:center;">
|
||||
<a href="../analysis/detail1.html">1. Analysis of cluster dynamics</a>
|
||||
</div>
|
||||
<div style="width:220px;float:left;text-align:center;font-weight:900;font-size:12px;">
|
||||
<a href="../analysis/detail2.html"> 2. Clusters in external fields</a>
|
||||
</div>
|
||||
<div style="width:220px;float:left;text-align:center;">
|
||||
<a href="../tddft-md/formal.html"> 3. Theoretical developments </a>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
</a>
|
||||
|
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<div id="WideContent">
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<div id="contentBoxWide">
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<div id="contentBoxHeader">
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<p> Clusters in strong external perturbations</p>
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</div>
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<div id="contentBoxContent">
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<!-- START CONTENT HERE -->
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<br>
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<p><img src="figs/na8_nacl_SHG.gif" align="right" width="300">
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Many experiments are done for clusters in contact with a
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substrate. The strong interface interaction modifies the cluster and
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theoretical simulations become more involved. However, some features
|
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can only be explored in connection with a substrate. E.g., the
|
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symmetry breaking through a surface gives access to second-harmonic
|
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generation (SHG). </br></br>
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|
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The figure beneath shows the results from a TDLDA
|
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simulation of SHG for Na<sub>8</sub> attached to a NaCl surface
|
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[<a href="../literatur.html#own1224">248</a>]. The spectra resulting
|
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from
|
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irradiation with a 1.4 eV pulse shows nicely the peaks at multiple
|
||||
frequencies. The SHG signal can be enhanced by increasing the laser
|
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intensity. This, however, breaks down at some point where the signals
|
||||
are substantially broadened. This is caused by a large ionization
|
||||
which spoils the otherwise clean dipole response of metal clusters.
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</p>
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<br>
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<br>
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<p>
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<img src="figs/na6_ar384d_deposit.gif" align="left" width="300">TDLDA
|
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coupled with molecular dynamics (MD) for ionic motion is a very
|
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powerfull tool to describe cluster dynamics. One application is
|
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cluster deposition which is illustrated in the figure on the left. It shows
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Na<sub>6</sub> impinging on an Ar surface (see [<a
|
||||
href="../literatur.html#own1303">328</a>] for further details). The substrate consists of
|
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six layers of Ar
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taken from an appropriate cut of the Ar fcc structure. The Na<sub>6</sub>
|
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cluster consist in a ring of 5 ions topped by one ion on the symmetry
|
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axix. The Na<sub>6</sub> approaches the surface with the symmetry axis
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in <i>z</i>
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direction (=perpendicular) and the top ion facing away from the
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surface. </br></br>
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The upper panel shows the evolution of the <i>z</i>
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coordinates,
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Na ions in red and Ar atoms in green. The cluster is immediately
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stopped by the surface. A large fraction of impact momentum is
|
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transferred at once to the substrate and propagates with velocity of
|
||||
light through the layers. The large dissipation through energy
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transfer and intrinsic cluster excitation leads to catching of the
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cluster by the subtrate. The kinetic energies in the lower panel
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confirm the dramatic and very fast energy exchange at the moment of
|
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first impact. Another fraction of energy, not shown in the figure, is
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turned into the large shape changes.
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</p>
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<br><br>
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<p>
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Clusters in the strong fields of extremely intense lasers show a much
|
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different dynamics. Core electrons can be released and contribute
|
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strongly to the process. The detailed description at the fully quantum
|
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mechanical level of TDLDA becomes untractable. However, the
|
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excitations involved validate classical approaches. </br></br>
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<img src="figs/MD_fig5.gif" align="right" width="300">
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The figure to the
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right shows the result of a molecular dynamics simulation of
|
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electronic and ionic dynamics of Na<sub>41</sub><sup>+</sup> under the
|
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influence of
|
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strong laser fields [<a href="../literatur.html#own1308">332</a>].
|
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Ionization is
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drawn as function of laser intensity. One sees a sharp kink at a
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critical intensity of I = 10<sup>16</sup> W/cm<sup>2</sup>. This
|
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threshold value is
|
||||
explained by the fact that the Coulomb force from the laser field
|
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at the threshold just equals the binding forces of the core electrons. The increase is
|
||||
due to the core electrons which now start to participate in the
|
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dynamics. This view is illustrated by separating the contributions from
|
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valence (green line) and core electrons (red line). There is indeed zero
|
||||
emission from core electrons up to I = 10<sup>16</sup> W/cm<sup>2</sup>
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and the
|
||||
strong increase above that critical intensity is exclusively due to
|
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the contribution from core electrons.
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<br>
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<br>
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<br>
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</p>
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<center>
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<div id="image">
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<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
|
||||
</font><font size="6">
|
||||
</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
|
||||
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</div>
|
||||
|
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<div id="content">
|
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|
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<div style="margin:15px;width:770px;border:1px solid gray;float:left;font-size:10px;">
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<div style="width:220px;float:left;text-align:center;">
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<a href="intro.html">1. What are clusters? </a>
|
||||
</div>
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||||
<div style="width:220px;float:left;text-align:center;font-weight:900;font-size:12px;">
|
||||
<a href="dynamics.html"> 2. Why study cluster dynamics?</a>
|
||||
</div>
|
||||
<div style="width:220px;float:left;text-align:center;">
|
||||
<a href="ourdynamics/our_dynamics.html"> 3. How we deal with cluster dynamics </a>
|
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</div>
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</div>
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<div id="WideContent">
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<div id="contentBoxWide">
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<div id="contentBoxHeader">
|
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Why study cluster dynamics ?
|
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</div>
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<div id="contentBoxContent">
|
||||
|
||||
|
||||
|
||||
|
||||
<br>
|
||||
Cluster dynamics represents a fast developing area of cluster
|
||||
physics. The field covers various phenomena with impact both
|
||||
on fundamental cluster research and on potential applications, for
|
||||
example in cluster engineering. We show in the following a few
|
||||
emblematic examples of the field. <br>
|
||||
<br>
|
||||
Clusters are made of electrons and ions. Both are charged particles
|
||||
which can then be excited by an electromagnetic field. A favorite and
|
||||
fashionable tool of investigation of cluster dynamics is thus provided
|
||||
by lasers. The latter deliver to the system electromagnetic
|
||||
pulses whose characteristics can be tailored
|
||||
almost at will both in terms of deposited energy and time profile. Some
|
||||
experiments are also performed by means of collisions between clusters
|
||||
and highly charged projectiles, as delivered by heavy-ion sources
|
||||
and facilities. <br> <br>
|
||||
In both cases (lasers, ions), electrons and ions
|
||||
strongly couple to the delivered
|
||||
electromagnetic field, but at different time scales. Indeed, electrons
|
||||
are light particles which thus react and evolve at short time scales,
|
||||
typically the fs (10<sup>-15</sup>s). In turn, the much heavier ions
|
||||
(several thousand times heavier than electrons) evolve on a much longer time scale of
|
||||
order 100-1000 fs. Of course these time
|
||||
scales are not fully independent of each other, through the natural
|
||||
coupling between electrons and ions, and the actual relation
|
||||
between these two time scales somewhat depends on the deposited energy.<br>
|
||||
Let us illustrate the two coupled electron and ion dynamics on a few
|
||||
examples. <br>
|
||||
<br>
|
||||
Metal clusters couple especially well to an electromagnetic
|
||||
perturbation because their electrons are only moderately bound to the
|
||||
ionic cores. They thus react strongly, for instance to a
|
||||
laser excitation. The response, called "optical response" (because the emitted light
|
||||
is to a large extent visible), is the
|
||||
fingerprint of this coupling. <br><br>
|
||||
The optical response
|
||||
is caused by the collective oscillations of the cluster electrons
|
||||
following an excitation by the electromagnetic pulse. The electron
|
||||
cloud, elastically bound to the ionic cores, oscillates around
|
||||
them, once displaced from its original position, and radiates
|
||||
visible light. This collective response provides a signature of the
|
||||
underlying structure of the irradiated cluster. The "color" of
|
||||
the irradiated cluster, for example, significantly depends on the size
|
||||
of the cluster. We thus have here an example where electron dynamics
|
||||
provides a direct means of investigation of
|
||||
structure properties. The case is illustrated on Figure 1 where the
|
||||
frequency (the color) of the optical response of mixed gold and silver
|
||||
clusters (embedded in an inert glass) is plotted as a function of
|
||||
cluster size. One can see that the cluster color significantly depends
|
||||
on size. It means that such golden inclusions in a glass (of course of
|
||||
various sizes) would deliver a variety of colors, as it was already
|
||||
well-known by ancient artcrafters (see in the cluster <a
|
||||
href="intro.html">introduction</a> page).<br>
|
||||
<br>
|
||||
<table style="width: 100%; text-align: left;" border="0" cellpadding="2"
|
||||
cellspacing="2">
|
||||
<tbody>
|
||||
<tr>
|
||||
<td style="vertical-align: top;"><img
|
||||
alt="silver optical response" src="images/opt2.jpg"
|
||||
style="width: 620px;"></td>
|
||||
<td style="vertical-align: middle;">Fig.1: Optical response of mixed
|
||||
gold and silver clusters, embedded in inert glass, as a function of
|
||||
size (<a href="javascript:lade(1)">details</a>).
|
||||
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
<br>
|
||||
<br>
|
||||
The optical response is a rather simple process, involving mostly electrons
|
||||
(although ions may also interfere, for example when temperatures
|
||||
are involved). Another interesting case is provided by cluster fission
|
||||
where ionic motion then plays a key role. When sufficiently charged
|
||||
(for example after a laser irradiation and escape of several electrons)
|
||||
a metal cluster may become unstable with respect to fission, exactly as
|
||||
massive atomic nuclei. It then becomes preferable for the
|
||||
system to break into two smaller clusters, the fission fragments. In
|
||||
such processes electrons play a relatively passive role (once the
|
||||
system is properly charged) and tend to follow the ions during the fission
|
||||
process. Fission is furthermore characterized by a potential barrier
|
||||
over which the system has to pass in order to evolve from one
|
||||
piece to two. This is illustrated in Figure 2 where the
|
||||
fission barrier of a small metal cluster is shown, together with the different shapes
|
||||
taken by the system at different deformations (from the smallest: 1
|
||||
piece, to the largest: 2 pieces). Figure 3 presents an example of
|
||||
fission dynamics for another cluster.<br>
|
||||
<br>
|
||||
<table style="width: 100%; text-align: left;" border="0" cellpadding="2"
|
||||
cellspacing="2">
|
||||
<tbody>
|
||||
<tr>
|
||||
<td style="vertical-align: middle;"><img alt="K12++ fission"
|
||||
src="images/fission2.jpg" style="width: 420px;"><br>
|
||||
</td>
|
||||
<td style="vertical-align: middle;">Fig.2: Potential energy of K<sub>12</sub><sup>++</sup>
|
||||
as a function of the extension of the cluster (<a
|
||||
href="javascript:lade(2)">details</a>).<br>
|
||||
<br>
|
||||
Fig.3: Movie of the fission of Na<sub>14</sub>, induced by a laser
|
||||
irradiation (<a href="javascript:lade(3)">details</a>).<br>
|
||||
<object data="images/film_fission.mpg" type="video/mpeg" width="300" height="300">
|
||||
<param name="src" value="film_fission.mpg">
|
||||
<object data="images/film_fission.mpg" type="video/mpeg" width="300" height="300">
|
||||
<param name="src" value="film_fission.mpg">
|
||||
<br><i>
|
||||
Your browser is unable to open the video. You need a suitable MPEG plugin to watch
|
||||
it inside this window.
