I collect spectra using a high purity germanium (HPGe) detector operating at liquid nitrogen temperature. The detector is run with a -1500 V bias and the detector preamplifier output (negative polarity) is passed through a NIM module amplifier. The output (positive polarity) is scaled to an appropriate range (0-8 V) for direct input into a Spectrum Techniques UCS-30 pulse-height analyzer (PHA) controlled by the USX software program, with the amplifier gain set so that a highest PHA channel corresponds to gamma of approximately 2800 keV (to ensure that I can comfortably identify gamma full energy peaks up to 2600 keV).
One spectrum is collected for each radioactive source and the spectrum is saved as both a *.spu file (for use with the USX software) and a *.tsv file (for reading data into an analysis program later.) I do not calibrate the x-axis in software; instead, all features are recorded in terms of channel and a calibration is performed post-experiment. (Gain and high voltage settings are not adjusted at any point, so the scale for each spectrum should be the same.)
Using either the plots produced within the PHA software or by re-plotting later using the saved *.tsv files, I identify all visible full energy peaks and make plausible identifications of associated Compton edges when possible. I determine peak centers and uncertainties “by-eye”; fitting the data to a Gaussian could potentially yield more precise values, but with the caveat that accuracy is affected greatly if the background is not well modeled in the fit. I determine Compton edges as the point approximately half-way down the slope of the edge; edge uncertainties are largely determined by how well defined the edge is. When many peaks are visible, a check is performed by estimating the Compton edge position from the Compton scattering formula for a given incident photon energy. Care it taken to still “measure” the edge position (and thus avoid a circular logic trap, since the Compton scattering formula can be derived from the special relativity relations we are trying to “discover”), but the formula is useful for determining to which peak a particular edge belongs.
Predicted gamma energies and branching ratios were taken from the online Table of Isotopes searchable database – http://nucleardata.nuclear.lu.se/toi/nucSearch.asp. Plausible connections between measured peaks and expected energies were made with help from catalogs of compiled spectra available online: for HPGe – http://www4vip.inl.gov/gammaray/catalogs/ge/catalog_ge.shtml – and for NaI – http://www4vip.inl.gov/gammaray/catalogs/nai/catalog_nai.shtml.
Raw Data
Peak and edge positions are tabulated in the tables below.
| Source | Predicted Energy | |||||
| Full Energy Peak | ||||||
| Compton edge | ||||||
| E_gamma (keV) | Branching Ratio | Eg(ch) | dEg(ch) | T (ch) | dT(ch) | |
| Co-57 | 121.94 | 85.60% | 94.9 | 1 | – | – |
| 136.31 | 10.70% | 83.9 | 1 | – | – | |
| Co-60 | 1173.23 | 100% | 867.8 | 1 | 714 | 2 |
| 1332.48 | 100% | 985.9 | 1 | 826 | 2 | |
| Cs-137 | 666.1 | 85.10% | 487.1 | 1 | 350 | 2 |
| Na-22 | 511 | 90% | 374.5 | 1 | 247 | 2 |
| 1274.6 | 100% | 942.1 | 1 | 784 | 2 | |
| Bi-207 | 72.8 and 75.0 | 57% | 47.9 | 1 | – | – |
| 84.5 and 84.9 | 12% | 56 | 1 | – | – | |
| 569.6 | 97.70% | 419.2 | 1 | 287 | 2 | |
| 1063.4 | 74.50% | 787.2 | 1 | 636 | 2 | |
| 1769.7 | 6.90% | 1311.9 | 2 | 1148 | 2 | |
| Ba-133 | 81 | 34.00% | 53 | 1 | – | – |
| 276.4 | 7.10% | 200 | 1 | – | – | |
| 302.9 | 18.30% | 220 | 1 | 115 | 3 | |
| 356 | 62% | 260 | 1 | 147 | 3 | |
| 383.8 | 8.90% | 280.7 | 1 | 165 | 3 | |
| 437 | (81+356 sum peak) | 320 | – | – | – | |
| In-116 | 138.3 | 3.30% | 96.1 | 1 | – | – |
| 416.9 | 27.70% | 305 | 1 | 190 | 3 | |
| 818.7 | 11.50% | 606.2 | 1 | 463 | 4 | |
| 1097.3 | 56.20% | 812.8 | 1 | 660 | 3 | |
| 1293.6 | 84.40% | 957 | 1 | 800 | 3 | |
| 1507.7 | 10.00% | 1117 | 1 | 958 | 4 | |
| 1752.7 | 2.50% | 1299 | 2 | – | – | |
| 2112.3 | 15.50% | 1565.2 | 1 | 1395 | 2 | |
| Neutron Howitzer | 511.0 | (1x annihilation) | 375.2 | 1 | – | – |
| 1202.7 | (double escape) | 889.2 | 1 | – | – | |
| 1713.7 | (single escape) | 1267 | 2 (v. weak) | – | – | |
| 2224.7 | (capture gamma) | 1646 | 1 | 1477 | 2 |
Table: Data collected using typical sources (plus the neutron howitzer).
