Quantitative Analysis of Alpha Decay
As noted in the Law of the conservation of mass page, potential energy has mass, which disappears when that energy is converted to kinetic energy, and the conversion factor is 1.11
10-17 kilograms per Joule. For any phenomenon other than a nuclear reaction, that mass is far too small to be measured, which is why this effect was not noticed until the discovery of radioactivity around 1900.
The table below shows quantitative measurements of the phenomenon for alpha decay. Alpha decay is by far the most straightforward interaction to measure, because there are only two resulting particles, and the kinetic energy of the alpha particle is easy to measure accurately with a mass spectrometer. By contrast, beta decay releases both an electron and an antineutrino, and the energy distribution between those two is indeterminate. This means that the beta particle (electron) energy is indeterminate. Alpha decay energies were historically the first observations that were made confirming this apparent mass loss.
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 [1]. The observed alpha particle energies are from [2], from [3].
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 4He2, 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 226Ra88 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.
This table is under construction. There are many more isotopes to come.
| Parent | Daughter | Parent mass | Daughter mass | Mass loss | Mass loss
times 931.494 |
Observed
alpha energy |
|---|---|---|---|---|---|---|
| 106Te52 | 102Sn50 | 105.9375 | 101.9303 | .0046 | 4.285 | 4.128 |
| 107Te52 | 103Sn50 | 106.9350 | 102.9281 | .0043 | 4.005 | 3.833 |
| 108I53 | 104Sb51 | 107.9435 | 103.9365 | .0044 | 4.099 | 3.947 |
| 110Xe54 | 106Te52 | 109.9443 | 105.9375 | .0042 | 3.912 | 3.745 |
| 144Nd60 | 140Ce58 | 143.9101 | 139.9054 | .0021 | 1.956 | 1.830 |
| 146Sm62 | 142Nd60 | 145.9130 | 141.9077 | .0027 | 2.515 | 2.455 |
| 147Sm62 | 143Nd60 | 146.9149 | 142.9098 | .0025 | 2.329 | 2.233 |
| 148Sm62 | 144Nd60 | 147.9148 | 143.9101 | .0021 | 1.956 | 1.960 |
| 148Gd64 | 144Sm62 | 147.9181 | 143.9120 | .0035 | 3.260 | 3.183 |
| 150Gd64 | 146Sm62 | 149.9187 | 145.9130 | .0031 | 2.886 | 2.726 |
| 152Gd64 | 148Sm62 | 151.9198 | 147.9148 | .0024 | 2.236 | 2.140 |
| 154Dy66 | 150Gd64 | 153.9244 | 149.9187 | .0031 | 2.886 | 2.872 |
| 152Er68 | 148Dy66 | 151.9351 | 147.9271 | .0054 | 5.030 | 4.799 |
| 153Er68 | 149Dy66 | 152.9351 | 148.9273 | .0052 | 4.843 | 4.674 |
| 154Yb70 | 150Er68 | 153.9464 | 149.9379 | .0059 | 5.496 | 5.325 |
| 155Yb70 | 151Er68 | 154.9458 | 150.9374 | .0058 | 5.403 | 5.200 |
| 157Hf72 | 153Yb70 | 156.9584 | 152.9495 | .0063 | 5.868 | 5.731 |
| 226Ra88 | 222Rn86 | 226.0254 | 222.0176 | .0052 | 4.844 | 4.784 |
The conversion factor, 1.1110-17 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.
