1. Physical constants
103
1. PHYSICAL CONSTANTS
Table 1.1. Reviewed 2007 by P.J. Mohr and B.N. Taylor (NIST). Based mainly on the “CODATA Recommended Values of the Fundamental
Physical Constants: 2006” by P.J. Mohr, B.N. Taylor, and D.B. Newell (to be published in Rev. Mod. Phys, and J. Phys. Chem. Ref. Data).
The last group of constants (beginning with the Fermi coupling constant) comes from the Particle Data Group. The figures in parentheses after
the values give the 1-standard-deviation uncertainties in the last digits; the corresponding fractional uncertainties in parts per 109 (ppb) are
given in the last column. This set of constants (aside from the last group) is recommended for international use by CODATA (the Committee
on Data for Science and Technology). The full 2006 CODATA set of constants may be found at http://physics.nist.gov/constants.
Quantity
Symbol, equation
Value
Uncertainty (ppb)
s−1
speed of light in vacuum
Planck constant
Planck constant, reduced
c
h
≡ h/2π
electron charge magnitude
conversion constant
conversion constant
e
c
(c)2
electron mass
proton mass
deuteron mass
unified atomic mass unit (u)
0.510 998 910(13) MeV/c2 = 9.109 382 15(45)×10−31 kg
938.272 013(23) MeV/c2 = 1.672 621 637(83)×10−27 kg
= 1.007 276 466 77(10) u = 1836.152 672 47(80) me
md
1875.612 793(47) MeV/c2
(mass 12 C atom)/12 = (1 g)/(NA mol) 931.494 028(23) MeV/c2 = 1.660 538 782(83)×10−27 kg
permittivity of free space
permeability of free space
0 = 1/μ0 c2
μ0
8.854 187 817 . . . ×10−12 F m−1
4π × 10−7 N A−2 = 12.566 370 614 . . . ×10−7 N A−2
fine-structure constant
classical electron radius
(e− Compton wavelength)/2π
Bohr radius (mnucleus = ∞)
wavelength of 1 eV/c particle
Rydberg energy
Thomson cross section
α = e2 /4π0 c
re = e2 /4π0 me c2
−
λe = /me c = re α−1
a∞ = 4π0 2 /me e2 = re α−2
hc/(1 eV)
hcR∞ = me e4 /2(4π0 )2 2 = me c2 α2 /2
σT = 8πre2 /3
7.297 352 5376(50)×10−3 = 1/137.035 999 679(94)†
2.817 940 2894(58)×10−15 m
3.861 592 6459(53)×10−13 m
0.529 177 208 59(36)×10−10 m
1.239 841 875(31)×10−6 m
13.605 691 93(34) eV
0.665 245 8558(27) barn
Bohr magneton
nuclear magneton
electron cyclotron freq./field
proton cyclotron freq./field
μB = e/2me
μN = e/2mp
e /B = e/m
ωcycl
e
p
/B = e/mp
ωcycl
5.788
3.152
1.758
9.578
gravitational constant‡
GN
6.674 28(67)×10−11 m3 kg−1 s−2
= 6.708 81(67)×10−39 c (GeV/c2 )−2
standard gravitational accel.
gN
9.806 65 m s−2
Avogadro constant
Boltzmann constant
NA
k
molar volume, ideal gas at STP
Wien displacement law constant
Stefan-Boltzmann constant
NA k(273.15 K)/(101 325 Pa)
b = λmax T
σ = π 2 k 4 /603 c2
6.022 141 79(30)×1023 mol−1
1.380 6504(24)×10−23 J K−1
= 8.617 343(15)×10−5 eV K−1
22.413 996(39)×10−3 m3 mol−1
2.897 7685(51)×10−3 m K
5.670 400(40)×10−8 W m−2 K−4
Fermi coupling constant∗∗
GF /(c)3
Z ) (MS)
sin2 θ(M
mW
mZ
αs (mZ )
me
mp
weak-mixing angle
W ± boson mass
Z 0 boson mass
strong coupling constant
π = 3.141 592 653 589 793 238
1 in ≡ 0.0254 m
1 Å ≡ 0.1 nm
1 barn ≡ 10−28 m2
∗
299 792 458 m
6.626 068 96(33)×10−34 J s
1.054 571 628(53)×10−34 J s
= 6.582 118 99(16)×10−22 MeV s
1.602 176 487(40)×10−19 C = 4.803 204 27(12)×10−10 esu
197.326 9631(49) MeV fm
0.389 379 304(19) GeV2 mbarn
1 G ≡ 10−4 T
381
451
820
833
7555(79)×10−11 MeV T−1
2326(45)×10−14 MeV T−1
150(44)×1011 rad s−1 T−1
92(24)×107 rad s−1 T−1
0.68, 0.68
2.1
1.4
0.68
25
25
4.1
1.4
1.4
25
25
1.0 × 105
1.0 × 105
50
1700
1700
1700
1700
7000
6.5 × 105
3.6 × 105
2.3 × 104
1.7 × 107
γ = 0.577 215 664 901 532 861
kT at 300 K = [38.681 685(68)]−1 eV
1 dyne ≡ 10−5 N
1 eV/c2 = 1.782 661 758(44) × 10−36 kg
0 ◦ C ≡ 273.15 K
−7
9
1
atmosphere
≡
760
Torr ≡ 101 325 Pa
1 erg ≡ 10 J 2.997 924 58 × 10 esu = 1 C
The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.
At Q2 = 0. At Q2 ≈ m2W the value is ∼ 1/128.
‡ Absolute lab measurements of G have been made only on scales of about 1 cm to 1 m.
N
∗∗ See the discussion in Sec. 10, “Electroweak model and constraints on new physics.”
†† The corresponding sin2 θ for the effective angle is 0.23149(13).
†
exact
exact
9000
0.231 19(14)††
80.398(25) GeV/c2
91.1876(21) GeV/c2
0.1176(20)
1 eV = 1.602 176 487(40) × 10−19 J
25, 50
25, 50
0.10, 0.43
25
25, 50
exact
1.166 37(1)×10−5 GeV−2
e = 2.718 281 828 459 045 235
exact∗
50
50
25
25, 25
25
50
104
2. Astrophysical constants
2. ASTROPHYSICAL CONSTANTS AND PARAMETERS
Table 2.1. Revised May 2008 by E. Bergren and D.E. Groom (LBNL). The figures in parentheses after some values give the one standard
deviation uncertainties in the last digit(s). Physical constants are from Ref. 1. While every effort has been made to obtain the most accurate
current values of the listed quantities, the table does not represent a critical review or adjustment of the constants, and is not intended as a
primary reference. The values and uncertainties for the cosmological parameters depend on the exact data sets, priors, and basis parameters
used in the fit. Many of the parameters reported in this table are derived parameters or have non-Gaussian likelihoods. The quoted errors
may be highly correlated with those of other parameters, so care must be taken in propagating them. Unless otherwise specified, cosmological
parameters are best fits of a spatially-flat ΛCDM cosmology with a power-law initial spectrum to WMAP 3-year data alone [2]. For more
information see Ref. 3 and the original papers.
