San José State University |
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Nuclear binding energies explained by shell-to-shell nucleonic interactions |
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When the incremental binding energies of neutron pairs are plotted against the number of proton pairs the relationships are linear over a range corresponding to the proton shells. For example, consider the case below:
The slope of the relationship corresponds to the binding energy associated with the interaction of the proton pairs with the last, in this case the 25th, neutron spin pair in the nuclide.
The same applies to the relationship between the incremental binding energies of neutron pairs and and proton pairs, as is illustrated below.
Furthermore when the incremental binding energies of neutron pairs are plotted versus the number of neutron pairs therre is also a more or less linear relationship, only that it has a negative slope.
The sharp drop after 41 neutron pairs corresponds to the magic number of 82 neutrons. The magic numbers correspond to the complete filling of a nucleon shell.
The conventional nucleonic magic numbers are {2, 8, 20, 28, 50, 82, 126}. These are based upon the number of stable nuclides. Those numbers also show up in terms of sharp drops in the levels of incremental binding energies. However 6 and 14 also show up in terms of drops in incremental binding energy. If the numbers corresponding to filled shells are taken to be {2, 6, 14, 28, 50, 82, 126} then there is a simple algorithm for their generation. The occupancies of the shells for this sequence are {2, 4, 8, 14, 22, 32, 44}. These numbers divided by 2 are {1, 2, 4, 7, 11, 16, 22}. Consider the sequence {0, 1, 2, 3, 4, 5, 6} and take the cumulative sums to get {0, 1, 3, 6, 10, 15, 21}. When 1 is added to each of these numbers the result is {1, 2, 4, 7, 11, 16, 22}, which is just the occupancies of the shells divided by 2, or equivalently the occupancies of the shells in terms of the number of nucleonic spin pairs. If sequence is extended one more level the occupancy of the next shell is 29 spin pairs or 58 nucleons. This would make the next magic number 126+58=184 nucleons or 92 spin pairs. In the work that follows the maximum occupancies of the shells in terms of spin pairs is taken to be {1, 2, 4, 7, 11, 16, 22, 29}.
The binding energy of a nuclide can be expressed as
where BEPP is the binding energy due to the interaction of proton spin pairs with other proton spin pairs and similarly for BEPN and BENN. The interaction binding energy of two nucleon pairs is a function of the shell numbers they are in.
The interaction binding energy due to a nucleon in Shell I with a nucleon in Shell J is
where q_{I} and q_{J} are the charges of the nucleons in Shell I and in Shell J, respectively. The average separation distance between the nucleons is denoted as S_{IJ} and F(S) is the function that gives the strong force as a function of separation distance.
The total binding energy due to the interaction of nucleons in Shell I with those in Shell J is then the number of such interactions times the binding energy amount for interaction, given above. If Shell I is different from Shell J then the number of interactions is equal to the product of the numbers of nucleons in the two shells, n_{I}n_{J}. If I is equal to J then the number of interactions is n_{I}(n_{I}-1).
Although the above selection of observations eliminates the effects of proton-proton and neutron-neutron spin pairs there are still the effects of neutron-proton spin pairs in the data. The coupling of a neutron-neutron spin pair with a proton-proton spin pair produces an alpha module; i.e., a chain of the form -n-p-p-n-, or equivalently, -p-n-n-p-. These alpha modules can account for a substantial portion of the binding energy of a nuclide. The number of alpha modules in a nuclide is simply the minimum of the number of neutron spin pairs and the number of proton spin pairs. The regression of binding energies on the number of alpha modules gives the following results
with a coefficient of determination (R²) of 0.9795 and a t-ratio for the coefficient of 187.5. The standard error of the estimate is 72.6 MeV.
Of course with there being zero alpha modules the binding energy should be zero. The regression without a constant is
with a standard error of the estimate of 74.4 MeV and a t-ratio for the coefficient of 432.9.
For the interactions between proton pairs and neutron pairs and between neutron pairs and neutron pairs The regression coefficient should represent
Thus if the nucleon charge of a proton is taken as unity and that of neutron as q then those for the pairs would be 2 and 2q, respectively. The regression coefficient for the interactions of proton pairs in the I-th shell with neutron pairs in the J-th shell would be 2*2q=4q while those for neutron pair interaction would be 4q²F(S_{IJ}). Therefore the ratio of the coefficient for neutron-neutron interactions to the proton-neutron interactions should be q.
