Question: localised mode, localisation effect of linear atomic chain

[jaykay]

urgent question: localised mode, localisation effect of linear atomic chain

Hello everyone, it is great place you have here, hope I can learn a lot from you

I am doing some readings and there are couple of concepts that I haven't been familiar with and if you spend a little time to help me with I would be really grateful:

Can you please give some explanation aobut the concepts of "localised modes", "localisation effects" and "localisation length" for a linear (atonmic) chain.

The definition I am given is "For the normalised eigenvectors q(wr) ‘localised’ modes may be thought of as those for which only a contiguous subset of the
components of q(wr) are significantly different from zero" which I find hard to understand.

Can you please explain abit more about localisation effect and how we would get an estimate of the "localisation length".


If it is too much trouble for you please point me to a source of information

Any help greatly appreciated


Thanks

 

[Nate810]

!Localised modes are those in which the magnitude of the eigenvector components is very large for a subset of the atoms in the chain, with the magnitude falling off sharply away from this subset. This subset of atoms is effectively "localised" around the mode's energy peak, and therefore the mode can be referred to as a localised mode. The localisation length is the number of atoms in the chain which contribute significantly to the localised mode. The localisation effect describes how the normal modes of vibration are affected by the interatomic interactions. If the interatomic interactions are strong enough, the normal modes become localised and the localisation length will be smaller than the size of the entire chain. On the other hand, if the interatomic interactions are weak, the normal modes will spread out across the entire chain and the localisation length will be equal to the size of the entire chain. The localisation length (or the localisation effect) can be estimated by looking at the magnitudes of the components of the eigenvector corresponding to the particular mode of interest. The magnitude of the components should decrease exponentially as you move away from the centre of the localised mode, and the localisation length can be estimated as the number of atoms where the magnitude drops below a certain threshold.

 

[wais]

for your question!

Localised modes refer to specific vibrational modes within a system that are confined to a certain region or subset of the system. In the case of a linear atomic chain, this would mean that there are certain atoms that are vibrating more significantly than others, creating a localized effect. This can be caused by various factors such as impurities, defects, or boundary conditions.

The localisation effect, on the other hand, is the phenomenon where energy or particles become confined to a specific region within a system. In the context of a linear atomic chain, this means that the vibrational energy is concentrated in certain atoms, leading to a localised mode.

The localisation length is a measure of how far the vibrational energy extends within the system. It can be estimated by looking at the decay of amplitude or energy as you move away from the localized region. The longer the localisation length, the further the energy will extend.

I hope this helps clarify the concepts for you. For further information, I suggest looking into solid state physics or condensed matter physics textbooks as they often cover these topics in more detail. Best of luck with your studies!