3
In the quantitative description of physical
phenomena and physical relationships, we
find constant parameters which appear to be
independent of the scale of the phenomena,
independent of the place where the phenomena
happen, and independent of the time when the
phenomena is observed. These parameters are
called fundamental constants. In Sect. 1.1.1, we give
a qualitative description of these basic parameters
and explain how “recommended values” for the
numerical values of the fundamental constants are
found. In Sect. 1.1.2, we present tables of the most
recently determined recommended numerical
values for a large number of those fundamental
constants which play a role in solid-state physics
and chemistry and in materials science.
1.1.1
1.1.2
What are the Fundamental Constants
and Who Takes Care of Them?..............
The CODATA Recommended Values
of the Fundamental Constants.............
1.1.2.1 The Most Frequently Used
Fundamental Constants ...........
1.1.2.2 Detailed Lists
of the Fundamental Constants
in Different Fields of Application
1.1.2.3 Constants from Atomic Physics
and Particle Physics .................
References ..................................................
3
4
4
5
7
9
1.1.1 What are the Fundamental Constants and Who Takes Care of Them?
The fundamental constants are constant parameters in
the laws of nature. They determine the size and strength
of the phenomena in the natural and technological
worlds. We conclude from observation that the numerical values of the fundamental constants are independent
of space and time; at least, we can say that if there is any
dependence of the fundamental constants on space and
time, then this dependence must be an extremely weak
one. Also, we observe that the numerical values are independent of the scale of the phenomena observed; for
example, they seem to be the same in astrophysics and
in atomic physics. In addition, the numerical values are
quite independent of the environmental conditions. So
we have confidence in the idea that the numerical values of the fundamental constants form a set of numbers
which are the same everywhere in the world, and which
have been the same in the past and will be the same in the
future. Whereas the properties of all material objects in
nature are more or less subject to continuous change, the
fundamental constants seem to represent a constituent of
the world which is absolutely permanent.
On the basis of this expected invariance of the fundamental constants in space and time, it appears reasonable
to relate the units of measurement for physical quanti-
ties to fundamental constants as far as possible. This
would guarantee that also the units of measurement
become independent of space and time and of environmental conditions. Within the frame work of the
International System of Units (Système International
d’Unités, abbreviated to SI), the International Committee for Weights and Measures (Comité International
des Poids et Mesures, CIPM) has succeeded in relating a large number of units of measurement for physical
quantities to the numerical values of selected fundamental constants; however, several units for physical
quantities are still represented by prototypes. For example, the unit of length 1 meter, is defined as the
distance light travels in vacuum during a fixed time; so
the unit of length is related to the fundamental constant c,
i. e. the speed of light, and the unit of time, 1 second. The
unit of mass, 1 kilogram, however, is still represented
by a prototype, the mass of a metal cylinder made of
a platinum–iridium alloy, which is carefully stored at the
International Office for Weights and Measures (Bureau
International des Poids et Mesures, BIPM), at Sèvres
near Paris. In a few years, however, it might become
possible also to relate the unit of mass to one or more
fundamental constants.
Part 1 1
The Fundamen
1.1. The Fundamental Constants
4
Part 1
General Tables
Part 1 1.2
The fundamental constants play an important role in
basic physics as well as in applied physics and technology; in fact, they have a key function in the development
of a system of reproducible and unchanging units for
physical quantities. Nevertheless, there is, at present, no
theory which would allow us to calculate the numerical
values of the fundamental constants. Therefore, national
institutes for metrology (NIMs), together with research
institutes and university laboratories, are making efforts
worldwide to determine the fundamental constants experimentally with the greatest possible accuracy and
reliability. This, of course, is a continuous process, with
hundreds of new publications every year.
The Committee on Data for Science and Technology
(CODATA), established in 1966 as an interdisciplinary,
international committee of the International Council of
the Scientific Unions (ICSU), has taken the responsibility for improving the quality, reliability, processing,
management, and accessibility of data of importance to
science and technology. The CODATA task group on
fundamental constants, established in 1969, has taken
on the job of periodically providing the scientific and
technological community with a self-consistent set of
internationally recommended values of the fundamental constants based on all relevant data available at given
points in time.
