Upper plot: β_eff vs. λ, as a function of E_g, where β_eff is the effective emissivity slope resulting from a combination of the a-Sil_Fe/a-C and a-C:H/a-C dust population optical properties. The green triple dot-dashed line shows the data for only the a-Sil_Fe/a-C grains, which do not depend on the band gap of the thin a-C(:H) coating. The λ-dependence and variations are therefore only due to the a-C(:H) dust properties. The line colour-coding is as per Fig. 11. The horizontal grey lines and wide yellow bands show β_eff = 1.80 ± 0.05 and 1.60 ± 0.05. Lower plot: the emissivity slope, β, for a-C(:H) dust, for λ = 100 μm, 300 μm, and 1 mm, as a function of E_g, for a = 100, 10 and 1 nm. The vertical grey dashed line shows the experimentally-derived “block” on band gap evolution (see Paper II). The grey area indicates the range of a-C materials that are consistent with many diffuse ISM dust properties (Sect. 6). The coloured squares at the bottom indicate the band gap colour coding scheme.