###  X.2 Boltzmann constant $\bm{k}$ and quotient $\bm{k/h}$

The following two sections discuss the two NIST experiments that have yielded
the last two entries of Table 16.

####  X.2.1 NIST 2007 value of k

This result was obtained by Schmidt _et al._ (2007) using the technique of
refractive index gas thermometry (RIGT), an approach similar to that of
dielectric constant gas thermometry (DCGT) discussed in CODATA-98, and to a
lesser extent in CODATA-02 and CODATA-06. The starting point of both DCGT and
RIGT is the virial expansion of the equation of state for a real gas of amount
of substance $n$ in a volume $V$ Schmidt _et al._ (2007),

$\displaystyle p=\rho RT\left[1+\rho
b(T)+\rho^{2}c(T)+\rho^{3}d(T)+\cdots\right],$ (271)

where $\rho=n/V$ is the amount of substance density of the gas at
thermodynamic temperature $T$, and $b(T)$ is the first virial coefficient,
$c(T)$ is the second, etc.; and the Clausius-Mossotti equation

$\displaystyle{\epsilon_{\rm r}-1\over\epsilon_{\rm r}+2}$ $\displaystyle=$
$\displaystyle\rho A_{\epsilon}\bigg{[}1+\rho B_{\epsilon}(T)$ (272)
$\displaystyle\qquad+\rho^{2}C_{\epsilon}(T)+\rho^{3}D_{\epsilon}(T)+\cdots\bigg{]},\qquad$

where $\epsilon_{\rm r}=\epsilon/\epsilon_{0}$ is the relative dielectric
constant (relative permittivity) of the gas, $\epsilon$ is its dielectric
constant, $\epsilon_{0}$ is the exactly known electric constant,
$A_{\epsilon}$ is the molar polarizability of the atoms, and
$B_{\epsilon}(T)$, $C_{\epsilon}(T)$, etc. are the dielectric virial
coefficients. The static electric polarizability of a gas atom $\alpha_{0}$,
$A_{\epsilon}$, $R$, and $k$ are related by
$A_{\epsilon}/R=\alpha_{0}/3{\epsilon_{0}}k$, which shows that if $\alpha_{0}$
is known sufficiently well from theory, which it currently is for ${}^{4}{\rm
He}$ Łach _et al._ (2004); Jentschura _et al._ (2011b); Puchalski _et al._
(2011), then a competitive value of $k$ can be obtained if the quotient
$A_{\epsilon}/R$ can be measured with a sufficiently small uncertainty.

In fact, by appropriately combining Eqs. (271) and (272), an expression is
obtained from which $A_{\epsilon}/R$ can be experimentally determined by
measuring $\epsilon_{\rm r}$ at a known constant temperature such as $T_{\rm
TPW}$ and at different pressures and extrapolating to zero pressure. This is
done in practice by measuring the fractional change in capacitance of a
specially constructed capacitor, first without helium gas and then with helium
gas at a known pressure. This is the DCGT technique.

In the RIGT technique of Schmidt _et al._ (2007), $A_{\epsilon}/R$ is
determined, and hence $k$, from measurements of $n^{2}(T,p)\equiv\epsilon_{\rm
r}\mu_{\rm r}$ of a gas of helium, where $n(T,p)$ is the index of refraction
of the gas, $\mu_{\rm r}=\mu/\mu_{0}$ is the relative magnetic permeability of
the gas, $\mu$ is its magnetic permeability, and $\mu_{0}$ is the exactly
known magnetic constant. Because ${}^{4}{\rm He}$ is slightly diamagnetic, the
quantity actually determined is $(A_{\epsilon}+A_{\mu})/R$, where
$A_{\mu}=4\mbox{{p}}\chi_{0}/3$ and $\chi_{0}$ is the diamagnetic
susceptibility of a ${}^{4}{\rm He}$ atom. The latter quantity is known from
theory and the theoretical value of $A_{\mu}$ was used to obtain
$A_{\epsilon}/R$ from the determined quantity.

Schmidt _et al._ (2007) obtained $n(T,p)$ by measuring the microwave resonant
frequencies from 2.7 GHz to 7.6 GHz of a quasispherical copper plated
resonator, either evacuated or filled with He at pressures of 0.1 MPa to 6.3
MPa. The temperature of the resonator was within a few millikelvin of $T_{\rm
TPW}$. A network analyzer was used to measure the resonant frequencies and a
calibrated pressure balance to measure $p$. The extrapolation to $p=0$
employed both theoretical and experimental values of the virial coefficients
$B,C,D,b$, and $c$ taken from the literature. The uncertainties of these
coefficients and of the pressure and temperature measurements, and the
uncertainty of the isothermal compressibility of the resonator, are the
largest components in the uncertainty budget.

####  X.2.2 NIST 2011 value of k/h

As discussed in CODATA-98, the Nyquist theorem predicts, with a fractional
error of less than one part in $10^{6}$ at frequencies less than 10 MHz and
temperatures greater than 250 K, that

$\displaystyle\langle U^{2}\rangle=4kTR_{\rm s}\Delta f\,.$ (273)

Here $\langle U^{2}\rangle$ is the mean-square-voltage, or Johnson noise
voltage, in a measurement bandwidth of frequency $\Delta f$ across the
terminals of a resistor of resistance $R_{\rm s}$ in thermal equilibrium at
thermodynamic temperature $T$. If $\langle U^{2}\rangle$ is measured in terms
of the Josephson constant $K_{\rm J}=2e/h$ and $R_{\rm s}$ in terms of the von
Klitzing constant $R_{\rm K}=h/e^{2}$, then this experiment yields a value of
$k/h$.

Such an experiment has been carried out at NIST, yielding the result in Table
16; see the paper by Benz _et al._ (2011) and references therein. In that
work, digitally synthesized pseudo-noise voltages are generated by means of a
pulse-biased Josephson junction array. These known voltages are compared to
the unknown thermal-noise voltages generated by a specially designed 100
$\Omega$ resistor in a well regulated thermal cell at or near $T_{\rm TPW}$.
Since the spectral density of the noise voltage of a 100 $\Omega$ resistor at
273.16 K is only 1.23 nV$\sqrt{\rm Hz}\,$, it is measured using a low-noise,
two-channel, cross-correlation technique that enables the resistor signal to
be extracted from uncorrelated amplifier noise of comparable amplitude and
spectral density. The bandwidths range from 10 kHz to 650 kHz. The final
result is based on two data runs, each of about 117 hours duration, separated
in time by about three months.

The dominant uncertainty component of the 12.1 parts in $10^{6}$ total
uncertainty is the 12.0 parts in $10^{6}$ component due to the measurement of
the ratio $\langle V^{2}_{\rm R}/V^{2}_{\rm Q}\rangle$, where $V_{\rm R}$ is
the resistor noise voltage and $V_{\rm Q}$ is the synthesized voltage. The
main uncertainty component contributing to the uncertainty of the ratio is
10.4 parts in $10^{6}$ due to spectral aberrations, that is, effects that lead
to variations of the ratio with bandwidth.

###  X.3 Other data

We note for completeness the following three results, each of which agrees
with its corresponding 2010 recommended value. The first has a non-competitive
uncertainty but is of interest because it is obtained from a relatively new
method that could yield a value with a competitive uncertainty in the future.
The other two became available only after the 31 December 2010 closing date of
the 2010 adjustment.

Lemarchand _et al._ (2011) find $R=8.314\,80(42)~{}{\rm
J~{}m^{-1}~{}K^{-1}}~{}[50\times 10^{-6}]$ determined by the method of Doppler
spectroscopy, in particular, by measuring near the ice point $T=273.15~{}{\rm
K}$ the absorption profile of a rovibrational line at $\nu=30~{}{\rm THz}$ of
ammonia molecules in an ammonium gas in thermal equilibrium. The width of the
line is mainly determined by the Doppler width due to the velocity
distribution of the ${}^{4}{\rm NH}_{3}$ molecules along the direction of the
incident laser beam. The relevant expression is

$\displaystyle{\Delta\omega_{\rm D}\over\omega_{0}}=\left({2kT\over m(^{4}{\rm
NH}_{3})c^{2}}\right)^{1/2}=\left({2RT\over A_{\rm r}(^{4}{\rm NH}_{3})M_{\rm
u}c^{2}}\right)^{1/2},$ (274)

where $\Delta\omega_{\rm D}$ is the e-fold angular frequency half-width of the
Doppler profile of the ammonium line at temperature $T$, $\omega_{0}$ is its
angular frequency, and $m(^{4}{\rm NH}_{3})$ and $A_{\rm r}(^{4}{\rm NH}_{3})$
are the mass and relative atomic mass of the ammonium molecule.

Zhang _et al._ (2011); Zhang (2011) obtain $R=8.314\,474(66)~{}{\rm
J~{}m^{-1}~{}K^{-1}}~{}[7.9\times 10^{-6}]$ using acoustic gas thermometry
with argon gas, more specifically, by measuring resonant frequencies of a
fixed-path-length cylindrical acoustic resonator at $T_{\rm TPW}$; its
approximate 129 mm length is measured by two-color optical interferometry.

Gaiser and Fellmuth (2012); Fellmuth _et al._ (2011) give
$k=1.380\,655(11)\times 10^{-23}~{}{\rm J/K}~{}[7.9\times 10^{-6}]$ measured
using dielectric gas thermometry (see Sec. X.2.1 above) and helium gas at
$T_{\rm TPW}$ and also at temperatures in the range 21 K to 27 K surrounding
the triple point of neon at $T\approx 25~{}{\rm K}$.

###  X.4 Stefan-Boltzmann constant $\sigma$

The Stefan-Boltzmann constant is related to $c$, $h$, and $k$ by
$\sigma=2\mbox{{p}}^{5}k^{4}/{15h^{3}c^{2}}$, which, with the aid of the
relations $k=R/N_{\rm A}$ and $N_{\rm A}h=cA_{\rm r}({\rm e})M_{\rm
u}\alpha^{2}/2R_{\infty}$, can be expressed in terms of the molar gas constant
and other adjusted constants as

$\displaystyle\sigma=\frac{32\mbox{{p}}^{5}h}{15c^{6}}\left(\frac{R_{\infty}R}{A_{\rm
r}({\rm e})M_{\rm u}\alpha^{2}}\right)^{4}.$ (275)

Since no competitive directly measured value of $\sigma$ is available for the
2010 adjustment, the 2010 recommended value is obtained from this equation.

Table 16:  Summary of thermal physical measurements relevant to the 2010
adjustment (see text for details). AGT: acoustic gas thermometry; RIGT:
refractive index gas thermometry; JNT: Johnson noise thermometry; cylindrical,
spherical, quasispherical: shape of resonator used; JE and QHE: Josephson
effect voltage and quantum Hall effect resistance standards.

Source  | Ident.${}^{\text{a}}$ | Quant.  | Method  | Value  | Rel. stand.
---|---|---|---|---|---
|  |  |  |  | uncert $u_{\rm r}$
Colclough _et al._ (1979) | NPL-79 |  $R$ | AGT, cylindrical, argon  |  $8.314\,504(70)$ J mol-1 K-1 | $8.4\times 10^{-6}$
Moldover _et al._ (1988) | NIST-88 |  $R$ | AGT, spherical, argon  |  $8.314\,471(15)$ J mol-1 K-1 | $1.8\times 10^{-6}$
Pitre _et al._ (2009) | LNE-09 |  $R$ | AGT, quasispherical, helium  |  $8.314\,467(22)$ J mol-1 K-1 | $2.7\times 10^{-6}$
Sutton _et al._ (2010) | NPL-10 |  $R$ | AGT, quasispherical, argon  |  $8.314\,468(26)$ J mol-1 K-1 | $3.1\times 10^{-6}$
Gavioso _et al._ (2010) | INRIM-10 |  $R$ | AGT, spherical, helium  |  $8.314\,412(63)$ J mol-1 K-1 | $7.5\times 10^{-6}$
Pitre _et al._ (2011) | LNE-11 |  $R$ | AGT, quasispherical, argon  |  $8.314\,456(10)$ J mol-1 K-1 | $1.2\times 10^{-6}$
Schmidt _et al._ (2007) | NIST-07 |  $k$ | RIGT, quasispherical, helium  |  $1.380\,653(13)\times 10^{-23}$ J K-1 | $9.1\times 10^{-6}$
Benz _et al._ (2011) | NIST-11 |  $k/h$ | JNT, JE and QHE  |  $2.083\,666(25)\times 10^{10}$ Hz K-1 | $1.2\times 10^{-5}$

${}^{\text{a}}$NPL: National Physical Laboratory, Teddington, UK; NIST:
National Institute of Standards and Technology, Gaithersburg, MD, and Boulder,
CO, USA; LNE: Laboritoire commun de métrologie (LCM), Saint-Denis, France, of
the Laboratoire national de métrologie et d’essais (LNE); INRIM: Istituto
Nazionale di Ricerca Metrologica, Torino, Italy.

##  XI Newtonian constant of gravitation $\bm{G}$

Table 17:  Summary of the results of measurements of the Newtonian constant of
gravitation relevant to the 2010 adjustment.

Source  | Identification${}^{\text{a}}$ | Method  |  $10^{11}\,G$ | Rel. stand.
---|---|---|---|---
|  |  | $\overline{{\rm m}^{3}\ {\rm kg}^{-1}\ {\rm s}^{-2}}$ | uncert $u_{\rm r}$
Luther and Towler (1982) | NIST-82 | Fiber torsion balance,  | $6.672\,48(43)$ | $6.4\times 10^{-5}$
|  | dynamic mode  |  |
Karagioz and Izmailov (1996) | TR&D-96 | Fiber torsion balance,  | $6.672\,9(5)$ | $7.5\times 10^{-5}$
|  | dynamic mode  |  |
Bagley and Luther (1997) | LANL-97 | Fiber torsion balance,  | $6.673\,98(70)$ | $1.0\times 10^{-4}$
|  | dynamic mode  |  |
Gundlach and Merkowitz (2000, 2002) | UWash-00 | Fiber torsion balance,  | $6.674\,255(92)$ | $1.4\times 10^{-5}$
|  | dynamic compensation  |  |
Quinn _et al._ (2001) | BIPM-01 | Strip torsion balance,  | $6.675\,59(27)$ | $4.0\times 10^{-5}$
|  | compensation mode, static deflection  |  |
Kleinevoß (2002); Kleinvoß _et al._ (2002) | UWup-02 | Suspended body,  | $6.674\,22(98)$ | $1.5\times 10^{-4}$
|  | displacement  |  |
Armstrong and Fitzgerald (2003) | MSL-03 | Strip torsion balance,  | $6.673\,87(27)$ | $4.0\times 10^{-5}$
|  | compensation mode  |  |
Hu _et al._ (2005) | HUST-05 | Fiber torsion balance,  | $6.672\,28(87)$ | $1.3\times 10^{-4}$
|  | dynamic mode  |  |
Schlamminger _et al._ (2006) | UZur-06 | Stationary body,  | $6.674\,25(12)$ | $1.9\times 10^{-5}$
|  | weight change  |  |
Luo _et al._ (2009); Tu _et al._ (2010) | HUST-09 | Fiber torsion balance,  | $6.673\,49(18)$ | $2.7\times 10^{-5}$
|  | dynamic mode  |  |
Parks and Faller (2010) | JILA-10 | Suspended body,  | $6.672\,34(14)$ | $2.1\times 10^{-5}$
|  | displacement  |  |

${}^{\text{a}}$NIST: National Institute of Standards and Technology,
Gaithersburg, MD, USA; TR&D: Tribotech Research and Development Company,
Moscow, Russian Federation; LANL: Los Alamos National Laboratory, Los Alamos,
New Mexico, USA; UWash: University of Washington, Seattle, Washington, USA;
BIPM: International Bureau of Weights and Measures, Sèvres, France; UWup:
University of Wuppertal, Wuppertal, Germany; MSL: Measurement Standards
Laboratory, Lower Hutt, New Zeland; HUST: Huazhong University of Science and
Technology, Wuhan, PRC; UZur: University of Zurich, Zurich, Switzerland; JILA:
JILA, University of Colorado and National Institute of Standards and
Technology, Boulder, Colorado, USA.

Table 17 summarizes the 11 values of the Newtonian constant of gravitation $G$
of interest in the 2010 adjustment. Because they are independent of the other
data relevant to the current adjustment, and because there is no known
quantitative theoretical relationship between $G$ and other fundamental
constants, they contribute only to the determination of the 2010 recommended
value of $G$. The calculation of this value is discussed in Sec. XIII.2.1.

The inconsistencies between different measurements of $G$ as discussed in the
reports of previous CODATA adjustments demonstrate the historic difficulty of
determining this most important constant. Unfortunately, this difficulty has
been demonstrated anew with the publication of two new competitive results for
$G$ during the past 4 years. The first is an improved value from the group at
the Huazhong University of Science and Technology (HUST), PRC, identified as
HUST-09 Luo _et al._ (2009); Tu _et al._ (2010); the second is a completely
new value from researchers at JILA, Boulder, Colorado, USA, identified as
JILA-10 Parks and Faller (2010). (JILA, formerly known as the Joint Institute
for Laboratory Astrophysics, is a joint institute of NIST and the University
of Colorado and is located on the University of Colorado campus, Boulder,
Colorado.)

The publication of the JILA value has led the Task Group to re-examine and
modify two earlier results. The first is that obtained at NIST (then known as
the National Bureau of Standards) by Luther and Towler (1982) in a NIST-
University of Virginia (UVa) collaboration, labeled NIST-82. This value was
the basis for the CODATA 1986 recommended value Cohen and Taylor (1987) and
was taken into account in determining the CODATA 1998 value Mohr and Taylor
(2000), but played no role in either the 2002 or 2006 adjustments. The second
is the Los Alamos National Laboratory (LANL), Los Alamos, USA, result of
Bagley and Luther (1997), labeled LANL-97; it was first included in the 1998
CODATA adjustment and in all subsequent adjustments. Details of the
modifications to NIST-82 and LANL-97 (quite minor for the latter), the reasons
for including NIST-82 in the 2010 adjustment, and discussions of the new
values HUST-09 and JILA-10 are given below. The 11 available values of $G$,
which are data items $G1$-$G11$ in Table 24, Sec. XIII, are the same as in
2006 with the exception of NIST-82, slightly modified LANL-97, and the two new
values. Thus, in keeping with our approach in this report, there is no
discussion of the other seven values since they have been covered in one or
more of the previous reports.

For simplicity, in the following text, we write $G$ as a numerical factor
multiplying $G_{0}$, where

$\displaystyle G_{0}=10^{-11}~{}{\rm m^{3}~{}kg^{-1}~{}s^{-2}}\ .$ (276)

###  XI.1 Updated values

####  XI.1.1 National Institute of Standards and Technology and University of
Virginia

As discussed in CODATA-98, the experiment of Luther and Towler (1982) used a
fiber-based torsion balance operated in the dynamic mode and the time-of-swing
method, thereby requiring measurement of a small change in the long
oscillation period of the balance. Ideally, the torsional spring constant of
the fiber should be independent of frequency at very low frequencies, for
example, at 3 mHz.

Long after the publication of the NIST-UVa result, Kuroda (1995) [see also
Matsumura _et al._ (1998) and Kuroda (1999)] pointed out that the anelasticity
of such fibers is sufficiently large to cause the value of $G$ determined in
this way to be biased. If $Q$ is the quality factor of the main torsional mode
of the fiber and it is assumed that the damping of the torsion balance is
solely due to the losses in the fiber, then the unbiased value of $G$ is
related to the experimentally observed value $G({\rm obs})$ by Kuroda (1995)

$\displaystyle G={G({\rm obs})\over 1+\mbox{{p}}Q\,}\,.$ (277)

Although the exact value of the $Q$ of the fiber used in the NIST-UVa
experiment is unknown, one of the researchers Luther (2010) has provided an
estimate, based on data obtained during the course of the experiment, of no
less than 10 000 and no greater than 30 000. Assuming a rectangular
probability density function for $Q$ with these lower and upper limits then
leads to $Q=2\times 10^{4}$ with a relative standard uncertainty of $4.6\times
10^{-6}$. Using these values, the result $G({\rm obs})=6.672~{}59(43)G_{0}$
$[64\times 10^{-6}]$ Luther and Towler (1982), Luther (1986), and Eq. (277) we
obtain

$\displaystyle G=6.672\,48(43)G_{0}\quad[6.4\times 10^{-5}]\,.$ (278)

In this case the correction $1/(1+\mbox{{p}}Q)$ reduced $G({\rm obs})$ by the
fractional amount $15.9(4.6)\times 10^{-6}$, but increased its $64\times
10^{-6}$ relative standard uncertainty by a negligible amount.