|
||||
</i>
|
||||
</object>
|
||||
|
||||
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
<br>
|
||||
<br>
|
||||
A most interesting situation is attained when both electron and
|
||||
ion dynamics explicitely couple to produce elaborate dynamical
|
||||
scenarios. This is illustrated on the third example we want to present
|
||||
here. We consider the case of embedded silver clusters, the shape of
|
||||
which can be tailored, as one can see on Figure 2. Metal clusters
|
||||
possess a specific frequency at which they couple to light (the optical
|
||||
response frequency seen above). If one shines a cluster with a laser
|
||||
precisely tuned at that frequency, one will so much excite the
|
||||
cluster that it will emit several electrons. This is a typical resonant
|
||||
behavior as is well known in any oscillating system. <!--Think for example to
|
||||
the case of noise associated to mechanical vibrations in a car or the
|
||||
search of a TV or a radio channel by tuning reception to the right
|
||||
frequency of the electromagnetic signal.--> <br> <br>
|
||||
In this case <!-- the case of irradiated
|
||||
clusters-->, if several electrons are stripped during the exposure to the
|
||||
laser, the cluster may become highly charged and will consequently
|
||||
expand because of the net charge acquired, as in fission. But full
|
||||
ionic expansion is hindered here by the fact that the cluster is
|
||||
included in a matrix. The final result is a somewhat expanded cluster.
|
||||
This expansion can be further analyzed by irradiating again the cluster
|
||||
and recording its optical response (pump and probe experiment). As seen above the optical response
|
||||
provides a signature of the cluster size. A variation in the optical
|
||||
response thus indicates a structure modification. This is exactly what
|
||||
one can see on Figure 4. The peak is at the same time broadened and
|
||||
shifted to a higher wavelength. Moreover the optical response depends
|
||||
on the laser polarization, reflecting a non-spherical shape of the
|
||||
cluster. The laser and its preferred coupling to the cluster has thus
|
||||
allowed to tailor the system shape. This allows to envision
|
||||
potential applications in laser assisted tailoring of materials at the
|
||||
nanometer scale. <br>
|
||||
<br>
|
||||
<table style="width: 100%; text-align: left;" border="0" cellpadding="1"
|
||||
cellspacing="0">
|
||||
<tbody>
|
||||
<tr>
|
||||
<td style="vertical-align: middle;" width="100"><img alt="Ag burning"
|
||||
src="images/burning2.jpg" width="600" ><br>
|
||||
</td>
|
||||
<td style="vertical-align: middle;" width="100">Fig.4: Optical response of
|
||||
silver clusters embedded in an inert glass (<a href="javascript:lade(4)">details</a>).<br>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
<br>
|
||||
|
||||
<center>
|
||||
<table width="70%">
|
||||
<tr>
|
||||
<td align="right">
|
||||
<a href="#top">Back to top </a>
|
||||
</td>
|
||||
</tr>
|
||||
</table>
|
||||
</center>
|
||||
|
||||
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="footer">
|
||||
<p></p>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,27 @@
|
|||
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<meta content="text/html; charset=ISO-8859-1"
|
||||
http-equiv="content-type">
|
||||
<title>Figure 1: Details</title>
|
||||
</head>
|
||||
<body>
|
||||
The left vertical axis, in eV, stands for the frequency of the optical
|
||||
response. The corresponding colors in visible light are depicted on the
|
||||
right of the
|
||||
figure. The large filled symbols are experimental data, while the small
|
||||
open symbols, joined
|
||||
by lines, come from theoretical calculations. From M.Gaudry <span
|
||||
style="font-style: italic;">et al.</span>, Phys. Rev. B <span
|
||||
style="font-weight: bold;">64</span>, 085407 (2001).<br>
|
||||
|
||||
<br>
|
||||
<center>
|
||||
<form>
|
||||
<input type="reset" value="Close window " onclick="window.close()">
|
||||
</form>
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,26 @@
|
|||
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<meta content="text/html; charset=ISO-8859-1"
|
||||
http-equiv="content-type">
|
||||
<title>Figure 2: Details</title>
|
||||
</head>
|
||||
<body>
|
||||
The extension is quantified by the distance between the fragment
|
||||
centers of mass. Three ionic configurations are depicted, corresponding
|
||||
to three stages in the fission process. The valence electron cloud is
|
||||
also represented for the last fissionning ionic configuration. From
|
||||
C. Brechignac <span style="font-style: italic;">et al.</span>, Phys.
|
||||
Rev. Lett. <span style="font-weight: bold;">72</span>, 1636 (1994).<br>
|
||||
|
||||
<br>
|
||||
<center>
|
||||
|
||||
<form>
|
||||
<input type="reset" value="Close window " onclick="window.close()">
|
||||
</form>
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,26 @@
|
|||
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<meta content="text/html; charset=ISO-8859-1"
|
||||
http-equiv="content-type">
|
||||
<title>Figure 3: Details</title>
|
||||
</head>
|
||||
<body>
|
||||
The laser (intensity of 5.10<sup>10</sup> W/cm<sup>2</sup>, frequency
|
||||
of 2.3 eV, FWHM of 36 fs) excites the Na<sub>14</sub> to the Na<sub>14</sub><sup>3+</sup>,
|
||||
which splits in Na<sub>8</sub><sup>+</sup> and Na<sub>6</sub><sup>2+</sup>
|
||||
after 1.2 ps. From P.M. Dinh, P.-G. Reinhard and E. Suraud<span
|
||||
style="font-style: italic;"></span>, J. Phys. B <span
|
||||
style="font-weight: bold;">38</span>, 1637 (2005).<br>
|
||||
|
||||
<br>
|
||||
<center>
|
||||
|
||||
<form>
|
||||
<input type="reset" value="Close window " onclick="window.close()">
|
||||
</form>
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,41 @@
|
|||
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<meta content="text/html; charset=ISO-8859-1"
|
||||
http-equiv="content-type">
|
||||
<title>
|
||||
Figure 4: Details
|
||||
</title>
|
||||
</head>
|
||||
<body>
|
||||
The initial embedded clusters are preferably spherical (yellow
|
||||
cartoon). After irradiation by a laser, the cluster expands and its
|
||||
shape changes to a
|
||||
non-spherical configuration (orange cartoon). The corresponding optical
|
||||
response splits in two parts (green and magenta dots) depending of the
|
||||
analyzing direction with respect to the laser polarization (in green:
|
||||
parallel direction; in magenta: perpendicular direction). From
|
||||
G. Seifert <span style="font-style: italic;">et al.</span>, Appl. Phys.
|
||||
B <span style="font-weight: bold;">71</span>, 795 (2000).<br>
|
||||
|
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|
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<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
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|
||||
Welcome to our website about the Theory of Cluster Dynamics. We
|
||||
are a collaboration between theoretical physics groups of the
|
||||
<a href="http://www.theorie2.physik.uni-erlangen.de" target="_blank">
|
||||
University of Erlangen</a> (Germany) and the
|
||||
<a href="http://www.lpt.ups-tlse.fr/" target="_blank">
|
||||
University Paul Sabatier
|
||||
of Toulouse</a> (France).
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|
||||
Here you can find
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||||
|
||||
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||||
<div id="infoBoxHeader"><p>Further Reading:</p>
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<div id="infoBoxContent">
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||||
<p>
|
||||
</p><center>
|
||||
<img src="images/buch.jpg" height="243px" width="170px"> </center><p></p>
|
||||
<center>
|
||||
<p>
|
||||
P.-G. Reinhard, E. Suraud<br>
|
||||
"Introduction to Cluster Dynamics"<br>
|
||||
Wiley-VCH (2003)<br>
|
||||
ISBN: 3527403450
|
||||
</p>
|
||||
</center>
|
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</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
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<div style="margin:15px;width:770px;border:1px solid gray;float:left;font-size:10px;">
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<div style="width:220px;float:left;text-align:center;font-weight:900;font-size:12px;">
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<a href="intro.html">1. What are clusters? </a>
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</div>
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||||
<div style="width:220px;float:left;text-align:center;">
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<a href="dynamics.html"> 2. Why study cluster dynamics?</a>
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</div>
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<div style="width:220px;float:left;text-align:center;">
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||||
<a href="ourdynamics/our_dynamics.html"> 3. How we deal with cluster dynamics </a>
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</div>
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</div>
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<div id="WideContent">
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<div id="contentBoxHeader">
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<p>Introduction to Clusters</p>
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</div>
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<div id="contentBoxContent">
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<p>
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||||
Clusters, or nanoparticles, are mesoscopically small pieces of a given material, typically consisting of 3 to 10<sup>6</sup> atoms or molecules of the same type. It took time to physicists to identify these "small particles", as they were called before, as objects with specific properties, between large molecules and small pieces of bulk.
|
||||
<br><br>
|
||||
<b> Clusters around us </b>
|
||||
|
||||
</p>
|
||||
<p>
|
||||
<img src="images/lycurguscup.gif" align="right" style="width: 175px; height: 165px;">
|
||||
Clusters, however, have been used since antiquity by artcraft workers. For instance, Romans used to add gold powder (thus dispersed particles of gold) in glass, producing red stained-glass.
|
||||
|
||||
</p>
|
||||
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||||
<p>
|
||||
An amazing example is given by the Lycurgus cup, a Roman vase from the fourth century A.D.: viewed in reflected light (as during the day), the cup appears green. However, in transmitted light, that is, with a light source in it, it appears red.
|
||||
</p>
|
||||
<p>
|
||||
Clusters played also a major role in photography: tiny clusters of Silver bromide AgBr on films, exposed to light, produced clusters of Silver Ag. The longer the exposure, the larger the number of silver clusters and the "darker" the regions of the negative film.
|
||||
</p>
|
||||
<p>
|
||||
<img src="images/fullerenes_nanotubes.gif" align="left" style="width: 475px; height: 449px;">
|
||||
More recently, the discovery of the C<sub>60</sub>, the so-called fullerene, has opened a wide field in cluster physics, in theory as well as in experiment. Other fullerenes and nanotubes, which rapidly followed this discovery, exhibit exceptional mechanical and electrical properties and look promising for many applications in industry.
|
||||
</p>
|
||||
<p>
|
||||
On the left are few examples of fullerenes and nanotubes, with various helicities. <br>
|
||||
<!-- (from <a href="http://www.imp.cnrs.fr/utilisateurs/guillard/" style="font-size:10px;">http://www.imp.cnrs.fr/utilisateurs/guillard/</a>)-->
|
||||
</p>
|
||||
<p>
|
||||
<b>Neither a molecule nor a piece of bulk</b>
|
||||
</p>
|
||||
<p>
|
||||
A cluster differs quantitatively from a large molecule in the sense that a molecule has usually a small number of isomers (that is, stable spatial configurations for the same number of constituents), whereas a cluster typically exhibits a large number of isomers. For instance, various theoretical models have demonstrated hundreds of isomers of the cluster Ar<sub>13</sub> .
|
||||
</p>
|
||||
<p>
|
||||
The difference between a cluster and a small piece of bulk is also significant: the ratio of atoms on the surface to those in the volume is generally not negligible. Indeed finite volume effects are often preponderant in cluster physics and are sources of complexity in the theoretical description of cluster dynamics.
|
||||
</p>
|
||||
<p>
|
||||
One could consider that cluster physics lies between molecular and solid state physics. This field, well identified since the last quarter of the 20th century, now booms, in close relation with quantum chemistry.
|
||||
</p>
|
||||
<p>
|
||||
<b>
|
||||
How to produce a cluster in a laboratory?
|
||||
</b>
|
||||
</p>
|
||||
<p>
|
||||
The first method consists in exposing a material to an external environment (vapor or salt) and inducing the exchange of atoms between the environment and the bulk or the surface. This is a way of manufacturing <b>embedded</b> clusters in glass or <b>deposited</b> on a surface.
|
||||
</p>
|
||||
<p>
|
||||
Since the 1980's, we know how to produce <b>free</b> clusters by a fast expansion of supersonic atomic jets. This experimental method has allowed a great development of cluster physics. Indeed, embedded and deposited clusters are much more involved theoretically than free clusters. This method can also be the first step in the production of deposited and embedded clusters, by colliding free clusters and a matrix.
|
||||
</p>
|
||||
<p>
|
||||
Another typical feature in cluster production is the large scalability in the number of constituents. This allows specific studies with respect to the size of the cluster.
