$\bar{\rm p}^{4}$He+: $(37,35)\rightarrow(38,34)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(37,35\\!\\!:\\!38,34)$
$\bar{\rm p}^{4}$He+: $(33,32)\rightarrow(31,30)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(33,32\\!\\!:\\!31,30)$
$\bar{\rm p}^{4}$He+: $(36,34)\rightarrow(34,32)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(36,34\\!\\!:\\!34,32)$
$\bar{\rm p}^{3}$He+: $(32,31)\rightarrow(31,30)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(32,31\\!\\!:\\!31,30)$
$\bar{\rm p}^{3}$He+: $(34,32)\rightarrow(33,31)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(34,32\\!\\!:\\!33,31)$
$\bar{\rm p}^{3}$He+: $(36,33)\rightarrow(35,32)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(36,33\\!\\!:\\!35,32)$
$\bar{\rm p}^{3}$He+: $(38,34)\rightarrow(37,33)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(38,34\\!\\!:\\!37,33)$
$\bar{\rm p}^{3}$He+: $(36,34)\rightarrow(37,33)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(36,34\\!\\!:\\!37,33)$
$\bar{\rm p}^{3}$He+: $(35,33)\rightarrow(33,31)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(35,33\\!\\!:\\!33,31)$
Table 35: Observational equations that express the input data related to antiprotonic helium in Table 22 as functions of adjusted constants in Tables 32 and 34. The numbers in the first column correspond to the numbers in the first column of Table 22. Definitions of the symbols and values of the parameters in these equations are given in Sec. IV.2. See Sec. XIII.2 for an explanation of the symbol $\doteq$.  Type of input  |  | Observational equation |
---|---|---|---
datum  |  |  |  |
$C1$–$C7\quad$ | $\delta_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})$ | $\doteq$ | $\delta_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})$ |
$C8$–$C12\quad$ | $\delta_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})$ | $\doteq$ | $\delta_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})$ |
$C13$–$C19\quad$ | $\Delta\nu_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})$ | $\doteq$ | $\Delta\nu_{\bar{\rm p}{\rm{}^{4}He^{+}}}^{(0)}(n,l:n^{\prime},l^{\prime})+a_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})\left[\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm p)}}}\right)^{\\!(0)}\\!\\!\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm p})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm e})}}\right)-1\right]$ |
|  |  | $+b_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})\left[\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\mbox{{a}})}}}\right)^{\\!(0)}\\!\\!\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\mbox{{a}}})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm e})}}\right)-1\right]+\delta_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})$ |
$C20$–$C24\quad$ | $\Delta\nu_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})$ | $\doteq$ | $\Delta\nu_{\bar{\rm p}{\rm{}^{3}He^{+}}}^{(0)}(n,l:n^{\prime},l^{\prime})+a_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})\left[\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm p)}}}\right)^{\\!(0)}\\!\\!\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm p})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm e})}}\right)-1\right]$ |
|  |  | $+b_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})\left[\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm h)}}}\right)^{\\!(0)}\\!\\!\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm h})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm e})}}\right)-1\right]+\delta_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})$ |
Table 36:  Summary of the results of some of the least-squares adjustments used to analyze the input data given in Tables 18-23. The values of $\alpha$ and $h$ are those obtained in the adjustment, $N$ is the number of input data, $M$ is the number of adjusted constants, $\nu=N-M$ is the degrees of freedom, and $R_{\rm B}=\sqrt{{\it\chi}^{2}/\nu}$ is the Birge ratio. See the text for an explanation and discussion of each adjustment, but in brief, adjustment 1 is all the data; 2 is the same as 1 except with the uncertainties of the key data that determine $h$ multiplied by 2; 3 is 2 with the low-weight input data deleted and is the adjustment on which the 2010 recommended values are based; 4 is 2 with the input data that provide the most accurate values of alpha deleted; and 5 is 1 with the input data that provide the most accurate values of $h$ deleted. Adj. |  $N$ |  $M$ |  $\nu$ |  ${\it\chi}^{2}$ |  $R_{\rm B}$ |  $\alpha^{-1}$ |  $u_{\rm r}(\alpha^{-1})$ |  $h$/(J s)  |  $u_{\rm r}(h)$
---|---|---|---|---|---|---|---|---|---
1 | 169  | 83  | 86  | 89.3  | 1.02  |  $137.035\,999\,075(44)$ |  $3.2\times 10^{-10}$ |  $6.626\,069\,58(15)\times 10^{-34}$ |  $2.2\times 10^{-8}$
2 | 169  | 83  | 86  | 75.7  | 0.94  |  $137.035\,999\,073(44)$ |  $3.2\times 10^{-10}$ |  $6.626\,069\,57(29)\times 10^{-34}$ |  $4.4\times 10^{-8}$
3 | 149  | 82  | 67  | 58.1  | 0.93  |  $137.035\,999\,074(44)$ |  $3.2\times 10^{-10}$ |  $6.626\,069\,57(29)\times 10^{-34}$ |  $4.4\times 10^{-8}$
4 | 161  | 81  | 80  | 69.4  | 0.93  |  $137.036\,0005(20)\phantom{\,44}$ |  $1.4\times 10^{-8~{}}$ |  $6.626\,069\,50(31)\times 10^{-34}$ |  $4.7\times 10^{-8}$
5 | 154  | 82  | 72  | 57.2  | 0.89  |  $137.035\,999\,074(44)$ |  $3.2\times 10^{-10}$ |  $6.626\,069\,48(80)\times 10^{-34}$ |  $1.2\times 10^{-7}$
Table 37:  Normalized residuals $r_{i}$ and self-sensitivity coefficients $S_{\rm c}$ that result from the five least-squares adjustments summarized in Table 36 for the three input data with the largest absolute values of $r_{i}$ in adjustment 1. $S_{\rm c}$ is a measure of how the least-squares estimated value of a given type of input datum depends on a particular measured or calculated value of that type of datum; see Appendix E of CODATA-98. See the text for an explanation and discussion of each adjustment; brief explanations are given at the end of the caption to the previous table. Item | Input | Identification | Adj. 1 | Adj. 2 | Adj. 3 | Adj. 4 | Adj. 5
---|---|---|---|---|---|---|---
number | quantity |  | $~{}r_{i}~{}~{}~{}~{}S_{\rm c}$ | $~{}r_{i}~{}~{}~{}~{}S_{\rm c}$ | $~{}r_{i}~{}~{}~{}~{}S_{\rm c}$ | $~{}r_{i}~{}~{}~{}~{}S_{\rm c}$ | $~{}r_{i}~{}~{}~{}~{}S_{\rm c}$
$B37.3$ | $K_{\rm J}^{2}R_{\rm K}$ | NIST-07 | $\phantom{-}2.83~{}~{}0.367~{}$ | $\phantom{-}1.39~{}~{}0.367~{}$ | $\phantom{-}1.39~{}~{}0.371~{}$ | $\phantom{-}1.23~{}~{}0.413~{}$ | Deleted
$B54$ | $N_{\rm A}$ | IAC-11 | $-2.57~{}~{}0.555~{}$ | $-1.32~{}~{}0.539~{}$ | $-1.31~{}~{}0.546~{}$ | $-1.16~{}~{}0.587~{}$ | Deleted
$B32.1$ | ${\it\Gamma}^{\prime}_{\rm p-90}({\rm lo})$ | NIST-89 | $\phantom{-}2.19~{}~{}0.010~{}$ | $\phantom{-}2.19~{}~{}0.010~{}$ | Deleted | $\phantom{-}2.46~{}~{}0.158~{}$ | Deleted
Table 38: Summary of the results of some of the least-squares adjustments used
to analyze the input data related to $R_{\infty}$. The values of $R_{\infty}$,
$r_{\rm p}$, and $r_{\rm d}$ are those obtained in the indicated adjustment,
$N$ is the number of input data, $M$ is the number of adjusted constants,
$\nu=N-M$ is the degrees of freedom, and $R_{\rm B}=\sqrt{{\it\chi}^{2}/\nu}$
is the Birge ratio. See the text for an explanation and discussion of each
adjustment, but in brief, adjustment 6 is 3, but the scattering data for the
nuclear radii are omitted; 7 is 3, but with only the hydrogen data included
(but not the isotope shift); 8 is 7 with the $r_{\rm p}$ data deleted; 9 and
10 are similar to 7 and 8, but for the deuterium data; 11 is 3 with the muonic
Lamb-shift value of $r_{\rm p}$ included; and 12 is 11, but without the
scattering values of $r_{\rm p}$ and $r_{\rm d}$.
Adj.  |  $N$ |  $M$ |  $\nu$ |  ${\it\chi}^{2}$ |  $R_{\rm B}$ |  $R_{\infty}/{\rm m}^{-1}$ |  $u_{\rm r}(R_{\infty})$ |  $r_{\rm p}$/fm  |  $r_{\rm d}$/fm
---|---|---|---|---|---|---|---|---|---
$\phantom{3}3$ |  $149$ |  $82$ | 67  | 58.1  | 0.93  |  $10\,973\,731.568\,539(55)$ |  $5.0\times 10^{-12}$ |  $0.8775(51)$ | $2.1424(21)$
$\phantom{3}6$ |  $146$ |  $82$ | 64  | 55.5  | 0.93  |  $10\,973\,731.568\,521(82)$ |  $7.4\times 10^{-12}$ |  $0.8758(77)$ | $2.1417(31)$
$\phantom{3}7$ |  $131$ |  $72$ | 59  | 53.4  | 0.95  |  $10\,973\,731.568\,561(60)$ |  $5.5\times 10^{-12}$ |  $0.8796(56)$ |
$\phantom{3}8$ |  $129$ |  $72$ | 57  | 52.5  | 0.96  |  $10\,973\,731.568\,528(94)$ |  $8.6\times 10^{-12}$ |  $0.8764(89)$ |
$\phantom{3}9$ |  $114$ |  $65$ | 49  | 46.9  | 0.98  |  $10\,973\,731.568\,37(13)$ |  $1.1\times 10^{-11}$ |  | $2.1288(93)$
$10$ |  $113$ |  $65$ | 48  | 46.8  | 0.99  |  $10\,973\,731.568\,28(30)$ |  $2.7\times 10^{-11}$ |  | $2.121(25)$
$11$ |  $150$ |  $82$ | 68  | 104.9  | 1.24  |  $10\,973\,731.568\,175(12)$ |  $1.1\times 10^{-12}$ |  $0.842\,25(65)$ | $2.128\,24(28)$
$12$ |  $147$ |  $82$ | 65  | 74.3  | 1.07  |  $10\,973\,731.568\,171(12)$ |  $1.1\times 10^{-12}$ |  $0.841\,93(66)$ | $2.128\,11(28)$
Table 39:  Generalized observational equations that express input data $B32$-$B38$ in Table LABEL:tab:pdata as functions of the adjusted constants in Tables 32 and 30 with the additional adjusted constants $\varepsilon_{\rm J}$ and $\varepsilon_{\rm K}$ as given in Eqs. (284) and (285). 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 |  | Generalized observational equation
---|---|---
datum |  |  |
$B32^{*}\quad$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ | $\doteq$ | $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{K_{\rm J-90}R_{\rm K-90}[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{3}}\over\vbox to9.0pt{}\textstyle{2\mu_{0}R_{\infty}(1+\varepsilon_{\rm J})(1+\varepsilon_{\rm K})}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}$
$B33^{*}\quad$ | ${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{K_{\rm J-90}R_{\rm K-90}[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{3}}\over\vbox to9.0pt{}\textstyle{2\mu_{0}R_{\infty}(1+\varepsilon_{\rm J})(1+\varepsilon_{\rm K})}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm h}^{\prime}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$
$B34^{*}\quad$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ | $\doteq$ | $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{c[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{2}}\over\vbox to9.0pt{}\textstyle{K_{\rm J-90}R_{\rm K-90}R_{\infty}h}}(1+\varepsilon_{\rm J})(1+\varepsilon_{\rm K})\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}$
$B35^{*}\quad$ | $R_{\rm K}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{0}c}\over\vbox to9.0pt{}\textstyle{2\alpha}}(1+\varepsilon_{\rm K})$
$B36^{*}\quad$ | $K_{\rm J}$ | $\doteq$ | $\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{8\alpha}\over\vbox to9.0pt{}\textstyle{\mu_{0}ch}}\right)^{1/2}(1+\varepsilon_{\rm J})$
$B37^{*}\quad$ | $K_{\rm J}^{2}R_{\rm K}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{4}\over\vbox to9.0pt{}\textstyle{h}}(1+\varepsilon_{\rm J})^{2}(1+\varepsilon_{\rm K})$
$B38^{*}\quad$ | ${\cal F}_{90}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{cM_{\rm u}A_{\rm r}({\rm e})\alpha^{2}}\over\vbox to9.0pt{}\textstyle{K_{\rm J-90}R_{\rm K-90}R_{\infty}h}}(1+\varepsilon_{\rm J})(1+\varepsilon_{\rm K})$

##  XIV The 2010 CODATA recommended values

###  XIV.1 Calculational details

The 168 input data and their correlation coefficients initially considered for
inclusion in the 2010 CODATA adjustment of the values of the constants are
given in Tables 18 to 23. The 2010 recommended values are based on adjustment
3, called the final adjustment, summarized in Tables 36 to 38 and discussed in
the associated text. Adjustment 3 omits 20 of the 168 initially considered
input data, namely, items $B10.1$, $B10.2$, $B13.1$, $B32.1$ to $B36.2$,
$B37.5$, $B38$, $B56$, $B59$, and $B56$, because of their low weight (self
sensitivity coefficient $S_{\rm c}<0.01$). However, because the observational
equation for $h/m(^{133}{\rm Cs})$, item $B56$, depends on $A_{\rm
r}(^{133}{\rm Cs})$ but item $B56$ is deleted because of its low weight, the
two values of $A_{\rm r}(^{133}{\rm Cs})$, items $B10.1$ and $B10.2$, are also
deleted and $A_{\rm r}(^{133}{\rm Cs})$ itself is deleted as an adjusted
constant. Further, the initial uncertainties of five input data, items $B37.1$
to $B37.4$ and $B56$, are multiplied by the factor 2, with the result that the
absolute values of the normalized residuals $|r_{i}|$ of the five data are
less than 1.4 and their disagreement is reduced to an acceptable level.

Each input datum in this final adjustment has a self sensitivity coefficient
$S_{c}$ greater than 0.01, or is a subset of the data of an experiment or
series of experiments that provide an input datum or input data with $S_{\rm
c}>0.01$. Not counting such input data with $S_{\rm c}<0.01$, the seven data
with $|r_{i}|>1.2$ are $A50$, $B11$, $B37.3$, $B54$, $C19$, $C21$, and $C28$;
their values of $r_{i}$ are $-1.24$, $1.43$, $1.39$, $-1.31$, $-1.60$,
$-1.83$, and $1.76$, respectively.

As discussed in Sec. XIII.2.1, the 2010 recommended value of $G$ is the
weighted mean of the 11 measured values in Table 24 after the uncertainty of
each is multiplied by the factor 14. Although these data can be treated
separately because they are independent of all of the other data, they could
have been included with the other data. For example, if the 11 values of $G$
with expanded uncertainties are added to the 148 input data of adjustment 3,
$G$ is taken as an additional adjusted constant so that these 11 values can be
included in a new adjustment using the observational equation $G\doteq G$, and
the so-modified adjustment 3 is repeated, then we find for this “grand final
adjustment” that $N=160$, $M=83$, $\nu=77$, $\chi^{2}=59.1$,
$p(59.1|77)=0.94$, and $R_{\rm B}=0.88$. Of course, the resulting values of
the adjusted constants, and of the normalized residuals and self sensitivity
coefficients of the input data, are exactly the same as those from adjustment
3 and the weighted mean of the 11 measured values of $G$ with expanded
uncertainties.

In any event, the 2010 recommended values are calculated from the set of best
estimated values, in the least-squares sense, of 82 adjusted constants,
including $G$, and their variances and covariances, together with (i) those
constants that have exact values such as $\mu_{0}$ and $c$; and (ii) the
values of $m_{\mbox{\scriptsize{{t}}}}$, $G_{\rm F}$, and $\sin^{2}\theta_{\rm
W}$ given in Sec. XII. See Sec. V.B of CODATA-98 for details.