The Task Group decided to include the value given in Eq. (278) as an input
datum in the 2010 adjustment even though it was not included in the 2002 and
2006 adjustments, because information provided by Luther (2010) allows the
original result to be corrected for the Kuroda effect. Further, although there
were plans to continue the NIST-UVa experiment Luther and Towler (1982),
recent conversations with Luther (2010) made clear that the measurements on
which the result is based were thorough and complete.

####  XI.1.2 Los Alamos National Laboratory

The experiment of Bagley and Luther (1997), also described in detail in
CODATA-98, is similar to the NIST-UVa experiment of Luther and Towler (1982),
and in fact used some of the same components including the tungsten source
masses. Its purpose was not only to determine $G$, but also to test the Kuroda
hypothesis by using two different fibers, one with $Q=950$ and the other with
$Q=490$. Because the value of $G$ resulting from this experiment is correlated
with the NIST-UVa value and both values are now being included in the
adjustment, we evaluated the correlation coefficient of the two results. This
was done with information from Bagley (2010), Luther (2010), and the Ph.D.
thesis of Bagley (1996). We take into account the uncertainties of the two $Q$
values (2 %) and the correlation coefficient of the two values of $G$ obtained
from the two fibers (0.147) when computing their weighted mean. The final
result is

$\displaystyle G=6.673\,98(70)\quad[1.0\times 10^{-4}]\,,$ (279)

which in fact is essentially the same as the value used in the 2002 and 2006
adjustments. The correlation coefficient of the NIST-UVa and LANL values of
$G$ is 0.351.

###  XI.2 New values

####  XI.2.1 Huazhong University of Science and Technology

The improved HUST-09 result for $G$ was first reported by Luo _et al._ (2009)
and subsequently described in detail by Tu _et al._ (2010); it represents a
reduction in uncertainty, compared to the previous Huazhong University result
HUST-05, of about a factor of five. As pointed out by Tu _et al._ (2010), a
number of changes in the earlier experiment contributed to this uncertainty
reduction, including (i) replacement of the two stainless steel cylindrical
source masses by spherical source masses with a more homogeneous density; (ii)
use of a rectangular quartz block as the principal portion of the torsion
balance’s pendulum, thereby improving the stability of the period of the
balance and reducing the uncertainty of the pendulum’s moment of inertia;
(iii) a single vacuum chamber for the source masses and pendulum leading to a
reduction of the uncertainty of their relative positions; (iv) a remotely
operated stepper motor to change the positions of the source masses, thereby
reducing environmental changes; and (v) measurement of the anelasticity of the
torsion fiber with the aid of a high-$Q$ quartz fiber.

The final result is the average of two values of $G$ that differ by 9 parts in
$10^{6}$ obtained from two partially correlated determinations using the same
apparatus. The dominant components of uncertainty, in parts in $10^{6}$, are
19 from the measurement of the fiber’s anelasticity, 14 (statistical) from the
measurement of the change in the square of the angular frequency of the
pendulum when the source masses are in their near and far positions, and 10
from the measured distance between the geometric centers of the source masses.
Although the uncertainty of HUST-05 is five times larger than that of HUST-09,
Luo (2010) and co-workers do not believe that HUST-09 supersedes HUST-05.
Thus, both are considered for inclusion in the 2010 adjustment. Based on
information provided to the Task Group by the researchers Luo (2010), their
correlation coefficient is estimated to be 0.234 and is used in the
calculations of Sec. XIII. The extra digits for the value and uncertainty of
HUST-05 were also provided by Luo (2011).

####  XI.2.2 JILA

As can be seen from Table 17, the $21\times 10^{-6}$ relative standard
uncertainty of the value of $G$ identified as JILA-10 and obtained at JILA by
Parks and Faller (2010) has the third smallest estimated uncertainty of the
values listed and is the second smallest of those values. It differs from the
value with the smallest uncertainty, identified as UWash-00, by $287(25)$
parts in $10^{6}$, which is $11$ times the standard uncertainty of their
difference $u_{\rm diff}$, or “$11\sigma$.” This disagreement is an example of
the “historic difficulty” referred to at the very beginning of this section.
The data on which the JILA researchers based their result was taken in 2004,
but being well aware of this inconsistency they hesitated to publish it until
they checked and rechecked their work Parks and Faller (2010). With this done,
they decided it was time to report their value for $G$.

The apparatus used in the JILA experiment of Parks and Faller (2010) consisted
of two $780~{}{\rm g}$ copper test masses (or “pendulum bobs”) separated by
$34~{}{\rm cm}$, each of which was suspended from a supporting bar by four
wires and together they formed a Fabry-Perot cavity. When the four $120~{}{\rm
kg}$ cylindrical tungsten source masses, two pairs with each member of the
pair on either side of the laser beam traversing the cavity, were periodically
moved parallel to the laser beam from their inner and outer positions (they
remained stationary for $80~{}{\rm s}$ in each position), the separation
between the bobs changed by about $90~{}{\rm nm}$. This change was observed as
a $125~{}{\rm MHz}$ beat frequency between the laser locked to the pendulum
cavity and the laser locked to a reference cavity that was part of the
supporting bar. The geometry of the experiment reduces the most difficult
aspect of determining the gravitational field of the source masses to six one
dimensional measurements: the distance between opposite source mass pairs in
the inner and outer positions and the distances between adjacent source masses
in the inner position. The most important relative standard uncertainty
components contributing to the uncertainty of $G$ are, in parts in $10^{6}$
Parks and Faller (2010), the six critical dimension measurements, 14; all
other dimension measurements and source mass density inhomogeneities, 8 each;
pendulum spring constants, 7; and total mass measurement and interferometer
misalignment, 6 each.

As already noted, we leave the calculation of the 2010 recommended value of
$G$ to Sec. XIII.2.1.

##  XII Electroweak quantities

As in previous adjustments, there are a few cases in the 2010 adjustment where
an inexact constant that is used in the analysis of input data is not treated
as an adjusted quantity, because the adjustment has a negligible effect on its
value. Three such constants, used in the calculation of the theoretical
expression for the electron magnetic moment anomaly $a_{\rm e}$, are the mass
of the tau lepton $m_{\mbox{\scriptsize{{t}}}}$, the Fermi coupling constant
$G_{\rm F}$, and sine squared of the weak mixing angle sin${}^{2}{\theta}_{\rm
W}$; they are obtained from the most recent report of the Particle Data Group
Nakamura _et al._ (2010):

$\displaystyle m_{\mbox{\scriptsize{{t}}}}c^{2}$ $\displaystyle=$
$\displaystyle 1776.82(16)\ {\rm MeV}\qquad[9.0\times 10^{-5}]\,,$ (280)
$\displaystyle{G_{\rm F}\over(\hbar c)^{3}}$ $\displaystyle=$ $\displaystyle
1.166\,364(5)\times 10^{-5}\ {\rm GeV}^{-2}\quad[4.3\times 10^{-6}]\,,$ (281)
$\displaystyle{\rm sin}^{2}{\theta}_{\rm W}$ $\displaystyle=$ $\displaystyle
0.2223(21)\qquad[9.5\times 10^{-3}]\,.$ (282)

The value for $G_{\rm F}/(\hbar c)^{3}$ is taken from p. 127 of Nakamura _et
al._ (2010). We use the definition sin${}^{2}{\theta}_{\rm W}=1-(m_{\rm
W}/m_{\rm Z})^{2}$, where $m_{\rm W}$ and $m_{\rm Z}$ are, respectively, the
masses of the ${\rm W}^{\pm}$ and ${\rm Z}^{0}$ bosons, because it is employed
in the calculation of the electroweak contributions to $a_{\rm e}$ Czarnecki
_et al._ (1996). The Particle Data Group’s recommended value for the mass
ratio of these bosons is $m_{\rm W}/m_{\rm Z}=0.8819(12)$, which leads to the
value of sin${}^{2}{\theta}_{\rm W}$ given above.

##  XIII Analysis of Data

We examine in this section the input data discussed in the previous sections
and, based upon that examination, select the data to be used in the least-
squares adjustment that determines the 2010 CODATA recommended values of the
constants. Tables 18, LABEL:tab:pdata, 22, and 24 give the input data,
including the $\delta$’s, which are corrections added to theoretical
expressions to account for the uncertainties of those expressions. The
covariances of the data are given as correlation coefficients in Tables 19,
21, 23, and 24. There are 14 types of input data for which there are two or
more experiments, and the data of the same type generally agree.

Table 18: Summary of principal input data for the determination of the 2010
recommended value of the Rydberg constant $R_{\infty}$.

Item  | Input datum |  Value |  Relative standard  | Identification  | Sec.
---|---|---|---|---|---
number  |  |  |  uncertainty1 $u_{\rm r}$ |  |
$A1$ |  $\delta_{\rm H}({\rm 1S_{1/2}})$ |  $0.0(2.5)$ kHz  |  $[7.5\times 10^{-13}]$ | theory  | IV.1.1
$A2$ |  $\delta_{\rm H}({\rm 2S_{1/2}})$ |  $0.00(31)$ kHz  |  $[3.8\times 10^{-13}]$ | theory  | IV.1.1
$A3$ |  $\delta_{\rm H}({\rm 3S_{1/2}})$ |  $0.000(91)$ kHz  |  $[2.5\times 10^{-13}]$ | theory  | IV.1.1
$A4$ |  $\delta_{\rm H}({\rm 4S_{1/2}})$ |  $0.000(39)$ kHz  |  $[1.9\times 10^{-13}]$ | theory  | IV.1.1
$A5$ |  $\delta_{\rm H}({\rm 6S_{1/2}})$ |  $0.000(15)$ kHz  |  $[1.6\times 10^{-13}]$ | theory  | IV.1.1
$A6$ |  $\delta_{\rm H}({\rm 8S_{1/2}})$ |  $0.0000(63)$ kHz  |  $[1.2\times 10^{-13}]$ | theory  | IV.1.1
$A7$ |  $\delta_{\rm H}({\rm 2P_{1/2}})$ |  $0.000(28)$ kHz  |  $[3.5\times 10^{-14}]$ | theory  | IV.1.1
$A8$ |  $\delta_{\rm H}({\rm 4P_{1/2}})$ |  $0.0000(38)$ kHz  |  $[1.9\times 10^{-14}]$ | theory  | IV.1.1
$A9$ |  $\delta_{\rm H}({\rm 2P_{3/2}})$ |  $0.000(28)$ kHz  |  $[3.5\times 10^{-14}]$ | theory  | IV.1.1
$A10$ |  $\delta_{\rm H}({\rm 4P_{3/2}})$ |  $0.0000(38)$ kHz  |  $[1.9\times 10^{-14}]$ | theory  | IV.1.1
$A11$ |  $\delta_{\rm H}({\rm 8D_{3/2}})$ |  $0.000\,00(44)$ kHz  |  $[8.5\times 10^{-15}]$ | theory  | IV.1.1
$A12$ |  $\delta_{\rm H}({\rm 12D_{3/2}})$ |  $0.000\,00(13)$ kHz  |  $[5.7\times 10^{-15}]$ | theory  | IV.1.1
$A13$ |  $\delta_{\rm H}({\rm 4D_{5/2}})$ |  $0.0000(35)$ kHz  |  $[1.7\times 10^{-14}]$ | theory  | IV.1.1
$A14$ |  $\delta_{\rm H}({\rm 6D_{5/2}})$ |  $0.0000(10)$ kHz  |  $[1.1\times 10^{-14}]$ | theory  | IV.1.1
$A15$ |  $\delta_{\rm H}({\rm 8D_{5/2}})$ |  $0.000\,00(44)$ kHz  |  $[8.5\times 10^{-15}]$ | theory  | IV.1.1
$A16$ |  $\delta_{\rm H}({\rm 12D_{5/2}})$ |  $0.000\,00(13)$ kHz  |  $[5.7\times 10^{-15}]$ | theory  | IV.1.1
$A17$ |  $\delta_{\rm D}({\rm 1S_{1/2}})$ |  $0.0(2.3)$ kHz  |  $[6.9\times 10^{-13}]$ | theory  | IV.1.1
$A18$ |  $\delta_{\rm D}({\rm 2S_{1/2}})$ |  $0.00(29)$ kHz  |  $[3.5\times 10^{-13}]$ | theory  | IV.1.1
$A19$ |  $\delta_{\rm D}({\rm 4S_{1/2}})$ |  $0.000(36)$ kHz  |  $[1.7\times 10^{-13}]$ | theory  | IV.1.1
$A20$ |  $\delta_{\rm D}({\rm 8S_{1/2}})$ |  $0.0000(60)$ kHz  |  $[1.2\times 10^{-13}]$ | theory  | IV.1.1
$A21$ |  $\delta_{\rm D}({\rm 8D_{3/2}})$ |  $0.000\,00(44)$ kHz  |  $[8.5\times 10^{-15}]$ | theory  | IV.1.1
$A22$ |  $\delta_{\rm D}({\rm 12D_{3/2}})$ |  $0.000\,00(13)$ kHz  |  $[5.6\times 10^{-15}]$ | theory  | IV.1.1
$A23$ |  $\delta_{\rm D}({\rm 4D_{5/2}})$ |  $0.0000(35)$ kHz  |  $[1.7\times 10^{-14}]$ | theory  | IV.1.1
$A24$ |  $\delta_{\rm D}({\rm 8D_{5/2}})$ |  $0.000\,00(44)$ kHz  |  $[8.5\times 10^{-15}]$ | theory  | IV.1.1
$A25$ |  $\delta_{\rm D}({\rm 12D_{5/2}})$ |  $0.000\,00(13)$ kHz  |  $[5.7\times 10^{-15}]$ | theory  | IV.1.1
$A26$ |  $\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ |  $2\,466\,061\,413\,187.080(34)$ kHz  |  $1.4\times 10^{-14}$ | MPQ-04  | IV.1.2
$A27$ |  $\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 3S_{1/2}})$ |  $2\,922\,743\,278\,678(13)$ kHz  |  $4.4\times 10^{-12}$ | LKB-10  | IV.1.2
$A28$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 8S_{1/2}})$ |  $770\,649\,350\,012.0(8.6)$ kHz  |  $1.1\times 10^{-11}$ | LK/SY-97  | IV.1.2
$A29$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 8D_{3/2}})$ |  $770\,649\,504\,450.0(8.3)$ kHz  |  $1.1\times 10^{-11}$ | LK/SY-97  | IV.1.2
$A30$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 8D_{5/2}})$ |  $770\,649\,561\,584.2(6.4)$ kHz  |  $8.3\times 10^{-12}$ | LK/SY-97  | IV.1.2
$A31$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 12D_{3/2}})$ |  $799\,191\,710\,472.7(9.4)$ kHz  |  $1.2\times 10^{-11}$ | LK/SY-98  | IV.1.2
$A32$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 12D_{5/2}})$ |  $799\,191\,727\,403.7(7.0)$ kHz  |  $8.7\times 10^{-12}$ | LK/SY-98  | IV.1.2
$A33$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4S_{1/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ |  $4\,797\,338(10)$ kHz  |  $2.1\times 10^{-6}$ | MPQ-95  | IV.1.2
$A34$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4D_{5/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ |  $6\,490\,144(24)$ kHz  |  $3.7\times 10^{-6}$ | MPQ-95  | IV.1.2
$A35$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 6S_{1/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 3S_{1/2}})$ |  $4\,197\,604(21)$ kHz  |  $4.9\times 10^{-6}$ | LKB-96  | IV.1.2
$A36$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 6D_{5/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 3S_{1/2}})$ |  $4\,699\,099(10)$ kHz  |  $2.2\times 10^{-6}$ | LKB-96  | IV.1.2
$A37$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4P_{1/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ |  $4\,664\,269(15)$ kHz  |  $3.2\times 10^{-6}$ | YaleU-95  | IV.1.2
$A38$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4P_{3/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ |  $6\,035\,373(10)$ kHz  |  $1.7\times 10^{-6}$ | YaleU-95  | IV.1.2
$A39$ |  $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 2P_{3/2}})$ |  $9\,911\,200(12)$ kHz  |  $1.2\times 10^{-6}$ | HarvU-94  | IV.1.2
$A40.1$ |  $\nu_{\rm H}({\rm 2P_{1/2}}-{\rm 2S_{1/2}})$ |  $1\,057\,845.0(9.0)$ kHz  |  $8.5\times 10^{-6}$ | HarvU-86  | IV.1.2
$A40.2$ |  $\nu_{\rm H}({\rm 2P_{1/2}}-{\rm 2S_{1/2}})$ |  $1\,057\,862(20)$ kHz  |  $1.9\times 10^{-5}$ | USus-79  | IV.1.2
$A41$ |  $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 8S_{1/2}})$ |  $770\,859\,041\,245.7(6.9)$ kHz  |  $8.9\times 10^{-12}$ | LK/SY-97  | IV.1.2
$A42$ |  $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 8D_{3/2}})$ |  $770\,859\,195\,701.8(6.3)$ kHz  |  $8.2\times 10^{-12}$ | LK/SY-97  | IV.1.2
$A43$ |  $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 8D_{5/2}})$ |  $770\,859\,252\,849.5(5.9)$ kHz  |  $7.7\times 10^{-12}$ | LK/SY-97  | IV.1.2
$A44$ |  $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 12D_{3/2}})$ |  $799\,409\,168\,038.0(8.6)$ kHz  |  $1.1\times 10^{-11}$ | LK/SY-98  | IV.1.2
$A45$ |  $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 12D_{5/2}})$ |  $799\,409\,184\,966.8(6.8)$ kHz  |  $8.5\times 10^{-12}$ | LK/SY-98  | IV.1.2
$A46$ |  $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 4S_{1/2}})-\frac{1}{4}\nu_{\rm D}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ |  $4\,801\,693(20)$ kHz  |  $4.2\times 10^{-6}$ | MPQ-95  | IV.1.2
$A47$ |  $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 4D_{5/2}})-\frac{1}{4}\nu_{\rm D}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ |  $6\,494\,841(41)$ kHz  |  $6.3\times 10^{-6}$ | MPQ-95  | IV.1.2
$A48$ |  $\nu_{\rm D}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})-\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ |  $670\,994\,334.606(15)$ kHz  |  $2.2\times 10^{-11}$ | MPQ-10  | IV.1.2
$A49.1$ |  $r_{\rm p}$ |  $0.895(18)$ fm  |  $2.0\times 10^{-2}$ | rp-03  | IV.1.3
$A49.2$ |  $r_{\rm p}$ |  $0.8791(79)$ fm  |  $9.0\times 10^{-3}$ | rp-10  | IV.1.3
$A50$ |  $r_{\rm d}$ |  $2.130(10)$ fm  |  $4.7\times 10^{-3}$ | rd-98  | IV.1.3

