The atomic masses of the proton, deuteron, triton, helion, and alpha particle (the nuclei of the atoms H, D, T,
3He, and
4He) are considered to be fundamental constants and are included in the recent CODATA tabulations, (e.g., references [
2,
3]). The ratio of the electron mass to the masses of the proton and deuteron are needed for the analysis of precision laser spectroscopy of hydrogen and deuterium used to obtain the Rydberg constant. The fractional uncertainty of
1s-
2s two-photon spectroscopy in hydrogen is currently 4 × 10
−15 [
62]. In principle, to extract the Rydberg constant to this precision requires that
me/mp be known to <8 × 10
−12. However, at present, uncertainty in the proton charge radius dominates and the CODATA14 value for
R∞ has an uncertainty of 5.9 × 10
−12 [
2], relaxing the precision required for
me/mp by three orders-of-magnitude. However, this situation could change if the so-called “proton radius puzzle” is resolved in such a way that confidence can be placed in the determination of the proton charge radius from the precise measurements of the Lamb-shift of muonic hydrogen [
63].
An interesting development is the proposal to obtain
me/mp and
me/md from precision spectroscopy of ro-vibrational transitions in H
2+ and HD
+, in combination with precise theoretical calculations including relativistic and QED corrections, (e.g., references [
64,
65,
66]). With further developments in the theory, this may lead to sub-10
−11 values for the mass ratios, but also, by using combinations of transitions, to competitive values for
R∞ and the charge radii of the proton and deuteron [
67]. Additionally, with a sufficiently developed theory, measurements of the Zeeman-hyperfine structure and cyclotron frequency of H
2+ can be used to obtain the electron-to-proton mass ratio, and the magnetic moment of the proton [
68]. All these projects motivate further measurements of
M[
p] and
M[
d] by Penning trap techniques.
The mass difference between tritium and helium-3 is closely related to the
Q-value (available energy) of the beta-decay of tritium. This is an important parameter for testing systematics in precision measurements of the beta-electron energy spectrum of tritium near its endpoint, which set upper limits on the mass of the electron neutrino [
69,
70]. For this reason, the masses of T
+ and
3He
+ are often measured together so that some systematics cancel in their difference. A precision atomic mass of
3He will also be needed to convert measurements of the ratio of the nuclear spin-flip frequency of
3He
2+ to its cyclotron frequency into a value for the nuclear
g-factor of the helion [
71]. The same will be true for the deuteron. A precise mass for
4He is needed for measurements of the electronic
g-factor of
4He
+, which can be used to obtain the electron mass and, possibly in the future, the fine-structure constant [
72] (but also see
Section 5).
In what follows we show those mass measurements that have sufficient precision to influence the results of least-squares adjustments such as the AME or CODATA, sometimes with additional, less precise values for context. In general, we show the atomic masses as published in the original papers. In some cases, the results are based on CFRs relative 12C, while others use atoms other than 12C as references. In the latter case, we attempt to show the effects of significant updates in the masses of the references. However, this is essentially illustrative and is not a substitute for a least-squares adjustment based on the relevant CFRs (or their equivalent mass difference equations) as is done by the AME.
3.1. Proton
Atomic mass results for the proton and the least-squares value from the AME2016 are shown in
Table 1. The first of these, and the first with an uncertainty of <10
−9 was by the MIT group. Here the CFR of
12CH
4+ versus
12C
+ was measured using a two-cyclotron-pulse (Ramsey) variant of the PnP technique [
73]. By mapping the final cyclotron phase onto cyclotron amplitude, both ions’ cyclotron frequencies could be measured at the same trap voltage, thereby eliminating the equilibrium position shift error discussed in
Section 2.3.3. The second is a result from the UW group [
74], who compared a proton against C
4+, using an exceptionally stable 6 T solenoid magnet and the swept cyclotron drive, axial-frequency-shift technique. Here, auxiliary measurements of C
4+ against C
5+ provided some check against the equilibrium-position and other systematic errors, and the error budget had contributions from anharmonicity (95 ppt), residual magnetic field drifts (70 ppt), statistics (50 ppt), and image charge shifts (40 ppt). Also notable was the use of the high axial frequency of 3.5 MHz, which reduces some anharmonic shifts and leads to a lower cyclotron temperature after cyclotron-to-axial coupling.
