\documentclass[twoside]{article}
\pagestyle{myheadings}
\markboth{QED and the fundamental constants}
{ Peter J. Mohr \& Barry N. Taylor }
\begin{document}
\setcounter{page}{231}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Mathematical Physics and Quantum Field Theory, \newline
Electronic Journal of Differential Equations, Conf. 04, 2000, pp. 231--242\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
\vspace{\bigskipamount} \\
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Quantum electrodynamics and \\ the fundamental constants
\thanks{ {\em Mathematics Subject Classifications:} 81V10.
\hfil\break\indent
{\em Key words:} Fundamental physical constants, quantum electrodynamics.
\hfil\break\indent
\copyright 2000 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Published August 28, 2000. \hfil\break\indent
Contribution of the National Institute of Standards and Technology (NIST),
not subject to copyright in the United States.
NIST is an agency of the Technology Administration,
U.S. Department of Commerce.} }
\date{}
\author{ Peter J. Mohr \& Barry N. Taylor }
\maketitle
\begin{abstract}
The fundamental constants are evaluated by comparison of
the results of critical experiments and the theoretical
expressions for these results written in terms of the
constants. Many of the theoretical expressions are based
on quantum electrodynamics (QED), so the consistency of
the comparison provides a critical test of the validity of
the theory.
\end{abstract}
\def\lbar{\lambda\hskip-5pt\vrule height4.7pt depth-4.1pt width5pt}
\section{Introduction}
This paper reviews the basic approach used in the
1998 adjustment of the values of the fundamental constants, which
is the subject of a long article by the authors written
under the auspices of the
Committee on Data for Science and Technology (CODATA)
Task Group on Fundamental Constants \cite{1999210,2000035}.
The purpose of the adjustment is to determine and
recommend values of various fundamental constants such as the \\
\def\iten{\item \vskip -5pt}
\vskip -20pt
\begin{list}{}
\null
\iten fine-structure constant $\alpha$
\iten Rydberg constant $R_\infty$
\iten Avogadro constant $N_{\rm A}$
\iten Planck constant $h$
\iten electron mass $m_{\rm e}$
\iten muon mass $m_\mu$
\end{list}
and many others, which provide the greatest consistency between
critical experiments and predictions based on
quantum electrodynamics (QED) theory and condensed matter theory.
\section{Background}
A pioneering comprehensive determination of constants was done
by Raymond T. Birge at the University of California, Berkeley,
and published in 1929.
Adjustments done since 1950, including those sponsored
by the National Research Council/National Academy of Sciences, USA
(NRC/NAS) and by CODATA, are the following:
\begin{list}{}
\null
\iten Bearden and Watts (1951) \cite{baw1951,baw1951a,1954005}
\iten NRC/NAS: DuMond and Cohen (1951) \cite{1951003,1953004,1955003};
\iten Bearden and Thomsen (1957) \cite{1957013};
\iten NRC/NAS: Cohen and DuMond (1965) \cite{1965008};
\iten Taylor, Parker, and Langenberg (1969) \cite{1969012};
\iten CODATA: Cohen and Taylor (1973) \cite{1973003};
\iten CODATA: Cohen and Taylor (1987) \cite{1987004}.
\end{list}
Prior to the 1998 adjustment of the constants
\cite{1999210,2000035}, the CODATA recommended values
were based on the 1986 adjustment \cite{1987004}.
The current CODATA recommended values for the constants,
based on the 1998 adjustment, are
available on the World Wide Web at
\begin{center}
physics.nist.gov/constants
\end{center}
\section{Method of least squares}
The method of least squares, as it is applied to the 1998 adjustment,
is summarized here.
Input data, which are the results of measurements,
or in some cases calculations,
are denoted by $q_1,\ q_2,\dots,q_N$.
The variables of the least-squares adjustment, which are members of
a suitable subset of the fundamental constants termed adjusted
constants, are denoted by
$z_1,\ z_2,\dots,z_M$, where $M \le N$.
The input data and the adjusted constants are related by
observational equations which are
theoretical expressions for the $q_i$ as functions
of the $z_j$ and are denoted by
\begin{eqnarray}
q_i \doteq f_i(z) \equiv f_i(z_1,z_2,\dots,z_M) \ .
