San José State University |
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applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
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Binding Energy and Number of Neutrons in Nuclides for the Various Nuclear Shells |
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 sharply 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 c_{1} for some of the elements which have data for the various nuclear shells. It is notable that for elements close in atomic number Z the parameters are close in values.
Regression Equation Slope
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 | Slope | Pair Formation Increment | Coefficient of Determination | Degrees of Freedom | |
The 50-to-82 Neutron Shell | ||||||
Neodymium | 60 | -0.19714 | 2.54643 | 0.990 | 13 | |
Cerium | 58 | -0.21482 | 2.53592 | 0.988 | 18 | |
Lanthanum | 57 | -0.21211 | 1.96302 | 0.982 | 19 | |
Barium | 56 | -0.21594 | 2.55961 | 0.988 | 21 | |
Cesium | 55 | -0.21010 | 1.98055 | 0.987 | 22 | |
Xenon | 54 | -0.20568 | 2.60039 | 0.985 | 23 | |
Iodine | 53 | -0.19513 | 2.22696 | 0.987 | 24 | |
Tellurium | 52 | -0.19705 | 2.69091 | 0.984 | 25 | |
Antimony | 51 | -0.19325 | 2.24661 | 0.986 | 24 | |
Tin | 50 | -0.19230 | 2.62740 | 0.983 | 29 | |
Indium | 49 | -0.1995 | 2.21273 | 0.983 | 23 | |
Cadmium | 48 | -0.20166 | 2.66403 | 0.984 | 29 | |
Silver | 47 | -0.20570 | 2.02970 | 0.984 | 27 | |
Palladium | 46 | -0.21352 | 2.58306 | 0.987 | 24 | |
Rhodium | 45 | -0.21447 | 2.08278 | 0.990 | 23 | |
Ruthenium | 44 | -0.22033 | 2.45778 | 0.990 | 21 | |
Technetium | 43 | -0.24322 | 2.21082 | 0.948 | 20 | |
Molybdenum | 42 | -0.22085 | 2.24318 | 0.987 | 18 | |
Niobium | 41 | -0.25928 | 1.88634 | 0.909 | 17 | |
Zirconium | 40 | -0.22605 | 1.67700 | 0.956 | 15 | |
Yttrium | 39 | -0.27304 | 1.34585 | 0.819 | 15 | |
Strontium | 38 | -0.22247 | 2.15995 | 0.953 | 13 | |
Rubidium | 37 | -0.27481 | 1.97722 | 0.962 | 9 | |
Krypton | 36 | -0.27656 | 1.76489 | 0.964 | 8 | |
Bromine | 35 | -0.32267 | 1.42492 | 0.964 | 5 | |
Selenium | 34 | -0.30068 | 1.75842 | 0.975 | 5 | |
Arsenic | 33 | -0.23000 | 1.37000 | 0.999 | 1 | |
Germanium | 32 | -0.31500 | 2.18500 | 0.993 | 1 |
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.
For the elements below molybdenum in atomic number the relationship to atomic number (number of protons) is irregular, but above molybdenum it is quite regular.
The parameter of interest is the amount by which the incremental binding energy is enhanced (increased) by neutron pair formation. The graph below shows irregularities below molybdenum but reasonable regularity above molybdenum, however with an odd-even alternation.
