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Concerning Binding Energy and Number of Neutrons in Nuclides |
The binding energies are known for about three thousand nuclides. When these values are used to construct profiles of binding energy versus the number of neutrons for various elements these profiles display some interesting characteristics. First the profiles are parabolic with binding energy increasing at a decreasing rate as neutrons are added. More interesting is the incremental increases which display a downward, almost linear, trend with fluctuations associated with the formation of neutron pairs. At particular numbers the incremental values decrease and the magnitude of the fluctuations due to pair formation changes. For example, here is the profile of incremental increases for bromine.
The break in the relationship occurs at the point where the number of neutrons is 50. Fifty is one of the so-called magic numbers of nuclear structure. For more on magic numbers in nuclear structure see Magic Numbers 0, Magic Numbers I, and Magic Numbers II.
The relationship between incremental binding energy ΔB and the number of additional neutrons n can be approximated by a function of the form
where if Z is the number of protons and N the number of neutrons for a nuclide then n=N-Z. The variable u reflecting neutron pair formation is equal to zero if N is odd and unity if N is evern. The regression coefficient c_{0} is called the intercept and c_{1} the slope.
There are indications that these regression parameters may reveal information about the structure of nuclei. For example, the magnitude of the slope may be inversely proportional to the radius of the shell that is being filled. For many elements there are two regression lines. For example, for the case of bromine shown above there is a regression for the data below the break and another one for the data above the break.
Given below are the regression parameter estimates for an arbitrary selection of elements. It is notable that for elements close in atomic number Z the parameters are close in values.
Regression Equation
Parameters for the Relationships Between Incremental Increases in Binding Energy and the Number of Neutrons in Excess of the Protons in Nuclides |
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Element | Number of Protons | Intercept | Slope | Pair Formation Increment | Coefficient of Determination | Degrees of Freedom |
The 82-to-126 Neutron Shell | ||||||
Uranium | 92 | 7.23582 | -0.11605 | 1.51121 | 0.963 | 21 |
Polonium | 84 | 9.09554 | -0.13461 | 1.92821 | 0.955 | 17 |
Bismuth | 83 | 12.26606 | -0.13101 | 1.47034 | 0.981 | 21 |
Lead | 82 | 11.66121 | -0.12915 | 2.13360 | 0.963 | 24 |
Mercury | 80 | 11.44383 | -0.14051 | 2.17044 | 0.979 | 28 |
Gold | 79 | 11.92086 | -0.14032 | 1.52679 | 0.985 | 33 |
Cesium | 55 | 7.96838 | -0.12169 | 1.28926 | 0.987 | 11 |
The 50-to-82 Neutron Shell | ||||||
Cesium | 55 | 11.80531 | -0.21010 | 1.98055 | 0.987 | 22 |
Xenon | 54 | 10.88332 | -0.20568 | 2.60039 | 0.985 | 23 |
Iodine | 53 | 10.79384 | -0.19513 | 2.22696 | 0.987 | 24 |
Tellurium | 52 | 10.45359 | -0.19705 | 2.69091 | 0.984 | 25 |
Antimony | 51 | 10.64351 | -0.19325 | 2.24661 | 0.986 | 24 |
Tin | 50 | 10.54740 | -0.19230 | 2.62740 | 0.983 | 29 |
Indium | 49 | 10.58613 | -0.1995 | 2.21273 | 0.983 | 23 |
Cadmium | 48 | 10.28301 | -0.20166 | 2.66403 | 0.984 | 29 |
Silver | 47 | 10.46401 | -0.20570 | 2.02970 | 0.984 | 27 |
Palladium | 46 | 10.11712 | -0.21352 | 2.58306 | 0.987 | 24 |
Rhodium | 45 | 10.20289 | -0.21447 | 2.08278 | 0.990 | 23 |
Ruthenium | 44 | 9.92022 | -0.22033 | 2.45778 | 0.990 | 21 |
Technetium | 43 | 10.36545 | -0.24322 | 2.21082 | 0.948 | 20 |
Molybdenum | 42 | 9.63716 | -0.22085 | 2.24318 | 0.987 | 18 |
Niobium | 41 | 10.46348 | -0.25928 | 1.88634 | 0.909 | 17 |
Zirconium | 40 | 9.59217 | -0.22605 | 1.67700 | 0.956 | 15 |
Yttrium | 39 | 10.61462 | -0.27304 | 1.34585 | 0.819 | 15 |
Rubidium | 37 | 11.45052 | -0.27481 | 1.97722 | 0.962 | 9 |
Krypton | 36 | 9.58940 | -0.27656 | 1.76489 | 0.964 | 8 |
Bromine | 35 | 10.99744 | -0.29832 | 2.36596 | 0.994 | 5 |
The 28-to-50 Neutron Shell | ||||||
Niobium | 41 | 11.73750 | -0.23750 | 2.23750 | 0.971 | 5 |
Yttrium | 39 | 11.32410 | -0.22375 | 2.28545 | 0.976 | 7 |
Strontium | 38 | 8.83274 | -0.22247 | 2.15995 | 0.979 | 8 |
Rubidium | 37 | 10.96541 | -0.22481 | 2.01149 | 0.962 | 9 |
Bromine | 35 | 12.81863 | -0.44703 | 1.79800 | 0.901 | 12 |
Selenium | 34 | 10.43328 | -0.32336 | 3.49824 | ||
Arsenic | 33 | 10.66552 | -0.31360 | 2.43748 | 0.992 | 12 |
Titanium | 22 | 9.11722 | -0.42475 | 1.97948 | 0.973 | |
20 to 28 Neutron Shell | ||||||
Titanium | 22 | 10.21436 | -0.45478 | 3.52321 | 0.982 | |
Calcium | 20 | 8.65859 | -0.22798 | 3.21944 | ||
Argon | 18 | 7.65135 | -0.27159 | 3.18186 | 0.984 | |
The 14-to-20 Neutron Shell | ||||||
Sulfur | 16 | 9.40556 | -0.79593 | 3.63517 | ||
Silicon | 14 | 9.16356 | -0.88301 | 3.48451 | ||
Magnesium | 12 | 7.29294 | -0.60462 | 3.30073 | 0.961 | 8 |
The 8-to-14 Neutron Shell | ||||||
Neon | 10 | 7.510346 | -0.76474 | 4.39858 | ||
Oxygen | 8 | 4.54168 | -0.19105 | 3.72418 | 3 | |
Carbon | 6 | 1.96729 | -0.25550 | 3.49186 | 0.995 | 4 |
The data in the table demonstrates the systematic variation in the regression parameters with the atomic number of the elements, the number of protons. The graph below shows the relationship between the magnitude of the slope and the proton number for the elements which reflect the filling of the 50-to-82 neutron shell. The data is shown only for the elements with an even number of protons to avoid the effect of the nonformation of alpha particles within the nucleus.
The slopes may inversely proportional to the radius of the shell. If so the above relationship indicates that the radius of the shell is larger for nuclides with more protons. This could reasonably be expected as a result of the electrostatic repulsion of the protons.
For shells of electrons the ratio of the regression intercept to the magnitude of the regression slope is related to the effective charge experienced the electrons in the shell. For the nuclear shells that ratio is correlated with the number of protons in the nucleus, as seen below.
(To be continued.)
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