###  XIV.2 Tables of values

Tables 40 to 47 give the 2010 CODATA recommended values of the basic constants
and conversion factors of physics and chemistry and related quantities.
Although very similar in form and content to their 2006 counterparts, several
new recommended values have been included in the 2010 tables and a few have
been deleted. The values of the four new constants, $m_{\rm n}-m_{\rm p}$ in
kg and u, and $(m_{\rm n}-m_{\rm p})c^{2}$ in J and MeV, are given in Table
LABEL:tab:constants under the heading “Neutron, n”; and the values of the four
new constants $\mu_{\rm h}$, $\mu_{\rm h}/\mu_{\rm B}$, $\mu_{\rm h}/\mu_{\rm
N}$, and $g_{\rm h}$ are given in the same table under the heading “Helion,
h.” The three constants deleted, $\mu_{\rm t}/\mu_{\rm e}$, $\mu_{\rm
t}/\mu_{\rm p}$, and $\mu_{\rm t}/\mu_{\rm n}$, were in the 2006 version of
Table LABEL:tab:constants under the heading “Triton, t.” It was decided that
these constants were of limited interest and the values can be calculated from
other constants in the table.

The values of the four new helion-related constants are calculated from the
adjusted constant $\mu_{\rm h}^{\prime}/\mu_{\rm p}^{\prime}$ and the
theoretically predicted shielding correction $\sigma_{\rm
h}=59.967\,43(10)\times 10^{-6}$ due to Rudziński _et al._ (2009) using the
relation $\mu_{\rm h}^{\prime}=\mu_{\rm h}(1-\sigma_{\rm h})$; see Sec.
VI.1.2.

Table 40: An abbreviated list of the CODATA recommended values of the fundamental constants of physics and chemistry based on the 2010 adjustment. |  |  |  | Relative std.
---|---|---|---|---
Quantity  |  Symbol  |  Numerical value  |  Unit  | uncert. $u_{\rm r}$
speed of light in vacuum  |  $c,c_{0}$ | 299 792 458  | m s-1 | exact
magnetic constant  |  $\mu_{0}$ |  $4\mbox{{p}}\times 10^{-7}$ | N A-2 |
|  |  $=12.566\,370\,614...\times 10^{-7}$ | N A-2 | exact
electric constant 1/$\mu_{0}c^{2}$ |  $\epsilon_{0}$ |  $8.854\,187\,817...\times 10^{-12}$ | F m-1 | exact
Newtonian constant of gravitation  |  $G$ |  $6.673\,84(80)\times 10^{-11}$ | m3 kg-1 s-2 | $1.2\times 10^{-4}$
Planck constant  |  $h$ |  $6.626\,069\,57(29)\times 10^{-34}$ | J s  | $4.4\times 10^{-8}$
$h/2\mbox{{p}}$ |  $\hbar$ |  $1.054\,571\,726(47)\times 10^{-34}$ | J s  | $4.4\times 10^{-8}$
elementary charge  |  $e$ |  $1.602\,176\,565(35)\times 10^{-19}$ | C  | $2.2\times 10^{-8}$
magnetic flux quantum $h$/2$e$ |  ${\it\Phi}_{0}$ |  $2.067\,833\,758(46)\times 10^{-15}$ | Wb  | $2.2\times 10^{-8}$
conductance quantum $2e^{2}\\!/h$ |  $G_{0}$ |  $7.748\,091\,7346(25)\times 10^{-5}$ | S  | $3.2\times 10^{-10}$
electron mass  |  $m_{\rm e}$ |  $9.109\,382\,91(40)\times 10^{-31}$ | kg  | $4.4\times 10^{-8}$
proton mass  |  $m_{\rm p}$ |  $1.672\,621\,777(74)\times 10^{-27}$ | kg  | $4.4\times 10^{-8}$
proton-electron mass ratio  |  $m_{\rm p}$/$m_{\rm e}$ |  $1836.152\,672\,45(75)$ |  | $4.1\times 10^{-10}$
fine-structure constant $e^{2}\\!/4\mbox{{p}}\epsilon_{0}\hbar c$ |  $\alpha$ |  $7.297\,352\,5698(24)\times 10^{-3}$ |  | $3.2\times 10^{-10}$
inverse fine-structure constant  |  $\alpha^{-1}$ |  $137.035\,999\,074(44)$ |  | $3.2\times 10^{-10}$
Rydberg constant $\alpha^{2}m_{\rm e}c/2h$ |  $R_{\infty}$ |  $10\,973\,731.568\,539(55)$ | m-1 | $5.0\times 10^{-12}$
Avogadro constant  |  $N_{\rm A},L$ |  $6.022\,141\,29(27)\times 10^{23}$ | mol-1 | $4.4\times 10^{-8}$
Faraday constant $N_{\rm A}e$ |  $F$ |  $96\,485.3365(21)$ | C mol-1 | $2.2\times 10^{-8}$
molar gas constant  |  $R$ |  $8.314\,4621(75)$ | J mol-1 K-1 | $9.1\times 10^{-7}$
Boltzmann constant $R$/$N_{\rm A}$ |  $k$ |  $1.380\,6488(13)\times 10^{-23}$ | J K-1 | $9.1\times 10^{-7}$
Stefan-Boltzmann constant  |  |  |  |
($\mbox{{p}}^{2}$/60)$k^{4}\\!/\hbar^{3}c^{2}$ |  $\sigma$ |  $5.670\,373(21)\times 10^{-8}$ | W m-2 K-4 | $3.6\times 10^{-6}$
Non-SI units accepted for use with the SI
electron volt ($e$/C) J  | eV  |  $1.602\,176\,565(35)\times 10^{-19}$ | J  | $2.2\times 10^{-8}$
(unified) atomic mass unit ${1\over 12}m(^{12}$C)  | u  |  $1.660\,538\,921(73)\times 10^{-27}$ | kg  |  $4.4\times 10^{-8}$ —

Table 40 is a highly-abbreviated list of the values of the constants and
conversion factors most commonly used. Table LABEL:tab:constants is a much
more extensive list of values categorized as follows: UNIVERSAL;
ELECTROMAGNETIC; ATOMIC AND NUCLEAR; and PHYSICOCHEMICAL. The ATOMIC AND
NUCLEAR category is subdivided into 11 subcategories: General; Electroweak;
Electron, ${\rm e}^{-}$; Muon, ${\mbox{{m}}}^{-}$; Tau, ${\mbox{{t}}}^{-}$;
Proton, ${\rm p}$; Neutron, ${\rm n}$; Deuteron, ${\rm d}$; Triton, ${\rm t}$;
Helion, ${\rm h}$; and Alpha particle, a. Table 42 gives the variances,
covariances, and correlation coefficients of a selected group of constants.
(Use of the covariance matrix is discussed in Appendix E of CODATA-98.) Table
43 gives the internationally adopted values of various quantities; Table 44
lists the values of a number of x-ray related quantities; Table 45 lists the
values of various non-SI units; and Tables 46 and 47 give the values of
various energy equivalents.

All of the values given in Tables 40 to 47 are available on the Web pages of
the Fundamental Constants Data Center of the NIST Physical Measurement
Laboratory at physics.nist.gov/constants. This electronic version of the 2010
CODATA recommended values of the constants also includes a much more extensive
correlation coefficient matrix. In fact, the correlation coefficient of any
two constants listed in the tables is accessible on the Web site, as well as
the automatic conversion of the value of an energy-related quantity expressed
in one unit to the corresponding value expressed in another unit (in essence,
an automated version of Tables 46 and 47).