Quantity
Symbol, equation
speed of light
Newtonian gravitational constant
Planck mass
c
G
N
c/GN
Planck length
standard gravitational acceleration
jansky (flux density)
GN /c3
gN
Jy
Value
Reference, footnote
s−1
299 792 458 m
6.674 3(7) × 10−11 m3 kg−1 s−2
1.220 89(6) × 1019 GeV/c2
= 2.176 44(11) × 10−8 kg
1.616 24(8) × 10−35 m
9.806 65 m s−2
10−26 W m−2 Hz−1
[1]
exact[1]
definition
tropical year (equinox to equinox) (2007)
yr
sidereal year (fixed star to fixed star) (2007)
mean sidereal day (2007) (time between vernal equinox transits)
31 556 925.2 s ≈ π × 107 s
31 558 149.8 s ≈ π × 107 s
23h 56m 04.s 090 53
[5]
[5]
[5]
astronomical unit
parsec (1 AU/1 arc sec)
light year (deprecated unit)
Schwarzschild radius of the Sun
Solar mass
Solar equatorial radius
Solar luminosity
Schwarzschild radius of the Earth
Earth mass
Earth mean equatorial radius
AU, A
pc
ly
2GN M /c2
M
R
L
2GN M⊕ /c2
M⊕
R⊕
149 597 870 700(3) m
3.085 677 6 × 1016 m = 3.262 . . . ly
0.306 6 . . . pc = 0.946 053 . . . × 1016 m
2.953 250 077 0(2) km
1.988 4(2) × 1030 kg
6.9551(3) × 108 m
3.842 7(1 4) × 1026 W
8.870 055 881 mm
5.972 2(6) × 1024 kg
6.378 137 × 106 m
[6]
[7]
luminosity conversion (deprecated)
L
[14]
flux conversion (deprecated)
F
ABsolute monochromatic magnitude
AB
3.02 × 1028 × 10−0.4 Mbol W
(Mbol = absolute bolometric magnitude
= bolometric magnitude at 10 pc)
2.52 × 10−8 × 10−0.4 mbol W m−2
(mbol = apparent bolometric magnitude)
−2.5 log10 fν −56.10 (for fν in W m−2 Hz−1 )
= −2.5 log10 fν + 8.90 (for fν in Jy)
Solar velocity around center of Galaxy
Solar distance from Galactic center
Θ0
R0
220(20) km s−1
8.0(5) kpc
[16]
[17]
local disk density
local halo density
present day CMB temperature
present day CMB dipole amplitude
Solar velocity with respect to CMB
3–12 ×10−24 g cm−3 ≈ 2–7 GeV/c2 cm−3
2–13 ×10−25 g cm−3 ≈ 0.1–0.7 GeV/c2 cm−3
2.725(1) K
3.358(17) mK
369(2) km/s
towards (, b) = (263.86(4)◦ , 48.24(10)◦)
vLG
627(22) km s−1
towards (, b) = (276(3)◦ , 30(3)◦
s/k
2 889.2 (T /2.725)3 cm−3
nγ
410.5(T /2.725)3 cm−3
H0
100 h km s−1 Mpc−1
= h × (9.777 752 Gyr)−1
h
0.73(3)
c/H0
0.925 063 × 1026 h−1 m ≈ 1.27 × 1026 m
c2 /3H02
2.852 × 1051 h−2 m2
ρc = 3H02 /8πGN
2.775 366 27 × 1011 h2 M Mpc−3
= 1.878 35(19) × 10−29 h2 g cm−3
= 1.053 68(11) × 10−5 h2 (GeV/c2 ) cm−3
Ωm = ρm /ρc
0.128(8) h−2 ≈ 0.24 (WMAP3)
0.132(4) h−2 ⇒ 0.27(2) (ALL mean)
Ωb = ρb /ρc
0.0223(7) h−2 ≈ 0.0425
Ωdm = Ωm − Ωb
0.105(8) h−2 ≈ 0.20
ΩΛ
0.73(3)
c/H0
0.925 063 × 1026 h−1 m ≈ 1.27 × 1026 m
Ωγ = ργ /ρc
2.471 × 10−5 (T /2.725)4 h−2 ≈ 4.6 ×10−5
Ων
0.0005 < Ων h−2 < 0.023 ⇒ 0.001 < Ων < 0.05
Ωtot = Ωm + . . . + ΩΛ 1.011(12)
Local Group velocity with respect to CMB
entropy density/Boltzmann constant
number density of CMB photons
present day Hubble expansion rate
present day normalized Hubble expansion rate‡
Hubble length
scale factor for cosmological constant
critical density of the Universe
pressureless matter density of the Universe‡
baryon density of the Universe‡
dark matter density of the universe‡
dark energy density of the Universe‡
Hubble length
radiation density of the Universe‡
neutrino density of the Universe‡
total energy density of the Universe‡
ρ disk
ρ halo
T0
exact[4]
[1]
[1]
[8]
[9]
[10]
[11]
[12]
[13]
[5]
from above
[15]
[18]
[19]
[20]
[21]
[21]
[22]
[14]
[23]
[24]
[2,3]
[2,3]
[2]
[2,3]
[2]
[25]
[23]
[26]
[2,27]
2. Astrophysical constants
Quantity
Symbol, equation
baryon-to-photon ratio‡
number density of baryons‡
dark energy equation of state parameter‡
fluctuation amplitude at 8h−1 Mpc scale‡
scalar spectral index from power-law fit to data‡
running spectral index slope at k0 = 0.05 Mpc−1 ‡
tensor-to-scalar field perturbations ratio
at k0 = 0.002 Mpc−1 ‡
reionization optical depth‡
age of Universe at reionization ‡
age of the Universe‡
Value
Reference, footnote
η = nb /nγ
nb
w
σ8
ns
dns /d ln k
6.12(19) × 10−10
[28]
(1.9 × 10−7 < nb < 2.7 × 10−7 ) cm−3 (95% CL) from η
−0.97(7)
[2]
0.76(5)
[2,3]
0.958(16)
[2,3]
−0.05 ± 0.03
[2,29]
r = T /S
τ
treion
t0
< 0.65 at 95% C.L.
0.09(3)
365 Myr
13.73(15) Gyr
‡ See caption for caveats.
References:
1. P.J. Mohr, B.N. Taylor, & D.B. Newell, CODATA Recommended
Values of the Fundamental Constants: 2006, Rev. Mod. Phys. (to
be published); physics.nist.gov/constants.
2. D.N. Spergel et al., Astrophys. J. Supp. 170, 377 (2007).
Post-deadline WMAP5 values have not been used. In any case,
they usually vary no more than 1σ from the WMAP3 values.
3. O. Lahav & A.R. Liddle, “The Cosmological Parameters,” this
Review.
4. B.W. Petley, Nature 303, 373 (1983).
5. The Astronomical Almanac for the year 2007, U.S. Government
Printing Office, Washington, and The Stationery Office, London
(2005).
6. With the range measurements of the Mars Global Surveyer and
Odyssey in 1999–2007 now added to the Viking ranges of 1976-82,
the value of the AU is determined to be 149 597 870 700±2 meters.
While the AU is approximately equal to the semi-major axis
of the Earth’s orbit, it is not exactly so. Nor is it exactly the
mean earth-sun distance. There are a number of reasons for this:
1) the Earth’s orbit is not exactly Keplerian due to relativity
and to perturbations from other planets; 2) the adopted value
for the Gaussian gravitational constant k is not exactly equal
to the earth’s mean motion; and 3) the mean distance in a
Keplerian orbit is not equal to the semi-major axis; instead, it is
r = a(1 + e2 /2), where e is the eccentricity.
For an observer far above Earth’s orbital plane at rest in the
inertial frame of the Solar System, terrestrial clocks would appear
to run slower than local clocks because of (a) time dilation
from Earth’s orbital motion and (b) gravitational redshift at
the Earth’s surface and in the Sun’s potential well. The last
contribution is twice as big as the time dilation. The clock
rates differ by 1.5 parts in 108 . These effects complicate the
measurement and definition of the AU and GN M (Discussion
courtesy of Myles Standish, JPL).
7. The distance at which 1 AU subtends 1 arc sec: 1 AU divided by
π/648 000.
8. Product of 2/c2 and the heliocentric gravitational constant
GN M = A3 k 2 /864002, where k is the Gaussian gravitational
constant, 0.017 202 098 95 (exact) [5]. The value and error for A
given in this table are used.
9. Obtained from the heliocentric gravitational constant [5] and
GN [1]. The error is the 100 ppm standard deviation of GN .
10. T. M. Brown & J. Christensen-Dalsgaard, Astrophys. J. 500,
L195 (1998). Many values for the Solar radius have been
published, most of which are consistent with this result.
