Temperature dependence of neutron scattering on He-4 gas Vladimir Ignatovich Frank Laboratory of Neutron Physics of Joint Institute for Nuclear Research, 141980, Dubna Moscow region, Russia (September 11, 2005) A suggestion [1] that temperature dependence of the total neutron- 4 He cross section can be proportional to T 3/2 is checked. The experiment is described. The temperature dependence was found to be T 1/2 in agreement with the standard scattering theory (SST). The consequence of this result for the scattering theory is discussed. 03.65.Nk, 28.20.Cz, 03.65.-w, 25.40.Dn I. INTRODUCTION We started to doubt the standard result for neutron-monatomic gas scattering cross section, because we found that all scattering theories (at least nonrelativistic ones) are contradictory. First we demonstrate the contradictions and try to overcome them. In this process we inevitably arrive at wave packet notion, and must ask ourselves of what size the wave packet is. It can be some fundamental constant or it can depend on neutron energy. These two possibilities lead to two different temperature dependencies of neutron transmission through 4 He gas. So we made an experiment to make a proper choice between them. A. Contradictions of scattering theories We have three scattering theories. 1. Theory of spherical harmonics For description of elastic scattering the neutron wave function is represented (see, for example, [2]) in the form Ψ = exp(ikr) − f (ϑ) r exp(ikr), (1) where plane exp(ikr) and spherical exp(ikr)/r waves describe the incident and scattered particles, respectively. The simplest process is elastic s-wave scattering from a fixed center, for which f (ϑ)= b =const. The approach with spherical waves is not a quantum scattering theory, because the spherical wave does not satisfy the free Schr¨odinger equation and therefore does not describe a free particle. To make it selfconsistent one has to find an asymptotic form of the spherical wave (see, for example, [3]). It satisfies the free Schr¨odinger equation and is a superposition of plane waves: Ψ = exp(ikr) −  F (Ω)dΩ exp(ik Ω r), (2) where |k Ω | = k, Ω is solid angle of the scattered particle, F (Ω) is dimensionless probability amplitude, and the intensity of scattering into an angle Ω is described by the dimensionless probability dw(Ω) = |F (Ω)| 2 dΩ, (3) and the total probability w of scattering is dimensionless integral w =  dw(Ω) =  |f (Ω)| 2 dΩ. In the case of the s-wave scattering from a single fixed center F (Ω) = bk/2π. In the case of many scattering centers F (Ω) = k 2π  d 3 r ′ n(r ′ )b(r ′ ) exp(iqr ′ ), (4) where q = k − k Ω is momentum transferred, n(r ′ ) is atomic number density, and b(r ′ ) is scattering amplitude of a center at the point r ′ . 1