Lindemann index

The Lindemann Index is a simple measure of thermally driven disorder. The local Lindemann index is defined as: [Melting and premelting of carbon nanotubes; Nanotechnology 18 (2007) 285703; Kaiwang Zhang, G Malcolm Stocks and Jianxin Zhong]

$q\_i\; =\; frac\{1\}\{N\; -\; 1\}\; sum\_\{j\; eq\; i\}\; frac\{sqrt\{langle\; r\_\{ij\}^2\; angle\; -\; langle\; r\_\{ij\}\; angle^2\; \{langle\; r\_\{ij\}\; angle\}$

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 atoms ( or micelles, particles, globules... ) is often taken as the indication that a solid-liquid phase transition has taken place.

;In Biomolecules

Biomolecules often possess separate regions with different order characteristics. In order to quantify or illustrate local disorder, the local Lindemann index can be used. [A Consensus View of Protein Dynamics; PNAS 104 (2007) 796; Manuel Rueda, Carles Ferrer-Costa, Tim Meyer, Alberto Perez, Jordi Camps, Adam Hospital, Josep Lluis Gelpi and Modesto Orozco]

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_ij only over a small number of neighbouring atoms to arrive at each q_i. A further variety of such modifications to the Lindemann index are available and have different merits, eg for the study of glassy vs crystalline materials. [The distance fluctuation criterion for melting: Comparison of square-well and Morse potential models for clusters and homopolymers;J Chem Phys 116 5 (2002) 2323;Yaoqi Zhou, Martin Karplus, Keith D. Ball and R. Stephen Berry]

References