San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
USA

The Determination of the Binding Energies
Associated with Spin Pair Formations

If a neutron could be added to a nucleus the binding energy would be affected in two general ways. The First there would be the effect due to the interaction of the additional neutron with the other nucleons. Second there could be formed neutron-neutron and/or neutron-proton spin pairs. Such spin pairs are exclusive in the sense that one neutron can form a spin pair with only one other neutron and with only one proton.

The effect of adding a neutron to a nucleus can be approximated by computing the incremental binding energy of neutrons. If a neutron is added to a nucleus with an even number of neutrons no additional neutron-neutron spin pair is formed. If a neutron is added to a nucleus in which there are more neutrons than protons then no neutron-proton spin pair is formed. This means there are four possibilities as displayed in the following table.

The Statistics of the Effects of Adding
a Neutron to Various Cases of Nuclides
Odd/evenness
of n
n versus pNumber
of cases
Average
Increment
(MeV)
Standard
Deviation of
Increments
(MeV)
Evenn≥p26137.4580632.642599
Evenn<p1810.584336.196072
Oddn≥p199.5656174.442570
Oddn<p17116.439262.520360

The case of n even and greater than or equal to p gives the extent of the binding energy due solely to the interaction of the additional neutron through the strong force; i.e., 7.458063 MeV. The case of n even and n less than p includes the effect of the formation of neutron-proton spin pair as well as the effect of the strong force interactions. Therefore the effect associated with the formation of the neutron-proton spin pair is about (10.58433−7.458063)=3.126370 MeV. The case of n odd and n greater than or equal to p includes the effect associated with the formation of a neutron-neutron spin pair along with the interactions involving the strong force. The effect solely associated with the formation of the neutron-neutron spin pair should be then be (9.565617−7.458063)=2.107557 MeV. The ratio of the effects of the two types of spin pairs is 0.674, notably close to 2/3.

The last case of n odd and n less than p includes the effects associated with the formation of both types of spin pairs as well as the interaction effects. The effects associated with the two types of spin pairs should be (16.43926−7.458063)=8.981197 MeV. This is greater than the sum of the separate effects of 5.233927 MeV. This might suggest that there is a substantial effect associated with the simultaneous formation of the two types of spin pairs. However the uncertainties in the estimates are such that the difference is not significanctly different from zero at the 95 percent level of confidence. Here are the computatations.

μ(nn+np)−nn−np = (16.43926 − 7.458063) − (10.58433−7.458063) − (9.565617−7.458063)
= 16.43926 − 10.58433−9.565617+7.458063 = 3.74727
 
σ²(nn+np)−nn−np = (2.520360)² + (6.196072)² + (4.442)² + (2.642599)² = 71.4633

Thus
σ(nn+np)−nn−np = 8.4536

The ratio of the deviation of the difference from zero to its standard deviation is 3.74727/8.4536=0.4433. For the difference to be significantly different from zero the ratio would have to be on the order of 2. The difference between the binding energies associated with the formations of a neutron-neutron spin pair and a neutron-proton spin pairs is also not significantly different from zero at the 95 percent level of confidence.''

(To be continued.)


HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins