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78. A test of the statistical assumption in nuclear reactions
III. MODELS
AND
76. T h e p e r t u r b a t i o n
THEORIES FOR ENERGIES
treatment
LOW
AND
MEDIUM
of the nuclear many-body
problem.
W . J. SWIATECKI. Gustav Werner I n s t i t u t , Universitet, Uppsala, Sverige. A re-examination of H. E u l e r ' s work 1) on the importance of correlations in the perturbation t r e a t m e n t of the nuclear many-body problem reveals a rapid damping out of correlations with increasing density of the zero-order Fermi gas (the starting point of independent particles). At observed nuclear densities the decrease is by a factor of 22 as compared with the zero-density limit (representing free two-body collisions). The decrease is due partly to the increasing average collision velocities and partly to the exclusion principle. The correlation corrections to the lowest order estimates of interaction energies (i.e. the expectation value of Jr N Z V~I) are of the order 7.6-12.2% for typical nucleon-nucleon interaction strengths. I t is concluded t h a t there is no evidence in the size of the correlation energy against the approximate validity of the single particle starting point, and that, within the limitations of a perturbation method, this very straightforward treatment should provide a quantitative basis for the study of the relation between the characteristics of the particle-particle force and the resulting properties of a composite system consisting of particles obeying the exclusion principle. 1) Euler, H., Z. Phys. 105 (1937) 553.
77. N u c l e a r m a n y - b o d y p r o b l e m . R. JASTROW. U.S. Naval Research Laboratory, Washington, D. C., U.S.A. A trial solution constructed from two-particle functions is applied to a variational t r e a t m e n t of the m a n y - b o d y problem with strong interactions of short range. The two-body function is found to satisfy a simple wave equation, which describes the motion of two particles in a field consisting of the actual two-body potential plus a n effective potential resulting from collisions with other particles. For the particular case of the hard sphere gas at low densities, the effective potential has the form of a Yukawa attraction whose range is of the order of the mean spacing in the gas.
78. A t e s t o f t h e s t a t i s t i c a l a s s u m p t i o n
in nuclear reactions.
R. M. EISBERG:
and N. M. HINTZ. Brookhaven National Laboratory, Upton, N.Y., U.S.A. The angular distributions of protons inelasticaUy scattered from discrete states of --
1161
--
1162
ABSTRACTS
nuclei will be symmetric about 90 ° providing: the reaction goes through the compound nucleus (compound nucleus assumption), m a n y overlapping levels are excited in the compound nucleus (continuum assumption), the phases of these levels are random (statistical assumption). The departure from symmetry of such angular distributions in situations where the continuum assumption is probably satisfied, is usually attributed to a violation of the compound nucleus assumption, b u t could also be due to a violation of the statistical assumption. I t may be possible to distinguish between these alternatives by measuring angular distributions at several closely spaced bombarding energies. A sensitive dependence of the angular disbutions on b o m b a r d i n g energy would be difficult to explain if the departure from symmetry were due to a violation of the compound nucleus assumption, b u t would be easy to explain if the statistical assumption were violated. We have measured the angular distribution for 40A(p, p') 40A*, to the 1.47 MeV level, at bombarding energies of 9.8, 9.0, and 8.5 MeV. Significant changes in the asymmetric angular distributions are observed.
79. T h e u s e o f z e r o e n e r g y s c a t t e r i n g c a l c u l a t i o n s i n t h e s e l e c t i o n of n u c l e a r o p t i c a l p o t e n t i a l s . H. J. AMSTER. Westinghouse Electric Corporation, Bettis Plant, Pittsburgh, Pa., U.S.A.
Most optical models for neutron scattering consist of a complex square well core enclosed by a surface layer in which the potential function is the same for all mass numbers. For these cases, = at -4- f l R J ( l
+ 7Re),
where 1/~ is the logarithmic derivative of the S-wave function at the outside of the surface layer and 1/~c is t h a t at the edge of the core. ct, 8, 7 depend only on the potential in the surface layer, and once found, they enable ~ to be readily calculated for all nuclear radii. The relationship has been used 1) to obtain certain optical restrictions l~etween measurable quantities at zero energy and to predict with analytic and interpretive simplicity some general properties of scattering existing even at higher energies. A natural explanation of the excessively decreasing amplitude in the fluctuations of the real part of the ~ from a complex square well as a function of mass n u m b e r is t h a t neutrons are preferentially absorbed near the nuclear surface. This effect may result either because the potential has a larger imaginary part so t h a t less of the wave penetrates to the core and/or because the imaginary part of the potential itself is larger near the surface t h a n toward the center. I n the first case, the surface must be diffused in order to prevent the fluctuations from being excessively decreased over the whole range of mass numbers; in doing so, the zero energy strength function, the imaginary part of ~R, is beneficially increased. I n the latter case 2), an increased strength function results when the absorbing surface is itself surrounded b y a thin non-absorbing layer. Above zero energy, both potentials give better total and compound formation cross sections than the simple square well, the overall fits being somewhat superior from a potential whose imaginary part is proportional to the derivative of a diffuse-edged real part. I) Amster, H. J., Zero Energy Scattering of Neutrons by Optical Models, Phys. Rev. (submitted). 2) Amster, H. J., Culpepper, L. M. and Emmerich, W. S., Bull. Am. phys. Soc. 1 (1956) 194.