Table 41: The CODATA recommended values of the fundamental constants of physics and chemistry based on the 2010 adjustment. |  |  |  | Relative std.
---|---|---|---|---
Quantity |  Symbol |  Numerical value |  Unit | uncert. $u_{\rm r}$
UNIVERSAL
speed of light in vacuum | $c,c_{0}$ |  $299\,792\,$458 | m s-1 | exact
magnetic constant | $\mu_{0}$ | 4$\mbox{{p}}\times 10^{-7}$ | N A-2 |
|  | $=12.566\,370\,614...\times 10^{-7}$ | N A-2 | exact
electric constant 1/$\mu_{0}c^{2}$ | $\epsilon_{0}$ | $8.854\,187\,817...\times 10^{-12}$ | F m-1 | exact
characteristic impedance of vacuum $\mu_{0}c$ | $Z_{0}$ | $376.730\,313\,461...$ | ${\rm\Omega}$ | exact
Newtonian constant of gravitation | $G$ | $6.673\,84(80)\times 10^{-11}$ | m3 kg-1 s-2 | $1.2\times 10^{-4}$
| $G/\hbar c$ | $6.708\,37(80)\times 10^{-39}$ | $({\rm GeV}/c^{2})^{-2}$ | $1.2\times 10^{-4}$
Planck constant | $h$ | $6.626\,069\,57(29)\times 10^{-34}$ | J s | $4.4\times 10^{-8}$
|  | $4.135\,667\,516(91)\times 10^{-15}$ | eV s | $2.2\times 10^{-8}$
$h/2\mbox{{p}}$ | $\hbar$ | $1.054\,571\,726(47)\times 10^{-34}$ | J s | $4.4\times 10^{-8}$
|  | $6.582\,119\,28(15)\times 10^{-16}$ | eV s | $2.2\times 10^{-8}$
| $\hbar c$ | $197.326\,9718(44)$ | MeV fm | $2.2\times 10^{-8}$
Planck mass $(\hbar c/G)^{1/2}$ | $m_{\rm P}$ | $2.176\,51(13)\times 10^{-8}$ | kg | $6.0\times 10^{-5}$
energy equivalent | $m_{\rm P}c^{2}$ | $1.220\,932(73)\times 10^{19}$ | GeV | $6.0\times 10^{-5}$
Planck temperature $(\hbar c^{5}/G)^{1/2}/k$ | $T_{\rm P}$ | $1.416\,833(85)\times 10^{32}$ | K | $6.0\times 10^{-5}$
Planck length $\hbar/m_{\rm P}c=(\hbar G/c^{3})^{1/2}$ | $l_{\rm P}$ | $1.616\,199(97)\times 10^{-35}$ | m | $6.0\times 10^{-5}$
Planck time $l_{\rm P}/c=(\hbar G/c^{5})^{1/2}$ | $t_{\rm P}$ | $5.391\,06(32)\times 10^{-44}$ | s | $6.0\times 10^{-5}$
ELECTROMAGNETIC
elementary charge | $e$ | $1.602\,176\,565(35)\times 10^{-19}$ | C | $2.2\times 10^{-8}$
| $e/h$ | $2.417\,989\,348(53)\times 10^{14}$ | A J-1 | $2.2\times 10^{-8}$
magnetic flux quantum $h/2e$ | ${\it\Phi}_{0}$ | $2.067\,833\,758(46)\times 10^{-15}$ | Wb | $2.2\times 10^{-8}$
conductance quantum $2e^{2}\\!/h$ | $G_{0}$ | $7.748\,091\,7346(25)\times 10^{-5}$ | S | $3.2\times 10^{-10}$
inverse of conductance quantum | $G_{0}^{-1}$ | $12\,906.403\,7217(42)$ | ${\rm\Omega}$ | $3.2\times 10^{-10}$
Josephson constant666See Table 43 for the conventional value adopted internationally for realizing representations of the volt using the Josephson effect. 2$e/h$ | $K_{\rm J}$ | $483\,597.870(11)\times 10^{9}$ | Hz V-1 | $2.2\times 10^{-8}$
von Klitzing constant777See Table 43 for the conventional value adopted internationally for realizing representations of the ohm using the quantum Hall effect. $h/e^{2}=\mu_{0}c/2\alpha$ | $R_{\rm K}$ | $25\,812.807\,4434(84)$ | ${\rm\Omega}$ | $3.2\times 10^{-10}$
Bohr magneton $e\hbar/2m_{\rm e}$ | $\mu_{\rm B}$ | $927.400\,968(20)\times 10^{-26}$ | J T-1 | $2.2\times 10^{-8}$
|  | $5.788\,381\,8066(38)\times 10^{-5}$ | eV T-1 | $6.5\times 10^{-10}$
| $\mu_{\rm B}/h$ | $13.996\,245\,55(31)\times 10^{9}$ | Hz T-1 | $2.2\times 10^{-8}$
| $\mu_{\rm B}/hc$ | $46.686\,4498(10)$ | m${}^{-1}~{}$T-1 | $2.2\times 10^{-8}$
| $\mu_{\rm B}/k$ | $0.671\,713\,88(61)$ | K T-1 | $9.1\times 10^{-7}$
nuclear magneton $e\hbar/2m_{\rm p}$ | $\mu_{\rm N}$ | $5.050\,783\,53(11)\times 10^{-27}$ | J T-1 | $2.2\times 10^{-8}$
|  | $3.152\,451\,2605(22)\times 10^{-8}$ | eV T-1 | $7.1\times 10^{-10}$
| $\mu_{\rm N}/h$ | $7.622\,593\,57(17)$ | MHz T-1 | $2.2\times 10^{-8}$
| $\mu_{\rm N}/hc$ | $2.542\,623\,527(56)\times 10^{-2}$ | m${}^{-1}~{}$T-1 | $2.2\times 10^{-8}$
| $\mu_{\rm N}/k$ | $3.658\,2682(33)\times 10^{-4}$ | K T-1 | $9.1\times 10^{-7}$
ATOMIC AND NUCLEAR
General
fine-structure constant $e^{2}\\!/4\mbox{{p}}\epsilon_{0}\hbar c$ | $\alpha$ | $7.297\,352\,5698(24)\times 10^{-3}$ |  | $3.2\times 10^{-10}$
inverse fine-structure constant | $\alpha^{-1}$ | $137.035\,999\,074(44)$ |  | $3.2\times 10^{-10}$
Rydberg constant $\alpha^{2}m_{\rm e}c/2h$ | $R_{\infty}$ | $10\,973\,731.568\,539(55)$ | m-1 | $5.0\times 10^{-12}$
| $R_{\infty}c$ | $3.289\,841\,960\,364(17)\times 10^{15}$ | Hz | $5.0\times 10^{-12}$
| $R_{\infty}hc$ | $2.179\,872\,171(96)\times 10^{-18}$ | J | $4.4\times 10^{-8}$
|  | $13.605\,692\,53(30)$ | eV | $2.2\times 10^{-8}$
Bohr radius $\alpha/4\mbox{{p}}R_{\infty}=4\mbox{{p}}\epsilon_{0}\hbar^{2}\\!/m_{\rm e}e^{2}$ | $a_{\rm 0}$ | $0.529\,177\,210\,92(17)\times 10^{-10}$ | m | $3.2\times 10^{-10}$
Hartree energy $e^{2}\\!/4\mbox{{p}}\epsilon_{\rm 0}a_{\rm 0}=2R_{\infty}hc=\alpha^{2}m_{\rm e}c^{2}$ | $E_{\rm h}$ | $4.359\,744\,34(19)\times 10^{-18}$ | J | $4.4\times 10^{-8}$
|  | $27.211\,385\,05(60)$ | eV | $2.2\times 10^{-8}$
quantum of circulation | $h/2m_{\rm e}$ | $3.636\,947\,5520(24)\times 10^{-4}$ | m${}^{2}~{}$s-1 | $6.5\times 10^{-10}$
| $h/m_{\rm e}$ | $7.273\,895\,1040(47)\times 10^{-4}$ | m${}^{2}~{}$s-1 | $6.5\times 10^{-10}$
Electroweak
Fermi coupling constant888Value recommended by the Particle Data Group Nakamura _et al._ (2010). | $G_{\rm F}/(\hbar c)^{3}$ | $1.166\,364(5)\times 10^{-5}$ | GeV-2 | $4.3\times 10^{-6}$
weak mixing angle999Based on the ratio of the masses of the W and Z bosons $m_{\rm W}/m_{\rm Z}$ recommended by the Particle Data Group Nakamura _et al._ (2010). The value for ${\rm sin}^{2}{\theta}_{\rm W}$ they recommend, which is based on a particular variant of the modified minimal subtraction $({\scriptstyle{\rm\overline{MS}}})$ scheme, is ${\rm sin}^{2}\hat{\theta}_{\rm W}(M_{\rm Z})=0.231\,16(13)$. $\theta_{\rm W}$ (on-shell scheme) |  |  |  |
$\sin^{2}\theta_{\rm W}=s^{2}_{\rm W}\equiv 1-(m_{\rm W}/m_{\rm Z})^{2}$ | $\sin^{2}\theta_{\rm W}$ | $0.2223(21)$ |  | $9.5\times 10^{-3}$
Electron, e-
electron mass | $m_{\rm e}$ | $9.109\,382\,91(40)\times 10^{-31}$ | kg | $4.4\times 10^{-8}$
|  | $5.485\,799\,0946(22)\times 10^{-4}$ | u | $4.0\times 10^{-10}$
energy equivalent | $m_{\rm e}c^{2}$ | $8.187\,105\,06(36)\times 10^{-14}$ | J | $4.4\times 10^{-8}$
|  | $0.510\,998\,928(11)$ | MeV | $2.2\times 10^{-8}$
electron-muon mass ratio | $m_{\rm e}/m_{\mbox{\scriptsize{{m}}}}$ | $4.836\,331\,66(12)\times 10^{-3}$ |  | $2.5\times 10^{-8}$
electron-tau mass ratio | $m_{\rm e}/m_{\mbox{\scriptsize{{t}}}}$ | $2.875\,92(26)\times 10^{-4}$ |  | $9.0\times 10^{-5}$
electron-proton mass ratio | $m_{\rm e}/m_{\rm p}$ | $5.446\,170\,2178(22)\times 10^{-4}$ |  | $4.1\times 10^{-10}$
electron-neutron mass ratio | $m_{\rm e}/m_{\rm n}$ | $5.438\,673\,4461(32)\times 10^{-4}$ |  | $5.8\times 10^{-10}$
electron-deuteron mass ratio | $m_{\rm e}/m_{\rm d}$ | $2.724\,437\,1095(11)\times 10^{-4}$ |  | $4.0\times 10^{-10}$
electron-triton mass ratio | $m_{\rm e}/m_{\rm t}$ | $1.819\,200\,0653(17)\times 10^{-4}$ |  | $9.1\times 10^{-10}$
electron-helion mass ratio | $m_{\rm e}/m_{\rm h}$ | $1.819\,543\,0761(17)\times 10^{-4}$ |  | $9.2\times 10^{-10}$
electron to alpha particle mass ratio | $m_{\rm e}/m_{\mbox{\scriptsize{{a}}}}$ | $1.370\,933\,555\,78(55)\times 10^{-4}$ |  | $4.0\times 10^{-10}$
electron charge to mass quotient | $-e/m_{\rm e}$ | $-1.758\,820\,088(39)\times 10^{11}$ | C kg-1 | $2.2\times 10^{-8}$
electron molar mass $N_{\rm A}m_{\rm e}$ | $M({\rm e}),M_{\rm e}$ | $5.485\,799\,0946(22)\times 10^{-7}$ | kg mol-1 | $4.0\times 10^{-10}$
Compton wavelength $h/m_{\rm e}c$ | $\lambda_{\rm C}$ | $2.426\,310\,2389(16)\times 10^{-12}$ | m | $6.5\times 10^{-10}$
$\lambda_{\rm C}/2\mbox{{p}}=\alpha a_{\rm 0}=\alpha^{2}\\!/4\mbox{{p}}R_{\infty}$ | $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm C}$ | $386.159\,268\,00(25)\times 10^{-15}$ | m | $6.5\times 10^{-10}$
classical electron radius $\alpha^{2}a_{\rm 0}$ | $r_{\rm e}$ | $2.817\,940\,3267(27)\times 10^{-15}$ | m | $9.7\times 10^{-10}$
Thomson cross section (8$\mbox{{p}}/3)r^{2}_{\rm e}$ | $\sigma_{\rm e}$ | $0.665\,245\,8734(13)\times 10^{-28}$ | m2 | $1.9\times 10^{-9}$
electron magnetic moment | $\mu_{\rm e}$ | $-928.476\,430(21)\times 10^{-26}$ | J T-1 | $2.2\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\rm e}/\mu_{\rm B}$ | $-1.001\,159\,652\,180\,76(27)$ |  | $2.6\times 10^{-13}$
to nuclear magneton ratio | $\mu_{\rm e}/\mu_{\rm N}$ | $-1838.281\,970\,90(75)$ |  | $4.1\times 10^{-10}$
electron magnetic moment |  |  |  |
anomaly $|\mu_{\rm e}|/\mu_{\rm B}-1$ | $a_{\rm e}$ | $1.159\,652\,180\,76(27)\times 10^{-3}$ |  | $2.3\times 10^{-10}$
electron $g$-factor $-2(1+a_{\rm e})$ | $g_{\rm e}$ | $-2.002\,319\,304\,361\,53(53)$ |  | $2.6\times 10^{-13}$
electron-muon magnetic moment ratio | $\mu_{\rm e}/\mu_{\mbox{\scriptsize{{m}}}}$ | $206.766\,9896(52)$ |  | $2.5\times 10^{-8}$
electron-proton magnetic moment ratio | $\mu_{\rm e}/\mu_{\rm p}$ | $-658.210\,6848(54)$ |  | $8.1\times 10^{-9}$
electron to shielded proton magnetic |  |  |  |
moment ratio (H2O, sphere, 25 ∘C) | $\mu_{\rm e}/\mu^{\prime}_{\rm p}$ | $-658.227\,5971(72)$ |  | $1.1\times 10^{-8}$
electron-neutron magnetic moment ratio | $\mu_{\rm e}/\mu_{\rm n}$ | $960.920\,50(23)$ |  | $2.4\times 10^{-7}$
electron-deuteron magnetic moment ratio | $\mu_{\rm e}/\mu_{\rm d}$ | $-2143.923\,498(18)$ |  | $8.4\times 10^{-9}$
electron to shielded helion magnetic |  |  |  |
moment ratio (gas, sphere, 25 ∘C) | $\mu_{\rm e}/\mu^{\prime}_{\rm h}$ | $864.058\,257(10)$ |  | $1.2\times 10^{-8}$
electron gyromagnetic ratio $2|\mu_{\rm e}|/\hbar$ | $\gamma_{\rm e}$ | $1.760\,859\,708(39)\times 10^{11}$ | s${}^{-1}~{}$T-1 | $2.2\times 10^{-8}$
| $\gamma_{\rm e}/2\mbox{{p}}$ | $28\,024.952\,66(62)$ | MHz T-1 | $2.2\times 10^{-8}$
Muon, ${\mbox{{m}}}^{-}$
muon mass | $m_{\mbox{\scriptsize{{m}}}}$ | $1.883\,531\,475(96)\times 10^{-28}$ | kg | $5.1\times 10^{-8}$
|  | $0.113\,428\,9267(29)$ | u | $2.5\times 10^{-8}$
energy equivalent | $m_{\mbox{\scriptsize{{m}}}}c^{2}$ | $1.692\,833\,667(86)\times 10^{-11}$ | J | $5.1\times 10^{-8}$
|  | $105.658\,3715(35)$ | MeV | $3.4\times 10^{-8}$
muon-electron mass ratio | $m_{\mbox{\scriptsize{{m}}}}/m_{\rm e}$ | $206.768\,2843(52)$ |  | $2.5\times 10^{-8}$
muon-tau mass ratio | $m_{\mbox{\scriptsize{{m}}}}/m_{\mbox{\scriptsize{{t}}}}$ | $5.946\,49(54)\times 10^{-2}$ |  | $9.0\times 10^{-5}$
muon-proton mass ratio | $m_{\mbox{\scriptsize{{m}}}}/m_{\rm p}$ | $0.112\,609\,5272(28)$ |  | $2.5\times 10^{-8}$
muon-neutron mass ratio | $m_{\mbox{\scriptsize{{m}}}}/m_{\rm n}$ | $0.112\,454\,5177(28)$ |  | $2.5\times 10^{-8}$
muon molar mass $N_{\rm A}m_{\mbox{\scriptsize{{m}}}}$ | $M({\mbox{{m}}}),M_{\mbox{\scriptsize{{m}}}}$ | $0.113\,428\,9267(29)\times 10^{-3}$ | kg mol-1 | $2.5\times 10^{-8}$
muon Compton wavelength $h/m_{\mbox{\scriptsize{{m}}}}c$ | $\lambda_{{\rm C},{\mbox{\scriptsize{{m}}}}}$ | $11.734\,441\,03(30)\times 10^{-15}$ | m | $2.5\times 10^{-8}$
$\lambda_{{\rm C},{\mbox{\scriptsize{{m}}}}}/2\mbox{{p}}$ | $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{{\rm C},{\mbox{\scriptsize{{m}}}}}$ | $1.867\,594\,294(47)\times 10^{-15}$ | m | $2.5\times 10^{-8}$
muon magnetic moment | $\mu_{\mbox{\scriptsize{{m}}}}$ | $-4.490\,448\,07(15)\times 10^{-26}$ | J T-1 | $3.4\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm B}$ | $-4.841\,970\,44(12)\times 10^{-3}$ |  | $2.5\times 10^{-8}$
to nuclear magneton ratio | $\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm N}$ | $-8.890\,596\,97(22)$ |  | $2.5\times 10^{-8}$
muon magnetic moment anomaly |  |  |  |
$|\mu_{\mbox{\scriptsize{{m}}}}|/(e\hbar/2m_{\mbox{\scriptsize{{m}}}})-1$ | $a_{\mbox{\scriptsize{{m}}}}$ | $1.165\,920\,91(63)\times 10^{-3}$ |  | $5.4\times 10^{-7}$
muon $g$-factor $-2(1+a_{\mbox{\scriptsize{{m}}}}$) | $g_{\mbox{\scriptsize{{m}}}}$ | $-2.002\,331\,8418(13)$ |  | $6.3\times 10^{-10}$
muon-proton magnetic moment ratio | $\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm p}$ | $-3.183\,345\,107(84)$ |  | $2.6\times 10^{-8}$
Tau, ${\mbox{{t}}}^{-}$
tau mass101010This and all other values involving $m_{\mbox{\scriptsize{{t}}}}$ are based on the value of $m_{\mbox{\scriptsize{{t}}}}c^{2}$ in MeV recommended by the Particle Data Group Nakamura _et al._ (2010). | $m_{\mbox{\scriptsize{{t}}}}$ | $3.167\,47(29)\times 10^{-27}$ | kg | $9.0\times 10^{-5}$
|  | $1.907\,49(17)$ | u | $9.0\times 10^{-5}$
energy equivalent | $m_{\mbox{\scriptsize{{t}}}}c^{2}$ | $2.846\,78(26)\times 10^{-10}$ | J | $9.0\times 10^{-5}$
|  | $1776.82(16)$ | MeV | $9.0\times 10^{-5}$
tau-electron mass ratio | $m_{\mbox{\scriptsize{{t}}}}/m_{\rm e}$ | $3477.15(31)$ |  | $9.0\times 10^{-5}$
tau-muon mass ratio | $m_{\mbox{\scriptsize{{t}}}}/m_{\mbox{\scriptsize{{m}}}}$ | $16.8167(15)$ |  | $9.0\times 10^{-5}$
tau-proton mass ratio | $m_{\mbox{\scriptsize{{t}}}}/m_{\rm p}$ | $1.893\,72(17)$ |  | $9.0\times 10^{-5}$
tau-neutron mass ratio | $m_{\mbox{\scriptsize{{t}}}}/m_{\rm n}$ | $1.891\,11(17)$ |  | $9.0\times 10^{-5}$
tau molar mass $N_{\rm A}m_{\mbox{\scriptsize{{t}}}}$ | $M({\mbox{{t}}}),M_{\mbox{\scriptsize{{t}}}}$ | $1.907\,49(17)\times 10^{-3}$ | kg mol-1 | $9.0\times 10^{-5}$
tau Compton wavelength $h/m_{\mbox{\scriptsize{{t}}}}c$ | $\lambda_{{\rm C},{\mbox{\scriptsize{{t}}}}}$ | $0.697\,787(63)\times 10^{-15}$ | m | $9.0\times 10^{-5}$
$\lambda_{{\rm C},{\mbox{\scriptsize{{t}}}}}/2\mbox{{p}}$ | $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{{\rm C},{\mbox{\scriptsize{{t}}}}}$ | $0.111\,056(10)\times 10^{-15}$ | m | $9.0\times 10^{-5}$
Proton, p
proton mass | $m_{\rm p}$ | $1.672\,621\,777(74)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
|  | $1.007\,276\,466\,812(90)$ | u | $8.9\times 10^{-11}$
energy equivalent | $m_{\rm p}c^{2}$ | $1.503\,277\,484(66)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
|  | $938.272\,046(21)$ | MeV | $2.2\times 10^{-8}$
proton-electron mass ratio | $m_{\rm p}/m_{\rm e}$ | $1836.152\,672\,45(75)$ |  | $4.1\times 10^{-10}$
proton-muon mass ratio | $m_{\rm p}/m_{\mbox{\scriptsize{{m}}}}$ | $8.880\,243\,31(22)$ |  | $2.5\times 10^{-8}$
proton-tau mass ratio | $m_{\rm p}/m_{\mbox{\scriptsize{{t}}}}$ | $0.528\,063(48)$ |  | $9.0\times 10^{-5}$
proton-neutron mass ratio | $m_{\rm p}/m_{\rm n}$ | $0.998\,623\,478\,26(45)$ |  | $4.5\times 10^{-10}$