|
||||
</p>
|
||||
|
||||
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<li style="margin-top:1px;border-top:1px solid #B0C4DE; "><a href="index.html">Home</a></li>
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<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
|
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</font><font size="6">
|
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</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
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<div id="contentBoxWide">
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<div id="contentBoxHeader">
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<p>References</p>
|
||||
</div>
|
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<div id="contentBoxContent">
|
||||
|
||||
<ul>
|
||||
<p>
|
||||
<li>
|
||||
<a name="own1307">[341] </a> <br><font color="#000000"><i>V.O. Nesterenko and P.-G. Reinhard and W. Kleinig</i></font><br><font color="#0000ff">Electron excitations in atomic clusters: beyond dipole plasmon</font><br>in: Atomic and Molecular Clusters: New Research, editors: F. Columbus, Nova Science, 2006
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1313">[338] </a> <br><font color="#000000"><i>M. Bär and B. Jakob and P.-G. Reinhard and C. Toepffer</i></font><br><font color="#0000ff">Excitation of atoms/molecules by highly relativistic ions</font><br>Phys. Rev. A <b>73</b>, 022719 (2006)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1291">[337] </a> <br><font color="#000000"><i>K. Andrae and P.M. Dinh and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Pump and probe analysis of metal cluster dynamics</font><br>to appear Phys. Rev. C <b>35</b>, 169 (2005)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1293">[336] </a> <br><font color="#000000"><i>M. Ma and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Dynamics of H<sub>9</sub><sup>+</sup> in intense laser pulses</font><br>preprint nucl-th/0510039; to appear Phys. Rev. C <b>33</b>, 49 (2005)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="1310">[334] </a> <br><font color="#000000"><i>V. O. Nesterenko and P.--G. Reinhard and Th. Halfmann and L. I. Pavlov</i></font><br><font color="#0000ff">Two-photon population of electronic infrared quadrupole statesin atomic clusters</font><br>subm. Eur. Phys. J. D <b></b>, (2005)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1309">[333] </a> <br><font color="#000000"><i>M. Belkacem and F. Megi and E. Suraud and P.-G. Reinhard and G. Zwicknagel</i></font><br><font color="#0000ff">A Molecular Dynamics description of clusters in strong fields</font><br>subm. Phys. Rev. A <b></b>, (2005)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1308">[332] </a> <br><font color="#000000"><i>M. Belkacem and F. Megi and P.-G. Reinhard and E. Suraud and G. Zwicknagel</i></font><br><font color="#0000ff">Coulomb explosion of simple metal clusters in intense laser fields</font><br> <b></b>, (2005)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1305">[330] </a> <br><font color="#000000"><i>P.M. Dinh and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Time resolved fission in metal clusters</font><br>J. Phys. B <b>38</b>, 1637 (2005)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1304">[329] </a> <br><font color="#000000"><i>F. Fehrer and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Coupled plasmon and phonon dynamics in embedded Na clusters</font><br>Appl. Phys. A <b>82</b>, 145 (2005)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1303">[328] </a> <br><font color="#000000"><i>F. Fehrer and P.-G. Reinhard and E. Suraud and E. Giglio and B. Gervais and A. Ipatov</i></font><br><font color="#0000ff">Linear and non-linear response of embedded Na clusters</font><br>Appl. Phys. A <b>82</b>, 151 (2005)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1302">[327] </a> <br><font color="#000000"><i>F. Fehrer and M. Mundt and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Modeling Na clusters in Ar matrices</font><br>Ann. Phys. (Leipzig) <b>14</b>, 411 (2005)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1300">[325] </a> <br><font color="#000000"><i>P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Dynamics of orientations in an ensemble of Na<sub>7</sub><sup>+</sup> clusters</font><br>to appear Eur. Phys. J. D <b></b>, (2005)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1299">[324] </a> <br><font color="#000000"><i>B. Gervais and and E. Giglio and E. Jaquet and A. Ipatov and P.-G. Reinhard and F. Fehrer and E. Suraud</i></font><br><font color="#0000ff">Spectroscopic properties of Na clusters embedded in a rare-gas matrix</font><br>Phys. Rev. A <b>71</b>, 015201 (2005)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1272">[323] </a> <br><font color="#000000"><i> A. Ipatov and P. G. Reinhard and and E. Suraud</i></font><br><font color="#0000ff">Velocity dependence of metal cluster deposition on an insulating surface</font><br>preprint nucl-th/0407036, Nucl. Phys. A <b>30</b>, 65 (2004)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1268">[322] </a> <br><font color="#000000"><i> P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Dynamics of metal nanoclusters</font><br>Nucl. Phys. A <b>2</b>, 717 (2004)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1294">[317] </a> <br><font color="#000000"><i> B. Gervais and E. Giglio and E. Jacquet and A. Ipatov and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Simple DFT model of clusters embedded in rare gas matrix:trapping sites and spectroscopic properties of Na embedded in Ar</font><br>Phys. Rev. A <b>121</b>, 8466 (2004)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1292">[316] </a> <br><font color="#000000"><i>V.O. Nesterenko and P.-G. Reinhard and W. Kleinig and D.S. Dolci</i></font><br><font color="#0000ff">Electron infrared quadrupole modes in deformed Na clusters</font><br>J. Comp. Mat. Sci. <b>70</b>, 023205 (2004)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1288">[313] </a> <br><font color="#000000"><i>A. Pohl and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Angular distribution of electrons emitted from Na clusters</font><br>Phys. Rev. A <b>70</b>, 023202 (2004)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1287">[312] </a> <br><font color="#000000"><i>A. Pohl and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Exponential Photoelectron spectra in Na clusters</font><br>J. Phys. B <b>37</b>, 3301 (2004)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1284">[309] </a> <br><font color="#000000"><i>K. Andrae and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Crossed beams pump and probe dynamics in metal clusters</font><br>Phys. Rev. Lett. <b>92</b>, 173402 (2004)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1269">[307] </a> <br><font color="#000000"><i> M. Belkacem and M.A. Bouchenne and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Photodynamics of nanoclusters</font><br> <b>8</b>, 575 (2004)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1265">[305] </a> <br><font color="#000000"><i> E. Giglio and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Influence of the dynamical correlations on the ionization of highly irradiated metal clusters </font><br> Phys. Rev. C <b>205</b>, 250 (2003)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1285">[304] </a> <br><font color="#000000"><i>A. Pohl and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Photoelectron spectra from K and Na clusters</font><br> Nucl. Inst. Meth. B <b>68</b>, 053202 (2003)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1281">[303] </a> <br>P.-G. Reinhard and E. Suraud,<br> Introduction to Cluster Dynamics,<br> Wiley, , 2003
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1278">[300] </a> <br><font color="#000000"><i> P.-G. Reinhard and E Suraud</i></font><br><font color="#0000ff">Metal clusters in strong fields</font><br> Phys. Rev. A <b>209</b>, 41 (2003)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1275">[297] </a> <br><font color="#000000"><i>K. Andrae and M. Belkacem and T.P.M. Dinh and E. Giglio and M. Ma and F. Megi and A. Pohl</i></font><br><font color="#0000ff">Analysis of cluster dynamics</font><br>in: Formation of Correlations - Nonequilibrium at short time scales, editors: K. Morawetz, Springer, 2003
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1273">[296] </a> <br><font color="#000000"><i> E. Giglio and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Angular distribution of emitted electrons in sodium clusters: A semi-classical approach </font><br>Eur. Phys. J. D <b>67</b>, 43202 (2003)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1264">[295] </a> <br><font color="#000000"><i> K. Andrae and A. Pohl and P.-G. Reinhard and C. Legrand and M. Ma and E. Suraud</i></font><br><font color="#0000ff">Time-dependent density functional theory from a practitioners perspective </font><br>in: Progress in Nonequilibrium Green's Functions II, editors: M. Bonitz and D. Semkat, World Scientific, 2003
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1271">[294] </a> <br><font color="#000000"><i> A. Ipatov and E. Suraud and and P. G. Reinhard</i></font><br><font color="#0000ff">A microscopic study of sodium cluster deposition on an insulating surface </font><br>Encycl. Nanosc. Nanotechn. <b>4</b>, 301 (2003)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1270">[293] </a> <br><font color="#000000"><i> V. O. Nesterenko and W. Kleinig and P.-G. Reinhard and N. Lo Iudice and F.F. de Souza Cruz and and J.R. Marinelli</i></font><br><font color="#0000ff">Orbital magnetism in axially deformed sodium clusters: From scissors mode to dia-para magnetic anisotropy </font><br> J. Phys. G <b>27</b>, 43 (2003)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1247">[291] </a> <br><font color="#000000"><i> P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">DFT studies of ionic vibrations in Na clusters</font><br> <b>21</b>, 315 (2002)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1246">[290] </a> <br><font color="#000000"><i> K. Andrae and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Theoretical exploration of pump and probe in medium size Na clusters</font><br> Nucl. Instr. Meth. B <b>35</b>, 4203 (2002)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1260">[285] </a> <br><font color="#000000"><i> P.-G. Reinhard and V.O. Nesterenko and E. Suraud and S. El Gammal and W. Kleinig</i></font><br><font color="#0000ff">Scissors modes in triaxial metal clusters</font><br> Phys. Rev. A <b>66</b>, 013206 (2002)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1257">[282] </a> <br><font color="#000000"><i> V. O. Nesterenko and W. Kleinig and P.--G. Reinhard</i></font><br><font color="#0000ff">Landau fragmentation and deformation effects in dipole response of sodium clusters </font><br> Eur. Phys. J. D <b>19</b>, 57 (2002)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1255">[280] </a> <br><font color="#000000"><i> T. Berkus and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Dynamical effects in the optical response of small carbon chains</font><br> Int. J. Mol. Sci. <b>3</b>, 69 (2002)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1252">[277] </a> <br><font color="#000000"><i> C. Legrand and E. Suraud and P.-G. Reinhard</i></font><br><font color="#0000ff">Comparison of Self-Interaction-Corrections for Metal Clusters</font><br> J. Phys. B <b>35</b>, 1115 (2002)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1251">[276] </a> <br><font color="#000000"><i> Ll. Serra and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Density functional calculations for shell closures in Mg clusters</font><br> Eur. Phys. J. D <b>18</b>, 327 (2002)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1244">[275] </a> <br><font color="#000000"><i> W. Kleinig and V. O. Nesterenko and P. -G. Reinhard</i></font><br><font color="#0000ff">Electric multipole oscillations in deformed sodium clusters</font><br> Appl. Phys. B <b>297</b>, 1 (2002)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1248">[273] </a> <br><font color="#000000"><i> E. Giglio and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Semi-classical description of ionic and electronic dynamics in metal clusters </font><br> Ann. Phys. (Leipzig) <b>11</b>, 291 (2002)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1237">[271] </a> <br><font color="#000000"><i> P.-G. Reinhard and E.Suraud</i></font><br><font color="#0000ff">Clusters in intense laser pulses</font><br> J. Phys.B <b>11</b>, 566 (2001)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1235">[270] </a> <br><font color="#000000"><i> L.M. Ma and E. Suraud and P-G. Reinhard</i></font><br><font color="#0000ff">Laser excitation and ionic motion in small clusters</font><br> Appl. Phys. B <b>14</b>, 217 (2001)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1250">[269] </a> <br><font color="#000000"><i></i></font><br><font color="#0000ff">Collectivity in the optical response of small metal clusters</font><br> Ann. Phys. (N.Y.) <b>73</b>, 1 (2001)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1249">[268] </a> <br><font color="#000000"><i> E. Giglio and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Hybrid ensemble method for the UU collision term</font><br> J. Phys. B <b>12</b>, 1439 (2001)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1245">[267] </a> <br><font color="#000000"><i> P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Dynamics of Na clusters in picosecond laser pulses</font><br> Phys. Rev. B <b>73</b>, 401 (2001)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1243">[266] </a> <br><font color="#000000"><i> A. Pohl and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Influence of intermediate states on photoelectron spectra</font><br> Nucl. Phys. A <b>34</b>, 4969 (2001)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1239">[263] </a> <br><font color="#000000"><i> E. Giglio and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Impact of two-body collisions on explosion dynamics of irradiated clusters </font><br> Nucl. Instr. Meth. A <b>34</b>, 1253 (2001)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1219">[262] </a> <br><font color="#000000"><i> S. Kümmel and T. Berkus and P.-G. Reinhard and M. Brack</i></font><br><font color="#0000ff">Static Electric Dipole Polarizabilities of Na Clusters</font><br> Laser Physics <b>11</b>, 239 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1212">[260] </a> <br><font color="#000000"><i> A. Domps and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Semi-classical electron dynamics in metal clusters beyond mean-field </font><br> Eur. Phys. J. D <b>280</b>, 211 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="proc1026">[259] </a> <br><font color="#000000"><i> A. Pohl and P.-G. Reinhard and E. Giglio and E. Suraud</i></font><br><font color="#0000ff">Electronic and Ionic Dynamics of Metal Clusters</font><br>proceedings of the Nobel Symposium on Cluster Physics, Visby 2000
