San José State University |
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The Diiferences in the NuclearBinding Energies Due to the Formation of the Three Types of Nucleonic Spin Pairs |
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The protons and neutrons in nuclei are held together primarily by the formation of the three types of spin pairs: neutron-neutron, proton-proton and neutron-proton. The binding energies associated with them are approximately equal. There are ways to establish estimates of the differences. If a neutron is added to a nucleus in which the number of protons is greater than the number of neutrons then binding energy is increased due to the formation of both a neutron-neutron spin pair and a neutron-proton spin pair plus the binding energy due to the net interaction of the additional neutron with all of the nucleons in the nucleus. If the incremental binding energies of nuclides are determined they will show essentially only the binding energy due to the formation of spin pairs.

If a neutron is added to a nuclide in which the number of protons is equal to the number of neutrons and the number of neutrons is odd then a neutron-neutron spin is formed but a neutron-proton spin pair is not formed. The difference in incremental binding energy for n=p+1 and its value for n=p gives the difference in binding energy for the formation of a neutron-neutron spin pair compared to that a neutron-proton spin pair.

to a Neutron-Proton Spin Pair

Here are the data for neutron-neutron spin pairs.

The Difference between the Binding Energy due to the Formation of a Neutron-Neutron Spin Pair and a Neutron-Proton Spin Pair |
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Nuclide | Protons | Neutrons | Binding Energy Difference (MeV) |

3H | 1 | 2 | 4.032675 |

7Li | 3 | 4 | 1.5853 |

11B | 5 | 6 | 3.0178 |

15N | 7 | 8 | 0.27998 |

19F | 9 | 10 | 1.28257 |

23Na | 11 | 12 | 1.34962 |

27Al | 13 | 14 | 1.69187 |

31P | 15 | 16 | 0.99189 |

35Cl | 17 | 18 | 1.13595 |

39K | 19 | 20 | 1.0026 |

43Sc | 21 | 22 | 0.5879 |

47V | 23 | 24 | -0.2704 |

51Mn | 25 | 26 | 0.605 |

55Co | 27 | 28 | 0.6525 |

59Cu | 29 | 30 | 0.3373 |

63Ga | 31 | 32 | 0.04 |

67As | 33 | 34 | 0.08 |

71Br | 35 | 36 | -0.2 |

75Rb | 37 | 38 | -0.075 |

79Y | 39 | 40 | 0 |

83Nb | 41 | 42 | 0.6 |

87Tc | 43 | 44 | 0.3 |

91Rh | 45 | 46 | -0.1 |

The graph of the above data shows the notable variation in the differences.

The cumulative frequency distribution

The irregularity of the outliers makes it inappropriate to include them in the determination of a mean value for the distribution. The regular part of the distribution between − 0.27 MeV and +1.69 MeV has an average binding energy difference of +0.5656 MeV. This is the best estimate of the binding energy difference between the formation of a neutron-neutron spin pair and a neutron-proton +spin pair. The standard deviation of this part of the distribution is 0.6050 MeV. There were 21 observations involved in computing the mean. This means that the standard deviation of the mean is 0.1320=0.6050/√21. The ratio of 0.5656 to that standard deviation is 4.28. This means that the value of 0.5656 MeV. is statistically significantly different from zero at the 95 percent level of confidence.

to a Neutron-Proton Spin Pair

Here are the data for proton-proton spin pairs.

The Difference between the Binding Energy due to the Formation of a Proton-Proton Spin Pair and a Neutron-Proton Spin Pair |
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Nuclide | Protons | Neutrons | Binding Energy Difference (MeV) |

3He | 2 | 1 | 3.268912 |

7Be | 4 | 3 | 1.0212 |

11C | 6 | 5 | 2.1034 |

15O | 8 | 7 | -0.253587 |

19Ne | 10 | 9 | 0.80476 |

23Mg | 12 | 11 | 0.84017 |

27Si | 14 | 13 | 1.15648 |

31S | 16 | 15 | 0.53894 |

35Ar | 18 | 17 | 0.75336 |

39Ca | 20 | 19 | 0.6208 |

43Ti | 22 | 21 | 0.2155 |

47Cr | 24 | 23 | -0.5887 |

51Fe | 26 | 25 | 0.3004 |

55Ni | 28 | 27 | 0.2608 |

59Zn | 30 | 29 | 0.009 |

63Ge | 32 | 31 | -0.747 |

67Se | 34 | 33 | -0.74 |

71Kr | 36 | 35 | -0.78 |

75Sr | 38 | 37 | 0.03 |

79Zr | 40 | 39 | 0.04 |

83Mo | 42 | 41 | 0.7 |

87Ru | 44 | 43 | 0 |

91Pd | 46 | 45 | 0.2 |

The average of the binding energies between the 6th largest and the 20th largest is 0.4682 MeV. This the best estimate of the difference in binding energy for the formation of a proton-proton spin pair compared to that for a neutron-proton spin pair. The standard deviation of this part of the distribution is 0.3836 MeV. The mean was computed using 16 observations.This means that the standard deviation of the mean is 0.09589=0.3836/√16. The value of the ratio of 0.4682 to that standard deviation is 4.88. This means the value of the mean of 0.4682 is statistically significantly different from zero at the 95 percent level of confidence.

to a Neutron-Neutron Spin Pair

Since the difference in binding energy for the formation of a neutron-neutron spin pair compared to that for the formation of a neutron-proton spin pair is 0.5656 MeV the difference between thehe binding energies for proton-proton and neutron-neutron spin pairs is 0.4682−0.5656=−0.0974MeV. Given the standard deviations of the two components of this difference clearly it is not significantly different from zero.

Thus there is no statistically significance difference in the binding energies associated with the formations of proton-proton, neuand neutron-neutron spin pairs at the usual level of confidence. However there is about 0.5 MeV between the binding energies of both neutron-neutron and proton-proton spin pairs and that of a neutron-proton spin pair. More precisely the difference is 0.5235 MeV.

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