What is the meaning of “recommended values” of
the fundamental constants?
Many fundamental constants are not independent of
one another; they are related to one another by equations
which allow one to calculate a numerical value for one
particular constant from the numerical values of other
constants. In consequence, the numerical value of a constant can be determined either by measuring it directly or
by calculating it from the measured values of other constants related to it. In addition, there are usually several
different experimental methods for measuring the value
of any particular fundamental constant. This allows one
to compute an adjustment on the basis of a least-squares
fit to the whole set of experimental data in order to determine a set of best-fitting fundamental constants from the
large set of all experimental data. Such an adjustment is
done today about every four years by the CODATA task
group mentioned above. The resulting set of best-fit values is then called the “CODATA recommended values
of the fundamental constants” based on the adjustment
of the appropriate year.
The tables in Sect. 1.1.2 show the CODATA recommended values of the fundamental constants of science
and technology based on the 2002 adjustment. This adjustment takes into account all data that became available
before 31 December 2002. A detailed description of
the adjustment has been published by Mohr and Taylor
of the National Institute of Standards and Technology,
Gaithersburg, in [1.1].
The editors of this Handbook, H. Warlimont and
W. Martienssen, would like to express their sincere
thanks to Mohr and Taylor for kindly putting their data at
our disposal prior to publication in Reviews of Modern
Physics. These data first became available in December
2003 on the web site of the NIST fundamental constants
data center [1.2].
1.1.2 The CODATA Recommended Values of the Fundamental Constants
1.1.2.1 The Most Frequently Used Fundamental Constants
Tables 1.1-1 – 1.1-9 list the CODATA recommended values of the fundamental constants based on the 2002
adjustment.
Table 1.1-1 Brief list of the most frequently used fundamental constants
Quantity
Symbol and relation
Numerical value
Units
Relative standard
uncertainty
Speed of light in vacuum
Magnetic constant
Electric constant
Newtonian constant
of gravitation
Planck constant
Reduced Planck constant
Elementary charge
c
µ0 = 4π × 10−7
ε0 = 1/(µ0 c2 )
G
299 792 458
12.566370614 . . . × 10−7
8.854187817 . . . × 10−12
6.6742(10) × 10−11
m/s
N/A2
F/m
m3 /(kg s2 )
Fixed by definition
Fixed by definition
Fixed by definition
1.5 × 10−4
h
4.13566743(35) × 10−15
6.58211915(56) × 10−16
1.60217653(14) × 10−19
eV s
eV s
C
8.5 × 10−8
8.5 × 10−8
8.5 × 10−8
~ = h/2π
e
The Fundamental Constants
1.2 The CODATA Recommended Values of the Fundamental Constants
Quantity
Symbol and relation
Numerical value
Fine-structure constant
α = (1/4πε0 )(e2 /~c)
7.297352568(24) × 10−3
Φ0 = h/2e
2.06783372(18) × 10−15
Wb
8.5 × 10−8
= 2e2 /h
7.748091733(26) × 10−5
S
3.3 × 10−9
= α2 m
10 973 731.568525(73)
1/m
6.6 × 10−12
me
9.1093826(16) × 10−31
kg
1.7 × 10−7
Proton mass
mp
1.67262171(29) × 10−27
kg
1.7 × 10−7
Proton–electron mass ratio
m p /m e
1836.15267261(85)
4.6 × 10−10
Avogadro number
NA , L
6.0221415(10) × 1023
1.7 × 10−7
Faraday constant
F = NA e
96 485.3383(83)
C
8.6 × 10−8
Molar gas constant
R
8.314472(15)
J/K
1.7 × 10−6
Boltzmann constant
k = R/NA
1.3806505(24) × 10−23
8.617343(15) × 10−5
J/K
eV/K
1.8 × 10−6
1.8 × 10−6
Stefan–Boltzmann constant
σ = (π 2 /60)(k4 /(~3 c2 ))
5.670400(40) × 10−8
W/(m2 K4 )
7.0 × 10−6
Magnetic flux quantum
Conductance quantum
Rydberg constant
Electron mass
G0
R∞
e c/2h
Units
Relative standard
uncertainty
3.3 × 10−9
1.1.2.2 Detailed Lists of the Fundamental Constants in Different Fields of Application
Table 1.1-2 Universal constants
Quantity
Symbol and relation
Numerical value
Units
Relative standard
uncertainty
Speed of light in vacuum
c
299 792 458
m/s
Fixed by definition
Magnetic constant
µ0 = 4π × 10−7
12.566370614 . . . × 10−7
N/A2
Fixed by definition
Electric constant
ε0 = 1/(µ0 c2 )
8.854187817 . . . × 10−12
F/m
Fixed by definition
Characteristic impedance
of vacuum
Z 0 = (µ0 /ε0 )1/2 = µ0 c
376.730313461 . . .