|
||||
A. Pohl and P.-G. Reinhard and E. Giglio and E. Suraud, Electronic and Ionic Dynamics of Metal Clusters, proceedings of the Nobel Symposium on Cluster Physics, Visby 2000
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1234">[258] </a> <br><font color="#000000"><i> J.R. Marinelli and V. Nesterenko and F.F. de Souza-Cruz and W. Kleinig and P.-G. Reinhard</i></font><br><font color="#0000ff">Twist mode in alcali metal clusters</font><br> Phys. Rev. Lett. <b>85</b>, 3141 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1233">[257] </a> <br><font color="#000000"><i> C.A. Ullrich and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Simplified implementation of self-interaction correction in sodium clusters </font><br> Phys. Rev. A <b>65</b>, 053202 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1232">[256] </a> <br><font color="#000000"><i> A. Domps and E. Giglio and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Semi-classical approach to electron dynamics in metal clusters</font><br> J. Phys. B <b>33</b>, L333 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1230">[254] </a> <br><font color="#000000"><i> F. Calvayrac and P.-G. Reinhard and E. Suraud and C. Ullrich</i></font><br><font color="#0000ff">Nonlinear electron dynamics in metal clusters</font><br> Phys. Rep. <b>337</b>, 493 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1228">[252] </a> <br><font color="#000000"><i> P. G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Cold versus hot ionization in metal clusters</font><br> Int. J. Mol. Sci. <b>1</b>, 92 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1227">[251] </a> <br><font color="#000000"><i> A. Pohl and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Towards single particle spectroscopy of small metal clusters</font><br> Phys. Rev. Lett. <b>84</b>, 5090 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1226">[250] </a> <br><font color="#000000"><i> E. Suraud and P.-G. Reinhard</i></font><br><font color="#0000ff">Impact of ionic motion on ionization of metal clusters under intense laser pulses </font><br> Phys. Rev. Lett. <b>85</b>, 2296 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1216">[249] </a> <br><font color="#000000"><i> S. Kümmel and M.Brack and P.-G. Reinhard</i></font><br><font color="#0000ff">Ionic and electronic structure of sodium clusters up to N=59 </font><br> Compte Rendu Acad. Sci. (Paris) <b>62</b>, 7602 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1224">[248] </a> <br><font color="#000000"><i> C. Kohl and E. Suraud and P. G. Reinhard</i></font><br><font color="#0000ff">Second Harmonic Generation in deposited clusters</font><br> Eur. Phys. J. D <b>11</b>, 115 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1220">[244] </a> <br><font color="#000000"><i> E. Giglio and P.-G. Reinhard and E.Suraud</i></font><br><font color="#0000ff">On violent excitations in metal clusters: a semi-classical approach</font><br> Comp. Mat. Science <b>17</b>, 534 (2000)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1202">[242] </a> <br><font color="#000000"><i> C. Kohl and S.M. El-Gammal and F. Calvayrac and E. Suraud and P.-G. Reinhard</i></font><br><font color="#0000ff">Towards spectral pattern of spin polarized sodium clusters: the example of Na<sub>12</sub></font><br> Eur.Phys.J D <b>5</b>, 271 (1999)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1225">[239] </a> <br><font color="#000000"><i> P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Excitation of metal clusters by intense lasers</font><br> Phys. Rev. B <b>327</b>, 893 (1999)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1213">[235] </a> <br><font color="#000000"><i> P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Resonance dynamics in metal clusters and nuclei</font><br>in: Cluster Physics, editors: W. Ekardt, Wiley, 1999
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1199">[234] </a> <br><font color="#000000"><i> P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">On electron dynamics in violent cluster excitations</font><br> Bull. Am. Phys. Soc. <b>10</b>, 239 (1999)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1208">[230] </a> <br><font color="#000000"><i> F. Calvayrac and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Dynamics of sodium clusters in a diabatic electron-ion model</font><br> Eur. Phys. J. D <b>9</b>, 389 (1999)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1207">[229] </a> <br><font color="#000000"><i> C. A. Ullrich and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Ionization dynamics of Na<sub>93</sub><sup>+</sup>: dependence on laser pulse length </font><br> Eur. Phys. J. D <b>9</b>, 407 (1999)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1206">[228] </a> <br><font color="#000000"><i> S. Kümmel and M. Brack and P.-G. Reinhard</i></font><br><font color="#0000ff">Ionic geometries and electronic excitations of Na<sub>9</sub><sup>+</sup> and Na<sub>55</sub><sup>+</sup></font><br> Eur. Phys. J. D <b>9</b>, 149 (1999)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1205">[227] </a> <br><font color="#000000"><i>P.-G. Reinhard and F. Calvayrac and C. Kohl and S. Kümmel and E. Suraud and C.A. Ullrich and M. Brack</i></font><br><font color="#0000ff">Frequencies, times, and forces in the dynamics of Na clusters</font><br> Eur. Phys. J. D <b>9</b>, 111 (1999)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1174">[224] </a> <br><font color="#000000"><i> A. Domps and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Time Dependent Thomas Fermi for Electron Dynamics in Metal Clusters</font><br> Eur. Phys. J. D <b>80</b>, 5520 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1172">[222] </a> <br><font color="#000000"><i> F. Calvayrac and A. Domps and S. El-Gammal and C. Kohl and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Large amplitude dynamics of clusters and nuclei</font><br> <b>48</b>, 715 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1196">[218] </a> <br><font color="#000000"><i></i></font><br><font color="#0000ff">Optical response of carbon chains</font><br> Nucl. Inst. Meth. B <b>146</b>, 29 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1195">[217] </a> <br><font color="#000000"><i></i></font><br><font color="#0000ff">Geometrical and quantal fragmentation of optical response in Na<sub>18</sub><sup>++</sup> </font><br> Eur. Phys. J. D <b> 2</b>, 191 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1194">[216] </a> <br><font color="#000000"><i> W. Kleinig and V.O. Nesterenko and P.-G. Reinhard and Ll. Serra</i></font><br><font color="#0000ff">Plasmon response in K, Na and Li clusters: systematics using the separable random-phase-approximation with pseudo-Hamiltonians </font><br> Eur. Phys. J. D <b>4</b>, 343 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1192">[214] </a> <br><font color="#000000"><i> F. Calvayrac and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Coulomb explosion of a Na<sub>12</sub> cluster in a diabatic electron--ion dynamical picture </font><br> J. Phys. B <b>31</b>, 5023 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1191">[213] </a> <br><font color="#000000"><i> P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Field amplification in Na clusters</font><br> Eur. Phys. J. D <b>3</b>, 175 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1190">[212] </a> <br><font color="#000000"><i> F. Spiegelmann and R. Poteau and B. Montag and P.-G. Reinhard</i></font><br><font color="#0000ff">Global structure of small Na clusters in different approaches</font><br> Phys. Lett. A <b>242</b>, 163 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1189">[211] </a> <br><font color="#000000"><i> S. Kümmel and M. Brack and P.-G. Reinhard</i></font><br><font color="#0000ff">Ionic structure and photoabsorption in medium sized sodium clusters</font><br> Phys. Rev. B <b>58</b>, 1774 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1188">[210] </a> <br><font color="#000000"><i> F. Calvayrac and A. Domps and P.-G. Reinhard and E. Suraud and C.A. Ullrich</i></font><br><font color="#0000ff">Ionization and energy deposit in metal clusters irradiated by intense lasers </font><br> Eur. Phys. J. D <b>4</b>, 207 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1187">[209] </a> <br><font color="#000000"><i> A. Domps and E. Suraud and P.-G. Reinhard</i></font><br><font color="#0000ff">Two body collisions and relaxation in metal clusters</font><br> Phys. Rev. Lett. <b>81</b>, 5524 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1186">[208] </a> <br><font color="#000000"><i> C.A. Ullrich and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Electron dynamics in strongly excited metal clusters: a density-functional study with self-interaction correction </font><br> J. Phys. B <b>31</b>, 1871 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1185">[207] </a> <br><font color="#000000"><i> P.-G. Reinhard and E. Suraud and C.A. Ullrich</i></font><br><font color="#0000ff">Ionization of metal clusters by ions in the Fermi velocity range </font><br> Eur. Phys. J. D <b> 1</b>, 303 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1184">[206] </a> <br><font color="#000000"><i> F. Calvayrac and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Ionic structure and plasmon response in sodium clusters</font><br> J. Phys. B <b>31</b>, 1367 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1171">[201] </a> <br><font color="#000000"><i> C. Kohl and F. Calvayrac and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Optical response of Na clusters on NaCl surfaces</font><br> <b>405</b>, 74 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1181">[200] </a> <br><font color="#000000"><i> R. Menegozzi and P.-G. Reinhard and M. Schulz</i></font><br><font color="#0000ff">Electron transport in ballistic electron emission microscopy</font><br> Appl. Phys. A <b>66</b>, S897 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1180">[199] </a> <br><font color="#000000"><i> C.A. Ullrich and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Electron emission from strongly excited metal clusters</font><br> Phys. Rev. A <b>57</b>, 1938 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1179">[198] </a> <br><font color="#000000"><i> R. Menegozzi and P.-G. Reinhard and M. Schulz</i></font><br><font color="#0000ff">Quantum mechanical transmission coefficient at interfaces and BEEM</font><br> Surf. Sci. Lett. <b>411</b>, L810 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1178">[197] </a> <br><font color="#000000"><i></i></font><br><font color="#0000ff">Nonlinear electronic dynamics in free and deposited sodium clusters: quantal and semi-classical approaches </font><br> Comp. Mat. Sci. <b>10</b>, 448 (1998)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1159">[193] </a> <br><font color="#000000"><i> P.-G. Reinhard and F. Calvayrac and E. Suraud</i></font><br><font color="#0000ff">Plasmons in fissioning metal clusters</font><br> Phys. Rev. Lett. <b> 41</b>, 151 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1156">[190] </a> <br><font color="#000000"><i> C. Kohl and P.-G. Reinhard</i></font><br><font color="#0000ff">Na clusters on Na-Cl surfaces - the impact of the interface potential</font><br> Surf. Science <b>39</b>, 605 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1176">[189] </a> <br><font color="#000000"><i> F. Calvayrac and S. El-Gammal and C. Kohl and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Nonlinear Dynamics of Nuclei and Metal Clusters</font><br> Phys. Rev. E <b>110</b>, 1175 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1168">[187] </a> <br><font color="#000000"><i> J. Babst and P.