San José State University

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Quantitative Analysis of Relationships
Concerning Binding Energy and Number of Neutrons
in Nuclides Which Span the 50-to-82 Neutron Shell

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

ΔB = c0 + c1n + c2u

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 c0 is called the intercept and c1 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 c1 for some of the elements which have data for the e 50-to-82 neutron shel. 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
ElementNumber
of Protons
    Slope     Pair
Formation
Increment
Coefficient of
Determination
Degrees of
Freedom
The 50-to-82 Neutron Shell
Cerium58-0.214822.535920.98818
Lanthanum57-0.212111.963020.98219
Barium56-0.215942.559610.98821
Cesium55-0.210101.980550.98722
Xenon54-0.205682.600390.98523
Iodine53-0.195132.226960.98724
Tellurium52-0.197052.690910.98425
Antimony51-0.193252.246610.98624
Tin50-0.192302.627400.98329
Indium49-0.19952.212730.98323
Cadmium48-0.201662.664030.98429
Silver47-0.205702.029700.98427
Palladium46-0.213522.583060.98724
Rhodium45-0.214472.082780.99023
Ruthenium44-0.220332.457780.99021
Technetium43-0.243222.210820.94820
Molybdenum42-0.220852.243180.98718
Niobium41-0.259281.886340.90917
Zirconium40-0.226051.677000.95615
Yttrium39-0.273041.345850.81915
Strontium38-0.222472.159950.95313
Rubidium37-0.274811.977220.9629
Krypton36-0.276561.764890.9648
Bromine35-0.322671.424920.9645
Selenium34-0.300681.758420.9755
Arsenic33-0.230001.370000.9991
Germanium32-0.315002.185000.9931

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.

(To be continued.)


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