11. 4π A2 ×(1366.4±0.5) W m−2 [30]. Assumes isotropic irradiance.
12. Schwarzschild radius of the Sun (above) scaled by the Earth/Sun
mass ratio given in Ref. 5.
13. Obtained from the geocentric gravitational constant [5] and
GN [1]. The error is the 100 ppm standard deviation of GN .
14. E.W. Kolb & M.S. Turner, The Early Universe, Addison-Wesley
(1990);
The IAU (Commission 36) has recommended 3.055 × 1028 W for
the zero point. Based on newer Solar measurements, the value
and significance given in the table seems more appropriate.
105
[2,3]
[2,3]
[2,3]
[2]
15. J. B. Oke & J. E. Gunn, Astrophys. J. 266, 713 (1983). Note
that in the definition of AB the sign of the constant is wrong.
16. F.J. Kerr & D. Lynden-Bell, Mon. Not. R. Astr. Soc. 221, 1023–
1038 (1985). “On the basis of this review these [R0 = 8.5±1.1 kpc
and Θ0 = 220 ± 20 km s−1 ] were adopted by resolution of IAU
Commission 33 on 1985 November 21 at Delhi.” We retain this
value for Θ0 but list a more modern value for R0 .
17. M.J. Reid, Annu. Rev. Astron. Astrophys. 31, 345–372 (1993);
M. Shen & Z. Zhu, Chin. Astron. Astrophys. 7, 120 (2007). In
Fig. 2 they present a summary of a dozen modern values for R0 .
All but one are within Reid’s error band.
18. G. Gilmore, R.F.G. Wyse, & K. Kuijken, Ann. Rev. Astron.
Astrophys. 27, 555 (1989).
19. E.I. Gates, G. Gyuk, & M.S. Turner (Astrophys. J. 449, L133
−25 g cm−3 ,
(1995)) find the local halo density to be 9.2+3.8
−3.1 × 10
but also comment that previously published estimates are in the
range 1–10 × 10−25 g cm−3 .
The value 0.3 GeV/c2 has been taken as “standard” in several
papers setting limits on WIMP mass limits, e.g. in M. Mori
et al., Phys. Lett. B289, 463 (1992).
20. J. Mather et al., Astrophys. J. 512, 511 (1999). This paper
gives T0 = (2.725 ± 0.002) K at 95%CL. We take 0.001 as the
one-standard deviation uncertainty.
21. G. Hinshaw, et al., Astrophys. J. Supp. 170, 288 (2007).
22. D. Scott & G.F. Smoot, “Cosmic Microwave Background,” this
Review.
2ζ(3) kT 3
π 2 (kT )4
23. nγ =
and ργ =
.
2
c
15 (c)3
π
24. Conversion using length of sidereal year.
25. ΩΛ from fits to various data sets is given in Table 12 in Ref. 2.
The (meaningless) weighted average from the not-independent
data sets is 0.727 ± 0.012. This is almost the WMAP + SDSS
LRG fit, which we quote with a 50% more conservative error.
The extended error band includes results obtained with all of the
data sets.
26. The lower limit follows from neutrino mixing results combined
with the assumptions that there are three light neutrinos
(m < 45 GeV) and that the lightest neutrino is substantially less
massive than the others. Limits set from analyses of WMAP,
large-scale structure, and other data are in the Ων < 0.02 range.
If the limit obtained from tritium decay experiments (mν < 2 eV)
is taken seriously, then one can only conclude that Ων < 0.1.
27. From WMAP 3-year + SNLS data. WMAP 3-year data
plus the HST Key Project constraint on H0 implies Ωtot =
1.014 ± 0.017 [2].
28. Calculated from ρc , Ωb , and nγ .
29. From WMAP 3-year data alone, assuming no tensors. If other
data are included, results from −0.058 to −0.066 are obtained.
Inclusion of tensors in the model results in values from −0.082 to
−0.090 [2].
30. R.C. Willson & A.V. Mordvinov, Geophys. Res. Lett. 30, 1119
(2003);
C. Frölich, Space Sci. Rev. 125, 53–65 (2006).
106
3. International system of units (SI)
3. INTERNATIONAL SYSTEM OF UNITS (SI)
See “The International System of Units (SI),” NIST Special Publication 330, B.N. Taylor, ed. (USGPO, Washington, DC, 1991); and “Guide for
the Use of the International System of Units (SI),” NIST Special Publication 811, 1995 edition, B.N. Taylor (USGPO, Washington, DC, 1995).
SI prefixes
Physical
quantity
Name
of unit
Symbol
Base units
1024
yotta
(Y)
1021
zetta
(Z)
1018
exa
(E)
length
meter
m
1015
peta
(P)
mass
kilogram
kg
1012
tera
(T)
time
second
s
electric current
ampere
A
109
giga
(G)
106
mega
(M)
103
thermodynamic
temperature
kelvin
amount of substance
mole
luminous intensity
candela
K
kilo
(k)
102
hecto
(h)
10
deca
(da)
10−1
deci
(d)
rad
10−2
centi
(c)
(m)
mol
cd
Derived units with special names
plane angle
radian
solid angle
steradian
sr
10−3
milli
frequency
hertz
Hz
energy
joule
J
10−6
micro (μ)
force
newton
N
10−9
nano
(n)
pressure
pascal
Pa
10−12
pico
(p)
W
10−15
femto
(f)
power
watt
electric charge
coulomb
C
10−18
atto
(a)
electric potential
volt
V
10−21
zepto
(z)
electric resistance
ohm
Ω
10−24
yocto
(y)
electric conductance
siemens
S
electric capacitance
farad
F
magnetic flux
weber
Wb
inductance
henry
H
magnetic flux density
tesla
T
luminous flux
lumen
lm
illuminance
lux
lx
celsius temperature
degree celsius
◦C
activity (of a
radioactive source)∗
absorbed dose (of
ionizing radiation)∗
dose equivalent∗
becquerel
Bq
gray
Gy
sievert
Sv
∗ See our section 29, on “Radioactivity and radiation
protection,” p. 312.
H
Beryllium
Sodium
22.98976928
Boron
B 6
N 8
Nitrogen
C 7
Carbon
15
VA
17
VIIA
Helium
Neon
4.002602
F 10
Ne
Fluorine
O 9
Oxygen
16
VIA
10.811
12.0107 14.0067 15.9994 18.9984032 20.1797
13
Al 14
Si 15
P 16
S 17
Cl 18
Ar
5
14
IVA
Calcium
Magnesium
Scandium Titanium Vanadium Chromium Manganese
Iron
Cobalt
Nickel
Copper
Zinc
Gallium
German.
Arsenic
Selenium
Bromine
Krypton
Yttrium Zirconium Niobium
Molybd.
Technet.
Ruthen.
Rhodium Palladium
Silver
Cadmium
Indium
Tin
Antimony Tellurium
Iodine
Xenon
89–103
Lanthanides
Iridium
Platinum
Gold
Actinide
series
Lanthanide
series
Ac 90
Th 91
U 93
144.242
Pa 92
140.90765
Pm 62
Sm 63
Np 94
Pu 95
Am 96
Lawrenc.
(262.110)
Lr
174.9668
No 103
173.054
Md 102
168.93421
Fm 101
167.259
Es 100
164.93032
Lu
Lutetium
Yb 71
Ytterbium
Tm 70
Thulium
Er 69
Bismuth Polonium Astatine
Radon
208.98040 (208.98243) (209.98715) (222.01758)
Erbium
Ho 68
Holmium
Cf 99
162.500
Bk 98
158.92535
Cm 97
157.25
Dyspros.
Tb 66
Terbium
Gd 65
Gadolin.
Eu 64
Prometh. Samarium Europium
(144.91275) 150.36
151.964
Nd 61
Lead
207.2
Actinium Thorium Protactin. Uranium Neptunium Plutonium Americ.
Curium Berkelium Californ.
Einstein.
