Recent progress in determination of fundamental constants and fundamental physics at low energies


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A brief overview on determination of the values of fundamental constants by means of atomic physics is given. Recommended values of CODATA-2010 least square adjustment are discussed as well as the related input data.

1 Introduction

Precision values of fundamental physical constants play an important role both in fundamental physics and in its practical applications. Some constants are purely macroscopic, such as the gravitational constant G, some are microscopic, such as the Planck constant h. However, while working with numerical values of physical constants, it is not so important what is the nature of the constant, microscopic or macroscopic. What is crucial is what kind of effects play the key role in the uncertainties of their determination.

We know most accurately the values which are possible to determine by means of frequency measurements of atomic or particle properties. The fundamental constants of this kind include the Rydberg constant inline image, the fine-structure constant α, the electron-to-proton mass ratio inline image, the Compton wavelength (of an electron) inline image, and some others.

Meanwhile, it happens that values of various important atomic or quantum constants, such as the Planck constant h, the elementary charge e, the Avogadro constant inline image, and the Boltzmann constant k are determined through macroscopic experiments. That is unavoidable since their values in the units of the International System of units, the SI, involve the kilogram and a certain comparison with the artefact, which has been kept at BIPM near Paris for more than a century, is in hidden form present in any determination of their values. Indeed, that requires a macroscopic experiment at a certain stage. (More accurately, that is correct for non-electrical values. Since the magnetic constant μ0, the value of which is exactly known in the SI, is a certain combination of the ampere and the kilogram, certain electrical quantities, such as the von Klitzing constant inline image, can be determined by microscopic means as far as the involvement of the kilogram can be expressed through μ0 or ε0.)

Usually, experiments with macroscopic classical objects have a substantially lower accuracy than that of pure atomic experiments. It is the latter which provide a base for verifying fundamental theories at low energies and they are briefly reviewed here. Since the accuracy of the values of the atomic constants, such as inline image, inline image, and α, is higher than the accuracy of determination of h, e, inline image, k, and G in macroscopic experiments, the determination of atomic constants is quite disentangled from macroscopic measurements and can be considered separately.

The most accurate value1 in the list of atomic constants is the one of the Rydberg constant inline image. Determination of inline image deals with spectra of atomic hydrogen and deuterium and thus is strongly correlated with our understanding of the proton structure and, in particular, with our knowledge of the rms proton charge radius inline image. The evaluation also deeply involves bound state quantum electrodynamics (QED) theory.

Next follows determination of various atomic masses in atomic units and, in particular, of the proton and electron masses in atomic units. Some of those experiments are experiments with trapped ions and particles. Some others deal with spectroscopy. Interpretation of the spectroscopic and of some ion trap measurements requires QED.

Determination of the electron mass in atomic units and the electron-to-proton mass ratio inline image has accuracy somewhat lower than that of the ‘best’ atomic masses. Accuracy of inline image is comparable presently with accuracy of determination of the fine-structure constant α. Determination of α is based on a broad variety of experiments; however, the most accurate of them requires high-accuracy QED calculations. A measurement of an atomic, nuclear, or particle mass in frequency units (inline image) or determination of the related Compton wavelength inline image has, at best, accuracy at the same level as for determination of α.

Below we discuss a determination of the values of the most important microscopic constants. As we have already mentioned, determination of most of them deeply involves quantum electrodynamics for free and bound particles and thus this activity is an application of low-energy fundamental physics. An evaluation of the world data on these fundamental constants is a kind of a cross-check of our understanding of both the fundamental background and the most advanced theoretical and experimental methods applied.

Basically we follow the CODATA-2010 least-square adjustment (LSA) results. The 2010 label means that the deadline for selection of data was the end of 2010; the evaluation took a few months and the results became available on-line at the NIST web site in early 2011; however, the full-size publication [1] appeared only in 2012.

The crucial atomic data and results of the CODATA-2010 least-square adjustment are reviewed as well as progress since 2010. We focus on the problems of the recent adjustments and on changes in comparison to the former CODATA evaluation [2]. To avoid a scatter of numerical values, in all the tables we present numerical values of the CODATA-2010 recommended values [1]. As for the plots, unless not stated explicitly otherwise, we present there input data points of the CODATA-2010 evaluation, a data point of the previously recommended value (CODATA-2006) and a band for the CODATA-2010 recommended value. More recent data, if there are any, are mentioned in the text, but they are not shown in the tables and plots.