1 The values in brackets are relative to the frequency equivalent of the
binding energy of the indicated level.

Table 19: Correlation coefficients $r(x_{i},x_{j})\geq 0.0001$ of the input data related to $R_{\infty}$ in Table 18. For simplicity, the two items of data to which a particular correlation coefficient corresponds are identified by their item numbers in Table 18. $r$($A1$, $A2$) | = $0.9905$ |  $r$($A6$, $A19$) | = $0.7404$ |  $r$($A28$, $A29$) | = $0.3478$ |  $r$($A31$, $A45$) | = $0.1136$
---|---|---|---|---|---|---|---
$r$($A1$, $A3$) | = $0.9900$ |  $r$($A6$, $A20$) | = $0.9851$ |  $r$($A28$, $A30$) | = $0.4532$ |  $r$($A32$, $A35$) | = $0.0278$
$r$($A1$, $A4$) | = $0.9873$ |  $r$($A7$, $A8$) | = $0.0237$ |  $r$($A28$, $A31$) | = $0.0899$ |  $r$($A32$, $A36$) | = $0.0553$
$r$($A1$, $A5$) | = $0.7640$ |  $r$($A9$, $A10$) | = $0.0237$ |  $r$($A28$, $A32$) | = $0.1206$ |  $r$($A32$, $A41$) | = $0.1512$
$r$($A1$, $A6$) | = $0.7627$ |  $r$($A11$, $A12$) | = $0.0006$ |  $r$($A28$, $A35$) | = $0.0225$ |  $r$($A32$, $A42$) | = $0.1647$
$r$($A1$, $A17$) | = $0.9754$ |  $r$($A11$, $A21$) | = $0.9999$ |  $r$($A28$, $A36$) | = $0.0448$ |  $r$($A32$, $A43$) | = $0.1750$
$r$($A1$, $A18$) | = $0.9656$ |  $r$($A11$, $A22$) | = $0.0003$ |  $r$($A28$, $A41$) | = $0.1225$ |  $r$($A32$, $A44$) | = $0.1209$
$r$($A1$, $A19$) | = $0.9619$ |  $r$($A12$, $A21$) | = $0.0003$ |  $r$($A28$, $A42$) | = $0.1335$ |  $r$($A32$, $A45$) | = $0.1524$
$r$($A1$, $A20$) | = $0.7189$ |  $r$($A12$, $A22$) | = $0.9999$ |  $r$($A28$, $A43$) | = $0.1419$ |  $r$($A33$, $A34$) | = $0.1049$
$r$($A2$, $A3$) | = $0.9897$ |  $r$($A13$, $A14$) | = $0.0006$ |  $r$($A28$, $A44$) | = $0.0980$ |  $r$($A33$, $A46$) | = $0.2095$
$r$($A2$, $A4$) | = $0.9870$ |  $r$($A13$, $A15$) | = $0.0006$ |  $r$($A28$, $A45$) | = $0.1235$ |  $r$($A33$, $A47$) | = $0.0404$
$r$($A2$, $A5$) | = $0.7638$ |  $r$($A13$, $A16$) | = $0.0006$ |  $r$($A29$, $A30$) | = $0.4696$ |  $r$($A34$, $A46$) | = $0.0271$
$r$($A2$, $A6$) | = $0.7625$ |  $r$($A13$, $A23$) | = $0.9999$ |  $r$($A29$, $A31$) | = $0.0934$ |  $r$($A34$, $A47$) | = $0.0467$
$r$($A2$, $A17$) | = $0.9656$ |  $r$($A13$, $A24$) | = $0.0003$ |  $r$($A29$, $A32$) | = $0.1253$ |  $r$($A35$, $A36$) | = $0.1412$
$r$($A2$, $A18$) | = $0.9754$ |  $r$($A13$, $A25$) | = $0.0003$ |  $r$($A29$, $A35$) | = $0.0234$ |  $r$($A35$, $A41$) | = $0.0282$
$r$($A2$, $A19$) | = $0.9616$ |  $r$($A14$, $A15$) | = $0.0006$ |  $r$($A29$, $A36$) | = $0.0466$ |  $r$($A35$, $A42$) | = $0.0307$
$r$($A2$, $A20$) | = $0.7187$ |  $r$($A14$, $A16$) | = $0.0006$ |  $r$($A29$, $A41$) | = $0.1273$ |  $r$($A35$, $A43$) | = $0.0327$
$r$($A3$, $A4$) | = $0.9864$ |  $r$($A14$, $A23$) | = $0.0003$ |  $r$($A29$, $A42$) | = $0.1387$ |  $r$($A35$, $A44$) | = $0.0226$
$r$($A3$, $A5$) | = $0.7633$ |  $r$($A14$, $A24$) | = $0.0003$ |  $r$($A29$, $A43$) | = $0.1475$ |  $r$($A35$, $A45$) | = $0.0284$
$r$($A3$, $A6$) | = $0.7620$ |  $r$($A14$, $A25$) | = $0.0003$ |  $r$($A29$, $A44$) | = $0.1019$ |  $r$($A36$, $A41$) | = $0.0561$
$r$($A3$, $A17$) | = $0.9651$ |  $r$($A15$, $A16$) | = $0.0006$ |  $r$($A29$, $A45$) | = $0.1284$ |  $r$($A36$, $A42$) | = $0.0612$
$r$($A3$, $A18$) | = $0.9648$ |  $r$($A15$, $A23$) | = $0.0003$ |  $r$($A30$, $A31$) | = $0.1209$ |  $r$($A36$, $A43$) | = $0.0650$
$r$($A3$, $A19$) | = $0.9611$ |  $r$($A15$, $A24$) | = $0.9999$ |  $r$($A30$, $A32$) | = $0.1622$ |  $r$($A36$, $A44$) | = $0.0449$
$r$($A3$, $A20$) | = $0.7183$ |  $r$($A15$, $A25$) | = $0.0003$ |  $r$($A30$, $A35$) | = $0.0303$ |  $r$($A36$, $A45$) | = $0.0566$
$r$($A4$, $A5$) | = $0.7613$ |  $r$($A16$, $A23$) | = $0.0003$ |  $r$($A30$, $A36$) | = $0.0602$ |  $r$($A37$, $A38$) | = $0.0834$
$r$($A4$, $A6$) | = $0.7600$ |  $r$($A16$, $A24$) | = $0.0003$ |  $r$($A30$, $A41$) | = $0.1648$ |  $r$($A41$, $A42$) | = $0.5699$
$r$($A4$, $A17$) | = $0.9625$ |  $r$($A16$, $A25$) | = $0.9999$ |  $r$($A30$, $A42$) | = $0.1795$ |  $r$($A41$, $A43$) | = $0.6117$
$r$($A4$, $A18$) | = $0.9622$ |  $r$($A17$, $A18$) | = $0.9897$ |  $r$($A30$, $A43$) | = $0.1908$ |  $r$($A41$, $A44$) | = $0.1229$
$r$($A4$, $A19$) | = $0.9755$ |  $r$($A17$, $A19$) | = $0.9859$ |  $r$($A30$, $A44$) | = $0.1319$ |  $r$($A41$, $A45$) | = $0.1548$
$r$($A4$, $A20$) | = $0.7163$ |  $r$($A17$, $A20$) | = $0.7368$ |  $r$($A30$, $A45$) | = $0.1662$ |  $r$($A42$, $A43$) | = $0.6667$
$r$($A5$, $A6$) | = $0.5881$ |  $r$($A18$, $A19$) | = $0.9856$ |  $r$($A31$, $A32$) | = $0.4750$ |  $r$($A42$, $A44$) | = $0.1339$
$r$($A5$, $A17$) | = $0.7448$ |  $r$($A18$, $A20$) | = $0.7366$ |  $r$($A31$, $A35$) | = $0.0207$ |  $r$($A42$, $A45$) | = $0.1687$
$r$($A5$, $A18$) | = $0.7445$ |  $r$($A19$, $A20$) | = $0.7338$ |  $r$($A31$, $A36$) | = $0.0412$ |  $r$($A43$, $A44$) | = $0.1423$
$r$($A5$, $A19$) | = $0.7417$ |  $r$($A21$, $A22$) | = $0.0002$ |  $r$($A31$, $A41$) | = $0.1127$ |  $r$($A43$, $A45$) | = $0.1793$
$r$($A5$, $A20$) | = $0.5543$ |  $r$($A23$, $A24$) | = $0.0001$ |  $r$($A31$, $A42$) | = $0.1228$ |  $r$($A44$, $A45$) | = $0.5224$
$r$($A6$, $A17$) | = $0.7435$ |  $r$($A23$, $A25$) | = $0.0001$ |  $r$($A31$, $A43$) | = $0.1305$ |  $r$($A46$, $A47$) | = $0.0110$
$r$($A6$, $A18$) | = $0.7433$ |  $r$($A24$, $A25$) | = $0.0002$ |  $r$($A31$, $A44$) | = $0.0901$ |  |

Table 20: Summary of principal input data for the determination of the 2010 recommended values of the fundamental constants ($R_{\infty}$ and $G$ excepted).  Item | Input datum |  Value |  Relative standard | Identification | Sec. and Eq.
---|---|---|---|---|---
number |  |  |  uncertainty1 $u_{\rm r}$ |  |
$B1$ |  $A_{\rm r}(^{1}{\rm H})$ | $\phantom{-}1.007\,825\,032\,07(10)$ | $1.0\times 10^{-10}$ | AMDC-03 | III.1
$B2.1$ |  $A_{\rm r}(^{2}{\rm H})$ | $\phantom{-}2.014\,101\,777\,85(36)$ | $1.8\times 10^{-10}$ | AMDC-03 | III.1
$B2.2$ |  $A_{\rm r}(^{2}{\rm H})$ | $\phantom{-}2.014\,101\,778\,040(80)$ | $4.0\times 10^{-11}$ | UWash-06 | III.1
$B3$ |  $A_{\rm r}(E_{\rm av})$ | $\phantom{-}0.794(79)$ | $1.0\times 10^{-1}$ | StockU-08 |  III.3 (13)
$B4$ |  $f_{\rm c}({\rm H}_{2}^{+*})/f_{\rm c}({\rm d})$ | $\phantom{-}0.999\,231\,659\,33(17)$ | $1.7\times 10^{-10}$ | StockU-08 |  III.3 (2)
$B5$ |  $f_{\rm c}({\rm t})/f_{\rm c}({\rm H}_{2}^{+*})$ | $\phantom{-}0.668\,247\,726\,86(55)$ | $8.2\times 10^{-10}$ | StockU-06 |  III.3 (14)
$B6$ |  $f_{\rm c}(^{3}{\rm He}^{+})/f_{\rm c}({\rm H}_{2}^{+*})$ | $\phantom{-}0.668\,252\,146\,82(55)$ | $8.2\times 10^{-10}$ | StockU-06 |  III.3 (15)
$B7$ |  $A_{\rm r}(^{4}{\rm He})$ | $\phantom{-}4.002\,603\,254\,131(62)$ | $1.5\times 10^{-11}$ | UWash-06 | III.1
$B8$ |  $A_{\rm r}(^{16}{\rm O})$ | $\phantom{-}15.994\,914\,619\,57(18)$ | $1.1\times 10^{-11}$ | UWash-06 | III.1
$B9.1$ |  $A_{\rm r}(^{87}{\rm Rb})$ | $\phantom{-}86.909\,180\,526(12)$ | $1.4\times 10^{-10}$ | AMDC-03 | III.1
$B9.2$ |  $A_{\rm r}(^{87}{\rm Rb})$ | $\phantom{-}86.909\,180\,535(10)$ | $1.2\times 10^{-10}$ | FSU-10 | III.1
$B10.1^{2}$ |  $A_{\rm r}(^{133}{\rm Cs})$ | $\phantom{-}132.905\,451\,932(24)$ | $1.8\times 10^{-10}$ | AMDC-03 | III.1
$B10.2^{2}$ |  $A_{\rm r}(^{133}{\rm Cs})$ | $\phantom{-}132.905\,451\,963(13)$ | $9.8\times 10^{-11}$ | FSU-10 | III.1
$B11$ |  $A_{\rm r}({\rm e})$ | $\phantom{-}0.000\,548\,579\,9111(12)$ | $2.1\times 10^{-9}$ | UWash-95 |  III.4 (20)
$B12$ |  $\delta_{\rm e}$ | $\phantom{-}0.00(33)\times 10^{-12}$ | [$2.8\times 10^{-10}$] | theory | V.1.1
$B13.1^{2}$ |  $a_{\rm e}$ | $\phantom{-}1.159\,652\,1883(42)\times 10^{-3}$ | $3.7\times 10^{-9}$ | UWash-87 |  V.1.2 (124)
$B13.2$ |  $a_{\rm e}$ | $\phantom{-}1.159\,652\,180\,73(28)\times 10^{-3}$ | $2.4\times 10^{-10}$ | HarvU-08 |  V.1.2 (125)
$B14$ |  $\overline{R}$ | $\phantom{-}0.003\,707\,2063(20)$ | $5.4\times 10^{-7}$ | BNL-06 |  V.2.2 (142)
$B15$ |  $\delta_{\rm C}$ | $\phantom{-}0.00(26)\times 10^{-10}$ | [$1.3\times 10^{-11}$] | theory | V.3.1
$B16$ |  $\delta_{\rm O}$ | $\phantom{-}0.0(1.1)\times 10^{-10}$ | [$5.3\times 10^{-11}$] | theory | V.3.1
$B17$ |  $f_{\rm s}({\rm{}^{12}C^{5+}})/f_{\rm c}({\rm{}^{12}C^{5+}})$ | $\phantom{-}4376.210\,4989(23)$ | $5.2\times 10^{-10}$ | GSI-02 |  V.3.2 (182)
$B18$ |  $f_{\rm s}({\rm{}^{16}O^{7+}})/f_{\rm c}({\rm{}^{16}O^{7+}})$ | $\phantom{-}4164.376\,1837(32)$ | $7.6\times 10^{-10}$ | GSI-02 |  V.3.2 (183)
$B19$ |  $\mu_{\rm e^{-}}({\rm H})/\mu_{\rm p}({\rm H})$ | $-658.210\,7058(66)$ | $1.0\times 10^{-8}$ | MIT-72 | VI.1.3
$B20$ |  $\mu_{\rm d}({\rm D})/\mu_{\rm e^{-}}({\rm D})$ | $-4.664\,345\,392(50)\times 10^{-4}$ | $1.1\times 10^{-8}$ | MIT-84 | VI.1.3
$B21$ |  $\mu_{\rm p}({\rm HD})/\mu_{\rm d}({\rm HD})$ | $\phantom{-}3.257\,199\,531(29)$ | $8.9\times 10^{-9}$ | StPtrsb-03 | VI.1.3
$B22$ |  $\sigma_{\rm dp}$ | $\phantom{-}15(2)\times 10^{-9}$ |  | StPtrsb-03 | VI.1.3
$B23$ |  $\mu_{\rm t}({\rm HT})/\mu_{\rm p}({\rm HT})$ | $\phantom{-}1.066\,639\,887(10)$ | $9.4\times 10^{-9}$ | StPtrsb-03 | VI.1.3
$B24$ |  $\sigma_{\rm tp}$ | $\phantom{-}20(3)\times 10^{-9}$ |  | StPtrsb-03 | VI.1.3
$B25$ |  $\mu_{\rm e^{-}}({\rm H})/\mu_{\rm p}^{\prime}$ | $-658.215\,9430(72)$ | $1.1\times 10^{-8}$ | MIT-77 | VI.1.3
$B26$ |  $\mu_{\rm h}^{\prime}/\mu_{\rm p}^{\prime}$ | $-0.761\,786\,1313(33)$ | $4.3\times 10^{-9}$ | NPL-93 | VI.1.3
$B27$ |  $\mu_{\rm n}/\mu_{\rm p}^{\prime}$ | $-0.684\,996\,94(16)$ | $2.4\times 10^{-7}$ | ILL-79 | VI.1.3
$B28$ |  $\delta_{\rm Mu}$ |  $\phantom{-}0(101)$ Hz | [$2.3\times 10^{-8}$] | theory | VI.2.1
$B29.1$ |  $\Delta\nu_{\rm Mu}$ |  $\phantom{-}4\,463\,302.88(16)$ kHz | $3.6\times 10^{-8}$ | LAMPF-82 |  VI.2.2 (221)
$B29.2$ |  $\Delta\nu_{\rm Mu}$ |  $\phantom{-}4\,463\,302\,765(53)$ Hz | $1.2\times 10^{-8}$ | LAMPF-99 |  VI.2.2 (224)
$B30$ |  $\nu(58~{}{\rm MHz})$ |  $\phantom{-}627\,994.77(14)$ kHz | $2.2\times 10^{-7}$ | LAMPF-82 |  VI.2.2 (222)
$B31$ |  $\nu(72~{}{\rm MHz})$ |  $\phantom{-}668\,223\,166(57)$ Hz | $8.6\times 10^{-8}$ | LAMPF-99 |  VI.2.2 (225)
$B32.1^{2}$ |  ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ |  $\phantom{-}2.675\,154\,05(30)\times 10^{8}$ s-1 T-1 | $1.1\times 10^{-7}$ | NIST-89 | VIII.2
$B32.2^{2}$ |  ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ |  $\phantom{-}2.675\,1530(18)\times 10^{8}$ s-1 T-1 | $6.6\times 10^{-7}$ | NIM-95 | VIII.2
$B33^{2}$ |  ${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$ |  $\phantom{-}2.037\,895\,37(37)\times 10^{8}$ s-1 T-1 | $1.8\times 10^{-7}$ | KR/VN-98 | VIII.2
$B34.1^{2}$ |  ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ |  $\phantom{-}2.675\,1525(43)\times 10^{8}$ s-1 T-1 | $1.6\times 10^{-6}$ | NIM-95 | VIII.2
$B34.2^{2}$ |  ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ |  $\phantom{-}2.675\,1518(27)\times 10^{8}$ s-1 T-1 | $1.0\times 10^{-6}$ | NPL-79 | VIII.2
$B35.1^{2}$ |  $R_{\rm K}$ | $\phantom{-}25\,812.808\,31(62)\ {\rm\Omega}$ | $2.4\times 10^{-8}$ | NIST-97 | VIII.2
$B35.2^{2}$ |  $R_{\rm K}$ | $\phantom{-}25\,812.8071(11)\ {\rm\Omega}$ | $4.4\times 10^{-8}$ | NMI-97 | VIII.2
$B35.3^{2}$ |  $R_{\rm K}$ | $\phantom{-}25\,812.8092(14)\ {\rm\Omega}$ | $5.4\times 10^{-8}$ | NPL-88 | VIII.2
$B35.4^{2}$ |  $R_{\rm K}$ | $\phantom{-}25\,812.8084(34)\ {\rm\Omega}$ | $1.3\times 10^{-7}$ | NIM-95 | VIII.2
$B35.5^{2}$ |  $R_{\rm K}$ | $\phantom{-}25\,812.8081(14)\ {\rm\Omega}$ | $5.3\times 10^{-8}$ | LNE-01 | VIII.2
$B36.1^{2}$ |  $K_{\rm J}$ | $\phantom{-}483\,597.91(13)\ {\rm GHz\ V^{-1}}$ | $2.7\times 10^{-7}$ | NMI-89 | VIII.2
$B36.2^{2}$ |  $K_{\rm J}$ | $\phantom{-}483\,597.96(15)\ {\rm GHz\ V^{-1}}$ | $3.1\times 10^{-7}$ | PTB-91 | VIII.2
$B37.1^{3}$ |  $K_{\rm J}^{2}R_{\rm K}$ | $\phantom{-}6.036\,7625(12)\times 10^{33}\ {\rm J^{-1}\ s^{-1}}$ | $2.0\times 10^{-7}$ | NPL-90 | VIII.2
$B37.2^{3}$ |  $K_{\rm J}^{2}R_{\rm K}$ | $\phantom{-}6.036\,761\,85(53)\times 10^{33}\ {\rm J^{-1}\ s^{-1}}$ | $8.7\times 10^{-8}$ | NIST-98 | VIII.2
$B37.3^{3}$ |  $K_{\rm J}^{2}R_{\rm K}$ | $\phantom{-}6.036\,761\,85(22)\times 10^{33}\ {\rm J^{-1}\ s^{-1}}$ | $3.6\times 10^{-8}$ | NIST-07 | VIII.2
$B37.4^{3}$ |  $K_{\rm J}^{2}R_{\rm K}$ | $\phantom{-}6.036\,7597(12)\times 10^{33}\ {\rm J^{-1}\ s^{-1}}$ | $2.0\times 10^{-7}$ | NPL-12 |  VIII.2.1 (244)
$B37.5^{2}$ |  $K_{\rm J}^{2}R_{\rm K}$ | $\phantom{-}6.036\,7617(18)\times 10^{33}\ {\rm J^{-1}\ s^{-1}}$ | $2.9\times 10^{-7}$ | METAS-11 |  VIII.2.2 (246)
$B38^{2}$ |  ${\cal F}_{90}$ | $\phantom{-}96\,485.39(13)\ {\rm C\ mol^{-1}}$ | $1.3\times 10^{-6}$ | NIST-80 | VIII.2
$B39$ |  $d_{220}({\rm{\scriptstyle{MO^{*}}}})$ |  $\phantom{-}192\,015.5508(42)$ fm | $2.2\times 10^{-8}$ | INRIM-08 |  IX.1 (249)
$B40$ |  $d_{220}({\rm{\scriptstyle{W04}}})$ |  $\phantom{-}192\,015.5702(29)$ fm | $1.5\times 10^{-8}$ | INRIM-09 |  IX.1 (250)
$B41.1$ |  $d_{220}({\rm{\scriptstyle{W4.2a}}})$ |  $\phantom{-}192\,015.563(12)$ fm | $6.2\times 10^{-8}$ | PTB-81 |  IX.1 (248)
$B41.2$ |  $d_{220}({\rm{\scriptstyle{W4.2a}}})$ |  $\phantom{-}192\,015.5691(29)$ fm | $1.5\times 10^{-8}$ | INRIM-09 |  IX.1 (251)
$B42$ |  $1-d_{220}({\rm{\scriptstyle{N}}})/d_{220}({\rm{\scriptstyle W17}})$ | $\phantom{-}7(22)\times 10^{-9}$ |  | NIST-97 | IX.2
$B43$ |  $1-d_{220}({\rm{\scriptstyle{W17}}})/d_{220}({\rm{\scriptstyle ILL}})$ | $-8(22)\times 10^{-9}$ |  | NIST-99 | IX.2
$B44$ |  $1-d_{220}({\rm{\scriptstyle{MO^{*}}}})/d_{220}({\rm{\scriptstyle ILL}})$ | $\phantom{-}86(27)\times 10^{-9}$ |  | NIST-99 | IX.2
$B45$ |  $1-d_{220}({\rm{\scriptstyle{NR3}}})/d_{220}({\rm{\scriptstyle ILL}})$ | $\phantom{-}33(22)\times 10^{-9}$ |  | NIST-99 | IX.2
$B46$ |  $d_{220}({\rm{\scriptstyle{NR3}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $-11(21)\times 10^{-9}$ |  | NIST-06 | IX.2
$B47$ |  $d_{220}({\rm{\scriptstyle{NR4}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $\phantom{-}25(21)\times 10^{-9}$ |  | NIST-06 | IX.2
$B48$ |  $d_{220}({\rm{\scriptstyle{W17}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $\phantom{-}11(21)\times 10^{-9}$ |  | NIST-06 | IX.2
$B49$ |  $d_{220}({\rm{\scriptstyle{W4.2a}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $-1(21)\times 10^{-9}$ |  | PTB-98 | IX.2
$B50$ |  $d_{220}({\rm{\scriptstyle{W17}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $\phantom{-}22(22)\times 10^{-9}$ |  | PTB-98 | IX.2
$B51$ |  $d_{220}({\rm{\scriptstyle{MO^{*}4}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $-103(28)\times 10^{-9}$ |  | PTB-98 | IX.2
$B52$ |  $d_{220}({\rm{\scriptstyle{NR3}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $-23(21)\times 10^{-9}$ |  | PTB-98 | IX.2
$B53$ |  $d_{220}/d_{220}({\rm{\scriptstyle W04}})-1$ | $\phantom{-}10(11)\times 10^{-9}$ |  | PTB-03 | IX.2
$B54^{3}$ |  $N_{\rm A}$ |  $\phantom{-}6.022\,140\,82(18)\times 10^{23}$ m3 mol-1 | $3.0\times 10^{-8}$ | IAC-11 |  IX.6 (267)
$B55$ |  $\lambda_{\rm meas}/d_{220}({\rm{\scriptstyle ILL}})$ |  $\phantom{-}0.002\,904\,302\,46(50)$ m s-1 | $1.7\times 10^{-7}$ | NIST-99 |  IX.3 (252)
$B56^{2}$ |  $h/m({\rm{}^{133}Cs})$ |  $\phantom{-}3.002\,369\,432(46)\times 10^{-9}$ m2 s-1 | $1.5\times 10^{-8}$ | StanfU-02 |  VII.1 (234)
$B57$ |  $h/m({\rm{}^{87}Rb})$ |  $\phantom{-}4.591\,359\,2729(57)\times 10^{-9}$ m2 s-1 | $1.2\times 10^{-9}$ | LKB-11 |  VII.2 (236)
$B58.1$ |  $R$ |  $\phantom{-}8.314\,504(70)$ J mol-1 K-1 | $8.4\times 10^{-6}$ | NPL-79 | X.1.1
$B58.2$ |  $R$ |  $\phantom{-}8.314\,471(15)$ J mol-1 K-1 | $1.8\times 10^{-6}$ | NIST-88 | X.1.1
$B58.3$ |  $R$ |  $\phantom{-}8.314\,467(22)$ J mol-1 K-1 | $2.7\times 10^{-6}$ | LNE-09 | X.1.2
$B58.4$ |  $R$ |  $\phantom{-}8.314\,468(26)$ J mol-1 K-1 | $3.1\times 10^{-6}$ | NPL-10 | X.1.3
$B58.5$ |  $R$ |  $\phantom{-}8.314\,412(63)$ J mol-1 K-1 | $7.5\times 10^{-6}$ | INRIM-10 | X.1.4
$B58.6$ |  $R$ |  $\phantom{-}8.314\,456(10)$ J mol-1 K-1 | $1.2\times 10^{-6}$ | LNE-11 | X.1.2
$B59^{2}$ |  $k$ |  $\phantom{-}1.380\,653(13)\times 10^{-23}$ J K-1 | $9.1\times 10^{-6}$ | NIST-07 | X.2.1
$B60^{2}$ |  $k/h$ |  $\phantom{-}2.083\,666(25)\times 10^{10}$ Hz K-1 | $1.2\times 10^{-5}$ | NIST-11 | X.2.2
$B61$ |  $\lambda({\rm CuK\alpha_{1}})/d_{220}({\rm{\scriptstyle{W4.2a}}})$ | $\phantom{-}0.802\,327\,11(24)$ | $3.0\times 10^{-7}$ | FSUJ/PTB-91 |  IX.4 (257)
$B62$ |  $\lambda({\rm WK\alpha_{1}})/d_{220}({\rm{\scriptstyle{N}}})$ | $\phantom{-}0.108\,852\,175(98)$ | $9.0\times 10^{-7}$ | NIST-79 |  IX.4 (258)
$B63$ |  $\lambda({\rm MoK\alpha_{1}})/d_{220}({\rm{\scriptstyle{N}}})$ | $\phantom{-}0.369\,406\,04(19)$ | $5.3\times 10^{-7}$ | NIST-73 |  IX.4 (259)
$B64$ |  $\lambda({\rm CuK\alpha_{1}})/d_{220}({\rm{\scriptstyle{N}}})$ | $\phantom{-}0.802\,328\,04(77)$ | $9.6\times 10^{-7}$ | NIST-73 |  IX.4 (260)
1 The values in brackets are relative to the quantities $a_{\rm e}$, $g_{\rm
e^{-}}(^{12}{\rm C}^{5+})$, $g_{\rm e^{-}}(^{16}{\rm O}^{7+})$, or
$\Delta\nu_{\rm Mu}$ as appropriate.
2 Datum not included in the final least-squares adjustment that provides the
recommended values of the constants.
3 Datum included in the final least-squares adjustment with an expanded
uncertainty.
Table 20: (Continued). Summary of principal input data for the determination of the 2006 recommended values of the fundamental constants ($R_{\infty}$ and $G$ excepted). Table 21: Non-negligible correlation coefficients $r(x_{i},x_{j})$ of the input data in Table LABEL:tab:pdata. For simplicity, the two items of data to which a particular correlation coefficient corresponds are identified by their item numbers in Table LABEL:tab:pdata. $r$($B1$, $B2.1$) | = $0.073$ |  $r$($B39$, $B40$) | = $0.023$ |  $r$($B43$, $B45$) | = $0.516$ |  $r$($B47$, $B48$) | = $0.509$
---|---|---|---|---|---|---|---
$r$($B2.2$, $B7$) | = $0.127$ |  $r$($B39$, $B41.2$) | = $0.023$ |  $r$($B43$, $B46$) | = $0.065$ |  $r$($B49$, $B50$) | = $0.469$
$r$($B2.2$, $B8$) | = $0.089$ |  $r$($B39$, $B54$) | = $-0.026$ |  $r$($B43$, $B47$) | = $0.065$ |  $r$($B49$, $B51$) | = $0.372$
$r$($B5$, $B6$) | = $0.876$ |  $r$($B40$, $B41.2$) | = $0.027$ |  $r$($B43$, $B48$) | = $-0.367$ |  $r$($B49$, $B52$) | = $0.502$
$r$($B7$, $B8$) | = $0.181$ |  $r$($B40$, $B54$) | = $-0.029$ |  $r$($B44$, $B45$) | = $0.421$ |  $r$($B50$, $B51$) | = $0.347$
$r$($B15$, $B16$) | = $0.994$ |  $r$($B41.2$, $B54$) | = $-0.029$ |  $r$($B44$, $B46$) | = $0.053$ |  $r$($B50$, $B52$) | = $0.469$
$r$($B17$, $B18$) | = $0.082$ |  $r$($B42$, $B43$) | = $-0.288$ |  $r$($B44$, $B47$) | = $0.053$ |  $r$($B51$, $B52$) | = $0.372$
$r$($B29.1$, $B30$) | = $0.227$ |  $r$($B42$, $B44$) | = $0.096$ |  $r$($B44$, $B48$) | = $0.053$ |  $r$($B58.3$, $B58.4$) | = $0.002$
$r$($B29.2$, $B31$) | = $0.195$ |  $r$($B42$, $B45$) | = $0.117$ |  $r$($B45$, $B46$) | = $-0.367$ |  $r$($B58.3$, $B58.5$) | = $0.001$
$r$($B32.2$, $B34.1$) | = $-0.014$ |  $r$($B42$, $B46$) | = $0.066$ |  $r$($B45$, $B47$) | = $0.065$ |  $r$($B58.3$, $B58.6$) | = $0.032$
$r$($B36.1$, $B58.2$) | = $0.068$ |  $r$($B42$, $B47$) | = $0.066$ |  $r$($B45$, $B48$) | = $0.065$ |  $r$($B58.4$, $B58.6$) | = $0.012$
$r$($B37.1$, $B37.4$) | = $0.003$ |  $r$($B42$, $B48$) | = $0.504$ |  $r$($B46$, $B47$) | = $0.509$ |  |
$r$($B37.2$, $B37.3$) | = $0.140$ |  $r$($B43$, $B44$) | = $0.421$ |  $r$($B46$, $B48$) | = $0.509$ |  |
Table 22: Summary of principal input data for the determination of the
relative atomic mass of the electron from antiprotonic helium transitions. The
numbers in parentheses $(n,l:n^{\prime},l^{\prime})$ denote the transition
$(n,l)\rightarrow(n^{\prime},l^{\prime})$.