| Source | Predicted Energy | |||||
| Full Energy Peak | ||||||
| Compton edge | ||||||
| E_gamma (keV) | Branching Ratio | Eg(ch) | dEg(ch) | T (ch) | dT(ch) | |
| Background (~121 hrs.) | 238.6 (Pb-212) | 43% | 171 | 1 | – | – |
| 295 (Pb-214) | ||||||
| 211.5 | 2 | – | – | |||
| 351.9 (Pb-214) | 38% | 256 | 5 (distorted) | – | – | |
| 510.77 (Tl-208) | 22% | 375 | 3 | – | – | |
| 583.20 (Tl-208) | 85% | 428 | 1.5 | – | – | |
| 609.3 (Bi-214) | 46% | 447.3 | 1 | – | – | |
| 661.6 (Cs-137) | 100% | 487 | 2 (v. weak) | – | – | |
| 911.204 (Ac-228) | 25.80% | 674 | 1.5 | – | – | |
| 968.971 (Ac-228) | 15.80% | 716 | 1 | – | – | |
| 1120.287 (Bi-214) | 15% | 829 | 1 | – | – | |
| 1332.48 (Co-60)? | 100% | 985 | 1 | – | – | |
| 1460.85 (K-40) | 100% | 1081.5 | 1 | 919 | 2 | |
| 1592.5 (Tl-208) | (double escape) | 1179 | 1 | – | – | |
| 1764.49 (Bi-214) | 16% | 1307 | 1 | – | – | |
| 2614.5 (Tl-208) | 99% | 1932.5 | 1.5 | 1761 | 10 |
Table: Data from a long background measurement (~121 hrs.). All peaks w/ E < 500 keV lie on a very steep background and therefore may be distorted (and will be dropped from subsequent analysis.) The exception is the 609.3 keV line which is v. strong.