-G. Reinhard</i></font><br><font color="#0000ff">A separable approach to linear response in Na clusters</font><br> Z. Phys. D <b> 42</b>, 209 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1167">[186] </a> <br><font color="#000000"><i> C.A. Ullrich and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Metallic clusters in strong femtosecond laser pulses</font><br> J. Phys. B <b>30</b>, 5043 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1166">[185] </a> <br><font color="#000000"><i> C. Kohl and B. Fischer and P.-G. Reinhard</i></font><br><font color="#0000ff">Polarized isomers of Na and anomalous magnetic response</font><br> Phys. Rev. B <b>56</b>, 11149 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1165">[184] </a> <br><font color="#000000"><i>A. Domps and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Fermionic Vlasov Propagation for Coulomb Interacting Systems</font><br>Ann. Phys. (N.Y.) <b>260</b>, 171 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1164">[183] </a> <br><font color="#000000"><i> A. Domps and A.S. Krepper and V. Savalli and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">On fermionic stability of Vlasov descriptions of finite Coulomb systems</font><br> Ann. Phys. (Leipzig) <b>6</b>, 468 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1163">[182] </a> <br><font color="#000000"><i> A. Domps and P. L'Eplattenier and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">The Vlasov equation for Coulomb systems and the Husimi picture</font><br> Ann. Phys. (Leipzig) <b>6</b>, 455 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1162">[181] </a> <br><font color="#000000"><i> T. Hirschmann and M. Brack and P.-G. Reinhard</i></font><br><font color="#0000ff">The collective response of deformed sodium clusters</font><br> Z. Phys. D <b> 40</b>, 254 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1161">[180] </a> <br><font color="#000000"><i>C. Ullrich and A. Domps and F. Calvayrac and E. Suraud and P. G. Reinhard</i></font><br><font color="#0000ff">Electron response of metallic clusters to strong laser pulses and energetic ion collisions </font><br> Z. Phys. D <b> 40</b>, 265 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1160">[179] </a> <br><font color="#000000"><i> P. G. Reinhard and J. Babst and B. Fischer and C. Kohl and F. Calvayrac and E. Suraud and T. Hirschmann and M. Brack</i></font><br><font color="#0000ff">Electron dynamics in metal clusters</font><br> Z. Phys. D <b> 40</b>, 314 (1997)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1142">[173] </a> <br><font color="#000000"><i> Th. Hirschmann and M. Brack and B. Montag and P.-G. Reinhard and J. Meyer</i></font><br><font color="#0000ff">Shape isomerism of sodium clusters with with quadrupole, octupole, and hexadecapole deformations in the structure averaged jellium model </font><br> Ann. Phys. <b>3</b>, 229 (1996)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1153">[171] </a> <br><font color="#000000"><i> P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Towards a Fermionic Vlasov Equation</font><br> Z. Phys. A <b> 355</b>, 339 (1996)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1151">[169] </a> <br><font color="#000000"><i> C. Kohl and B. Montag and P.-G. Reinhard</i></font><br><font color="#0000ff">Shell effects in planar electron clusters</font><br> Z. Phys. D <b>38</b>, 81 (1996)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1150">[168] </a> <br><font color="#000000"><i> P.-G. Reinhard and O. Genzken and M. Brack</i></font><br><font color="#0000ff">From sum rules to RPA: 3. metal clusters</font><br> Ann. Phys. (Leipzig) <b>5</b>, 576 (1996)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1149">[167] </a> <br><font color="#000000"><i> L. Feret and E. Suraud and F. Calvayrac and P.-G. Reinhard</i></font><br><font color="#0000ff">On the electron dynamics in metal clusters: A Vlasov approach</font><br> J. Phys. B <b> 29</b>, 4477 (1996)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1148">[166] </a> <br><font color="#000000"><i> L. Mornas and F. Calvayrac and P.-G. Reinhard and E. Suraud</i></font><br><font color="#0000ff">Spin dynamics in sodium clusters</font><br> Z. Phys. D <b>38</b>, 73 (1996)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1132">[164] </a> <br><font color="#000000"><i> B. Montag and P.-G. Reinhard</i></font><br><font color="#0000ff">Ionic structure and global deformation of axially symmetric simple metal clusters </font><br> Phys. Rev. C <b>33</b>, 265 (1995)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1126">[161] </a> <br><font color="#000000"><i> B. Montag and P.-G. Reinhard</i></font><br><font color="#0000ff">On the width of plasmon resonances in metal clusters</font><br> Phys. Rev. C <b>51</b>, 14686 (1995)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1140">[158] </a> <br><font color="#000000"><i> B. Montag and P.-G. Reinhard</i></font><br><font color="#0000ff">Symmetric and asymmetric fission of metal clusters</font><br> Phys. Rev. B <b> 52</b>, 16365 (1995)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1139">[157] </a> <br><font color="#000000"><i> F. Calvayrac and E. Suraud and P.-G. Reinhard</i></font><br><font color="#0000ff">Non linear plasmon response in highly excited metallic clusters</font><br> Phys. Rev. B <b>52</b>, R17056 (1995)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1138">[156] </a> <br><font color="#000000"><i> C. Kohl and B. Montag and P.-G. Reinhard</i></font><br><font color="#0000ff">Spin effects in small sodium clusters</font><br> Z. Phys. D <b>35</b>, 57 (1995)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1136">[153] </a> <br><font color="#000000"><i> S. Kasperl and C. Kohl and P.-G. Reinhard</i></font><br><font color="#0000ff">Plasmon resonances and triaxial deformations in small Na clusters </font><br> Phys. Lett. A <b> 206</b>, 81 (1995)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1135">[152] </a> <br><font color="#000000"><i> B. Montag and Th. Hirschmann and J. Meyer and P.-G. Reinhard and M. Brack</i></font><br><font color="#0000ff"></font><br> Phys. Rev. B <b> 52</b>, 4775 (1995)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1129">[143] </a> <br><font color="#000000"><i> P.-G. Reinhard and S. Weisgerber and O. Genzken and M. Brack</i></font><br><font color="#0000ff">RPA in nuclei and metal clusters</font><br> Phys. Rev. B <b>349</b>, 219 (1994)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1123">[140] </a> <br><font color="#000000"><i> B. Montag and P.-G. Reinhard</i></font><br><font color="#0000ff">Small metal clusters in a cylindrically averaged pseudopotential scheme</font><br> Phys. Lett. A <b>193</b>, 380 (1994)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1122">[139] </a> <br><font color="#000000"><i> B. Montag and P.-G. Reinhard and J. Meyer</i></font><br><font color="#0000ff">The Structure-Averaged Jellium Model for Metal Clusters</font><br> Z. Phys. D <b>32</b>, 125 (1994)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1112">[135] </a> <br><font color="#000000"><i> S. Weisgerber and P.-G. Reinhard</i></font><br><font color="#0000ff">From sum rules to RPA: 2. <sup>3</sup>He droplets</font><br> Int. J. Mod. Phys. E <b>2</b>, 666 (1993)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1102">[128] </a> <br><font color="#000000"><i> P.-G. Reinhard</i></font><br><font color="#0000ff">Correlations and local density approximation</font><br> <b>169</b>, 281 (1992)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1105">[120] </a> <br><font color="#000000"><i> S. Weisgerber and P.-G. Reinhard</i></font><br><font color="#0000ff">The shell structure of <sup>3</sup>He droplets</font><br> Z. Phys. D <b>23</b>, 275 (1992)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1098">[117] </a> <br><font color="#000000"><i> S. Weisgerber and P.-G. Reinhard</i></font><br><font color="#0000ff">A density functional with finite range for liquid <sup>3</sup>He systems</font><br> Phys. Lett. A <b>158</b>, 407 (1991)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1097">[114] </a> <br><font color="#000000"><i> G. Lauritsch and P.-G. Reinhard and J. Meyer and M. Brack</i></font><br><font color="#0000ff">Triaxially Deformed Sodium Clusters in a Selfconsistent Microscopic Description </font><br> Ann. Phys. (N.Y.) <b>160</b>, 179 (1991)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1091">[110] </a> <br><font color="#000000"><i> P.-G. Reinhard and M. Brack</i></font><br><font color="#0000ff">Random-Phase Approximation in a Local Representation</font><br> <b>41</b>, 5568 (1990)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="proc1011">[107] </a> <br><font color="#000000"><i> S. Weisgerber and P.-G. Reinhard and C. Toepffer</i></font><br><font color="#0000ff">Effective Forces with Zero Range for Liquid <sup>3</sup>He</font><br>in "Spin Polarized Quantum Systems" (Ed. S. Stringari), p. 211,Singapore 1989
|
||||
S. Weisgerber and P.-G. Reinhard and C. Toepffer, Effective Forces with Zero Range for Liquid <sup>3</sup>He, in "Spin Polarized Quantum Systems" (Ed. S. Stringari), p. 211,Singapore 1989
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1003">[16] </a> <br><font color="#000000"><i> P.-G. Reinhard and W. Greiner and H. Arenhövel</i></font><br><font color="#0000ff">Electrons in strong external fields</font><br> Fizika <b>166</b>, 173 (1971)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1002">[15] </a> <br><font color="#000000"><i> P.-G. Reinhard</i></font><br><font color="#0000ff">Quantum electrodynamics for strong fields and superheavy nuclei</font><br> Z. Phys. A <b>I.3</b>, 313 (1970)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="proc1019">[10] </a> <br><font color="#000000"><i> F. Calvayrac and P. G. Reinhard and E. Suraud and C.Ullrich</i></font><br><font color="#0000ff">Nonlinear electron dynamics in metal clusters</font><br>Proceedings of PC'96 conference 163-166 (ed. CYFRONET KRAKOW ISBN 83-902363-3-8)
|
||||
F. Calvayrac and P. G. Reinhard and E. Suraud and C.Ullrich, Nonlinear electron dynamics in metal clusters, Proceedings of PC'96 conference 163-166 (ed. CYFRONET KRAKOW ISBN 83-902363-3-8)
|
||||
</li>
|
||||
</p><p>
|
||||
<li>
|
||||
<a name="own1155">[9] </a> <br><font color="#000000"><i> F. Calvayrac and E. Suraud and P.-G. Reinhard</i></font><br><font color="#0000ff">Spectral signals from electronic dynamics in sodium clusters</font><br> Phys. Lett B <b>254</b>, (1997) 125 (N.Y.)
|
||||
</li>
|
||||
</p></ul>
|
||||
<!-- START CONTENT HERE -->
|
||||
|
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|
||||
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<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
|
||||
</font><font size="6">
|
||||
</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
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<p>Laser excitation</p>
|
||||
</div>
|
||||
<div id="contentBoxContent">
|
||||
|
||||
<div style="text-align: center;"><big><big><span
|
||||
style="font-weight: bold;">Irradiation of Na</span><sub
|
||||
style="font-weight: bold;">9</sub><sup style="font-weight: bold;">+</sup><span
|
||||
style="font-weight: bold;"> by a laser pulse</span></big></big><br>
|
||||
</div>
|
||||
<br>
|
||||
<div style="text-align: center;"><img alt="laser"
|
||||
src="laser_slide_small.jpg" style="width: 750px;"><br>
|
||||
<br>
|
||||
<br>
|
||||
<div style="text-align: left;">To watch the movie, please click on
|
||||
the images below:</div>
|
||||
<table
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style="margin-left: auto; margin-right: auto; text-align: left; width: 804px; height: 243px;"
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<td style="vertical-align: top; text-align: center;">
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||||
<a href="film_1.mpg">
|
||||
<img
|
||||
alt="laser1" src="laser1.jpg"
|
||||
style="width: 320px; height: 214px;"></a><br>
|
||||
movie in mpg<br>
|
||||
</td>
|
||||
<td style="vertical-align: top; text-align: center;">
|
||||
<a href="F.avi">
|
||||
<img
|
||||
alt="laser1" src="laser1.jpg"
|
||||
style="width: 320px; height: 214px;"></a><br>
|
||||
movie in avi<br>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
<br>
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|
||||
<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
|
||||
</font><font size="6">
|
||||
</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
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<div style="width:220px;float:left;text-align:center;">
|
||||
<a href="../intro.html">1. What are clusters? </a>
|
||||
</div>
|
||||
<div style="width:220px;float:left;text-align:center;">
|
||||
<a href="../dynamics.html"> 2. Why study cluster dynamics?</a>
|
||||
</div>
|
||||
<div style="width:220px;float:left;text-align:center;font-weight:900;font-size:12px;">
|
||||
<a href="our_dynamics.html"> 3. How we deal with cluster dynamics </a>
|
||||
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|
||||
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
<div id="WideContent">
|
||||
<div id="contentBoxWide">
|
||||
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|
||||
<p>Dynamics </p>
|
||||
</div>
|
||||
<div id="contentBoxContent">
|
||||
|
||||
<div style="text-align: center;">
|
||||
<div style="text-align: justify;">
|
||||
<h1>How we deal with cluster dynamics<br>
|
||||
</h1>
|
||||
The understanding of the complicated dynamical scenarios such as the
|
||||
ones described previously requires dedicated theoretical modelling.