Fermium Mendelev. Nobelium
(227.02775) 232.03806 231.03588 238.02891 (237.04817) (244.06420) (243.06138) (247.07035) (247.07031) (251.07959) (252.0830) (257.09510) (258.09843) (259.1010)
89
140.116
138.90547
Pr 60
Praseodym. Neodym.
Ce 59
Cerium
La 58
Lanthan.
57
Dy 67
Thallium
Osmium
204.3833
Rhenium
Mercury
Tantalum Tungsten
178.49 180.94788 183.84
186.207
190.23
192.217 195.084 196.966569 200.59
104
Rf 105 Db 106
Sg 107 Bh 108 Hs 109 Mt 110 Ds 111 Rg 112
Hafnium
Francium
Radium Actinides Rutherford. Dubnium Seaborg. Bohrium Hassium Meitner. Darmstadt. Roentgen.
(223.01974) (226.02541)
(267.122) (268.125) (271.133) (270.134) (269.134) (276.151) (281.162) (280.164) (285.174)
87
Barium
137.327
Fr 88
Ra
132.9054519
Cesium
85.4678
87.62
88.90585 91.224 92.90638
95.96 (97.90722) 101.07 102.90550 106.42 107.8682 112.411 114.818 118.710 121.760
127.60 126.90447 131.293
55
Cs 56
Ba 57–71 72
Hf 73
Ta 74
W 75
Re 76
Os 77
Ir 78
Pt 79
Au 80
Hg 81
Tl 82
Pb 83
Bi 84
Po 85
At 86
Rn
Rubidium Strontium
39.0983
40.078 44.955912 47.867
50.9415 51.9961 54.938045 55.845 58.933195 58.6934
63.546
65.38
69.723
72.64
74.92160
78.96
79.904
83.798
37
Rb 38
Sr 39
Y 40
Zr 41
Nb 42
Mo 43
Tc 44
Ru 45
Rh 46
Pd 47
Ag 48
Cd 49
In 50
Sn 51
Sb 52
Te 53
I 54
Xe
Potassium
19
PERIODIC TABLE OF THE ELEMENTS
13
IIIA
18
VIIIA
2
He
9
3
4
5
6
7
8
10
11
12
Aluminum Silicon
Phosph.
Sulfur
Chlorine
Argon
VIII
IIIB
IVB
VB
VIB
VIIB
IB
IIB
24.3050
26.9815386 28.0855 30.973762
32.065
35.453
39.948
K 20
Ca 21
Sc 22
Ti 23
V 24
Cr 25
Mn 26
Fe 27
Co 28
Ni 29
Cu 30
Zn 31
Ga 32
Ge 33
As 34
Se 35
Br 36
Kr
6.941
9.012182
11
Na 12
Mg
Lithium
Hydrogen
2
IIA
1.00794
3
Li 4
Be
1
1
IA
Table 4.1. Revised 2008 by C.G. Wohl (LBNL), D.E. Groom (LBNL), and E. Bergren. Atomic weights of stable elements are adapted from the Commission
on Isotopic Abundances and Atomic Weights, “Atomic Weights of the Elements 2007,” http://www.chem.qmul.ac.uk/iupac/AtWt/. The atomic number (top
left) is the number of protons in the nucleus. The atomic mass (bottom) of a stable elements is weighted by isotopic abundances in the Earth’s surface.
If the element has no stable isotope, the atomic mass (in parentheses) of the most stable isotope currently known is given. In this case the mass is from
http://www.nndc.bnl.gov/amdc/masstables/Ame2003/mass.mas03 and the longest-lived isotope is from www.nndc.bnl.gov/ensdf/za form.jsp. The exceptions
are Th, Pa, and U, which do have characteristic terrestrial compositions. Atomic masses are relative to the mass of 12 C, defined to be exactly 12 unified atomic mass
units (u) (approx. g/mole). Relative isotopic abundances often vary considerably, both in natural and commercial samples; this is reflected in the number of significant
figures given. As of early 2008 element 112 has not been assigned a name, and there are no confirmed elements with Z > 112.
4. Periodic table of the elements
4. PERIODIC TABLE OF THE ELEMENTS
107
108
5. Electronic structure of the elements
5. ELECTRONIC STRUCTURE OF THE ELEMENTS
Table 5.1. Reviewed 2005 by C.G. Wohl (LBNL). The electronic configurations and the ionization energies are from the NIST database,
“Ground Levels and Ionization Energies for the Neutral Atoms,” W.C. Martin, A. Musgrove, S. Kotochigova, and J.E. Sansonetti (2003),
http://physics.nist.gov (select “Physical Reference Data”). The electron configuration for, say, iron indicates an argon electronic core (see
argon) plus six 3d electrons and two 4s electrons. The ionization energy is the least energy necessary to remove to infinity one electron from an
atom of the element.
Element
Electron configuration
(3d5 = five 3d electrons, etc.)
1
2
H
He
Hydrogen
Helium
1s
1s2
3
4
5
6
7
8
9
10
Li
Be
B
C
N
O
F
Ne
Lithium
Beryllium
Boron
Carbon
Nitrogen
Oxygen
Fluorine
Neon
(He)2s
(He)2s2
(He)2s2
(He)2s2
(He)2s2
(He)2s2
(He)2s2
(He)2s2
11
12
13
14
15
16
17
18
Na
Mg
Al
Si
P
S
Cl
Ar
Sodium
Magnesium
Aluminum
Silicon
Phosphorus
Sulfur
Chlorine
Argon
(Ne)3s
(Ne)3s2
(Ne)3s2
(Ne)3s2
(Ne)3s2
(Ne)3s2
(Ne)3s2
(Ne)3s2
Ionization
energy
(eV)
13.5984
24.5874
2S
1/2
1S
0
2S
1/2
1S
0
2P
1/2
3P
0
4S
3/2
3P
2
2P
3/2
1S
0
2p
2p2
2p3
2p4
2p5
2p6
2S
1/2
1S
0
2P
1/2
3P
0
4S
3/2
3P
2
2P
3/2
1S
0
3p
3p2
3p3
3p4
3p5
3p6
19
K
Potassium
(Ar)
4s
20
Ca Calcium
(Ar)
4s2
- - - - - - - - - - - - - - - - - - - - - - - - 21
Sc
Scandium
(Ar) 3d 4s2
22
Ti
Titanium
(Ar) 3d2 4s2
23
V
Vanadium
(Ar) 3d3 4s2
24
Cr Chromium
(Ar) 3d5 4s
25
Mn Manganese
(Ar) 3d5 4s2
26
Fe
Iron
(Ar) 3d6 4s2
27
Co Cobalt
(Ar) 3d7 4s2
28
Ni
Nickel
(Ar) 3d8 4s2
29
Cu Copper
(Ar) 3d10 4s
30
Zn Zinc
(Ar) 3d10 4s2
- - - - - - - - - - - - - - - - - - - - - - - - 31
Ga Gallium
(Ar) 3d10 4s2 4p
32
Ge Germanium
(Ar) 3d10 4s2 4p2
33
As Arsenic
(Ar) 3d10 4s2 4p3
34
Se
