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We show that a one-component variable-emissivity-spectral-index model (thefree-{\alpha} model) provides more physically motivated estimates of dusttemperature at the Galactic polar caps than one- or two-componentfixed-emissivity-spectral-index models (fixed-{\alpha} models) for interstellardust thermal emission at far-infrared and millimeter wavelengths. For thecomparison we have fit all-sky one-component dust models with fixed or variableemissivity spectral index to a new and improved version of the 210-channel dustspectra from the COBE-FIRAS, the 100 - 240 {\mu}m maps from the COBE-DIRBE andthe 94 GHz dust map from the WMAP. The best model, the free-{\alpha} model, iswell constrained by data at 60-3000 GHz over 86 per cent of the total sky area.It predicts dust temperature (Tdust) to be 13.7-22.7 (+/-1.3) K, the emissivityspectral index ({\alpha}) to be 1.2 - 3.1 (+/-0.3) and the optical depth({\tau}) to range 0.6 - 46 x 10^(-5) at \nu 0 = 900 GHz (\lambda 0 = 333{\mu}m) with a 23 per cent uncertainty. Using these estimates, we presentall-sky evidence for an inverse correlation between the emissivity spectralindex and dust temperature, which fits the relation {\alpha} =1/({\delta}+{\omega} Tdust) with {\delta} = -0.510 +/- 0.011 and {\omega} =0.059 +/- 0.001. This best model will be useful to cosmic microwave backgroundexperiments for removing foreground dust contamination and it can serve as anall-sky extended-frequency reference for future higher resolution dust models.

All-sky Observational Evidence for An Inverse Correlation between Dust Temperature and Emissivity Spectral Index