\label{eq:eq1}
\end{eqnarray}
We employ the symbol $\doteq$ to indicate the unsymmetric relation between the item
of input data on the left-hand side and the corresponding theoretical
expression for that item as a function of the adjusted constants on the
right-hand side. In general, this set of equations is overdetermined,
so the left- and right-sides will not be equal, even for optimized values
of the constants.
Almost all of the observational equations are non-linear,
but they can be linearized by expanding about less-accurately
known starting values $s$ for the variables $z$ by writing
\begin{eqnarray}
q_i \doteq f_i(s)
+ \sum_{j=1}^{M} {\partial
f_i(s)
\over\partial s_j}
(z_j - s_j) + \cdots
\end{eqnarray}
or
\begin{eqnarray}
y_i \doteq \sum_{j=1}^{M}A_{ij}x_j + \cdots \ ,
\label{eq:2}
\end{eqnarray}
where $y_i = q_i - f_i(s)$,
$x_j = z_j - s_j$, and $A_{ij}
= {\textstyle{\partial f_i(s)}\over\textstyle{\partial s_j}}$.
Equation (\ref{eq:2})
can be written compactly in terms of matrices as
\begin{eqnarray}
Y \doteq AX
\end{eqnarray}
with the standard uncertainties and covariances of the observational data
expressed as elements of the covariance matrix
\begin{eqnarray}
V = {\rm cov}(Y) \ .
\label{eq:covy}
\end{eqnarray}
The least-squares adjustment is based on
the solution $\hat{X}$ for $X$ that minimizes
\begin{eqnarray}
(Y-AX)^\top V^{-1}(Y-AX) \ .
\label{eq:min}
\end{eqnarray}
The solution is \cite{1934003}
\begin{eqnarray}
\hat{X} &=& (A^\top V^{-1}A)^{-1}A^\top V^{-1}Y
\label{eq:sol} \\
{\rm cov}(\hat{X}) &=& (A^\top V^{-1}A)^{-1} \ ,
\end{eqnarray}
so the constants and their covariance matrix are given by
\begin{eqnarray}
\hat{Z} &=& S + \hat{X} \\
{\rm cov}(\hat{Z}) &=& {\rm cov}(\hat{X}) \ ,
\end{eqnarray}
and, as a corollary,
the best estimates of the measured quantities are given by
\begin{eqnarray}
\hat Y &=& A \hat X
\label{eq:cor} \\
\hat q_i &=& f_i(s) + \hat y_i \ .
\end{eqnarray}
It should be remarked that the initial linear correction described above
is usually not sufficient. In most trials and variants of the 1998
least-squares adjustment, several iterations were needed in which the
resulting values of the constants of one iteration become the starting
values in the next.
Motivation for using the solution $\hat X$ obtained by minimization
of Eq.~(\ref{eq:min}) is provided by the following considerations.
In the case where the observations $Y$ are uncorrelated,
$V$ is diagonal, and Eq.~(\ref{eq:min})
reduces to the simple weighted sum of differences given by
\begin{eqnarray}
(Y-AX)^\top V^{-1}(Y-AX) = \sum_{i=1}^{N}{(Y-AX)_i^2\over (\delta Y)_i^2} \ ,
\label{eq:wtav}
\end{eqnarray}
where $(\delta Y)_i^2 = V_{ii}$ is the square of the standard uncertainty
associated with the input datum $q_i$.
In this case, the solution $\hat X$ provides the best match of the theoretical
expressions to the input data as depicted in Eq.~(\ref{eq:wtav}).
However, in general $V$ is not diagonal, and we seek to optimize the constants
taking covariances of the input data into account. In this case,
there is an alternative criterion that may be applied.
From Eqs.~(\ref{eq:sol}) and (\ref{eq:cor}) we have
\begin{eqnarray}
\hat{Y} = C\,Y \ ,
\label{eq:yhaty}
\end{eqnarray}
where
\begin{eqnarray}
C = A(A^\top V^{-1} A)^{-1}
A^\top V^{-1} \ .
\label{eq:cmatr}
\end{eqnarray}
The elements of $\hat{Y}$ so obtained are the best estimates
for the quantities represented by $Y$ in the following sense:
If we consider an estimate of the quantities represented by $Y$
of the form
$Y^\prime = DY$ such that the sum of the squares of the
uncertainties of $Y^\prime$ as given by the trace of the covariance matrix
${\rm cov}(Y^\prime) = DVD^\top$
is a minimum, subject to the condition
that the matrix $D$ reproduces any set of data
of the form $AX$ (that is, $DAX=AX$
for any $X$), then $D = C$, where $C$ is just the
matrix in Eq.~(\ref{eq:cmatr}),
and hence $Y^\prime =\hat{Y}$ \cite{1934003}.
