San José State University |
---|
applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
---|
of a Neutron Relative to that of a Proton |
Two previous studies (1 and 2) developed information that can be used to estimate the strong force charge of a neutron relative to that of a proton. Those studies estimated the interaction binding energy of neutrons with protons and of neutrons with the previous neutron and likewise for protons.
Let q_{1} and q_{2} be the strong force charges of two particles. The force between them is of the form
where s is the separation distance of the centers of the two particles.
This dependence carries over into the potential energy of the two particles. The binding energy of nuclides is based upon the mass deficits of the nuclides and appears to be much like the potential energy loss involved in the formation of nuclei.
Conventional theory takes the force between any two nucleons to be an attraction. In addition to the strong force per se there are forces associated with the formation of the three types of spin pairs. These are all attractions. But spin pair formation is exclusive in the sense that a neutron can form a pair with one proton and with one neutron. The energies involved in pair formations are on the order of about two million electron volts (MeV). This is far larger in magnitude than the energies involved in the strong force interaction of two nucleons. However in nuclei involving a larger number of nucleons the strong force interaction becomes dominant. The effects of the strong force cannot be discerned in scattering experiments. They are perceptible only in the binding energies of larger nuclides.
Let the strong force charge of a proton be taken to be +1 and denote that of the neutron as q, where q may be positive or negative. The binding energies associated with the three types of interactions are then:
Interaction of neutron and proton | Interaction of neutron and neutron | Interaction of proton and proton |
1·qH | q·qH | 1·1H |
qH | q²H | H |
where H is a constant. Thus the ratio of the interaction of a neutron and a neutron to the interaction of a neutron and a proton should give the value q²H/qH or q. The ratio of the interaction of a neutron and a proton to the interaction of a proton and a proton should also give the value q. But the ratio of the interaction of a neutron and a neutron to the interaction of a proton and a proton should give the value of q² and thus the square root of the ratio should give the value of q.
The sources of the data for the following displays are the tables given in the Appendix.
First the data based upon the incremental binding energies of neutrons will be considered. The higher shells have more data points and are more accurate. These will be displayed first. There are no enough data points to obtain accurate values for the first and second shells.
Neutron Shell Number | Proton Shell Number | Interaction of neutron and proton(qH) | Interaction of neutron and neutron | Ratio (q) |
7 | 7 | 0.18118 | -0.14081 | -0.7772 |
6 | 6 | 0.29800 | -0.20134 | -0.6756 |
5 | 5 | 0.48214 | -0.32096 | -0.6657 |
4 | 4 | 0.75146 | -0.56127 | -0.7459 |
3 | 3 | 1.21772 | -0.87970 | -0.7224 |
Second come the results based upon the data for the incremental binding energies of protons.
Neutron Shell Number | Proton Shell Number | Interaction of neutron and proton (qH) | Interaction of proton and proton (H) | Ratio (q) |
7 | 7 | 0.19607 | -0.39708 | -0.4938 |
6 | 6 | 0.35983 | -0.50648 | -0.7105 |
5 | 5 | 0.55628 | -0.71125 | -0.7821 |
4 | 4 | 0.96481 | -1.1158 | -0.8647 |
3 | 3 | 1.54063 | -1.62933 | -0.9456 |
Third comes the results combining elements of the two approaches.
Neutron Shell Number | Proton Shell Number | Interaction of neutron and neutron (q²H) | Interaction of proton and proton (H) | Ratio (q²) | Square Root of Ratio (q) |
7 | 7 | -0.14081 | -0.39708 | 0.3546 | -0.5955 |
6 | 6 | -0.20134 | -0.50648 | 0.3975 | -0.6305 |
5 | 5 | -0.32096 | -0.71125 | 0.4513 | -0.6718 |
4 | 4 | -0.56127 | -1.1158 | 0.5030 | -0.7092 |
3 | 3 | -0.8797 | -1.62933 | 0.5399 | -0.7348 |
The averages of the estimates of q in the three above tables are, respectively, −0.7174, −0.7593 and −0.6684. The value of q undoubtedly must be a ratio of whole numbers. The value of −2/3 is what came out of previous investigations.
The strong force charges of the neutron and the proton should be derived from such charges of the quarks that make them up. A proton consists of two up quarks and one down quark, whereas a neutron consists of two down quarks and one up quark. Let u and d denote the strong force charges of the up and down quarks respectively. Let q_{p} and q_{n} be the strong force charges of the proton and neutron, respectively. Then
Adding these two equation together gives
Subtracting this last equation from the first equation above gives
Now if q_{p}=3 and q_{n}=−2 then u is equal to +(8/3) and d is −(7/3). Or scaled up, u would be +8 and d would be −7. These values are plausible.
The strong force charge of a neutron is opposite in sign to that of a proton. This is consistent with the force between a neutron and a proton being an attraction. The magnitude of the strong force charge of the neutron is less than that of a proton. The values found are consistent with the magnitude of the strong force charge of a neutron being two-thirds that of a proton.
