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The 13C (α,n)16O reaction cross-section is important data for nuclear physics, astrophysical, and neutrino physics experiments, however, they exhibit uncertainties due to the discrepancies in the experimental data. In this study, using the Nedis-2m program code, the energy spectrum of α-induced neutrons in a thin carbon target was calculated and the corresponding reaction cross-section was refined in the alpha particle energy range of 5-8 MeV. The results were used to calculate the intensity and energy spectrum of background neutrons produced in the liquid scintillator of KamLAND. The results will be useful in a variety of astrophysical and neutrino experiments especially those based on LS or Gd-LS detectors.