Far-Infrared Studies of the Development of the Electronic Lattices in La$_2$CuO$_{4+0.0315}$

La$_{2}$CuO$_{4+0.0315}$ undergoes a superconducting transition with an onset at T$_{c}$ = 30 K when it is slow-cooled at a rate of 1 K/min from the room temperature but it becomes a T$_{c}$ = 15 K superconductor when quenched down to 77 K in less than 1 minute. The corresponding far-infrared reflectivity measurements reveal that there exists only one collective mode at $\omega _{G1} \quad \sim $ 24 cm$^{-1}$ in the T$_{c}$ = 15 K phase [1]. However, when slow-cooled, there develop a series of collective modes at $\omega _{G1} \quad \sim $ 24 cm$^{-1}$, $\omega _{G2} \quad \sim $ 37 cm$^{-1}$, $\omega _{G3} \quad \sim $ 46 cm$^{-1}$ , and $\omega _{G4}$ $\sim $ 60 cm$^{-1}$ with a complex temperature dependences. Upon cooling, $\omega _{G1}$ and $\omega _{G3}$ show the similar behaviors as the Sr/O co-doped LSCO at the hole concentration p = 0.07 [1] while the $\omega _{G2}$ mode loses its oscillator strength for T $<$ 150 K and the $\omega _{G4}$ mode shows no change in its strength at all temperatures. We explain the implications of our findings within the 2D electronic lattice model [1, 2]. \begin{enumerate} \item Y. H. Kim and P.H. Hor, Mod. Phys. Lett. B\textbf{15}, 497 (2001). \item P.H. Hor and Y.H. Kim, J. Phys.: Condens. Matter \textbf{14}, 10377 (2002). \end{enumerate}