The third is a measurement by the SMILETRAP group [
75] using their time-of-flight detection technique to measure the CFRs of H
2+ relative to several charge states of
12C, but also of
28Si,
14N,
20Ne, and
40Ar, making use of measurements of these atoms at the 10
−10 level by MIT [
41]. The fourth is another measurement by the SMILETRAP group [
38], but making use of the Ramsey variant of their TOF method, and measuring H
2+ against D
+ as a reference. To obtain the proton mass they assumed an average vibrational excitation energy of 0.74(7) eV for H
2+ and a value for
M[
d] = 2.013 553 212 73(4) u, as provided in a private communication from the UW group. (Using the updated mass for D from the UW group, see
Table 2, increases this value for
M[
p] by only 7.5 ppt.)
The fifth is a recent and advanced measurement with a fractional uncertainty of 3.3 × 10
−11 by the MPIK-Heidelberg/Mainz/GSI/RIKEN collaboration, using a purpose built, cylindrical, multi-Penning trap system for light ions, now called LIONTRAP [
46]. Here the cyclotron frequencies of a proton and a
12C
6+ ion were measured, mainly using the PnA technique, but with checks by the double-dip technique, and by alternately shuttling the two ions into a “measurement” trap from reservoir traps on either side. A quadratic potential could be produced over a relatively large volume in the measurement trap by incorporating two sets of compensation electrodes. The equilibrium position shift was avoided by using two axial detectors, one resonant at about 525 kHz for
12C
6+ and the other at 740 kHz for the proton, so the measurements of
fc’,
fz and
fm could be carried out at the same trap voltage. By using the PnA technique, and reducing the effective temperature of the axial motion using feedback cooling, a proton cyclotron radius as small as 9 μm could be used. Hence relativistic shifts to the cyclotron frequency were reduced and the estimated systematic uncertainty due to special relativity on the CFR was 7 ppt. Overall, the largest uncertainty was from the finite axial motion, mainly of the proton, combined with the quadratic magnetic field inhomogeneity
B2, which gave 27 ppt. The model of systematics was also checked by comparing
12C
3+ to
12C
6+ and the proton to
16O
8+. As
Table 1 shows, the MPIK* result is in a 2 standard-deviation disagreement with both the UW and the second SMILETRAP result and indicates a lighter proton. However, the SMILETRAP–MPIK* disagreement might also indicate that the deuteron is lighter than as measured by the UW group, see next section.
New measurements of H
2+/D
+ using two simultaneously trapped ions are currently in progress at FSU [
55]. Unlike the SMILETRAP TOF technique in which the measurements were carried out on H
2+ within a few seconds after leaving the ion source, the H
2+ is trapped for many days and can be Stark-quenched to the vibrational ground state [
76]. This removes the uncertainty due to the vibrational distribution of the H
2+ that had to be assumed in [
38].
3.3. Tritium and Helium-3
In
Table 4 we show values for the atomic masses of tritium and helium-3 as given in the respective publications. (The atomic mass of the triton can be obtained from the mass of the tritium atom by subtracting 5.485 653 052 57(16) × 10
−4 u, and that of the helion from helium-3 by subtracting 1.097 075 006 640(32) × 10
−3 u).
The first row gives the results of the SMILETRAP group from CFRs of
3He
+,
3He
2+, and T
+ measured against H
2+ using the TOF technique [
79]. Here the H
2+ ions, allowing for vibrational excitation energy, were assumed to have an average mass of 2.015 101 497 03(27) u as given in reference [
34].
The second row gives the results from the FSU group of CFR measurements of
3He
+ and T
+ against HD
+ [
32]. For both HD
+/
3He
+ and HD
+/T
+, both ions were simultaneously trapped, but alternated between large and small cyclotron radii, the cyclotron frequency measurement being carried out on the inner ion using the PnP technique. The results were obtained from 34 measurements of HD
+/
3He
+ and 31 of HD
+/T
+, each lasting 8 to 10 h, resulting in a statistical uncertainty of 17 ppt for each ratio. The HD
+ mass was calculated using the CODATA2010 masses for the proton and deuteron [
80], and the very precise non-relativistic HD
+ binding energy given in reference [
81]. In addition, the CFR was corrected by 9.4 × 10
−11 to allow for the polarizability of the HD
+ molecular ion [
58,
81], with the assumption it remains in its ground rotational and vibrational level. Because the two CFRs differ from 1 by only 0.2%, most systematic errors were estimated to be negligible. However, a surprisingly large 3% imbalance was observed in the cyclotron radii, which was assumed to be due to a strong frequency dependence in the cyclotron drive transfer function. This led to a common systematic uncertainty of 45 ppt being assigned to both ratios.
