Lattice Waves: Atom Chain Fixed at Ends or Free is Space?

[dimensionless]

My textbook treats lattice waves as if they are waves propagating through a mass-spring chain. Kind of like this:

M -/\/\/- M -/\/\/- M -/\/\/- M

The book talks about the motion of the nth atom. It says that the motion of the nth atom has been simplified by only considering the motion of its nearest neighbors. Should I treat the chain of atoms as though it is floating free in space(as shown above)? Should I instead consider the chain as if the ends were fixed to walls by springs(as shown below)?

|||-/\/\/- M -/\/\/- M -/\/\/- M -/\/\/- M -/\/\/-|||

Should I not use either of these simplifications? I am investigating the dispersion curves.

 

[Dr Transport]

Usually the chain is infinite and the ends are not considered. If they are, you need to worry about the BC's and then you get standing waves.

 

[charng]

I would suggest considering both scenarios as they can provide valuable insights into the behavior of lattice waves. Treating the chain as if it is floating free in space allows for a simplified analysis of the motion of individual atoms and their nearest neighbors. This can be useful in understanding the fundamental properties of lattice waves, such as their frequency and wavelength.

On the other hand, considering the chain as if the ends were fixed to walls by springs allows for a more realistic representation of lattice waves in a physical system. This can help in understanding how external forces and constraints affect the propagation of lattice waves.

Ultimately, the choice between these two simplifications will depend on the specific research question and the level of complexity needed to answer it. It is important to carefully consider the assumptions and limitations of each approach and use them appropriately in your investigation.

It is also worth noting that there may be other factors to consider, such as the type of material and the type of lattice structure, which may affect the behavior of lattice waves. Therefore, it may be beneficial to explore and compare different scenarios to gain a more comprehensive understanding of lattice waves and their dispersion curves.