The torsional Raman spectra of two astrophysically detected isotopologues of dimethyl-ether, ($^{12}$CH$_3$O$^{12}$CH$_3$ and $^{13}$CH$_3$O$^{12}$CH$_3$), have been recorded at room temperature and cooled in supersonic jet, and interpreted with the help of highly correlated ab initio calculations. Dimethyl-ether displays excited torsional and vibrational levels at low energy that can be populated at the temperatures of the star forming regions, obliging to extend the analysis of the rotational spectrum over the ground state. Its spectrum in the THz region is rather complex due to the coupling of the torsional overtones $2 u_{11}$ and $2 u_{15}$ with the COC bending mode, and the presence of many hot bands. The torsional overtones are set here at $2 u_{11}=385.2$~cm$^{-1}$ and $2 u_{15}=482.0$~cm$^{-1}$ for $^{12}$CH$_3$O$^{12}$CH$_3$, and $2 u_{11}=385.0$~cm$^{-1}$ and $2 u_{15}=481.1$~cm$^{-1}$ for $^{13}$CH$_3$O$^{12}$CH$_3$. The new assignment of $2 u_{11}$ is downshifted around $sim 10$~cm$^{-1}$ with respect to the literature. All the other (hot) bands have been re-assigned consistently. In addition, the infrared-forbidden torsional fundamental band $ u_{11}$ is observed here at 197.8~cm$^{-1}$. The new spectral characterization in the THz region reported here provides improved values of the Hamiltonian parameters, to be used in the analysis of the rotational spectra of DME isotopologues for further astrophysical detections.
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