Regression Equation Slope
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 | Slope | Pair Formation Increment | Coefficient of Determination | Degrees of Freedom | |
The 28-to-50 Neutron Shell | ||||||
Silver | 47 | -0.30000 | 2.10000 | 1.000 | 0 | |
Palladium | 46 | -0.26750 | 2.7325 | 0.997 | 1 | |
Rhodium | 45 | -0.14800 | 1.75333 | 0.990 | 2 | |
Ruthenium | 44 | -0.20637 | 2.56938 | 0.986 | 3 | |
Technetium | 43 | -0.17482 | 1.70375 | 0.928 | 4 | |
Molybdenum | 42 | -0.17670 | 2.51820 | 0.974 | 5 | |
Niobium | 41 | -0.23750 | 2.23750 | 0.971 | 5 | |
Zirconium | 40 | -0.21875 | 2.81494 | 0.976 | 7 | |
Yttrium | 39 | -0.22375 | 2.28545 | 0.976 | 7 | |
Strontium | 38 | -0.26903 | 2.95510 | 0.979 | 8 | |
Rubidium | 37 | -0.22481 | 2.01149 | 0.962 | 8 | |
Krypton | 36 | -0.30385 | 3.16542 | 0.991 | 11 | |
Bromine | 35 | -0.29832 | 2.36596 | 0.994 | 12 | |
Selenium | 34 | -0.32336 | 3.49824 | 0.988 | 13 | |
Arsenic | 33 | -0.31360 | 2.43748 | 0.992 | 14 | |
Germanium | 32 | -0.33123 | 3.32932 | 0.986 | 15 | |
Gallium | 31 | -0.32708 | 2.30623 | 0.979 | 16 | |
Zinc | 30 | -0.33913 | 3.07915 | 0.979 | 17 | |
Copper | 29 | -0.34692 | 2.47208 | 0.981 | 17 | |
Nickel | 28 | -0.35029 | 2.81549 | 0.980 | 19 | |
Cobalt | 27 | -0.37556 | 2.29749 | 0.966 | 14 | |
Iron | 26 | -0.40661 | 2.581489 | 0.969 | 12 | |
Manganese | 25 | -0.40149 | 1.83550 | 0.962 | 11 | |
Chromium | 24 | -0.41716 | 2.90811 | 0.964 | 10 | |
Vanadium | 23 | -0.41227 | 1.927 | 0.964 | 9 | |
Titanium | 22 | -0.42475 | 1.97948 | 0.973 | 8 | |
Scandium | 21 | -0.43728 | 0.85808 | 0.934 | 7 | |
Calcium | 20 | -0.44908 | 1.03105 | 0.970 | 6 | |
Potassium | 19 | -0.49870 | 2.07820 | 0.976 | 5 | |
Argon | 18 | -0.54000 | 2.58333 | 0.917 | 4 | |
Chlorine | 17 | -0.63750 | 2.00417 | 0.915 | 3 | |
Sulfur | 16 | -0.52000 | 2.66667 | 0.982 | 2 | |
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 (the slope is negative) and the proton number for the elements which reflect the filling of the 28-to-50 neutron shell.
Except for the few data points at the ends of the range the relationship appears to be linear. The regression equation for the range from Z=20 (calcium) to Z=45 (rhodium) is
However close examination of the graph reveal that there are breaks after Z=25 (iron) and Z=36 (krypton). At Z=36 there is not only a reduction in the level of the slope magnitude but the amplitude of the fluctuations in the slope increases.
The other parameter of interest is the amount by which the incremental binding energy is enhanced (increased) by neutron pair formation. The graph below shows irregularities below titanium but reasonable regularity above titanium, however with an odd-even alternation.
Regression Equation Slope
Parameters for the Relationships Between Incremental Increases in Binding Energy and the Number of Neutrons in Excess of the Protons in Nuclides |
||||||
---|---|---|---|---|---|---|
Element | Number of Protons | Slope | Pair Formation Increment | Coefficient of Determination | Degrees of Freedom | |
Manganese | 25 | -0.81650 | 2.33480 | 1.00- | 0 | |
Chromium | 24 | -0.57027 | 3.16812 | 0.996 | 1 | |
Vanadium | 23 | -0.51095 | 1.93112 | 0.976 | 2 | |
Titanium | 22 | -0.45478 | 3.5321 | 0.982 | 3 | |
Scandium | 21 | -0.34412 | 2.16192 | 0.995 | 4 | |
Calcium | 20 | -0.22798 | 3.21944 | 0.994 | 5 | |
Potassium | 19 | -0.22524 | 2.09017 | 0.973 | 5 | |
Argon | 18 | -0.25372 | 3.22346 | 0.983 | 5 | |
Chlorine | 17 | -0.29461 | 2.25992 | 0.938 | 4 | |
Sulfur | 16 | -0.23065 | 3.52239 | 0.985 | 5 | |
Phosphorus | 15 | -0.19361 | 2.36176 | 0.844 | 5 | |
Silicon | 14 | -0.16883 | 3.33108 | 0.984 | 5 | |
Aluminum | 13 | -0.29875 | 2.50208 | 0.951 | 3 | |
Magnesium | 12 | -0.56800 | 3.44667 | 0.976 | 2 | |
Sodium | 11 | -0.69000 | -0.99000 | 1.000 | 1 | |
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 (the slope is negative) and the proton number for the elements which reflect the filling of the 20-to-28 neutron shell.
Except for the three data points at the lower end of the range and the last data point at the upper end of the range the relationship could be considered to be linear.