The tables below were developed in the two webpages, Incremental Binding Energies of Neutrons and Incremental Binding Energies of Protons
The Regression Equation Results for the Incremental Binding Energies of Neutrons in Various Shells as a Function of the Number of Neutrons and Protons | ||||||||
---|---|---|---|---|---|---|---|---|
Neutron Shell Number | Proton Shell Number |
Constant (MeV) | P (MeV) | e(P) (MeV) |
d(P≥N) (MeV) | N (MeV) | e(N) (MeV) | R² |
4 | 5 | 10.83982 | 0.65935 | -0.10170 | 0 | -0.55997 | 2.65900 | 0.98373 |
5 | 5 | 5.29675 | 0.48214 | 0.02049 | 2.1583 | -0.32096 | 2.63296 | 0.9798 |
6 | 5 | 0.56093 | 0.41939 | -0.01998 | 0 | -0.20786 | 2.06865 | 0.97313 |
7 | 5 | 7.14857 | 0 | 0.30571 | 0 | -0.06 | 1.25429 | 0.92345 |
6 | 6 | 6.15082 | 0.298 | 0.00826 | 0 | -0.20134 | 2.21852 | 0.97626 |
7 | 6 | 1.86393 | 0.29000 | -0.00710 | 0 | -0.15658 | 1.72517 | 0.97026 |
8 | 6 | -8.86265 | 0.21937 | -0.04632 | 0 | -0.04070 | 1.33403 | 0.99110 |
3 | 4 | 10.73053 | 0.93339 | 0.15810 | 0 | -0.8046 | 4.6504 | 0.97940 |
4 | 4 | 6.15224893 | 0.75146 | 0.070873 | 2.54022 | -0.56127 | 3.10121 | 0.97270 |
5 | 4 | 2.15020 | 0.68663 | -0.02306 | 0 | -0.38942 | 2.17326 | 0.96051 |
2 | 3 | 5.5553 | 2.21881 | -0.00434 | 0 | -1.16881 | 6.43827 | 0.98815 |
3 | 3 | 4.00449 | 1.21772 | 0.17548 | 4.12186 | -0.8797 | 0.96625 | |
4 | 3 | 1.26803 | 1.14448 | 0.06639 | 0 | -0.60993 | 2.3985 | 0.88805 |
2 | 2 | 3.07600 | 2.3221 | 1.38288 | 4.99037 | -2.33799 | 8.94596 | 0.92193 |
3 | 2 | -1.30201 | 1.66479 | -0.03685 | 0 | -0.60993 | 2.77570 | 0.88051 |
7 | 7 | 9.06369 | 0.18118 | -0.01552 | 0 | -0.14081 | 1.57530 | 0.94902 |
8 | 7 | 2.46658 | 0.22826 | 0.0010277 | 0 | -0.12820 | 1.33960 | 0.91938 |
Where 0 appears for the coefficient of d(P≥N) it means that there was no variation in d(P≥N) for that data set. Either P≥N for all of the cases or none of the cases in the data set.
The regression coefficients for N represent the interaction binding energy of a neutron with the previous neutron in the shell.
The same statistical analysis was carried out for the incremental binding energies of protons. The results are displayed in the same sort of table that was used for the analysis of neutrons. However the roles of N and P are reversed in the table.
The Regression Equation Results for the Incremental Binding Energies of Protons in Various Shells as a Function of the Number of Neutrons and Protons | ||||||||
---|---|---|---|---|---|---|---|---|
Neutron Shell Number | Proton Shell Number |
Constant (MeV) | N (MeV) | e(N) (MeV) |
d(P≥N) (MeV) | P (MeV) | e(P) (MeV) | R² |
5 | 4 | 15.33727 | 0.6614 | 0.00290 | 0 | -1.08327 | 3.01353 | 0.96599 |
5 | 5 | 7.52963 | 0.55628 | -0.03131 | -1.64371 | -0.71125 | 2.53724 | 0.98716 |
6 | 5 | 14.52240 | 0.39424 | 0.06526 | 0 | -0.69040 | 2.65958 | 0.98976 |
7 | 6 | 14.10400 | 0.2787 | 0.03977 | 0 | -0.5253 | 2.00687 | 0.98591 |
7 | 7 | 12.09002 | 0.19607 | -0.03737 | 0 | -0.39708 | 1.96627 | 0.97660 |
8 | 7 | 11.52883 | 0.25765 | 0.021940 | 0 | -0.47352 | 1.67387 | 0.97432 |
4 | 3 | 16.10828 | 1.03216 | 0.17652 | -1.72723 | -1.72723 | 4.68060 | 0.90126 |
4 | 4 | 8.21452 | 0.96481 | -0.05981 | -1.3428 | -1.1158 | 2.94595 | 0.98066 |
4 | 5 | 8.89236 | 0.59585 | -0.02981 | 0 | -0.85299 | 2.50624 | 0.96942 |
3 | 2 | 7.80413 | 1.39912 | 0 | -0.80572 | 5.7504 | 0.97358 | |
3 | 3 | 8.18032 | 1.54063 | 0.18163 | -2.98841 | -1.62933 | 4.47439 | 0.96823 |
3 | 4 | 5.28637 | 0.88169 | -0.02836 | 0 | -0.99469 | 2.37122 | 0.94190 |
2 | 2 | 4.55387 | 2.61442 | 0.52651 | -5.21684 | -1.98458 | 7.11596 | 0.91530 |
2 | 3 | 3.54834 | 1.98532 | -0.04742 | 0 | -1.87876 | 3.6384 | 0.98178 |
6 | 6 | 6.47405 | 0.35983 | -0.11103 | 0 | -0.50648 | 2.38749 | 0.98479 |
HOME PAGE OF Thayer Watkins |