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An Investigation of the Changes in Binding
Energy that would Result from the Conversion
of a Neutronin a Nucleus into a Proton and an Electron

In investigating why neutrons, which as free particles decay with a half-life of slightly over ten minutes, are so stable within nuclei statistics were compiled on the binding energy that would result from the conversion of a neutron into a proton and an electron . Outside of a nucleus the decay of a neutron into a proton and an electron releases energy.

Mass energy of a proton = 938.27231 MeV

Mass energy of a proton = 0.510999906 MeV

Combined mass energy of
a proton and an electron = 937.7613101 MeV

Mass energy of a neutron = 939.56563 MeV

Difference in mass energy =1.804319906 MeV

The binding energy difference D that would result from a conversion of a neutron into a proton and electron was computed as

D = BE(n-1, p+1) − BE(n, p)

If D is positive then energy would be liberated as, a result of the decay of a neutron into a proton and electron, not counting the mass of the neutron in excess of the mass of the proton and electron. If D is negative the conversion would require an input of energy and thus its nonoccurrence is explainable in energy terms.

There were 2657 nuclides for which D could be computed. The average energy difference was −1.439 million electron volts (MeV). This is on the order of the energy liberated when a free neutron decays into a proton and an electron.

The standard deviation of these difference was 7.37 MeV. The histogram for the differences is shown below.

The range of this distribution of D is a strong indication that the stability of neutrons in nuclei has nothing to do with energy balance. The near-normality of the distribution suggests that the values of D might be the result of a large number of possibly random effects.

Below is a scatter diagram of the changes D plotted versus the number of nucleons in the nuclide (n+p).

While there seems to be no apparent relationship between the average of D and (n+p) there is obviously a relationship between (n+p) and the variance of D. However when the values of D are regressed on the values of (n+p) there is a different story to be told.

D = 1.08883 −0.01887(n+p)

,

The coefficient of determination (R²) of the regression is a pathetically small 0.00213 but the t-ratio (ratio of the coefficient to its standard deviation) for the regression coefficient is −8.9, a small but definite effect of nuclide size (n+p) on the change in binding energy D which would result from the conversion of a neutron

When the number of neutrons and the number of protons are included in the regression as separate variables the coefficient of determination (R²) rises to a respectable 0.6569. The regression equation obtained is

D = 4.47347 + 0.63393n − 1.03452p
     [23.0]    [67.1]    [-70.8]

The numbers in brackets below the coefficients are their t-ratios. The magnitude of the t-ratio for a coefficient must be 2 or greater for the coefficient to be significantly different from zero at the 95 percent level of confidence.

Spin Paired and Non-Spin-Paired Neutrons

The value of D for a nuclide may be affected by whether there are any unpaired neutrons or unpaired protons. The following tables give the number of the various cases and the average value of D, the change in binding energy that would result from the decay of a neutron.

The Number of Cases of the Computed Change
in Binding Energy that would result from a Conversion
of a Neutron into a Proton and Electron
Neutron number
Proton
number
OddEven
Odd668662
Even663665

The Average Change in Binding Energy
that would result from a Conversion of a
Neutron into a Proton and Electron
(MeV)
Neutron number
Proton
number
OddEven
Odd-3.66233-1.52701
Even-1.05903+0.65682

Apparently the change in binding energy that would occur for a conversion of a neutron in a nucleus to a proton and electron does depend upon whether there is an unpaired neutron and/or an unpaired proton. For the odd-odd case a neutron conversion would result in the disappearance of a neutron-proton spin pair and the creation of a proton-proton spin pair. If there is an unpaired neutron and no unpaired proton the conversion would result only in the creation of an unpaired proton. If there is no unpaired neutron and an unpaired proton the conversion of a neutron would result in the disappearance of a neutron-neutron spin pair and the creation of a neutron-proton spin pair. For the even-even case the conversion of a neutron would result in the disappearance of of a neutron-neutron spin pair and the creation of an unpaired proton.

When the odd-even-nesses of n and p are included in the regression there is a small but definite improvement in the coefficient of determination; from 0.6569 to 0.70734. The regression coeficients and their t-ratios are given in the following table. The odd-even-ness of a variable z is denoted as z%2 where z%2 equals 1 if z is odd and 0 if it is even.

Regression Results
VariableCoefficientt-Ratio
n0.634572.7
p-1.03525-76.6
n%2-2.19877-15.9
p%2-2.46511-14.2
Const..8012132.3

Conclusions

The binding energy change that would result from a conversion of a neutron into a proton and electron definitely depends upon the number of neutrons and protons in a nuclide and their odd-even-ness but it also depends upon some other functions of the their numbers.

Further Analysis

In lieu of any other theoretically justified additional variables the quadratic terms n², p² and n*p may be used.

-13.8
Regression Results
VariableCoefficientt-Ratio
n1.5454486.5
p-2.28098--82.0
n%2-2.1682-20.6
p%2-2.46803-23.4
0.004256.5
p²'0.03059-19.8
n*p-0.02749
Const.4.1583119.2

The coefficient of determination for this equation is a satisfying 0.8646.


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