Low energy spin wave excitations in bilayered magnetic manganite   La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7}$ ($0.30\leq{x}\leq{0.50}$)
  We have studied the low-temperature behavior of the magnetization and the specific heat of the bilayered perovskite system $% La_{2-2x}Sr_{1+2x}Mn_{2}O_{7}$, for $0.30\leq x\leq 0.50$. Our analysis reveals that below 30 K the temperature dependence of the magnetization of the ferromagnetic samples, $x=0.30$, 0.32, and 0.36, in a field of 1T can be interpreted in terms of the thermal excitations of a two-dimensional gas of ferromagnetic magnons. The specific heat in zero field for these samples as for the $x=0.50$ antiferromagnetic one, is linear with temperature between the range of 1.8K $\leq $ T $\leq $ 10K. This behavior can be also explained by the magnon gas model. By comparing specific heat measurements in zero field with those taken in a field of 9T we are able to extract the lattice and electronic contributions and determine the in-plane exchange interactions. That are found to be in reasonable agreement with the values inferred from the analysis of the magnetization data and also with the values reported by inelastic neutron scattering studies. In addition, we found that the electronic density of states obtained for the $x=0.50$ sample is in agreement with previous band structure calculations. 
H. Martinho, C. Rettori, D. L. Huber, S. B. Oseroff and J. F. Mitchell