San José State University

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The Increment in Binding Energy Due to
the Formation of Nucleon Pairs

This is a continuation of studies on the effect of nucleon pair formation on the binding energies of nuclides. One previous study examined the level and constancy of the enhancement of binding energy due to the formation of neutron pairs and proton pairs by looking at the values for the isotopes of Calcium (proton number 20) and for the istones of neutron number 28. It was found that the enhancements are nearly constant within shells but differ between shells. Also the values for neutron pairs differed in some shells from the values for proton pairs. Two other studies, pairenhancement I and pairenhancement II looked at the effect of neutron pairs and neutron-proton pairs on the excess binding energies of nuclides which could contain an integral number of alpha particles. (These are herefater called alpha nuclides.)

In this study again the alpha nuclides are used as a test case. First the binding energies of the alpha nuclides are compiled. Then the nuclides which could contain an integral number of alpha particles plus one proton. These are denoted as the α+p nuclides. Likewise the binding energies for the α+n, α+2n, α+2p and the α+n+p nuclides are compiled. The values are shown below.


Binding Energies in millions of electron volts (MeV)

#a	 α		α+p		α+n		α+2n		α+n+p		α+2p
0	0		0		0		0		2.224573	0		
1	28.295674	26.33	        27.41		29.2691	    	31.9946 	26.924
2	56.49951	56.3144		58.1649		64.9772		64.7507		60.3205
3	92.161728	94.1053		97.108063	105.284508	104.658625	98.73323
4	127.619336	128.21961	131.76266	139.807		137.3692	132.1535
5	160.644859	163.0762	167.40597	177.76991	174.1453	168.5776
6	198.25689	200.5282	205.58756	216.68063	211.89414	206.046
7	236.53689	239.285		245.01044	255.61963	250.6049	243.685
8	271.78066	274.0572	280.42222	291.83916	285.56553	278.721
9	306.7157	308.5735	315.5046	327.3427	320.6473	313.122
10	342.052	        343.137	        350.4147	 361.8953	354.6871	346.905
11	375.4747	377.089		385.0047	398.1944	390.3607	381.975
12	411.462		413.547		422.044		435.0441	426.6288	417.7
13	447.697		449.296		458.3802	471.7587	462.7332	453.15
14	483.988		484.682		494.235		506.4538	497.108		486.96
15	514.992		515.45		525.223		538.119		528.17		517.63
16	545.95		545.9		556.01		569.29		558.71		547.8
17	576.4		575.9		586.62		600.33		589.2		577.8
18	607.1		606.5		617.93		631.28		620.1		608.2
19	638.1		637.9		649.74		663.008		651.7	
20	669.8		669.2		681.3		694.7		682.7	
21	700.9		700.0		712.3		725.8		713.7	
22	731.4		730.3		743.4		757.4		744.4	
23	762.1				774.3		789.1		775.2	
24	793.4				806.0		820.9		806.5	
25	824.9				835.6		848.9	


From these values the enhancements due to the addition of various nucleon combinations are computed. For example, the enhancement due to the addition of a proton is computed by subtracting from the binding energy of an α+p nuclide the binding energy of the corresponding α nuclide; i.e., BE(α+p)−BE(α).


#a	p	n	2n  		n+p		np		nn		pp		nn-pp
0	0	0	0		2.224573	2.224573	0		
1	-1.965674	-0.885674	0.973426	3.698926	6.550274	2.744774	2.559674	0.1851
2	-0.18511	1.66539		8.47769		8.25119		6.77091		5.14691		4.19121		0.9557
3	1.943572	4.946335	13.12278	12.496897	5.60699		3.23011		2.684358	0.545752
4	0.600274	4.143324	12.187664	9.749864	5.006266	3.901016	3.333616	0.5674
5	2.431341	6.761111	17.125051	13.500441	4.307989	3.602829	3.070059	0.53277
6	2.27131		7.33067		18.42374	13.63725	4.03527		3.7624		3.24649		0.51591
7	2.74811		8.47355		19.08274	14.06801	2.84635		2.13564		1.65189		0.48375
8	2.27654		8.64156		20.0585		13.78487	2.86677		2.77538		2.38726		0.38812
9	1.8578		8.7889		20.627		13.9316		3.2849		3.0492		2.6907		0.3585
10	1.085		8.3627		19.8433		12.6351		3.1874		3.1179		2.683		0.4349
11	1.6143		9.53		22.7197		14.886		3.7417		3.6597		3.2717		0.388
12	2.085		10.582		23.5821		15.1668		2.4998		2.4181		2.068		0.3501
13	1.599		10.6832		24.0617		15.0362		2.754		2.6953		2.255		0.4403
14	0.694		10.247		22.4658		13.12		2.179		1.9718		1.584		0.3878
15	0.458		10.231		23.127		13.178		2.489		2.665		1.722		0.943
16	-0.05		10.06		23.34		12.76		2.75		3.22		1.95		1.27
17	-0.5		10.22		23.93		12.8		3.08		3.49		2.4		1.09
18	-0.6		10.83		24.18		13.0		2.77		2.52		2.3		0.22
19	-0.2		11.64		24.908		13.6		2.16		1.628		
20	-0.6		11.5		24.9		12.9		2.0		1.9		
21	-0.9		11.4		24.9		12.8		2.3		2.1
22	-1.1		12.0		26.0		13.0		2.1		2.0
23			12.2		27.0		13.1		
24			12.6		27.5		13.1		
25			10.7		24.0			

The graph of the enhancement due to an additional neutron is shown below.

In this graph the effect of an additional neutron increases with the size of the nuclide which it augments because the more other nucleons the more nucleon-nucleon interactions. There are two roughly linear portions in the display. The first goes from 1 to 3 alpha particles. This corresponds to 2 to 6 neutrons. This is the second shell for neutrons. The second linear portion goes from 8 neutrons up to 100. The slope of this second portion is lower than that of the first portion. This reflects the greater separation distance in the higher shells than in the lower shell. The strength of the nucleon-nucleon interaction is greater in the lower shell. The relative minima in the display correspond to the nuclear magic numbers.

The graph for the effect of an additional proton appears to be quite different from that for a neutron.

Again there are two linear portions corresponding to 1 to 3 alpha particles (2 to 6 neutrons) and 4 and above alpha particles. The second portion has a negative slope. This is because at the greater separation distance of the higher shells the electrostatic repulsion of the protons is greater than the nucleonic attraction.

The effect of just the pairing of neutrons can be estimated by deducting from the effect of two additional neutrons double the effect of one additional neutron. The graph of these values is shown below.

Although the effect is not constant it decreases from about 4 MeV at 2 alphas to 2 MeV at 22 alphas. It is thus about 3 MeV over the range of alphas.

The effect of proton pairing has a similar pattern.

Again the values go from about 4 MeV at 2 alphas to 2 MeV at 22 alphas. The similarity is easier to comprehend when the two effects are plotted in the same graph, as below.

Over a good deal of the range the difference is nearly constant.

The same procedure can be used to estimate the effect of a pairing of a neutron and a proton into a deuteron; i.e., BE(α+n+p)−BE(α) less the effect of a single neutron and a single proton.

Here the level for the small nuclides is larger but the level for the larger nuclides is about 2 MeV, the same as for the neutron pair and the proton pair. The effects for all three nucleon pairs are displayed together in the graph below.

The strength of the effect of neutron-proton pairings for the small nuclides is striking given that the binding energy of the deutron, neutron-proton pairing, is only about 2.225 MeV.


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