Lindemann index

The Lindemann index^[1] is a simple measure of thermally driven disorder in atoms or molecules. The local Lindemann index is defined as:^[2]

$q_{i}={\frac {1}{N-1}}\sum _{j\neq i}{\frac {\sqrt {\langle r_{ij}^{2}\rangle -\langle r_{ij}\rangle ^{2}}}{\langle r_{ij}\rangle }}$

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, etc.) is often taken as the indication that a solid-liquid phase transition has taken place. See Lindemann melting criterion.

Biomolecules: often possess separate regions with different order characteristics. In order to quantify or illustrate local disorder, the local Lindemann index can be used.^[3]

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, e.g. for the study of glassy vs crystalline materials.^[4]

References

↑ Lindemann FA (1910). "The calculation of molecular vibration frequencies". Physik. Z. 11: 609–612.
↑ Zhang K, Stocks GM, Zhong J (2007). "Melting and premelting of carbon nanotubes". Nanotechnology. 18 (285703): 285703. doi:10.1088/0957-4484/18/28/285703.
↑ 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. doi:10.1073/pnas.0605534104. PMC 1783393. PMID 17215349.
↑ 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. doi:10.1063/1.1426419.

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