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Quantitative Analysis of Alpha Decay (view source)
Revision as of 05:55, June 18, 2016
, 05:55, June 18, 2016Put in accurate data for Radium.
In the table below, the alpha-active isotopes were selected from information from the National Nuclear Data Center, Brookhaven National Laboratory. The atomic weights are from the Wolfram web site<ref>http://m.wolframalpha.com/</ref>. The observed alpha particle energies are from the Lawrence Berkely National Laboratory<ref>http://dbserv.pnpi.spb.ru/elbib/tablisot/toi98/www/decay/table3.pdf</ref>, from <ref>http://www.wiley-vch.de/books/info/0-471-35633-6/toi99/www/decay/table3.pdf</ref>.
In the table below, the alpha-active isotopes were selected from information from the National Nuclear Data Center, Brookhaven National Laboratory. The atomic weights are from the Wolfram web site<ref>http://m.wolframalpha.com/</ref>. The observed alpha particle energies are from the Lawrence Berkely National Laboratory<ref>http://dbserv.pnpi.spb.ru/elbib/tablisot/toi98/www/decay/table3.pdf</ref>, from <ref>http://www.wiley-vch.de/books/info/0-471-35633-6/toi99/www/decay/table3.pdf</ref>.
The atomic weights are in the usual amu (atomic mass units). The "mass loss" column is obtained by subtracting the Daughter mass from the Parent mass, and then subtracting 4.0026 amu, which is the atomic weight of <sup>4</sup>He<sub>2</sub>, that is, an alpha particle. All atomic weights, by convention, include the electrons, so they are not "nuclear weights", but electrons are preserved, so it makes no difference. The Mass loss is then multiplied by the conversion factor of 931.494 MeV per amu to get the expected potential energy converted to kinetic energy. The observed alpha emission energies (column 7) closely match the converted potential energy loss (column 6). The discrepancy is explained by the recoil energy of the daughter nucleus. The released kinetic energy is divided between the daughter nucleus and the alpha particle. Since this is a two-body problem, it can be solved, using conservation of energy and conservation of momentum, to get the alpha energy and recoil energy. For example, the alpha decay of <sup>226</sup>Ra<sub>88</sub> shows a total kinetic energy release of 4.844 MeV. The observed alpha particle kinetic energy is 4.784 MeV, and the recoil energy of the Radon atom is about 0.06 Mev, for a total kinetic energy of 4.844 MeV.
The atomic weights are in the usual amu (atomic mass units). The "mass loss" column is obtained by subtracting the Daughter mass from the Parent mass, and then subtracting 4.0026033 amu, which is the atomic weight of <sup>4</sup>He<sub>2</sub>, that is, an alpha particle. All atomic weights, by convention, include the electrons, so they are "atomic weights" and not "nuclear weights". But electrons are preserved, so it makes no difference when doing the subtractions. The Mass loss is then multiplied by the conversion factor of 931.494095 MeV per amu to get the expected potential energy converted to kinetic energy. The observed alpha emission energies (column 7) roughly match the converted potential energy loss (column 6).
{| class="wikitable"
{| class="wikitable"
|<sup>226</sup>Ra<sub>88</sub>
|<sup>226</sup>Ra<sub>88</sub>
|<sup>222</sup>Rn<sub>86</sub>
|<sup>222</sup>Rn<sub>86</sub>
|226.0254
|226.0254098
|222.0176
|222.0175777
|.0052
|.0052288
|4.844
|4.870596
|4.784
|4.784
|-
|-
|}
|}
Many of the alpha decays in the table are difficult to observe, and the atomic weights have only been calculated to 4 digits after the decimal point. This means that the mass defect is only accurate to about two significant digits. Aside from this inaccuracy, the observed alpha particle energy is less than the total energy released because the recoil of the parent nucleus takes some of the energy.
The important thing to note is not that the 6th and 7th columns track each other accurately, but that they track very energetic alpha decays (over 7 MeV) and very weak ones (about 2 MeV).
The <sup>226</sup>Ra<sub>88</sub> decay, however has been characterized very accurately. Because only two particles are involved, it is possible to do a quick calculation, using the laws of conservation of momentum and of energy, of the recoil energy. The ratio of the alpha energy to the daughter energy is just the ratio of the daughter mass to the alpha mass, which in this case is 55.5 to 1. This means that the total kinetic energy of 4.870596 should be divided into .0868 Mev for the Radon daughter and 4.7838 MeV for the alpha particle, which is the observed alpha energy.
The conversion factor, 1.11<math>\times</math>10<sup>-17</sup> kilograms per Joule, has the dimensions of seconds-squared per meter-squared. (A Joule is a Newton-meter; by Newton's formula F=ma a Newton is a kilogram-meter-per-second-squared; so the conversion factor is kilograms per Joule, or seconds-squared per meter-squared.) Hence the conversion factor is the reciprocal of the square of a speed. That speed is 299,792,458 meters per second.
The conversion factor, 1.11<math>\times</math>10<sup>-17</sup> kilograms per Joule, has the dimensions of seconds-squared per meter-squared. (A Joule is a Newton-meter; by Newton's formula F=ma a Newton is a kilogram-meter-per-second-squared; so the conversion factor is kilograms per Joule, or seconds-squared per meter-squared.) Hence the conversion factor is the reciprocal of the square of a speed. That speed is 299,792,458 meters per second.
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[[Category:Relativity]]
[[Category:Physics]]
[[Category:Physics]]