Ω
Fixed by definition
Newtonian constant
of gravitation
G
6.6742(10) × 10−11
m3 /(kg s2 )
1.5 × 10−4
Reduced Planck constant
G/~c
6.7087(10) × 10−39
(GeV/c2 )−2
1.5 × 10−4
Planck constant
h
6.6260693(11) × 10−34
4.13566743(35) × 10−15
Js
eV s
1.7 × 10−7
8.5 × 10−8
(Ratio)
~ = h/2π
1.05457168(18) × 10−34
6.58211915(56) × 10−16
Js
eV s
1.7 × 10−7
8.5 × 10−8
(Product)
~c
197.326968(17)
MeV fm
8.5 × 10−8
3.74177138(64) × 10−16
W m2
1.7 × 10−7
1.19104282(20) × 10−16
W m2 /sr
1.7 × 10−7
1.4387752(25) × 10−2
mK
1.7 × 10−6
5.670400(40) × 10−8
W/(m2 K4 )
7.0 × 10−6
mK
1.7 × 10−6
(Product)
(Product)
(Product)
Stefan–Boltzmann constant
= 2πhc2
c1
(1/π)c1
= 2hc2
c2 = h(c/k)
σ
= (π 2 /60)(k4 /(~3 c2 ))
Wien displacement law
constant
b = λmax T = c2 /4.965114231
2.8977685(51) × 10−3
Planck mass
m P = (~c/G)1/2
2.17645(16) × 10−8
kg
7.5 × 10−5
1.41679(11) × 1032
K
7.5 × 10−5
m
7.5 × 10−5
s
7.5 × 10−5
= (1/k)(~c5 /G)1/2
Planck temperature
TP
Planck length
lP = ~/m P c
= (~G/c3 )1/2
1.61624(12) × 10−35
Planck time
tP = lP /c = (~G/c5 )1/2
5.39121(40) × 10−44
Part 1 1.2
Table 1.1-1 Brief list of the most frequently used fundamental constants, cont.
5
6
Part 1
General Tables
Part 1 1.2
Table 1.1-3 Electromagnetic constants
Quantity
Symbol and relation
Numerical value
Units
Relative standard
uncertainty
Elementary charge
(Ratio)
Fine-structure constant
Inverse fine-structure constant
Magnetic flux quantum
Conductance quantum
Inverse
of conductance quantum
Josephson constant a
von Klitzing constant b
Bohr magneton
e
e/h
α = (1/4πε0 )(e2 /~c)
1/α
Φ0 = h/2e
G 0 = 2e2 /h
1/G 0
1.60217653(14) × 10−19
2.41798940(21) × 1014
7.297352568(24) × 10−3
137.03599911(46)
2.06783372(18) × 10−15
7.748091733(26) × 10−5
12 906.403725(43)
C
A/J
Wb
S
Ω
8.5 × 10−8
8.5 × 10−8
3.3 × 10−9
3.3 × 10−9
8.5 × 10−8
3.3 × 10−9
3.3 × 10−9
K J = 2e/h
RK = h/e2 = µ0 c/2α
µB = e~/2m e
(Ratio)
(Ratio)
(Ratio)
Nuclear magneton
µB /h
µB /hc
µB /k
µN = e~/2m p
(Ratio)
(Ratio)
(Ratio)
µN /h
µN /hc
µN /k
483 597.879(41) × 109
25 812.807449(86)
927.400949(80) × 10−26
5.788381804(39) × 10−5
13.9962458(12) × 109
46.6864507(40)
0.6717131(12)
5.05078343(43) × 10−27
3.152451259(21) × 10−8
7.62259371(65)
2.54262358(22) × 10−2
3.6582637(64) × 10−4
Hz/V
Ω
J/T
eV/T
Hz/T
1/(m T)
K/T
J/T
eV/T
MHz/T
1/(m T)
K/T
8.5 × 10−8
3.3 × 10−9
8.6 × 10−8
6.7 × 10−9
8.6 × 10−8
8.6 × 10−8
1.8 × 10−6
8.6 × 10−8
6.7 × 10−9
8.6 × 10−8
8.6 × 10−8
1.8 × 10−6
a
b
See Table 1.2-16 for the conventional value adopted internationally for realizing representations of the volt using the Josephson effect.
