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Energy of Neutrons When the Number of Protons Equals or Exceeds the Number of Neutrons |
The incremental binding energy for a neutron in a nuclide is the binding energy for that nuclide less the binding energy of a nuclide having one less neutron. Binding energies are available for 2931 nuclides but the incremental binding energy (IBE) can be computed for only 2816 nuclides. These are the ones for which a nuclide with one less neutron exists.
The incremental binding energies of nuclides as functions of their proton number have a degree of regularity. The data are shown in the graph for the case of the 28th neutron.
Shown below are those relations for the 27th, 28th and 29th neutrons.
There are jumps which occur when the proton number becomes equal to the neutron number.
When the relations are shifted so that the points of the sharp increases coincide the graph is as follows.
When the differences in incremental binding energies are computed from that rearrangement of the data the result is as shown below.
The above procedure can now be used to display the jump in IBE at other neutron numbers. The corresponding graphs for the 13th, 14th and 15th neutrons are shown below.
As in the case of the 27th, 28th and 29th neutrons there are sharp increase at the points where the proton number is equal to the neutron number. When the data is plotted versus the difference between the proton and the neutron number the result is:
Below are two graphs displaying the information for neutrons 29, 30 and 31.
As can be seen the phenomenon of the jump in IBE when the proton number equals or exceeds the neutron number is displayed in all cases.
The analysis was worthwhile in identifying the relative large increases in incremental binding energy which occur when the proton number becomes equal to the neutron number.
In a previous study it was found that the vast majority of the variation in the binding energies of 2931 nuclides is explained in terms of the number of interactions of protons and neutrons. The interactions are of two types: 1. The nuclear strong force interactions 2. The formation of spin pairs. The strong force interactions are of three types: 1. neutron-proton interactions 2. neutron-neutron interactions 3. proton-proton interactions. Likewise there are these same three type of spin pair formation. If a neutron is added to a nuclide containing #n neutrons and #p protons it interacts with all of them. If a neutron is added to a nuclide that has fewer protons than neutrons then the additional neutron cannot form a neutron-proton pair. It can form a neutron-neutron pair only if #n is an odd number; i.e., if there is an unpaired neutron in the nuclide.
Thus the explanation for the jump in the IBE of a neutron when #p equals or exceeds #n is that only in this case can the additional neutron from a neutron-proton pair. This explanation then suggests that the size of the jump should be independent of the number of neutrons. It also suggests that a similar phenomenon should occur for the IBE of protons as a function of the neutron number.
Below are shown the graphs of the IBE of protons as a function of the numberof neutrons is the nuclide.
In each case there a relatively large jump in the IBE. Although it is difficult to determine from the graphs, the jump occurs when the neutron number equals the proton number.
Regression equations were obtained for the IBE of neutrons for the cases of #n=27, 28 and 29 with four explanatory variables: #p, d(#p-#n), u(#p-#n) and e(#p) where
The results were:
The Incremental Binding Energy of
Neutrons as a Function of the Number of Protons in the Nuclide |
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#n | Const. | #p | d(#p-#n) | u(#p-#n) | e(#p) | R² |
27 | -8.11627 | 0.755511 | 1.33281 | -0.08547 | -0.41617 | 0.99323 |
28 | -5.19984 | 0.70913 | 1.58057 | -0.13153 | 0.38497 | 0.99654 |
29 | -7.85267 | 0.66542 | 1.13892 | -0.05462 | -0.31482 | 0.99653 |
The coefficients high-lighted in white are not statistically significant at the 95 percent level of confidence. The statistical insignificance of the coefficients for u(#p-#n) indicates the slope of the relationship above the jump at #p=#n is not different from the slope below that value. The coefficient for d(#p-#n) is value of the jump in IBE when #p≥#n. It is about 1.3 MeV for the range of #p for 27 to 29.
The corresponding regression equations for the IBE of protons were obtained for #p=27, 28 and 29.
The Incremental Binding Energy of
Protons as a Function of the Number of Neutrons in the Nuclide |
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#p | Const. | #n | d(#n-#p) | u(#n-#p) | e(#n) | R² |
27 | -13.34592 | 0.57498 | 2.350557 | 0.05448 | -0.20373 | 0.99728 |
28 | -13.44934 | 0.65810 | 1.88214 | -0.01534 | 0.33421 | 0.99396 |
29 | -14.68974 | 0.56285 | 1.74263 | 0.01721 | -0.27360 | 0.99811 |
Again the coefficients high-lighted in white are not statistically significant at the 95 percent level of confidence. The statistical insignificance of the coefficient for u(#n-#p) indicates that there is a jump in the level of IBE at #n=#p but no change in the slope. The magnitude of the jump is greater for the IBE for protons than for neutrons although in both cases the value is supposed to be due to the formation of a neutron-proton pair. However the general level of of the IBE, as evidenced by the value of the constant, is lower for the IBE of protons than that for neutrons.
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