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Energy

An Alternative view of Nuclear Energy
Aran David Stubbs

While it has been a long-standing convention to calculate the energy of isotopes based on how much energy would be released by merging n neutrons and z protons to form the atom, this does not jibe with the actual initial conditions.  The early universe had a large quantity of hydrogen gas, which is basically a proton and an electron, and trivial amounts of other nuclei which contained neutrons.

A better alternative is to determine the energy content of the nuclei as a group of baryons and find the energy per baryon.  An immediate problem arises due to lack of accurate data about nuclear mass.  A large quantity of data has been generated about atomic mass – and almost all the isotopes have such data – but that includes the masking values of electrons orbiting the nucleus.

A method of getting around this is to estimate the energy content of the electrons and subtract that from the total energy content of the atom.  For the small isotopes that can be done.  However the total energy content of the electrons in large isotopes is not readily available.  Since the total energy content of the electrons is primarily the rest mass of the electrons, a reasonable approximation can be done by assuming some value for the kinetic energy of the electrons.

The simplest method is to use a reasonable model of electron energy and test against known values.  In this case the Bohr model is sufficient.  Kinetic energy of the inner electrons of the heaviest known elements is less than half the mass equivalent.  Using a simple formula, a reasonable approximation can be generated: the energy of electrons in shell n each have 13.606 eV *(z-(electrons in shells <n)-(half the electrons in shell n))2.

The results from this method differ at the margin from results generated using the conventional method.  In this case Iron-56 is the lowest energy nuclei, easily beating Nickel 62. In general, those with a low neutron fraction improved their ranking, while those with a high neutron fraction dropped.  The lowest energy nuclei (those with less than 930.25 MeV per baryon) are shown below.

(Note – first set of columns is alphabetic by element name, second set is numeric by baryon count, third set is by energy per baryon. Each relevant isotope occurs in all 3 sets.)

Chromium 50 930.24312 Titanium 48 930.24929 Iron 56 930.17407
Chromium 51 930.24433 Titanium 50 930.24030 Nickel 60 930.18050
Chromium 52 930.19205 Chromium 50 930.24312 Nickel 62 930.18620
Chromium 53 930.21911 Vanadium 51 930.23963 Chromium 52 930.19205
Chromium 54 930.21220 Chromium 51 930.24433 Iron 58 930.19289
Cobalt 56 930.24643 Chromium 52 930.19205 Iron 57 930.20468
Cobalt 57 930.21034 Chromium 53 930.21911 Cobalt 59 930.20497
Cobalt 58 930.22382 Manganese 53 930.22069 Iron 54 930.20579
Cobalt 59 930.20497 Iron 54 930.20579 Nickel 61 930.20615
Cobalt 60 930.23611 Chromium 54 930.21220 Iron 55 930.20690
Cobalt 61 930.23625 Manganese 54 930.22820 Nickel 58 930.20839
Copper 61 930.23441 Iron 55 930.20690 Cobalt 57 930.21034
Copper 62 930.24160 Manganese 55 930.21203 Manganese 55 930.21203
Copper 63 930.21732 Iron 56 930.17407 Chromium 54 930.21220
Copper 64 930.23970 Cobalt 56 930.24643 Nickel 59 930.21445
Copper 65 930.23069 Manganese 56 930.24923 Zinc 66 930.21727
Iron 54 930.20579 Iron 57 930.20468 Copper 63 930.21732
Iron 55 930.20690 Cobalt 57 930.21034 Chromium 53 930.21911
Iron 56 930.17407 Iron 58 930.19289 Manganese 53 930.22069
Iron 57 930.20468 Nickel 58 930.20839 Nickel 64 930.22155
Iron 58 930.19289 Cobalt 58 930.22382 Zinc 64 930.22262
Iron 59 930.24020 Cobalt 59 930.20497 Cobalt 58 930.22382
Iron 60 930.24863 Nickel 59 930.21445 Nickel 63 930.22654
Manganese 53 930.22069 Iron 59 930.24020 Manganese 54 930.22820
Manganese 54 930.22820 Nickel 60 930.18050 Copper 65 930.23069
Manganese 55 930.21203 Cobalt 60 930.23611 Copper 61 930.23441
Manganese 56 930.24923 Iron 60 930.24863 Cobalt 60 930.23611
Nickel 58 930.20839 Nickel 61 930.20615 Cobalt 61 930.23625
Nickel 59 930.21445 Copper 61 930.23441 Zinc 68 930.23853
Nickel 60 930.18050 Cobalt 61 930.23625 Vanadium 51 930.23963
Nickel 61 930.20615 Nickel 62 930.18620 Copper 64 930.23970
Nickel 62 930.18620 Copper 62 930.24160 Iron 59 930.24020
Nickel 63 930.22654 Copper 63 930.21732 Titanium 50 930.24030
Nickel 64 930.22155 Nickel 63 930.22654 Copper 62 930.24160
Titanium 48 930.24929 Nickel 64 930.22155 Chromium 50 930.24312
Titanium 50 930.24030 Zinc 64 930.22262 Zinc 65 930.24359
Vanadium 51 930.23963 Copper 64 930.23970 Chromium 51 930.24433
Zinc 64 930.22262 Copper 65 930.23069 Cobalt 56 930.24643
Zinc 65 930.24359 Zinc 65 930.24359 Iron 60 930.24863
Zinc 66 930.21727 Zinc 66 930.21727 Manganese 56 930.24923
Zinc 68 930.23853 Zinc 68 930.23853 Titanium 48 930.24929

