Electron density variations in the interstellar medium and the average frequency profile of a scintle from pulsar scintillation spectra

We observed the scintillation pattern of nine bright pulsars at 324 MHz and three at 1.68 GHz and analyzed the wavenumber spectrum which is related to electron density variations of the plasma turbulence of the interstellar medium. For all pulsars the frequency section of the autocorrelation function of the dynamic spectra to at least 45\% of the maximum corresponds to predictions of scattering theories with a range of power-law exponents of the wavenumber spectrum of $3.56 \leq \alpha \leq 3.97$ with errors $\leq 0.05$ and a mean with standard deviation of $3.76\pm0.13$. The range includes $\alpha=3.67$ for the Kolmogorov spectrum. Similar results although with larger errors were found from the Fourier transform of the autocorrelation functions down to $\sim 10^{-3}$ of the maximum. No clear case of a distinction between thin-screen and extended-medium scattering models was found. The average frequency profile of the scintles can be characterized for steep wavenumber spectra with $\alpha\lesssim4$ by a cusp with a somewhat rounded peak. For flatter spectra, down to at least $\alpha\sim 3.56$ the cusp with its peak becomes more pronounced and its decay steepens. We discuss our findings in the context of scattering characteristics of the interstellar medium.