San José State University

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Thayer Watkins
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 The Potential Function for the Nuclear Force

A central force between two particles, such as nucleons, which is carried by decaying particles, such as π mesons, will have the form

#### −H*exp(−λr)/r²,

where r is the separation distance between the two particles. The negative sign indicates that it is an attractive force. The parameter λ will depend upon the time rate of decay of the force-carrying particles and their velocity. It is convenient to express λ as the reciprocal of a scale distance r0. The purpose of this page is obtain an estimate of the constant H* using the properties of the deuteron.

An estimate of the value of r0 is available from the mass of the π mesons. Hideki Yukawa preposed a relationship between the mass of the force-carrying particle and the scale distance of the force. From Yukawa's relation the scale distance for the nuclear force should be 1.522 fermi; i.e., 1.522×10−15 meters.

The potential energy function V(r) is defined to be such that (∂V/∂r)=−F(r), where F is the force function.

For the above formula for the nuclear force the potential energy for one nucleon separated from another by a distance of r is

#### V(r) = ∫r∞H*(exp(−z/r0)/z²)dz = H*∫r∞(exp(−z/r0)/z²)dz

This may be simplified by the substitution z/r0=s and thus z=r0s and dz=r0ds; i.e.,

#### V(r) = −(H*/d0)∫r/r0∞((exp(−s)/s²)ds

Consider the integral W(x) = ∫x((exp(−s)/s²)ds and the integration-by-parts formula of ∫GdF=GF−∫FdG, with G=exp(-s) and F=−1/s. Thus dF=1/s² and dG=−exp(−s). Therefore, GdF=exp(−s)/s², FG=−exp(−s)/s and FdG=+exp(−s)/s. Thus,