Item  | Input Datum  |  Value  |  Relative standard  | Identification | Sec.
---|---|---|---|---|---
number  |  |  |  uncertainty1 $u_{\rm r}$ |  |
$C1$ |  $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(32,31:31,30)$ |  $0.00(82)$ MHz  |  $[7.3\times 10^{-10}]$ | JINR-06 | IV.2.1
$C2$ |  $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(35,33:34,32)$ |  $0.0(1.0)$ MHz  |  $[1.3\times 10^{-9}]$ | JINR-06 | IV.2.1
$C3$ |  $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(36,34:35,33)$ |  $0.0(1.1)$ MHz  |  $[1.6\times 10^{-9}]$ | JINR-06 | IV.2.1
$C4$ |  $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(37,34:36,33)$ |  $0.0(1.1)$ MHz  |  $[1.8\times 10^{-9}]$ | JINR-06 | IV.2.1
$C5$ |  $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(39,35:38,34)$ |  $0.0(1.2)$ MHz  |  $[2.3\times 10^{-9}]$ | JINR-06 | IV.2.1
$C6$ |  $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(40,35:39,34)$ |  $0.0(1.3)$ MHz  |  $[2.9\times 10^{-9}]$ | JINR-06 | IV.2.1
$C7$ |  $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(37,35:38,34)$ |  $0.0(1.8)$ MHz  |  $[4.5\times 10^{-9}]$ | JINR-06 | IV.2.1
$C8$ |  $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(33,32:31,30)$ |  $0.0(1.6)$ MHz  |  $[7.6\times 10^{-10}]$ | JINR-10 | IV.2.1
$C9$ |  $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(36,34:34,32)$ |  $0.0(2.1)$ MHz  |  $[1.4\times 10^{-9}]$ | JINR-10 | IV.2.1
$C10$ |  $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(32,31:31,30)$ |  $0.00(91)$ MHz  |  $[8.7\times 10^{-10}]$ | JINR-06 | IV.2.1
$C11$ |  $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(34,32:33,31)$ |  $0.0(1.1)$ MHz  |  $[1.4\times 10^{-9}]$ | JINR-06 | IV.2.1
$C12$ |  $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(36,33:35,32)$ |  $0.0(1.2)$ MHz  |  $[1.8\times 10^{-9}]$ | JINR-06 | IV.2.1
$C13$ |  $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(38,34:37,33)$ |  $0.0(1.1)$ MHz  |  $[2.3\times 10^{-9}]$ | JINR-06 | IV.2.1
$C14$ |  $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(36,34:37,33)$ |  $0.0(1.8)$ MHz  |  $[4.4\times 10^{-9}]$ | JINR-06 | IV.2.1
$C15$ |  $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(35,33:33,31)$ |  $0.0(2.2)$ MHz  |  $[1.4\times 10^{-9}]$ | JINR-10 | IV.2.1
$C16$ |  $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(32,31:31,30)$ |  $1\,132\,609\,209(15)$ MHz  |  $1.4\times 10^{-8}$ | CERN-06 | IV.2.2
$C17$ |  $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(35,33:34,32)$ |  $804\,633\,059.0(8.2)$ MHz  |  $1.0\times 10^{-8}$ | CERN-06 | IV.2.2
$C18$ |  $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(36,34:35,33)$ |  $717\,474\,004(10)$ MHz  |  $1.4\times 10^{-8}$ | CERN-06 | IV.2.2
$C19$ |  $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(37,34:36,33)$ |  $636\,878\,139.4(7.7)$ MHz  |  $1.2\times 10^{-8}$ | CERN-06 | IV.2.2
$C20$ |  $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(39,35:38,34)$ |  $501\,948\,751.6(4.4)$ MHz  |  $8.8\times 10^{-9}$ | CERN-06 | IV.2.2
$C21$ |  $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(40,35:39,34)$ |  $445\,608\,557.6(6.3)$ MHz  |  $1.4\times 10^{-8}$ | CERN-06 | IV.2.2
$C22$ |  $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(37,35:38,34)$ |  $412\,885\,132.2(3.9)$ MHz  |  $9.4\times 10^{-9}$ | CERN-06 | IV.2.2
$C23$ |  $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(33,32:31,30)$ |  $2\,145\,054\,858.2(5.1)$ MHz  |  $2.4\times 10^{-9}$ | CERN-10 | IV.2.2
$C24$ |  $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(36,34:34,32)$ |  $1\,522\,107\,061.8(3.5)$ MHz  |  $2.3\times 10^{-9}$ | CERN-10 | IV.2.2
$C25$ |  $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(32,31:31,30)$ |  $1\,043\,128\,608(13)$ MHz  |  $1.3\times 10^{-8}$ | CERN-06 | IV.2.2
$C26$ |  $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(34,32:33,31)$ |  $822\,809\,190(12)$ MHz  |  $1.5\times 10^{-8}$ | CERN-06 | IV.2.2
$C27$ |  $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(36,33:35,32)$ |  $646\,180\,434(12)$ MHz  |  $1.9\times 10^{-8}$ | CERN-06 | IV.2.2
$C28$ |  $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(38,34:37,33)$ |  $505\,222\,295.7(8.2)$ MHz  |  $1.6\times 10^{-8}$ | CERN-06 | IV.2.2
$C29$ |  $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(36,34:37,33)$ |  $414\,147\,507.8(4.0)$ MHz  |  $9.7\times 10^{-9}$ | CERN-06 | IV.2.2
$C30$ |  $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(35,33:33,31)$ |  $1\,553\,643\,099.6(7.1)$ MHz  |  $4.6\times 10^{-9}$ | CERN-10 | IV.2.2

