Herman Feshbach

Unified theory of nuclear reactions

Abstract A new formulation of the theory of nuclear reactions based on the properties of a generalized “optical” potential is presented. The real and imaginary part of this potential satisfy a dispersion type relation while its poles give rise to resonances in nuclear reactions. A new derivation of the Breit-Wigner formula is given in which the concept of channel radius is not employed. This derivation is extended to the case of overlapping resonances. These results can then be employed to obtain the complex potential well model for pure elastic scattering. This potential well is shown to become real as the average width of the resonances increases. Reactions as well as elastic scattering are treated. Considering the former process in an isolated resonance, we obtain a nonresonant term analogous to the familiar potential scattering term of elastic scattering. This is just the direct interaction term which thus appears automatically in this formalism. Upon performing the appropriate energy averages over resonances, the complex potential well model is generalized so as to include inelastic scattering. The effects of the identity of nucleons is investigated. It is shown that our formalism is valid as long as the exit channels can at most contain one nucleon.

A unified theory of nuclear reactions. II

Abstract The effective Hamiltonian method for nuclear reactions described in an earlier paper with the same title, part I, is generalized so as to include all possible reaction types, as well as the effects arising from the identity of particles. The principal device employed, as in part I, is the projection operator which selects the open channel components of the wave function. It is found that the formal structure of part I providing a unified description for direct and compound nuclear reactions including the coupled equation description for direct reactions remains valid in this wider context. A Kapur-Peierls expansion may also be readily obtained. The concept of channel radii is not needed nor is any decomposition of the wave function for the system into angular momentum eigenstates required, so that the expressions for transition amplitudes and widths are invariant with respect to the angular momentum coupling scheme. Since the open channels can only be defined in an asymptotic sense, the corresponding projection operators are not unique. As a consequence the projection operator method has a flexibility which in the first place is consonant with the wide range of phenomena which can occur in nuclear reactions and in the second place can effectively exploit an intuitive understanding of the phenomena. Example of projection operators are obtained including one which leads to the Wigner-Eisenbud formalism, another which is appropriate for the stripping reaction, and, finally, one which takes the Pauli exclusion principle into account. Note that explicit representations of the projection operators are not required for the development of general formal results but are necessary if, eventually, quantitative calculations are made.

Pions and Nuclei

The pion has emerges as the main feature of nuclear structure beyond the traditional description in terms of neutrons and protons. It manifests itself in a number of areas which are normally only loosely interlinked, but intimately related to the pion-nucleon and pion-nuclear interactions: the nucleon-nucleon force; the nuclear many-body problem; nuclear electromagnetic and weak interactions; nuclear spinisospin interactions; pion-nucleus scattering and reactions; etc. This book is a systematic introduction to and survey of nuclear pion physics, a major sub-field of nuclear pion physics. The theoretical foundations are padagogically developed and the physical picture is illustrated with supporting experimental examples.

Theoretical Nuclear Physics: Nuclear Reactions

Multiple scattering formal theory of nuclear reactions elastic and inelastic scattering transfer reactions multistep reactions heavy ions high energy nuclear phenomena pion and kaon interactions with nuclei.

Intermediate structure and doorway states in nuclear reactions

Abstract Intermediate structure in the energy dependence of cross sections is interpreted in terms of an intermediate model, which describes the transition amplitude averaged over an energy interval which is much larger than the width and spacing of the compound nuclear levels, but which is much smaller than the widths of the structure in the optical model cross sections. The intermediate structure is ascribed to resonances in the intermediate amplitude, which in turn are assumed to be the consequence of the existence of doorway states which are described and defined. A formalism is developed which takes the existence of these states explicitly into account. The resultant amplitude is then averaged and the intermediate model obtained. For an isolated doorway state this amplitude can be divided into a slowly varying part and a rapidly varying resonant part. The energy width of the resonant term equals the width of the doorway state, which is composed of an escape width into open channels, and a decay width into more complex excitations. Strength functions are shown to exhibit intermediate structure. These results are generalized to the case of overlapping doorways. Reactions as well as elastic scattering are discussed. The results are put in a form which permits the analysis of experimental data in terms of intermediate model resonances, and the quantum numbers of the doorway states extracted. Experimental situations which would tend to exhibit intermediate structure, and the identification of such structure as an intermediate model resonance, are outlined. Some examples of intermediate structure are discussed.

