_{1}

^{*}

Using an exact and complete Yukawa Potential Energy (YPE), the neutron-triton and neutron-helium elastic total scattering cross section is estimated and extrapolated to zero energy. The estimated value agrees pretty well with the experimental value and the procedure can be extended to any neutron-nucleus total scattering. The results are extended to the case of
^{4}He,
^{7}Li,
^{9}Be and
^{27}Al nuclei also.

The theoretical description of A = 4 systems still constitutes a challenging problem from the stand-point of nuclear few body theory. Only recently the study of alpha particle bound state reached a satisfactory level of accuracy because of new numerical methods. The 4N bound state has been calculated to a few tenths of KeV [

Instead of the P-^{3}He elastic scattering at low energies, the situation is simpler for the n-^{3}H elastic scattering. There is no Coulomb complication with the latter. Experimentally, only the total cross section

The above value is a very precisely measured value for the neutron-triton elastic scattering. There is another parameter called the coherent scattering length

Estimation is also obtained from P-^{3}He scattering data for the coherent scattering length through Coulomb- corrected R-matrix theory. This gives [

The above value is also an approximate experimental value. The theoretically obtained values are [

In this paper we estimate the above experimental values with an exact Yukawa Potential Energy function and by using the methods of ordinary quantum mechanics. In Section 2, we present the relevant theory and apply it to the nucleus triton. In Section 3 the estimations are extended to many more nuclei with ease. We present our conclusions in Section 4.

The complete Yukawa Potential Energy function operative for any nuclear process is given by,

where,

The other unknown is a of the Yukawa factor. For One Pion Exchange Potential (OPEP),

and,

The parameter

We now have the entire YPE for any nucleus. For example for the nucleus ^{3}H, the parameter

We can now calculate the scattering amplitude for the scattering of a neutron of mass

where, the reduced mass of the neutron is given by,

Here, the wave number,

scattering cross section can be obtained by squaring Equation (2.6). The total elastic scattering cross-section can then be obtained by integrating the differential cross-section. The Yukawa potential has the advantage relative to the Coulomb potential that certain integrals converge to finite numbers. The total elastic scattering-cross section is given by,

The above expression gives the total elastic scattering-cross section for a neutron by any nucleus. It can be used for any neutron-nucleus scattering provided there are no resonances and there is no absorption. The total elastic scattering cross-section extrapolated to zero energy is given by,

where,

The above Equations (2.9) and (2.10) can be used to compute the total scattering cross section at zero energy and the coherent scattering length. The required masses and other data are given in the table given in the next section. For ^{3}H,

When

And when,

The above values obtained should be compared with the experimental value quoted in Equations (1.1) and (1.2). The average value of (2.11) and Equation (2.13) is 1.7285 barn. This should be compared with the value given by Equation (1.1). Similarly the average coherent radius is given by, 3.70846 fm.

Below we tabulate the estimated results for different nuclei.

Nucleus | Mass | Average | Mean | |||||
---|---|---|---|---|---|---|---|---|

5.00830 | 1.103243 | 1.6839 | 1.7731 | 1.7285 | 3.661 | 3.76 | 3.71 | |

5.00824 | 1.142021 | 1.57205 | 1.65499 | 1.61352 | 3.536 | 3.63 | 3.568 | |

6.64648 | 0.935301 | 4.6875 | 4.9361 | 4.8118 | 6.11 | 6.27 | 6.19 | |

11.65036 | 0.872391 | 19.8359 | 20.8872 | 20.3615 | 12.56 | 12.9 | 12.73 | |

14.96509 | 0.973006 | 27.8388 | 29.3141 | 28.5765 | 14.88 | 15.3 | 15.09 | |

44.80393 | 1.094011 | 226.768 | 238.785 | 232.777 | 42.48 | 43.6 | 42.54 |

For completeness, we have also included the estimation of the Triton nucleus as well. The ease with which the bulk nuclear properties can be obtained with the YPE can be seen from

The scattering of a neutron by Helium-3 or Helium-4 is quite difficult to measure. But scattering of neutrons of different energy by an Aluminum foil must not be very difficult to conduct. If measurements are made for different neutron energies for the elastic total scattering off Aluminum target our results can be put to test. This will pin down the non-relativistic YPE for any nucleus. Once it is confirmed, many bulk properties of a nucleus can be predicted with ease.

In this paper an exact YPE function is used to predict the total elastic scattering-cross section of a neutron by any nucleus. For all nuclei for which the parameter

other interactions. The interaction constant

The author is very grateful to V. G. Krishnan, Prudhvi R. Chintalapati and G. Prasad.