Selenium
(Ar) 3d10 4s2 4p4
35
Br Bromine
(Ar) 3d10 4s2 4p5
36
Kr Krypton
(Ar) 3d10 4s2 4p6
Ground
state
2S+1 L
J
2S
5.3917
9.3227
8.2980
11.2603
14.5341
13.6181
17.4228
21.5645
5.1391
7.6462
5.9858
8.1517
10.4867
10.3600
12.9676
15.7596
4.3407
6.1132
0
- - - - - - - - - - - - - 2D
6.5615
3/2
3F
6.8281
2
e
4F
6.7462
3/2
l
7S
6.7665
e
3
6S
7.4340
m
5/2
5D
e
7.9024
4
4F
n
7.8810
9/2
3F
t
7.6398
4
s
2S
7.7264
1/2
1S
9.3942
0
- - - - - - - - - - - - - 2P
5.9993
1/2
3P
7.8994
0
4S
9.7886
3/2
3P
9.7524
2
2P
11.8138
3/2
1S
13.9996
0
1/2
1S
T
r
a
n
s
i
t
i
o
n
-
2S
37
Rb Rubidium
(Kr)
5s
4.1771
1/2
1S
38
Sr
Strontium
(Kr)
5s2
5.6949
0
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2D
6.2173
39
Y
Yttrium
(Kr) 4d 5s2
T
3/2
3F
r
40
Zr
Zirconium
(Kr) 4d2 5s2
6.6339
2
e
6D
a
6.7589
41
Nb Niobium
(Kr) 4d4 5s
1/2
l
n
7S
42
Mo Molybdenum
(Kr) 4d5 5s
7.0924
e
3
s
6S
7.28
43
Tc Technetium
(Kr) 4d5 5s2
m
5/2
i
5F
e
44
Ru Ruthenium
(Kr) 4d7 5s
7.3605
5
t
4F
n
7.4589
45
Rh Rhodium
(Kr) 4d8 5s
9/2
i
1S
t
46
Pd Palladium
(Kr) 4d10
8.3369
0
o
s
2S
7.5762
47
Ag Silver
(Kr) 4d10 5s
n
1/2
1S
48
Cd Cadmium
(Kr) 4d10 5s2
8.9938
0
5. Electronic structure of the elements
49
50
51
52
53
54
In
Sn
Sb
Te
I
Xe
Indium
Tin
Antimony
Tellurium
Iodine
Xenon
(Kr) 4d10 5s2
(Kr) 4d10 5s2
(Kr) 4d10 5s2
(Kr) 4d10 5s2
(Kr) 4d10 5s2
(Kr) 4d10 5s2
5p
5p2
5p3
5p4
5p5
5p6
2P
1/2
3P
0
4S
3/2
3P
2
2P
3/2
1S
0
5.7864
7.3439
8.6084
9.0096
10.4513
12.1298
2S
55
Cs Cesium
(Xe)
6s
3.8939
1/2
2
1
56
Ba Barium
(Xe)
6s
S0
5.2117
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2
2
D3/2
5.5769
57
La Lanthanum
(Xe)
5d 6s
1G
58
Ce Cerium
(Xe)4f 5d 6s2
5.5387
4
4I
6s2
5.473
59
Pr
Praseodymium
(Xe)4f 3
L
9/2
5I
a
60
Nd Neodymium
(Xe)4f 4
6s2
5.5250
4
6H
n
6s2
5.582
61
Pm Promethium
(Xe)4f 5
5/2
t
7F
62
Sm Samarium
(Xe)4f 6
6s2
5.6437
0
h
8S
6s2
5.6704
63
Eu Europium
(Xe)4f 7
7/2
a
9D
64
Gd Gadolinium
(Xe)4f 7 5d 6s2
6.1498
2
n
6H
6s2
5.8638
65
Tb Terbium
(Xe)4f 9
15/2
i
5I
66
Dy Dysprosium
(Xe)4f 10
6s2
5.9389
8
d
4I
6s2
6.0215
67
Ho Holmium
(Xe)4f 11
e
15/2
3H
68
Er
Erbium
(Xe)4f 12
6s2
6.1077
s
6
2F
6s2
6.1843
69
Tm Thulium
(Xe)4f 13
7/2
1S
70
Yb Ytterbium
(Xe)4f 14
6s2
6.2542
0
2D
5.4259
71
Lu Lutetium
(Xe)4f 14 5d 6s2
3/2
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 3F
72
Hf
Hafnium
(Xe)4f 14 5d2 6s2
6.8251
T
2
4F
r
7.5496
73
Ta Tantalum
(Xe)4f 14 5d3 6s2
3/2
e
5D
a
74
W
Tungsten
(Xe)4f 14 5d4 6s2
7.8640
0
l
6S
n
7.8335
75
Re Rhenium
(Xe)4f 14 5d5 6s2
e
5/2
s m
5D
76
Os Osmium
(Xe)4f 14 5d6 6s2
8.4382
4
i
4F
e
8.9670
77
Ir
Iridium
(Xe)4f 14 5d7 6s2
9/2
t
3D
n
78
Pt
Platinum
(Xe)4f 14 5d9 6s
8.9588
3
i
t
2S1/2
9.2255
79
Au Gold
(Xe)4f 14 5d10 6s
o
s
1S
80
Hg Mercury
(Xe)4f 14 5d10 6s2
10.4375
0
n
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2P
6.1082
81
Tl
Thallium
(Xe)4f 14 5d10 6s2 6p
1/2
3P
82
Pb Lead
(Xe)4f 14 5d10 6s2 6p2
7.4167
0
4S
7.2855
83
Bi
Bismuth
(Xe)4f 14 5d10 6s2 6p3
3/2
3P
84
Po Polonium
(Xe)4f 14 5d10 6s2 6p4
8.414
2
2P
85
At Astatine
(Xe)4f 14 5d10 6s2 6p5
3/2
1S
86
Rn Radon
(Xe)4f 14 5d10 6s2 6p6
10.7485
0
87
Fr
Francium
(Rn)
88
Ra Radium
(Rn)
- - - - - - - - - - - - - - - - - - 89
Ac Actinium
(Rn)
6d
90
Th Thorium
(Rn)
6d2
91
Pa Protactinium
(Rn)5f 2 6d
92
U
Uranium
(Rn)5f 3 6d
93
Np Neptunium
(Rn)5f 4 6d
94
Pu Plutonium
(Rn)5f 6
95
Am Americium
(Rn)5f 7
96
Cm Curium
(Rn)5f 7 6d
97
Bk Berkelium
(Rn)5f 9
98
Cf
Californium
(Rn)5f 10
99
Es
Einsteinium
(Rn)5f 11
100
Fm Fermium
(Rn)5f 12
101
Md Mendelevium
(Rn)5f 13
102
No Nobelium
(Rn)5f 14
103
Lr
Lawrencium
(Rn)5f 14
- - - - - - - - - - - - - - - - - - 104
Rf
Rutherfordium
(Rn)5f 14 6d2
2S
7s
4.0727
1/2
2
1
7s
S0
5.2784
- - - - - - - - - - - - - - - - - - - - 2
2
D3/2
5.17
7s
3F
7s2
6.3067
2
∗
4K
7s2
5.89
A
11/2
5L ∗
c
7s2
6.1941
6
∗
6L
t
7s2
6.2657
11/2
i
7F
7s2
6.0260
0
n
8S
7s2
5.9738
7/2
i
9D
7s2
5.9914
2
d
6H
7s2
6.1979
15/2
e
5I
7s2
6.2817
8
s
4I
7s2
6.42
15/2
3H
7s2
6.50
6
2F
7s2
6.58
7/2
1S
7s2
6.65
0
2P
7s2 7p?
4.9?
1/2 ?
- - - - - - - - - - - - - - - - - - - - 3F ?
7s2 ?
6.0?
2
∗ The usual LS coupling scheme does not apply for these three elements. See the introductory
note to the NIST table from which this table is taken.
109
110
6. Atomic and nuclear properties of materials
6. ATOMIC AND NUCLEAR PROPERTIES OF MATERIALS
Table 6.1 Abridged from pdg.lbl.gov/AtomicNuclearProperties by D. E. Groom (2007). See web pages for more detail about entries in
this table including chemical formulae, and for several hundred other entries. Quantities in parentheses are for NTP (20◦ C and 1 atm), and
square brackets indicate quantities evaluated at STP. Boiling points are at 1 atm. Refractive indices n are evaluated at the sodium D line blend
(589.2 nm); values 1 in brackets are for (n − 1) × 106 (gases).
