Proton and carbon ion therapy is an emerging technique used for the treatment of solid cancers. The monitoring of the dose delivered during such treatments and the on-line knowledge of the Bragg peak position is still a matter of research. A possible technique exploits the collinear $511 kiloelectronvolt$ photons produced by positrons annihilation from $beta^+$ emitters created by the beam. This paper reports rate measurements of the $511 kiloelectronvolt$ photons emitted after the interactions of a $80 megaelectronvolt / u$ fully stripped carbon ion beam at the Laboratori Nazionali del Sud (LNS) of INFN, with a Poly-methyl methacrylate target. The time evolution of the $beta^+$ rate was parametrized and the dominance of $^{11}C$ emitters over the other species ($^{13}N$, $^{15}O$, $^{14}O$) was observed, measuring the fraction of carbon ions activating $beta^+$ emitters $A_0=(10.3pm0.7)cdot10^{-3}$. The average depth in the PMMA of the positron annihilation from $beta^+$ emitters was also measured, $D_{beta^+}=5.3pm1.1 millimeter$, to be compared to the expected Bragg peak depth $D_{Bragg}=11.0pm 0.5 millimeter$ obtained from simulations.
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