An accurate calibration of the JET neutron diagnostics with a 14 MeV neutron generator was performed in the first half of 2017 in order to provide a reliable measurement of the fusion power during the next JET deuterium-tritium (DT) campaign. In order to meet the target accuracy, the chosen neutron generator has been fully characterized at the Neutron Metrology Laboratory of the National Physical Laboratory (NPL), Teddington, United Kingdom. The present paper describes the measurements of the neutron energy spectra obtained using a high-resolution single-crystal diamond detector (SCD). The measurements, together with a new neutron source routine 'ad hoc' developed for the MCNP code, allowed the complex features of the neutron energy spectra resulting from the mixed D/T beam ions interacting with the T/D target nuclei to be resolved for the first time. From the spectral analysis a quantitative estimation of the beam ion composition has been made. The unprecedented intrinsic energy resolution (<1% full width at half maximum (FWHM) at 14 MeV) of diamond detectors opens up new prospects for diagnosing DT plasmas, such as, for instance, the possibility to study non-classical slowing down of the beam ions by neutron spectroscopy on ITER.