proton charge to mass quotient | $e/m_{\rm p}$ | $9.578\,833\,58(21)\times 10^{7}$ | C kg-1 | $2.2\times 10^{-8}$
proton molar mass $N_{\rm A}m_{\rm p}$ |  $M$(p), $M_{\rm p}$ | $1.007\,276\,466\,812(90)\times 10^{-3}$ | kg mol-1 | $8.9\times 10^{-11}$
proton Compton wavelength $h/m_{\rm p}c$ | $\lambda_{\rm C,p}$ | $1.321\,409\,856\,23(94)\times 10^{-15}$ | m | $7.1\times 10^{-10}$
$\lambda_{\rm C,p}/2\mbox{{p}}$ | $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm C,p}$ | $0.210\,308\,910\,47(15)\times 10^{-15}$ | m | $7.1\times 10^{-10}$
proton rms charge radius | $r_{\rm p}$ | $0.8775(51)\times 10^{-15}$ | m | $5.9\times 10^{-3}$
proton magnetic moment | $\mu_{\rm p}$ | $1.410\,606\,743(33)\times 10^{-26}$ | J T-1 | $2.4\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\rm p}/\mu_{\rm B}$ | $1.521\,032\,210(12)\times 10^{-3}$ |  | $8.1\times 10^{-9}$
to nuclear magneton ratio | $\mu_{\rm p}/\mu_{\rm N}$ | $2.792\,847\,356(23)$ |  | $8.2\times 10^{-9}$
proton $g$-factor $2\mu_{\rm p}/\mu_{\rm N}$ | $g_{\rm p}$ | $5.585\,694\,713(46)$ |  | $8.2\times 10^{-9}$
proton-neutron magnetic moment ratio | $\mu_{\rm p}/\mu_{\rm n}$ | $-1.459\,898\,06(34)$ |  | $2.4\times 10^{-7}$
shielded proton magnetic moment | $\mu^{\prime}_{\rm p}$ | $1.410\,570\,499(35)\times 10^{-26}$ | J T-1 | $2.5\times 10^{-8}$
(H2O, sphere, 25 ∘C) |  |  |  |
to Bohr magneton ratio | $\mu^{\prime}_{\rm p}/\mu_{\rm B}$ | $1.520\,993\,128(17)\times 10^{-3}$ |  | $1.1\times 10^{-8}$
to nuclear magneton ratio | $\mu^{\prime}_{\rm p}/\mu_{\rm N}$ | $2.792\,775\,598(30)$ |  | $1.1\times 10^{-8}$
proton magnetic shielding correction |  |  |  |
$1-\mu^{\prime}_{\rm p}/\mu_{\rm p}$ (H2O, sphere, 25 ∘C) | $\sigma^{\prime}_{\rm p}$ | $25.694(14)\times 10^{-6}$ |  | $5.3\times 10^{-4}$
proton gyromagnetic ratio $2\mu_{\rm p}/\hbar$ | $\gamma_{\rm p}$ | $2.675\,222\,005(63)\times 10^{8}$ | s${}^{-1}~{}$T-1 | $2.4\times 10^{-8}$
| $\gamma_{\rm p}/2\mbox{{p}}$ | $42.577\,4806(10)$ | MHz T-1 | $2.4\times 10^{-8}$
shielded proton gyromagnetic ratio |  |  |  |
$2\mu^{\prime}_{\rm p}/\hbar$ (H2O, sphere, 25 ∘C) | $\gamma^{\prime}_{\rm p}$ | $2.675\,153\,268(66)\times 10^{8}$ | s${}^{-1}~{}$T-1 | $2.5\times 10^{-8}$
| $\gamma^{\prime}_{\rm p}/2\mbox{{p}}$ | $42.576\,3866(10)$ | MHz T-1 | $2.5\times 10^{-8}$
Neutron, n
neutron mass | $m_{\rm n}$ | $1.674\,927\,351(74)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
|  | $1.008\,664\,916\,00(43)$ | u | $4.2\times 10^{-10}$
energy equivalent | $m_{\rm n}c^{2}$ | $1.505\,349\,631(66)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
|  | $939.565\,379(21)$ | MeV | $2.2\times 10^{-8}$
neutron-electron mass ratio | $m_{\rm n}/m_{\rm e}$ | $1838.683\,6605(11)$ |  | $5.8\times 10^{-10}$
neutron-muon mass ratio | $m_{\rm n}/m_{\mbox{\scriptsize{{m}}}}$ | $8.892\,484\,00(22)$ |  | $2.5\times 10^{-8}$
neutron-tau mass ratio | $m_{\rm n}/m_{\mbox{\scriptsize{{t}}}}$ | $0.528\,790(48)$ |  | $9.0\times 10^{-5}$
neutron-proton mass ratio | $m_{\rm n}/m_{\rm p}$ | $1.001\,378\,419\,17(45)$ |  | $4.5\times 10^{-10}$
neutron-proton mass difference | $m_{\rm n}-m_{\rm p}$ | $2.305\,573\,92(76)\times 10^{-30}$ | kg | $3.3\times 10^{-7}$
|  | $0.001\,388\,449\,19(45)$ | u | $3.3\times 10^{-7}$
energy equivalent | ($m_{\rm n}-m_{\rm p})c^{2}$ | $2.072\,146\,50(68)\times 10^{-13}$ | J | $3.3\times 10^{-7}$
|  | $1.293\,332\,17(42)$ | MeV | $3.3\times 10^{-7}$
neutron molar mass $N_{\rm A}m_{\rm n}$ | $M({\rm n}),M_{\rm n}$ | $1.008\,664\,916\,00(43)\times 10^{-3}$ | kg mol-1 | $4.2\times 10^{-10}$
neutron Compton wavelength $h/m_{\rm n}c$ | $\lambda_{\rm C,n}$ | $1.319\,590\,9068(11)\times 10^{-15}$ | m | $8.2\times 10^{-10}$
$\lambda_{\rm C,n}/2\mbox{{p}}$ | $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm C,n}$ | $0.210\,019\,415\,68(17)\times 10^{-15}$ | m | $8.2\times 10^{-10}$
neutron magnetic moment | $\mu_{\rm n}$ | $-0.966\,236\,47(23)\times 10^{-26}$ | J T-1 | $2.4\times 10^{-7}$
to Bohr magneton ratio | $\mu_{\rm n}/\mu_{\rm B}$ | $-1.041\,875\,63(25)\times 10^{-3}$ |  | $2.4\times 10^{-7}$
to nuclear magneton ratio | $\mu_{\rm n}/\mu_{\rm N}$ | $-1.913\,042\,72(45)$ |  | $2.4\times 10^{-7}$
neutron $g$-factor $2\mu_{\rm n}/\mu_{\rm N}$ | $g_{\rm n}$ | $-3.826\,085\,45(90)$ |  | $2.4\times 10^{-7}$
neutron-electron magnetic moment ratio | $\mu_{\rm n}/\mu_{\rm e}$ | $1.040\,668\,82(25)\times 10^{-3}$ |  | $2.4\times 10^{-7}$
neutron-proton magnetic moment ratio | $\mu_{\rm n}/\mu_{\rm p}$ | $-0.684\,979\,34(16)$ |  | $2.4\times 10^{-7}$
neutron to shielded proton magnetic |  |  |  |
moment ratio (H2O, sphere, 25 ∘C) | $\mu_{\rm n}/\mu_{\rm p}^{\prime}$ | $-0.684\,996\,94(16)$ |  | $2.4\times 10^{-7}$
neutron gyromagnetic ratio $2|\mu_{\rm n}|/\hbar$ | $\gamma_{\rm n}$ | $1.832\,471\,79(43)\times 10^{8}$ | s${}^{-1}~{}$T-1 | $2.4\times 10^{-7}$
| $\gamma_{\rm n}/2\mbox{{p}}$ | $29.164\,6943(69)$ | MHz T-1 | $2.4\times 10^{-7}$
Deuteron, d
deuteron mass | $m_{\rm d}$ | $3.343\,583\,48(15)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
|  | $2.013\,553\,212\,712(77)$ | u | $3.8\times 10^{-11}$
energy equivalent | $m_{\rm d}c^{2}$ | $3.005\,062\,97(13)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
|  | $1875.612\,859(41)$ | MeV | $2.2\times 10^{-8}$
deuteron-electron mass ratio | $m_{\rm d}/m_{\rm e}$ | $3670.482\,9652(15)$ |  | $4.0\times 10^{-10}$
deuteron-proton mass ratio | $m_{\rm d}/m_{\rm p}$ | $1.999\,007\,500\,97(18)$ |  | $9.2\times 10^{-11}$
deuteron molar mass $N_{\rm A}m_{\rm d}$ | $M({\rm d}),M_{\rm d}$ | $2.013\,553\,212\,712(77)\times 10^{-3}$ | kg mol-1 | $3.8\times 10^{-11}$
deuteron rms charge radius | $r_{\rm d}$ | $2.1424(21)\times 10^{-15}$ | m | $9.8\times 10^{-4}$
deuteron magnetic moment | $\mu_{\rm d}$ | $0.433\,073\,489(10)\times 10^{-26}$ | J T-1 | $2.4\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\rm d}/\mu_{\rm B}$ | $0.466\,975\,4556(39)\times 10^{-3}$ |  | $8.4\times 10^{-9}$
to nuclear magneton ratio | $\mu_{\rm d}/\mu_{\rm N}$ | $0.857\,438\,2308(72)$ |  | $8.4\times 10^{-9}$
deuteron $g$-factor $\mu_{\rm d}/\mu_{\rm N}$ | $g_{\rm d}$ | $0.857\,438\,2308(72)$ |  | $8.4\times 10^{-9}$
deuteron-electron magnetic moment ratio | $\mu_{\rm d}/\mu_{\rm e}$ | $-4.664\,345\,537(39)\times 10^{-4}$ |  | $8.4\times 10^{-9}$
deuteron-proton magnetic moment ratio | $\mu_{\rm d}/\mu_{\rm p}$ | $0.307\,012\,2070(24)$ |  | $7.7\times 10^{-9}$
deuteron-neutron magnetic moment ratio | $\mu_{\rm d}/\mu_{\rm n}$ | $-0.448\,206\,52(11)$ |  | $2.4\times 10^{-7}$
Triton, t
triton mass | $m_{\rm t}$ | $5.007\,356\,30(22)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
|  | $3.015\,500\,7134(25)$ | u | $8.2\times 10^{-10}$
energy equivalent | $m_{\rm t}c^{2}$ | $4.500\,387\,41(20)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
|  | $2808.921\,005(62)$ | MeV | $2.2\times 10^{-8}$
triton-electron mass ratio | $m_{\rm t}/m_{\rm e}$ | $5496.921\,5267(50)$ |  | $9.1\times 10^{-10}$
triton-proton mass ratio | $m_{\rm t}/m_{\rm p}$ | $2.993\,717\,0308(25)$ |  | $8.2\times 10^{-10}$
triton molar mass $N_{\rm A}m_{\rm t}$ | $M({\rm t}),M_{\rm t}$ | $3.015\,500\,7134(25)\times 10^{-3}$ | kg mol-1 | $8.2\times 10^{-10}$
triton magnetic moment | $\mu_{\rm t}$ | $1.504\,609\,447(38)\times 10^{-26}$ | J T-1 | $2.6\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\rm t}/\mu_{\rm B}$ | $1.622\,393\,657(21)\times 10^{-3}$ |  | $1.3\times 10^{-8}$
to nuclear magneton ratio | $\mu_{\rm t}/\mu_{\rm N}$ | $2.978\,962\,448(38)$ |  | $1.3\times 10^{-8}$
triton $g$-factor $2\mu_{\rm t}/\mu_{\rm N}$ | $g_{\rm t}$ | $5.957\,924\,896(76)$ |  | $1.3\times 10^{-8}$
Helion, h
helion mass | $m_{\rm h}$ | $5.006\,412\,34(22)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
|  | $3.014\,932\,2468(25)$ | u | $8.3\times 10^{-10}$
energy equivalent | $m_{\rm h}c^{2}$ | $4.499\,539\,02(20)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
|  | $2808.391\,482(62)$ | MeV | $2.2\times 10^{-8}$
helion-electron mass ratio | $m_{\rm h}/m_{\rm e}$ | $5495.885\,2754(50)$ |  | $9.2\times 10^{-10}$
helion-proton mass ratio | $m_{\rm h}/m_{\rm p}$ | $2.993\,152\,6707(25)$ |  | $8.2\times 10^{-10}$
helion molar mass $N_{\rm A}m_{\rm h}$ | $M({\rm h}),M_{\rm h}$ | $3.014\,932\,2468(25)\times 10^{-3}$ | kg mol-1 | $8.3\times 10^{-10}$
helion magnetic moment | $\mu_{\rm h}$ | $-1.074\,617\,486(27)\times 10^{-26}$ | J T-1 | $2.5\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\rm h}/\mu_{\rm B}$ | $-1.158\,740\,958(14)\times 10^{-3}$ |  | $1.2\times 10^{-8}$
to nuclear magneton ratio | $\mu_{\rm h}/\mu_{\rm N}$ | $-2.127\,625\,306(25)$ |  | $1.2\times 10^{-8}$
helion $g$-factor $2\mu_{\rm h}/\mu_{\rm N}$ | $g_{\rm h}$ | $-4.255\,250\,613(50)$ |  | $1.2\times 10^{-8}$
shielded helion magnetic moment | $\mu^{\prime}_{\rm h}$ | $-1.074\,553\,044(27)\times 10^{-26}$ | J T-1 | $2.5\times 10^{-8}$
(gas, sphere, 25 ∘C) |  |  |  |
to Bohr magneton ratio | $\mu^{\prime}_{\rm h}/\mu_{\rm B}$ | $-1.158\,671\,471(14)\times 10^{-3}$ |  | $1.2\times 10^{-8}$
to nuclear magneton ratio | $\mu^{\prime}_{\rm h}/\mu_{\rm N}$ | $-2.127\,497\,718(25)$ |  | $1.2\times 10^{-8}$
shielded helion to proton magnetic |  |  |  |
moment ratio (gas, sphere, 25 ∘C) | $\mu^{\prime}_{\rm h}/\mu_{\rm p}$ | $-0.761\,766\,558(11)$ |  | $1.4\times 10^{-8}$
shielded helion to shielded proton magnetic |  |  |  |
moment ratio (gas/H2O, spheres, 25 ∘C) | $\mu^{\prime}_{\rm h}/\mu^{\prime}_{\rm p}$ | $-0.761\,786\,1313(33)$ |  | $4.3\times 10^{-9}$
shielded helion gyromagnetic ratio |  |  |  |
$2|\mu^{\prime}_{\rm h}|/\hbar$ (gas, sphere, 25 ∘C) | $\gamma^{\prime}_{\rm h}$ | $2.037\,894\,659(51)\times 10^{8}$ | s-1 T-1 | $2.5\times 10^{-8}$
| $\gamma^{\prime}_{\rm h}/2\mbox{{p}}$ | $32.434\,100\,84(81)$ | MHz T-1 | $2.5\times 10^{-8}$
Alpha particle, a
alpha particle mass | $m_{\mbox{\scriptsize{{a}}}}$ | $6.644\,656\,75(29)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
|  | $4.001\,506\,179\,125(62)$ | u | $1.5\times 10^{-11}$
energy equivalent | $m_{\mbox{\scriptsize{{a}}}}c^{2}$ | $5.971\,919\,67(26)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
|  | $3727.379\,240(82)$ | MeV | $2.2\times 10^{-8}$
alpha particle to electron mass ratio | $m_{\mbox{\scriptsize{{a}}}}/m_{\rm e}$ | $7294.299\,5361(29)$ |  | $4.0\times 10^{-10}$
alpha particle to proton mass ratio | $m_{\mbox{\scriptsize{{a}}}}/m_{\rm p}$ | $3.972\,599\,689\,33(36)$ |  | $9.0\times 10^{-11}$
alpha particle molar mass $N_{\rm A}m_{\mbox{\scriptsize{{a}}}}$ | $M({\mbox{{a}}}),M_{\mbox{\scriptsize{{a}}}}$ | $4.001\,506\,179\,125(62)\times 10^{-3}$ | kg mol-1 | $1.5\times 10^{-11}$
PHYSICOCHEMICAL
Avogadro constant | $N_{\rm A},L$ | $6.022\,141\,29(27)\times 10^{23}$ | mol-1 | $4.4\times 10^{-8}$
atomic mass constant |  |  |  |
$m_{\rm u}=\frac{1}{12}m(^{12}{\rm C})=1$ u | $m_{\rm u}$ | $1.660\,538\,921(73)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
energy equivalent | $m_{\rm u}c^{2}$ | $1.492\,417\,954(66)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
|  | $931.494\,061(21)$ | MeV | $2.2\times 10^{-8}$
Faraday constant111111The numerical value of $F$ to be used in coulometric chemical measurements is $96\,485.3321(43)$ [$4.4\times 10^{-8}$] when the relevant current is measured in terms of representations of the volt and ohm based on the Josephson and quantum Hall effects and the internationally adopted conventional values of the Josephson and von Klitzing constants $K_{\rm J-90}$ and $R_{\rm K-90}$ given in Table 43.  $N_{\rm A}e$ | $F$ | $96\,485.3365(21)$ | C mol-1 | $2.2\times 10^{-8}$
molar Planck constant | $N_{\rm A}h$ | $3.990\,312\,7176(28)\times 10^{-10}$ | J s mol-1 | $7.0\times 10^{-10}$
| $N_{\rm A}hc$ | $0.119\,626\,565\,779(84)$ | J m mol-1 | $7.0\times 10^{-10}$
molar gas constant | $R$ | $8.314\,4621(75)$ | J mol${}^{-1}~{}$K-1 | $9.1\times 10^{-7}$
Boltzmann constant $R/N_{\rm A}$ | $k$ | $1.380\,6488(13)\times 10^{-23}$ | J K-1 | $9.1\times 10^{-7}$
|  | $8.617\,3324(78)\times 10^{-5}$ | eV K-1 | $9.1\times 10^{-7}$
| $k/h$ | $2.083\,6618(19)\times 10^{10}$ | Hz K-1 | $9.1\times 10^{-7}$
| $k/hc$ | $69.503\,476(63)$ | m${}^{-1}~{}$K-1 | $9.1\times 10^{-7}$
molar volume of ideal gas $RT/p$ |  |  |  |
$T=273.15\ {\rm K},\,p=100\ {\rm kPa}$ | $V_{\rm m}$ | $22.710\,953(21)\times 10^{-3}$ | m${}^{3}~{}$mol-1 | $9.1\times 10^{-7}$
Loschmidt constant $N_{\rm A}/V_{\rm m}$ | $n_{0}$ | $2.651\,6462(24)\times 10^{25}$ | m-3 | $9.1\times 10^{-7}$
molar volume of ideal gas $RT/p$ |  |  |  |
$T=273.15\ {\rm K},\,p=101.325\ {\rm kPa}$ | $V_{\rm m}$ | $22.413\,968(20)\times 10^{-3}$ | m${}^{3}~{}$mol-1 | $9.1\times 10^{-7}$
Loschmidt constant $N_{\rm A}/V_{\rm m}$ | $n_{0}$ | $2.686\,7805(24)\times 10^{25}$ | m-3 | $9.1\times 10^{-7}$
Sackur-Tetrode (absolute entropy) constant121212The entropy of an ideal monatomic gas of relative atomic mass $A_{\rm r}$ is given by $S=S_{0}+{3\over 2}R\,\ln A_{\rm r}-R\,\ln(p/p_{0})+{5\over 2}R\,\ln(T/{\rm K}).$ |  |  |  |
$\frac{5}{2}+\ln[(2\mbox{{p}}m_{\rm u}kT_{1}/h^{2})^{3/2}kT_{1}/p_{0}]$ |  |  |  |
$T_{1}=1\ {\rm K},\,p_{0}\,=\,100\ {\rm kPa}$ | $S_{0}/R$ | $-1.151\,7078(23)$ |  | $2.0\times 10^{-6}$
$T_{1}=1\ {\rm K},\,p_{0}\,=\,101.325\ {\rm kPa}$ |  | $-1.164\,8708(23)$ |  | $1.9\times 10^{-6}$
Stefan-Boltzmann constant |  |  |  |
($\mbox{{p}}^{2}/60)k^{4}\\!/\hbar^{3}c^{2}$ | $\sigma$ | $5.670\,373(21)\times 10^{-8}$ | W m${}^{-2}~{}$K-4 | $3.6\times 10^{-6}$
first radiation constant 2$\mbox{{p}}hc^{2}$ | $c_{1}$ | $3.741\,771\,53(17)\times 10^{-16}$ | W m2 | $4.4\times 10^{-8}$
first radiation constant for spectral radiance 2$hc^{2}$ | $c_{\rm 1L}$ | $1.191\,042\,869(53)\times 10^{-16}$ | W m2 sr-1 | $4.4\times 10^{-8}$
second radiation constant $hc/k$ | $c_{2}$ | $1.438\,7770(13)\times 10^{-2}$ | m K | $9.1\times 10^{-7}$
Wien displacement law constants |  |  |  |
$b=\lambda_{\rm max}T=c_{2}/4.965\,114\,231...$ | $b$ | $2.897\,7721(26)\times 10^{-3}$ | m K | $9.1\times 10^{-7}$
$b^{\prime}=\nu_{\rm max}/T=2.821\,439\,372...\,c/c_{2}$ | $b^{\prime}$ | $5.878\,9254(53)\times 10^{10}$ | Hz K-1 | $9.1\times 10^{-7}$
Table 41: (Continued).