See Table 1.2-16 for the conventional value adopted internationally for realizing representations of the ohm using the quantum Hall effect.
Table 1.1-4 Thermodynamic constants
Quantity
Symbol and relation
Numerical value
Avogadro number
Atomic mass constant
NA , L
u = (1/12)m(12 C)
= (1/NA ) × 10−3 kg
m u c2
6.0221415(10) × 1023
1.66053886(28) × 10−27
kg
1.7 × 10−7
1.7 × 10−7
1.49241790(26) × 10−10
931.494043(80)
96 485.3383(83)
3.990312716(27) × 10−10
0.11962656572(80)
8.314472(15)
1.3806505(24) × 10−23
8.617343(15) × 10−5
2.0836644(36) × 1010
69.50356(12)
22.413996(39) × 10−3
J
MeV
C
Js
Jm
J/K
J/K
eV/K
Hz/K
1/(m K)
m3
1.7 × 10−7
8.6 × 10−8
8.6 × 10−8
6.7 × 10−9
6.7 × 10−9
1.7 × 10−6
1.8 × 10−6
1.8 × 10−6
1.7 × 10−6
1.7 × 10−6
1.7 × 10−6
2.6867773(47) × 1025
5.670400(40) × 10−8
1/m3
W/(m2 K4 )
1.8 × 10−6
7.0 × 10−6
2.8977685(51) × 10−3
mK
1.7 × 10−6
Energy equivalent
of atomic mass constant
Faraday constant
Molar Planck constant
(Product)
Molar gas constant
Boltzmann constant
(Ratio)
(Ratio)
Molar volume of ideal gas
at STP
Loschmidt constant
Stefan–Boltzmann
constant
Wien displacement law
constant
F = NA e
NA h
NA hc
R
k = R/NA
k/h
k/hc
Vm = RT/ p
at T = 273.15 K
and p = 101.325 kPa
n 0 = NA /Vm
σ = (π 2 /60)(k4 /(~3 c2 ))
b = λmax T = c2 /4.965114231
Units
Relative standard
uncertainty
The Fundamental Constants
1.2 The CODATA Recommended Values of the Fundamental Constants
Table 1.1-5 Constants from atomic physics
Quantity
Symbol and relation
Numerical value
Units
Relative standard
uncertainty
Rydberg constant
R∞ = α2 m e c/2h
10 973 731.568525(73)
1/m
6.6 × 10−12
R∞ c
3.289841960360(22) × 1015
Hz
6.6 × 10−12
R∞ hc
2.17987209(37) × 10−18
13.6056923(12)
J
eV
1.7 × 10−7
8.5 × 10−8
(Product)
(Product)
Bohr radius
a0 = α/4πR∞
= 4πε0 ~2 /m e e2
0.5291772108(18) × 10−10
m
3.3 × 10−9
Hartree energy
E H = e2 /4πε0 a0
= 2R∞ hc = α2 m e c2
4.35974417(75) × 10−18
27.2113845(23)
J
eV
1.7 × 10−7
8.5 × 10−8
Quantum of circulation
h/2m e
3.636947550(24) × 10−4
m2 /s
6.7 × 10−9
h/m e
7.273895101(48) × 10−4
m2 /s
6.7 × 10−9
(Product)
Table 1.1-6 Properties of the electron
Quantity
Symbol and relation
Numerical value
Units
Relative standard
uncertainty
Electron mass
me
9.1093826(16) × 10−31
5.4857990945(24) × 10−4
kg
u
1.7 × 10−7
4.4 × 10−10
Energy equivalent
of electron mass
m e c2
8.1871047(14) × 10−14
0.510998918(44)
J
MeV
1.7 × 10−7
8.6 × 10−8
Electron–proton mass ratio
m e /m p
5.4461702173(25) × 10−4
4.6 × 10−10
