Contribution of acoustic modes to the density of vibrational states measured by inelastic scattering techniques

We consider the contribution of acoustic modes to the density of vibrational states measured by nuclear inelastic, inelastic x-ray, and inelastic neutron scattering. In nuclear and x-ray scattering, the low-energy part of the density of states (DOS) is compared with the contribution of acoustic modes to the generalized density of states. Different to that, in neutron scattering the DOS is compared with the contribution of acoustic modes to the true density of states. We argue that in general this is not correct and that similar to nuclear and x-ray scattering, the neutron data in most cases must also be compared with the contribution of acoustic modes to the generalized DOS. For neutron scattering, this contribution usually is smaller than the contribution to the true DOS. Thus, the comparison of the neutron data with the contribution of acoustic modes to the true DOS systematically overestimates the level of acoustic modes. However, an extrapolation of the neutron DOS to zero energy often exceeds even this overestimated level. In our eyes, even for glasses the manifold excess of the extrapolation of the neutron DOS to zero energy over the expected level of acoustic modes seems to be unreasonable even though in this case one can still argue on existing of additional soft modes. However, a similar excess observed also for crystalline samples clearly indicates an uncertainty of the absolute scale of the DOS measured by neutron scattering.