Table 42: The variances, covariances, and correlation coefficients of the
values of a selected group of constants based on the 2010 CODATA adjustment.
The numbers in bold above the main diagonal are $10^{16}$ times the numerical
values of the relative covariances; the numbers in bold on the main diagonal
are $10^{16}$ times the numerical values of the relative variances; and the
numbers in italics below the main diagonal are the correlation coefficients.1

$\alpha$ $h$ $e$ $m_{\rm e}$ $N_{\rm A}$ $m_{\rm e}/m_{\rm\mu}$ $F$ $\alpha$
${\bf 0.0010}$ ${\bf 0.0010}$ ${\bf 0.0010}$ ${\bf-0.0011}$ ${\bf 0.0009}$
${\bf-0.0021}$ ${\bf 0.0019}$ $h$ ${\it 0.0072}$ ${\bf 19.4939}$ ${\bf
9.7475}$ ${\bf 19.4918}$ ${\bf-19.4912}$ ${\bf-0.0020}$ ${\bf-9.7437}$ $e$
${\it 0.0145}$ ${\it 1.0000}$ ${\bf 4.8742}$ ${\bf 9.7454}$ ${\bf-9.7452}$
${\bf-0.0020}$ ${\bf-4.8709}$ $m_{\rm e}$ ${\it-0.0075}$ ${\it 0.9999}$ ${\it
0.9998}$ ${\bf 19.4940}$ ${\bf-19.4929}$ ${\bf 0.0021}$ ${\bf-9.7475}$ $N_{\rm
A}$ ${\it 0.0060}$ ${\it-0.9999}$ ${\it-0.9997}$ ${\it-1.0000}$ ${\bf
19.4934}$ ${\bf-0.0017}$ ${\bf 9.7483}$ $m_{\rm e}/m_{\rm\mu}$ ${\it-0.0251}$
${\it-0.0002}$ ${\it-0.0004}$ ${\it 0.0002}$ ${\it-0.0002}$ ${\bf 6.3872}$
${\bf-0.0037}$ $F$ ${\it 0.0265}$ ${\it-0.9993}$ ${\it-0.9990}$ ${\it-0.9997}$
${\it 0.9997}$ ${\it-0.0007}$ ${\bf 4.8774}$ 1 The relative covariance is
$u_{\rm r}(x_{i},x_{j})=u(x_{i},x_{j})/(x_{i}x_{j})$, where $u(x_{i},x_{j})$
is the covariance of $x_{i}$ and $x_{j}$; the relative variance is $u_{\rm
r}^{2}(x_{i})=u_{\rm r}(x_{i},x_{i})$: and the correlation coefficient is
$r(x_{i},x_{j})=u(x_{i},x_{j})/[u(x_{i})u(x_{j})]$.

Table 43: Internationally adopted values of various quantities.

|  |  |  | Relative std.
---|---|---|---|---
Quantity  | Symbol  | Numerical value  | Unit  | uncert. $u_{\rm r}$
relative atomic mass1 of 12C  |  $A_{\rm r}(^{12}$C)  |  $12$ |  | exact
molar mass constant  |  $M_{\rm u}$ |  $1\times 10^{-3}$ | kg mol-1 | exact
molar mass2 of 12C  |  $M(^{12}$C)  |  $12\times 10^{-3}$ | kg mol-1 | exact
conventional value of Josephson constant3 |  $K_{\rm J-90}$ | 483 597.9  | GHz V-1 | exact
conventional value of von Klitzing constant4 |  $R_{\rm K-90}$ | 25 812.807  |  ${\rm\Omega}$ | exact
standard-state pressure  |  |  $100$ | kPa  | exact
standard atmosphere  |  |  $101.325$ | kPa  | exact

1 The relative atomic mass $A_{\rm r}(X)$ of particle $X$ with mass $m(X)$ is
defined by $A_{\rm r}(X)=m(X)/m_{\rm u}$, where $m_{\rm u}=m(^{12}{\rm
C})/12=M_{\rm u}/N_{\rm A}=1~{}{\rm u}$ is the atomic mass constant, $M_{\rm
u}$ is the molar mass constant, $N_{\rm A}$ is the Avogadro constant, and u is
the unified atomic mass unit. Thus the mass of particle $X$ is $m(X)=A_{\rm
r}(X)$ u and the molar mass of $X$ is $M(X)=A_{\rm r}(X)M_{\rm u}$.
2 Value fixed by the SI definition of the mole.
3 This is the value adopted internationally for realizing representations of
the volt using the Josephson effect.
4 This is the value adopted internationally for realizing representations of
the ohm using the quantum Hall effect.

Table 44: Values of some x-ray-related quantities based on the 2010 CODATA
adjustment of the values of the constants.

Relative std. Quantity Symbol Numerical value Unit uncert. $u_{\rm r}$ Cu x
unit: $\lambda({\rm CuK}{\rm\alpha}_{\rm 1})/1\,537.400$ ${\rm xu}({\rm
CuK}{\rm\alpha}_{\rm 1})$ $1.002\,076\,97(28)\times 10^{-13}$ m $2.8\times
10^{-7}$ Mo x unit: $\lambda({\rm MoK}{\rm\alpha}_{\rm 1})/707.831$ ${\rm
xu}({\rm MoK}{\rm\alpha}_{\rm 1})$ $1.002\,099\,52(53)\times 10^{-13}$ m
$5.3\times 10^{-7}$ ångstrom star$:\lambda({\rm WK}{\rm\alpha}_{\rm
1})/0.209\,010\,0$ Å∗ $1.000\,014\,95(90)\times 10^{-10}$ m $9.0\times
10^{-7}$ lattice parameter1 of Si (in vacuum, 22.5 ∘C) $a$
$543.102\,0504(89)\times 10^{-12}$ m $1.6\times 10^{-8}$ {220} lattice spacing
of Si $a/\sqrt{8}$ $d_{\rm 220}$ $192.015\,5714(32)\times 10^{-12}$ m
$1.6\times 10^{-8}$ (in vacuum, 22.5 ∘C) molar volume of Si $M({\rm
Si})/\rho({\rm Si})=N_{\rm A}a^{3}\\!/8$ $V_{\rm m}$(Si)
$12.058\,833\,01(80)\times 10^{-6}$ m3 mol-1 $6.6\times 10^{-8}$ (in vacuum,
22.5 ∘C) 1 This is the lattice parameter (unit cell edge length) of an ideal
single crystal of naturally occurring Si free of impurities and imperfections,
and is deduced from measurements on extremely pure and nearly perfect single
crystals of Si by correcting for the effects of impurities.

