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The Relationship Between the Effects on Binding Energy
of Adding and Subtracting Neutrons and Protons

Let BE(n,p) be the binding energy of a nuclide as a function of the number of neutrons n and the number of protons p which it contains. The alpha nuclides are the ones in which n and p are equal and are an even number so that they could contain an integral number of alpha particles. This means that n=p=2a where a is the number of alpha particles.

The effect of subtracting one neutron from each alpha nuclide is then

SBE(2a, 2a) = BE(2a, 2a) − BE(2a-1, 2a)

The effect on binding energy of adding one neutron is

ABE(2a, 2a) = BE(2a+1, 2a) − BE(2a, 2a)

The difference of these two quantities is then

DNBE(2a, 2a) = (BE(2a, 2a) − BE(2a-1, 2a)) − (BE(2a+1, 2a) − BE(2a, 2a))
which reduces to
DNBE(2a, 2a) = 2BE(2a, 2a) − (BE(2a+1, 2a) + BE(2a-1, 2a))

The analogous quantity for the subtraction and addition of a proton to each alpha nuclide is

DPBE(2a, 2a) = 2BE(2a, 2a) − (BE(2a, 2a+1) + BE(2a, 2a-1))

The graphs of these two quantities, DNBE and DPBE, as functions of the number of alpha particles a are shown below.

The close correspondence of the shapes is remarkable. The points where the curves show a sharp drop at 7, 10 and 14 alpha particles at where the numbers of neutrons and the numbers of protons are equal to 14, 20 and 28, all magic numbers.

Although the curves virtually coincide over some ranges the values, as shown below, are not identical.

Number
of Alpha
Particlels
DNBE
(MeV)
DPBE
(MeV)
1 21.779527 21.46329
2 17.44012 17.23372
3 14.013356 13.775493
4 11.527132 11.520412
5 10.412158 10.103248
6 9.42156 9.20142
7 8.83683 8.70614
8 6.58743 6.4008
9 6.64809 6.4646
10 7.2433 7.2785
11 7.0353 6.7697
12 6.015 5.752
13 5.7824 5.4998
14 6.4711 6.394
15 4.6627 4.771
16 5.07 5.52
17 5.3 5.51
18 5.4 5.27
19 4.675 4.16
20 4.9 4.6
21 5.0 4.7
22 4.8 4.3
23 4.3

The implication of the close correspondence of the two curves is that each twist and turn is of significance in the physics of nuclei.


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