2 Disentanglement in determinations of atomic constants

First, we note that the determination of the values of various atomic fundamental constants looks correlated. Apparently, some methods are rather straightforward and isolated from the others, such as a determination of the fine-structure constant from the anomalous magnetic moment of the electron and the electron-to-proton mass ratio from comparing their cyclotron frequencies. Meanwhile, there are many correlated determinations.

For instance, a set of equations

display math(1)


  1. a determination of the Rydberg constant inline image by combining a number of atomic hydrogen transition frequencies inline image measured and the related coefficients inline image calculated;
  2. a determination of the fine-structure constant through recoil spectroscopy, which allows one to obtain a value of inline image for certain atoms;
  3. a determination of the electron-to-proton mass ratio from a ratio rion of an ion cyclotron frequency and its spin precession frequency taking into account a theoretical correction with coefficients inline image (where Z is the nuclear charge);
  4. a determination of the electron-to-proton mass ratio derived from antiprotonic helium transition frequencies inline image with the constants c5 obtained theoretically (their values can be much smaller than unity depending on the state under investigation).

Looking more carefully, we note that actually these equations do not expose too many real correlations on the data. That is because the less accurately known constants, such as α and inline image, enter expressions for a determination of more accurate constants, such as inline image, through small additive corrections. Indeed, we measure α and inline image with a lower fractional accuracy than inline image. However, we know inline image and inline image with a smaller absolute uncertainty than inline image and eventually the uncertainties of inline image and inline image are practically negligible for a determination of inline image. Meanwhile, while determining values of α and inline image, we see that inline image enters equations as a multiplicative factor with the lowest fractional uncertainty, which in its turn can be neglected.

The only surviving correlation is the one in a determination of α and inline image. However, as we see below, it is of a somewhat reduced importance now, because the correlated values are not the most accurate values of their types. That is rather the second in accuracy values of α and inline image. Such values are crucial for reliability of the final evaluation, but as far as the data are consistent, they remain ‘beyond the scene’ and affect the LSA results marginally (see Section 5 for details).

3 Determining the Rydberg constant and the proton electric charge radius

The main source of information about the value of the Rydberg constant

display math(2)

is a study of the hydrogen and deuterium spectra. In principle, measurement of any transition in any atom with a high accuracy, accompanied by a theoretical calculation with comparable accuracy, could contribute. However, the number of atoms which are simple enough and enable a calculation with an appropriate accuracy is rather limited to two-body atomic systems and among those atoms only hydrogen and deuterium can be measured with appropriate accuracy (see, e.g., [3]).

Many transitions in gross structure of hydrogen and deuterium have been measured with a fractional uncertainty at the level of a part in 1011 [4-6] (for a complete list of transitions with relevant accuracy, see [1]).

Theory of those transitions is very complicated and involves various uncertainties. The problem has been addressed by a theoretical investigation of the n dependence of the crucial uncalculated corrections for the inline image states. The problem was discussed for the first time in [7, 8]. That allows us to present all the theoretical values in terms of two quantities to determine, namely, the 1s Lamb shift and the Rydberg constant. While the Rydberg constant (after taking into account the reduced mass corrections) is the same for hydrogen and deuterium, their Lamb shifts are correlated only in part.

A certain QED theory of the 1s Lamb shift indeed exists (see, e.g., [1]). It has a computational theoretical uncertainty and an uncertainty in determination of the value of the proton or deuteron charge radius, which cannot be calculated ab initio in contrast to pure QED calculations. Thus, a determination of the charge radii and the Rydberg constant from the world data should be done simultaneously. The results [1] are summarized in Table 1.

Table 1. The CODATA-2010 recommended values of the Rydberg constant and some related quantities; inline image is the fractional uncertainty
QuantitySymbolValueinline image
Rydberg constantinline imageinline imagem−1inline image
Proton rms charge radiusinline imageinline imagefminline image
Deuteron rms charge radiusinline imageinline imagefminline image

A set of the data to determine the Rydberg constant is formed by the gross-structure transitions in hydrogen and deuterium [4-6], by the measured values of their fine structure (inline image) [10] and the Lamb splitting (inline image) [9], various electron–proton [11, 12] and electron–deuteron [13] scattering data, and the result for the Lamb shift in muonic hydrogen [14].