Thus, the solution $\hat X$ obtained by minimizing Eq.~(\ref{eq:min})
provides the set of constants for which the
observational equations make the most accurate theoretical predictions.
As mentioned above, a subset of the constants is adjusted by minimizing
Eq.~(\ref{eq:min}); the rest are calculated from this subset.
The choice of the subset
is arbitrary, provided the constants in it are independent. The results
of the adjustment will be the same for any such choice, as can be seen from
the form of the solution. If an alternate subset $W$ were selected, then
since both $X$ and $W$ are independent sets, we can write $W = BX$, where $B$
has an inverse, and $Y\doteq AB^{-1}W$. The solution would then be
\begin{eqnarray}
\hat W
&=& ((AB^{-1})^\top V^{-1}AB^{-1})^{-1}(AB^{-1})^\top V^{-1} Y
= B \hat X \\
{\rm cov}(\hat W) &=& ((A B^{-1})^\top V^{-1} A B^{-1})^{-1}
= B {\rm cov}(\hat X) B^\top \ ,
\end{eqnarray}
where the second line is just the standard formula for the propagation of
uncertainty. Hence, we reproduce the same optimized constants
$\hat X$ by finding the least-squares solution for any independent
set $W$, with its corresponding matrix $B$, and forming
\begin{eqnarray}
\hat X
&=& B^{-1}\hat W \\
{\rm cov}(\hat X) &=& B^{-1} {\rm cov}(\hat W) (B^{-1})^\top \ .
\end{eqnarray}
\def\h{\vbox to 10 pt {} }
\begin{center}
\begin{table}
\caption{Variables of the 1998 least-squares adjustment.}
\label{tab:vars}
\begin{tabular}{lc@{\hbox to 50 pt {}}c@{\hbox to 5 pt {}}c}
\hline
\hline
\vbox to 10 pt {} Variable name && Symbol & \\
\hline
\h electron relative atomic mass && $A_{\rm r}({\rm e})$ & \\
\h proton relative atomic mass && $A_{\rm r}({\rm p})$ & \\
\h neutron relative atomic mass && $A_{\rm r}({\rm n})$ & \\
\h deuteron relative atomic mass && $A_{\rm r}({\rm d})$ & \\
\h helion relative atomic mass && $A_{\rm r}({\rm h})$ & \\
\h alpha particle relative atomic mass && $A_{\rm r}(\alpha)$ & \\
\h electron-muon mass ratio && $m_{\rm e}/m_\mu$ & \\
\h fine-structure constant && $\alpha$ & \\
\h Planck constant && $ h $ & \\
\h Rydberg constant && $R_\infty$ & \\
\h proton rms charge radius && $R_{\rm p}$ & \\
\h deuteron rms charge radius && $R_{\rm d}$ & \\
\h molar gas constant && $R$ & \\
\h Newtonian constant of gravitation && $ G $ & \\
\h electron-proton magnetic moment ratio && $\mu_{\rm e}/\mu_{\rm p}$ & \\
\h deuteron-electron magnetic moment ratio \hbox to -0.6 in {} && $\mu_{\rm d}/\mu_
{\rm e}$ & \\
\h electron to shielded proton && & \\
\h \ \ magnetic moment ratio && $\mu_{\rm e}/\mu^\prime_{\rm p}$ & \\
\h shielded helion to shielded proton && & \\
\h \ \ magnetic moment ratio && $\mu^\prime_{\rm h}/\mu^\prime_{\rm p}$ & \\
\h neutron to shielded proton && & \\
\h \ \ magnetic moment ratio && $\mu_{\rm n}/\mu_{\rm p}^\prime$ & \\
\h \{220\} lattice spacing of Si crystal X && $d_{220}({\rm X})$ & \\
\h \{220\} lattice spacing of an ideal Si crystal && $d_{220}$ & \\
\h Cu x unit && xu(Cu\,K$\alpha_1$) & \\
\h Mo x unit && xu(Mo\,K$\alpha_1$) & \\
\h \aa ngstrom star && \AA$^*$ & \\
\h corrections to hydrogen level theory && $\delta_{\rm H}(n{\rm L}_j)$ & \\
\h corrections to deuterium level theory && $\delta_{\rm D}(n{\rm L}_j)$ & \\
\h correction to $a_{\rm e}({\rm th})$ && $\delta_{\rm e}$ & \\
\h correction to $a_\mu({\rm th})$ && $\delta_\mu$ & \\
\h correction to ${\rm \Delta}\nu_{\rm Mu}({\rm th})$ && $\delta_{\rm Mu}$ & \\
\hline
\hline
\end{tabular}
\end{table}
\end{center}
\section{Adjusted Constants}
The subset of constants used in the 1998 adjustment is given
in Table~\ref{tab:vars}.
