deletions | additions       diff --git a/The Peierls Transition.tex b/The Peierls Transition.tex  index 8238263..ca2dc6d 100644  --- a/The Peierls Transition.tex  +++ b/The Peierls Transition.tex 
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When the lattice is cooled down below certain critical temperature, also called Peierls temperature $T_P$, the transition from the state (a) to state (b) will take place, which is known as the Peierls transition \cite{Peierls_2001}. The Peierls temperatures for NbSe$_3$ are $T_P = T_1 = 142K$ and $T_P = T_2 = 58K$, below which the CDW as a whole is pinned on the sites of atoms, resulting in increase in resistance at these two temperatures. The behavior and detailed physical models will be discussed later.   The CDW state is the ground state of quasi-1D materials below Peierls temperature. In an absolute 1D crystal, there is no CDW state, i.e. $T_P \rightarrow 0K$ since the effects of random distortion cancel within the crystal. If the distortion in the lattice is coherent, for example, with displacement of the ith atom $\delta u_i$ being   \begin{equation}  \delta u_i = (-1)^{i} \delta a \sin{\omega t}  \end{equation}  where $\delta a$ represents the displacement amplitude, the CDW state will then take place. Such coherence can be achieved by 2D or 3D materials with 1D conducting chains of weak interactions among neighboring chains.