1 The values in brackets are relative to the corresponding transition
frequency.

Table 23: Non-negligible correlation coefficients $r(x_{i},x_{j})$ of the input data in Table 22. For simplicity, the two items of data to which a particular correlation coefficient corresponds are identified by their item numbers in Table 22. $r$($C1$, $C2$) | = $0.929$ |  $r$($C4$, $C10$) | = $0.959$ |  $r$($C9$, $C14$) | = $-0.976$ |  $r$($C18$, $C27$) | = $0.141$
---|---|---|---|---|---|---|---
$r$($C1$, $C3$) | = $0.936$ |  $r$($C4$, $C11$) | = $0.949$ |  $r$($C9$, $C15$) | = $0.986$ |  $r$($C18$, $C28$) | = $0.106$
$r$($C1$, $C4$) | = $0.936$ |  $r$($C4$, $C12$) | = $0.907$ |  $r$($C10$, $C11$) | = $0.978$ |  $r$($C18$, $C29$) | = $0.217$
$r$($C1$, $C5$) | = $0.912$ |  $r$($C4$, $C13$) | = $0.931$ |  $r$($C10$, $C12$) | = $0.934$ |  $r$($C19$, $C20$) | = $0.268$
$r$($C1$, $C6$) | = $0.758$ |  $r$($C4$, $C14$) | = $-0.952$ |  $r$($C10$, $C13$) | = $0.959$ |  $r$($C19$, $C21$) | = $0.193$
$r$($C1$, $C7$) | = $-0.947$ |  $r$($C4$, $C15$) | = $0.961$ |  $r$($C10$, $C14$) | = $-0.980$ |  $r$($C19$, $C22$) | = $0.302$
$r$($C1$, $C8$) | = $0.954$ |  $r$($C5$, $C6$) | = $0.734$ |  $r$($C10$, $C15$) | = $0.990$ |  $r$($C19$, $C25$) | = $0.172$
$r$($C1$, $C9$) | = $0.960$ |  $r$($C5$, $C7$) | = $-0.917$ |  $r$($C11$, $C12$) | = $0.925$ |  $r$($C19$, $C26$) | = $0.190$
$r$($C1$, $C10$) | = $0.964$ |  $r$($C5$, $C8$) | = $0.924$ |  $r$($C11$, $C13$) | = $0.949$ |  $r$($C19$, $C27$) | = $0.189$
$r$($C1$, $C11$) | = $0.954$ |  $r$($C5$, $C9$) | = $0.930$ |  $r$($C11$, $C14$) | = $-0.970$ |  $r$($C19$, $C28$) | = $0.144$
$r$($C1$, $C12$) | = $0.912$ |  $r$($C5$, $C10$) | = $0.934$ |  $r$($C11$, $C15$) | = $0.980$ |  $r$($C19$, $C29$) | = $0.294$
$r$($C1$, $C13$) | = $0.936$ |  $r$($C5$, $C11$) | = $0.925$ |  $r$($C12$, $C13$) | = $0.907$ |  $r$($C20$, $C21$) | = $0.210$
$r$($C1$, $C14$) | = $-0.957$ |  $r$($C5$, $C12$) | = $0.883$ |  $r$($C12$, $C14$) | = $-0.927$ |  $r$($C20$, $C22$) | = $0.295$
$r$($C1$, $C15$) | = $0.966$ |  $r$($C5$, $C13$) | = $0.907$ |  $r$($C12$, $C15$) | = $0.936$ |  $r$($C20$, $C25$) | = $0.152$
$r$($C2$, $C3$) | = $0.924$ |  $r$($C5$, $C14$) | = $-0.927$ |  $r$($C13$, $C14$) | = $-0.952$ |  $r$($C20$, $C26$) | = $0.167$
$r$($C2$, $C4$) | = $0.924$ |  $r$($C5$, $C15$) | = $0.936$ |  $r$($C13$, $C15$) | = $0.961$ |  $r$($C20$, $C27$) | = $0.169$
$r$($C2$, $C5$) | = $0.900$ |  $r$($C6$, $C7$) | = $-0.762$ |  $r$($C14$, $C15$) | = $-0.982$ |  $r$($C20$, $C28$) | = $0.141$
$r$($C2$, $C6$) | = $0.748$ |  $r$($C6$, $C8$) | = $0.767$ |  $r$($C16$, $C17$) | = $0.210$ |  $r$($C20$, $C29$) | = $0.287$
$r$($C2$, $C7$) | = $-0.935$ |  $r$($C6$, $C9$) | = $0.773$ |  $r$($C16$, $C18$) | = $0.167$ |  $r$($C21$, $C22$) | = $0.235$
$r$($C2$, $C8$) | = $0.941$ |  $r$($C6$, $C10$) | = $0.776$ |  $r$($C16$, $C19$) | = $0.224$ |  $r$($C21$, $C25$) | = $0.107$
$r$($C2$, $C9$) | = $0.948$ |  $r$($C6$, $C11$) | = $0.768$ |  $r$($C16$, $C20$) | = $0.197$ |  $r$($C21$, $C26$) | = $0.118$
$r$($C2$, $C10$) | = $0.952$ |  $r$($C6$, $C12$) | = $0.734$ |  $r$($C16$, $C21$) | = $0.138$ |  $r$($C21$, $C27$) | = $0.122$
$r$($C2$, $C11$) | = $0.942$ |  $r$($C6$, $C13$) | = $0.753$ |  $r$($C16$, $C22$) | = $0.222$ |  $r$($C21$, $C28$) | = $0.112$
$r$($C2$, $C12$) | = $0.900$ |  $r$($C6$, $C14$) | = $-0.770$ |  $r$($C16$, $C25$) | = $0.129$ |  $r$($C21$, $C29$) | = $0.229$
$r$($C2$, $C13$) | = $0.924$ |  $r$($C6$, $C15$) | = $0.778$ |  $r$($C16$, $C26$) | = $0.142$ |  $r$($C22$, $C25$) | = $0.170$
$r$($C2$, $C14$) | = $-0.945$ |  $r$($C7$, $C8$) | = $-0.959$ |  $r$($C16$, $C27$) | = $0.141$ |  $r$($C22$, $C26$) | = $0.188$
$r$($C2$, $C15$) | = $0.954$ |  $r$($C7$, $C9$) | = $-0.966$ |  $r$($C16$, $C28$) | = $0.106$ |  $r$($C22$, $C27$) | = $0.191$
$r$($C3$, $C4$) | = $0.931$ |  $r$($C7$, $C10$) | = $-0.970$ |  $r$($C16$, $C29$) | = $0.216$ |  $r$($C22$, $C28$) | = $0.158$
$r$($C3$, $C5$) | = $0.907$ |  $r$($C7$, $C11$) | = $-0.960$ |  $r$($C17$, $C18$) | = $0.209$ |  $r$($C22$, $C29$) | = $0.324$
$r$($C3$, $C6$) | = $0.753$ |  $r$($C7$, $C12$) | = $-0.917$ |  $r$($C17$, $C19$) | = $0.280$ |  $r$($C23$, $C24$) | = $0.155$
$r$($C3$, $C7$) | = $-0.942$ |  $r$($C7$, $C13$) | = $-0.942$ |  $r$($C17$, $C20$) | = $0.247$ |  $r$($C23$, $C30$) | = $0.104$
$r$($C3$, $C8$) | = $0.948$ |  $r$($C7$, $C14$) | = $0.963$ |  $r$($C17$, $C21$) | = $0.174$ |  $r$($C24$, $C30$) | = $0.167$
$r$($C3$, $C9$) | = $0.955$ |  $r$($C7$, $C15$) | = $-0.972$ |  $r$($C17$, $C22$) | = $0.278$ |  $r$($C25$, $C26$) | = $0.109$
$r$($C3$, $C10$) | = $0.959$ |  $r$($C8$, $C9$) | = $0.973$ |  $r$($C17$, $C25$) | = $0.161$ |  $r$($C25$, $C27$) | = $0.108$
$r$($C3$, $C11$) | = $0.949$ |  $r$($C8$, $C10$) | = $0.977$ |  $r$($C17$, $C26$) | = $0.178$ |  $r$($C25$, $C28$) | = $0.081$
$r$($C3$, $C12$) | = $0.907$ |  $r$($C8$, $C11$) | = $0.967$ |  $r$($C17$, $C27$) | = $0.177$ |  $r$($C25$, $C29$) | = $0.166$
$r$($C3$, $C13$) | = $0.931$ |  $r$($C8$, $C12$) | = $0.924$ |  $r$($C17$, $C28$) | = $0.132$ |  $r$($C26$, $C27$) | = $0.120$
$r$($C3$, $C14$) | = $-0.952$ |  $r$($C8$, $C13$) | = $0.948$ |  $r$($C17$, $C29$) | = $0.271$ |  $r$($C26$, $C28$) | = $0.090$
$r$($C3$, $C15$) | = $0.961$ |  $r$($C8$, $C14$) | = $-0.969$ |  $r$($C18$, $C19$) | = $0.223$ |  $r$($C26$, $C29$) | = $0.184$
$r$($C4$, $C5$) | = $0.907$ |  $r$($C8$, $C15$) | = $0.979$ |  $r$($C18$, $C20$) | = $0.198$ |  $r$($C27$, $C28$) | = $0.091$
$r$($C4$, $C6$) | = $0.753$ |  $r$($C9$, $C10$) | = $0.984$ |  $r$($C18$, $C21$) | = $0.140$ |  $r$($C27$, $C29$) | = $0.186$
$r$($C4$, $C7$) | = $-0.942$ |  $r$($C9$, $C11$) | = $0.974$ |  $r$($C18$, $C22$) | = $0.223$ |  $r$($C28$, $C29$) | = $0.154$
$r$($C4$, $C8$) | = $0.948$ |  $r$($C9$, $C12$) | = $0.930$ |  $r$($C18$, $C25$) | = $0.128$ |  |
$r$($C4$, $C9$) | = $0.955$ |  $r$($C9$, $C13$) | = $0.955$ |  $r$($C18$, $C26$) | = $0.142$ |  |
Table 24: Summary of values of $G$ used to determine the 2010 recommended
value (see also Table 17, Sec. XI).

|  | Relative  |
---|---|---|---
Item  | Value1 | standard  |
number  | ($10^{-11}$ m3 kg-1 s-2)  | uncertainty $u_{\rm r}$ | Identification
$G1$ |  $6.672\,48(43)$ |  $6.4\times 10^{-5}$ | NIST-82
$G2$ |  $6.672\,9(5)$ |  $7.5\times 10^{-5}$ | TR&D-96
$G3$ |  $6.673\,98(70)$ |  $1.0\times 10^{-4}$ | LANL-97
$G4$ |  $6.674\,255(92)$ |  $1.4\times 10^{-5}$ | UWash-00
$G5$ |  $6.675\,59(27)$ |  $4.0\times 10^{-5}$ | BIPM-01
$G6$ |  $6.674\,22(98)$ |  $1.5\times 10^{-4}$ | UWup-02
$G7$ |  $6.673\,87(27)$ |  $4.0\times 10^{-5}$ | MSL-03
$G8$ |  $6.672\,28(87)$ |  $1.3\times 10^{-4}$ | HUST-05
$G9$ |  $6.674\,25(12)$ |  $1.9\times 10^{-5}$ | UZur-06
$G10$ |  $6.673\,49(18)$ |  $2.7\times 10^{-5}$ | HUST-09
$G11$ |  $6.672\,34(14)$ |  $2.1\times 10^{-5}$ | JILA-10

1Correlation coefficients: $r(G1,G3)=0.351$; $r(G8,G10)=0.234$.

###  XIII.1 Comparison of data through inferred values of $\bm{\alpha}$,
$\bm{h}$, $\bm{k}$ and $\bm{A_{\rm r}({\rm e})}$

Here the level of consistency of the data is shown by comparing values of
$\alpha$, $h$, $k$ and $A_{\rm r}({\rm e})$ that can be inferred from
different types of experiments. Note, however, that the inferred value is for
comparison purposes only; the datum from which it is obtained, not the
inferred value, is used as the input datum in the least-squares calculations.

Table 25: Inferred values of the fine-structure constant $\alpha$ in order of increasing standard uncertainty obtained from the indicated experimental data in Table LABEL:tab:pdata. Primary  | Item  | Identification  | Sec. and Eq.  |  $\alpha^{-1}$ |  Relative standard
---|---|---|---|---|---
source  | number  |  |  |  |  uncertainty $u_{\rm r}$
$a_{\rm e}$ |  $B13.2$ | HarvU-08  |  V.1.3 (127)  |  $137.035\,999\,084(51)$ |  $3.7\times 10^{-10}$
$h/m(^{87}{\rm Rb})$ |  $B57$ | LKB-11  |  VII.2 |  $137.035\,999\,049(90)$ |  $6.6\times 10^{-10}$
$a_{\rm e}$ |  $B11$ | UWash-87  |  V.1.3 (126)  |  $137.035\,998\,19(50)$ |  $3.7\times 10^{-9}$
$h/m(^{133}{\rm Cs})$ |  $B56$ | StanfU-02  |  VII.1 |  $137.036\,0000(11)$ |  $7.7\times 10^{-9}$
$R_{\rm K}$ |  $B35.1$ | NIST-97  |  VIII.2 |  $137.036\,0037(33)$ |  $2.4\times 10^{-8}$
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ |  $B32.1$ | NIST-89  |  VIII.2 |  $137.035\,9879(51)$ |  $3.7\times 10^{-8}$
$R_{\rm K}$ |  $B35.2$ | NMI-97  |  VIII.2 |  $137.035\,9973(61)$ |  $4.4\times 10^{-8}$
$R_{\rm K}$ |  $B35.5$ | LNE-01  |  VIII.2 |  $137.036\,0023(73)$ |  $5.3\times 10^{-8}$
$R_{\rm K}$ |  $B35.3$ | NPL-88  |  VIII.2 |  $137.036\,0083(73)$ |  $5.4\times 10^{-8}$
$\Delta\nu_{\rm Mu}$ |  $B29.1,B29.2$ | LAMPF  |  VI.2.2 (232)  |  $137.036\,0018(80)$ |  $5.8\times 10^{-8}$
${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$ |  $B33$ | KR/VN-98  |  VIII.2 |  $137.035\,9852(82)$ |  $6.0\times 10^{-8}$
$R_{\rm K}$ |  $B35.4$ | NIM-95  |  VIII.2 |  $137.036\,004(18)$ |  $1.3\times 10^{-7}$
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ |  $B32.2$ | NIM-95  |  VIII.2 |  $137.036\,006(30)$ |  $2.2\times 10^{-7}$
$\nu_{\rm H},\nu_{\rm D}$ |  |  |  IV.1.1 (88)  |  $137.036\,003(41)$ |  $3.0\times 10^{-7}$

Figure 1:  Values of the fine-structure constant $\alpha$ with $u_{\rm
r}<10^{-7}$ implied by the input data in Table LABEL:tab:pdata, in order of
decreasing uncertainty from top to bottom (see Table 25).

Figure 2:  Values of the fine-structure constant $\alpha$ with $u_{\rm
r}<10^{-8}$ implied by the input data in Table LABEL:tab:pdata and the 2006
and 2010 CODATA recommended values in chronological order from top to bottom
(see Table 25).

Table 25 and Figs. 1 and 2 compare values of $\alpha$ obtained from the
indicated input data. These values are calculated using the appropriate
observational equation for each input datum as given in Table 33 and the 2010
recommended values of the constants other than $\alpha$ that enter that
equation. (Some inferred values have also been given in the portion of the
paper where the relevant datum is discussed.) Inspection of the Table and
figures shows that there is agreement among the vast majority of the various
values of $\alpha$, and hence the data from which they are obtained, to the
extent that the difference between any two values of $\alpha$ is less than
$2u_{\rm diff}$, the standard uncertainty of the difference.

The two exceptions are the values of $\alpha$ from the NIST-89 result for
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ and, to a lesser extent, the
KR/VN-98 result for ${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$; of the 91
differences, six involving $\alpha$ from NIST-89 and two involving $\alpha$
from KR/VN-98 are greater than $2u_{\rm diff}$. The inconsistency of these
data has in fact been discussed in previous CODATA reports but, as in 2006,
because their self-sensitivity coefficients $S_{\rm c}$ (see Sec. XIII.2
below) are less than $0.01$, they are not included in the final adjustment on
which the 2010 recommended values are based. Hence, their disagreement is not
a serious issue. Examination of the table and figures also shows that even if
all of the data from which these values of $\alpha$ have been inferred were to
be included in the final adjustment, the recommended value of $\alpha$ would
still be determined mainly by the HarvU-08 $a_{\rm e}$ and LKB-10
$h/m(^{87}{\rm Rb})$ data. Indeed, the comparatively large uncertainties of
some of the values of $\alpha$ means that the data from which they are
obtained will have values of $S_{\rm c}<0.01$ and will not be included in the
final adjustment.

Table 26: Inferred values of the Planck constant $h$ in order of increasing standard uncertainty obtained from the indicated experimental data in Table LABEL:tab:pdata. Primary  | Item  | Identification  | Sec. and Eq.  |  $h/({\rm J\ s})$ | Relative standard
---|---|---|---|---|---
source  | number  |  |  |  |  uncertainty $u_{\rm r}$
$N_{\rm A}$(28Si)  |  $B54$ | IAC-11  |  IX.6 (267)  |  $6.626\,070\,09(20)\times 10^{-34}$ | $3.0\times 10^{-8}$
$K_{\rm J}^{2}R_{\rm K}$ |  $B37.3$ | NIST-07  |  VIII.2 |  $6.626\,068\,91(24)\times 10^{-34}$ | $3.6\times 10^{-8}$
$K_{\rm J}^{2}R_{\rm K}$ |  $B37.2$ | NIST-98  |  VIII.2 |  $6.626\,068\,91(58)\times 10^{-34}$ | $8.7\times 10^{-8}$
$K_{\rm J}^{2}R_{\rm K}$ |  $B37.1$ | NPL-90  |  VIII.2 |  $6.626\,0682(13)\times 10^{-34}$ | $2.0\times 10^{-7}$
$K_{\rm J}^{2}R_{\rm K}$ |  $B37.4$ | NPL-12  |  VIII.2.1 (243)  |  $6.626\,0712(13)\times 10^{-34}$ | $2.0\times 10^{-7}$
$K_{\rm J}^{2}R_{\rm K}$ |  $B37.5$ | METAS-11  |  VIII.2.2 (245)  |  $6.626\,0691(20)\times 10^{-34}$ | $2.9\times 10^{-7}$
$K_{\rm J}$ |  $B36.1$ | NMI-89  |  VIII.2 |  $6.626\,0684(36)\times 10^{-34}$ | $5.4\times 10^{-7}$
$K_{\rm J}$ |  $B36.2$ | PTB-91  |  VIII.2 |  $6.626\,0670(42)\times 10^{-34}$ | $6.3\times 10^{-7}$
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ |  $B34.2$ | NPL-79  |  VIII.2 |  $6.626\,0730(67)\times 10^{-34}$ | $1.0\times 10^{-6}$
${\cal F}_{90}$ |  $B38$ | NIST-80  |  VIII.2 |  $6.626\,0657(88)\times 10^{-34}$ | $1.3\times 10^{-6}$
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ |  $B34.1$ | NIM-95  |  VIII.2 |  $6.626\,071(11)\times 10^{-34}$ | $1.6\times 10^{-6}$

Figure 3:  Values of the Planck constant $h$ with $u_{{\rm r}}<10^{-6}$
implied by the input data in Table LABEL:tab:pdata and the 2006 and 2010
CODATA recommended values in chronological order from top to bottom (see Table
26).

Table 26 and Fig. 3 compare values of $h$ obtained from the indicated input
data. The various values of $h$, and hence the data from which they are
calculated, agree to the extent that the 55 differences between any two values
of $h$ is less than $2u_{\rm diff}$, except for the difference between the
NIST-07 and IAC-11 values. In this case, the difference is $3.8u_{\rm diff}$.

Because the uncertainties of these two values of $h$ are smaller than other
values and are comparable, they play the dominant role in the determination of
the recommended value of $h$. This discrepancy is dealt with before carrying
out the final adjustment. The relatively large uncertainties of many of the
other values of $h$ means that the data from which they are calculated will
not be included in the final adjustment.

Table 27: Inferred values of the Boltzmann constant $k$ in order of increasing standard uncertainty obtained from the indicated experimental data in Table LABEL:tab:pdata. Primary  | Item  | Identification  | Section  |  $k/({\rm J\ K^{-1}})$ | Relative standard
---|---|---|---|---|---
source  | number  |  |  |  |  uncertainty $u_{\rm r}$
$R$ |  $B58.6$ | LNE-11  |  X.1.2 |  $1.380\,6477(17)\times 10^{-23}$ | $1.2\times 10^{-6}$
$R$ |  $B58.2$ | NIST-88  |  X.1.1 |  $1.380\,6503(25)\times 10^{-23}$ | $1.8\times 10^{-6}$
$R$ |  $B58.3$ | LNE-09  |  X.1.2 |  $1.380\,6495(37)\times 10^{-23}$ | $2.7\times 10^{-6}$
$R$ |  $B58.4$ | NPL-10  |  X.1.3 |  $1.380\,6496(43)\times 10^{-23}$ | $3.1\times 10^{-6}$
$R$ |  $B58.5$ | INRIM-10  |  X.1.4 |  $1.380\,640(10)\times 10^{-23}$ | $7.5\times 10^{-6}$
$R$ |  $B58.1$ | NPL-79  |  X.1.1 |  $1.380\,656(12)\times 10^{-23}$ | $8.4\times 10^{-6}$
$k$ |  $B59$ | NIST-07  |  X.2.1 |  $1.380\,653(13)\times 10^{-23}$ | $9.1\times 10^{-6}$
$k/h$ |  $B60$ | NIST-11  |  X.2.2 |  $1.380\,652(17)\times 10^{-23}$ | $1.2\times 10^{-5}$

Figure 4:  Values of the Boltzmann constant $k$ implied by the input data in
Table LABEL:tab:pdata and the 2006 and 2010 CODATA recommended values in
chronological order from top to bottom (see Table 27). AGT: acoustic gas
thermometry; RIGT: refractive index gas thermometry; JNT: Johnson noise
thermometry.

Table 27 and Fig. 4 compare values of $k$ obtained from the indicated input
data. Although most of the source data are values of $R$, values of
$k=R/N_{\rm A}$ are compared, because that is the constant used to define the
kelvin in the “New” SI; see, for example, Mills _et al._ (2011). All of these
values are in general agreement, with none of the 28 differences exceeding
$2u_{\rm diff}$. However, some of the input data from which they are
calculated have uncertainties so large that they will not be included in the
final adjustment.

