Lindemann index

<h2 id="definition">Definition</h2>
<p>The local Lindemann index is defined as:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
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    {\displaystyle q_{i}={\frac {1}{N-1}}\sum _{j\neq i}{\frac {\sqrt {\langle r_{ij}^{2}\rangle -\langle r_{ij}\rangle ^{2}}}{\langle r_{ij}\rangle }}}

</p><p>where angle brackets indicate a time average. The global Lindemann index is a system average of this quantity.
</p>
In <a href="/facts/Condensed_matter_physics/EBVI5nmL">condensed matter physics</a>
a departure from <a href="/facts/Linearity/fChtg3KC">linearity</a> in the behaviour of the global Lindemann index or an increase above a threshold value related to the  spacing between <a href="/facts/Atom/FoQ4kGN5">atoms</a> (or <a href="/facts/Micelle/eMG9whea">micelles</a>, particles, globules, etc.) is often taken as the indication that a solid-liquid <a href="/facts/Phase_transition/eLP2BY8R">phase transition</a> has taken place. See <a href="/facts/Lindemann_melting_criterion/zhhDBo5V">Lindemann melting criterion</a>.
<a href="/facts/Biomolecule/DGvFzN7R">Biomolecules</a>
often possess separate regions with different order characteristics. In order to quantify or illustrate local disorder, the local Lindemann index can be used.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
<h2 id="factors-when-using-the-lindemann-index">Factors when using the Lindemann index</h2>
<p>Care must be taken if the molecule possesses globally defined dynamics, such as about a hinge or pivot, because these motions will obscure the local motions which the Lindemann index is designed to quantify. An appropriate tactic in this circumstance is to sum the rij only over a small number of neighbouring atoms to arrive at each qi. A further variety of such modifications to the Lindemann index are available and have different merits, e.g. for the study of <a href="/facts/Glassy_phase/YTIzmrR0">glassy</a> vs <a href="/facts/Crystalline/e2R2mk3E">crystalline</a> materials.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Lindemann FA (1910). "The calculation of molecular vibration frequencies". Phys. Z. 11: 609–612. <a href="/wiki/Frederick_Lindemann" target="_blank">/wiki/Frederick_Lindemann</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Zhang K, Stocks GM, Zhong J (2007). "Melting and premelting of carbon nanotubes". Nanotechnology. 18 (285703): 285703. Bibcode:2007Nanot..18B5703Z. doi:10.1088/0957-4484/18/28/285703. S2CID 104003958. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Rueda M, Ferrer-Costa C, Meyer T, Perez A, Camps J, Hospital A, Gelpi JL, Orozco M (2007-01-16). "A consensus view of protein dynamics". PNAS. 104 (3): 796–801. Bibcode:2007PNAS..104..796R. doi:10.1073/pnas.0605534104. PMC 1783393. PMID 17215349. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1783393" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1783393</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Zhou Y, Karplus M, Ball KD, Berry RS (2002). "The distance fluctuation criterion for melting: Comparison of square-well and Morse potential models for clusters and homopolymers". J. Chem. Phys. 116 (5): 2323. Bibcode:2002JChPh.116.2323Z. doi:10.1063/1.1426419. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Lindemann index open-in-new

Lindemann index