In the table, the symbol $A_{\rm r}({\rm x})$ denotes the
relative atomic mass of particle x, i.e., the ratio $m_{\rm x}/m_{\rm u}$,
where the atomic mass constant $m_{\rm u}$ is 1/12 times
the mass of a $^{12}$C atom.
The $^3$He nucleus (helion) is denoted by h.
Also in that table, $d_{220}({\rm X})$ refers to the
lattice spacing of any of 7 different Si crystals,
with names represented by X, used
in various experiments that provide information on the constants.
The last five entries termed ``corrections'' are correction
terms that are added to the theoretical expressions for the
corresponding quantities (hydrogen and deuterium energy levels,
anomalous magnetic moment of the electron and muon, and
muonium hyperfine splitting)
to take into account the uncertainty
of the theory. There are a total of 25 correction terms for the
hydrogen and deuterium energy levels.
Other quantities can be derived from the adjusted constants.
For example,
based on familiar formulas, we can express the elementary charge
in terms of constants contained in the adjusted subset:
\begin{eqnarray}
\alpha = {e^2\over 4 \pi \epsilon_0\hbar c}
\quad \Rightarrow \quad
e = \sqrt{2\epsilon_0hc\alpha} \ .
\end{eqnarray}
Similarly, for the electron mass, we have
\begin{eqnarray}
R_\infty = {\alpha^2m_{\rm e} c\over 2 h}
\quad \Rightarrow \quad
m_{\rm e} = {2hR_\infty\over
\alpha^2 c} \ .
\end{eqnarray}
It should be noted that it is not necessary to select the most accurately known
constants for the adjusted subset. Since we calculate covariances and use them
to calculate values of the derived constants, it is possible to obtain values
for the derived constants
with relative uncertainties that are smaller than the relative uncertainties of the
values of the constants from which they are derived.
The final result of the adjustment is a set of values for over 300 constants that are
either in the adjusted subset or calculated as above.
\section{Observational Equations}
Some of the observational equations, represented by Eq.~(\ref{eq:eq1}),
employed in the 1998 least-squares adjustment
are given in Table~\ref{tab:obseqs}.
In each observational equation, the symbol on the left-hand side represents
one of the 93 measured, or in some cases calculated, input data of the
adjustment. The right-hand side expresses that quantity as a function of the
constants that are taken as variables of the adjustment. (This distinction
explains the appearance of equations such as $\delta_{\rm e} \doteq \delta_{\rm e}$,
which expresses the relation between an input datum and the corresponding
adjusted constant, even though in general their values will not be equal.)
For example, in the
first equation in Table~\ref{tab:obseqs}, the quantity $A_{\rm r}(^1{\rm H})$ represents
the measured value of the relative atomic mass of the hydrogen atom.
The right-hand side is the
sum of the relative atomic masses of the proton and electron, both of which are
variables of the adjustment, minus the binding energy equivalent mass in the same units,
which is relatively small and sufficiently well known to be taken as exact.