AND
76. T h e p e r t u r b a t i o n
THEORIES FOR ENERGIES
treatment
LOW
AND
MEDIUM
of the nuclear many-body
problem.
W . J. SWIATECKI. Gustav Werner I n s t i t u t , Universitet, Uppsala, Sverige. A re-examination of H. E u l e r ' s work 1) on the importance of correlations in the perturbation t r e a t m e n t of the nuclear many-body problem reveals a rapid damping out of correlations with increasing density of the zero-order Fermi gas (the starting point of independent particles). At observed nuclear densities the decrease is by a factor of 22 as compared with the zero-density limit (representing free two-body collisions). The decrease is due partly to the increasing average collision velocities and partly to the exclusion principle. The correlation corrections to the lowest order estimates of interaction energies (i.e. the expectation value of Jr N Z V~I) are of the order 7.6-12.2% for typical nucleon-nucleon interaction strengths. I t is concluded t h a t there is no evidence in the size of the correlation energy against the approximate validity of the single particle starting point, and that, within the limitations of a perturbation method, this very straightforward treatment should provide a quantitative basis for the study of the relation between the characteristics of the particle-particle force and the resulting properties of a composite system consisting of particles obeying the exclusion principle. 1) Euler, H., Z. Phys. 105 (1937) 553.
77. N u c l e a r m a n y - b o d y p r o b l e m . R. JASTROW. U.S. Naval Research Laboratory, Washington, D. C., U.S.A. A trial solution constructed from two-particle functions is applied to a variational t r e a t m e n t of the m a n y - b o d y problem with strong interactions of short range. The two-body function is found to satisfy a simple wave equation, which describes the motion of two particles in a field consisting of the actual two-body potential plus a n effective potential resulting from collisions with other particles. For the particular case of the hard sphere gas at low densities, the effective potential has the form of a Yukawa attraction whose range is of the order of the mean spacing in the gas.
78. A t e s t o f t h e s t a t i s t i c a l a s s u m p t i o n
in nuclear reactions.
R. M. EISBERG:
and N. M. HINTZ. Brookhaven National Laboratory, Upton, N.Y., U.S.A. The angular distributions of protons inelasticaUy scattered from discrete states of --
1161
--
1162
ABSTRACTS
nuclei will be symmetric about 90 ° providing: the reaction goes through the compound nucleus (compound nucleus assumption), m a n y overlapping levels are excited in the compound nucleus (continuum assumption), the phases of these levels are random (statistical assumption). The departure from symmetry of such angular distributions in situations where the continuum assumption is probably satisfied, is usually attributed to a violation of the compound nucleus assumption, b u t could also be due to a violation of the statistical assumption. I t may be possible to distinguish between these alternatives by measuring angular distributions at several closely spaced bombarding energies. A sensitive dependence of the angular disbutions on b o m b a r d i n g energy would be difficult to explain if the departure from symmetry were due to a violation of the compound nucleus assumption, b u t would be easy to explain if the statistical assumption were violated. We have measured the angular distribution for 40A(p, p') 40A*, to the 1.47 MeV level, at bombarding energies of 9.8, 9.0, and 8.5 MeV. Significant changes in the asymmetric angular distributions are observed.
79. T h e u s e o f z e r o e n e r g y s c a t t e r i n g c a l c u l a t i o n s i n t h e s e l e c t i o n of n u c l e a r o p t i c a l p o t e n t i a l s . H. J. AMSTER. Westinghouse Electric Corporation, Bettis Plant, Pittsburgh, Pa., U.S.A.
Most optical models for neutron scattering consist of a complex square well core enclosed by a surface layer in which the potential function is the same for all mass numbers. For these cases, = at -4- f l R J ( l
+ 7Re),
where 1/~ is the logarithmic derivative of the S-wave function at the outside of the surface layer and 1/~c is t h a t at the edge of the core. ct, 8, 7 depend only on the potential in the surface layer, and once found, they enable ~ to be readily calculated for all nuclear radii. The relationship has been used 1) to obtain certain optical restrictions l~etween measurable quantities at zero energy and to predict with analytic and interpretive simplicity some general properties of scattering existing even at higher energies. A natural explanation of the excessively decreasing amplitude in the fluctuations of the real part of the ~ from a complex square well as a function of mass n u m b e r is t h a t neutrons are preferentially absorbed near the nuclear surface. This effect may result either because the potential has a larger imaginary part so t h a t less of the wave penetrates to the core and/or because the imaginary part of the potential itself is larger near the surface t h a n toward the center. I n the first case, the surface must be diffused in order to prevent the fluctuations from being excessively decreased over the whole range of mass numbers; in doing so, the zero energy strength function, the imaginary part of ~R, is beneficially increased. I n the latter case 2), an increased strength function results when the absorbing surface is itself surrounded b y a thin non-absorbing layer. Above zero energy, both potentials give better total and compound formation cross sections than the simple square well, the overall fits being somewhat superior from a potential whose imaginary part is proportional to the derivative of a diffuse-edged real part. I) Amster, H. J., Zero Energy Scattering of Neutrons by Optical Models, Phys. Rev. (submitted). 2) Amster, H. J., Culpepper, L. M. and Emmerich, W. S., Bull. Am. phys. Soc. 1 (1956) 194.