The data on gross-structure transitions [4-6] by themselves allow one to find the Rydberg constant and the proton and deuteron charge radii. The rest of the data by themselves are used to determine the proton or deuteron charge radius only; however, either radius in a combination with the gross-structure data leads to a value of the Rydberg constant.

The gross-structure transitions [4-6] have been measured with a comparable accuracy for hydrogen and deuterium. The scattering data are also compatible for the proton and deuteron [11, 12]. The accuracy for the Lamb splitting and fine structure achieved for hydrogen [9, 10] has been higher than for related transitions in deuterium. Results on the Lamb shift in muonic hydrogen [14] have been published, while the muonic deuterium results are still in preparation.

Inconsistency in the world data

Unfortunately, the data are not consistent. That can be seen, e.g., from results on determination of the proton charge radius in Fig. 1. We follow there the same notation as in [1].

Figure 1.

Determination of the rms charge radius of a proton inline image. The vertical band is for the CODATA-2010 recommended value [1].

The most accurate value is deduced from the Lamb shift in muonic hydrogen measured at PSI. It disagrees with the values from hydrogen and deuterium spectroscopy (Hinline imageD point)2 and with electron–proton scattering results, obtained either from a recent MAMI experiment [12] or from the evaluations of the previous world data by Sick [11]. Those two results are consistent with each other. One more result is derived from the hydrogen–deuterium isotopic shift [5] and the deuteron radius [13] from electron–deuteron data. The latter has a somewhat lower accuracy and cannot help to resolve the controversy.

The CODATA-2010 recommended value is based on the evaluation of all the data, except for the datum from muonic hydrogen, which was dismissed from the adjustment.

The Lamb shift in hydrogen: higher-order two-loop corrections

The hydrogen and deuterium spectroscopy has the longest history among methods used to determine the Rydberg constant. The result is based on a number of experiments, but those, which are statistically dominant in the uncertainty of the determination of inline image and inline image, were performed in a single laboratory [6].

The computational theoretical uncertainty for the 1s Lamb shift is substantially smaller than the experimental one; however, there is a controversy in the calculations of a so-called B60 coefficient for the 1s Lamb shift (see [1] for details on notation):

display math(3)
display math(4)

The nature of theoretical errors is such that after the correction the new result can be completely different from the previous one. Because of that, the controversy of the two theoretical calculations should be clarified; even so, the departure is small in absolute units.

Meanwhile, the contradicting results are correlated. While a direct calculation of B60 is presented in [15], the other result comes from the extrapolation of numerical calculations [17]. The numerical evaluation [17] is done without expansion in inline image, where Z is the nuclear charge in a hydrogen-like atom. Unfortunately, it is not possible to perform the calculation directly for hydrogen (inline image). Instead it is done for some higher values. The B60 contribution is a certain term in the inline image expansion. To perform an extrapolation to inline image or, which is about the same, to restore the inline image expansion at low Z, is directly not possible with a required accuracy. To successfully determine B60, one has to input all known information on lower-order coefficients, such as B40, B50 [18, 19], B63 [20], B62 [16], and B61 [21]. The calculation [15] is correlated with the calculation of B62 [16] and B61 [16, 21], which eventually makes the B60 values in Eqs. (3) and (4) correlated.

Electron–proton scattering data: fitting procedure

Electron–proton scattering plays an important role in a determination of the Rydberg constant. Not only the value of the proton electric charge radius is crucial. For a theoretical evaluation of the Lamb shift in muonic hydrogen, one has to take into account higher-order effects of proton structure. (Similar effects are also important for the Lamb shift in ordinary and muonic deuterium.)

Scattering does not deliver directly the value of inline image (or any other single value of interest). Instead, cross sections are measured. They are recorded as functions of two parameters, the momentum transfer and the scattering angle. Fitting the data, one can extract the values of interest, e.g., the electric and magnetic radii. The results of their determination by various means are summarized in Fig. 2.

Figure 2.

Determination of the rms electric (charge) and magnetic radii of a proton. In contrast to Fig. 1, we denote here the proton radii as inline image and inline image in order to distinguish the electric and magnetic values. A certain correlation in the determination of two radii from the same scattering data set is possible but seems marginal. The magnetic radius of the proton has not been considered in the CODATA-2010 evaluation.