Material
H2
D2
He
Li
Be
C diamond
C graphite
N2
O2
F2
Ne
Al
Si
Cl2
Ar
Ti
Fe
Cu
Ge
Sn
Xe
W
Pt
Au
Pb
U
Z
1
1
2
3
4
6
6
7
8
9
10
13
14
17
18
22
26
29
32
50
54
74
78
79
82
92
A
1.00794(7)
2.01410177803(8)
4.002602(2)
6.941(2)
9.012182(3)
12.0107(8)
12.0107(8)
14.0067(2)
15.9994(3)
18.9984032(5)
20.1797(6)
26.9815386(8)
28.0855(3)
35.453(2)
39.948(1)
47.867(1)
55.845(2)
63.546(3)
72.64(1)
118.710(7)
131.293(6)
183.84(1)
195.084(9)
196.966569(4)
207.2(1)
[238.02891(3)]
Z/A
Nucl.coll. Nucl.inter. Rad.len. dE/dx|min Density
X0
length λT length λI
{ MeV {g cm−3 }
{g cm−2 } {g cm−2 } {g cm−2 } g−1 cm2 } ({g−1 })
Melting
point
(K)
Boiling
point
(K)
Refract.
index
(@ Na D)
13.81
18.7
20.28
23.65
4.220
1615.
2744.
1.11[132.]
1.11[138.]
1.02[35.0]
0.99212
0.49650
0.49967
0.43221
0.44384
0.49955
0.49955
0.49976
0.50002
0.47372
0.49555
0.48181
0.49848
0.47951
0.45059
0.45961
0.46557
0.45636
0.44053
0.42119
0.41129
0.40252
0.39983
0.40108
0.39575
0.38651
42.8
51.3
51.8
52.2
55.3
59.2
59.2
61.1
61.3
65.0
65.7
69.7
70.2
73.8
75.7
78.8
81.7
84.2
86.9
98.2
100.8
110.4
112.2
112.5
114.1
118.6
52.0
71.8
71.0
71.3
77.8
85.8
85.8
89.7
90.2
97.4
99.0
107.2
108.4
115.7
119.7
126.2
132.1
137.3
143.0
166.7
172.1
191.9
195.7
196.3
199.6
209.0
63.04
125.97
94.32
82.78
65.19
42.70
42.70
37.99
34.24
32.93
28.93
24.01
21.82
19.28
19.55
16.16
13.84
12.86
12.25
8.82
8.48
6.76
6.54
6.46
6.37
6.00
(4.103)
(2.053)
(1.937)
1.639
1.595
1.725
1.742
(1.825)
(1.801)
(1.676)
(1.724)
1.615
1.664
(1.630)
(1.519)
1.477
1.451
1.403
1.370
1.263
(1.255)
1.145
1.128
1.134
1.122
1.081
0.071(0.084)
0.169(0.168)
0.125(0.166)
0.534
1.848
3.520
2.210
0.807(1.165)
1.141(1.332)
1.507(1.580)
1.204(0.839)
2.699
2.329
1.574(2.980)
1.396(1.662)
4.540
7.874
8.960
5.323
7.310
2.953(5.483)
19.300
21.450
19.320
11.350
18.950
Air (dry, 1 atm)
Shielding concrete
Borosilicate glass (Pyrex)
Lead glass
Standard rock
0.49919
0.50274
0.49707
0.42101
0.50000
61.3
65.1
64.6
95.9
66.8
90.1
97.5
96.5
158.0
101.3
36.62
26.57
28.17
7.87
26.54
(1.815)
1.711
1.696
1.255
1.688
(1.205)
2.300
2.230
6.220
2.650
78.80
Methane (CH4 )
Ethane (C2 H6 )
Propane (C3 H8 )
Butane (C4 H10 )
Octane (C8 H18 )
Paraffin (CH3 (CH2 )n≈23 CH3 )
Nylon (type 6, 6/6)
Polycarbonate (Lexan)
Polyethylene ([CH2 CH2 ]n )
Polyethylene terephthalate (Mylar)
Polyimide film (Kapton)
Polymethylmethacrylate (acrylic)
Polypropylene
Polystyrene ([C6 H5 CHCH2 ]n )
Polytetrafluoroethylene (Teflon)
Polyvinyltoluene
0.62334
0.59861
0.58962
0.59497
0.57778
0.57275
0.54790
0.52697
0.57034
0.52037
0.51264
0.53937
0.55998
0.53768
0.47992
0.54141
54.0
55.0
55.3
55.5
55.8
56.0
57.5
58.3
56.1
58.9
59.2
58.1
56.1
57.5
63.5
57.3
73.8
75.9
76.7
77.1
77.8
78.3
81.6
83.6
78.5
84.9
85.5
82.8
78.5
81.7
94.4
81.3
46.47
45.66
45.37
45.23
45.00
44.85
41.92
41.50
44.77
39.95
40.58
40.55
44.77
43.79
34.84
43.90
(2.417)
(0.667)
90.68
(2.304)
(1.263)
90.36
(2.262) 0.493(1.868) 85.52
(2.278)
(2.489)
134.9
2.123
0.703
214.4
2.088
0.930
1.973
1.18
1.886
1.20
2.079
0.89
1.848
1.40
1.820
1.42
1.929
1.19
2.041
0.90
1.936
1.06
1.671
2.20
1.956
1.03
111.7
184.5
231.0
272.6
398.8
Aluminum oxide (sapphire)
Barium flouride (BaF2 )
Bismuth germanate (BGO)
Carbon dioxide gas (CO2 )
Solid carbon dioxide (dry ice)
Cesium iodide (CsI)
Lithium fluoride (LiF)
Lithium hydride (LiH)
Lead tungstate (PbWO4 )
Silicon dioxide (SiO2 , fused quartz)
Sodium chloride (NaCl)
Sodium iodide (NaI)
Water (H2 O)
0.49038
0.42207
0.42065
0.49989
0.49989
0.41569
0.46262
0.50321
0.41315
0.49930
0.55509
0.42697
0.55509
65.5
90.8
96.2
60.7
60.7
100.6
61.0
50.8
100.6
65.2
71.2
93.1
58.5
98.4
149.0
159.1
88.9
88.9
171.5
88.7
68.1
168.3
97.8
110.1
154.6
83.3
27.94
9.91
7.97
36.20
36.20
8.39
39.26
79.62
7.39
27.05
21.91
9.49
36.08
1.647
1.303
1.251
1.819
1.787
1.243
1.614
1.897
1.229
1.699
1.847
1.305
1.992
3.970
4.893
7.130
(1.842)
1.563
4.510
2.635
0.820
8.300
2.200
2.170
3.667
1.000(0.756)
Silica aerogel
0.50093
65.0
97.3
27.25
1.740
0.200
453.6
1560.
2.42
63.15
54.36
53.53
24.56
933.5
1687.
171.6
83.81
1941.
1811.
1358.
1211.
505.1
161.4
3695.
2042.
1337.
600.6
1408.
2327.
1641.
1317.
Sublimes
894.2
1121.
965.
1403.
1986.
1075.
933.2
273.1
77.29
90.20
85.03
27.07
2792.
3538.
239.1
87.26
3560.
3134.
2835.
3106.
2875.
165.1
5828.
4098.
3129.
2022.
4404.
1.20[298.]
1.22[271.]
[195.]
1.09[67.1]
3.95
[773.]
1.23[281.]
1.39[701.]
[444.]
1.49
1.59
1.58
3273.
2533.
1.77
1.47
2.15
[449.]
at 194.7 K
1553.
1.79
1946.
1.39
3223.
1738.
1577.
