One-dimensional “monatomic” crystals

The simplest model is the one-dimensional monatomic crystal consisting of N atoms, each with mass M and separated by lattice spacing a along the... 

The nature of the problem. Consider a monatomic one-dimensional crystal (Fig. 2). Imagine the state of the crystal under static conditions, in which the temperature is zero and zero-point motion is absent. Note that this state is inaccessible in the laboratory whereas temperatures close to absolute zero may be achieved, zero-point motion cannot be eliminated. In this idealized situation, the atoms are stationary and occupy ideal crystallographic sites. The resulting structure possesses high symmetry and a small unit cell. From a theoretical point of view, these are desirable properties. We are able to apply periodic boundary conditions, and the period is small. These properties are unique to the static crystal and account for the importance of this idealization in theoretical studies. 

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