We see that, despite the fact that the results of the evaluation of the scattering data produce a consistent value of the charge radius, they disagree on the magnetic one. As followed from the exchange [22, 23], the data are rather consistent, and the difference in the results should be attributed to a different treatment of the two-photon contributions. That opens a question of the reliability and the model independence of the results of various evaluations.

Muonic hydrogen Lamb shift

In contrast to hydrogen spectroscopy and electron–proton scattering, the method of determination of the proton radius from the Lamb shift in muonic hydrogen is a brand new one in terms of results achieved. After decades of experimental efforts the Lamb shift has been measured for the first time only recently [14].

Usually, when such a new result contradicts a few results, obtained by traditional methods, we should rather doubt it. However, the doubts should be ‘measured’ in terms of the value of the standard uncertainty. While the contradiction of the Hinline imageD value and the MAMI, Scat ones from the muonic value is about five times their standard deviations for each, the departure of the μH value from them is about 100 standard deviations for the muonic Lamb shift experiment and about 70 theoretical ones [14]. Briefly speaking, the spectroscopic discrepancy is a small portion of the line width for ordinary hydrogen and a few line widths for the muonic one. That makes the muonic value rather reliable at the level of discrepancy.

A certain theoretical progress took place recently [25-28]. However, the contributions under question, although important for pure theoretical reasons, have a marginal effect on a final numerical value, which is rather well established.

Progress on the experimental side is a publication of the second muonic hydrogen transition [24]. Following CODATA-2010 analysis we have not included in our presentation the results obtained after 2010. Two muonic hydrogen values are quite consistent with each other, while the difference between former [14] and new [24] results is marginal at the level of discrepancy.

4 Weighing atoms, nuclei, and particles in the atomic mass units

As we mentioned above, speaking about the values of fundamental constants one has to distinguish between microscopic and macroscopic units. That is essentially important while speaking about the mass of atoms, nuclei, and particles (see, e.g., [29]). The most accurate values of the masses for microscopic objects can be indeed achieved in atomic mass units and some results on a determination of atomic, nuclear, and particle masses are collected in Table 2.

Table 2. Some atomic masses, used as input data for CODATA-2010 adjustment, and the recommended values of particle and nuclear masses in atomic units [1]
QuantitySymbolValueinline image
Atomic mass of 16Om(16O)inline imageuinline image
Atomic mass of 87Rbm(87Rb)inline imageuinline image
Proton massinline imageinline imageuinline image
Deuteron massinline imageinline imageuinline image
Triton massinline imageinline imageuinline image
Helion massinline imageinline imageuinline image
Alpha particle massinline imageinline imageuinline image
Electron massinline imageinline imageinline image

The atomic mass of 16O is an important ingredient in finding a value of inline image (see Section 5 for details); it was obtained in [30]. The atomic mass of 87Rb, measured in [31], is required for a determination of α by means of recoil spectroscopy [32] (see Section 5 for details).

In principle, the original results [30] and [31] should be used as input data and the final values are to result from the adjustment. However, the adjustment can affect those values only marginally. The evaluation of masses is performed by the Atomic mass data center. The previous atomic mass adjustment (AME) was published in 2003 [33] and some more recent data such as masses of oxygen and rubidium were taken in [1] from the original papers. After the CODATA-2010 adjustment was completed and published [1], new AME-2012 results appeared [34].

Since the proton mass in atomic units is determined much more accurately than the electron mass, there is no difference in determination of inline image and inline image. Indeed, the proton-to-electron mass ratio is one of the most important atomic constants related to the mass. Is it known with a fractional accuracy inline image and it plays an important role in determination of the fine-structure constant. For this reason we consider it in the next section due to determination of α.

5 Determining the fine-structure constant and related values

The fine-structure constant

display math(5)

can be determined by many different means. A summary of different methods in plotted in Fig. 3.

Figure 3.

Precision determination of the fine-structure constant α [1]. The vertical band is for the CODATA-2010 recommended value.

The determination includes microscopic and macroscopic experiments. The macroscopic results are based on electrical measurements.

  • RK Determination through the quantum Hall effect involves a comparison of the quantum Hall resistance with an impedance of such a macroscopic classical device as a calculable capacitor.
  • inline image Gyromagnetic ratio of a proton in water, measured in presence of a low magnetic field, assumes a calculation of the strength of that field. The electrical current, applied to induce the field, is calibrated versus quantum electrical standards, such as quantum Hall resistance and Josephson-junction voltage standard.

