Four-body System Research Articles

Calculation on nuclear three- and four-body systems with Jastrow-type correlated wave functions

The nuclear three- and four-body systems are investigated with a variational method and various nucleon-nucleon potentials. The trial wave functions used are products of a central-field wave function and a correlation factor symmetric in the interparticle distances. The results show that i) the use of a fixed-centre reference frame or the centre-of-mass reference frame yield very nearly the same values for the binding energies and r.m.s. radii of3H and4He and ii) the trial functions, although containing few variational parameters, allowed for accurate evaluations. Moreover it is shown that the correlations between more than three particles give a disregardable contribution to the various calculated quantities.

Investigation of Nuclear Three- and Four-Body Systems with Soft-Core Nucleon-Nucleon Potentials

The nuclear three- and four-body systems are investigated with a variational method and various soft-core nucleon-nucleon potentials. A Monte Carlo method is used to compute the multidimensional integrals involved. The results show that (i) a soft-core potential and a hard-core potential which give the same effective-range parameters and $S$-wave phase shifts yield very nearly the same values for the binding energies and rms radii of ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$, and (ii) as far as the ground-state properties of ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$ are concerned, the fitting of the high-energy ($\ensuremath{\gtrsim}100$-MeV) $S$-wave phase shifts need not be seriously considered in choosing a nucleon-nucleon potential. Also, it is shown that by paying less attention to the higher-energy ($\ensuremath{\gtrsim}50$-MeV) nucleon-nucleon scattering data, a soft-core potential can be found which yields nearly correct values for the binding energies, rms radii, and form factors up to about 10 ${\mathrm{F}}^{\ensuremath{-}2}$ for both ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$. The soft-core potentials proposed by Eikemeier and Hackenbroich and by Volkov have also been briefly considered. Here the results indicate that the mathematical approximations used by Hackenbroich et al. to investigate the properties of light nuclei are inaccurate, while the binding energy of ${\mathrm{He}}^{4}$ calculated by Volkov is too small by about 2 MeV.

Independent-pair wave functions for the three- and four-body nuclei

A method which was described previously for generating independent-pair wave functions for three-nucleon systems is generalized so that the S′ state with mixed symmetry can be included in three-body calculations and so that the method can be used to generate independent pair wave functions for the alpha particle. This procedure is used to find approximate ground-state functions for several three- and four-body Hamiltonians. The functions so obtained are used to calculate an upper and, in some cases, a lower bound to the ground-state energy and to calculate the charge form factors for these three- and four-body systems. When the interactions are chosen so that both low- and high-energy two-nucleon data are correctly predicted, then the calculated binding energies are slightly too large, the ratio of the form factors for 3H to that of 3He is consistent with the experimental ratio, and the form factors for all three nuclei are too large at large momentum transfers. By increasing the radius of the hard core, the calculated form factor functions are decreased closer to the experimental ones.

Investigation of Nuclear Three- and Four-Body Systems with Soft-Core Nucleon-Nucleon Potentials

The nuclear three- and four-body systems are investigated with a variational method and various soft-core nucleon-nucleon potentials. A Monte Carlo method is used to compute the multidimensional integrals involved. The results show that (i) a soft-core potential and a hard-core potential which give the same effective-range parameters and $S$-wave phase shifts yield very nearly the same values for the binding energies and rms radii of ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$, and (ii) as far as the ground-state properties of ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$ are concerned, the fitting of the high-energy ($\ensuremath{\gtrsim}100$-MeV) $S$-wave phase shifts need not be seriously considered in choosing a nucleon-nucleon potential. Also, it is shown that by paying less attention to the higher-energy ($\ensuremath{\gtrsim}50$-MeV) nucleon-nucleon scattering data, a soft-core potential can be found which yields nearly correct values for the binding energies, rms radii, and form factors up to about 10 ${\mathrm{F}}^{\ensuremath{-}2}$ for both ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$. The soft-core potentials proposed by Eikemeier and Hackenbroich and by Volkov have also been briefly considered. Here the results indicate that the mathematical approximations used by Hackenbroich et al. to investigate the properties of light nuclei are inaccurate, while the binding energy of ${\mathrm{He}}^{4}$ calculated by Volkov is too small by about 2 MeV.

Upper and lower bounds of the eigenvalue of a four-body system

The upper and lower bounds of the ground state eigenvalue of a four-body system are determined. The two-body nuclear potential employed is a simple Gaussian potential which has been used by a number of authors in their calculations on a three-body system. With a four-parameter trial function, the two bounds are calculated to be −38.98±0.08 and −39.52±0.19 MeV, respectively. Using these values, it is estimated that the eigenvalue of this particular four-body system is −39.1±0.1 MeV.

Double Octet Model of Bosons

The properties of the eight dimensional representation of a four-body system which are expressed in terms of the tensor T[C~'_:j (or Tt~:}?) are investigated. Since these bosons are nol self-conjugate for the charge co0jugate transformation, they are not eigenstates of G and A parities, but the linear combination states of the two multiplets T(~':j and T(:;;p are. Hence, there appear two boson' octets with opposite G and A parities. The double octet scheme is applied to the vector and axial vector bosons. In the former case, (!), p, K* and K are assumed to belong to the double octets and then it is required that there exist an isoscalar vector boson with positive G parity and an isovector one with negative G parity whose masses are about 850 and 830 MeV respectively. In the latter case, supposing A 1, B, D, Knrc (1175 MeV) and Care members of the double octets, it is predicted that there exists an isoscalar axial vector boson with negative G parity whose mass is about 1140 MeV. In addition it is noticed that vector bosons with abnormal G parity have no local Yukawa interactions and some topics related to this fact are discussed.

