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 the Wolfram web site.[1] The observed alpha particle energies are from the Lawrence Berkely National Laboratory,[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 4.0026033 amu, which is the atomic weight of 4He2, 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). As discussed below, the alpha energies are smaller than the potential energy loss because the recoiling parent atom's energy is not considered.
| 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 |
| 159W74 | 155Hf72 | 158.9729 | 154.9634 | .0069 | 6.427 | 6.299 |
| 160W74 | 156Hf72 | 159.9685 | 155.9594 | .0065 | 6.055 | 5.912 |
| 162Os76 | 158W74 | 161.9844 | 157.9746 | .0072 | 6.707 | 6.611 |
| 163Os76 | 159W74 | 162.9827 | 158.9729 | .0072 | 6.707 | 6.510 |
| 164Os76 | 160W74 | 163.9780 | 159.9685 | .0069 | 6.427 | 6.321 |
| 165Os76 | 161W74 | 164.9768 | 160.9674 | .0068 | 6.334 | 6.188 |
| 166Os76 | 162W74 | 165.9727 | 161.9635 | .0066 | 6.147 | 5.983 |
| 167Os76 | 163W74 | 166.9715 | 162.9625 | .0064 | 5.962 | 5.838 |
| 173Os76 | 169W74 | 172.9598 | 168.9518 | .0054 | 5.030 | 4.940 |
| 186Os76 | 182W74 | 185.9538 | 181.9482 | .0030 | 2.795 | 2.758 |
| 168Pt78 | 164Os76 | 167.9882 | 163.9780 | .0076 | 7.079 | 6.832 |
| 169Pt78 | 165Os76 | 168.9867 | 164.9768 | .0073 | 6.800 | 6.678 |
| 170Pt78 | 166Os76 | 169.9825 | 165.9727 | .0072 | 6.707 | 6.545 |
| 171Pt78 | 167Os76 | 170.9812 | 166.9715 | .0071 | 6.614 | 6.540 |
| 172Pt78 | 168Os76 | 171.9773 | 167.9678 | .0069 | 6.427 | 6.314 |
| 173Pt78 | 169Os76 | 172.9764 | 168.9670 | .0068 | 6.334 | 6.213 |
| 174Pt78 | 170Os76 | 173.9728 | 169.9636 | .0066 | 6.148 | 6.038 |
| 175Pt78 | 171Os76 | 174.9724 | 170.9632 | .0066 | 6.148 | 6.038 |
| 190Pt78 | 186Os76 | 189.9599 | 185.9538 | .0035 | 3.260 | 3.180 |
| 174Au79 | 170Ir77 | 173.9848 | 169.9750 | .0072 | 6.707 | 6.530 |
| 175Hg80 | 171Pt78 | 174.9914 | 170.9812 | .0076 | 7.079 | 6.860 |
| 176Hg80 | 172Pt78 | 175.9874 | 171.9773 | .0075 | 6.986 | 6.767 |
| 177Hg80 | 173Pt78 | 176.9863 | 172.9764 | .0073 | 6.800 | 6.580 |
| 178Hg80 | 174Pt78 | 177.9825 | 173.9728 | .0071 | 6.614 | 6.430 |
| 179Hg80 | 175Pt78 | 178.9818 | 174.9724 | .0068 | 6.334 | 6.288 |
| 180Hg80 | 176Pt78 | 179.9783 | 175.9689 | .0068 | 6.334 | 6.120 |
| 182Pb82 | 178Hg80 | 181.9927 | 177.9825 | .0076 | 7.079 | 6.921 |
| 211Bi83 | 207Tl81 | 210.9873 | 206.9774 | .0073 | 6.800 | 6.278 |
| 190Po84 | 186Pb82 | 189.9951 | 185.9842 | .0083 | 7.731 | 7.490 |
| 192Po84 | 188Pb82 | 191.9913 | 187.9808 | .0079 | 7.359 | 6.610 |
| 194Po84 | 190Pb82 | 193.9882 | 189.9781 | .0075 | 6.986 | 6.843 |
| 196Po84 | 192Pb82 | 195.9855 | 191.9758 | .0071 | 6.614 | 5.769 |
| 198Po84 | 194Pb82 | 197.9834 | 193.9740 | .0068 | 6.334 | 6.182 |
| 208Po84 | 204Pb82 | 207.9812 | 203.9730 | .0056 | 5.216 | 4.220 |
| 209Po84 | 205Pb82 | 208.9824 | 204.9745 | .0053 | 4.937 | 4.110 |
| 210Po84 | 206Pb82 | 209.9828 | 205.9745 | .0057 | 5.310 | 4.517 |
| 213Po84 | 209Pb82 | 212.9929 | 208.9811 | .0092 | 8.570 | 7.614 |
| 214Po84 | 210Pb82 | 213.9952 | 209.9842 | .0084 | 7.825 | 7.687 |
| 215Po84 | 211Pb82 | 214.9994 | 210.9887 | .0081 | 7.545 | 7.386 |
| 216Po84 | 212Pb82 | 216.0019 | 211.9919 | .0074 | 6.893 | 5.985 |
| 217Po84 | 213Pb82 | 217.0063 | 212.9966 | .0071 | 6.614 | 6.537 |
| 218Po84 | 214Pb82 | 218.0090 | 213.9998 | .0066 | 6.148 | 5.181 |
| 226Ra88 | 222Rn86 | 226.0254098 | 222.0175777 | .0052288 | 4.870596 | 4.784 |
Many of the alpha decays in the table are difficult to observe, and, for the purposes of this table, 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 226Ra88 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, 1.1110−17 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.