| Source | Predicted Energy | ||||
| Full Energy Peak | |||||
| Compton edge | |||||
| E_gamma (keV) | Branching Ratio | Eg(ch) | dEg(ch) | T (ch) | dT(ch) |
| Ra-226 | Bi x-rays (many) | ||||
| 49.6 | 0.5 | – | – | ||
| Bi x-rays (many) | |||||
| 58.2 | 0.5 | – | – | ||
| 186/187 | |||||
| 132.3 | 0.5 | – | – | ||
| 242.00 (Pb-214) | 7.40% | 174.1 | 0.5 | – | – |
| 258.9 (Pb-214) | 0.50% | 187.2 | 1 | – | – |
| 274.8 (Pb-214) | 0.50% | 200 | 1 | – | – |
| 295.2 (Pb-214) | 19.30% | 214.1 | 0.5 | 111.5 | 3 |
| 351.9 (Pb-214) | 37.60% | 257.5 | 1.5 (distorted) | 146 | 2 |
| 609.3 (Bi-214) | 46.10% | 449.1 | 0.5 | 315.5 | 2 |
| 665.4 (Bi-214) | 1.50% | 491 | 0.5 | – | – |
| 703.1 (Bi-214) | 0.40% | 521 | 2 | – | – |
| 719.9 (Bi-214) | 0.40% | 534 | 1.5 | – | – |
| 742 (Bi-214) | (double escape of 1764 keV) | 550 | 2 | – | – |
| 768.4 (Bi-214) | 4.90% | 569.3 | 1 | – | – |
| 786 (Pb-214) | 1.10% | 582 | 2 | – | – |
| 806.2 (Bi-214) | 1.20% | 691.7 | 1 | – | – |
| 934.1 (Bi-214) | 3.30% | 691.7 | 1 | – | – |
| 1120.3 (Bi-214) | 15.10% | 830.9 | 1 | 678 | 3 |
| ? | |||||
| 857 | 2 | – | – | ||
| ? | |||||
| 877 | 2 | – | – | ||
| 1238.1 (Bi-214) | 5.80% | 918 | 1 | – | – |
| 1280.96 (Bi-214) | 1.40% | 950 | 1 | – | – |
| 1377.7 (Bi-214) | 4.00% | 1021 | 1 | – | – |
| 1729.6 (Bi-214) | 2.90% | 1283 | 3 | – | – |
| 1764 (Bi-214) | 15.40% | 1309 | 1 | 1144 | 3 |
| 1847.4 (Bi-214) | 2.10% | 1371 | 2 | – | – |
| 2118.6 (Bi-214) | 1.40% | 1570 | 3 | – | – |
| 2204.2 (Bi-214) | 5.10% | 1633.2 | 1 | 1469 | 4 |
| 2447.9 (Bi-214) | 1.60% | 1813 | 1 | – | – |
Table: Data from took a spectrum (~18 hrs.) with two spinthariscopes (which contain radium bromide). The spectrum was weak, so I subtracted off the background rate before making peak identifications. The resulting spectrum is consistent with the decay chain of Ra-226.
For the above data, clear peaks without edges were used as calibration points, whereas clear peak and edge pairs were used as test data (See Tables 4 and 5). Some trial and error was needed – misidentifications are common, but appear as points far from the expected lines – but ultimately the best set of calibration and test data was chosen and is shown below.
| source | ch | dch | E (keV) |
| Co-57 | 83.9 | 1 | 121.94 |
| Co-57 | 94.9 | 1 | 135.31 |
| Neutron | 375.2 | 1 | 511.00 |
| Neutron | 889.2 | 1 | 1202.7 |
| Ba-133 | 53 | 1 | 81 |
| Ba-133 | 200 | 1 | 276.4 |
| Bkgd | 172 | 2 | 238.6 |
| Bkgd | 374 | 2 | 510.77 |
| Bkgd | 428 | 2 | 583.2 |
| Bkgd | 448 | 2 | 609.3 |
| Bkgd | 1307 | 1 | 1764.49 |
| Bkgd | 1179 | 1 | 1592.5 |
| In-116 | 96.1 | 1 | 138.32 |
| Ra-226 | 174.1 | 0.5 | 242 |
| Ra-226 | 569.3 | 1 | 768.4 |
| Ra-226 | 691.7 | 1 | 934.1 |
| Ra-226 | 918 | 1 | 1238.1 |
| Ra-226 | 1570 | 2 | 2118.6 |
| Ra-226 | 1813 | 1 | 2447.9 |
Table: Calibration data
{
${/download/attachments/164084021/calibration.png?version=1&modificationDate=1502736782000&api=v2}$
| source | Eg | dEg | T | dT |
| Bkgd | 1081.5 | 1 | 919 | 2 |
| Bkgd | 1932.5 | 1.5 | 1761 | 10 |
| In-116 | 305 | 1 | 190 | 3 |