\def\tfrac#1#2{{\vbox to 0pt {}\phantom{_I}\textstyle{#1}
\over\vbox to 0pt {}\textstyle{#2}}}
\def\vh{\vbox to 18pt {}}
\def\vhh{\vbox to 23pt {}}
\begin{table}
\caption{Some observational equations used in the 1998 adjustment.}
\label{tab:obseqs}
\begin{tabular}{r@{\hskip 5pt}c@{\hskip 5pt}l}
\hline
\hline
$\vbox to 11 pt {}
A_{\rm r}(^1{\rm H})$ &$\doteq$&$ A_{\rm r}({\rm p}) + A_{\rm r}({\rm e})
- E_{\rm b}(^1{\rm H})/m_{\rm u}c^2$ \\
\vh$\tfrac{\lambda_{\rm meas}}{d_{220}({\rm {\scriptstyle ILL}})}
$&$\doteq$&$ \tfrac{\alpha^2 A_{\rm r}({\rm e})
}{ R_\infty d_{220}({\rm {\scriptstyle ILL}})}
\tfrac{A_{\rm r}({\rm n}) + A_{\rm r}({\rm p})
}{ \left[A_{\rm r}({\rm n}) + A_{\rm r}({\rm p})\right]^2
- A_{\rm r}^2({\rm d})}$ \\
\vh$ \tfrac{m_{\rm e}}{m(^{12}{\rm C}^{6+})} $&$\doteq$&
$\tfrac{6A_{\rm r}({\rm e})}{12-6A_{\rm r}({\rm e}) + E_{\rm b}(^{12}{\rm C})
/m_{\rm u}c^2}$ \\
\vh$ a_{\rm e} $&$\doteq$&$ a_{\rm e}(\alpha,\delta_e)$ \\
\vh$ \delta_{\rm e}$&$\doteq$&$ \delta_{\rm e}$ \\
\vh$ \tfrac{\mu_{\rm e^-}({\rm H})}{\mu_{\rm p}({\rm H})} $&$\doteq$&$
\tfrac{g_{\rm e^-}({\rm H})}{g_{\rm e^-}}
\left(\tfrac{g_{\rm p}({\rm H})}{g_{\rm p}}\right)^{-1}
\tfrac{\mu_{\rm e^-}}{\mu_{\rm p}} $ \\
\vh$ \nu(f_{\rm p}) $&$\doteq$&$
\nu\!\left(f_{\rm p};R_\infty,\alpha,\tfrac{m_{\rm e}}{m_\mu},
\tfrac{\mu_{\rm e^-}}{\mu_{\rm p}},
\delta_{\rm e}, \delta_\mu, \delta_{\rm Mu}\right) $ \\
\vh$ \tfrac{\mu_{\mu^+}}{\mu_{\rm p}} $&$\doteq$&$
- \tfrac{1+a_\mu(\alpha,\delta_\mu)}
{1+a_{\rm e}(\alpha,\delta_{\rm e})}
\tfrac{m_{\rm e}}{m_\mu}
\tfrac{\mu_{\rm e^-}}{\mu_{\rm p}} $ \\
\vh$ \Delta \nu_{\rm Mu} $&$\doteq$&$
\Delta \nu_{\rm Mu}\!\!\left(R_\infty,\alpha,\tfrac{m_{\rm e}}{m_\mu},
\delta_\mu, \delta_{\rm Mu}\right)$ \\
\vh$ \delta_{\rm Mu}$&$\doteq$&$ \delta_{\rm Mu}$ \\
\vh$ K_{\rm J} $&$\doteq$&$
\left(\tfrac{8\alpha}{\mu_0ch}\right)^{1/2} $ \\
\vh$ R_{\rm K} $&$\doteq$&$ \tfrac{\mu_0c}{2\alpha} $ \\
\vh$ K_{\rm J}^2R_{\rm K} $&$\doteq$&$ \tfrac{4}{h} $ \\
\hbox to -200 pt {}
\vh$ \tfrac{h}{m_{\rm n}d_{220}({\rm {\scriptstyle W04}})} $&$\doteq$&
$ \tfrac{ c A_{\rm r}({\rm e}) \alpha^2 }
{ 2 R_\infty A_{\rm r}({\rm n})d_{220}({\rm {\scriptstyle W04}}) } $ \\
\hbox to -200 pt {}
\vhh$ \tfrac{d_{220}({\rm {\scriptstyle X}}) -
d_{220}({\rm {\scriptstyle Y}})}{
d_{220}({\rm {\scriptstyle Y}})} $&$\doteq$&$
\tfrac{d_{220}({\rm {\scriptstyle X}}) -
d_{220}({\rm {\scriptstyle Y}})}{
d_{220}({\rm {\scriptstyle Y}})}$ \\
\vh$ d_{220}({\rm {\scriptstyle X}}) $&$\doteq$&$ d_{220}({\rm {\scriptstyle X}}) $
\\
\vhh$\tfrac{\lambda({\rm W\,K\alpha_1})}{d_{220}({\rm {\scriptstyle N}})}
$&$\doteq$&$ \tfrac{0.209\,010\,0 \ {\rm \AA^*}}{d_{220}
({\rm {\scriptstyle N}})} $ \\
\vh$ \nu_{\rm H}(n_1{\rm L_1}_{j_1}-n_2{\rm L_2}_{j_2}) $&$\doteq$&$
\bigg[E_{\rm H}\Big(n_2{\rm L_2}_{j_2};R_\infty,\alpha,A_{\rm r}({\rm e}),
