In the present paper we shall attempt to collate the results of four separate lines of research which, taken together, appear to provide some interesting checks between theory and experiment. The investigations to be considered are (1) the discussion by Waller* and by Wentzel,† on the basis of the quantum (wave) mechanics, of the scattering of radiation by an atom ; (2) the calculation by Hartree of the Schrödinger distribution of charge in the atoms of chlorine and sodium ; (3) the measurements of James and Miss Firth‡ of the scattering power of the sodium and chlorine atoms in the rock-salt crystal for X-rays at a series of temperatures extending as low as the temperature of liquid air ; and (4) the theoretical discussion of the temperature factor of X-ray reflexion by Debye§ and by Waller.∥ Application of the laws of scattering to the distribution of charge calculated for the sodium and chlorine atoms, enables us to calculate the coherent atomic scattering for X-radiation, as a function of the angle of scattering and of the wave-length, for these atoms in a state of rest, assuming that the frequency of the X-radiation is higher than, and not too near the frequency of the K - absorption edge for the atom.¶ From the observed scattering power at the temperature of liquid air, and from the measured value of the temperature factor, we can, by applying the theory of the temperature effect, calculate the scattering power at the absolute zero, or rather for the atom reduced to a state of rest. The extrapolation to a state of rest will differ according to whether we assume the existence or absence of zero point energy in the crystal lattice. Hence we may hope, in the first place to test the agreement between the observed scattering power and that calculated from the atomic model, and in the second place to see whether the experimental results indicate the presence of zero-point energy or no.
Hubbard model for ultracold bosonic atoms interacting via zero-point-energy–induced three-body interactions