Table 45: The values in SI units of some non-SI units based on the 2010 CODATA adjustment of the values of the constants. |  |  |  | Relative std.
---|---|---|---|---
Quantity  |  Symbol  |  Numerical value  |  Unit  | uncert. $u_{\rm r}$
Non-SI units accepted for use with the SI
electron volt: ($e/{\rm C}$) J  | eV  |  $1.602\,176\,565(35)\times 10^{-19}$ | J  |  $2.2\times 10^{-8}$
(unified) atomic mass unit: ${1\over 12}m(^{12}$C)  | u  |  $1.660\,538\,921(73)\times 10^{-27}$ | kg  |  $4.4\times 10^{-8}$
Natural units (n.u.)
n.u. of velocity  |  $c,c_{0}$ | 299 792 458  | m s-1 | exact
n.u. of action: $h/2\mbox{{p}}$ |  $\hbar$ |  $1.054\,571\,726(47)\times 10^{-34}$ | J s  |  $4.4\times 10^{-8}$
|  |  $6.582\,119\,28(15)\times 10^{-16}$ | eV s  |  $2.2\times 10^{-8}$
|  $\hbar c$ |  $197.326\,9718(44)$ | MeV fm  |  $2.2\times 10^{-8}$
n.u. of mass  |  $m_{\rm e}$ |  $9.109\,382\,91(40)\times 10^{-31}$ | kg  |  $4.4\times 10^{-8}$
n.u. of energy  |  $m_{\rm e}c^{2}$ |  $8.187\,105\,06(36)\times 10^{-14}$ | J  |  $4.4\times 10^{-8}$
|  |  $0.510\,998\,928(11)$ | MeV  |  $2.2\times 10^{-8}$
n.u. of momentum  |  $m_{\rm e}c$ |  $2.730\,924\,29(12)\times 10^{-22}$ | kg m s-1 |  $4.4\times 10^{-8}$
|  |  $0.510\,998\,928(11)$ | MeV/$c$ |  $2.2\times 10^{-8}$
n.u. of length: $\hbar/m_{\rm e}c$ |  $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm C}$ |  $386.159\,268\,00(25)\times 10^{-15}$ | m  |  $6.5\times 10^{-10}$
n.u. of time  |  $\hbar/m_{\rm e}c^{2}$ |  $1.288\,088\,668\,33(83)\times 10^{-21}$ | s  |  $6.5\times 10^{-10}$
Atomic units (a.u.)
a.u. of charge  |  $e$ |  $1.602\,176\,565(35)\times 10^{-19}$ | C  |  $2.2\times 10^{-8}$
a.u. of mass  |  $m_{\rm e}$ |  $9.109\,382\,91(40)\times 10^{-31}$ | kg  |  $4.4\times 10^{-8}$
a.u. of action: $h/2\mbox{{p}}$ |  $\hbar$ |  $1.054\,571\,726(47)\times 10^{-34}$ | J s  |  $4.4\times 10^{-8}$
a.u. of length: Bohr radius (bohr)  |  |  |  |
$\alpha/4\mbox{{p}}R_{\infty}$ |  $a_{0}$ |  $0.529\,177\,210\,92(17)\times 10^{-10}$ | m  |  $3.2\times 10^{-10}$
a.u. of energy: Hartree energy (hartree)  |  |  |  |
$e^{2}\\!/4\mbox{{p}}\epsilon_{0}a_{0}=2R_{\infty}hc=\alpha^{2}m_{\rm e}c^{2}$ |  $E_{\rm h}$ |  $4.359\,744\,34(19)\times 10^{-18}$ | J  |  $4.4\times 10^{-8}$
a.u. of time  |  $\hbar/E_{\rm h}$ |  $2.418\,884\,326\,502(12)\times 10^{-17}$ | s  |  $5.0\times 10^{-12}$
a.u. of force  |  $E_{\rm h}/a_{0}$ |  $8.238\,722\,78(36)\times 10^{-8}$ | N  |  $4.4\times 10^{-8}$
a.u. of velocity: $\alpha c$ |  $a_{0}E_{\rm h}/\hbar$ |  $2.187\,691\,263\,79(71)\times 10^{6}$ | m s-1 |  $3.2\times 10^{-10}$
a.u. of momentum  |  $\hbar/a_{0}$ |  $1.992\,851\,740(88)\times 10^{-24}$ | kg m s-1 |  $4.4\times 10^{-8}$
a.u. of current  |  $eE_{\rm h}/\hbar$ |  $6.623\,617\,95(15)\times 10^{-3}$ | A  |  $2.2\times 10^{-8}$
a.u. of charge density  |  $e/a_{0}^{3}$ |  $1.081\,202\,338(24)\times 10^{12}$ | C m-3 |  $2.2\times 10^{-8}$
a.u. of electric potential  |  $E_{\rm h}/e$ |  $27.211\,385\,05(60)$ | V  |  $2.2\times 10^{-8}$
a.u. of electric field  |  $E_{\rm h}/ea_{0}$ |  $5.142\,206\,52(11)\times 10^{11}$ | V m-1 |  $2.2\times 10^{-8}$
a.u. of electric field gradient  |  $E_{\rm h}/ea_{0}^{2}$ |  $9.717\,362\,00(21)\times 10^{21}$ | V m-2 |  $2.2\times 10^{-8}$
a.u. of electric dipole moment  |  $ea_{0}$ |  $8.478\,353\,26(19)\times 10^{-30}$ | C m  |  $2.2\times 10^{-8}$
a.u. of electric quadrupole moment  |  $ea_{0}^{2}$ |  $4.486\,551\,331(99)\times 10^{-40}$ | C m2 |  $2.2\times 10^{-8}$
a.u. of electric polarizability  |  $e^{2}a_{0}^{2}/E_{\rm h}$ |  $1.648\,777\,2754(16)\times 10^{-41}$ | C2 m2 J-1 |  $9.7\times 10^{-10}$
a.u. of 1st hyperpolarizability  |  $e^{3}a_{0}^{3}/E_{\rm h}^{2}$ |  $3.206\,361\,449(71)\times 10^{-53}$ | C3 m3 J-2 |  $2.2\times 10^{-8}$
a.u. of 2nd hyperpolarizability  |  $e^{4}a_{0}^{4}/E_{\rm h}^{3}$ |  $6.235\,380\,54(28)\times 10^{-65}$ | C4 m4 J-3 |  $4.4\times 10^{-8}$
a.u. of magnetic flux density  |  $\hbar/ea_{0}^{2}$ |  $2.350\,517\,464(52)\times 10^{5}$ | T  |  $2.2\times 10^{-8}$
a.u. of magnetic dipole moment: $2\mu_{\rm B}$ |  $\hbar e/m_{\rm e}$ |  $1.854\,801\,936(41)\times 10^{-23}$ | J T-1 |  $2.2\times 10^{-8}$
a.u. of magnetizability  |  $e^{2}a_{0}^{2}/m_{\rm e}$ |  $7.891\,036\,607(13)\times 10^{-29}$ | J T-2 |  $1.6\times 10^{-9}$
a.u. of permittivity: $10^{7}/c^{2}$ |  $e^{2}/a_{0}E_{\rm h}$ |  $1.112\,650\,056\ldots\times 10^{-10}$ | F m-1 | exact
Table 46: The values of some energy equivalents derived from the relations
$E=mc^{2}=hc/\lambda=h\nu=kT$, and based on the 2010 CODATA adjustment of the
values of the constants; 1 eV $=(e/{\rm C})$ J, 1 u $=m_{\rm
u}=\textstyle{1\over 12}m(^{12}{\rm C})=10^{-3}$ kg mol${}^{-1}\\!/N_{\rm A}$,
and $E_{\rm h}=2R_{\rm\infty}hc=\alpha^{2}m_{\rm e}c^{2}$ is the Hartree
energy (hartree). Relevant unit
---
|  J  |  kg  |  m-1 |  Hz
1 J  |  $(1\ {\rm J})=$ | (1 J)/$c^{2}=$ | (1 J)/$hc=$ | (1 J)/$h=$
| 1 J  |  $1.112\,650\,056\ldots\times 10^{-17}$ kg  |  $5.034\,117\,01(22)\times 10^{24}$ m-1 |  $1.509\,190\,311(67)\times 10^{33}$ Hz
1 kg  | (1 kg)$c^{2}=$ |  $(1\ {\rm kg})=$ | (1 kg)$c/h=$ | (1 kg)$c^{2}/h=$
|  $8.987\,551\,787\ldots\times 10^{16}$ J  | 1 kg  |  $4.524\,438\,73(20)\times 10^{41}$ m-1 |  $1.356\,392\,608(60)\times 10^{50}$ Hz
1 m-1 | (1 m${}^{-1})hc=$ | (1 m${}^{-1})h/c=$ |  $(1$ m${}^{-1})=$ | (1 m${}^{-1})c=$
|  $1.986\,445\,684(88)\times 10^{-25}$ J  |  $2.210\,218\,902(98)\times 10^{-42}$ kg  | 1 m-1 |  $299\,792\,458$ Hz
1 Hz  | (1 Hz)$h=$ | (1 Hz)$h/c^{2}=$ | (1 Hz)/$c=$ |  $(1$ Hz$)=$
|  $6.626\,069\,57(29)\times 10^{-34}$ J  |  $7.372\,496\,68(33)\times 10^{-51}$ kg  |  $3.335\,640\,951\ldots\times 10^{-9}$ m-1 | 1 Hz
1 K  | (1 K)$k=$ | (1 K)$k/c^{2}=$ | (1 K)$k/hc=$ | (1 K)$k/h=$
|  $1.380\,6488(13)\times 10^{-23}$ J  |  $1.536\,1790(14)\times 10^{-40}$ kg  |  $69.503\,476(63)$ m-1 |  $2.083\,6618(19)\times 10^{10}$ Hz
1 eV  | (1 eV) =  |  $(1~{}{\rm eV})/c^{2}=$ |  $(1~{}{\rm eV})/hc=$ | $(1~{}{\rm eV})/h=$
|  $1.602\,176\,565(35)\times 10^{-19}$ J  |  $1.782\,661\,845(39)\times 10^{-36}$ kg  |  $8.065\,544\,29(18)\times 10^{5}$ m-1 |  $2.417\,989\,348(53)\times 10^{14}$ Hz
1 u  |  $(1~{}{\rm u})c^{2}=$ | (1 u) =  |  $(1~{}{\rm u})c/h=$ | $(1~{}{\rm u})c^{2}/h=$
|  $1.492\,417\,954(66)\times 10^{-10}$ J  |  $1.660\,538\,921(73)\times 10^{-27}$ kg  |  $7.513\,006\,6042(53)\times 10^{14}$ m-1 |  $2.252\,342\,7168(16)\times 10^{23}$ Hz
1 $E_{\rm h}$ |  $(1~{}E_{\rm h})=$ |  $(1~{}E_{\rm h})/c^{2}=$ |  $(1~{}E_{\rm h})/hc=$ | $(1~{}E_{\rm h})/h=$
|  $4.359\,744\,34(19)\times 10^{-18}$ J  |  $4.850\,869\,79(21)\times 10^{-35}$ kg  |  $2.194\,746\,313\,708(11)\times 10^{7}$ m-1 |  $6.579\,683\,920\,729(33)\times 10^{15}$ Hz
Table 47: The values of some energy equivalents derived from the relations
$E=mc^{2}=hc/\lambda=h\nu=kT$, and based on the 2010 CODATA adjustment of the
values of the constants; 1 eV $=(e/{\rm C})$ J, 1 u $=m_{\rm
u}=\textstyle{1\over 12}m(^{12}{\rm C})=10^{-3}$ kg mol${}^{-1}\\!/N_{\rm A}$,
and $E_{\rm h}=2R_{\rm\infty}hc=\alpha^{2}m_{\rm e}c^{2}$ is the Hartree
energy (hartree). Relevant unit
---
|  K  |  eV  |  u  |  $E_{\rm h}$
1 J  | (1 J)/$k=$ | (1 J) =  | (1 J)/$c^{2}$ =  | (1 J) =
|  $7.242\,9716(66)\times 10^{22}$ K  |  $6.241\,509\,34(14)\times 10^{18}$ eV  |  $6.700\,535\,85(30)\times 10^{9}$ u  |  $2.293\,712\,48(10)\times 10^{17}$ $E_{\rm h}$
1 kg  | (1 kg)$c^{2}/k=$ | (1 kg)$c^{2}$ =  | (1 kg) =  | (1 kg)$c^{2}=$
|  $6.509\,6582(59)\times 10^{39}$ K  |  $5.609\,588\,85(12)\times 10^{35}$ eV  |  $6.022\,141\,29(27)\times 10^{26}$ u  |  $2.061\,485\,968(91)\times 10^{34}$ $E_{\rm h}$
1 m-1 | (1 m${}^{-1})hc/k=$ | (1 m${}^{-1})hc=$ | (1 m${}^{-1})h/c$ =  | (1 m${}^{-1})hc=$
|  $1.438\,7770(13)\times 10^{-2}$ K  |  $1.239\,841\,930(27)\times 10^{-6}$ eV  |  $1.331\,025\,051\,20(94)\times 10^{-15}$ u  |  $4.556\,335\,252\,755(23)\times 10^{-8}$ $E_{\rm h}$
1 Hz  | (1 Hz)$h/k=$ | (1 Hz)$h=$ | (1 Hz)$h/c^{2}$ =  | (1 Hz)$h=$
|  $4.799\,2434(44)\times 10^{-11}$ K  |  $4.135\,667\,516(91)\times 10^{-15}$ eV  |  $4.439\,821\,6689(31)\times 10^{-24}$ u  |  $1.519\,829\,846\,0045(76)\times 10^{-16}$ $E_{\rm h}$
1 K  |  $(1$ K$)=$ | (1 K)$k=$ | (1 K)$k/c^{2}=$ | (1 K)$k=$
| 1 K  |  $8.617\,3324(78)\times 10^{-5}$ eV  |  $9.251\,0868(84)\times 10^{-14}$ u  |  $3.166\,8114(29)\times 10^{-6}$ $E_{\rm h}$
1 eV  | (1 eV)/$k=$ |  $(1$ eV$)=$ |  $(1~{}{\rm eV})/c^{2}=$ | $(1~{}{\rm eV})=$
|  $1.160\,4519(11)\times 10^{4}$ K  | 1 eV  |  $1.073\,544\,150(24)\times 10^{-9}$ u  |  $3.674\,932\,379(81)\times 10^{-2}$ $E_{\rm h}$
1 u  |  $(1~{}{\rm u})c^{2}/k=$ |  $(1~{}{\rm u})c^{2}=$ |  $(1$ u$)=$ | $(1~{}{\rm u})c^{2}=$
|  $1.080\,954\,08(98)\times 10^{13}$ K  |  $931.494\,061(21)\times 10^{6}$ eV  | 1 u  |  $3.423\,177\,6845(24)\times 10^{7}$ $E_{\rm h}$
1 $E_{\rm h}$ |  $(1~{}E_{\rm h})/k=$ |  $(1~{}E_{\rm h})=$ |  $(1~{}E_{\rm h})/c^{2}=$ | $(1~{}E_{\rm h})=$
|  $3.157\,7504(29)\times 10^{5}$ K  |  $27.211\,385\,05(60)$ eV  |  $2.921\,262\,3246(21)\times 10^{-8}$ u  | $1~{}E_{\rm h}$

##  XV Summary and Conclusion

The focus of this section is (i) comparison of the 2010 and 2006 recommended
values of the constants and identification of those new results that have
contributed most to the changes in the 2006 values; (ii) presentation of
several conclusions that can be drawn from the 2010 recommended values and the
input data on which they are based; and (iii) identification of new
experimental and theoretical work that can advance our knowledge of the values
of the constants.

Topic (iii) is of special importance in light of the adoption by the 24th
General Conference on Weights and Measures (CGPM) at its meeting in Paris in
October 2011 of Resolution 1 entitled “On the possible future revision of the
International System of Units, the SI,” available on the BIPM Web site at
bipm.org/utils/common/pdf/24_CGPM_Resolutions.pdf.

In brief, this resolution notes the intention of the CIPM to propose, possibly
to the 25th CGPM in 2014, a revision of the SI. The “New SI,” as it is called
to distinguish it from the current SI, will be the system of units in which
seven reference constants, including the Planck constant $h$, elementary
charge $e$, Boltzmann constant $k$, and Avogadro constant $N_{\rm A}$, have
exact assigned values. Resolution 1 also looks to CODATA to provide the
necessary values of these four constants for the new definition. Details of
the proposed New SI may be found in Mills _et al._ (2011) and the references
cited therein; see also Taylor (2011); Mohr and Newell (2010).

###  XV.1 Comparison of 2010 and 2006 CODATA recommended values

Table 48: Comparison of the 2010 and 2006 CODATA adjustments of the values of the constants by the comparison of the corresponding recommended values of a representative group of constants. Here $D_{\rm r}$ is the 2010 value minus the 2006 value divided by the standard uncertainty $u$ of the 2006 value (i.e., $D_{\rm r}$ is the change in the value of the constant from 2006 to 2010 relative to its 2006 standard uncertainty). Quantity  | 2010 rel. std.  | Ratio 2006 $u_{\rm r}$ |  $D_{\rm r}$
---|---|---|---
| uncert. $u_{\rm r}$ | to 2010 $u_{\rm r}$ |
$\alpha$ |  $3.2\times 10^{-10}$ |  $2.1$ |  $6.5$
$R_{\rm K}$ |  $3.2\times 10^{-10}$ |  $2.1$ |  $-6.5$
$a_{\rm 0}$ |  $3.2\times 10^{-10}$ |  $2.1$ |  $6.5$
$\lambda_{\rm C}$ |  $6.5\times 10^{-10}$ |  $2.1$ |  $6.5$
$r_{\rm e}$ |  $9.7\times 10^{-10}$ |  $2.1$ |  $6.5$
$\sigma_{\rm e}$ |  $1.9\times 10^{-9}$ |  $2.1$ |  $6.5$
$h$ |  $4.4\times 10^{-8}$ |  $1.1$ |  $1.9$
$m_{\rm e}$ |  $4.4\times 10^{-8}$ |  $1.1$ |  $1.7$
$m_{\rm h}$ |  $4.4\times 10^{-8}$ |  $1.1$ |  $1.7$
$m_{\mbox{\scriptsize{{a}}}}$ |  $4.4\times 10^{-8}$ |  $1.1$ |  $1.7$
$N_{\rm A}$ |  $4.4\times 10^{-8}$ |  $1.1$ |  $-1.7$
$E_{\rm h}$ |  $4.4\times 10^{-8}$ |  $1.1$ |  $1.9$
$c_{1}$ |  $4.4\times 10^{-8}$ |  $1.1$ |  $1.9$
$e$ |  $2.2\times 10^{-8}$ |  $1.1$ |  $1.9$
$K_{\rm J}$ |  $2.2\times 10^{-8}$ |  $1.1$ |  $-1.8$
$F$ |  $2.2\times 10^{-8}$ |  $1.1$ |  $-1.4$
$\gamma^{\,\prime}_{\rm p}$ |  $2.5\times 10^{-8}$ |  $1.1$ |  $-1.3$
$\mu_{\rm B}$ |  $2.2\times 10^{-8}$ |  $1.1$ |  $2.3$
$\mu_{\rm N}$ |  $2.2\times 10^{-8}$ |  $1.1$ |  $2.3$
$\mu_{\rm e}$ |  $2.2\times 10^{-8}$ |  $1.1$ |  $-2.3$
$\mu_{\rm p}$ |  $2.4\times 10^{-8}$ |  $1.1$ |  $2.2$
$R$ |  $9.1\times 10^{-7}$ |  $1.9$ |  $-0.7$
$k$ |  $9.1\times 10^{-7}$ |  $1.9$ |  $-0.7$
$V_{\rm m}$ |  $9.1\times 10^{-7}$ |  $1.9$ |  $-0.7$
$c_{2}$ |  $9.1\times 10^{-7}$ |  $1.9$ |  $0.7$
$\sigma$ |  $3.6\times 10^{-6}$ |  $1.9$ |  $-0.7$
$G$ |  $1.2\times 10^{-4}$ |  $0.8$ |  $-0.7$
$R_{\infty}$ |  $5.0\times 10^{-12}$ |  $1.3$ |  $0.2$
$m_{\rm e}/m_{\rm p}$ |  $4.1\times 10^{-10}$ |  $1.1$ |  $0.0$
$m_{\rm e}/m_{\mbox{\scriptsize{{m}}}}$ |  $2.5\times 10^{-8}$ |  $1.0$ |  $-0.4$
$A_{\rm r}({\rm e})$ |  $4.0\times 10^{-10}$ |  $1.1$ |  $0.1$
$A_{\rm r}({\rm p})$ |  $8.9\times 10^{-11}$ |  $1.2$ |  $0.4$
$A_{\rm r}({\rm n})$ |  $4.2\times 10^{-10}$ |  $1.0$ |  $0.1$
$A_{\rm r}({\rm d})$ |  $3.8\times 10^{-11}$ |  $1.0$ |  $-0.2$
$A_{\rm r}({\rm t})$ |  $8.2\times 10^{-10}$ |  $1.0$ |  $0.0$
$A_{\rm r}({\rm h})$ |  $8.3\times 10^{-10}$ |  $1.0$ |  $-0.2$
$A_{\rm r}({\mbox{{a}}})$ |  $1.5\times 10^{-11}$ |  $1.0$ |  $0.0$
$d_{\rm 220}$ |  $1.6\times 10^{-8}$ |  $1.6$ |  $-1.0$
$g_{\rm e}$ |  $2.6\times 10^{-13}$ |  $2.8$ |  $0.5$
$g_{\mbox{\scriptsize{{m}}}}$ |  $6.3\times 10^{-10}$ |  $1.0$ |  $-0.3$
$\mu_{\rm p}/\mu_{\rm B}$ |  $8.1\times 10^{-9}$ |  $1.0$ |  $0.0$
$\mu_{\rm p}/\mu_{\rm N}$ |  $8.2\times 10^{-9}$ |  $1.0$ |  $0.0$
$\mu_{\rm n}/\mu_{\rm N}$ |  $2.4\times 10^{-7}$ |  $1.0$ |  $0.0$
$\mu_{\rm d}/\mu_{\rm N}$ |  $8.4\times 10^{-9}$ |  $1.0$ |  $0.0$
$\mu_{\rm e}/\mu_{\rm p}$ |  $8.1\times 10^{-9}$ |  $1.0$ |  $0.0$
$\mu_{\rm n}/\mu_{\rm p}$ |  $2.4\times 10^{-7}$ |  $1.0$ |  $0.0$
$\mu_{\rm d}/\mu_{\rm p}$ |  $7.7\times 10^{-9}$ |  $1.0$ |  $0.0$

Table 48 compares the 2010 and 2006 recommended values of a representative
group of constants. The fact that the values of many constants are obtained
from expressions proportional to the fine-structure constant $\alpha$, Planck
constant $h$, or molar gas constant $R$ raised to various powers leads to the
regularities observed in the numbers in columns 2 to 4. For example, the first
six quantities are obtained from expressions proportional to $\alpha^{a}$,
where $|a|=1,~{}2,~{}3$, or $6$. The next 15 quantities, $h$ through the
magnetic moment of the proton $\mu_{\rm p}$, are calculated from expressions
containing the factor $h^{a}$, where $|a|=1$ or $1/2$. And the five quantities
$R$ through the Stefan Boltzmann constant $\sigma$ are proportional to
$R^{a}$, where $|a|=1$ or $4$.