A_{\rm r}({\rm p}),
R_{\rm p}, \delta_{\rm H}(n_2{\rm L_2}_{j_2})\Big) $ \\
\vh&&$
- E_{\rm H}\Big(n_1{\rm L_1}_{j_1};R_\infty,\alpha,A_{\rm r}({\rm e}),
A_{\rm r}({\rm p}),
R_{\rm p}, \delta_{\rm H}(n_1{\rm L_1}_{j_1})\Big)\bigg]\Big/h$ \\
\vh$ \delta_{\rm H}(n{\rm L}_j) $&$\doteq$&$ \delta_{\rm H}(n{\rm L}_j)$ \\
\vh$ R_{\rm p} $&$\doteq$&$ R_{\rm p} $ \\
\hline
\hline
\end{tabular}
\end{table}
\section{Electron magnetic moment anomaly}
It is well known that the magnetic moment of the electron is not
equal to the value predicted by the Dirac equation
$g_{\rm e}({\rm Dirac})=-2$, and the deviation from that value
is given in terms of the electron magnetic moment anomaly $a_{\rm e}$
by
\begin{eqnarray}
g_{\rm e}=-2(1+a_{\rm e}) \ .
\nonumber\end{eqnarray}
The anomaly has been measured for the electron and positron at the
University of Washington with the results
%(Van Dyck, Schwinberg and Dehmelt, 1987)
\cite{1987003}
\begin{eqnarray}
a_{{\rm e}^-}({\rm exp}) &=& 1 \, 159 \, 652 \, 188.4(4.3) \ \times \
10^{-12} \nonumber \\
a_{{\rm e}^+}({\rm exp}) &=& 1 \, 159 \, 652 \, 187.9(4.3) \ \times \
10^{-12} \ .
\end{eqnarray}
The average of these values (assuming as we do that the value of the anomaly
is $CPT$ invariant) yields the input datum corresponding to $a_{\rm e}$ on
the left-hand side of the fourth equation in Table~\ref{tab:obseqs}.
The theoretical expression on the right-hand side of that equation is
\begin{eqnarray}
a_{\rm e}(\alpha,\delta_{\rm e}) &=& C_{\rm e}^{(2)}\left({\alpha\over\pi}\right)
+C_{\rm e}^{(4)}\left({\alpha
\over\pi}\right)^2
+ \, C_{\rm e}^{(6)}\left({\alpha\over\pi}\right)^3
\nonumber\\&&+ \, C_{\rm e}^{(8)}\left({\alpha
\over\pi}\right)^4 + a_{\rm e}({\rm had}) + a_{\rm e}({\rm weak})
\nonumber\\&&+ \, \delta_{\rm e} \ ,
\end{eqnarray}
which gives the anomaly as a function of the variables $\alpha$
and $\delta_{\rm e}$. The coefficients $C_{\rm e}^{(n)}$, the
strong interaction correction
$a_{\rm e}({\rm had})$, and the weak interaction correction $a_{\rm e}({\rm weak})$
are calculated from theory.
The largest uncertainty in the theory arises from numerical integration uncertainty
in the massive calculation of $C_{\rm e}^{(8)}$ \cite{1999026}.
The total uncertainty of the theory is estimated to be $1.1\times 10^{-12}$, so the
observational equation for $\delta_{\rm e}$ is
\begin{eqnarray}
0.0(1.1)\times 10^{-12} \doteq \delta_{\rm e} \ .
\label{eq:inde}
\end{eqnarray}
The electron anomalous magnetic moment data provide the most influential
information on the value of the fine-structure constant $\alpha$.
When considered alone they yield
\begin{eqnarray}
\alpha^{-1} = 137.035\,999\,58(52) \ .
\end{eqnarray}
Of course, in the final adjustment,
all sources of information on $\alpha$ contribute to the 1998 recommended value:
\begin{eqnarray}
\alpha^{-1} = 137.035\,999\,76(50) \ .
