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The '''Lindemann index'''<ref name="Lindemann1910">{{cite journal |author=[[Frederick Lindemann|Lindemann FA]] |title=The calculation of molecular vibration frequencies |journal=Physik. Z. |volume=11 |year=1910 |pages=609–612}}</ref> is a simple measure of thermally driven [[Entropy|disorder]] in atoms or molecules. The local Lindemann index is defined as:<ref name="Zhang2007">{{cite journal |author=Zhang K, Stocks GM, Zhong J |title=Melting and premelting of carbon nanotubes |journal=Nanotechnology |volume=18 |issue=285703 |year=2007 |doi=10.1088/0957-4484/18/28/285703 |pages=285703}}</ref>
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<math>
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}
</math>
 
Where angle brackets indicate a time average. The global Lindemann index is a system average of this quantity.
 
;In [[condensed matter physics]] : a departure from [[linearity]] in the behaviour of the global Lindemann index or an increase above a threshold value related to the  spacing between [[atom]]s (or [[micelle]]s, particles, globules, etc.) is often taken as the indication that a solid-liquid [[phase transition]] has taken place. See [[Lindemann melting criterion]].
 
;[[Biomolecule]]s : often possess separate regions with different order characteristics. In order to quantify or illustrate local disorder, the local Lindemann index can be used.<ref name="Rueda2007">{{cite journal |author=Rueda M, Ferrer-Costa C, Meyer T, Perez A, Camps J, Hospital A, Gelpi JL, Orozco M |title=A consensus view of protein dynamics |journal=PNAS |date=2007-01-16|volume=104 |pages=796–801 |doi=10.1073/pnas.0605534104 |pmid=17215349 |issue=3 |pmc=1783393}}</ref>
 
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 '''r<sub>ij</sub>''' only over a small number of neighbouring atoms to arrive at each '''q<sub>i</sub>'''. A further variety of such modifications to the Lindemann index are available and have different merits, e.g. for the study of [[glassy]] vs [[crystalline]] materials.<ref name="Zhou2002">{{cite journal |author=Zhou Y, Karplus M, Ball KD, Berry RS |title=The distance fluctuation criterion for melting: Comparison of square-well and Morse potential models for clusters and homopolymers |journal=J. Chem. Phys. |volume=116 |issue=5 |page=2323 |year=2002 |doi=10.1063/1.1426419}}</ref>
 
==References==
{{reflist}}
 
[[Category:Molecular physics]]
[[Category:Condensed matter physics]]
[[Category:Dimensionless numbers]]

Revision as of 02:07, 25 February 2014

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