Comparison of Approximate Translational—Vibrational Energy-Transfer Formulas with Exact Classical Calculations

The Hamiltonian equations of motion for collinear-colliding diatomic-molecule—atom and diatomic-molecule—diatomic-molecule systems are numerically integrated to obtain vibrational energy-transfer data. The diatomic molecules are taken to be either harmonic or Morse-type oscillators, and the collision potential is either an exponentially repulsive potential or a Lennard-Jones 6–12 potential. The results of the exact classical three-body calculations are compared to results predicted by classical and semiclassical approximations. The ratios of approximate to exact energy-transfer values are related to the value of a parameter which depends only upon the masses of the particles involved in the collision; this correlation is found to be relatively insensitive to the values of the collision energy and of the potential parameters over wide ranges. The exact classical energy-transfer data for the four-body systems are compared to the approximate and exact three-body data. The validity of using three-body approximations in treating four-body collisions is discussed.

Λ−ΛPotential from Analysis ofBe10ΛΛ

The strength of the $\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda}$ interaction in the $^{1}S_{0}$ state is determined by analyzing the double hypernucleus $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{Be}^{10}$ treated as a four-body system of $2\ensuremath{\alpha}+2\ensuremath{\Lambda}$. The $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ potential chosen fits the $^{1}S_{0}$ phase shifts up to a c.m. energy of about 12 MeV. The $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\Lambda}$ potential is determined from the binding energy of $_{\ensuremath{\Lambda}}\mathrm{He}^{5}$, taking into proper account the size of the alpha particle and the range of the $\ensuremath{\Lambda}$-nucleon interaction. With these potentials, the binding energy of $_{\ensuremath{\Lambda}}\mathrm{Be}^{9}$ is also given correctly. For the $\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda}$ part, a potential is used which has a hard core of radius 0.4 F and an attractive well of exponential shape. The intrinsic range is chosen as 1.5 F, corresponding to the mechanism of 2-pion exchange. Using a 12-parameter variational function, the $\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda} ^{1}S_{0}$ potential which yields the observed separation energy of the two $\ensuremath{\Lambda}$ particles in $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{Be}^{10}$ is found to have a well-depth parameter of ${{0.732}_{\ensuremath{-}0.034}}^{+0.027}$, which is about 10% smaller than the value obtained by treating $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{Be}^{10}$ as a three-body system of ${\mathrm{Be}}^{8}+2\ensuremath{\Lambda}$. The scattering length and effective range are equal to --- (${{1.04}_{\ensuremath{-}0.22}}^{+0.20}$) and ${{2.91}_{\ensuremath{-}0.27}}^{+0.46}$ F, respectively. Also, we found that the ${\mathrm{Be}}^{8}$ core in $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{Be}^{10}$ is quite compressed, with the $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ separation about 12% smaller than the corresponding separation in $_{\ensuremath{\Lambda}}\mathrm{Be}^{9}$.

Binding energies of nuclear two, three and four-body systems

The binding energies of the nuclear three and four-body problems are calculated using a variational method with hard-core central potentials of various core radii, which fit the two-nucleon low-energy scattering data. Both the upper and the lower bound are computed. The results show that when the core radius has a value between 0.35 and 0.55 fm, reasonable agreement with experimental data on the binding energies and rms radii of the triton and the alpha particle, and the Coulomb energy of He 3 can be obtained.

P-He3Scattering at 9.75-Mev Proton Energy

The differential cross section for the elastic scattering of 9.75-Mev protons from ${\mathrm{He}}^{3}$ has been measured in the range of center-of-mass angles from 30 to 150 degrees. The cross section varies from 345 millibarns/steradian at 30 degrees, through a minimum of 18.2 mb at 110 degrees, and rises to 84.5 mb at 150 degrees. Comparison of the data with the distributions calculated theoretically by Swan gives only rough agreement, the closest being for ordinary forces in the four-body system. This conflicts with the good agreement obtained by Swan with the 14-Mev $n$-T data of Coon et al., in which a symmetric exchange force scheme was clearly preferable.

Normal States of Nuclear Three- and Four-Body Systems

A variational method employing orthogonal (Hermite) functions in linear combination is used for calculating the binding energies of ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$, with the following choice of nuclear constants: $A=35.60 \mathrm{Mev}; a=2.25\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13} \mathrm{cm}; g=0.20,$ and an error function potential. Two types of coordinates are used for ${\mathrm{H}}^{3}$; (1) normal, (2) individual particle coordinates. Their advantages and disadvantages are discussed. With the use of a certain limited set of functions of normal coordinates the energy has been depressed from -6.21 Mev in zeroth approximation to -7.21 Mev, while a suitable set of functions involving only individual particle coordinates reduced it from -6.16 Mev to -6.84 Mev. The second Schr\"odinger perturbation is less effective than the latter scheme by 0.22 Mev. Functions of different symmetry, called into play by the Heisenberg operators, are found to contribute 0.07 Mev on the basis of a modified variation method. ${\mathrm{He}}^{4}$ has been treated only with the use of individual particle coordinates (Hartree method). A similar group of functions lowers the energy from -24.81 Mev in zeroth approximation to -25.85 Mev, which is better than the effect of Schr\"odinger's perturbation theory by 0.25 Mev. General estimates of convergence limits are given.