Details on laboratories, where these experiments were performed, can be found in [1]. Macroscopic experiments are in general more vulnerable to various systematic effects. More accurate results are obtained by the microscopic methods.

  • inline image The hyperfine splitting of the ground state in muonium, a bound system of a muon and an electron, allows one a precision test of bound-state QED. The developed theory (see, e.g., [3, 39] and references therein) is more advanced that that for the hyperfine structure in hydrogen, which, as a QED property, is ‘contaminated’ by the proton-structure effects. Besides, a muon is nine times lighter than a proton, which makes the recoil effect enhanced. Unfortunately, at the present time this method is hardly compatible for α, but it produces the best value of the electron-to-muon mass ratio.
  • He, fs The helium fine structure provides us with a precision QED theory of three-body systems, but the related α value [40-42] is also not competitive with the most accurate ones.
  • inline image The recoil Raman spectroscopy allows one to determine the value of inline image for certain atoms. That is one of the two most precise methods of α determination.
  • inline image The most accurate value is from the anomalous magnetic moment of an electron.

Summarizing the list of methods, we have to mention that for a pure α determination only the last two methods are important. The related results for CODATA-2006 and CODATA-2010 are presented in Fig. 4 and discussed below in detail. The results on determination of α and related constants are collected in Table 3.

Figure 4.

Most accurate results for α determination for CODATA-2006 (left) [2] and CODATA-2010 (right) [1] adjustments. The vertical band is for the related recommended values.

Table 3. Recommended values of α-related fundamental constants
QuantitySymbolValueinline image
Inverse fine-structure constantinline imageinline imageinline image
Molar Planck constantinline imageinline imageinline image
Compton wavelengthinline imageinline imageminline image
von Klitzing constantinline imageinline imageinline image
Muon–electron mass ratioinline image206.768 2843(52)inline image
Muon–proton magnetic moment ratioinline imageinline imageinline image

All other values, except for the helium one, were nevertheless used as input data in [1]. The electric values allow us to check self-consistency of quantum electric standards, which are important for determination of other fundamental constants, such as h, e, and inline image. The muonium value is our main key for determination of inline image and inline image. The helium value, which is one of the best tests of three-body QED calculations, is of no direct importance for a determination of the values of any other fundamental constant, except for the fact that another three-body system, antiprotonic helium, studied in part by the same theoretical methods, is important for a determination of inline image (see Section 5 for details).

Still, only a few data pieces are important to determine α. The results with relevant accuracy are obtained either by means of Raman scattering recoil spectroscopy or by studying the anomalous magnetic moment of an electron.

A substantial progress took place since the CODATA-2006 adjustment in both methods (see Fig. 4).

The recoil values presented in the 2006 evaluation were by experiments on caesium [43] and rubidium [44]. The accuracy of the new rubidium result [32] is substantially higher (inline image). To obtain a value of the fine-structure constant, one has to rely on the identity

display math(6)

The Raman-scattering experiments are designed to determine inline image, while the other constants, inline image, mRb[u], and inline image, are known with a sufficient accuracy, as discussed above. Here, inline image stands for the electron mass in atomic mass units.

The progress on the anomalous magnetic moment of an electron consists of two parts. On the experimental side, there are two measurements (from the University of Washington (UW) [45] and from Harvard University). After a long domination of the UW value, already in the CODATA-2006 adjustment [2], a new and more accurate value from Harvard appeared [46]. Since that time the experiment delivered even a more accurate value [47], which dominates in the 2010 evaluation.

Meanwhile, the anomalous magnetic moment of an electron, which is in the first approximation equal to inline image, requires indeed a very advanced and complicated theory for higher-order QED contributions. The 2006 value of α was based on [48]. Later on, an error in numerical calculations was discovered and the theory was corrected and improved [49].

The situation can be clearly seen in the right-hand plot in Fig. 4. Roughly speaking, the CODATA-2010 value (the grey belt) is equal to the Harvard value with the corrected theory, while the former CODATA value (CODATA-2006) is based on the earlier version of the Harvard experiment and the un-corrected theory. We note that the former result on the Raman spectroscopy [43, 44] does not really distinguish between those two inline image results, while the new rubidium result [32] does. In other words, the new rubidium Raman spectroscopy result has a sufficient accuracy to check QED theory at the fourth order of magnitude.