Likewise the square of the coefficent for the proton-neutron interaction to the coefficient for the neutron-neutron interactions should be F(S_{IJ}).
The proton pair interactions are affected by the electrostatic repulsion between protons. The ratio of the electrostatic force to the strong force is dependent upon the separation distance between the proton pairs. The distances from the I-th proton shell to some shell J may be taken to be equal to the distance from he I-th neutorn shell to J. Under this assumption the values of the ratio of the electrostatic force between proton pairs to that of the strong force can be calculated. If those ratios are found to be positive numbers then that would be a confirmation that the nature of the strong force between proton is of the same character as the electrostatic force between protons; namely repulsion.
Since EXCEL can accept only about sixty independent variable it is not possible at this point to get the regression coefficients of the full model. However in lieu of the full model results regressions were run on subsets of the independent variables. When binding energy was regressed on the proton pair to proton pair interaction and neutron pair to neutron pair interaction the result obtained are the ones shown Appendix I. There are many cases in which EXCEL did not include a variable in the regression that was supposed to be there. This is because of multicollinearity; i.e. one variable is equal to a linear combination of the other variables in the regression.
About the only meaningful bit of information to come out of that regression is that its coefficient of determination (R²) is 0.991258. This is impressive but a far cry from the values of 0.99998 and so forth obtained for related models.
If the binding energies are regressed on the interactions between proton shells and neutron shells the coefficient of determination is slightly higher at 0.991628. The results for this regression are given in Appendix II. Again there is no particular pattern and there is no reason for expecting there to be.
If the required regression could have been carried out and the model vindicated it have had coefficients for the interactions in definite ratios. If the nucleonic charge of the neutron relative to that of the proton is q then the interactrion of the proton spin pairs in shell I with the neutron spin pairs in shell J is proportional to q whereas the interaction of the neutron spin pairs in shell I with the neutron spin pairs in shell J is proportional to q². If the electrostatic repulsion between protons were not operating the interaction of the proton spin pairs in shell I with proton spin pairs in shell J would be proportional to 1, but because of the electrostatic repulsion it is proportional to (1+d_{IJ}) where d_{IJ} is a function of the distance between the spin pairs in shell I and shell J.
A regression was carried out using q equal to −2/3 and all the d_{IJ} equal to 0.25. The results as displayed in Appendix III. The coefficient of determination for this regression is 0.99068.
(To be continued.)
Regression of Binding Energy on the Interactions within Same Nucleon Type Shells |
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Shell x Shell | Coefficient | t-Ratio |
N92x92 | 1.369522519 | 1.650391709 |
N63x92 | 2.66122983 | 9.672005546 |
N63x63 | -3.236125482 | -7.153324704 |
N41x92 | 0 | |
N41x63 | 8.926973227 | 20.2081161 |
N41x41 | -2.318449003 | -2.693539883 |
N25x92 | 0 | |
N25x63 | 0 | |
N25x41 | 10.02336271 | 12.50297996 |
N25x25 | 3.607280704 | 2.639640496 |
N14x92 | 0 | |