Electron–neutron mass ratio
m e /m n
5.4386734481(38) × 10−4
7.0 × 10−10
Electron–muon mass ratio
m e /m µ
4.83633167(13) × 10−3
Electron molar mass
M(e) = NA m e
5.4857990945(24) × 10−7
kg
4.4 × 10−10
−e/m e
− 1.75882012(15) × 1011
C/kg
8.6 × 10−8
λC = h/m e c
2.426310238(16) × 10−12
m
6.7 × 10−9
(Ratio)
λC /2π = αa0 = α2 /(4πR∞ )
386.1592678(26) × 10−15
m
6.7 × 10−9
Classical electron radius
re = α2 a0
2.817940325(28) × 10−15
m
1.0 × 10−8
Thomson cross section
σe = (8π/3)re2
0.665245873(13) × 10−28
m2
2.0 × 10−8
Magnetic moment
µe
− 928.476412(80) × 10−26
J/T
8.6 × 10−8
Ratio of magnetic moment
to Bohr magneton
µe /µB
− 1.0011596521859(38)
3.8 × 10−12
Ratio of magnetic moment
to nuclear magneton
µe /µN
− 1838.28197107(85)
4.6 × 10−10
Ratio of magnetic moment
to proton magnetic moment
µe /µp
− 658.2106862(66)
1.0 × 10−8
Ratio of magnetic moment
to neutron magnetic moment
µe /µn
960.92050(23)
2.4 × 10−7
Electron magnetic-moment
anomaly
ae = |µe |/(µB − 1)
1.1596521859(38) × 10−3
3.2 × 10−9
g-factor
ge = −2(1 + ae )
− 2.0023193043718(75)
3.8 × 10−12
Gyromagnetic ratio
γe = 2|µe |/~
1.76085974(15) × 1011
1/(s T)
8.6 × 10−8
(Ratio)
γe /2π
28 024.9532(24)
MHz/T
8.6 × 10−8
Charge-to-mass ratio
Compton wavelength
2.6 × 10−8
Part 1 1.2
1.1.2.3 Constants from Atomic Physics and Particle Physics
7
8
Part 1
General Tables
Part 1 1.2
Table 1.1-7 Properties of the proton
Quantity
Symbol and relation
Numerical value
Units
Relative standard
uncertainty
Proton mass
mp
Energy equivalent
of proton mass
Proton–electron mass ratio
Proton–neutron mass ratio
Proton molar mass
Charge-to-mass ratio
Compton wavelength
(Ratio)
rms charge radius
Magnetic moment
Ratio of magnetic moment
to Bohr magneton
Ratio of magnetic moment
to nuclear magneton
Ratio of magnetic moment
to neutron magnetic moment
g-factor
Gyromagnetic ratio
(Ratio)
m p c2
kg
u
J
MeV
m p /m e
m p /m n
M(p) = NA m p
e/m p
λC,p = h/m p c
(1/2π)λC,p
Rp
µp
µp /µB
1.67262171(29) × 10−27
1.00727646688(13)
1.50327743(26) × 10−10
938.272029(80)
1836.15267261(85)
0.99862347872(58)
1.00727646688(13) × 10−3
9.57883376(82) × 107
1.3214098555(88) × 10−15
0.2103089104(14) × 10−15
0.8750(68) × 10−15
1.41060671(12) × 10−26
1.521032206(15) × 10−3
1.7 × 10−7
1.3 × 10−10
1.7 × 10−7
8.6 × 10−8
4.6 × 10−10
5.8 × 10−10
1.3 × 10−10
8.6 × 10−8
6.7 × 10−9
6.7 × 10−9
7.8 × 10−3
8.7 × 10−8
1.0 × 10−8
µp /µN
2.792847351(28)
1.0 × 10−8
µp /µn
−1.45989805(34)
2.4 × 10−7
gp = 2µp /µN
