## Abstract

The Committee on Data for Science and Technology has recently recommended a new self-consistent set of values of basic constants and conversion factors of physics and chemistry. These values are based on a least-squares analysis that takes into account all of the latest relevant experimental and theoretical information in a consistent framework. Theory plays a role, because the experimental data are compared to the corresponding theoretical predictions which are functions of the fundamental constants. The best values of the constants are taken to be those that give the best agreement between the data and these predictions, in the least-squares sense. An overview of the calculations that influence the recommended values of the constants will be given.

## 1. Introduction

The fundamental constants appear as parameters in theoretical expressions for measurable physical quantities. In order to test a theory, one generally evaluates the theoretical expression for some quantity with constants from a table and compares the result to the experimentally measured value. However, in order to create such a table, we reverse the procedure by finding the values of constants for which the theory and experiment agree. The values are obtained by a least-squares adjustment. This process entails assumptions about the validity of the theoretical expressions.

The discussion in this paper is based on the 2002 Committee on Data for Science and Technology (CODATA) least-squares adjustment of the fundamental constants (Mohr & Taylor 2005). For the sake of brevity, the many citations to the original theoretical calculations are omitted from this paper; the reader is referred to Mohr & Taylor (2000, 2005) for a detailed account.

A number of questions to be addressed in this paper are the following:

How does theory enter into the CODATA evaluation of the fundamental constants?

What are the most questionable theoretical assumptions used in the CODATA evaluation of the fundamental constants?

To what extent do the values of the constants depend on these assumptions?

How well are these assumptions checked by the consistency of the results for the fundamental constants?

What theoretical assumptions might be called into question by the discrepancy between the measurement of the Avogadro constant and the measurement of the Planck constant?

## 2. Fundamental constants

Some of the fundamental constants relevant for the present discussion are:

Newtonian constant of gravitation:

Avogadro constant:

electron mass:

Planck constant:

fine-structure constant:

electron mass (in u):

Rydberg constant:These constants are listed in order of decreasing relative uncertainty, which is given in square brackets. The Newtonian constant of gravitation is substantially less precise than the rest given here, due to the difficulty of carrying out this kind of experiment. The next three listed, the Avogadro constant, the electron mass and the Planck constant, all have essentially the same relative uncertainty. This is not a coincidence, as will be shown below. The fine-structure constant has a nearly two order-of-magnitude smaller relative uncertainty than any of the other constants given above in the list. The value of the electron mass in atomic mass units u, is significantly more precise than the value of the electron mass in kilograms. This is a reflection of the fact that the accuracy of a fundamental constant can depend on the units in which it is expressed. Physically, the difference is due to a greater difficulty of relating the mass of an electron to a one kilogram mass, as compared to the difficulty of relating the mass of an electron to the mass of an atom. The Rydberg constant, which has a substantially smaller relative uncertainty than the rest, is determined by precise frequency metrology on hydrogen and deuterium atoms.

## 3. Least-squares adjustment overview

The least-squares adjustment is summarized by Aitken (1934)(3.1)where * Q* is a vector of input data,

*is a vector of constants to be adjusted and*

**Z***is a vector of functional relations between the data and the constants. The dot over the equals sign is meant to indicate that the equation is only satisfied in the least-squares sense. In general, the two sides are not equal, because the set of equations is overdetermined.*

**F**The adjustment is made by minimizing(3.2)where *V*=cov(*Q*) is the covariance matrix of the input data * Q*.

If is the solution for the minimum, then the ‘optimal’ prediction for the input data quantities is(3.3)The predicted values are optimal in the sense that the sum of the squares of the uncertainties is a minimum.

### (a) Input data for the adjustment

In the 2002 CODATA least-squares adjustment, a total of 113 items of data were included. Of these, 83 are measured values with 13 redundant measurements. Examples of the various types of measurements are:

hydrogen transition frequencies

electron anomalous magnetic moment

muon anomalous magnetic moment

relative atomic masses of H, D,

^{4}He,^{3}Hebound particle magnetic moments

silicon lattice spacings (absolute and relative)

neutron diffraction wavelength (relative to silicon lattice spacing)

Josephson constant

von Klitzing constant

Faraday Constant

molar gas constant

The remaining 30 items of input data are estimated uncertainties of the theoretical expressions, as described in §4.