Table 28: Inferred values of the electron relative atomic mass $A_{\rm r}({\rm e})$ in order of increasing standard uncertainty obtained from the indicated experimental data in Table LABEL:tab:pdata. Primary | Item  | Identification  | Sec. and Eq.  |  $A_{\rm r}({\rm e})$ |  Relative standard
---|---|---|---|---|---
source | number  |  |  |  |  uncertainty $u_{\rm r}$
$f_{\rm s}({\rm C})/f_{\rm c}({\rm C})$ |  $B17$ | GSI-02  |  V.3.2 (187)  |  $0.000\,548\,579\,909\,32(29)$ | $5.2\times 10^{-10}$
$f_{\rm s}({\rm O})/f_{\rm c}({\rm O})$ |  $B18$ | GSI-02  |  V.3.2 (188)  |  $0.000\,548\,579\,909\,57(42)$ | $7.6\times 10^{-10}$
$\Delta\nu_{\bar{\rm p}\,{\rm He^{+}}}$ |  $C16-C30$ | CERN-06/10  |  IV.2.3 (101)  |  $0.000\,548\,579\,909\,14(75)$ | $1.4\times 10^{-9}$
$A_{\rm r}({\rm e})$ |  $B11$ | UWash-95  |  III.4 (20)  |  $0.000\,548\,579\,9111(12)$ |  $2.1\times 10^{-9}$

Figure 5:  Values of the electron relative atomic mass $A_{\rm r}({\rm e})$
implied by the input data in Tables LABEL:tab:pdata and 22 and the 2006 and
2010 CODATA recommended values in chronological order from top to bottom (see
Table 28).

Finally, in Table 28 and Fig. 5 we compare four values of $A_{\rm r}({\rm e})$
calculated from different input data as indicated. They are in agreement, with
all six differences less than $2u_{\rm diff}$. Further, since the four
uncertainties are comparable, all four of the source data are included in the
final adjustment.

###  XIII.2 Multivariate analysis of data

Our multivariate analysis of the data employs a well known least-squares
method that takes correlations among the input data into account. Used in the
three previous adjustments, it is described in Appendix E of CODATA-98 and
references cited therein. We recall from that appendix that a least-squares
adjustment is characterized by the number of input data $N$, number of
variables or adjusted constants $M$, degrees of freedom $\nu=N-M$, measure
$\chi^{2}$, probability $p\,({\chi^{2}|\nu})$ of obtaining an observed value
of $\chi^{2}$ that large or larger for the given value of $\nu$, Birge ratio
$R_{\rm B}=\sqrt{\chi^{2}/{\nu}}$, and normalized residual of the $i$th input
datum $r_{i}=(x_{i}-\langle x_{i}\rangle)/u_{i}$, where $x_{i}$ is the input
datum, $\langle x_{i}\rangle$ its adjusted value, and $u_{i}$ its standard
uncertainty.

The observational equations for the input data are given in Tables 31, 33, and
35. These equations are written in terms of a particular independent subset of
constants (broadly interpreted) called, as already noted, _adjusted
constants_. These are the variables (or unknowns) of the adjustment. The
least-squares calculation yields values of the adjusted constants that predict
values of the input data through their observational equations that best agree
with the data themselves in the least squares sense. The adjusted constants
used in the 2010 calculations are given in Tables 30, 32, and 34.

The symbol $\doteq$ in an observational equation indicates that an input datum
of the type on the left-hand side is ideally given by the expression on the
right-hand side containing adjusted constants. But because the equation is one
of an overdetermined set that relates a datum to adjusted constants, the two
sides are not necessarily equal. The best estimate of the value of an input
datum is its observational equation evaluated with the least-squares adjusted
values of the adjusted constants on which its observational equation depends.
For some input data such as $\delta_{\rm e}$ and $R$, the observational
equation is simply $\delta_{\rm e}\doteq\delta_{\rm e}$ and $R\doteq R$.

The binding energies $E_{\rm b}(X)/m_{\rm u}c^{2}$ in the observational
equations of Table 33 are treated as fixed quantities with negligible
uncertainties, as are the bound-state $g$-factor ratios. The frequency $f_{\rm
p}$ is not an adjusted constant but is included in the equation for data items
$B30$ and $B31$ to indicate that they are functions of $f_{\rm p}$. Finally,
the observational equations for items $B30$ and $B31$, which are based on Eqs.
(LABEL:eq:murat)-(229) of Sec. VI.2.2, include the function $a_{\rm
e}(\alpha,\delta_{\rm e})$, as well as the theoretical expression for input
data of type $B29$, $\Delta\nu_{\rm Mu}$. The latter expression is discussed
in Sec. VI.2.1 and is a function of $R_{\infty}$, $\alpha$, $m_{\rm
e}/m_{\mbox{\scriptsize{{m}}}}$, and $a_{\mbox{\scriptsize{{m}}}}$.

The self-sensitivity coefficient $S_{\rm c}$ for an input datum is a measure
of the influence of a particular item of data on its corresponding adjusted
value. As in previous adjustments, in general, for an input datum to be
included in the final adjustment on which the 2010 recommended values are
based, its value of $S_{\rm c}$ must be greater than 0.01, or $1\,\%$, which
means that its uncertainty must be no more than about a factor of 10 larger
than the uncertainty of the adjusted value of that quantity; see Sec. I.D of
CODATA-98 for the justification of this $1\,\%$ cutoff. However, the exclusion
of a datum is not followed if, for example, a datum with $S_{\rm c}<0.01$ is
part of a group of data obtained in a given experiment, or series of
experiments, where most of the other data have self-sensitivity coefficients
greater than $0.01$. It is also not followed for $G$, because in this case
there is substantial disagreement of some of the data with the smallest
uncertainties and hence relatively greater significance of the data with
larger uncertainties.

In summary, there is one major discrepancy among the data discussed in this
section: the disagreement of the NIST-07 watt balance value of $K_{\rm
J}^{2}R_{\rm K}$ and the IAC-11 enriched 28Si XRCD value of $N_{\rm A}$, items
$B37.3$ and $B54$ of Table LABEL:tab:pdata.

####  XIII.2.1 Data related to the Newtonian constant of gravitation $G$

Figure 6:  Values of the Newtonian constant of gravitation $G$ in Table 24 and
the 2006 and 2010 CODATA recommended values in chronological order from top to
bottom

Our least-squares analysis of the input data begins with the 11 values of $G$
in Table 24, which are graphically compared in Fig. 6. (Because the $G$ data
are independent of all other data, they can be treated separately.) As
discussed in Secs. XI.1.2 and XI.2.1, there are two correlation coefficients
associated with these data: $r(G1,G3)=0.351$ and $r(G8,G10)=0.234$. It is
clear from both the table and figure that the data are highly inconsistent. Of
the 55 differences among the 11 values, the three largest, $11.4u_{\rm diff}$,
$10.7u_{\rm diff}$, and $10.2u_{\rm diff}$ are between JILA-10 and UWash-00,
BIPM-01, and UZur-06, respectively. Further, eight range from $4u_{\rm diff}$
to $7u_{\rm diff}$. The weighted mean of the 11 values has a relative standard
uncertainty of $8.6\times 10^{-6}$. For this calculation, with $\nu=11-1=10$,
we have $\chi^{2}=209.6$, $p\,(209.6|10)\approx 0$, and $R_{\rm B}=4.58$.
(Recall that a multivariate least-squares calculation with only one variable
is a weighted mean with covariances.) Five data have normalized residuals
$r_{i}>2.0$: JILA-10, BIPM-01, UWash-00, NIST-82, and UZur-06; their
respective values are $-10.8$, $6.4$, $4.4$, $-3.2$ and $3.2$.

Repeating the calculation using only the six values of $G$ with relative
uncertainties $\leq 4.0\times 10^{-5}$, namely, UWash-00, BIPM-01, MSL-03,
UZur-06, HUST-09, and JILA-10, has little impact: the value of $G$ increases
by the fractional amount $5.0\times 10^{-6}$ and the relative uncertainty
increases to $8.8\times 10^{-6}$; for this calculation $\nu=6-1=5$,
$\chi^{2}=191.4$, $p\,(191.4|5)\approx 0$, and $R_{\rm B}=6.19$; the values of
$r_{i}$ are $4.0$, $6.3$, $-0.05$, $3.0$, $-2.2$, and $-11.0$, respectively.

Taking into account the historic difficulty in measuring $G$ and the fact that
all $11$ values of $G$ have no apparent issue besides the disagreement among
them, the Task Group decided to take as the 2010 recommended value the
weighted mean of the 11 values in Table 24 after each of their uncertainties
is multiplied by the factor 14. This yields

$\displaystyle
G=6.673\,84(80)~{}\mbox{m}^{3}~{}\mbox{kg}^{-1}~{}\mbox{s}^{-2}\quad[1.2\times
10^{-4}]\,.$ (283)

The largest normalized residual, that of JILA-10, is now 0.77, and the largest
difference between values of $G$, that between JILA-10 and UWash-00, is
$0.82u_{\rm diff}$. For the calculation yielding the recommended value,
$\nu=11-1=10$, $\chi^{2}=1.07$, $p\,(1.07|10)=1.00$, and $R_{\rm B}=0.33$. In
view of the significant scatter of the measured values of $G$, the factor of
14 was chosen so that the smallest and largest values would differ from the
recommended value by about twice its uncertainty; see Fig. 6. The 2010
recommended value represents a fractional decrease in the 2006 value of
$0.66\times 10^{-4}$ and an increase in uncertainty of $20\,\%$.

####  XIII.2.2 Data related to all other constants

Tables 36, 37, and 38 summarize 12 least-squares analyses, discussed in the
following paragraphs, of the input data and correlation coefficients in Tables
18 to 23. Because the adjusted value of $R_{\infty}$ is essentially the same
for all five adjustments summarized in Table 36 and equal to that of
adjustment 3 of Table 38, the values are not listed in Table 36. (Note that
adjustment 3 in Tables 36 and 38 is the same adjustment.)

_Adjustment 1_. The initial adjustment includes all of the input data, three
of which have normalized residuals whose absolute magnitudes are
problematically greater than 2; see Table 37. They are the 2007 NIST watt-
balance result for $K^{2}_{\rm J}R_{\rm K}$, the 2011 IAC enriched silicon
XRCD result for $N_{\rm A}$, and the 1989 NIST result for
${\it\Gamma}^{\prime}_{\rm p-90}$(lo). All other input data have values of
$|r_{i}|$ less than 2, except those for two antiprotonic ${}^{3}{\rm He}$
transitions, data items $C25$ and $C27$ in Table 22, for which $r_{25}=2.12$
and $r_{27}=2.10$. However, the fact that their normalized residuals are
somewhat greater than 2 is not a major concern, because their self-sensitivity
coefficients $S_{\rm c}$ are considerably less than 0.01. In this regard, we
see from Table 37 that two of the three inconsistent data have values of
$S_{\rm c}$ considerably larger than 0.01; the exception is
${\it\Gamma}^{\prime}_{\rm p-90}$(lo) with $S_{\rm c}=0.0096$, which is
rounded to 0.010 in the table.

_Adjustment 2_. The difference in the IAC-11 and NIST-07 values of $h$ (see
first two lines of Table 26) is 3.8$u_{\rm diff}$, where as before $u_{\rm
diff}$ is the standard uncertainty of the difference. To reduce the difference
between these two highly credible results to an acceptable level, that is, to
2$u_{\rm diff}$ or slightly below, the Task Group decided that the
uncertainties used in the adjustment for these data would be those in the
Table LABEL:tab:pdata multiplied by a factor of two. It was also decided to
apply the same factor to the uncertainties of all the data that contribute in
a significant way to the determination of $h$, so that the relative weights of
this set of data are unchanged. (Recall that if the difference between two
values of the same quantity is $au_{\rm diff}$ and the uncertainty of each is
increased by a factor $b$, the difference is reduced to $(a/b)u_{\rm diff}$.)
Thus, adjustment 2 differs from adjustment 1 in that the uncertainties of data
items $B36.1$, $B36.2$, $B37.1$ to $B37.5$, and $B54$ in Table
LABEL:tab:pdata, which are the two values of $K_{\rm J}$, the five values of
$K^{2}_{\rm J}R_{\rm K}$, and the value of $N_{\rm A}$, are increased by a
factor of 2. (Although items $B31.1$, $B31.2$, and $B38$, the two values of
${\it\Gamma}^{\prime}_{\rm p-90}$(hi) and ${\cal F}_{90}$, also contribute to
the determination of $h$, their contribution is small and no multiplicative
factor is applied.)

From Tables 36 and 37 we see that the values of $\alpha$ and $h$ from
adjustment 2 are very nearly the same as from adjustment 1, that $|r_{i}|$ for
both $B37.3$ and $B54$ have been reduced to below 1.4, and that the residual
for ${\it\Gamma}^{\prime}_{\rm p-90}$(lo) is unchanged.

_Adjustment 3_. Adjustment 3 is the adjustment on which the 2010 CODATA
recommended values are based, and as such it is referred to as the “final
adjustment.”. It differs from adjustment 2 in that, following the prescription
described above, 18 input data with values of $S_{\rm c}$ less than 0.01 are
deleted. These are data items $B13.1$, $B32.1$ to $B36.2$, $B37.5$, $B38$,
$B56$, $B59$, and $B60$ in Table LABEL:tab:pdata. (The range in values of
$S_{\rm c}$ for the deleted data is 0.0003 to 0.0097, and no datum with a
value of $S_{\rm c}>1$ was “converted” to a value with $S_{\rm c}<1$ due to
the multiplicative factor.) Further, because $h/m(^{133}{\rm Cs})$, item
$B56$, is deleted as an input datum due to its low weight, the two values of
$A_{\rm r}(^{133}{\rm Cs})$, items $B10.1$ and $10.2$, which are not relevant
to any other input datum, are also deleted and $A_{\rm r}(^{133}{\rm Cs})$ is
omitted as an adjusted constant. This brings the total number of omitted data
items to 20. Table 36 shows that deleting them has virtually no impact on the
values of $\alpha$ and $h$ and Birge ratio $R_{\rm B}$. The data for the final
adjustment are quite consistent, as demonstrated by the value of $\chi^{2}$:
$p\,(58.1|67)=0.77$.

_Adjustments 4 and 5_. The purpose of these adjustments is to test the
robustness of the 2010 recommended values of $\alpha$ and $h$ by omitting the
most accurate data relevant to these constants. Adjustment 4 differs from
adjustment 2 in that the four data that provide values of $\alpha$ with the
smallest uncertainties are deleted, namely, items $B13.1$, $B13.2$, $B56$ and
$B57$, the two values of $a_{\rm e}$ and the values of $h/m(^{133}{\rm Cs})$
and $h/m(^{87}{\rm Rb})$; see the first four entries of Table 25. (For the
same reason as in adjustment 3, in adjustment 4 the two values of $A_{\rm
r}(^{133}{\rm Cs})$ are also deleted as input data and $A_{\rm r}(^{133}{\rm
Cs})$ is omitted as an adjusted constant; the same applies to $A_{\rm
r}(^{87}{\rm Rb})$.) Adjustment 5 differs from adjustment 1 in that the three
data that provide values of $h$ with the smallest uncertainties are deleted,
namely, items $B37.2$, $B37.3$, and $B54$, the two NIST values of $K^{2}_{\rm
J}R_{\rm K}$ and the IAC value of $N_{\rm A}$; see the first three entries of
Table 26. Also deleted are the data with $S_{\rm c}<0.01$ that contribute in a
minimal way to the determination of $\alpha$ and are deleted in the final
adjustment. Table 36 shows that the value of $\alpha$ from the less accurate
$\alpha$-related data used in adjustment 4, and the value of $h$ from the less
accurate $h$-related data used in adjustment 5, agree with the corresponding
recommended values from adjustment 3. This agreement provides a consistency
check on the 2010 recommended values.

_Adjustments 6 to 12_. The aim of the seven adjustments summarized in Table 38
is to investigate the data that determine the recommended values of
$R_{\infty}$, $r_{\rm p}$, and $r_{\rm d}$. Results from adjustment 3, the
final adjustment, are included in the table for reference purposes. We begin
with a discussion of adjustments 6 to 10, which are derived from adjustment 3
by deleting selected input data. We then discuss adjustments 11 and 12, which
examine the impact of the value of the proton rms charge radius derived from
the measurement of the Lamb shift in muonic hydrogen discussed in Sec. IV.1.3
and given in Eq. (97). Note that the value of $R_{\infty}$ depends only weakly
on the data in Tables LABEL:tab:pdata and 22.

In adjustment 6, the electron scattering values of $r_{\rm p}$ and $r_{\rm
d}$, data items $A49.1$, $A49.2$, and $A50$ in Table 18, are not included.
Thus, the values of these two quantities from adjustment 6 are based solely on
H and D spectroscopic data. It is evident from a comparison of the results of
this adjustment and adjustment 3 that the scattering values of the radii play
a smaller role than the spectroscopic data in determining the 2010 recommended
values of $R_{\infty}$, $r_{\rm p}$ and $r_{\rm d}$.

Adjustment 7 is based on only hydrogen data, including the two scattering
values of $r_{\rm p}$ but not the difference between the $1{\rm S_{1/2}}-2{\rm
S_{1/2}}$ transition frequencies in H and D, item $A48$ in Table 18, hereafter
referred to as the “isotope shift.” Adjustment 8 differs from adjustment 7 in
that the two scattering values of $r_{\rm p}$ are deleted. Adjustments 9 and
10 are similar to 7 and 8 but are based on only deuterium data; that is,
adjustment 9 includes the scattering value of $r_{\rm d}$ but not the isotope
shift, while for adjustment 10 the scattering value is deleted. The results of
these four adjustments show the dominant role of the hydrogen data and the
importance of the isotope shift in determining the recommended value of
$r_{\rm d}$. Further, the four values of $R_{\infty}$ from these adjustments
agree with the 2010 recommended value, and the two values of $r_{\rm p}$ and
of $r_{\rm d}$ also agree with their respective recommended values: the
largest difference from the recommended value for the eight results is
$1.4u_{\rm diff}$.

Adjustment 11 differs from adjustment 3 in that it includes the muonic
hydrogen value $r_{\rm p}=0.841\,69(66)~{}{\rm fm}$, and adjustment 12 differs
from adjustment 11 in that the three scattering values of the nuclear radii
are deleted. Because the muonic hydrogen value is significantly smaller and
has a significantly smaller uncertainty than the purely spectroscopic value of
adjustment 6 and the two scattering values, it has a major impact on the
results of adjustments 11 and 12, as can be seen from Table 38: for both
adjustments the value of $R_{\infty}$ shifts down by over 6 standard
deviations and its uncertainty is reduced by a factor of 4.6. Moreover, and
not surprisingly, the values of $r_{\rm p}$ and of $r_{\rm d}$ from both
adjustments are significantly smaller than the recommended values and have
significantly smaller uncertainties. The inconsistencies between the muonic
hydrogen result for $r_{\rm p}$ and the spectroscopic and scattering results
is demonstrated by the large value and low probability of $\chi^{2}$ for
adjustment 11; $p\,(104.9|68)=0.0027$.

The impact of the muonic hydrogen value of $r_{\rm p}$ can also be seen by
examining for adjustments 3, 11, and 12 the normalized residuals and self-
sensitivity coefficients of the principal experimental data that determine
$R_{\infty}$, namely, items $A26$ to $A50$ of Table 18. In brief, $|r_{i}|$
for these data in the final adjustment range from near 0 to 1.24 for item
$A50$, the $r_{\rm d}$ scattering result, with the vast majority being less
than 1. For the three greater than 1, $|r_{i}|$ is 1.03, 1.08, and 1.04. The
value of $S_{\rm c}$ is 1.00 for items $A26$ and $A48$, the hydrogen $1{\rm
S}_{1/2}-2{\rm S}_{1/2}$ transition frequency and the H-D isotope shift; and
0.42 for item $A49.2$, which is the more accurate of the two scattering values
of $r_{\rm p}$. Most others are a few percent, although some values of $S_{\rm
c}$ are near 0. The situation is markedly different for adjustment 12. First,
$|r_{i}|$ for item $A30$, the hydrogen transition frequency involving the
$8{\rm D}_{5/2}$ state, is 3.06 compared to 0.87 in adjustment 3; and items
$A41$, $A42$, and $A43$, deuterium transitions involving the $8{\rm S}_{1/2}$,
$8{\rm D}_{3/2}$, and $8{\rm D}_{5/2}$ states, are now 2.5, 2.4, and 3.0,
respectively, compared to 0.40, 0.17, and 0.68. Further, ten other transitions
have residuals in the range 1.02 to 1.76. As a result, with this proton
radius, the predictions of the theory for hydrogen and deuterium transition
frequencies are not generally consistent with the experiments. Equally
noteworthy is the fact that although $S_{\rm c}$ for items $A26$ and $A48$
remain equal to 1.00, for all other transition frequencies $S_{\rm c}$ is less
than 0.01, which means that they play an inconsequential role in determining
$R_{\infty}$. The results for adjustment 11, which includes the scattering
values of the nuclear radii as well as the muonic hydrogen value, are similar.