\end{eqnarray}
This value has an uncertainty that is about 1/12 times
the uncertainty of the 1986 recommended value, and differs from the earlier
value by about 1.7 times the uncertainty of the earlier value.
The adjusted value of $\delta_{\rm e}$ is $0.1(1.1)\times 10^{-12}$, very nearly
its input value in Eq.~(\ref{eq:inde}).
\section{Planck constant from the watt balance}
The Planck constant can be measured by comparing a watt
of mechanical power expressed in terms of the meter, kilogram, and second
to a watt of electrical power expressed in terms of
the Josephson constant $K_{\rm J} = 2e/h$ and
von Klitzing constant $R_{\rm K} = h/e^2$ in the combination
\begin{eqnarray}
K_{\rm J}^2\,R_{\rm K} = {4\over h} \ .
\end{eqnarray}
The apparatus that makes the comparison is called a watt balance
\cite{1990057,1998071}. A remarkable aspect of the experiment is that
it provides a value of the Planck constant through measurements that involve
classical mechanics and classical electromagnetism in a two-story-high
apparatus. The Planck constant enters through the current and voltage
calibration.
The basic principle of the watt balance is illustrated by one of
its implementations \cite{1998071}.
A vertical solenoid is wound in such a way as to produce a radial
magnetic flux density in a region outside the solenoid. A horizontal coil of wire
enclosing the solenoid is suspended in this region from bands that
extend over a balance wheel to a counterweight on the other side.
In one phase of the experiment, a precisely known mass is added to the
side of the balance with the horizontal coil, and a measurement is made of
the current in the coil needed to produce an upward force that balances
the weight of the mass.
The force is given by the equation
\begin{eqnarray}
F_z= \int {\rm d}^3 \!x \ (\vec J\times \vec B )\cdot \hat z
= I \int ( {\rm d}\vec\ell \times \vec B)\cdot\hat z \ ,
\label{eq:phase1}
\end{eqnarray}
where $\vec J$ is the current density in the coil, $I$ is the current
in the coil, and d$\vec\ell$ is an element of length of the wire in the
coil. In the other
phase of the experiment, the coil is slowly moved through the flux density
and the voltage induced in the coil is measured.
This induced voltage is given by
\begin{eqnarray}
U_{\rm v}= \int {\rm d}\vec\ell \cdot (\vec v\times \vec B )
= -\, v \int ( {\rm d}\vec\ell \times \vec B)\cdot\hat z \ ,
\label{eq:phase2}
\end{eqnarray}
where $\vec v$ is the (vertical) velocity of the coil measured by laser
interferometry and, as a first
approximation, it is assumed that the vertical force on the coil in the
first phase is in exactly the same direction as the vertical motion in the
second phase.
Equations~(\ref{eq:phase1}) and (\ref{eq:phase2}) can be combined to obtain
\begin{eqnarray}
F_z &=& - mg = - \,{I\over v}\ U_{\rm v} \ ,
\end{eqnarray}
where $m$ is the precise mass used in the experiment, and $g$ is the local
acceleration of free fall, which is accurately measured with a gravimeter.
The key to all versions of the
experiment is the fact that the flux density and geometry
of the coil drop out of this relation.
Since the voltage $U_{\rm v}$ and the voltage and resistance
that determine $I$ are calibrated in terms of the Josephson
and von Klitzing constants, we have
\begin{eqnarray}
mgv = I\,U_{\rm v} = {Af_1f_2 \over K_{\rm J}^2 R_{\rm K}}
= Af_1f_2\,{h\over 4} \ ,
\label{eq:wbal}
\end{eqnarray}
where $A$ is an exactly known constant and $f_1$ and $f_2$ are the
precisely known frequencies applied to the Josephson junctions in
the two phases of the experiment. Equation (\ref{eq:wbal})
gives $h$ in terms of simple quantities measured in the experiment.