### (b) Theoretical relations between data and constants

The data, such as those listed in §2, are related to the fundamental constants through theoretical expressions that are functions of the constants. As an example, for the Josephson constant, one item of input data is (Clothier *et al*. 1989)(3.4)and the corresponding theoretical expression is(3.5)where *μ*_{0} is the magnetic constant and *c* is the speed of light in vacuum.

Another example is the transition frequency *ν*_{H}(1S_{1/2}–2S_{1/2}) in the hydrogen atom. In this case, the input datum is (Niering *et al*. 2000)(3.6)and the theoretical expression is(3.7)where(3.8)(3.9)(3.10)In equation (3.7), *m*_{p} is the mass of the proton, *R*_{p} is the proton rms charge radius and is the Compton wavelength of the electron divided by 2*π*. The quantities *δ*_{2S} and *δ*_{1S} are the uncertainties associated with the theoretical expressions for the 2S and 1S energy levels. The estimated values for *δ*_{2S} and *δ*_{1S} are entered as input data in the least-squares adjustment, and *δ*_{2S} and *δ*_{1S} are taken to be adjusted constants. These uncertainties are highly correlated as shown by the value of their correlation coefficient given by equation (3.10).

Other theoretical expressions used in the least-squares adjustment include the relation between the relative atomic mass of the hydrogen atom *A*_{r}(^{1}H) and the relative atomic mass of the proton *A*_{r}(p) and electron *A*_{r}(e) and the electron binding energy *E*_{b}(^{1}H); *m*_{u} is the mass associated with the relative atomic mass unit(3.11)the relation between the electron magnetic moment anomaly *a*_{e} and *α*(3.12)where is calculated from quantum electrodynamics (QED), *a*_{e}(had) and *a*_{e}(weak) follow from the standard model and electroweak theory, respectively; the relation between the von Klitzing constant *R*_{K} and *α*(3.13)the relation between the combination of constants measured in the watt balance experiments (Kibble *et al*. 1990; Williams *et al*. 1998) and *h*(3.14)and the relation between the ratio of *h* to the product of the neutron mass *m*_{n} and the silicon lattice spacing *d*_{220}(W04) (Krüger *et al*. 1999) and the relevant combination of constants(3.15)where *A*_{r}(n) is the relative atomic mass of the neutron.

### (c) Adjusted constants

Some of the 62 adjusted constants in the 2002 CODATA least-squares adjustment are:

electron relative atomic mass:

*A*_{r}(e)proton relative atomic mass:

*A*_{r}(p)neutron relative atomic mass:

*A*_{r}(n)alpha particle relative atomic mass:

*A*_{r}(α)electron–muon mass ratio:

*m*_{e}/*m*_{μ}fine-structure constant:

*α*Planck constant:

*h*Rydberg constant:

*R*_{∞}proton rms charge radius:

*R*_{p}molar gas constant:

*R*Newtonian constant of gravitation:

*G*

Also included as adjusted constants are the estimated theoretical uncertainties, such as *δ*_{1S} mentioned above.

### (d) Other quantities

Fundamental constants not included as adjusted variables in the least-squares adjustment are derived from the adjusted constants by using theoretical identities. For example, the electron charge *e* follows from *α*(3.16)where *ϵ*_{0} is the electric constant and . For the electron mass, one has(3.17)

Similarly, for the Avogadro constant(3.18)where *m*_{u}=*m*(^{12}C)/12 is the atomic mass constant and *m*(^{12}C) is the mass of the carbon 12 atom.

