San José State University

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

Estimates of Incremental Binding
Energy for Proton-Proton and
Neutron-Proton Pair Formation

This is an examination of the incremental binding energy for protons to see if the same phenomena prevails as do for neutrons. In particular do the same critical levels, known as magic numbers, occur for protons as for neutron. The incremental binding energy for a nucleon is the difference in binding energy for nuclide compared with the binding energy of the nuclide with one less of that nucleon but the same number of the other nucleon. Here is the graph of the incremental binding energies of the neutrons in the isotopes of Chromium (atomic number 24).

The sawtooth pattern is due to the formation of nucleon spin pairs. The number of protons in chromium nuclei is 24. If the number of neutrons is less than 24 then the addition of another neutron forms a neutron-proton spin pair. When the number of neutrons is odd then the addition of another neutron also forms a neutron spin pair. When the number of neutrons exceeds the number of protons the addition of another neutron does not form a neutron-proton spin pair. When the number of neutrons reaches 28 a neutron shell is filled and any additional neutrons must go into a higher shell involving less energy. In this case there is very little in the way of a sharper drop at 28. More noticable is a change in the character of the odd-even fluctuations.

Here is the graph of the incremental binding energies of the neutrons in the isotopes of Silver (atomic number 47).

Here there is a noticably sharper drop at the magic number of 50.

For comparison here is the graph of the incremental binding energies of the neutrons in the isotopes of tin (atomic number 50).

Again when the number of neutrons reaches a magic number, in this case 82, a neutron shell is filled and any additional neutrons must go into a higher shell involving less energy. A change in the amplitude of the odd-even fluctuations after 50 indicates the magnitude of the binding energy associated with the formation of neutron-proton spin pairs.

Below is shown the graph of the incremental binding energies of protons in nuclides with 36 neutrons. There is a discernably sharper drop after the magic number of 28. Number 20 is a magic number but it is not always evidenced in the data. Here it is uncertain whether there is a sharper drop after 20 than for the other proton numbers. There is also a sharper drop after 36 but this has to do with there not being additional neutron-proton pairs being formed.

Here is a case in which there are no magic numbers over the range of protons. It is the case of nuclides with 46 neutrons.

The odd-even fluctuation has entirely to do with the formation of proton pairs. The average fluctuation for the above case is 3.268 MeV. There are also no magic numbers for the case of 48 neutrons and the average fluctuation is significantly smaller at 2.948 MeV. A comparison of the two cases shows that the number of neutrons does affect the binding energy involved in the formation of proton pairs.

The drop in incremental binding energy when the number of protons exceeds the number of neutrons provides a measure of the binding energy involved in the formation of neutron-proton pairs. Here is the case for 44 neutrons.

There is a sharper drop when the number of protons goes beyond 44, but there would have been some drop as a continuation of the trend. The projected level in the above graph is shown in red. The projected value is

Proj IBE(p) = IBE(p-2) + (IBE(p-2)-IBE(p-4)
= 2*IBE(p-2) − IBE(p-4)

The difference between the actual value and the projected value is 1.95 MeV. This is the value of the binding energy involved in the formation of a neutron-proton pair, at least for nuclides having 44 neutrons. The value for the case of 42 neutrons is 2.1 MeV. For the case of nuclides with 40 neutrons the value is 2.448 MeV. However for some cases, such at the ones for 33, 35 and 37 neutrons shown below, something different is occurring at the point where the number of protons equals the number of neutrons.



In the above cases, all involving odd numbers of neutrons, the effect of proton pair formation is offsetting the effect due to the number of protons exceeding the number of neutrons.

However, the cases for nuclides having 34 and 36 neutrons, which give values of 1.1777 MeV and 1.52 MeV, are also suspect.

Beyond 45 neutrons the number of protons never exceeds the number of neutrons. The table below gives the estimates for the cases which do not involve any magic numbers at or near the level at which the number of protons equals the number of neutrons.

Estimates of the Binding
Energy Involved in a
Neutron-Proton Pair Formation
Number of
Neutrons
np Pair drop
(MeV)
26 2.1858
38 1.96
40 2.448
42 2.101
44 1.95

The average of the above values is 2.129 MeV. There is of course a direct estimate of the binding energy of a neutron-proton pair, the binding energy for the formation of a deuteron, 2.295 MeV.

(To be continued.)


HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins