Nuclear Many-Body Problem

Phys. Rev. 103, 1353 – Published 1 September 1956
H. A. Bethe


A self-contained and largely new description is given of Brueckner's method for studying the nucleus as a system of strongly interacting particles. The aim is to develop a method which is applicable to a nucleus of finite size and to present the theory in sufficient detail that there are no ambiguities of interpretation and the nature of the approximations required for actual computation is clear.

It is shown how to construct a model of the nucleus in which each nucleon moves in a self-consistent potential matrix of the form (r|V|r) (Sec. II). The potential is obtained by calculating the reaction matrix for two nucleons in the nucleus from scattering theory. Some complications arise in the definition of the energy levels of excited nucleons (Sec. III). The actual wave function is obtained from the model function by an operator which takes into account multiple scattering of the nucleons by each other (Sec. IV).

The method of Brueckner is a vast improvement over the normal Hartree-Fock method since, in calculating the self-consistent potential acting on an individual particle in the model, account is already taken of the correlations between paris of nucleons which arise from the strong internucleon forces (Sec. V). Although the actual wave function is derivable from a wave function which corresponds essentially to the shell model, the probability of finding a large nucleus of mass number A "actually" in its shell model state is small (of order e-αA, where α is a constant) (Sec. VI). The influence of spin is investigated (Sec. VIII). In the case of an infinite nucleus, an integral equation is obtained for the reaction matrix, just as in the theory of Brueckner and Levinson (Sec. IX).

The exclusion principle must be applied in intermediate states in solving the integral equation for the reaction matrix. This makes an enormous difference for the solution. When the exclusion principle is used, the scattering matrix is very nearly given by the Born approximation, for any well-behaved potential (Sec. X). Numerical results are given for the case when nucleons interact only in S states, an assumption which leads to saturation without a repulsive core. The agreement with observation is fair to poor, owing to the poor assumption for the interaction (Sec. XI). Brueckner's result that three-particle clusters give a small contribution to the energy is confirmed, although the numerical value is many times his result; the calculation is then extended to the case of a repulsive core (Sec. XII). The dependence of the binding energy on the mass number A is investigated for saturating and nonsaturating interactions (Sec. XIII). Terms of relative order 1A are calculated, and it is shown that these terms are much smaller than Brueckner and Levinson found, making the method also applicable to relatively small nuclei (Sec. XIV). Some aspects of the problem of the finite nucleus are discussed, including that of degeneracy (Sec. XVI).


  • Received 12 March 1956
  • Published in the issue dated September 1956

© 1956 The American Physical Society

Authors & Affiliations

H. A. Bethe*

  • Cavendish Laboratory, Cambridge, England

  • *Permanent address: Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, New York.

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