373.1
(0.03 H2 O, 0.97 SiO2 )
2.20
1.46
1.54
1.77
1.33
6. Atomic and nuclear properties of materials
Material
Dielectric
constant (κ = /0 )
() is (κ–1)×106
for gas
Young’s
modulus
[106 psi]
Coeff. of
thermal
expansion
[10−6 cm/cm-◦ C]
Specific
heat
[cal/g-◦ C]
Electrical
Thermal
resistivity
conductivity
[μΩcm(@◦ C)] [cal/cm-◦ C-sec]
H2
He
Li
Be
(253.9)
(64)
—
—
—
—
—
37
—
—
56
12.4
—
—
0.86
0.436
—
—
8.55(0◦ )
5.885(0◦ )
—
—
0.17
0.38
C
N2
O2
Ne
Al
Si
Ar
Ti
—
(548.5)
(495)
(127)
—
11.9
(517)
—
0.7
—
—
—
10
16
—
16.8
0.6–4.3
—
—
—
23.9
2.8–7.3
—
8.5
0.165
—
—
—
0.215
0.162
—
0.126
1375(0◦ )
—
—
—
2.65(20◦ )
—
—
50(0◦ )
0.057
—
—
—
0.53
0.20
—
—
Fe
Cu
Ge
Sn
Xe
W
Pt
Pb
U
—
—
16.0
—
—
—
—
—
—
28.5
16
—
6
—
50
21
2.6
—
11.7
16.5
5.75
20
—
4.4
8.9
29.3
36.1
0.11
0.092
0.073
0.052
—
0.032
0.032
0.038
0.028
9.71(20◦ )
1.67(20◦ )
—
11.5(20◦ )
—
5.5(20◦ )
9.83(0◦ )
20.65(20◦ )
29(20◦ )
0.18
0.94
0.14
0.16
—
0.48
0.17
0.083
0.064
111
112
7. Electromagnetic relations
7. ELECTROMAGNETIC RELATIONS
Revised September 2005 by H.G. Spieler (LBNL).
Quantity
Conversion factors:
Charge:
Gaussian CGS
SI
2.997 924 58 × 109 esu
=1C=1As
Potential:
(1/299.792 458) statvolt (ergs/esu)
= 1 V = 1 J C−1
Magnetic field:
104 gauss = 104 dyne/esu
v
F = q (E + × B)
c
= 1 T = 1 N A−1 m−1
∇ D = 4πρ
4π
1 ∂D
=
J
∇×H−
c ∂t
c
∇ B=0
1 ∂B
=0
∇×E+
c ∂t
∇ D=ρ
.
D = E + 4πP,
Linear media:
D = E,
1
.
∇×H−
.
Constitutive relations:
F = q (E + v × B)
.
∂D
=J
∂t
∇ B=0
∂B
=0
∇×E+
∂t
H = B − 4πM
H = B/μ
D = 0 E + P,
H = B/μ0 − M
D = E, H = B/μ
0 = 8.854 187 . . . × 10−12 F m−1
μ0 = 4π × 10−7 N A−2
1
E = −∇V −
B=∇×A
1 ∂A
c ∂t
E = −∇V −
B=∇×A
qi
ρ (r ) 3 d x
V =
=
ri
|r − r |
charges
1
A=
c
1
I d
=
|r − r |
c
J(r ) 3 d x
|r − r |
V =
∂A
∂t
ρ (r ) 3 1 qi
1
d x
=
4π0
ri
4π0 |r − r |
charges
μ0
A=
4π
μ0
I d
=
|r − r |
4π
E = E
E = E
1
E⊥ = γ(E⊥ + v × B)
c
B = B
E⊥ = γ(E⊥ + v × B)
1
B⊥ = γ(B⊥ − v × E)
c
1
B⊥ = γ(B⊥ − 2 v × E)
c
J(r ) 3 d x
|r − r |
B = B
1
μ0
1
= 10−7 N A−2 ; c = √
= c2 × 10−7 N A−2 = 8.987 55 . . . × 109 m F−1 ;
= 2.997 924 58 × 108 m s−1
4π0
4π
μ0 0
7. Electromagnetic relations
7.1. Impedances (SI units)
7.3. Synchrotron radiation (CGS units)
For a particle of charge e, velocity v = βc, and energy E = γmc2 ,
traveling in a circular orbit of radius R, the classical energy loss per
revolution δE is
4π e2 3 4
δE =
β γ .
(7.10)
3 R
For high-energy electrons or positrons (β ≈ 1), this becomes
ρ = resistivity at room temperature in 10−8 Ω m:
∼ 1.7 for Cu
∼ 5.5 for W
∼ 2.4 for Au
∼ 73 for SS 304
∼ 2.8 for Al
∼ 100 for Nichrome
(Al alloys may have double the Al value.)
δE (in MeV) ≈ 0.0885 [E(in GeV)]4 /R(in m) .
For alternating currents, instantaneous current I, voltage V ,
angular frequency ω:
V = V0 ejωt = ZI .
(7.1)
Impedance of self-inductance L: Z = jωL .
Impedance of capacitance C: Z = 1/jωC .
Impedance of free space: Z = μ0 /0 = 376.7 Ω .
(1 + j) ρ
,
where δ = skin depth ;
δ
ρ
6.6 cm
≈ δ=
for Cu .
πνμ
ν (Hz)
(7.2)
(7.3)
(7.11)
For γ 1, the energy radiated per revolution into the photon energy
interval d(ω) is
8π
α γ F (ω/ωc ) d(ω) ,
(7.12)
dI =
9
where α = e2 /c is the fine-structure constant and
3γ 3 c
2R
is the critical frequency. The normalized function F (y) is
∞
9 √
F (y) =
3y
K5/3 (x) dx ,
8π
y
ωc =
High-frequency surface impedance of a good conductor:
Z=
113
(7.13)
(7.14)
where K5/3 (x) is a modified Bessel function of the third kind. For
electrons or positrons,
ωc (in keV) ≈ 2.22 [E(in GeV)]3 /R(in m) .
(7.15)
Fig. 7.1 shows F (y) over the important range of y.
7.2. Capacitors, inductors, and transmission Lines
The capacitance between two parallel plates of area A spaced by the
distance d and enclosing a medium with the dielectric constant ε is
0.6
C = KεA/d ,
0.5
(7.4)
For very short wires, representative of vias in a printed circuit board,
the inductance is
L(in nH) ≈ /d .
(7.6)
A transmission line is a pair of conductors with inductance
L and
capacitance C. The characteristic
impedance
Z
=
L/C
and
the
√
√
phase velocity vp = 1/ LC = 1/ με, which decreases with the
inverse square root of the dielectric constant of the medium. Typical
coaxial and ribbon cables have a propagation delay of about 5 ns/cm.
The impedance of a coaxial cable with outer diameter D and inner
diameter d is
1
D
Z = 60 Ω · √ ln ,
(7.7)
εr
d
where the relative dielectric constant εr = ε/ε0 . A pair of parallel
wires of diameter d and spacing a > 2.5 d has the impedance
1
2a
Z = 120 Ω · √ ln
.
εr
d
(7.8)
This yields the impedance of a wire at a spacing h above a ground
plane,
1
4h
Z = 60 Ω · √ ln
.
(7.9)
d
εr
A common configuration utilizes a thin rectangular conductor above
a ground plane with an intermediate dielectric (microstrip). Detailed
calculations for this and other transmission line configurations are
given by Gunston.*
* M.A.R. Gunston. Microwave Transmission Line Data, Noble Publishing Corp., Atlanta (1997) ISBN 1-884932-57-6, TK6565.T73G85.
0.4
F(y)
where the correction factor K depends on the extent of the fringing
field. If the dielectric fills the capacitor volume without extending
beyond the electrodes. the correction factor K ≈ 0.8 for capacitors of
typical geometry.
The inductance at high frequencies of a straight wire whose length is much greater than the wire diameter d is
nH
4
L ≈ 2.0
· ln
−1 .
(7.5)
cm
d
0.3
File [deg.pdg]synch.top
0.2
0.1
0.0
0.01
0.1
y
1.0
10
Figure 7.1: The normalized synchrotron radiation spectrum F (y).
For γ 1 and ω ωc ,
dI
≈ 3.3α (ωR/c)1/3 ,
(7.16)
d(ω)
whereas for
γ 1 and ω 3ωc ,
1/2
dI
3π
ω
55 ωc
≈
αγ
+ ... .