### (e) Consistency of the relations

It is of interest to consider the consistency of the various relations between the data and constants by comparing the values for a particular constant obtained from different sources of information. In the case of the Planck constant *h*, the values implied by equations (3.5), (3.14) and (3.18), among others, are compared in figure 1. In that figure, the molar volume of silicon *V*_{m}(Si) indicates the value of *h* that follows from its value recommended by the Working Group on the Avogadro constant of the CIPM Consultative Committee for Mass and Related Quantities (2003, private communication). That value, together with the relatively accurately known lattice spacing in silicon, yields a value for the Avogadro constant, which in turn yields a value for *h* from equation (3.18). These equations are interpreted as a consistency check on the values of *h*, rather than the rest of the constants that may appear in them, because the other constants have significantly smaller uncertainties and are therefore assumed to be relatively well known. In particular, the other constants are *α*, *A*_{r}(e) and *R*_{∞}. In figure 1 there is an apparent discrepancy between the value of *h* obtained from *V*_{m}(Si) and the values obtained from *K*_{J} and from . In view of this difference, a review of the assumptions and theoretical relations that are needed to obtain the values of *α*, *A*_{r}(e) and *R*_{∞}, which are necessary in order to compare the values of *h*, is warranted and is given in the following.

## 4. Theoretical assumptions

The various areas of physics that play a role in the determination of the values of the fundamental constants can, for the present purpose, be roughly described by the following list:

classical mechanics

thermodynamics

classical electromagnetism

quantum mechanics

condensed matter theory relations for

*K*_{J}and*R*_{K}relativistic kinematics

QED

electroweak theory

Standard Model of particle physics

This categorization is motivated by the theoretical analysis of the various experiments considered in the least-squares adjustment. In the following, the primary information that yields values of *α*, *A*_{r}(e) and *R*_{∞} is reviewed.

### (a) Electron anomalous magnetic moment *a*_{e}

The deviation of the electron magnetic moment from the value predicted by the Dirac equation *g*_{e}(Dirac)=−2 is given in terms of the electron magnetic moment anomaly *a*_{e} by *g*_{e}=−2(1+*a*_{e}). The anomaly has been measured for the electron and positron with the results (Van Dyck *et al*. 1987)(4.1)

(4.2)The average of these values yields the input datum for the least-squares adjustment corresponding to *a*_{e}. The theoretical expression for *a*_{e} is(4.3)which gives the anomaly as a function of the variables *α* and *δ*_{e}. The term *δ*_{e} represents the total uncertainty of the theoretical expression.

The coefficients , the strong interaction correction *a*_{e}(had) and the weak interaction correction *a*_{e}(weak) are also calculated from theory. In particular, the coefficients are based entirely on QED. The largest uncertainty in the theory arises from numerical integration uncertainty in the massive calculation of (T. Kinoshita 2002, private communication) and an uncertainty from the essentially unknown coefficient , for a total theoretical uncertainty of *u*(δ_{e})=1.15×10^{−12}.

The 2002 recommended value of *α* is mainly derived from *a*_{e}.

### (b) ^{133}Cs-recoil photon frequency shift

The atomic recoil frequency shift of photons absorbed and emitted by caesium atoms has been measured to determine the quotient *h*/*m*(^{133}Cs), where *m*(^{133}Cs) is the mass of the cesium 133 atom. If a photon of frequency *ν*_{1} is absorbed by an atom initially at rest, part of the energy of the photon will go into the kinetic energy of the atom, which recoils from the momentum of the photon. Similarly if a photon is emitted by stimulated emission, part of the energy of the transition in the atom will be converted into kinetic energy of the atom, with the result that the photon will be emitted with its frequency red-shifted relative to the frequency associated with the energy difference of the atom. These processes are depicted schematically in figure 2.

Conservation of energy and momentum applied to the combined processes in figure 2 yields(4.4)where(4.5)(4.6)In equation (4.5), the quantity Δ*ν*_{Cs} is the frequency shift due to recoil, and *m*_{1} and *m*_{2} are the two atomic masses corresponding to the two hyperfine levels of the atom used in the experiment. The fine-structure constant can be determined from the relations above together with the definition of the Rydberg constant (see equation (3.17)) by writing(4.7)

The schematic description given above is not sufficient for a realistic experiment, because the atoms are not initially at rest, although the basic formulation is still valid. The photon recoil frequency shift has been measured by Wicht *et al*. (2002), who utilized atom interferometry to make a precision measurement. The result of the measurement is(4.8)