|
||||
Cluster physics and even more so cluster dynamics lays at the interface
|
||||
of several fields of science, especially chemistry and physics. The
|
||||
theory of cluster dynamics has thus borrowed inspiration from these
|
||||
various domains to develop its own and original methods. <br><br>
|
||||
|
||||
Not surprisingly, a direct transposition of methods well developed in a
|
||||
given field only provides guidelines and a starter for further
|
||||
developments. Still, it also allows to benchmark new developments on
|
||||
well established test cases. Cluster dynamics has thus benefited a lot
|
||||
from experience gained in chemistry, especially at the side of moderate
|
||||
excitation, and in physics for more violent scenarios, especially from
|
||||
solid state and nuclear physics. The description of cluster dynamics is
|
||||
made difficult by two basic problems: The fact that one would like to
|
||||
deal with large (although finite) systems and the fact that electrons
|
||||
and ions move at awfully different time scales (typically a factor
|
||||
100). This implies huge simulation times to be able to resolve
|
||||
simultaneously electronic and ionic dynamics. One thus needs both
|
||||
robust and simple approaches to overcome these two difficulties of time
|
||||
scales and system size.<br>
|
||||
<br>
|
||||
<table style="border: 0px; float: left;">
|
||||
<tr>
|
||||
<td>
|
||||
<a href="laser.html">
|
||||
<img alt="laser"
|
||||
src="laser_slide_small.jpg"
|
||||
style="border: 0px solid ; width: 527px; height: 275px; float: left;">
|
||||
</a>
|
||||
</td>
|
||||
<tr>
|
||||
<td align="center">
|
||||
<b>
|
||||
Fig.1:
|
||||
Irradiation of Na<sub>9</sub><sup>+</sup> by a laser pulse.
|
||||
</b>
|
||||
</td>
|
||||
</tr>
|
||||
</table>
|
||||
In the case of violent excitation, the most robust and simple approaches
|
||||
rely on Density Functional Theory, a theory developed since the mid
|
||||
60's for electronic systems and which has met impressive successes, in bulk materials
|
||||
as well as in finite molecules. <br><br>
|
||||
<font color="red"><b>???In density functional
|
||||
theory, the complicated many-body electronic problem is simplified as
|
||||
it can be shown that the one body electronic density constitutes a key
|
||||
ingredient, espcially for computing the energy of the system. The
|
||||
extension of this theory to truly time-dependent processes is more
|
||||
recent and still in development and cluster dynamics offers here a
|
||||
fascinating domain of applications and testing.???</b></font> <br><br>
|
||||
In order to illustrate
|
||||
the capabilities of such methods we present here two examples of
|
||||
cluster response to violent external excitation. Figure 1 shows the
|
||||
irradiation of Na<sub>9</sub><sup>+</sup> by a laser pulse, while Figure 2
|
||||
displays a collision of Na<sub>9</sub><sup>+</sup> with Ar<sup>8+</sup>
|
||||
considered as an energetic projectile. The actual dynamical scenarios can be
|
||||
visualized through the two movies below (click on the image to download
|
||||
the corresponding movie). Various characteristics of the dynamics,
|
||||
especially in
|
||||
terms of time scales, are presented in both figures. These cartoons
|
||||
demonstrate strong interactions between electrons and ions and a
|
||||
complex non-adiabatic dynamics.<br>
|
||||
<table>
|
||||
<tr>
|
||||
<td>
|
||||
<a href="projectile.html"><img alt="projectile"
|
||||
src="proj_slide_small.jpg"
|
||||
style="border: 0px solid ; width: 518px; height: 276px;">
|
||||
</a>
|
||||
</td>
|
||||
<td align="left">
|
||||
<b>
|
||||
Fig.2:
|
||||
Collision of Na<sub>9</sub><sup>+</sup> with Ar<sup>8+</sup>
|
||||
</b>
|
||||
</td>
|
||||
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|
||||
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|
||||
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|
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|
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<a href="#top">Back to top </a>
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<td style="vertical-align: top; text-align: center;"><a
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|
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<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
|
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</font><font size="6">
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</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
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<p>Ionic impact</p>
|
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|
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|
||||
|
||||
<div style="text-align: center;"><big><big><span
|
||||
style="font-weight: bold;">Collision of Na</span><sub
|
||||
style="font-weight: bold;">9</sub><sup style="font-weight: bold;">+</sup><span
|
||||
style="font-weight: bold;"> with an energetic ion,<big><big> </big></big></span><span
|
||||
style="font-weight: bold;">Ar</span><sup style="font-weight: bold;">8+</sup></big></big></div>
|
||||
<br>
|
||||
<br>
|
||||
<div style="text-align: center;"><img alt="projectile"
|
||||
src="proj_slide_small.jpg" style="width: 750px; height: 409px;"><br>
|
||||
<br>
|
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<div style="text-align: left;"><br>
|
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To watch the movie, please click on the images below:<br>
|
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|
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<br>
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<table
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style="margin-left: auto; margin-right: auto; text-align: left; height: 227px; width: 526px;"
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<td style="vertical-align: top; text-align: center;"><a href="fproj.mpg"><img
|
||||
alt="proj1" src="proj1.jpg" style="width: 154px; height: 157px;"></a><br>
|
||||
movie in mpg<br>
|
||||
</td>
|
||||
<td style="vertical-align: top; text-align: center;"><a href="fProj.avi"><img
|
||||
alt="proj1" src="proj1.jpg" style="width: 154px; height: 157px;"></a><br>
|
||||
movie in avi<br>
|
||||
</td>
|
||||
</tr>
|
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<div style="text-align: center;"><br>
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|
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<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
|
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</font><font size="6">
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</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
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<p>Our Research Activities</p>
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--> </div>
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<div style="width:250px;float:left;">
|
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|
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A popular guide
|
||||
</div>
|
||||
<div id="cBoxContent">
|
||||
|
||||
<p>
|
||||
For non-experts we provide some interesting basic and popular information on
|
||||
our research activities: </p>
|
||||
|
||||
|
||||
<ul>
|
||||
<li>
|
||||
<p><a href="intro.html">What are clusters?</a></p>
|
||||
</li>
|
||||
<li>
|
||||
<p><a href="dynamics.html">Why studying cluster dynamics?</a></p>
|
||||
</li>
|
||||
<li>
|
||||
<p><a href="ourdynamics/our_dynamics.html">How we deal with cluster dynamics</a></p>
|
||||
</li>
|
||||
</ul>
|
||||
|
||||
|
||||
|
||||
|
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</div>
|
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</div>
|
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</div>
|
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-->
|
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<div style="width: 770px; float: left;">
|
||||
<div id="cBoxEnv">
|
||||
<div id="cBoxHeader"> A scientific guide </div>
|
||||
<div id="cBoxContent">
|
||||
<div style="text-align: justify;"> </div>
|
||||
<p style="text-align: justify;">The core of our activities concerns the
|
||||
dynamics of clusters. One can sort the various explored paths along
|
||||
three major directions of research. In the first place, we focus on
|
||||
intrinsic dynamical properties of clusters as revealed by moderate
|
||||
external excitations. The second research axis deals with the response
|
||||
of clusters when subjected to a possibly intense external field which, to a
|
||||
large extent, shapes the response of the system. The third
|
||||
aspect covers the numerous theoretical developments motivated by the
|
||||
description of cluster dynamics in the various situations and domains
|
||||
of excitations explored in the two previous items.
|
||||
</p>
|
||||
<p>
|
||||
<b><a href="analysis/detail1.html">Time and energy resolved analysis
|
||||
(intrinsic cluster dynamics)</a></b>
|
||||
</p>
|
||||
<p style="text-align: justify;">At moderate perturbations, the cluster
|
||||
response dominantly reflects its own (structure and dynamical)
|
||||
properties. This first item covers such situations (which for the
|
||||
simplest ones can also be addressed in purely static pictures). The
|
||||
optical response in metal clusters provides a typical example. </!--of such
|
||||
situations--> But we also pursue detailed investigations of photoelectron
|
||||
spectroscopy (energy and/or angle resolved) and of pump and probe
|
||||
scenarios at moderate excitations.
|
||||
</p>
|
||||
<p>
|
||||
<b><a href="analysis/detail2.html">
|
||||
Free clusters in external fields</a></b></p>
|
||||
<p style="text-align: justify;">
|
||||
This general title covers several aspects of our activities sharing the
|
||||
common denominator that the observed dynamics is a result of the
|
||||
cluster in interaction with an external (static or time dependent)
|
||||
field and not only of the cluster itself. The related phenomena lie in
|
||||
the adiabatic regime (plasmon, harmonic generation) as well as
|
||||
strongly non adiabatic situations. Extensive studies have thus been led
|
||||
on the various scenarios encountered by clusters irradiated by intense
|
||||
laser beams or hit by energetic highly charged projectiles.<br>
|
||||
</p>
|
||||
<p><b><a href="tddft-md/detailQMMM.html">Molecules and clusters in
|
||||
contact with a polarizable environment</a></b>
|
||||
</p>
|
||||
<p style="text-align: justify;">Clusters can be more
|
||||
easily handled experimentally when they are produced in contact with an environment
|
||||
(deposited on a surface or embedded in a matrix). This concerns various experiments and a large
|
||||
amount of experimental data. We have thus developed a generalized
|
||||
Quantum Mechanics / Molecular Mechanics (QM/MM) method in the sense
|
||||
that electronic degrees of freedom of the environment can be
|
||||
explicitely treated dynamically. This hierarchical approach allows us
|
||||
to explore various dynamical scenarios, as optical response of
|
||||
deposited clusters, deposition processes, irradiation of embedded
|
||||
clusters by an intense laser field, etc.<br>
|
||||
</p>
|
||||
<p><b><a href="tddft-md/formal.html">Theoretical developments</a> </b></p>
|
||||
<p style="text-align: justify;">Understanding of cluster dynamics
|
||||
represents a complex task which requires elaborate theoretical tools.