Further comments on some of the entries in Table 48 are as follows.

(i) The large shift in the 2006 recommended value of $\alpha$ is mainly due to
the discovery and correction of an error in the numerically calculated value
of the eighth-order coefficient $A_{1}^{(8)}$ in the theoretical expression
for $a_{\rm e}$; see Sec. V.1.1. Its reduction in uncertainty is due to two
new results. The first is the 2008 improved value of $a_{\rm e}$ obtained at
Harvard University with a relative standard uncertainty of $2.4\times
10^{-10}$ compared to the $7.0\times 10^{-10}$ uncertainty of the earlier
Harvard result used in the 2006 adjustment. The second result is the 2011
improved LKB atom-recoil value of $h/m(^{87}{\rm Rb})$ with an uncertainty of
$1.2\times 10^{-9}$ compared to the $1.3\times 10^{-8}$ uncertainty of the
earlier LKB result used in 2006. The much reduced uncertainty of $g_{\rm e}$
is also due to the improved value of $\alpha$.

(ii) The change in the 2006 recommended value of $h$ is due to the 2011 IAC
result for $N_{\rm A}$ with a relative standard uncertainty of $3.0\times
10^{-8}$ obtained using ${}^{28}{\rm Si}$ enriched single crystals. It
provides a value of $h$ with the same uncertainty, which is smaller than the
$3.6\times 10^{-8}$ uncertainty of the value of $h$ from the 2007 NIST watt-
balance measurement of $K_{\rm J}^{2}R_{\rm K}$; the latter played the
dominant role in determining the 2006 recommended value. The two differ by
about 18 parts in $10^{8}$, resulting in a shift of the 2006 recommended value
by nearly twice its uncertainty. In the 2006 adjustment inconsistencies among
some of the electrical and silicon crystal data (all involving natural
silicon) led the Task Group to increase the uncertainties of these data by the
multiplicative factor 1.5 to reduce the inconsistencies to an acceptable
level. In the 2010 adjustment, inconsistencies among the data that determine
$h$ are reduced to an acceptable level by using a multiplicative factor of 2.
Consequently the uncertainties of the 2006 and 2010 recommended values of $h$
do not differ significantly.

(iii) The 2006 recommended value of the molar gas constant $R$ was determined
by the 1988 NIST speed-of-sound result with a relative standard uncertainty of
$1.8\times 10^{-6}$, and to a much lesser extent the 1979 NPL speed-of-sound
result with an uncertainty of $8.4\times 10^{-6}$ obtained with a rather
different type of apparatus. The six new data of potential interest related to
$R$ that became available during the 4 years between the 2006 and 2010
adjustments have uncertainties ranging from $1.2\times 10^{-6}$ to $12\times
10^{-6}$ and agree with each other as well as with the NIST and NPL values.
Further, the self-sensitivity coefficients of four of the six were
sufficiently large for them to be included in the 2010 final adjustment, and
they are responsible for the small shift in the 2006 recommended value and the
reduction of its uncertainty by nearly a factor of 2.

(iv) Other constants in Table 48 whose changes are worth noting are the
Rydberg constant $R_{\infty}$, proton relative atomic mass $A_{\rm r}({\rm
p})$, and {220} natural Si lattice spacing $d_{220}$. The reduction in
uncertainty of $R_{\infty}$ is due to improvements in the theory of H and D
energy levels and the 2010 LKB result for the $1{\rm S_{1/2}}-3{\rm S_{1/2}}$
transition frequency in hydrogen with a relative standard uncertainty of
$4.4\times 10^{-12}$. For $A_{\rm r}({\rm p})$, the reduction in uncertainty
is due to the 2008 Stockholm University (SMILETRAP) result for the ratio of
the cyclotron frequency of the excited hydrogen molecular ion to that of the
deuteron, $f_{\rm c}({\rm H_{2}^{+*}})/f_{\rm c}({\rm d})$, with a relative
uncertainty of $1.7\times 10^{-10}$. The changes in $d_{220}$ arise from the
omission of the 1999 PTB result for $h/m_{\rm n}d_{220}({\scriptstyle{\rm
W}04})$, the 2004 NMIJ result for $d_{220}({\scriptstyle{\rm NR3}})$, the 2007
INRIM results for $d_{220}({\scriptstyle{\rm W4.2a}})$, and
$d_{220}({\scriptstyle{\rm MO^{*}}})$, and the inclusion of the new 2008 INRIM
result for $d_{220}({\scriptstyle{\rm MO^{*}}})$ as well as the new 2009 INRIM
results for $d_{220}({\scriptstyle{\rm W}04})$ and $d_{220}({\scriptstyle{\rm
W4.2a}})$.

###  XV.2 Some implications of the 2010 CODATA recommended values and
adjustment for metrology and physics

_Conventional electric units_. The adoption of the conventional values $K_{\rm
J-90}=483\,597.9~{}{\rm GHz/V}$ and $R_{\rm K-90}=25\,812.807~{}{\Omega}$ for
the Josephson and von Klitzing constants in 1990 can be viewed as establishing
conventional, practical units of voltage and resistance, $V_{90}$ and
${\it\Omega}_{90}$, given by $V_{90}=(K_{\rm J-90}/K_{\rm J})$ V and
${\it\Omega}_{90}=(R_{\rm K}/R_{\rm K-90})~{}{\rm\Omega}$. Other conventional
electric units follow from $V_{90}$ and ${\it\Omega}_{90}$, for example,
$A_{90}=V_{90}/{\it\Omega}_{90}$, $C_{90}=A_{90}$ s, $W_{90}=A_{90}V_{90}$,
$F_{90}=C_{90}/V_{90}$, and $H_{90}={\it\Omega}_{90}$ s, which are the
conventional, practical units of current, charge, power, capacitance, and
inductance, respectively Taylor and Mohr (2001). For the relations between
$K_{\rm J}$ and $K_{\rm J-90}$, and $R_{\rm K}$ and $R_{\rm K-90}$, the 2010
adjustment gives

$\displaystyle K_{\rm J}$ $\displaystyle=$ $\displaystyle K_{\rm
J-90}[1-6.3(2.2)\times 10^{-8}]\,,$ (286) $\displaystyle R_{\rm K}$
$\displaystyle=$ $\displaystyle R_{\rm K-90}[1+1.718(32)\times 10^{-8}]\,,$
(287)

which lead to

$\displaystyle V_{90}$ $\displaystyle=$ $\displaystyle[1+6.3(2.2)\times
10^{-8}]~{}{\rm V},$ (288) $\displaystyle{\it\Omega}_{90}$ $\displaystyle=$
$\displaystyle[1+1.718(32)\times 10^{-8}]~{}{\rm\Omega},$ (289) $\displaystyle
A_{90}$ $\displaystyle=$ $\displaystyle[1-4.6(2.2)\times 10^{-8}]~{}{\rm
A}\,,$ (290) $\displaystyle C_{90}$ $\displaystyle=$
$\displaystyle[1-4.6(2.2)\times 10^{-8}]~{}{\rm C}\,,$ (291) $\displaystyle
W_{90}$ $\displaystyle=$ $\displaystyle[1+10.8(5.0)\times 10^{-8}]~{}{\rm
W}\,,$ (292) $\displaystyle F_{90}$ $\displaystyle=$
$\displaystyle[1-1.718(32)\times 10^{-8}]~{}{\rm F}\,,$ (293) $\displaystyle
H_{90}$ $\displaystyle=$ $\displaystyle[1+1.718(32)\times 10^{-8}]~{}{\rm
H}\,.$ (294)

Equations (288) and (289) show that $V_{90}$ exceeds V and ${\it\Omega}_{90}$
exceeds $\Omega$ by $6.3(2.2)\times 10^{-8}$ and $1.718(32)\times 10^{-8}$,
respectively. This means that measured voltages and resistances traceable to
the Josephson effect and $K_{\rm J-90}$ and the quantum Hall effect and
$R_{\rm K-90}$, respectively, are too small relative to the SI by these same
fractional amounts. However, these differences are well within the $40\times
10^{-8}$ uncertainty assigned to $V_{90}/$V and the $10\times 10^{-8}$
uncertainty assigned to ${\it\Omega}_{90}/\Omega$ by the Consultative
Committee for Electricity and Magnetism (CCEM) of the CIPM Quinn (2001, 1989).

_Josephson and quantum Hall effects_. Although there is extensive theoretical
and experimental evidence for the exactness of the Josephson and quantum-Hall-
effect relations $K_{\rm J}=2e/h$ and $R_{\rm K}=h/e^{2}$, and some of the
input data available for the 2010 adjustment provide additional supportive
evidence for these expressions, some other data are not supportive. This
dichotomy reflects the rather significant inconsistencies among a few key
data, particularly the highly accurate IAC enriched silicon XRCD result for
$N_{\rm A}$, and the comparably accurate NIST watt-balance result for $K_{\rm
J}^{2}R_{\rm K}$, and will only be fully resolved when the inconsistencies are
reconciled.

_The New SI_. Implementation of the New SI requires that the four reference
constants $h$, $e$, $k$, and $N_{\rm A}$ must be known with sufficiently small
uncertainties to meet current and future measurement needs. However, of equal
if not greater importance, the causes of any inconsistencies among the data
that provide their values must be understood. Although the key data that
provide the 2010 recommended value of $k$ would appear to be close to meeting
both requirements, this is not the case for $h$, $e$, and $N_{\rm A}$, which
are in fact interrelated. We have

$\displaystyle N_{\rm A}h$ $\displaystyle=$ $\displaystyle{cA_{\rm r}({\rm
e})M_{\rm u}\alpha^{2}\over 2R_{\infty}}\,,$ (295) $\displaystyle e$
$\displaystyle=$ $\displaystyle\left({2\alpha h\over\mu_{0}c}\right)^{1/2}\,.$
(296)

Since the combined relative standard uncertainty of the 2010 recommended
values of the constants on the right-hand-side of Eq. (295) is only $7.0\times
10^{-10}$, a measurement of $h$ with a given relative uncertainty, even as
small as $5\times 10^{-9}$, determines $N_{\rm A}$ with essentially the same
relative uncertainty. Further, since the recommended value of $\alpha$ has a
relative uncertainty of only $3.2\times 10^{-10}$, based on Eq. (296) the
relative uncertainty of $e$ will be half that of $h$ or $N_{\rm A}$. For these
reasons, the 2010 recommended values of $h$ and $N_{\rm A}$ have the same
$4.4\times 10^{-8}$ relative uncertainty, and the uncertainty of the
recommended value of $e$ is $2.2\times 10^{-8}$. However, these uncertainties
are twice as large as they would have been if there were no disagreement
between the watt-balance values of $h$ and the enriched silicon XRCD value of
$N_{\rm A}$. This disagreement led to an increase in the uncertainties of the
relevant data by a factor of 2. More specifically, if the data had been
consistent the uncertainties of the recommended values of $h$ and $N_{\rm A}$
would be $2.2\times 10^{-8}$ and $1.1\times 10^{-8}$ for $e$. Because these
should be sufficiently small for the New SI to be implemented, the
significance of the disagreement and the importance of measurements of $h$ and
$N_{\rm A}$ are apparent.

_Proton radius_. The proton rms charge radius $r_{\rm p}$ determined from the
Lamb shift in muonic hydrogen disagrees significantly with values determined
from H and D transition frequencies as well as from electron-proton scattering
experiments. Although the uncertainty of the muonic hydrogen value is
significantly smaller than the uncertainties of these other values, its
negative impact on the internal consistency of the theoretically predicted and
experimentally measured frequencies, as well as on the value of the Rydberg
constant, was deemed so severe that the only recourse was to not include it in
the final least-squares adjustment on which the 2010 recommended values are
based.

_Muon magnetic moment anomaly_. Despite extensive new theoretical work, the
long-standing significant difference between the theoretically predicted,
standard-model value of $a_{\mbox{\scriptsize{{m}}}}$ and the experimentally
determined value remains unresolved. Because the difference is from 3.3 to
possibly 4.5 times the standard uncertainty of the difference, depending on
the way the all-important hadronic contribution to the theoretical expression
for $a_{\mbox{\scriptsize{{m}}}}$ is evaluated, the theory was not
incorporated in the 2010 adjustment. The recommended values of
$a_{\mbox{\scriptsize{{m}}}}$ and those of other constants that depend on it
are, therefore, based on experiment.

_Electron magnetic moment anomaly, fine-structure constant, and QED_. The most
accurate value of the fine-structure constant $\alpha$ currently available
from a single experiment has a relative standard uncertainty of $3.7\times
10^{-10}$; it is obtained by equating the QED theoretical expression for the
electron magnetic moment anomaly $a_{\rm e}$ and the most accurate
experimental value of $a_{\rm e}$, obtained from measurements on a single
electron in a Penning trap. This value of $\alpha$ is in excellent agreement
with a competitive experimental value with an uncertainty of $6.6\times
10^{-10}$. Because the latter is obtained from the atom-recoil determination
of the quotient $h/m(^{87}{\rm Rb})$ using atom-interferometry and is only
weakly dependent on QED theory, the agreement provides one of the most
significant confirmations of quantum electrodynamics.

_Newtonian constant of gravitation_. The situation regarding measurements of
$G$ continues to be problematic and has become more so in the past 4 years.
Two new results with comparatively small uncertainties have become available
for the 2010 adjustment, leading to an increase in the scatter among the now
11 values of $G$. This has resulted in a $20\,\%$ increase in the uncertainty
of the 2010 recommended value compared to that of its 2006 predecessor.
Clearly, there is a continuing problem for the determination of this
important, but poorly-known, fundamental constant; the uncertainty of the 2010
recommended value is now 120 parts in $10^{6}$.

###  XV.3 Suggestions for future work

For evaluation of the fundamental constants, it is desirable not only to have
multiple results with competitive uncertainties for a given quantity, but also
to have one or more results obtained by a different method. If the term
“redundant” is used to describe such an ideal set of data, there is usually
only limited redundancy among the key data available for any given CODATA
adjustment.

With this in mind, based on the preceding discussion, our suggestions are as
follows.

(i) Resolution of the disagreement between the most accurate watt-balance
result for $K_{\rm J}^{2}R_{\rm K}$ and the XRCD result for $N_{\rm A}$.
Approaches to solving this problem might include new measurements of $K_{\rm
J}^{2}R_{\rm K}$ using watt balances of different design (or their equivalent)
with uncertainties at the 2 to 3 parts in $10^{8}$ level, a thorough review by
the researchers involved of their existing measurements of this quantity,
tests of the exactness of the relations $K_{\rm J}=2e/h$ and $R_{\rm
K}=h/e^{2}$, independent measurements of the isotopic composition of the
enriched silicon crystals and their $d_{220}$ lattice spacing used in the
determination of $N_{\rm A}$ (these are the two principal quantities for which
only one measurement exists), and a thorough review by the researchers
involved of the many corrections required to obtain $N_{\rm A}$ from the
principal quantities measured.

(ii) Measurements of $k$ (and related quantities such as $k/h$) with
uncertainties at the 1 to 3 parts in $10^{6}$ level using the techniques of
dielectric gas thermometry, refractive index gas thermometry, noise
thermometry, and Doppler broadening, because these methods are so very
different from acoustic gas thermometry, which is the dominant method used to
date.

(iii) Resolution of the discrepancy between the muonic hydrogen inferred value
of $r_{\rm p}$ and the spectroscopic value from H and D transition
frequencies. Work underway on frequency measurements in hydrogen as well as
the analysis of $\mbox{{m}}^{-}{\rm p}$ and $\mbox{{m}}^{-}{\rm d}$ data and
possible measurements in $\mbox{{m}}^{-}{\rm h}$ and
$\mbox{{m}}^{-}\mbox{{a}}$ should provide additional useful information.
Independent evaluations of electron scattering data to determine $r_{\rm p}$
are encouraged as well as verification of the theory of H, D, and muonic
hydrogen-like energy levels.