Meanwhile, the updated theory (after the 2010 evaluation) includes now the fifth-order contributions [50], which is important theoretical progress.

Measuring α and ratio inline image

Above, we discussed the chain of values in the identity (6), which produced a value of the fine-structure constant α from the rubidium Compton wavelength inline image [32], electron [37, 38] and rubidium-87 [31] masses in atomic mass units, and the Rydgerg constant.

We can rearrange the identity and read it in a different way. Combining the rubidium Compton wavelength inline image [32], the fine-structure constant α form inline image [47, 49], the rubidium mass in atomic mass units [31], and the Rydberg constant, one finds the new value of the electron-to-proton mass ratio

display math(7)

A summary of the determination of inline image is given in Fig. 5. All the results are consistent with each other.

Figure 5.

Determination of the proton-to-electron mass ratio inline image [1]. The vertical band is for the CODATA-2010 recommended value inline image.

The results are obtained by four different methods. We follow [1] in the related labeling.

  • The earliest result in Fig. 5 is from the University of Washington; it was obtained by a comparison of the electron and proton cyclotron frequencies [35]. In contrast to the other methods, this result is more or less straightforward in a theoretical sense.
  • The UW result was dominant until a new method based on comparison of the ion cyclotron frequency and the spin precession one for a hydrogen-like ion with a spinless nucleus was realized. The first accurate inline image result was obtained for carbon-12 [36]. The input data used in CODATA-2010 evaluations originate from carbon-12 [37] and oxygen-16 [38] experiments, performed at the University of Mainz by an UM-GSI team3. They need a sophisticated QED theory [52].
  • A result from the antiprotonic helium, a three-body atomic system, which consists of a nucleus, a helion or an alpha particle depending on the isotope, an antiproton, and an electron, allows one to find a mass ratio for an antiproton and an electron. Precision measurements [53, 54], performed by the ASACUSA collaboration at CERN, and three-body QED calculations [55] deliver us a competitive result.
  • One more result, the one from rubidium, has been already discussed in the text. It requires a number of precision measurements and still needs a QED theory for the anomalous magnetic moment of an electron.

All the results are consistent. Experimentally they are completely independent. From the theoretical side, they are independent as well. Indeed, in three of the four methods we rely on QED; however, from a practical point of view, the crucial points are different. In the case of the anomalous magnetic moment we have to deal with QED of a free particle at the four-loop (and now at the five-loop) level. Two other problems are for bound-state QED. In the case of hydrogen-like ions, it is important to calculate two-loop diagrams only; however, it is necessary to do that without inline image expansion. In the near future, it will be necessary to deal exactly in inline image with three-loop diagrams. The three-body problem, in contrast to the two-body one, does not allow any ‘good’ first approximation to be obtained analytically. One has to start from a problem of minimization of the energy by numerical means. Thence, all the crucial details in these three QED calculations are completely different and independent.

Measuring atomic masses in the frequency units

The interpretation of the Raman-spectroscopy value of α has shown that the determination of α is closely related to a measurement of any mass in the frequency units. In its turn, in order to obtain a numerical value of a certain mass m in the frequency units, it is sufficient to find its ratio to the electron mass and further to apply values of α and inline image for the conversion

display math

Instead of α we can use another conversion factor, the so-called molar Planck constant inline image. It is clear that inline image is a conversion factor from the atomic mass units to kilograms, while the inline image value is to convert from the kilograms to the frequency units. Thence, the accuracy in the numerical value of inline image is determined by the accuracy of α2 and inline image. We recall that the fractional uncertainty of the conversion factor from atomic mass units to hertz, inline image, is inline image.

In contrast to the atomic mass units, the frequency units are not used frequently. In fact, in the CODATA paper [1], the masses of the most important particles and nuclei are given in various units, but not in the frequency ones. More often the energy units are applied and, in particular, a non-SI unit, such as the electronvolt. To convert hertz to electronvolts, one has to find an SI value of inline image. This ratio tells us about a comparison of the energies of a photon with a given frequency and an electron at a given difference of electric potentials. Dealing with the energy we always need to rely on the kilogram's artefact at BIPM. If we really need it, the measurement is unavoidably of macroscopic nature and cannot be done by means of atomic physics only. Despite the microscopic area of the application, the units, eV for the energy and eV/c2 for the mass, are units of macroscopic nature (see [29] for details). For the electron or proton masses, those units do not allow a sufficient accuracy. The conversion factor from inline image is known [1] with a fractional uncertainty of inline image and no masses in eV/c2 can be known better. That is to be compared with a fractional uncertainty of inline image in the value of h, which determines the accuracy of atomic and particle masses in kilograms.

