lec08 - 22.101 Applied Nuclear Physics(Fall 2006 Lecture 8...

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_______________________________________________________________________ ________________________________________________________________________ 22.101 Applied Nuclear Physics (Fall 2006) Lecture 8 (10/4/06) Neutron-Proton Scattering References : M. A. Preston, Physics of the Nucleus (Addison-Wesley, Reading, 1962). E. Segre, Nuclei and Particles (W. A. Benjamin, New York, 1965), Chap. X. We continue the study of the neutron-proton system by taking up the well-known problem of neutron scattering in hydrogen. The scattering cross section has been carefully measured to be 20.4 barns over a wide energy range. Our intent is to apply the method of phase shifts summarized in the preceding lecture to this problem. We see very quickly that the s-wave approximation (the condition of interaction at low energy) is very well justified in the neutron energy range of 1 - 1000 eV. The scattering-state solution, with E > 0, gives us the phase shift or equivalently the scattering length. This calculation yields a cross section of 2.3 barns which is considerably different from the experimental value. The reason for the discrepancy lies in the fact that we have not taken into account the spin-dependent nature of the n-p interaction. The neutron and proton spins can form two distinct spin configurations, the two spins being parallel (triplet state) or anti-parallel (singlet), each giving rise to a scattering length. When this is taken into account, the new estimate is quite close to the experimental value. The conclusion is therefore that n-p interaction is spin-dependent and that the anomalously large value of the hydrogen scattering cross section for neutrons is really due to this aspect of the nuclear force. For the scattering problem our task is to solve the radial wave equation for s-wave for solutions with E > 0. The interior and exterior solutions have the form ( ) = B sin( K r ' ) u r , r < r o (8.1) u r (8.2) and ( ) = C sin( kr + δ o ) , r > r o 1
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( o h and k = where K ' = m V + E ) / mE / h . Applying the interface condition we obtain K 'cot( K r ' o ) = k cot( kr o + δ o ) (8.3)) which is the relation that allows the phase shift to be determined in terms of the potential parameters and the incoming energy E. We can simplify the task of estimating the phase shift by recalling that the phase shift is simply related to the scattering length by δ o = − ak (cf. (7.22)). Assuming the scattering length a is larger than r
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  • Spring '13
  • AyhanYılmazer
  • Proton, Neutron, Fundamental physics concepts, Nuclear physics

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