| In-116 | 606.2 | 1 | 463 | 4 |
| In-116 | 812.8 | 1 | 659 | 2 |
| In-116 | 957 | 1 | 800 | 3 |
| In-116 | 1117 | 1 | 958 | 4 |
| In-116 | 1565.2 | 1 | 1395 | 2 |
| Bi-207 | 419.2 | 1 | 287 | 2 |
| Bi-207 | 787.9 | 1 | 636 | 2 |
| Bi-207 | 1311.9 | 1 | 1148 | 2 |
| Neutron | 1646.0 | 1 | 1477 | 2 |
| Na-22 | 374.5 | 1 | 247 | 2 |
| Na-22 | 942.1 | 1 | 784 | 2 |
| Cs-137 | 487.1 | 1 | 350 | 2 |
| Co-60 | 867.8 | 1 | 714 | 2 |
| Co-60 | 985.0 | 1 | 826 | 2 |
| Ba-133 | 281 | 1 | 165 | 3 |
| Ba-133 | 260 | 1 | 147 | 3 |
| Ba-133 | 220 | 1 | 115 | 3 |
| Ra-226 | 214.1 | 0.5 | 111.5 | 3 |
| Ra-226 | 257.5 | 1.5 | 146 | 2 |
| Ra-226 | 449.1 | 0.5 | 315.5 | 2 |
| Ra-226 | 830.9 | 1 | 678 | 3 |
| Ra-226 | 1309 | 1 | 1144 | 3 |
| Ra-226 | 1633 | 1 | 1469 | 4 |
Table: Test data
Spectra
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Analysis
Using the calibrated data for Eg and T, I compute the electron momentum p and “test” the classical relation, p_2/2_T = _m_nr. The non-relativistic prediction is that _m_nr = _m_0 is a constant, but plotting the data reveals a very different dependence on T. The data appear to increase linearly with kinetic energy, so I attempt a fit of the form p_2/2_T = AT + B. The fit matches the data well, and I make a plausible identification for the fit parameters that result using only numbers and constants related to electron properties; the value A = (0.497 +/- 0.004)/c2 is consistent with A = 1/2c2, and the value B = (512 +/-5) keV/c2 is consistent with B = _m_0, the rest mass of the electron. Thus we have “discovered the dispersion relation, p_2/2_T = T/2_c_2 + _m_0.
{ ${/download/attachments/164084021/classical.png?version=1&modificationDate=1502740185000&api=v2}$
Using the fact that the total energy is the sum of the kinetic and rest energies, E = T + _m_0_c_2, we can rearrange the dispersion relation to find the more well-known special relativity relation:
| p_2/2_T = T/2_c_2 + _m_0 | |
| (multiply by 2_Tc_2) | _ p_2_c_2 = _T_2 + 2_Tm_0_c_2 |
| (complete the square) | = [_T_2 + 2_Tm_0_c_2 + (m_0_c_2)2] - (m_0_c_2)2 |
| (use E = T + _m_0_c_2) | = E_2 - _(m_0_c_2)_2 |
| (rearrange) | E2 = (pc)2 + (m_0_c_2)_2 |
From this relation, we can compute our best estimate of the rest mass, _m_0 = [(pc)2 - _T_2)/(2_Tc_2).
{ ${/download/attachments/164084021/mass.png?version=1&modificationDate=1502740191000&api=v2}$
In addition, we can plot our data against the expected forms to see how the measured dispersion relation compares to the nonrelativistic and massless forms, and to see how E and p each vary with velocity, β. (NOTE: performing additional fits will yield no new information or better/different results. The data set used in the mass fit and the subsequent fits is the same, so later plots use “expected forms”, not fits.)
{ ${/download/attachments/164084021/energy.png?version=1&modificationDate=1502740189000&api=v2}${ ${/download/attachments/164084021/momentum.png?version=1&modificationDate=1502740194000&api=v2}$
{ ${/download/attachments/164084021/E_vs_p.png?version=1&modificationDate=1502740187000&api=v2}$