|
||||
Density Functional Theory (DFT) represents here a robust starting point
|
||||
which allows to address various situations. We use DFT at various
|
||||
levels of sophistications (Local Density Approximation, <a
|
||||
href="tddft-md/detailTDSIC.html">Self Interaction Correction</a>) in
|
||||
our time dependent approach. The basic tool is <a
|
||||
href="tddft-md/formal.html">
|
||||
Time Dependent LDA</a> in the quantal Kohn Sham picture, but we have
|
||||
also developed semi-classical schemes, in terms of the Vlasov-LDA approximation, possibly
|
||||
complemented by dynamical correlations. Exploratory investigations are
|
||||
also led to account for fluctuations by means of stochastic extensions
|
||||
of time dependent mean field theories. </p>
|
||||
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|
||||
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|
||||
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|
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|
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|
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<div id="image">
|
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<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
|
||||
</font><font size="6">
|
||||
</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
|
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|
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|
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|
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<b>Permanent staff:</b>
|
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|
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|
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|
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</p>
|
||||
<p>
|
||||
<b>PhD Students:</b>
|
||||
<ul>
|
||||
<li>Philipp Wopperer</li>
|
||||
<li>Bernhard Faber</li>
|
||||
<li>
|
||||
<a href="http://www.theorie2.physik.uni-erlangen.de/~baer">
|
||||
Matthias B<>r*</a></li>
|
||||
<li>Frank Fehrer*</li>
|
||||
<li>Andreas Pohl*</li>
|
||||
<li>Karsten Andrae*</li>
|
||||
<li>Michael Mundt*</li>
|
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<li>Christian Kohl*</li>
|
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<li>M. Montag*</li>
|
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</ul>
|
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|
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<p>
|
||||
<b>Diploma Students:</b>
|
||||
<ul>
|
||||
<li>Daniel Berger</li>
|
||||
</ul>
|
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|
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|
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|
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|
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|
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<p><b>Permanent staff:</b>
|
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<ul>
|
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<li><a href="ES.html">Prof. Dr. E. Suraud</a></li>
|
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<li><a href="MD.html">Dr. P. M. Dinh</a></li>
|
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</ul>
|
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</p>
|
||||
<p>
|
||||
<b>PhD Students:</b>
|
||||
<ul>
|
||||
<li>Gaspard Bousquet</li>
|
||||
<li>F. M<>gi*</li>
|
||||
<li>L. M. Ma*</li>
|
||||
<li>E. Giglio*</li>
|
||||
<li>A. Domps*</li>
|
||||
<li>F. Calvayrac*</li>
|
||||
</ul>
|
||||
</p>
|
||||
</div>
|
||||
|
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</div>
|
||||
</div>
|
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</div>
|
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|
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<div id="footer">
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<p>* former members</p>
|
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|
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</body>
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|
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|
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text-decoration:none;
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padding-left:10px;
|
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color:#333333;
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}
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#container
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@ -0,0 +1,155 @@
|
|||
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
|
||||
<html xmlns="http://www.w3.org/1999/xhtml">
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
|
||||
<title>Theory of Cluster Dynamics</title>
|
||||
<link href="../style.css" rel="stylesheet" type="text/css" />
|
||||
</head>
|
||||
|
||||
<body>
|
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<div id="container">
|
||||
<div id="header">
|
||||
<div id="menu">
|
||||
<div id="navMenu">
|
||||
|
||||
<ul>
|
||||
<li style="margin-top:7px;border-top:1px solid #B0C4DE; "><a href="../index.html">Home</a></li>
|
||||
<li><a href="../intro.html">Introductory Overview</a></li>
|
||||
<li><a href="../research.html">Scientific Information</a></li>
|
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<li><a href="../staff.html">Staff</a></li>
|
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</ul>
|
||||
</div>
|
||||
</div>
|
||||
<div id="image">
|
||||
<p><a name="top"<font size="5" color="white">Theory of Cluster Dynamics</font><font size="5"></a><br/>
|
||||
</font><font size="6">
|
||||
</font><font size="4">The Toulouse - Erlangen Collaboration</font></p>
|
||||
|
||||
</div>
|
||||
|
||||
<a name="oben">
|
||||
<div id="content">
|
||||
|
||||
<div style="margin:15px;width:770px;border:1px solid gray;float:left;font-size:10px;">
|
||||
<div style="width:220px;float:left;text-align:center;">
|
||||
<a href="../analysis/detail1.html">1. Analysis of cluster dynamics</a>
|
||||
</div>
|
||||
<div style="width:220px;float:left;text-align:center;font-weight:900;font-size:12px;">
|
||||
<a href="../analysis/detail2.html"> 2. Clusters in external fields</a>
|
||||
</div>
|
||||
<div style="width:220px;float:left;text-align:center;">
|
||||
<a href="../tddft-md/formal.html"> 3. Theoretical developments </a>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
</a>
|
||||
|
||||
|
||||
|
||||
<div id="WideContent">
|
||||
<div id="contentBoxWide">
|
||||
<div id="contentBoxHeader">
|
||||
<p> Clusters in strong external perturbations</p>
|
||||
</div>
|
||||
<div id="contentBoxContent">
|
||||
|
||||
<!-- START CONTENT HERE -->
|
||||
|
||||
<p><img src="na8_nacl_SHG.gif" align="right" width="300">
|
||||
Many experiments are done for clusters in contact with a
|
||||
substrate. The strong interface interaction modifies the cluster and
|
||||
theoretical simulations become more involved. However, some feature
|
||||
can only be explored in connection with a substrate. E.g., the
|
||||
symmetry breaking through a surface gives access to second-harmonic
|
||||
generation (SHG). The figure benaeth shows the results from a TDLDA
|
||||
simulation of SHG for Na<sub>8</sub> attached to a NaCl surface
|
||||
[<a href="../literatur.html#own1224">248</a>]. The spectra resulting
|
||||
from
|
||||
irradiation with a 1.4 eV pulse shows nicely the peaks at multiple
|
||||
frequencies. The SHG signal can be enhanced by increasing the laser
|
||||
intensity. This, however, breaks down at some point where the signals
|
||||
are substantially broadened. This is caused by a large ionization
|
||||
which spoils the elsewise clean dipole response of metal clusters.
|
||||
</p>
|
||||
<br>
|
||||
<br>
|
||||
<p>
|
||||
<img src="na6_ar384d_deposit.gif" align="left" width="300">TDLDA
|
||||
coupled with molecular dynamics (MD) for ionic motion is a very
|
||||
powerfull tool to describe cluster dynamics. One application is
|
||||
cluster deposition which is illustrated in the left figure. It shows
|
||||
Na<sub>6</sub> impinging on an Ar surface (for the modeling [<a
|
||||
href="../literatur.html#own1303">328</a>]). The substrate consists of
|
||||
six layers of Ar
|
||||
taken from an appropriate cut of the Ar fcc structure. The Na<sub>6</sub>
|
||||
cluster consist in a ring of 5 ions topped by one ion on the symmetry
|
||||
axix. The Na<sub>6</sub> approaches the surface with the symmetry axis
|
||||
in <i>z</i>
|
||||
direction (=perpendicular) and the top ion facing away from the
|
||||
surface. The upper panel shows the evolution of the <i>z</i>
|
||||
coordinates,
|
||||
Na ions in red and Ar atoms in green. The cluster is immediately
|
||||
stopped by the surface. A large fraction of impact momentum is
|
||||
transferred at once to the substrate and propagates with velocity of
|
||||
light through the layers. The large dissipation through energy
|
||||
transfer and intrinsic cluster excitation leads to catching of the
|
||||
cluster by the subtrate. The kinetic energies in the lower panel
|
||||
confirm the dramatic and very fast energy exchange at the moment of
|
||||
first impact. Another fraction of energy, missing in that figure, is
|
||||
turned into the large shape changes.
|
||||
</p>
|
||||
<p>
|
||||
Clusters in the strong fields of extremely intense lasers show a much
|
||||
different dynamics. The core electrons can be released and contribute
|
||||
strongly to the process. The detailed description at the fully quantum
|
||||
mechanical level of TDLDA becomes untractable. However, the
|
||||
excitations involved validate classical approaches. <img
|
||||
src="MD_fig5.gif" align="right" width="300">
|
||||
The figure to the
|
||||
right shows the result of a molecular dynamics simulation of
|
||||
electronic and ionic dynamics of Na<sub>41</sub><sup>+</sup> under the
|
||||
influence of
|
||||
strong laser fields [<a href="../literatur.html#own1308">332</a>].
|
||||
Ionization is
|
||||
drawn as function of laser intensity. One sees a sharp kink at a
|
||||
critical intensity of <i>I</i>=10<sup>16</sup> W/cm<sup>2</sup>. The
|
||||
critical value is
|
||||
diistinguished by the fact that the Coulomb force from the laser field
|
||||
just equals the binding forces of the core electrons. The increase is
|
||||
due to the core electrons which now start to participate in the
|
||||
process. This view is checked by sepparating the contributions from
|
||||
valence (green) and core electrons (red line). There is indeed zero
|
||||
emission from core electrons up to <i>I</i>=10<sup>16</sup> W/cm<sup>2</sup>
|
||||
and the
|
||||
strong increase above that critical intensity is exclusively due to
|
||||
the core contribution.
|
||||
<br>
|
||||
<br>
|
||||
<br>
|
||||
<br>
|
||||
</p>
|
||||
<table>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<a href="#oben">back to top</a>
|
||||
</td>
|
||||
</tr>
|
||||
</table>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
<div id="footer">
|
||||
<p></p>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,469 @@
|
|||
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
|
||||
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|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
|
||||
<title>Theory of Cluster Dynamics</title>
|
||||
<link href="../style.css" rel="stylesheet" type="text/css" />
|
||||
</head>
|
||||
|
||||
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|
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|
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|
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<div id="navMenu">
|
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|
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<ul>
|
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<li style="margin-top:1px;border-top:1px solid #B0C4DE; "><a href="../index.html">Home</a></li>
|
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|
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|
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|
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|
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|
||||
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|
||||
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|
||||
</div>
|
||||
<div id="image">
|
||||
<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
|
||||
</font><font size="6">
|
||||
</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
|
||||
|
||||
</div>
|
||||
<div id="content">
|
||||
|
||||
<a name="oben">
|
||||
<div style="margin:15px;width:770px;border:1px solid gray;float:left;font-size:10px;">
|
||||
<div style="width:220px;float:left;text-align:center;">
|
||||
<a href="../analysis/detail1.html">1. Analysis of cluster dynamics</a>
|
||||
</div>
|
||||
<div style="width:220px;float:left;text-align:center;font-size:10px;">
|
||||
<a href="../analysis/detail2.html"> 2. Clusters in external fields</a>
|
||||
</div>
|
||||
<div style="width:220px;float:left;text-align:center;font-weight:900;font-size:12px;">
|
||||
<a href="formal.html"> 3. Theoretical developments </a>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</a>
|
||||
|
||||
|
||||
<div id="WideContent">
|
||||
<div id="contentBoxWide">
|
||||
<div id="contentBoxHeader">
|
||||
<p>Time Dependent Density Functional Theory with Molecular Dynamics </p>
|
||||
</div>
|
||||
<div id="contentBoxContent">
|
||||
|
||||
|
||||
<P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<FONT SIZE="+2"><B> TDLDA-MD:</B></FONT>
|
||||
<BR>
|
||||
<BR><FONT SIZE="+1"><B>Time-dependent local-density approximation
|
||||
plus ionic molecular dynamics</B></FONT>
|
||||
<BR>
|
||||
</DIV>
|
||||
|
||||
<P>
|
||||
(<EM>This is a very short summary of our formal scheme. A most
|
||||
detailed description is found in </EM>[<a href="../literatur.html#own1281">303</a>].)
|
||||
|
||||
<P>
|
||||
The
|
||||
<FONT COLOR="#ff0000"> electron cloud</FONT> is described by density functional theory at
|
||||
the level of TDLDA. The dynamical degrees of freedom are the set of
|
||||
occupied
|
||||
<FONT COLOR="#ff0000"> single-electron wavefunctions
|
||||
<FONT COLOR="#ff0000"><!-- MATH
|
||||
$\varphi_\alpha$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img1.png"
|
||||
ALT="\bgroup\color{red}$ \varphi_\alpha$\egroup"></FONT></FONT>. The
|
||||
|
||||
<FONT COLOR="#00b300"> ions</FONT> are treated by classical MD and their degrees of freedom are
|
||||
the
|
||||
<FONT COLOR="#00b300"> positions <i><b>R<sub>I</sub></b></i> and momenta <i><b>P<sub>I</sub></b></i>
|
||||
<FONT COLOR="#00b300"><!-- MATH
|
||||
$({R}_I,{P}_I)$
|
||||
-->
|
||||
<!-- <IMG
|
||||
WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="\bgroup\color{dgreen}$ ({R}_I,{P}_I)$\egroup"></FONT></FONT>.--></FONT></FONT>. The starting
|
||||
point is the total energy given by:
|
||||
<BR>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{eqnarray*}
|
||||
E_{\rm total}
|
||||
&=&
|
||||
{\color{red}
|
||||
E_{\rm kin}(\{\varphi_\alpha\})
|
||||
+
|
||||
E_{\rm C}(\rho)
|
||||
+
|
||||
E_{\rm xc}^{\rm (LDA)}(\rho_\uparrow,\rho_\downarrow)
|
||||
}
|
||||
+
|
||||
E_{\rm el,ion}({\color{red} \rho},{\color{dgreen} \{{R}_I\}})
|
||||
+
|
||||
{\color{dgreen} E_{\rm ion}(\{{R}_I,{P}_I\})}
|
||||
+
|
||||
E_{\rm ext}({\color{red} \rho},{\color{dgreen} {R}_I},t)
|
||||
\quad.
|
||||
\end{eqnarray*}
|
||||
-->
|
||||
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
|
||||
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
|
||||
WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img3.png"
|
||||
ALT="$\displaystyle E_{\rm total}$"></TD>
|
||||
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
|
||||
WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img4.png"
|
||||
ALT="$\displaystyle =$"></TD>
|
||||
<TD ALIGN="LEFT" NOWRAP><IMG
|
||||
WIDTH="690" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img5.png"
|
||||
ALT="$\displaystyle {\color{red}
|
||||
E_{\rm kin}(\{\varphi_\alpha\})
|
||||
+
|
||||
E_{\rm C}(\rho)
|
||||
+
|
||||
...