N14x63 | 0 | |
N14x41 | 0 | |
N14x25 | 10.94324306 | 6.818957218 |
N14x14 | 0.966120799 | 0.24942747 |
N7x92 | 0 | |
N7x63 | 0 | |
N7x41 | 0 | |
N7x25 | 0 | |
N7x14 | 21.31585159 | 4.41678173 |
N7x7 | 0.700953811 | 0.05328248 |
N3x92 | 0 | |
N3x63 | 0 | |
N3x41 | 0 | |
N3x25 | 0 | |
N3x14 | 0 | |
N3x7 | 31.14743126 | 2.148407888 |
N3x3 | 19.85164667 | 0.269256961 |
N1x92 | 0 | |
N1x63 | 0 | |
N1x41 | 0 | |
N1x25 | 0 | |
N1x14 | 0 | |
N1x7 | 0 | |
N1X3 | 0.00145 | 2.45291E-05 |
63X63 | 0 | |
41X63 | -8.454654798 | -9.676399101 |
41X41 | 0 | |
25X63 | 0 | |
25X41 | -10.89493197 | -1.987752397 |
25X25 | 0 | |
14X63 | 4.068375226 | 5.21362348 |
14X41 | -1.243179256 | -2.286647778 |
14X25 | -1.180021913 | -1.189049237 |
14X14 | 0.636221022 | 0.249560308 |
7X63 | 0 | |
7X41 | 0 | |
7X25 | -22.27726106 | -5.619254055 |
7X14 | -18.06066559 | -3.362931381 |
7X7 | -4.488162878 | -0.625941075 |
3X63 | 0 | |
3X41 | 0 | |
3X25 | 0 | |
3X14 | 0 | |
3X7 | -13.99698714 | -0.885489684 |
3X3 | -17.2703895 | -0.493119308 |
1x63 | 0 | |
1x41 | 2.042876 | 0.034558523 |
1x25 | 0 | |
1x14 | 0 | |
1x7 | 0 | |
1x3 | 19.12690333 | 0.343189577 |
C0 | 28.295674 | 0.586244577 |
Regression of Binding Energy on the Interactions between Different Nucleon Type Shells |
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Shell x Shell | Coefficient | t-Ratio |
63xN92 | 5.392641285 | 18.18788826 |
63xN63 | 5.459407022 | 0.99911904 |
63xN41 | -17.79613215 | -2.385244247 |
63xN25 | 0 | |
63xN14 | 0 | |
63xN7 | 0 | |
63xN3 | 0 | |
41xN92 | 3.375503551 | 14.6345205 |
41xN63 | -2.889962754 | -11.53438779 |
41xN41 | 6.917856873 | 3.830767217 |
41xN25 | -25.20833895 | -3.841513614 |
41xN14 | 0 | |
41xN7 | 0 | |
41xN3 | 23.36429244 | 0.710379457 |
25xN92 | 0 | |
25xN63 | 13.30724203 | 22.5224223 |
25xN41 | -4.094898491 | -4.591351437 |
25xN25 | 2.769570943 | 2.326275983 |
25xN14 | -17.3826679 | -8.356858323 |
25xN7 | 0 | |
25xN3 | 0 | |
14xN92 | 0 | |
14Xn63 | 0 | |
14Xn41 | 17.5472573 | 10.83205647 |
14Xn25 | -4.880877845 | -0.659792463 |
14Xn14 | 0.353517785 | 0.165014049 |
14Xn7 | -18.04503989 | -3.882961208 |
14Xn3 | 0 | |
7Xn92 | 0 | |
7Xn63 | 0 | |
7Xn41 | 0 | |
7Xn25 | 37.13616959 | 2.822521877 |
7XN14 | 22.24271459 | 0.628054406 |
7xN7 | -3.312801787 | -0.497058232 |
7xN3 | -19.30529779 | -1.223953906 |
3xN92 | 0 |
Regression Results for q=−2/3 and (1+d)=1.25 | ||
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Inter | coeff | t-ratio |
C0 | 28.295674 | 0.568423845 |
N1X3 | 91.97057709 | 1.394024678 |
N1x7 | 0 | |
N1x14 | 0 | |
N1x25 | 0 | |
N1x41 | 1.702799445 | 0.035549019 |
N1x63 | 0 | |
N1x92 | 19.85309667 | 0.436889015 |
N3x3 | -14.48018088 | -0.58189229 |
N3x7 | 43.68935801 | 2.245240693 |
N3x14 | 0 | |
N3x25 | 0 | |
N3x41 | -76.03068543 | -1.342897085 |
N3x63 | 0 | |
N3x92 | -0.221937088 | -0.016358404 |
N7x7 | -2.673806032 | -0.434025811 |
N7x14 | -8.418036242 | -1.867602068 |
N7x25 | -20.69175682 | -4.270307771 |
N7x41 | -111.3882562 | -3.331241187 |
N7x63 | 0 | |
N7x92 | -8.237202631 | -2.197147771 |
N14x14 | 5.495422501 | 2.666636646 |
N14x25 | -54.38130454 | -9.713853777 |
N14x41 | -30.2379953 | -20.49320931 |
N14x63 | -6.32595053 | -8.643700015 |
N14x92 | 10.06845621 | 9.832638931 |
N25x25 | 50.88809535 | 9.258946485 |
N25x41 | 43.8541431 | 2.623215316 |
N25x63 | 0 | |
N25x92 | -0.343038632 | -0.409895095 |
N41x41 | 29.73122539 | 18.42753552 |
N41x63 | -6.858892708 | -9.14004255 |
N41x92 | -3.744947919 | -8.135849156 |
N63x63 | 9.624229708 | 10.11141484 |
N63x92 | 1.315017946 | 1.53660688 |
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