The differential and total scattering cross sections have been calculated for slow neutrons impinging on crystalline spheres whose radii are very much greater than the neutron wavelength. The results for the differential cross section show that when the difference, $\ensuremath{\rho}$, between the neutron phase shift in traversing the particle diameter and the phase shift in traversing the same distance in vacuum is small, the result agrees with the Born approximation, while if $\ensuremath{\rho}$ is large, the result agrees with that obtained by geometrical optics. Also, the total cross section for $\ensuremath{\rho}$ small is $\frac{\ensuremath{\pi}{R}^{2}{\ensuremath{\rho}}^{2}}{2}$ and for $\ensuremath{\rho}$ large is $2\ensuremath{\pi}{R}^{2}$, where $R$ is the particle radius.Using the Oak Ridge reactor, the broadening of a neutron beam in traversing finely divided materials was measured as a function of neutron wavelength, particle size, and coherent scattering cross section, and the results verified these considerations.A new method for determining coherent cross sections of nuclei and the phases of slow neutron scattering was demonstrated through the use of a liquid of known index of refraction into which the finely divided materials were immersed. In this manner, the phases of Mo, Cb, Pd, Ru, Si, P, and Te were shown to be positive.