The other parameter of interest is the amount by which the incremental binding energy is enhanced (increased) by neutron pair formation. The graph below shows an anomaly for Z=11 (sodium) but reasonable regularity for the other data points, however with an odd-even alternation. The lower points, except for sodium, fall nearly along a straight line.
Regression Equation Slope
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 | Slope | Pair Formation Increment | Coefficient of Determination | Degrees of Freedom | |
Chlorine | 17 | -1.16665 | 2.89794 | 1.000 | 0 | |
Sulfur | 16 | -0.79593 | 3.63517 | 1.000 | 1 | |
Phosphorus | 15 | -0.96320 | 3.13130 | 0.996 | 1 | |
Silicon | 14 | -0.93876 | 3.39168 | 0.994 | 2 | |
Aluminum | 13 | -0.93329 | 2.42748 | 0.992 | 3 | |
Magnesium | 12 | -0.86086 | 3.48675 | 0.952 | 3 | |
Sodium | 11 | -0.78318 | 2.09404 | 0.937 | 3 | |
Neon | 10 | -0.56925 | 2.72125 | 0.859 | 3 | |
Fluorine | 9 | -0.91375 | 1.51708 | 0.882 | 3 | |
Oxygen | 8 | -1.42000 | 2.41000 | 1.000 | 0 | |
Nitrogen | 7 | -1.21000 | 2.38000 | 1.000 | 0 |
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 (the slope is negative) and the proton number for the elements which reflect the filling of the 14-to-20 neutron shell.
Except for the four data points at the lower end of the range and the last two data points at the upper end of the range the relationship could be considered to be linear.
The other parameter of interest is the amount by which the incremental binding energy is enhanced (increased) by neutron pair formation. The graph below shows a reasonable regularity for the data points, however with an odd-even alternation. The lower points, except for Z=7 (nitrogen) and Z=17 (chlorine), fall nearly along a straight line.
Regression Equation Slope
Parameters for the Relationships Between Incremental Increases in Binding Energy and the Number of Neutrons in Excess of the Protons in Nuclides |
||||||
---|---|---|---|---|---|---|
Element | Number of Protons | Slope | Pair Formation Increment | Coefficient of Determination | Degrees of Freedom | |
Magnesium | 12 | -0.86086 | 3.48675 | 0.952 | 3 | |
Sodium | 11 | -1.70379 | 3.75548 | 1.000 | 0 | |
Neon | 10 | -0.76474 | 4.39858 | 1.000 | 1 | |
Fluorine | 9 | -0.71638 | 2.77434 | 0.965 | 2 | |
Oxygen | 8 | -0.22610 | 3.57530 | 0.992 | 1 | |
Nitrogen | 7 | -0.34525 | 2.82075 | 1.000 | 1 |
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 (the slope is negative) and the proton number for the elements which reflect the filling of the 20-to-28 neutron shell.
In this case the data consists mostly of the anomalous end points and little regularity can be perceived.
The other parameter of interest is the amount by which the incremental binding energy is enhanced (increased) by neutron pair formation. The graph below shows reasonable regularity for the other data points, however with an odd-even alternation.
The tabulation of the results for the 82-to-126 neutron shell is not complete but some values are given in the table below to show that the results generally fit into the pattern observed for the other shells. There is also a shell for more than 126 neutrons, but no results are available yet for it.
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 | Slope | Pair Formation Increment | Coefficient of Determination | Degrees of Freedom | |
The 82-to-126 Neutron Shell | ||||||
Uranium | 92 | -0.11605 | 1.51121 | 0.963 | 21 | |
Polonium | 84 | -0.13461 | 1.92821 | 0.955 | 17 | |
Bismuth | 83 | -0.13101 | 1.47034 | 0.981 | 21 | |
Lead | 82 | -0.12915 | 2.13360 | 0.963 | 24 | |
Mercury | 80 | -0.14051 | 2.17044 | 0.979 | 28 | |
Gold | 79 | -0.14032 | 1.52679 | 0.985 | 33 | |
Lanthanum | 57 | -0.13140 | 1.54865 | 0.964 | 13 | |
Barium | 56 | -0.12834 | 2.55961 | 0.988 | 21 | |
Cesium | 55 | -0.12169 | 1.28926 | 0.918 | 12 |
(To be continued.)
The magnitudes of the regression slopes are larger for the lower shells, indicating a possible relationship between the magnitude of the slope and the radius of the shell. The shell radii for each shell type appear to to depend upon the number of protons in the nucleus, its atomic number.
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