|
||||
...}_I,{P}_I\})}
|
||||
+
|
||||
E_{\rm ext}({\color{red} \rho},{\color{dgreen} {R}_I},t)
|
||||
\quad.$"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
</TD></TR>
|
||||
</TABLE></DIV>
|
||||
<BR CLEAR="ALL">
|
||||
|
||||
<P>
|
||||
The electronic kinetic energy
|
||||
<FONT COLOR="#00b300"><!-- MATH
|
||||
${\color{red} E_{\rm kin}}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img6.png"
|
||||
ALT="\bgroup\color{dgreen}$ {\color{red} E_{\rm kin}}$\egroup"></FONT> employs the
|
||||
single-electron wavefunctions
|
||||
<FONT COLOR="#00b300"><!-- MATH
|
||||
${\color{red} \varphi_\alpha}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img7.png"
|
||||
ALT="\bgroup\color{dgreen}$ {\color{red} \varphi_\alpha}$\egroup"></FONT> which maintains
|
||||
the quantum mechanical shell effects. All other electronic energies
|
||||
refer only to the local spin-densities or total density
|
||||
|
||||
<!-- MATH
|
||||
${\color{red} \rho=\rho_\uparrow+\rho_\downarrow}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="92" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img8.png"
|
||||
ALT="\bgroup\color{dgreen}$ {\color{red} \rho=\rho_\uparrow+\rho_\downarrow}$\egroup">; the Coulomb energy
|
||||
|
||||
<FONT COLOR="#00b300"><!-- MATH
|
||||
${\color{red} E_{\rm C}}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="\bgroup\color{dgreen}$ {\color{red} E_{\rm C}}$\egroup"></FONT> naturally, and the exchange-correlation energy
|
||||
|
||||
<FONT COLOR="#00b300"><!-- MATH
|
||||
${\color{red} E_{\rm xc}}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="\bgroup\color{dgreen}$ {\color{red} E_{\rm xc}}$\egroup"></FONT> by virtue of the LDA (often augmented by a
|
||||
self-interaction correction (SIC) <a href="../literatur.html#own1252">[277]</a>). The electron-ion coupling
|
||||
|
||||
<FONT COLOR="#00b300"><!-- MATH
|
||||
$E_{\rm el,ion}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="\bgroup\color{dgreen}$ E_{\rm el,ion}$\egroup"></FONT> is realized by pseudo-potentials, mostly soft local
|
||||
ones <a href="../literatur.html#own1216">[249]</a>. The ionic part
|
||||
<FONT COLOR="#00b300"><!-- MATH
|
||||
${\color{dgreen} E_{\rm ion}}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img12.png"
|
||||
ALT="\bgroup\color{dgreen}$ {\color{dgreen} E_{\rm ion}}$\egroup"></FONT> is composed of Coulomb
|
||||
interaction and kinetic energy. Excitation mechanisms (laser, ionic
|
||||
collisions) are described in
|
||||
<FONT COLOR="#00b300"><!-- MATH
|
||||
$E_{\rm ext}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img13.png"
|
||||
ALT="\bgroup\color{dgreen}$ E_{\rm ext}$\egroup"></FONT> as external time-dependent
|
||||
potentials.
|
||||
|
||||
<P>
|
||||
The coupled equations of motion are obtained in standard manner by
|
||||
variation. They read
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
{\color{red}
|
||||
\imath\partial_t\varphi_\alpha
|
||||
=
|
||||
\Big(\frac{\hat{p}^2}{2m}
|
||||
+
|
||||
\frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}}\Big)
|
||||
\varphi_\alpha
|
||||
}
|
||||
\qquad,\qquad
|
||||
{\color{dgreen} \partial_t{R}_I
|
||||
=
|
||||
\frac{{P}_I}{M_I}
|
||||
\quad,\quad
|
||||
\partial_t{P}_I
|
||||
=
|
||||
-\nabla_{{R}_I}E_{\rm total}}
|
||||
\quad.
|
||||
\end{displaymath}
|
||||
-->
|
||||
<P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<IMG
|
||||
WIDTH="618" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img14.png"
|
||||
ALT="\bgroup\color{dgreen}$\displaystyle {\color{red}
|
||||
\imath\partial_t\varphi_\alpha...
|
||||
...M_I}
|
||||
\quad,\quad
|
||||
\partial_t{P}_I
|
||||
=
|
||||
-\nabla_{{R}_I}E_{\rm total}}
|
||||
\quad.
|
||||
$\egroup">
|
||||
</DIV><P>
|
||||
where
|
||||
<FONT COLOR="#00b300"><!-- MATH
|
||||
${\color{red} \sigma_\alpha}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img15.png"
|
||||
ALT="\bgroup\color{dgreen}$ {\color{red} \sigma_\alpha}$\egroup"></FONT> is the spin orientation of the state
|
||||
|
||||
<FONT COLOR="#00b300"><!-- MATH
|
||||
${\color{red} \alpha}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img16.png"
|
||||
ALT="\bgroup\color{dgreen}$ {\color{red} \alpha}$\egroup"></FONT>. The equations imply a non-adiabatic coupling which
|
||||
goes beyond usual Born-Oppenheimer approach. Non-adiabatic effects
|
||||
become crucial in cluster dynamics induced by strong fields. The
|
||||
numerical solution involves the representation of the wavefunctions on
|
||||
a spatial grid, time-splitting for the electronic propagation and the
|
||||
Verlet algorithm for MD, for details see [<a href="../literatur.html#own1230">254</a>]. The obtained
|
||||
wavefunctions, densities, and ionic coordinates allow to compute a
|
||||
wide variety of observables, <!-- at the side of the electrons -->e.g.
|
||||
|
||||
<FONT COLOR="#ff0000"> optical absorption spectra</FONT> [<a href="../literatur.html#own1155">9</a>],
|
||||
<FONT COLOR="#ff0000"> angular distributions</FONT>
|
||||
[<a href="../literatur.html#own1288">313</a>],
|
||||
<FONT COLOR="#ff0000"> emission spectra</FONT> [<a href="../literatur.html#own1285">304</a>],
|
||||
or
|
||||
<FONT COLOR="#ff0000"> ionization</FONT> [<a href="../literatur.html#own1186">208</a>] for electronic degrees of freedom.
|
||||
The
|
||||
<FONT COLOR="#00b300"> ionic configurations</FONT> can be measured indirectly through optical
|
||||
response and its dynamics with various pump and probe scenarios
|
||||
[<a href="../literatur.html#own1246">290</a>].
|
||||
|
||||
<P></P>
|
||||
<P>
|
||||
|
||||
Often, we use a
|
||||
<FONT COLOR="#ff0000"> semi-classical description for the electronic
|
||||
dynamics</FONT> at the level of Vlasov-LDA, particularly for energetic
|
||||
processes and/or large clusters. Instead of the
|
||||
<FONT COLOR="#ff0000"> wavefunctions</FONT>,
|
||||
the key ingredient becomes here the
|
||||
<FONT COLOR="#ff0000"> one-electron phase-space
|
||||
distribution
|
||||
<FONT COLOR="#ff0000"><!-- MATH
|
||||
$f({r},{p},t)$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img17.png"
|
||||
ALT="\bgroup\color{red}$ f({r},{p},t)$\egroup"></FONT></FONT>. The quantum-mechanical
|
||||
propagation for the electrons is replaced by the Vlasov equation
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
{\color{red}
|
||||
\partial_t f
|
||||
=
|
||||
\frac{{p}}{m}\nabla_{r}f
|
||||
-
|
||||
\Big(
|
||||
\nabla_{r}\frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}}
|
||||
\Big)
|
||||
\nabla_{p}f
|
||||
}
|
||||
\end{displaymath}
|
||||
-->
|
||||
<P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<IMG
|
||||
WIDTH="270" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img18.png"
|
||||
ALT="\bgroup\color{red}$\displaystyle {\color{red}
|
||||
\partial_t f
|
||||
=
|
||||
\frac{{p}}{m}\nabl...
|
||||
...{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}}
|
||||
\Big)
|
||||
\nabla_{p}f
|
||||
}
|
||||
$\egroup">
|
||||
</DIV><P></P>
|
||||
<P>
|
||||
|
||||
again non-adiabatically coupled to ionic motion as above.
|
||||
Note that formally the same Kohn-Sham potential
|
||||
|
||||
<FONT COLOR="#ff0000"><!-- MATH
|
||||
${\color{red} {\delta E_{\rm total}}\big/{\delta\rho_{\sigma_\alpha}}}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="102" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img19.png"
|
||||
ALT="\bgroup\color{red}$ {\color{red} {\delta E_{\rm total}}\big/{\delta\rho_{\sigma_\alpha}}}$\egroup"></FONT>
|
||||
is employed. For a derivation and justification from TDLDA see
|
||||
[<a href="../literatur.html#own1163">182</a>]. The Vlasov-LDA equation is solved with the
|
||||
test-particle method where the distribution function
|
||||
<FONT COLOR="#ff0000"><!-- MATH
|
||||
${\color{red} f}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="\bgroup\color{red}$ {\color{red} f}$\egroup"></FONT> is
|
||||
represented as a sum of Gaussian test-particles which are propagated
|
||||
again by the Verlet algorithm [<a href="../literatur.html#own1248">273</a>].
|
||||
|
||||
<P>
|
||||
The semi-classical description makes it feasible to include dynamical
|
||||
correlations from electron-electron collisions. This is achieved by
|
||||
adding an Ühling-Uhlenbeck collision term leading to
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
{\color{red}
|
||||
\partial_t f
|
||||
=
|
||||
\frac{{p}}{m}\nabla_{r}f
|
||||
-
|
||||
\Big(
|
||||
\nabla_{r}\frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}}
|
||||
\Big)
|
||||
\nabla_{p}f
|
||||
+
|
||||
I_{\rm UU}(f)
|
||||
}
|
||||
\quad.
|
||||
\end{displaymath}
|
||||
-->
|
||||
<P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<IMG
|
||||
WIDTH="372" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="\bgroup\color{red}$\displaystyle {\color{red}
|
||||
\partial_t f
|
||||
=
|
||||
\frac{{p}}{m}\nabl...
|
||||
...\delta\rho_{\sigma_\alpha}}
|
||||
\Big)
|
||||
\nabla_{p}f
|
||||
+
|
||||
I_{\rm UU}(f)
|
||||
}
|
||||
\quad.
|
||||
$\egroup">
|
||||
</DIV><P>
|
||||
|
||||
The collision term
|
||||
<FONT COLOR="#ff0000"><!-- MATH
|
||||
${\color{red} I_{\rm UU}}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img22.png"
|
||||
ALT="\bgroup\color{red}$ {\color{red} I_{\rm UU}}$\egroup"></FONT> is a non-linear functional of
|
||||
the distribution function
|
||||
<FONT COLOR="#ff0000"><!-- MATH
|
||||
${\color{red} f}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="\bgroup\color{red}$ {\color{red} f}$\egroup"></FONT>. It contains terms up to third
|
||||
power in
|
||||
<FONT COLOR="#ff0000"><!-- MATH
|
||||
${\color{red} f}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="\bgroup\color{red}$ {\color{red} f}$\egroup"></FONT>. It is constructed from local and instantaneous
|
||||
collisions which obey energy conservation, momentum conservation, and
|
||||
the Pauli principle [<a href="../literatur.html#own1248">273</a>]. The resulting equation is called the
|
||||
Vlasov-Ühling-Uhlenbeck approach (VUU).
|
||||
|
||||
<P>
|
||||
<center>
|
||||
<table width="70%">
|
||||
<tr>
|
||||
<td align="right">
|
||||
<a href="#top">Back to top </a>
|
||||
</td>
|
||||
</tr>
|
||||
</table>
|
||||
</center>
|
||||
<!-- <HR> --></HR>
|
||||
|
||||
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"></A>
|
||||
|
||||
<!--End of Table of Child-Links-->
|
||||
<!-- <HR>
|
||||
<ADDRESS>
|
||||
Paul-Gerhard Reinhard
|
||||
2006-03-18
|
||||
</ADDRESS> -->
|
||||
<!-- START CONTENT HERE -->
|
||||
|
||||
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
<div id="footer">
|
||||
<p></p>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
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Reference in New Issue