\section{Kilogram definition}
This experiment suggests a possible new definition of the kilogram
\cite{1999074}:
%(Taylor and Mohr, 1999)
\begin{list}{}
\item
{\it The kilogram is the mass of a body at rest whose equivalent energy
equals the energy of a collection of photons whose frequencies
sum to $135\,639\,274 \times 10^{42}$ hertz.}
\end{list}
\vskip 5 pt
\noindent
This definition has the consequence that the Planck constant
is exactly defined, because the
relations $E=mc^2$ and $E=h\nu_1 +h\nu_2+\dots$
imply that
\begin{eqnarray}
h &=& {mc^2\over \nu_1 + \nu_2 + \dots}
= {(1 \ {\rm kg})(299\,792\,458 \ {\rm m \ s}^{-1})^2
\over 135\,639\,274 \times 10 ^{42} \ {\rm Hz}} \\ \\
&=& 6.626\,068\,9 \dots \times 10^{-34} \ {\rm J \ s} \ ,
\end{eqnarray}
exactly.
Since the value of the Planck constant
would be exactly defined by this definition,
the watt balance apparatus could be viewed as a precise scale.
In the present-day mode where $h$ is measured,
a precise mass is employed to determine the mechanical energy
expenditure. However,
if $h$ were exact, than it would be the mass that is being measured instead.
Of course, the utility of such a definition depends on the accuracy to
which the watt balance apparatus can be developed. If it approaches
the current long-term stability of the international prototype of the
kilogram, then it becomes an attractive means of
defining the kilogram and measuring mass.
\section{Rydberg Constant}
The Rydberg constant is determined primarily by comparison of
theory and experiment for energy levels in hydrogen and deuterium.
For example, the observational equation corresponding to the
the $1{\rm S} - 2{\rm S}$ transition frequency of
hydrogen is given approximately by the expression
\begin{eqnarray} \label{eq:rth}
\lefteqn{ \nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}}) }\\
&\doteq& \frac{3}{4}R_{\rm \infty}c \bigg[
1 - {m_{\rm e}\over m_{\rm p}} + \frac{11}{48}\,\alpha^2
- \frac{28}{9}{\alpha^3\over\pi}\ln{\alpha^{-2}}
- \frac{14}{9}\left({\alpha R_{\rm p}\over \lbar_{\rm C}} \right)^2
+ \cdots \bigg] \,, \nonumber
\end{eqnarray}
where the input datum for this transition is
%(Udem, Huber, Gross, Reichert, Prevedelli, Weitz and H\"ansch, 1997)
\cite{1997131}
\begin{eqnarray}
\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})
= 2\,466\,061\,413\,187.34(84) \ {\rm kHz} \ ,
\end{eqnarray}
which has a relative uncertainty of $3.4\times10^{-13}$.
The expression on the right-hand side of Eq.~(\ref{eq:rth})
is approximate and only indicates the leading term of each of
several contributions. In particular, the four terms beyond the ``1''
correspond to contributions from reduced mass, relativistic, radiative, and
finite nuclear size effects, respectively.
However, it is evident that this expression gives information on the
value of the Rydberg constant $R_\infty$.
In the 1998 adjustment, 23 transition frequencies or frequency differences
in hydrogen or deuterium
were included. The theory was carefully reviewed and the expressions for the
energy levels employed in the adjustment were based on
many more terms and precise numerical evaluations than are indicated in
Eq.~(\ref{eq:rth}).
The result for the 1998 recommended value for the Rydberg constant is
\begin{eqnarray}
R_\infty = 10\,973\,731.568\,549(83) \ {\rm m}^{-1} \ ,
\end{eqnarray}
where the relative uncertainty is $7.6\times10^{-12}$. This uncertainty
is about 1/160 times the uncertainty of the 1986 recommended value. In addition,
the 1998
recommended value for $R_\infty$ differs from the 1986 recommended value
by about 2.7 times the uncertainty of the latter value, due to an incorrect
experimental result that strongly influenced the 1986 value.
\section{Future Adjustments}
The availability of the World Wide Web as a resource for making
results immediately available to the public raises
the possibility of issuing recommended values of the constants
more frequently than the 13 year periods that preceded each of the
last two adjustments.
The present thinking is to recommend new values of the constants every four
years, and possibly after two years if developments in theory or experiment
warrant it. With this in mind, the computation of the 1998 constants
has been automated, so that new information can be incorporated
and new values produced with virtually no delay.
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\noindent{\sc Peter J. Mohr} (e-mail: mohr@nist.gov) \\
{\sc Barry N. Taylor} (e-mail: barry.taylor@nist.gov) \\
National Institute of Standards and Technology \\
Gaithersburg, MD 20899-8401, USA.
\end{document}