An accurate value of the effective atomic transition frequency is given by (Udem *et al*. 1999),(4.9)and the relative atomic mass of caesium is given by (Bradley *et al*. 1999)(4.10)The values of the fine-structure constant *α* derived from the electron anomalous magnetic moment measurement and the atomic recoil measurement are included in figure 3. While the most accurate value for *α* is given by comparison of theory and experiment for the anomalous magnetic moment of the electron, it is clear from the figure that the value derived from the cesium recoil experiment is in agreement and still sufficiently accurate to provide confirmation of the value of *α* at a slightly lower level of precision. This can be considered a completely independent confirmation, because the recoil experiment is based simply on kinematics together with atom interferometry, and does not rely on QED.

### (c) *g*-factor of ^{12}C^{5+} and ^{16}O^{7+}

The value of the relative atomic mass of the electron *A*_{r}(e) with the smallest uncertainty follows from high-precision Penning trap measurements of frequency ratios for hydrogenic ions (Beier *et al*. 2002; Verdú *et al*. 2002, 2003; Häffner *et al*. 2003; G. Werth, private communication). The measured ratios are(4.11)(4.12)where in the case of carbon, for example, the frequencies are the spin-flip frequency, given by(4.13)and the cyclotron frequency, given by(4.14)In equation (4.13), *g*_{e}(^{12}C^{5+}) is the bound-electron *g*-factor in the carbon ion. The magnetic flux density *B* drops out of the ratio, which is(4.15)where *A*_{r}(^{12}C^{5+}) is the relative atomic mass of the carbon ion which can be accurately related to the neutral carbon relative atomic mass *A*_{r}(^{12}C)=12. Evidently from this equation, the relative atomic mass of the electron can be determined from the frequency ratio, provided the *g*-factor is sufficiently well known.

In fact, the *g*-factor of ^{12}C^{5+} and ^{16}O^{7+} can be accurately calculated from theory. The calculation is formulated by writing the energy splitting between the two spin states of the carbon ion in a magnetic flux density as(4.16)where *μ*_{B} is the Bohr magneton. The *g*-factor can be written in terms of the various contributions as(4.17)where *g*_{D} is the Dirac equation contribution, Δ*g*_{rad} is the radiative correction, Δ*g*_{rec} is the recoil contribution, Δ*g*_{ns} is the nuclear size contribution and the dots represent smaller uncalculated terms. The numerical values of these various terms are given in table 1.

It is clear from the table that the value of the *g*-factor is strongly dependent on theory. In fact, the leading term requires a relativistic formulation, because it is zero in the Schrödinger approximation. The QED contribution, termed the radiative correction, is quite significant, so this may be regarded as a QED-dependent result, which yields the most accurate value for the relative atomic mass of the electron(4.18)On the other hand, the second most accurate value for *A*_{r}(e) is obtained by a direct comparison of the electron and carbon nucleus masses in a trap, which is based on classical electromagnetism. This result is (Farnham *et al*. 1995)(4.19)which has a factor of four larger uncertainty.

### (d) Rydberg constant *R*_{∞}

The Rydberg constant is determined primarily by comparison of theoretical and experimental values of energy levels of hydrogen and deuterium. For example, the theoretical expression for the 1S–2S transition frequency of hydrogen is given approximately by the expression(4.20)where the terms beyond 1 are corrections due to reduced mass, relativity, QED, and the finite charge radius of the proton, respectively. There are numerous higher-order terms necessary for the theoretical value to have a precision comparable to the experimental value which are not shown here. Equating the measured value for this transition (Niering *et al*. 2000)(4.21)to the theoretical expression fixes only the particular combination of these constants that appears in the equation, while additional input data from other transitions further specify the individual values of these constants.

In the 2002 adjustment, the 23 transition frequencies or frequency differences in hydrogen or deuterium listed in table 2 were included, as in the 1998 adjustment. However, the 2002 theory is based on new results for the one- and two-photon QED contributions, with a sufficient improvement in precision that it was possible to recommend a value of the proton charge radius(4.22)which has a smaller uncertainty than the electron-scattering value *R*_{p}=0.895(18) fm (Sick 2003). A value for the deuteron radius *R*_{d} is also determined by the least-squares analysis.