γp = 2µp /~
(1/2π)γp
5.585694701(56)
2.67522205(23) × 108
42.5774813(37)
1/(s T)
MHz/T
kg
C/kg
m
m
m
J/T
1.0 × 10−8
8.6 × 10−8
8.6 × 10−8
Table 1.1-8 Properties of the neutron
Quantity
Symbol and relation
Numerical value
Units
Relative standard
uncertainty
Neutron mass
mn
Energy equivalent
m n c2
kg
u
J
MeV
Neutron–electron mass ratio
Neutron–proton mass ratio
Molar mass
Compton wavelength
(Ratio)
Magnetic moment
Ratio of magnetic moment
to Bohr magneton
Ratio of magnetic moment
to nuclear magneton
Ratio of magnetic moment
to electron magnetic moment
Ratio of magnetic moment
to proton magnetic moment
g-factor
Gyromagnetic ratio
(Ratio)
m n /m e
m n /m p
M(n) = NA m n
λC,n = h/(m n c)
(1/2π)λC,n
µn
µn /µB
1.67492728(29) × 10−27
1.00866491560(55)
1.50534957(26) × 10−10
939.565360(81)
1838.6836598(13)
1.00137841870(58)
1.00866491560(55) × 10−3
1.3195909067(88) × 10−15
0.2100194157(14) × 10−15
−0.96623645(24) × 10−26
−1.04187563(25) × 10−3
1.7 × 10−7
5.5 × 10−10
1.7 × 10−7
8.6 × 10−8
7.0 × 10−10
5.8 × 10−10
5.5 × 10−10
6.7 × 10−9
6.7 × 10−9
2.5 × 10−7
2.4 × 10−7
µn /µN
−1.91304273(45)
2.4 × 10−7
µn /µe
1.04066882(25) × 10−3
2.4 × 10−7
µn /µp
−0.68497934(16)
2.4 × 10−7
gn = 2µn /µN
γn = 2|µn |/~
(1/2π)γn
−3.82608546(90)
1.83247183(46) × 108
29.1646950(73)
kg
m
m
J/T
1/(s T)
MHz/T
2.4 × 10−7
2.5 × 10−7
2.5 × 10−7
The Fundamental Constants
References
Quantity
Symbol and relation
Numerical value
Units
Relative standard
uncertainty
Alpha particle mass a
mα
Energy equivalent
of alpha particle mass
Ratio of alpha particle mass
to electron mass
Ratio of alpha particle mass
to proton mass
Alpha particle molar mass
m α c2
kg
u
J
MeV
m α /m e
6.6446565(11) × 10−27
4.001506179149(56)
5.9719194(10) × 10−10
3727.37917(32)
7294.2995363(32)
1.7 × 10−7
1.4 × 10−11
1.7 × 10−7
8.6 × 10−8
4.4 × 10−10
m α /m p
3.97259968907(52)
M(α) = NA m α
4.001506179149(56) × 10−3
a
1.3 × 10−10
kg/mol
1.4 × 10−11
The mass of the alpha particle in units of the atomic mass unit u is given by m α = Ar (α) u; in words, the alpha particle mass is
given by the relative atomic mass Ar (α) of the alpha particle, multiplied by the atomic mass unit u
References
1.1
P. J. Mohr, B. N. Taylor: CODATA recommended values
of the fundamental physical constants, Rev. Mod.
Phys. (2004) (in press)
1.2
NIST Physics Laboratory: Web pages of the Fundamental Constants Data Center,
http:/ /physics.nist.gov/constants
Part 1 1
Table 1.1-9 Properties of the alpha particle
9