In view of the impact of the latter value on the internal consistency of the
$R_{\infty}$ data and its disagreement with the spectroscopic and scattering
values, the Task Group decided that it was premature to include it as an input
datum in the 2010 CODATA adjustment; it was deemed more prudent to wait to see
if further research can resolve the discrepancy. See Sec. IV.1.3 for
additional discussion.

####  XIII.2.3 Test of the Josephson and quantum Hall effect relations

As in CODATA-02 and CODATA-06, the exactness of the relations $K_{\rm J}=2e/h$
and $R_{\rm K}=h/e^{2}$ is investigated by writing

$\displaystyle K_{\rm J}$ $\displaystyle=$
$\displaystyle\frac{2e}{h}\left({1+\varepsilon_{\rm
J}}\right)=\left({\frac{8\alpha}{\mu_{0}ch}}\right)^{1/2}\left({1+\varepsilon_{\rm
J}}\right),$ (284) $\displaystyle R_{\rm K}$ $\displaystyle=$
$\displaystyle\frac{h}{e^{2}}\left({1+\varepsilon_{\rm
K}}\right)=\frac{\mu_{0}c}{2\alpha}\left({1+\varepsilon_{\rm K}}\right),$
(285)

where $\varepsilon_{\rm J}$ and $\varepsilon_{\rm K}$ are unknown correction
factors taken to be additional adjusted constants. Replacing the relations
$K_{\rm J}=2e/h$ and $R_{\rm K}=h/e^{2}$ in the analysis leading to the
observational equations in Table 33 with the generalizations in Eqs. (284) and
(285) leads to the modified observational equations given in Table 39.

Although the NIST value of $k/h$, item $B60$, was obtained using the Josephson
and quantum Hall effects, it is not included in the tests of the relations
$K_{\rm J}=2e/h$ and $R_{\rm K}=h/e^{2}$, because of its large uncertainty.

The results of seven different adjustments are summarized in Table 29. An
entry of 0 in the $\varepsilon_{\rm K}$ column means that it is assumed that
$R_{\rm K}=h/e^{2}$ in the corresponding adjustment; similarly, an entry of 0
in the $\varepsilon_{\rm J}$ column means that it is assumed that $K_{\rm
J}=2e/h$ in the corresponding adjustment. The following comments apply to the
adjustments of Table 29.

Adjustment (i) uses all of the data and thus differs from adjustment 1 of
Table 36 discussed in the previous section only in that the assumption $K_{\rm
J}=2e/h$ and $R_{\rm K}=h/e^{2}$ is relaxed. For this adjustment, $\nu=86$,
$\chi^{2}=78.1$, and $R_{\rm B}=1.02$. The normalized residuals $r_{i}$ for
the three inconsistent data items in Table 37, the companion table to Table
36, are $0.75$, $-0.56$, and $2.88$. Examination of Table 29 shows that
$\epsilon_{\rm K}$ is consistent with 0 within 1.2 times its uncertainty of
$1.8\times 10^{-8}$, while $\epsilon_{\rm J}$ is consistent with 0 within 2.4
times its uncertainty of $5.7\times 10^{-8}$.

It is important to recognize that any conclusions that can be drawn from the
values of $\varepsilon_{\rm K}$ and $\varepsilon_{\rm J}$ of adjustment (i)
must be tempered, because not all of the individual values of
$\varepsilon_{\rm K}$ and $\varepsilon_{\rm J}$ that contribute to their
determination are consistent. This is demonstrated by adjustments (ii) to
(vii) and Figs. 7 and 8. (Because of their comparatively small uncertainties,
it is possible in these adjustments to take the 2010 recommended values for
the constants $a_{\rm e}$, $\alpha$, $R_{\infty}$, and $A_{\rm r}(\rm e)$,
which appear in the observational equations of Table 39, and assume that they
are exactly known.)

Adjustments (ii) and (iii) focus on $\epsilon_{\rm K}$: $\epsilon_{\rm J}$ is
set equal to 0 and values of $\epsilon_{\rm K}$ are obtained from data whose
observational equations are independent of $h$. These data are the five values
of $R_{\rm K}$, items $B35.1$ to $B35.5$; and the three low-field gyromagnetic
ratios, items $B32.1$, $B32.2$, and $B33$. We see from Table 29 that the two
values of $\epsilon_{\rm K}$ resulting from the two adjustments not only have
opposite signs but their difference is $3.0u_{\rm diff}$. Figure 7 compares
the combined value of $\epsilon_{\rm K}$ obtained from the five values of
$R_{\rm K}$ with the five individual values, while Fig. 8 does the same for
the results obtained from the three gyromagnetic ratios.

Figure 7:  Comparison of the five individual values of $\epsilon_{\rm K}$
obtained from the five values of $R_{\rm K}$, data items $B35.1$ to $B35.5$,
and the combined value (open circle) from adjustment (ii) given in Table 29.
The applicable observational equation in Table 39 is $B35^{*}$.

Figure 8:  Comparison of the three individual values of $\epsilon_{\rm K}$
obtained from the three low-field gyromagnetic ratios, data items $B32.1$,
$B32.2$, and $B33$, and the combined value (open circle) from adjustment (iii)
given in Table 29. The applicable observational equations in Table 39 are
$B32^{*}$ and $B33^{*}$. Because of the form of these equations, the value of
$\epsilon_{\rm K}$ when $\epsilon_{\rm J}=0$ is identical to the value of
$\epsilon_{\rm J}$ when $\epsilon_{\rm K}=0$, hence the label at the bottom of
the figure.

Adjustments (iv) to (vii) focus on $\epsilon_{\rm J}$: $\epsilon_{\rm K}$ is
set equal to 0 and values of $\epsilon_{\rm J}$ are, with the exception of
adjustment (iv), obtained from data whose observational equations are
dependent on $h$. Examination of Table 29 shows that although the values of
$\epsilon_{\rm J}$ from adjustments (iv) and (v) are of opposite sign, their
difference of $49.1\times 10^{-8}$ is less than the $72.0\times 10^{-8}$
uncertainty of the difference. However, the difference between the values of
$\epsilon_{\rm J}$ from adjustments (iv) and (vi) is $3.6u_{\rm diff}$, and is
$3.3u_{\rm diff}$ even for the value of $\epsilon_{\rm J}$ from adjustment
(vii), in which the uncertainties of the most accurate data have been
increased by the factor 2. (The multiplicative factor 2 is that used in
adjustment 2 and the final adjustment; see Tables 37, 36, and their associated
text.) On the other hand, we see that the value of $\epsilon_{\rm J}$ from
adjustment (vi) is consistent with 0 only to within 3.9 times its uncertainty,
but that this is reduced to 2.0 for the value of $\epsilon_{\rm J}$ from
adjustment (vii) which uses expanded uncertainties.

The results of the adjustments discussed above reflect the disagreement of the
NIST-07 watt-balance value for $K_{\rm J}^{2}R_{\rm K}$, and to a lesser
extent that of the similar NIST-98 value, items $B37.2$ and $B37.3$, with the
IAC-11 enriched silicon value of $N_{\rm A}$, item $B54$; and the disagreement
of the NIST-89 result for ${\it\Gamma}^{\prime}_{p-90}$(lo), and to a lesser
extent the KR/VN-98 result for ${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$,
items $B32.1$ and $B33$, with the highly accurate values of $\alpha$. If
adjustment 1 is repeated with these five data deleted, we find
$\varepsilon_{\rm K}=2.8(1.8)\times 10^{-8}$ and $\varepsilon_{\rm
J}=15(49)\times 10^{-8}$. These values can be interpreted as confirming that
$\varepsilon_{\rm K}$ is consistent with 0 to within 1.6 times its uncertainty
of $1.8\times 10^{-8}$ and that $\varepsilon_{\rm J}$ is consistent with 0
well within its uncertainty of $49\times 10^{-8}$.

We conclude this section by briefly discussing recent efforts to close what is
called the “metrology triangle.” Although there are variants, the basic idea
is to use a single electron tunneling (SET) device that generates a quantized
current $I=ef$ when an alternating voltage of frequency $f$ is applied to it,
where as usual $e$ is the elementary charge. The current $I$ is then compared
to a current derived from Josephson and quantum Hall effect devices. In view
of quantization of charge in units of $e$ and conservation of charge, the
equality of the currents shows that $K_{\rm J}R_{\rm K}e=2$, as expected,
within the uncertainty of the measurements Keller (2008); Keller _et al._
(2008); Feltin and Piquemal (2009). Although there is no indication from the
results reported to date that this relation is not valid, the uncertainties of
the results are at best at the 1 to 2 parts in $10^{6}$ level Keller _et al._
(2007, 2008); Feltin _et al._ (2011); Camarota _et al._ (2012).

Table 29:  Summary of the results of several least-squares adjustments to
investigate the relations $K_{\rm J}=(2e/h)(1+\epsilon_{\rm J})$ and $R_{\rm
K}=(h/e^{2})(1+\epsilon_{\rm K})$. See the text for an explanation and
discussion of each adjustment, but in brief, adjustment (i) uses all the data,
(ii) assumes $K_{\rm J}=2e/h$ (that is, $\epsilon_{\rm J}=0$) and obtains
$\epsilon_{\rm K}$ from the five measured values of $R_{\rm K}$, (iii) is
based on the same assumption and obtains $\epsilon_{\rm K}$ from the two
values of the proton gyromagnetic ratio and one value of the helion
gyromagnetic ratio, (iv) is (iii) but assumes $R_{\rm K}=h/e^{2}$ (that is,
$\epsilon_{\rm K}=0$) and obtains $\epsilon_{\rm J}$ in place of
$\epsilon_{\rm K}$, (v) to (vii) are based on the same assumption and obtain
$\epsilon_{\rm J}$ from all the measured values given in Table LABEL:tab:pdata
for the quantities indicated.

Adj.  | Data included1 |  $10^{8}\varepsilon_{\rm K}$ | $10^{8}\varepsilon_{\rm J}$
---|---|---|---
(i)  | All  |  $2.2(1.8)$ | $5.7(2.4)$
(ii)  |  $R_{\rm K}$ |  $2.6(1.8)$ | $0$
(iii)  |  ${\it\Gamma}^{\prime}_{\rm p,h-90}({\rm lo})$ |  $-25.4(9.3)$ | $0$
(iv)  |  ${\it\Gamma}^{\prime}_{\rm p,h-90}({\rm lo})$ |  $0$ | $-25.4(9.3)$
(v)  |  ${\it\Gamma}^{\prime}_{\rm p-90}({\rm hi}),K_{\rm J},K_{\rm J}^{\rm 2}\\!R_{\rm K},{\cal F}_{{\rm 90}}$ |  $0$ | $23.7(72.0)$
(vi)  |  ${\it\Gamma}^{\prime}_{\rm p-90}({\rm hi}),K_{\rm J},K_{\rm J}^{\rm 2}\\!R_{\rm K},{\cal F}_{{\rm 90}},\\!N_{\rm A}$ |  $0$ | $8.6(2.2)$
(vii)  |  ${\it\Gamma}^{\prime}_{\rm p-90}({\rm hi}),[K_{\rm J}],[K_{\rm J}^{\rm 2}\\!R_{\rm K}],{\cal F}_{{\rm 90}},\\![N_{\rm A}]$ |  $0$ | $8.6(4.4)$

1The data items in brackets have their uncertainties expanded by a factor of
two.

Table 30: The 28 adjusted constants (variables) used in the least-squares multivariate analysis of the Rydberg-constant data given in Table 18. These adjusted constants appear as arguments of the functions on the right-hand side of the observational equations of Table 31.  Adjusted constant  | Symbol
---|---
Rydberg constant  | $R_{\infty}$
bound-state proton rms charge radius  | $r_{\rm p}$
bound-state deuteron rms charge radius  | $r_{\rm d}$
additive correction to $E_{\rm H}(1{\rm S}_{1/2})/h$ | $\delta_{\rm H}(1{\rm S}_{1/2})$
additive correction to $E_{\rm H}(2{\rm S}_{1/2})/h$ | $\delta_{\rm H}(2{\rm S}_{1/2})$
additive correction to $E_{\rm H}(3{\rm S}_{1/2})/h$ | $\delta_{\rm H}(3{\rm S}_{1/2})$
additive correction to $E_{\rm H}(4{\rm S}_{1/2})/h$ | $\delta_{\rm H}(4{\rm S}_{1/2})$
additive correction to $E_{\rm H}(6{\rm S}_{1/2})/h$ | $\delta_{\rm H}(6{\rm S}_{1/2})$
additive correction to $E_{\rm H}(8{\rm S}_{1/2})/h$ | $\delta_{\rm H}(8{\rm S}_{1/2})$
additive correction to $E_{\rm H}(2{\rm P}_{1/2})/h$ | $\delta_{\rm H}(2{\rm P}_{1/2})$
additive correction to $E_{\rm H}(4{\rm P}_{1/2})/h$ | $\delta_{\rm H}(4{\rm P}_{1/2})$
additive correction to $E_{\rm H}(2{\rm P}_{3/2})/h$ | $\delta_{\rm H}(2{\rm P}_{3/2})$
additive correction to $E_{\rm H}(4{\rm P}_{3/2})/h$ | $\delta_{\rm H}(4{\rm P}_{3/2})$
additive correction to $E_{\rm H}(8{\rm D}_{3/2})/h$ | $\delta_{\rm H}(8{\rm D}_{3/2})$
additive correction to $E_{\rm H}(12{\rm D}_{3/2})/h$ | $\delta_{\rm H}(12{\rm D}_{3/2})$
additive correction to $E_{\rm H}(4{\rm D}_{5/2})/h$ | $\delta_{\rm H}(4{\rm D}_{5/2})$
additive correction to $E_{\rm H}(6{\rm D}_{5/2})/h$ | $\delta_{\rm H}(6{\rm D}_{5/2})$
additive correction to $E_{\rm H}(8{\rm D}_{5/2})/h$ | $\delta_{\rm H}(8{\rm D}_{5/2})$
additive correction to $E_{\rm H}(12{\rm D}_{5/2})/h$ | $\delta_{\rm H}(12{\rm D}_{5/2})$
additive correction to $E_{\rm D}(1{\rm S}_{1/2})/h$ | $\delta_{\rm D}(1{\rm S}_{1/2})$
additive correction to $E_{\rm D}(2{\rm S}_{1/2})/h$ | $\delta_{\rm D}(2{\rm S}_{1/2})$
additive correction to $E_{\rm D}(4{\rm S}_{1/2})/h$ | $\delta_{\rm D}(4{\rm S}_{1/2})$
additive correction to $E_{\rm D}(8{\rm S}_{1/2})/h$ | $\delta_{\rm D}(8{\rm S}_{1/2})$
additive correction to $E_{\rm D}(8{\rm D}_{3/2})/h$ | $\delta_{\rm D}(8{\rm D}_{3/2})$
additive correction to $E_{\rm D}(12{\rm D}_{3/2})/h$ | $\delta_{\rm D}(12{\rm D}_{3/2})$
additive correction to $E_{\rm D}(4{\rm D}_{5/2})/h$ | $\delta_{\rm D}(4{\rm D}_{5/2})$
additive correction to $E_{\rm D}(8{\rm D}_{5/2})/h$ | $\delta_{\rm D}(8{\rm D}_{5/2})$
additive correction to $E_{\rm D}(12{\rm D}_{5/2})/h$ | $\delta_{\rm D}(12{\rm D}_{5/2})$
Table 31: Observational equations that express the input data related to $R_{\infty}$ in Table 18 as functions of the adjusted constants in Table 30. The numbers in the first column correspond to the numbers in the first column of Table 18. Energy levels of hydrogenic atoms are discussed in Sec. IV.1. As pointed out at the beginning of that section, $E_{X}(n{\rm L}_{j})/h$ is in fact proportional to $cR_{\infty}$ and independent of $h$, hence $h$ is not an adjusted constant in these equations. See Sec. XIII.2 for an explanation of the symbol $\doteq$. Type of input  |  | Observational equation |
---|---|---|---
datum  |  |  |  |
$A1$–$A16\quad$ | $\delta_{\rm H}(n{\rm L}_{j})$ | $\doteq$ | $\delta_{\rm H}(n{\rm L}_{j})$ |
$A17$–$A25\quad$ | $\delta_{\rm D}(n{\rm L}_{j})$ | $\doteq$ | $\delta_{\rm D}(n{\rm L}_{j})$ |
$A26$–$A31\quad$ | $\nu_{\rm H}(n_{1}{\rm L_{1}}_{j_{1}}-n_{2}{\rm L_{2}}_{j_{2}})$ | $\doteq$ | $\big{[}E_{\rm H}\big{(}n_{2}{\rm L_{2}}_{j_{2}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{2}{\rm L_{2}}_{j_{2}})\big{)}$ |
$A38,A39$ |  |  | $-E_{\rm H}\big{(}n_{1}{\rm L_{1}}_{j_{1}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{1}{\rm L_{1}}_{j_{1}})\big{)}\big{]}/h$ |
$A32$–$A37\quad$ | $\nu_{\rm H}(n_{1}{\rm L_{1}}_{j_{1}}-n_{2}{\rm L_{2}}_{j_{2}})-{\textstyle{1\over 4}}\nu_{\rm H}(n_{3}{\rm L_{3}}_{j_{3}}-n_{4}{\rm L_{4}}_{j_{4}})$ | $\doteq$ | $\Big{\\{}E_{\rm H}\big{(}n_{2}{\rm L_{2}}_{j_{2}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{2}{\rm L_{2}}_{j_{2}})\big{)}$ |
|  |  | $\ -E_{\rm H}\big{(}n_{1}{\rm L_{1}}_{j_{1}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{1}{\rm L_{1}}_{j_{1}})\big{)}$ |
|  |  | $\ -{\textstyle{1\over 4}}\big{[}E_{\rm H}\big{(}n_{4}{\rm L_{4}}_{j_{4}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{4}{\rm L_{4}}_{j_{4}})\big{)}$ |
|  |  | $\quad-E_{\rm H}\big{(}n_{3}{\rm L_{3}}_{j_{3}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{3}{\rm L_{3}}_{j_{3}})\big{)}\big{]}\Big{\\}}/h$ |
$A40$–$A44\quad$ | $\nu_{\rm D}(n_{1}{\rm L_{1}}_{j_{1}}-n_{2}{\rm L_{2}}_{j_{2}})$ | $\doteq$ | $\big{[}E_{\rm D}\big{(}n_{2}{\rm L_{2}}_{j_{2}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{2}{\rm L_{2}}_{j_{2}})\big{)}$ |
|  |  | $-E_{\rm D}\big{(}n_{1}{\rm L_{1}}_{j_{1}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{1}{\rm L_{1}}_{j_{1}})\big{)}\big{]}/h$ |
$A45$–$A46\quad$ | $\nu_{\rm D}(n_{1}{\rm L_{1}}_{j_{1}}-n_{2}{\rm L_{2}}_{j_{2}})-{\textstyle{1\over 4}}\nu_{\rm D}(n_{3}{\rm L_{3}}_{j_{3}}-n_{4}{\rm L_{4}}_{j_{4}})$ | $\doteq$ | $\Big{\\{}E_{\rm D}\big{(}n_{2}{\rm L_{2}}_{j_{2}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{2}{\rm L_{2}}_{j_{2}})\big{)}$ |
|  |  | $\ -E_{\rm D}\big{(}n_{1}{\rm L_{1}}_{j_{1}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{1}{\rm L_{1}}_{j_{1}})\big{)}$ |
|  |  | $\ -{\textstyle{1\over 4}}\big{[}E_{\rm D}\big{(}n_{4}{\rm L_{4}}_{j_{4}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{4}{\rm L_{4}}_{j_{4}})\big{)}$ |
|  |  | $\quad-E_{\rm D}\big{(}n_{3}{\rm L_{3}}_{j_{3}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{3}{\rm L_{3}}_{j_{3}})\big{)}\big{]}\Big{\\}}/h$ |
$A47\quad$ | $\nu_{\rm D}(1{\rm S}_{1/2}-2{\rm S}_{1/2})-\nu_{\rm H}(1{\rm S}_{1/2}-2{\rm S}_{1/2})$ | $\doteq$ | $\Big{\\{}E_{\rm D}\big{(}2{\rm S}_{1/2};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(2{\rm S}_{1/2})\big{)}$ |
|  |  | $\ -E_{\rm D}\big{(}1{\rm S}_{1/2};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(1{\rm S}_{1/2})\big{)}$ |
|  |  | $\ -\big{[}E_{\rm H}\big{(}2{\rm S}_{1/2};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(2{\rm S}_{1/2})\big{)}$ |
|  |  | $\quad-E_{\rm H}\big{(}1{\rm S}_{1/2};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(1{\rm S}_{1/2})\big{)}\big{]}\Big{\\}}/h$ |
$A48\quad$ | $r_{\rm p}$ | $\doteq$ | $r_{\rm p}$ |
$A49\quad$ | $r_{\rm d}$ | $\doteq$ | $r_{\rm d}$ |
Table 32: The 39 adjusted constants (variables) used in the least-squares multivariate analysis of the input data in Table LABEL:tab:pdata. These adjusted constants appear as arguments of the functions on the right-hand side of the observational equations of Table 33.  Adjusted constant  | Symbol
---|---
electron relative atomic mass  | $A_{\rm r}({\rm e})$
proton relative atomic mass  | $A_{\rm r}({\rm p})$
neutron relative atomic mass  | $A_{\rm r}({\rm n})$
deuteron relative atomic mass  | $A_{\rm r}({\rm d})$
triton relative atomic mass  | $A_{\rm r}({\rm t})$
helion relative atomic mass  | $A_{\rm r}({\rm h})$
alpha particle relative atomic mass  | $A_{\rm r}(\mbox{{a}})$
16O7+ relative atomic mass  | $A_{\rm r}(^{16}{\rm O}^{7+})$
87Rb relative atomic mass  | $A_{\rm r}(^{87}{\rm Rb})$
133Cs relative atomic mass  | $A_{\rm r}(^{133}{\rm Cs})$
average vibrational excitation energy  | $A_{\rm r}(E_{\rm av})$
fine-structure constant  | $\alpha$
additive correction to $a_{\rm e}$(th)  | $\delta_{\rm e}$
muon magnetic moment anomaly  | $a_{\mbox{\scriptsize{{m}}}}$
additive correction to $g_{\rm C}$(th)  | $\delta_{\rm C}$
additive correction to $g_{\rm O}$(th)  | $\delta_{\rm O}$
electron-proton magnetic moment ratio  | $\mu_{\rm e^{-}}/\mu_{\rm p}$
deuteron-electron magnetic moment ratio  | $\mu_{\rm d}/\mu_{\rm e^{-}}$
triton-proton magnetic moment ratio  | $\mu_{\rm t}/\mu_{\rm p}$
shielding difference of d and p in HD  | $\sigma_{\rm dp}$
shielding difference of t and p in HT  | $\sigma_{\rm tp}$
electron to shielded proton  |
magnetic moment ratio  | $\mu_{\rm e^{-}}/\mu^{\prime}_{\rm p}$
shielded helion to shielded proton  |
magnetic moment ratio  | $\mu^{\prime}_{\rm h}/\mu^{\prime}_{\rm p}$
neutron to shielded proton  |
magnetic moment ratio  | $\mu_{\rm n}/\mu_{\rm p}^{\prime}$
electron-muon mass ratio  | $m_{\rm e}/m_{\mbox{\scriptsize{{m}}}}$
additive correction to $\Delta\nu_{\rm Mu}({\rm th})$ | $\delta_{\rm Mu}$
Planck constant  | $h$
molar gas constant  | $R$
copper K${\rm\alpha}_{1}$ x unit  | xu(CuK${\rm\alpha}_{1})$
molybdenum K${\rm\alpha}_{1}$ x unit  | xu(MoK${\rm\alpha}_{1})$
ångstrom star  | Å∗
$d_{220}$ of Si crystal ILL  | $d_{220}({\rm{\scriptstyle ILL}})$
$d_{220}$ of Si crystal N  | $d_{220}({\rm{\scriptstyle N}})$
$d_{220}$ of Si crystal WASO 17  | $d_{220}({\rm{\scriptstyle W17}})$
$d_{220}$ of Si crystal WASO 04  | $d_{220}({\rm{\scriptstyle W04}})$
$d_{220}$ of Si crystal WASO 4.2a  | $d_{220}({\rm{\scriptstyle W4.2a}})$
$d_{220}$ of Si crystal MO∗ | $d_{220}({\rm{\scriptstyle MO^{*}}})$
$d_{220}$ of Si crystal NR3  | $d_{220}({\rm{\scriptstyle NR3}})$
$d_{220}$ of Si crystal NR4  | $d_{220}({\rm{\scriptstyle NR4}})$
$d_{220}$ of an ideal Si crystal  | $d_{220}$