6 Concluding remarks

Progress in the determination of the values of fundamental constants is summarized in Fig. 6. The atomic constants (inline image, inline image, α, inline image, and inline image) as we discussed in the introduction are much more accurate than macroscopically determined quantities (h, inline image, k, and G) and the accuracy in their determination is substantially higher than that for constants measured in macroscopic experiments.

Figure 6.

Fractional uncertainty of CODATA recommended values of fundamental constants [1, 2, 56-59].

A comparison of the CODATA-2006 and the CODATA-2010 values for the key atomic constants in Table 4 shows that accuracy was somewhat improved. It has also demonstrated that the reliability of the recommended values is not beyond question, since there was a significant jump in the value of the fine-structure constant. Notably a currently good-looking situation with the Rydberg constant is a result of ignoring the muonic hydrogen experiment, which strongly disagrees with the rest of the data.

Table 4. Recent progress in a determination of the values of atomic fundamental constants in 2006–2010 as follows from comparison of the CODATA-2006 and CODATA-2010 evaluations [1, 2]. Here, inline image is a fractional shift of the value A: inline image inline image
Quantityinline imageΔinline imageinline imageinline image
inline imageinline imageinline image0.17inline image0.76
inline imageinline imageinline image0.03inline image0.95
αinline imageinline image6.50inline image0.47

In general, we see significant progress in the last decade due to establishing and improving new methods for inline image, α, and the proton radius (and consequently for the Rydberg constant); however, certain problems remain. One of the problems is the reliability of the α determination, since the second most accurate method, Raman spectroscopy, is still too far behind the most accurate one, based on the anomalous magnetic moment of an electron. We need a tool to cross-check. The other problem is a puzzle with the proton radius.

As for the test of fundamental theories, such as QED, we concluded that most accurate atomic physics data produce a rather consistent picture. Experiment and theory of muonium hyperfine structure, helium fine structure, gross-structure transitions in antiprotonic helium, g-factor of a bound electron in a hydrogen-like ion, and anomalous magnetic moment of an electron are in perfect agreement. A certain 5-sigma disagreement between results for hydrogen and muonic hydrogen is rather a motivation to improve experimental and theoretical techniques than to change theory.

A zoo of useful simple atoms has been broadened. A certain new trend of applications of a QED system for a determination of fundamental constants is a successful use of three-body atoms, such as ordinary and antiprotonic helium. The latter is an exotic atom and we have to acknowledge the role played by exotic atoms, also including muonium and muonic hydrogen, in precision measurements to determine fundamental constants. While in theory ‘simple’ atoms become more and more complex, in experiments exotic systems become less and less exotic. Eventually that builds a broadened base for the precision QED tests.


The author is a member of the CODATA task group on fundamental physical constants and is grateful to his colleagues for fruitful discussions. The data are officially recommended by the group and published in [1]; the interpretation of certain details does not necessarily coincide with the group's opinion and presents a personal author's view. The work was supported in part by DFG.

  1. 1

    Indeed, it is suggested that within the SI system certain numerical values, such as that of the speed of light, c, are known exactly, as follows from the definition of the related units. Those adopted values are not a subject of any experimental determination.

  2. 2

    The CODATA-2006 value [2] is basically the same as the Hinline imageD one, since spectroscopy was dominant then and the progress in the hydrogen and deuterium spectroscopy since 2006 was marginal.

  3. 3

    There is a systematic effect due to the interaction with image charges, which shifts the bound-electron g-factor of carbon [37] and oxygen [38]. Recently [51] it was realized that this shift, already mentioned in [37], was not considered in [38]. Eventually, that shifts up the proton-electron mass ratio by approximately, but somewhat below, one standard deviation and does not break the overall consistency of the data in Fig. 5.


  • Image of creator

    Savely G. Karshenboim graduated from Leningrad State Univestity in 1983, got his Ph. D. (1992) and Dr. Sc. (1999) degree from St. Petersburg State University. At present days he works for Pulkovo Observatory (St. Petersburg, Russia) and Max-Planck-Institut für Quantenoptik (Garching, Germany). He is a member of CODATA task group on fundamental constants.