The third row gives the result of a measurement of
3He
2+ against
12C
6+ and
12C
5+ by the UW group, which was reported in the same publication as the measurements on D
+ [
77]. The data consisted of 4 runs of
3He
2+ versus
12C
6+, of which two involved an extra contaminant
12C
6+ ion, and 3 of
3He
2+ versus
12C
5+, all of which are non-doublets. The error budget contains similar contributions as for the measurements on the deuteron, with a common systematic uncertainty of 9 ppt on an estimated image charge shift of 220 ppt.
Since the SMILETRAP publication [
79] and the FSU publication [
32], improved values for the mass of the proton and deuteron have been published. In the case of the SMILETRAP result, it is possible to bypass uncertainty due to the mass of H
2+, both due to the proton mass and the average excitation energy, by using their H
2+/D
+ measurement in reference [
38] to relate T
+ and
3He
+ to D
+. Hence, the latest UW result for D
+ [
77] can be used as the reference. Making these adjustments, the corrected T and
3He masses are shown in the first row of
Table 5. Further, and significantly, there has been a re-measurement of HD
+/
3He
+ by the FSU group using a substantially rebuilt apparatus, with confirmation by taking the ratio of the two CFRs
3He
+/H
3+ and HD
+/H
3+ [
61]. The result is in good agreement with the 2015 result [
32], but a factor of two more precise. Because systematic errors in both the 2017 and 2015 measurements are likely to be common, the new HD
+/
3He
+ ratio supersedes the previous one. However, the 2015 result can still be used to relate the mass of T to the new mass of
3He. Hence, using the new HD
+/
3He
+ ratio from reference [
61], the T
+/
3He
+ ratio from reference [
32], the deuteron mass from reference [
77], and the proton mass from reference [
46], we obtain the results given in the third row of
Table 5. For ease of comparison, in the second row, we have copied the UW result for
3He [
77] from
Table 4.
As can be seen from
Table 4 and
Table 5, the FSU mass of
3He, while being consistent with the lower precision SMILETRAP result, is significantly higher than the UW result. And, although allowing for the updated proton mass has reduced the discrepancy from 0.76(0.19) nu to 0.47(0.1) nu, because of the increased precision, the discrepancy is still 4 standard deviations. Additional evidence for a heavier
3He than that reported in reference [
77] is provided by referring the mass of
3He to that of the proton only, using the H
3+ ion [
56]. In this case, because the H
3+ may have significant stored rotational energy, only an upper limit to 3
M[
p] −
M[
h] can be given. However, if this is done, using the proton mass of reference [
46] as a reference, the result is
M[
3He] > 3.016 029 322 00(14) u, which is inconsistent with the UW result [
77] by (more than) 3.3(1.5) × 10
−10 u. We note that the discrepancy would be nearly removed if the FSU CFRs involving HD
+ were not corrected for the polarizability of the HD
+ ion, which would be appropriate if the HD
+ did not spend the majority of time in its ground-state during the measurement. However, this is inconsistent with the fact that the first excited rotational level of HD
+, which has a mean lifetime of 150 s, has an excitation energy corresponding to 63 K, while the trap electrodes are at 4.2 K. The AME2016 results are 0.5 nu higher than the updated FSU results because they were based on the 2015 FSU results and lacked the updated proton mass of reference [
46].
In
Table 6, we show the mass differences between atoms of tritium and helium-3 expressed in eV. It is important to note that several important systematic errors cancel for this difference, so the uncertainty is smaller than that obtained from the uncertainties in the individual masses. The first row gives this difference as measured by the UW group [
82], using an earlier apparatus than that used for the other measurements discussed in this review. (The absolute masses for
3He and T reported in reference [
82] were found to be in error by several standard deviations by the SMILETRAP group, but the relevant systematic error at least partly cancels for the T-
3He difference.) The second row gives the result from the 2006 SMILETRAP measurements [
79], and the third gives the more than a factor of 10 more precise result from the FSU group [
32]. This result is higher than the SMILETRAP result by 2.2(1.2) eV and its uncertainty is slightly less than that of the absolute calibration of the retarding voltage in KATRIN, which is now 90 meV [
70].