This energy mapping does have some drawbacks.  In many cases capturing an electron increases the stability of the resulting nucleus while also increasing the energy per baryon, for example Iron 55 versus Manganese 55 (which is still less than a third the mismatch of the standard form).  On the plus side, it provides insight into the relative long life of the heavier isotopes.  Uranium 238 only has 931.493 MeV per baryon, far less than the 931.845 of Helium 4.  A complete table is available online giving energy for the 3200 known isotopes. This is in n-2z (y-axis) by a/3, a useful form for such data. Lowest energy for a baryon count is outlined in blue.  The 2 lowest ratios for an element are outlined in black (for even N) and then orange (odd N).

Here is a comparison between this form of the data and the binding energy per baryon classically.  Sum approximates the weighted average of protons and neutrons the classic would use if the electrons effect were taken into account.  It tracks closely to the ratio of neutrons to protons, with rising z a secondary effect.

Element Weight Energy/ Baryon Classic Energy Sum N/Z
Helium 4 931.84473 7.073915 938.91864 1.000
Nickel 58 930.20839 8.732038 938.94043 1.071
Cobalt 56 930.24643 8.694824 938.94125 1.074
Iron 54 930.20579 8.736343 938.94213 1.077
Chromium 50 930.24312 8.700981 938.94410 1.083
Copper 61 930.23441 8.715505 938.94992 1.103
Nickel 59 930.21445 8.736568 938.95102 1.107
Cobalt 57 930.21034 8.741856 938.95220 1.111
Iron 55 930.20690 8.746559 938.95346 1.115
Manganese 53 930.22069 8.734132 938.95482 1.120
Chromium 51 930.24433 8.711955 938.95628 1.125
Zinc 64 930.22262 8.735897 938.95852 1.133
Copper 62 930.24160 8.718074 938.95967 1.138
Nickel 60 930.18050 8.780754 938.96125 1.143
Cobalt 58 930.22382 8.738944 938.96276 1.148
Iron 56 930.17407 8.790322 938.96439 1.154
Manganese 54 930.22820 8.737922 938.96612 1.160
Chromium 52 930.19205 8.775944 938.96799 1.167
Zinc 65 930.24359 8.724257 938.96785 1.167
Copper 63 930.21732 8.752131 938.96945 1.172
Nickel 61 930.20615 8.765006 938.96998 1.179
Titanium 48 930.24929 8.722905 938.97220 1.182
Cobalt 59 930.20497 8.768010 938.97298 1.185
Iron 57 930.20468 8.770248 938.97493 1.192
Manganese 55 930.21203 8.764988 938.97702 1.200
Zinc 66 930.21727 8.759631 938.97690 1.200
Copper 64 930.23970 8.739067 938.97877 1.207
Chromium 53 930.21911 8.760155 938.97926 1.208
Nickel 62 930.18620 8.794546 938.98075 1.214
Vanadium 51 930.23963 8.742051 938.98168 1.217
Cobalt 60 930.23611 8.746742 938.98285 1.222
Iron 58 930.19289 8.792220 938.98511 1.231
Manganese 56 930.24923 8.738299 938.98753 1.240
Copper 65 930.23069 8.757094 938.98778 1.241
Chromium 54 930.21220 8.777913 938.99011 1.250
Nickel 63 930.22654 8.763486 938.99003 1.250
Cobalt 61 930.23625 8.756149 938.99240 1.259
Zinc 68 930.23853 8.755677 938.99421 1.267
Iron 59 930.24020 8.754742 938.99494 1.269
Titanium 50 930.24030 8.755621 938.99592 1.273
Nickel 64 930.22155 8.777461 938.99901 1.286
Iron 60 930.24863 8.755836 939.00447 1.308
Uranium 238 931.49291 7.570120 939.06303 1.587

 

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