#### W(x) = [−exp(−s)/s]x∞ − (∫x∞(−exp(−s)/s)ds) or, equivalently W(x) = −exp(−x)/x + (∫x∞(exp(−s)/s)ds)

The integral on the right is closely related to the function called the exponential integral and its values have been tabulated. However, an acceptable approximation of the function W(x) is easily obtained numerically. Some of the relevant values are given below where F(x) denotes exp(−x)/x². F(x)dx stands for the area of the trapezoid between x and x−dx. The computations started with W(30) taken to be zero.

```  x	     F(x)	        F(x)dx	        W(x)

3.00	0.005531896	5.57841E-05	0.003547917
2.99	0.005624930	5.67229E-05	0.003603702
2.98	0.005719656	5.76788E-05	0.003660424
2.97	0.005816108	5.86521E-05	0.003718103
2.96	0.005914321	5.96433E-05	0.003776755
2.95	0.006014330	6.06525E-05	0.003836399
2.94	0.006116170	6.16802E-05	0.003897051
2.93	0.006219879	6.27269E-05	0.003958731
2.92	0.006325493	6.37927E-05	0.004021458
2.91	0.006433052	6.48782E-05	0.004085251
2.90	0.006542595	6.59838E-05	0.004150129
2.89	0.006654160	6.71098E-05	0.004216113
2.88	0.006767791	6.82566E-05	0.004283223
2.87	0.006883527	6.94247E-05	0.004351479
2.86	0.007001413	7.06145E-05	0.004420904
2.85	0.007121492	7.18265E-05	0.004491519
2.84	0.007243809	7.30611E-05	0.004563345
2.83	0.007368409	7.43187E-05	0.004636406
2.82	0.007495340	7.55999E-05	0.004710725
2.81	0.007624649	7.69052E-05	0.004786325
2.80	0.007756386	7.82349E-05	0.004863230
2.79	0.007890599	7.95897E-05	0.004941465
2.78	0.008027342	8.09700E-05	0.005021055
2.77	0.008166665	8.23764E-05	0.005102025
2.76	0.008308623	8.38095E-05	0.005184401
2.75	0.008453271	8.52697E-05	0.005268211
2.74	0.008600664	8.67576E-05	0.005353480
2.73	0.008750861	8.82739E-05	0.005440238
2.72	0.008903919	8.98191E-05	0.005528512
2.71	0.009059899	9.13938E-05	0.005618331
2.70	0.009218863	9.29987E-05	0.005709725
2.69	0.009380874	9.46343E-05	0.005802723
2.68	0.009545995	9.63014E-05	0.005897358
2.67	0.009714293	9.80006E-05	0.005993659
2.66	0.009885836	9.97326E-05	0.006091660
2.65	0.010060692	0.000101498	0.006191393
2.64	0.010238933	0.000103298	0.006292891
2.63	0.010420631	0.000105132	0.006396188
2.62	0.010605860	0.000107003	0.006501321
2.61	0.010794695	0.000108910	0.006608324
2.60	0.010987216	0.000110854	0.006717233
2.59	0.011183501	0.000112836	0.006828087
2.58	0.011383631	0.000114857	0.006940923
2.57	0.011587692	0.000116917	0.007055779
2.56	0.011795767	0.000119019	0.007172696
2.55	0.012007946	0.000121161	0.007291715
2.54	0.012224316	0.000123346	0.007412876
2.53	0.012444972	0.000125575	0.007536223
2.52	0.012670006	0.000127848	0.007661798
2.51	0.012899516	0.000130166	0.007789645
2.50	0.013133600	0.000132530	0.007919811
2.49	0.013372360	0.000134941	0.008052341
2.48	0.013615899	0.000137401	0.008187282
2.47	0.013864325	0.000139910	0.008324683
2.46	0.014117746	0.000142470	0.008464593
2.45	0.014376274	0.000145081	0.008607063
2.44	0.014640025	0.000147746	0.008752145
2.43	0.014909115	0.000150464	0.008899891
2.42	0.015183665	0.000153237	0.009050355
2.41	0.015463800	0.000156067	0.009203592
2.40	0.015749645	0.000158955	0.009359659
2.39	0.016041330	0.000161902	0.009518614
2.38	0.016338990	0.000164909	0.009680516
2.37	0.016642761	0.000167978	0.009845424
2.36	0.016952784	0.000171110	0.010013402
2.35	0.017269201	0.000174307	0.010184512
2.34	0.017592161	0.000177570	0.010358819
2.33	0.017921816	0.000180901	0.010536389
2.32	0.018258321	0.000184301	0.010717289
2.31	0.018601835	0.000187772	0.010901590
2.30	0.018952522	0.000191315	0.011089362
2.29	0.019310551	0.000194933	0.011280677
2.28	0.019676094	0.000198627	0.011475611
2.27	0.020049328	0.000202399	0.011674238
2.26	0.020430434	0.000206250	0.011876636
2.25	0.020819600	0.000210183	0.012082887
2.24	0.021217017	0.000214199	0.012293070
2.23	0.021622882	0.000218301	0.012507269
2.22	0.022037397	0.000222491	0.012725571
2.21	0.022460770	0.000226770	0.012948061
2.20	0.022893215	0.000231141	0.013174831
2.19	0.023334949	0.000235606	0.013405972
2.18	0.023786199	0.000240167	0.013641578
2.17	0.024247195	0.000244827	0.013881745
2.16	0.024718176	0.000249588	0.014126572
2.15	0.025199385	0.000254452	0.014376160
2.14	0.025691074	0.000259423	0.014630612
2.13	0.026193501	0.000264502	0.014890035
2.12	0.026706931	0.000269693	0.015154537
2.11	0.027231636	0.000274998	0.015424230
2.10	0.027767898	0.000280420	0.015699227
2.09	0.028316004	0.000285961	0.015979647
2.08	0.028876251	0.000291626	0.016265608
2.07	0.029448944	0.000297417	0.016557234
2.06	0.030034398	0.000303337	0.016854651
2.05	0.030632934	0.000309389	0.017157988
2.04	0.031244884	0.000315577	0.017467377
2.03	0.031870592	0.000321905	0.017782954
2.02	0.032510407	0.000328375	0.018104859
2.01	0.033164693	0.000334993	0.018433234
2.00	0.033833821	0.000169169	0.018768227

```

The dimensions of the ratio (H/d0) are those of energy. The disassociation and binding energy of the deuteron is 2.22457 MeV. The accepted diameter of the deuteron is 4.2 fermi.1 The nucleons themselves are believed to have diameters of 1 fermi. Thus the diameter of the deuteron would include the two half diameters of the nucleons so the separation distance of the centers of the two nucleons is 3.2 fermi. The ratio of this distance to r0=1.522 fermi is 2.1. The value of W(2.1) is 0.015699227. Thus, this would mean that the ratio (H/r0) would have the value of 141.7 MeV. In joules this is 2.27×10−11. This figure multiplied by r0=1.522×10−15 gives H*=3.455×10−26 kg m³/s².

A value of 1 fermi for the diameters of the nucleons is conceptually and numerically uncertain. Therefore it is not advisable to rely too strongly on this figure. If the diameters of the nucleons is ignored and the separation distance of the nucleons is taken to be 4.2 fermi then the ratio to r0 is 2.76 and W(2.76) is 0.005184401. Thus the estimate of H*/r0 would be 429.0 MeV or 6.874×10−11 joules. This implies a value of H* of 1.0462×10−25. This suggests that H* has a value on the order of 1×10−25 kg m³/s².

An estimate of the separation distance of 4.13 fermi of the nucleons in the deuteron arose in another attempt to estimate the dimensions of the deuteron. This separation distance gives a value for (H/r0) of 397.8567 MeV or 6.3737×10−11 joules and a value of H* of 9.7007×10−26 kg m³/s².

(To be continued.)

1 I. Sick, "Precise Radii of Light Nuclei from Electron Scattering," in Savely G. Karshenboim (ed.), Precision Physics of Simple Atoms and Molecules,, (Springer, Berlin Heidelberg 2008), page 70 gives the root-mean-square charge radius of the deuteron as 2.13±0.01 fermi.