Table 33:  Observational equations that express the input data in Table LABEL:tab:pdata as functions of the adjusted constants in Table 32. The numbers in the first column correspond to the numbers in the first column of Table LABEL:tab:pdata. For simplicity, the lengthier functions are not explicitly given. See Sec. XIII.2 for an explanation of the symbol $\doteq$. Type of input  |  | Observational equation | Sec.
---|---|---|---
datum  |  |  |  |
$B1\quad$ | $A_{\rm r}(^{1}{\rm H})$ | $\doteq$ |  $A_{\rm r}({\rm p})+A_{\rm r}({\rm e})-E_{\rm b}(^{1}{\rm H})/m_{\rm u}c^{2}$ | III.2
$B2\quad$ | $A_{\rm r}(^{2}{\rm H})$ | $\doteq$ |  $A_{\rm r}({\rm d})+A_{\rm r}({\rm e})-E_{\rm b}(^{2}{\rm H})/m_{\rm u}c^{2}$ | III.2
$B3\quad$ | $A_{\rm r}(E_{\rm av})$ | $\doteq$ |  $A_{\rm r}(E_{\rm av})$ | III.3
$B4\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{f_{\rm c}({\rm H}_{2}^{+*})}\over\vbox to9.0pt{}\textstyle{f_{\rm c}({\rm d})}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm d})}\over\vbox to9.0pt{}\textstyle{2A_{\rm r}({\rm p})+A_{\rm r}({\rm e})-\left[\,2E_{\rm I}({\rm H})+E_{\rm B}({\rm H}_{2})-E_{\rm I}({\rm H}_{2})-E_{\rm av}\,\right]/m_{\rm u}c^{2}}}$ | III.3
$B5\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{f_{\rm c}({\rm t})}\over\vbox to9.0pt{}\textstyle{f_{\rm c}({\rm H}_{2}^{+*})}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{2A_{\rm r}({\rm p})+A_{\rm r}({\rm e})-\left[\,2E_{\rm I}({\rm H})+E_{\rm B}({\rm H}_{2})-E_{\rm I}({\rm H}_{2})-E_{\rm av}\,\right]/m_{\rm u}c^{2}}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm t})}}$ | III.3
$B6\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{f_{\rm c}(^{3}{\rm He}^{+})}\over\vbox to9.0pt{}\textstyle{f_{\rm c}({\rm H}_{2}^{+*})}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{2A_{\rm r}({\rm p})+A_{\rm r}({\rm e})-\left[\,2E_{\rm I}({\rm H})+E_{\rm B}({\rm H}_{2})-E_{\rm I}({\rm H}_{2})-E_{\rm av}\,\right]/m_{\rm u}c^{2}}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm h})+A_{\rm r}({\rm e})-E_{\rm I}(^{3}{\rm He}^{+})/m_{\rm u}c^{2}}}$ | III.3
$B7\quad$ | $A_{\rm r}(^{4}{\rm He})$ | $\doteq$ |  $A_{\rm r}({\mbox{{a}}})+2A_{\rm r}({\rm e})-E_{\rm b}(^{4}{\rm He})/m_{\rm u}c^{2}$ | III.2
$B8\quad$ | $A_{\rm r}(^{16}{\rm O})$ | $\doteq$ |  $A_{\rm r}(^{16}{\rm O}^{7+})+7A_{\rm r}({\rm e})-\left[E_{\rm b}(^{16}{\rm O})-E_{\rm b}(^{16}{\rm O}^{7+})\right]\\!/m_{\rm u}c^{2}$ | III.2
$B9\quad$ | $A_{\rm r}(^{87}{\rm Rb})$ | $\doteq$ |  $A_{\rm r}(^{87}{\rm Rb})$ |
$B10\quad$ | $A_{\rm r}(^{133}{\rm Cs})$ | $\doteq$ |  $A_{\rm r}(^{133}{\rm Cs})$ |
$B11\quad$ | $A_{\rm r}({\rm e})$ | $\doteq$ |  $A_{\rm r}({\rm e})$ |
$B12\quad$ | $\delta_{\rm e}$ | $\doteq$ |  $\delta_{\rm e}$ |
$B13\quad$ | $a_{\rm e}$ | $\doteq$ |  $a_{\rm e}(\alpha,\delta_{e})$ | V.1.1
$B14\quad$ | $\overline{R}$ | $\doteq$ |  $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{a_{\mbox{\scriptsize{{m}}}}}\over\vbox to9.0pt{}\textstyle{1+a_{\rm e}(\alpha,\delta_{\rm e})}}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{m_{\rm e}}\over\vbox to9.0pt{}\textstyle{m_{\mbox{\scriptsize{{m}}}}}}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}}}$ | V.2.2
$B15\quad$ | $\delta_{\rm C}$ | $\doteq$ |  $\delta_{\rm C}$ |
$B16\quad$ | $\delta_{\rm O}$ | $\doteq$ |  $\delta_{\rm O}$ |
$B17\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{f_{\rm s}\left({}^{12}{\rm C}^{5+}\right)}\over\vbox to9.0pt{}\textstyle{f_{\rm c}\left({}^{12}{\rm C}^{5+}\right)}}$ | $\doteq$ |  $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm C}(\alpha,\delta_{\rm C})}\over\vbox to9.0pt{}\textstyle{10A_{\rm r}({\rm e})}}\left[12-5A_{\rm r}({\rm e})+{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{E_{\rm b}\left({}^{12}{\rm C}\right)-E_{\rm b}\left({}^{12}{\rm C}^{5+}\right)}\over\vbox to9.0pt{}\textstyle{m_{\rm u}c^{2}}}\right]$ | V.3.2
$B18\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{f_{\rm s}\left({}^{16}{\rm O}^{7+}\right)}\over\vbox to9.0pt{}\textstyle{f_{\rm c}\left({}^{16}{\rm O}^{7+}\right)}}$ | $\doteq$ |  $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm O}(\alpha,\delta_{\rm O})}\over\vbox to9.0pt{}\textstyle{14A_{\rm r}({\rm e})}}A_{\rm r}(^{16}{\rm O}^{7+})$ | V.3.2
$B19\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}({\rm H})}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}({\rm H})}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm e^{-}}({\rm H})}\over\vbox to9.0pt{}\textstyle{g_{\rm e^{-}}}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm p}({\rm H})}\over\vbox to9.0pt{}\textstyle{g_{\rm p}}}\right)^{-1}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}}}$ |
$B20\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm d}({\rm D})}\over\vbox to9.0pt{}\textstyle{\mu_{\rm e^{-}}({\rm D})}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm d}({\rm D})}\over\vbox to9.0pt{}\textstyle{{g_{\rm d}}}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm e^{-}}({\rm D})}\over\vbox to9.0pt{}\textstyle{g_{\rm e^{-}}}}\right)^{-1}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm d}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm e^{-}}}}$ |
$B21\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm p}({\rm HD})}\over\vbox to9.0pt{}\textstyle{\mu_{\rm d}({\rm HD})}}$ | $\doteq$ |  $\left[1+\sigma_{\rm dp}\right]{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm p}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm e^{-}}}}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm d}}}$ |
$B22\quad$ | $\sigma_{\rm dp}$ | $\doteq$ |  $\sigma_{\rm dp}$ |
$B23\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm t}({\rm HT})}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}({\rm HT})}}$ | $\doteq$ |  $\left[1-\sigma_{\rm tp}\right]{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm t}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}}}$ |
$B24\quad$ | $\sigma_{\rm tp}$ | $\doteq$ |  $\sigma_{\rm tp}$ |
$B25\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}({\rm H})}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm e^{-}}({\rm H})}\over\vbox to9.0pt{}\textstyle{g_{\rm e^{-}}}}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ |
$B26\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm h}^{\prime}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm h}^{\prime}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ |
Table 33: (Continued). Observational equations that express the input data in Table LABEL:tab:pdata as functions of the adjusted constants in Table 32. The numbers in the first column correspond to the numbers in the first column of Table LABEL:tab:pdata. For simplicity, the lengthier functions are not explicitly given. See Sec. XIII.2 for an explanation of the symbol $\doteq$. Type of input  |  | Observational equation |
---|---|---|---
datum  |  |  |  | Sec.
$B27\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm n}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm n}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ |
$B28\quad$ | $\delta_{\rm Mu}$ | $\doteq$ |  $\delta_{\rm Mu}$ |
$B29\quad$ | $\Delta\nu_{\rm Mu}$ | $\doteq$ |  $\Delta\nu_{\rm Mu}\\!\\!\left(R_{\infty},\alpha,{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{m_{\rm e}}\over\vbox to9.0pt{}\textstyle{m_{\mbox{\scriptsize{{m}}}}}},\delta_{\rm Mu}\right)$ | VI.2.1
$B30,B31\quad$ | $\nu(f_{\rm p})$ | $\doteq$ |  $\nu\\!\left(f_{\rm p};R_{\infty},\alpha,{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{m_{\rm e}}\over\vbox to9.0pt{}\textstyle{m_{\mbox{\scriptsize{{m}}}}}},{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}}},\delta_{\rm e},\delta_{\rm Mu}\right)$ | VI.2.2
$B32\quad$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ | $\doteq$ |  $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{K_{\rm J-90}R_{\rm K-90}[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{3}}\over\vbox to9.0pt{}\textstyle{2\mu_{0}R_{\infty}}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}$ |
$B33\quad$ | ${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{K_{\rm J-90}R_{\rm K-90}[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{3}}\over\vbox to9.0pt{}\textstyle{2\mu_{0}R_{\infty}}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm h}^{\prime}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ |
$B34\quad$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ | $\doteq$ |  $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{c[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{2}}\over\vbox to9.0pt{}\textstyle{K_{\rm J-90}R_{\rm K-90}R_{\infty}h}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}$ |
$B35\quad$ | $R_{\rm K}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{0}c}\over\vbox to9.0pt{}\textstyle{2\alpha}}$ |
$B36\quad$ | $K_{\rm J}$ | $\doteq$ |  $\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{8\alpha}\over\vbox to9.0pt{}\textstyle{\mu_{0}ch}}\right)^{1/2}$ |
$B37\quad$ | $K_{\rm J}^{2}R_{\rm K}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{4}\over\vbox to9.0pt{}\textstyle{h}}$ |
$B38\quad$ | ${\cal F}_{90}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{cM_{\rm u}A_{\rm r}({\rm e})\alpha^{2}}\over\vbox to9.0pt{}\textstyle{K_{\rm J-90}R_{\rm K-90}R_{\infty}h}}$ |
$B39$-$B41\quad$ | $d_{220}({{\scriptstyle X}})$ | $\doteq$ |  $d_{220}({{\scriptstyle X}})$ |
$B42$-$B53\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{d_{220}({{\scriptstyle X}})}\over\vbox to9.0pt{}\textstyle{d_{220}({{\scriptstyle Y}})}}-1$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{d_{220}({{\scriptstyle X}})}\over\vbox to9.0pt{}\textstyle{d_{220}({{\scriptstyle Y}})}}-1$ |
$B54\quad$ | $N_{\rm A}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{cM_{\rm u}A_{\rm r}({\rm e})\alpha^{2}}\over\vbox to9.0pt{}\textstyle{2R_{\infty}h}}$ |
$B55\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\lambda_{\rm meas}}\over\vbox to9.0pt{}\textstyle{d_{220}({\rm{\scriptstyle ILL}})}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\alpha^{2}A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{R_{\infty}d_{220}({\rm{\scriptstyle ILL}})}}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm n})+A_{\rm r}({\rm p})}\over\vbox to9.0pt{}\textstyle{\left[A_{\rm r}({\rm n})+A_{\rm r}({\rm p})\right]^{2}-A_{\rm r}^{2}({\rm d})}}$ | IX.3
$B56,B57\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{h}\over\vbox to9.0pt{}\textstyle{m(X)}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}(X)}}$ ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{c\alpha^{2}}\over\vbox to9.0pt{}\textstyle{2R_{\infty}}}$ | VII.1
$B58\quad$ | $R$ | $\doteq$ |  $R$ |
$B59\quad$ | $k$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{2R_{\infty}hR}\over\vbox to9.0pt{}\textstyle{cM_{\rm u}A_{\rm r}({\rm e})\alpha^{2}}}$ |
$B60\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{k}\over\vbox to9.0pt{}\textstyle{h}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{2R_{\infty}R}\over\vbox to9.0pt{}\textstyle{cM_{\rm u}A_{\rm r}({\rm e})\alpha^{2}}}$ |
$B61,B64\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\lambda({\rm CuK\mbox{{a}}_{1}})}\over\vbox to9.0pt{}\textstyle{d_{220}({{\scriptstyle X}})}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\rm 1\,537.400~{}xu(CuK\mbox{{a}}_{1})}\over\vbox to9.0pt{}\textstyle{d_{220}({{\scriptstyle X}})}}$ | IX.4
$B62\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\lambda({\rm WK\mbox{{a}}_{1}})}\over\vbox to9.0pt{}\textstyle{d_{220}({\rm{\scriptstyle N}})}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\rm 0.209\,010\,0~{}\AA^{*}}\over\vbox to9.0pt{}\textstyle{d_{220}({\rm{\scriptstyle N}})}}$ | IX.4
$B63\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\lambda({\rm MoK\mbox{{a}}_{1}})}\over\vbox to9.0pt{}\textstyle{d_{220}({\rm{\scriptstyle N}})}}$ | $\doteq$ |  ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\rm 707.831~{}xu(MoK\mbox{{a}}_{1})}\over\vbox to9.0pt{}\textstyle{d_{220}({\rm{\scriptstyle N}})}}$ | IX.4
Table 34: The 15 adjusted constants relevant to the antiprotonic helium data given in Table 22. These adjusted constants appear as arguments of the theoretical expressions on the right-hand side of the observational equations of Table 35. Transition  | Adjusted constant
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$\bar{\rm p}^{4}$He+: $(32,31)\rightarrow(31,30)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(32,31\\!\\!:\\!31,30)$
$\bar{\rm p}^{4}$He+: $(35,33)\rightarrow(34,32)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(35,33\\!\\!:\\!34,32)$
$\bar{\rm p}^{4}$He+: $(36,34)\rightarrow(35,33)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(36,34\\!\\!:\\!35,33)$
$\bar{\rm p}^{4}$He+: $(37,34)\rightarrow(36,33)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(37,34\\!\\!:\\!36,33)$
$\bar{\rm p}^{4}$He+: $(39,35)\rightarrow(38,34)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(39,35\\!\\!:\\!38,34)$
$\bar{\rm p}^{4}$He+: $(40,35)\rightarrow(39,34)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(40,35\\!\\!:\\!39,34)$