deletions | additions       diff --git a/The Peierls Transition.tex b/The Peierls Transition.tex  index 73c3511..95ab639 100644  --- a/The Peierls Transition.tex  +++ b/The Peierls Transition.tex 
...
\section{The Peierls Transition}  The Fermi surface of one-dimensional lattice is not stable at zero temperature due to its geometry and lattice deformation. Considering an one-dimensional lattice with spacing a at zero temperature. Without the electron-phonon interaction, the electrons will fill half of the band to Fermi energy $\epsilon_F$ with Fermi wave vector $k_F = \frac{\pi}{2a}$ (Fig.2a). The empty states above the Fermi energy suggest the metallic conductance of the lattice. This state is unstable in the presence of lattice deformation and phonon. If we now introduce lattice distortion with $\lambda = 2a = \frac{\pi}{k_F}$, transforming the lattice spacing from a to 2a, the period of electric potential doubles and the strength of the potential increases accordingly. This distortion opens a band gap at Fermi surface (Fig.2b) and the lattice becomes insulator. The total energy of the distorted lattice is determined by the increase of elastic deformation energy in the lattice and the decrease in electric energy of electrons. The piled up electrons form an energetically favorable state corresponding to the CDW as an analogy to the bonding state of molecules.   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. 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. $T_P \arrow 0K$