(iv) Independent calculation of the eighth- and tenth-order coefficients in
the QED expression for $a_{\rm e}$, in order to increase confidence in the
value of $\alpha$ from $a_{\rm e}$.

(v) Resolution of the disagreement between the theoretical expression for
$a_{\mbox{\scriptsize{{m}}}}$ and its experimental value. This discrepancy
along with the discrepancy between theory and experiment in muonic hydrogen
are two important problems in muon-related physics.

(vi) Determinations of $G$ with an uncertainty of one part in $10^{5}$ using
new and innovative approaches that might resolve the disagreements among the
measurements made within the past three decades.

##  XVI Acknowledgments

We gratefully acknowledge the help of our many colleagues throughout the world
who provided the CODATA Task Group on Fundamental Constants with results prior
to formal publication and for promptly and patiently answering our many
questions about their work. We wish to thank Barry M. Wood, the current chair
of the Task Group, as well as our fellow Task-Group members, for their
invaluable guidance and suggestions during the course of the 2010 adjustment
effort.

## Nomenclature

AMDC |  Atomic Mass Data Center, Centre de Spectrométrie Nucléaire et de Spectrométrie de Masse (CSNSM), Orsay, France
---|---
$A_{\rm r}(X)$ |  Relative atomic mass of $X$: $A_{\rm r}(X)=m(X)/m_{\rm u}$
$A_{90}$ |  Conventional unit of electric current:
$A_{90}=V_{90}/{\it\Omega}_{90}$
Å∗ |  Ångström-star: ${\rm\lambda}({\rm WK}{\mbox{{a}}}_{1})=0.209\,010\,0\ {\rm\AA}^{\ast}$
$a_{\rm e}$ |  Electron magnetic moment anomaly:
$a_{\rm e}=(|g_{\rm e}|-2$)/2
$a_{\mbox{\scriptsize{{m}}}}$ |  Muon magnetic moment anomaly:
$a_{\mbox{\scriptsize{{m}}}}=(|g_{\mbox{\scriptsize{{m}}}}|-2$)/2
BIPM |  International Bureau of Weights and Measures, Sèvres, France
BNL |  Brookhaven National Laboratory, Upton, New York, USA
CERN |  European Organization for Nuclear Research, Geneva, Switzerland
CIPM |  International Committee for Weights and Measures
CODATA |  Committee on Data for Science and Technology of the International Council for Science
$CPT$ |  Combined charge conjugation, parity inversion, and time reversal
$c$ |  Speed of light in vacuum
d |  Deuteron (nucleus of deuterium D, or 2H)
$d_{220}$ |  $\\{220\\}$ lattice spacing of an ideal crystal of naturally occurring silicon
$d_{220}({\scriptstyle X})$ |  $\\{220\\}$ lattice spacing of crystal $X$ of naturally occurring silicon
$E_{\rm b}$ |  Binding energy
e |  Symbol for either member of the electron-positron pair; when necessary, e- or e+ is used to indicate the electron or positron
$e$ |  Elementary charge: absolute value of the charge of the electron
$F$ |  Faraday constant: $F$ = $N_{\rm A}e$
FSU |  Florida State University, Tallahassee, Florida, USA
FSUJ |  Friedrich-Schiller University, Jena, Germany
${\cal F}_{90}$ |  ${\cal F}_{90}=(F/A_{90})$ A
$G$ |  Newtonian constant of gravitation
$g$ |  Local acceleration of free fall
$g_{\rm d}$ |  Deuteron $g$-factor: $g_{\rm d}=\mu_{\rm d}/\mu_{\rm N}$
$g_{\rm e}$ |  Electron $g$-factor: $g_{\rm e}=2\mu_{\rm e}/\mu_{\rm B}$
$g_{\rm p}$ |  Proton $g$-factor: $g_{\rm p}=2\mu_{\rm p}/\mu_{\rm N}$
$g^{\prime}_{\rm p}$ |  Shielded proton $g$-factor: $g_{\rm p}^{\prime}=2\mu_{\rm p}^{\prime}/\mu_{\rm N}$
$g_{\rm t}$ |  Triton $g$-factor: $g_{\rm t}=2\mu_{\rm t}/\mu_{\rm N}$
$g_{X}(Y)$ |  $g$-factor of particle $X$ in the ground (1S) state of hydrogenic atom $Y$
$g_{\mbox{\scriptsize{{m}}}}$ |  Muon $g$-factor: $g_{\mbox{\scriptsize{{m}}}}=2\mu_{\mbox{\scriptsize{{m}}}}/(e\hbar/2m_{\mbox{\scriptsize{{m}}}})$
GSI |  Gesellschaft für Schweironenforschung, Darmstadt, Germany
HD |  HD molecule (bound state of hydrogen and deuterium atoms)
HT |  HT molecule (bound state of hydrogen and tritium atoms)
h |  Helion (nucleus of 3He)
$h$ |  Planck constant; $\hbar=h/2\mbox{{p}}$
HarvU |  Harvard University, Cambridge, Massachusetts, USA
IAC |  International Avogadro Coordination
ILL |  Institut Max von Laue-Paul Langevin, Grenoble, France
INRIM |  Istituto Nazionale di Ricerca Metrologica, Torino, Italy
IRMM |  Institute for Reference Materials and Measurements, Geel, Belgium
KRISS |  Korea Research Institute of Standards and Science, Taedok Science Town, Republic of Korea
KR/VN |  KRISS-VNIIM collaboration
$K_{\rm J}$ |  Josephson constant: $K_{\rm J}=2e/h$
$K_{\rm J-90}$ |  Conventional value of the Josephson constant $K_{\rm J}$: $K_{\rm J-90}=483\,597.9$ GHz V-1
$k$ |  Boltzmann constant: $k=R/N_{\rm A}$
LAMPF |  Clinton P. Anderson Meson Physics Facility at Los Alamos National Laboratory, Los Alamos, New Mexico, USA
LKB |  Laboratoire Kastler-Brossel, Paris, France
LK/SY |  LKB and SYRTE collaboration
LNE |  Laboratoire national de métrologie et d’essais, Trappes, France
METAS |  Federal Office of Metrology, Bern-Wabern, Switzerland
MIT |  Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
MPQ |  Max-Planck-Institut für Quantenoptik, Garching, Germany
$M(X)$ |  Molar mass of $X$: $M(X)=A_{\rm r}(X)M_{\rm u}$
Mu |  Muonium (${\mbox{{m}}}^{+}{\rm e}^{-}$ atom)
$M_{\rm u}$ |  Molar mass constant: $M_{\rm u}=10^{-3}~{}{\rm kg~{}mol^{-1}}$
$m_{\rm u}$ |  Unified atomic mass constant: $m_{\rm u}=m(^{12}{\rm C})/12$
$m_{X}$, $m(X)$ |  Mass of $X$ (for the electron e, proton p, and other elementary particles, the first symbol is used, i.e., $m_{\rm e}$, $m_{\rm p}$, etc.)
$N_{\rm A}$ |  Avogadro constant
NIM |  National Institute of Metrology, Beijing, China (People’s Republic of)
NIST |  National Institute of Standards and Technology, Gaithersburg, Maryland and Boulder, Colorado, USA
NMI |  National Metrology Institute, Lindfield, Australia
NMIJ |  National Metrology Institute of Japan, Tsukuba, Japan
NPL |  National Physical Laboratory, Teddington, UK
n |  Neutron
PTB |  Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin, Germany
p |  Proton
$\overline{\rm p}\,^{A}$He+ |  Antiprotonic helium (AHe+ \+ $\overline{\rm p}$ atom, $A=3\mbox{ or }4$)
QED |  Quantum electrodynamics
$p(\chi^{2}|\nu)$ |  Probability that an observed value of chi-square for $\nu$ degrees of freedom would exceed $\chi^{2}$
$R$ |  Molar gas constant
$\overline{R}$ |  Ratio of muon anomaly difference frequency to free proton NMR frequency
$R_{\rm B}$ |  Birge ratio: $R_{\rm B}=(\chi^{2}/\nu)^{\frac{1}{2}}$
$r_{\rm d}$ |  Bound-state rms charge radius of the deuteron
$R_{\rm K}$ |  von Klitzing constant: $R_{\rm K}=h/e^{2}$
$R_{\rm K-90}$ |  Conventional value of the von Klitzing constant $R_{\rm K}$: $R_{\rm K-90}=25\,812.807~{}{\rm\Omega}$
$r_{\rm p}$ |  Bound-state rms charge radius of the proton
$R_{\infty}$ |  Rydberg constant: $R_{\infty}=m_{\rm e}c\alpha^{2}/2h$
$r(x_{i},x_{j})$ |  Correlation coefficient of estimated values $x_{i}$ and $x_{j}$: $r(x_{i},x_{j})=u(x_{i},x_{j})/[u(x_{i})u(x_{j})]$
$S_{\rm c}$ |  Self-sensitivity coefficient
SI |  Système international d’unités (International System of Units)
StanfU |  Stanford University, Stanford, California, USA
StockU |  Stockholm University, Stockholm, Sweden
StPtrsb |  St. Petersburg, Russian Federation
SYRTE |  Systèmes de référence Temps Espace, Paris, France
$T$ |  Thermodynamic temperature
Type A |  Uncertainty evaluation by the statistical analysis of series of observations
Type B |  Uncertainty evaluation by means other than the statistical analysis of series of observations
$t_{90}$ |  Celsius temperature on the International Temperature Scale of 1990 (ITS-90)
t |  Triton (nucleus of tritium T, or 3H)
USus |  University of Sussex, Sussex, UK
UWash |  University of Washington, Seattle, Washington, USA
u |  Unified atomic mass unit (also called the dalton, Da): 1 u = $m_{\rm u}$ = $m(^{12}$C)/12
$u(x_{i})$ |  Standard uncertainty (i.e., estimated standard deviation) of an estimated value $x_{i}$ of a quantity $X_{i}$ (also simply $u$)
$u_{\rm r}(x_{i})$ |  Relative standard uncertainty of an estimated value $x_{i}$ of a quantity $X_{i}$:
$u_{\rm r}(x_{i})=u(x_{i})/|x_{i}|,\ x_{i}\neq 0$ (also simply $u_{\rm r}$)
$u(x_{i},x_{j})$ |  Covariance of estimated values $x_{i}$ and $x_{j}$
$u_{\rm r}(x_{i},x_{j})$ |  Relative covariance of estimated values $x_{i}$ and $x_{j}$: $u_{\rm r}(x_{i},x_{j})=u(x_{i},x_{j})/(x_{i}x_{j})$
$V_{\rm m}({\rm Si)}$ |  Molar volume of naturally occurring silicon
VNIIM |  D. I. Mendeleyev All-Russian Research Institute for Metrology, St. Petersburg, Russian Federation
$V_{90}$ |  Conventional unit of voltage based on the Josephson effect and $K_{\rm J-90}$: $V_{90}=(K_{\rm J-90}/K_{\rm J}$) V
$W_{90}$ |  Conventional unit of power: $W_{90}=V^{2}_{90}/{\it\Omega}_{90}$
XROI |  Combined x-ray and optical interferometer
xu(CuK${\mbox{{a}}}_{1}$) |  Cu x unit: ${\rm\lambda}$(CuK${\mbox{{a}}}_{1}$) = 1 537.400 xu(CuK${\mbox{{a}}}_{1}$)
xu(MoK${\mbox{{a}}}_{1}$) |  Mo x unit: ${\rm\lambda}$(MoK${\mbox{{a}}}_{1}$) = 707.831 xu(MoK${\mbox{{a}}}_{1}$)
$x(X)$ |  Amount-of-substance fraction of $X$
YaleU |  Yale University, New Haven, Connecticut, USA
$\alpha$ |  Fine-structure constant: $\alpha=e^{2}/4\mbox{{p}}\epsilon_{0}\hbar c\approx 1/137$
a |  Alpha particle (nucleus of 4He)
$\it\Gamma^{\prime}_{X-{\rm 90}}$(lo) |  $\it\Gamma^{\prime}_{X-{\rm 90}}({\rm lo})=(\gamma_{X}^{\prime}\,A_{\rm 90})$ A-1, $X$ = p or h
$\it\Gamma^{\prime}_{\rm p-90}$(hi) |  $\it\Gamma^{\prime}_{\rm p-90}({\rm hi})=(\gamma_{\rm p}^{\prime}/A_{\rm 90})$ A
$\gamma_{\rm p}$ |  Proton gyromagnetic ratio: $\gamma_{\rm p}=2\mu_{\rm p}/\hbar$
$\gamma_{\rm p}^{\prime}$ |  Shielded proton gyromagnetic ratio: $\gamma_{\rm p}^{\prime}=2\mu^{\prime}_{\rm p}/\hbar$
$\gamma^{\prime}_{\rm h}$ |  Shielded helion gyromagnetic ratio: $\gamma^{\prime}_{\rm h}=2|\mu^{\prime}_{\rm h}|/\hbar$
$\Delta\nu_{\rm Mu}$ |  Muonium ground-state hyperfine splitting
$\delta_{\rm e}$ |  Additive correction to the theoretical expression for the electron magnetic moment anomaly $a_{\rm e}$
$\delta_{\rm Mu}$ |  Additive correction to the theoretical expression for the ground-state hyperfine splitting of muonium ${\rm\Delta\nu}_{\rm Mu}$
$\delta_{\overline{\rm p}\,{\rm He}}$ |  Additive correction to the theoretical expression for a particular transition frequency of antiprotonic helium
$\delta_{X}(n{\rm L}_{j})$ |  Additive correction to the theoretical expression for an energy level of either hydrogen H or deuterium D with quantum numbers $n$, L, and $j$
$\delta_{\mbox{\scriptsize{{m}}}}$ |  Additive correction to the theoretical expression for the muon magnetic moment anomaly $a_{\mbox{\scriptsize{{m}}}}$
$\epsilon_{\rm 0}$ |  Electric constant: $\epsilon_{\rm 0}=1/\mu_{\rm 0}c^{2}$
$\doteq$ |  Symbol used to relate an input datum to its observational equation
${\rm\lambda}({X\,{\rm K}}{\mbox{{a}}}_{1})$ |  Wavelength of K${\mbox{{a}}}_{1}$ x-ray line of element $X$
${\rm\lambda}_{\rm meas}$ |  Measured wavelength of the 2.2 MeV capture g-ray emitted in the reaction n + p $\rightarrow$ d + g
m |  Symbol for either member of the muon-antimuon pair; when necessary, ${\mbox{{m}}}^{-}$ or ${\mbox{{m}}}^{+}$ is used to indicate the negative muon or positive muon
$\mu_{\rm B}$ |  Bohr magneton: $\mu_{\rm B}=e\hbar/2m_{\rm e}$
$\mu_{\rm N}$ |  Nuclear magneton: $\mu_{\rm N}=e\hbar/2m_{\rm p}$
$\mu_{X}(Y)$ |  Magnetic moment of particle $X$ in atom or molecule $Y$.
$\mu_{\rm 0}$ |  Magnetic constant: $\mu_{\rm 0}=4{\mbox{{p}}}\times 10^{-7}$ N/A2
$\mu_{X}$, $\mu^{\prime}_{X}$ |  Magnetic moment, or shielded magnetic moment, of particle $X$
$\nu$ |  Degrees of freedom of a particular adjustment
$\nu(f_{\rm p})$ |  Difference between muonium hyperfine splitting Zeeman transition frequencies $\nu_{34}$ and $\nu_{12}$ at a magnetic flux density $B$ corresponding to the free proton NMR frequency $f_{\rm p}$
$\sigma$ |  Stefan-Boltzmann constant: $\sigma=2{\mbox{{p}}}^{5}k^{4}/(15h^{3}c^{2})$
t |  Symbol for either member of the tau-antitau pair; when necessary, ${\mbox{{t}}}^{-}$ or ${\mbox{{t}}}^{+}$ is used to indicate the negative tau or positive tau
${\rm\chi}^{2}$ |  The statistic “chi square”
${\it\Omega}_{90}$ |  Conventional unit of resistance based on the quantum Hall effect and $R_{\rm K-90}:{\it\Omega}_{90}=(R_{\rm K}/R_{\rm K-90})~{}{\rm\Omega}$

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