The theoretical values generated by the 2002 CODATA least-squares adjustment are compared to the measured values in figure 4. The vertical bar at the right-hand end of the graph indicates the uncertainty of the Rydberg constant determined by the adjustment. The uncertainty is larger than might be expected from the rest of the data in the graph, because it includes uncertainty from the charge radii of the proton and deuteron. If the relative uncertainty of the proton charge radius were reduced to 0.001, as may be expected from an experiment on muonic hydrogen underway at the Paul Scherrer Institute in Switzerland, then the uncertainty of the Rydberg constant would be smaller by a factor of about six.

## 5. Check of *K*_{J} and *R*_{K} relations

Unlike the experiment that measures the Avogadro constant by the crystal density method, the watt balance experiment does not directly measure the Planck constant. Rather, it measures the combination of constants which is equal to 4/*h*, if the standard relations between the Josephson and von Klitzing constants, and the constants *e* and *h*, namely *K*_{J}=2*e*/*h* and *R*_{K}=*h*/*e*^{2}, are valid. While there is every reason, both theoretically and experimentally, to believe these relations are valid, this validity ultimately rests on the consistency of the results based on this assumption. Since we are faced with an apparent inconsistency, it is worth rechecking if relaxing the assumption that these relations are valid improves the situation.

Such an investigation was done as part of the 2002 CODATA adjustment of the constants. It was implemented by carrying out extra adjustments in which the standard relations were not assumed to be true and introducing two new adjusted constants *ϵ*_{J} and *ϵ*_{K} defined by(5.1)

(5.2)

As a result of these generalizations, seven theoretical relations between the input data and the adjusted constants are modified. For example, equation (3.14), the relation relevant to the watt balance experiment, is replaced by the relation(5.3)

The results of the test least-squares adjustments are given in table 3, where eight adjustments are listed. For each adjustment, the second column gives the Birge ratio, *R*_{B}, the third and fourth columns give *ϵ*_{K} and *ϵ*_{J}, and the fifth to seventh columns give the normalized residuals for the three items of data *q*_{i} listed in the header. Here is the estimated value of the input datum *q*_{i} resulting from the least squares adjustment (see equation (3.3)) and *u*(*q*_{i}) is the standard uncertainty of *q*_{i}. The three items of data are those for which the absolute value of *r*_{i} is the largest. In the *ϵ*_{K} and *ϵ*_{J} columns, the entry 0(0) indicates that the corresponding variable was fixed to be exactly zero, which amounts to imposing the conventional relations to *e* and *h*. In the adjustments number (v) to (viii), the first of the three items of data was not included, as in the final 2002 adjustment. From the column of residuals for *V*_{m}(Si), which determines a value for the Avogadro constant, it is clear that allowing *ϵ*_{K} and *ϵ*_{J} to vary freely does not significantly reduce the discrepancy between *V*_{m}(Si) and the rest of the data in the adjustment.

## 6. Conclusion

Since the least-squares adjustment is based on letting the adjusted constants vary as free parameters, any comparison of only one experimental value to a theoretical expression is assured to be in agreement. The tests of the consistency of various aspects of physics are made when more than one source of information on the value of a constant is available. Then the consistency of the comparisons is the measure of the validity of the theory.

On the other hand, the theory and constants that relate the value of the Planck constant to the Avogadro constant appear to be well founded, despite the discrepancy in the corresponding values of the Planck constant, which suggests that the problem is attributable to a problem with the data.

On the positive side, the set of transition frequencies in hydrogen and deuterium provide a particularly stringent test of the consistency of the theory and the data, because there are 23 accurate measurements and only three constants (*R*_{∞}, *R*_{p} and *R*_{d}) that are effective in determining the values of the frequencies. There is excellent agreement between the measured and derived values; one frequency has slightly more than one standard deviation difference, and the rest are within one standard deviation.

The good level of agreement between values of the fine structure constant deduced from the various measurements shows consistency of physical theory over a very wide range of phenomena.

## Footnotes

One contribution of 14 to a Discussion Meeting ‘The fundamental constants of physics, precision measurements and the base units of the SI’.

- © 2005 The Royal Society