On high-energy scattering by nuclei—II

Abstract A coupled equation description of the scattering of fast particles is formulated. For elastic scattering two coupled channels are employed. One wavefunction describes the elastic channel. The other is an “average” wavefunction which takes into account the collisions which remove the incident particles from the elastic channel. Explicit formulas are given for the coupling and diagonal potentials as obtained from the Kerman, McManus, and Thaler multiple scattering theory which includes the effects of single and double scattering in their nonlocal optical potential. The properties of the target nucleus involved are its density and pair correlation function. These are described in terms of intrinsic coordinates, the relation to model wavefunctions being briefly discussed. Three kinds of correlations are considered; the center-of-mass correlation required of the target nucleons in order that the target recoil as a unit, the exclusion principle or Pauli correlation and the short-range dynamical correlations. The coupling potential is directly related to correlation function vanishing if the correlation funciton vanishes. Numerical calculations are performed for protons of 1.69 Gev/c momentum incident on 4 He and 16 O. It is found that the Pauli correlation has little effect, that the center-of-mass correlation is the dominant correlation effect for the 4 He target but not important for 16 O. The effect of the short-range correlation is measured by the parameter ( r c R )/[1 + 2 r v 2 r c 2 ] where r c is the correlation length, R the nuclear radius, r v the nuclear “force range.” In the heavier nuclei, the effect of short-range correlations is proportional to this parameter; in 4 He, because of interference with the center-of-mass correlation terms, the effect is proportional to its square root. Short-range correlations are thus more difficult to observe for heavier nuclei and when their range is small compared to the ranges of nuclear forces. Absence of required nucleon-nucleon data prevents a definitive statement with respect to the presence of short range correlations. However the present nucleon-nucleon and proton- 4 He data are not inconsistent with the presence of a repulsive short-range correlation.

The unified theory of nuclear reactions: III. Overlapping resonances

Abstract The unified theory of nuclear reactions is employed to derive general expressions which: (i) are valid in the presence of direct interactions as well as well separated or overlapping compound nuclear resonances; (ii) explicitly satisfy the requirements of unitarity; (iii) involve the minimal number of real parameters.

A nucleon-nucleon potential consistent with experiment and the boson exchange theory of nuclear forces

Abstract The nucleon-nucleon data are fitted by a boundary condition model interaction determined largely by field-theoretic forms. One and two pions, ϱ, ω, and η meson exchange adiabatic local potentials determine th interaction for r ⩾ 1 2 μ where (1/μ) is the pion Compton wavelength. The two-pion-exchange contribution contains a degree of ambiguity measured by the parameters ξ and λ for the pion-ladder and nucleonpair diagrams respectively. The interaction at r0 is given by an energy-independent boundary condition for each partial wave sensitive to the short-range interaction. A very good fit to the pp data and a good fit to the np data below 350-MeV-nucleon laboratory energy comparable to the better phenomenological fits are obtained. The optimum or fixed value of exchange-particle masses and coupling constants correspond to their available measured values, and the λ and ξ parameters optimize in their theoretical range. The value obtained for r0 corresponds to that predicted by the theory of the boundary-condition model. There remain 19 boundary-condition parameters freely fitted to which the data is sensitive. Rescattering and long-range recoil corrections remain to be investigated, as does the effect of coupling to inelastic channels.

Radiation of low energy quanta in nuclear reactions

Abstract The production of radiation concomitant with the elastic scattering of nucleons by nuclei is considered. It is shown that the first term of a series expansion of the differential radiation cross-section in the energy of the radiated quantum may be expressed in terms of the elastic scattering amplitudes at the incident and final energies. This result is valid even when there are resonances in the scattering whose widths are a small fraction of the energy loss. The interference between these 2 amplitudes permits the determination of the shape elastic amplitude, and therefore the cross-section for compound elastic scattering as well as the correlation distance W defined as the energy separation over which there is a substantial correlation in the fluctuations in the elastic scattering amplitude; its inverse measures the time delay in the reaction. When a resonance has a width of the order of the energy loss, the interference can be employed to evaluate the change of phase of the elastic scattering amplitude across the resonance and may therefore serve as an indicator of whether or not a fluctuation in the elastic cross-section is a resonance. These results when applied to reactions lead to a method for the measurement of the relative amount of direct and compound nuclear processes and to a direct test of the random phase approximation of the statistical hypothesis in nuclear reactions. A derivation of our principal result valid for relativistic particles is also given.

On Scattering by Nuclei at High Energies

Abstract We study how nucleon-nucleon correlations influence high-energy scattering by nuclei. Using the multiple scattering theory of Kerman, McManus, and Thaler, we formulate the scattering problem in terms of an infinite system of coupled equations. For the calculation of elastic scattering, we propose replacing the infinite system by a pair of coupled equations, containing the elastic channel, and another effective one which carries the inelastic strength. The coupling potential is proportional to the nuclear pair correlation function, and a first approximation to the potential is obtained. These equations are accurate to the extent that the pair correlations and not higher-order correlations are important. There are no restrictions to forward scattering, nor are any “on the energy shell” approximations made. In order to obtain insight into the structure of the many-channel S-matrix for high energies, the infinite system of coupled equations is solved by semi-classical methods, and explicit formulae for the S-matrix elements as a function of various potentials and the nuclear pair correlation function are obtained. We show how properties of excited target states may complicate a reliable extraction of correlations in high-energy scattering by nuclei. A close relation between our semi-classical solution and Glaubers multiple scattering is established. A numerical study of high-energy nucleon-nucleus scattering using the methods developed in this paper is under way.