(7.17)
e−ω/ωc 1 +
d(ω)
2
ωc
72 ω
The radiation is confined to angles 1/γ relative to the instantaneous
direction of motion. For γ 1, where Eq. (7.12) applies, the mean
number of photons emitted per revolution is
5π
Nγ = √ αγ ,
(7.18)
3
and the mean energy per photon is
8
(7.19)
ω = √ ωc .
15 3
When ω O(E), quantum corrections are important.
See J.D. Jackson, Classical Electrodynamics, 3rd edition (John Wiley
& Sons, New York, 1998) for more formulae and details. (Note that
earlier editions had ωc twice as large as Eq. (7.13).
114
8. Naming scheme for hadrons
8. NAMING SCHEME FOR HADRONS
Revised 2004 by M. Roos (University of Finland) and C.G. Wohl
(LBNL).
8.1. Introduction
We introduced in the 1986 edition [1] a new naming scheme for the
hadrons. Changes from older terminology affected mainly the heavier
mesons made of the light (u, d, and s) quarks. Old and new names
were listed alongside until 1994. Names also change from edition to
edition because some characteristic like mass or spin changes. The
Summary Tables give both the new and old names whenever a change
occurred.
8.2. “Neutral-flavor” mesons (S = C = B = T = 0)
Table 8.1 shows the names for mesons having the strangeness
and all heavy-flavor quantum numbers equal to zero. The scheme is
designed for all ordinary non-exotic mesons, but it will work for many
exotic types too, if needed.
Table 8.1: Symbols for mesons with the strangeness and all
heavy-flavor quantum numbers equal to zero.
JPC
qq content
2S+1 L
J
=
⎧
⎪
⎨ 0−+
2−+
⎪
⎩ ...
0++
1++
..
.
= 1 (L even)J 1 (L odd)J 3 (L even)J 3 (L odd)J
ud, uu − dd, du (I = 1)
(I = 0)
dd + uu
and/or ss
cc
bb
tt
† The
1−−
2−−
..
.
1+−
3+−
..
.
π
b
ρ
a
η, η h, h
ω, φ
f, f ηc
ηb
ηt
hc
hb
ht
ψ†
Υ
θ
χc
χb
χt
J/ψ remains the J/ψ.
First, we assign names to those states with quantum numbers
compatible with being qq states. The rows of the Table give the
possible qq content. The columns give the possible parity/chargeconjugation states,
P C = −+, +−, −−, and ++ ;
these combinations correspond one-to-one with the angular-momentum
state 2S+1 LJ of the qq system being
1 (L
even)J ,
1 (L
odd)J ,
3 (L
even)J , or
3 (L
odd)J .
Here S, L, and J are the spin, orbital, and total angular momenta of
the qq system. The quantum numbers are related by P = (−1)L+1 ,
C = (−1)L+S , and G parity = (−1)L+S+I ,
where of course the C quantum number is only relevant to neutral
mesons.
The entries in the Table give the meson names. The spin J is added
as a subscript except for pseudoscalar and vector mesons, and the
mass is added in parentheses for mesons that decay strongly. However,
for the lightest meson resonances, we omit the mass.
Measurements of the mass, quark content (where relevant), and
quantum numbers I, J, P , and C (or G) of a meson thus fix its
symbol. Conversely, these properties may be inferred unambiguously
from the symbol.
If the main symbol cannot be assigned because the quantum
numbers are unknown, X is used. Sometimes it is not known whether
a meson is mainly the isospin-0 mix of uu and dd or is mainly ss.
A prime (or pair ω, φ) may be used to distinguish two such mixing
states.
We follow custom and use spectroscopic names such as Υ (1S) as the
primary name for most of those ψ, Υ , and χ states whose spectroscopic
identity is known. We use the form Υ (9460) as an alternative, and as
the primary name when the spectroscopic identity is not known.
Names are assigned for tt mesons, although the top quark is
evidently so heavy that it is expected to decay too rapidly for bound
states to form.
Gluonium states or other mesons that are not qq states are, if
the quantum numbers are not exotic, to be named just as are the
qq mesons. Such states will probably be difficult to distinguish from
qq states and will likely mix with them, and we make no attempt to
distinguish those “mostly gluonium” from those “mostly qq.”
An “exotic” meson with J P C quantum numbers that a qq
system cannot have, namely J P C = 0−− , 0+− , 1−+ , 2+− , 3−+ , · · · ,
would use the same symbol as does an ordinary meson with all
the same quantum numbers as the exotic meson except for the
C parity. But then the J subscript may still distinguish it; for
example, an isospin-0 1−+ meson could be denoted ω1 .
8.3. Mesons with nonzero S, C, B, and/or T
Since the strangeness or a heavy flavor of these mesons is nonzero,
none of them are eigenstates of charge conjugation, and in each of
them one of the quarks is heavier than the other. The rules are:
1. The main symbol is an upper-case italic letter indicating the
heavier quark as follows:
s→K
c→D
b→B
t→T .
We use the convention that the flavor and the charge of a quark
have the same sign. Thus the strangeness of the s quark is
negative, the charm of the c quark is positive, and the bottom
of the b quark is negative. In addition, I3 of the u and d
quarks are positive and negative, respectively. The effect of this
convention is as follows: Any flavor carried by a charged meson
has the same sign as its charge. Thus the K + , D+ , and B + have
positive strangeness, charm, and bottom, respectively, and all
have positive I3 . The Ds+ has positive charm and strangeness.
Furthermore, the Δ(flavor) = ΔQ rule, best known for the kaons,
applies to every flavor.
2. If the lighter quark is not a u or a d quark, its identity is given
by a subscript. The Ds+ is an example.
3. If the spin-parity is in the “normal” series, J P = 0+ , 1− , 2+ , · · ·,
a superscript “∗ ” is added.
4. The spin is added as a subscript except for pseudoscalar or vector
mesons.
8.4. Ordinary (3-quark) baryons
The symbols N , Δ, Λ, Σ, Ξ, and Ω used for more than 30 years
for the baryons made of light quarks (u, d, and s quarks) tell the
isospin and quark content, and the same information is conveyed by
the symbols used for the baryons containing one or more heavy quarks
(c and b quarks). The rules are:
1. Baryons with three u and/or d quarks are N ’s (isospin 1/2) or
Δ’s (isospin 3/2).
2. Baryons with two u and/or d quarks are Λ’s (isospin 0) or Σ’s
(isospin 1). If the third quark is a c, b, or t quark, its identity is
given by a subscript.
3. Baryons with one u or d quark are Ξ’s (isospin 1/2). One or two
subscripts are used if one or both of the remaining quarks are
heavy: thus Ξc , Ξcc , Ξb , etc.∗
4. Baryons with no u or d quarks are Ω’s (isospin 0), and subscripts
indicate any heavy-quark content.
5. A baryon that decays strongly has its mass as part of its name.
0
−
+
Thus p, Σ − , Ω − , Λ+
c , etc., but Δ(1232) , Σ(1385) , Ξc (2645) ,
etc.
In short, the number of u plus d quarks together with the isospin
determine the main symbol, and subscripts indicate any content of
heavy quarks. A Σ always has isospin 1, an Ω always has isospin 0,
etc.
8. Naming scheme for hadrons
115
8.5. Exotic baryons
Footnote and Reference:
In 2003, several experiments reported finding a strangeness S = +1,
charge Q = +1 baryon, and one experiment reported finding an
S = −2, Q = −2 baryon; see the “Exotic Baryons” section of the Data
Listings. Baryons with such quantum numbers cannot be made from
three quarks, and thus they are exotic. The S = +1 baryon, which
once would have been called a Z, was quickly dubbed the Θ(1540)+ ,
and we propose to name the S = −2 baryon the Φ(1860).
Sometimes a prime is necessary to distinguish two Ξc ’s in the
same SU(n) multiplet. See the “Note on Charmed Baryons” in
the Charmed Baryon Listings.
1. Particle Data Group: M. Aguilar-Benitez et al., Phys. Lett. 170B
(1986).
∗