Sp.-V/AQuan/1999/10/08:06:45 Page 7 Chapter 2 General Constants and Units Arthur N. Cox 2.1 Mathematical Constants ................. 7 2.2 Physical Constants .................... 8 2.3 General Astronomical Constants ............ 12 2.4 Astronomical Constants Involving Time ....... 13 2.5 Units ............................. 17 2.6 Electric and Magnetic Unit Relations .......... 22 2.1 MATHEMATICAL CONSTANTS [1–3] Constant Number Log π 3.141 592 653 6 0.497 149 872 7 2π 6.283 185 307 2 0.798 179 868 4 4π 12.566 370 614 4 1.099 209 864 0 2 √π 9.869 604 401 1 0.994 299 745 4 π 1.772 453 850 9 0.248 574 936 3 eore 2.718 281 828 5 0.434 294 481 9 mod = M = log e 0.434 294 481 9 0.637 784 311 3 − 1 1/M = ln 10 2.302 585 093 0 0.362 215 688 7 √2 2.000 000 000 0 0.301 029 995 7 √2 1.414 213 562 4 0.150 514 997 8 √ 3 1.732 050 807 6 0.238 560 627 4 10 3.162 277 660 2 0.500 000 000 0 7 Sp.-V/AQuan/1999/10/08:06:45 Page 8 8/2 GENERAL CONSTANTS AND UNITS Constant Number Log ln π 1.144 729 885 8 0.058 703 021 2 eπ 23.140 692 632 8 1.364 376 353 8 Euler constant γ 0.577 215 664 9 0.761 338 108 8 − 1 1 radian rad = 57.◦295 779 513 1 1.758 122 632 4 = 3 437.746 770 78 3.536 273 882 8 = 206 264.806 25 5.314 425 133 2 1◦ = 0.rad017 453 292 5 0.241 877 367 6 − 2 1 = 0.rad000 290 888 2 0.463 726 117 2 − 4 1 = 0.rad000 004 848 1 0.685 574 866 8 − 6 Square degrees on a sphere = 129 600/π = 41 252.961 25. Square degrees in a steradian = 32 400/π 2 = 32 82.806 35. 1 x2 for Gaussian distribution √ exp − . σ 2π 2σ 2 Probable error/Standard error = r/σ = 0.674 489 750 2. Probable error/Average error = r/η = 0.845 347 539 4. σ/η =√1.253 314 137. ρ = (r/σ )/ 2 = 0.476 936 276 2. 2.2 PHYSICAL CONSTANTS [4, 5] These fundamental physical constants, mostly in SI units from [5], are the latest available. A revision by Cohen and Taylor is expected by the end of 1998. For many values, the standard error of the last digits follows in parentheses. In the formulations the electron charge e is in esu and e in emu = e/c. Fundamental constants Speed of light (exact) c = 2.997 924 58 × 108 ms−1 c2 = 8.987 551 79 × 1016 m2 s−2 Gravitation constant G = 6.672 59(85) × 10−11 m3 kg−1 s−2 −1 Standard acceleration of gravity (exact) gn = 9.806 65 m s Planck constant 2π = h = 6.626 075 5(40) × 10−34 Js = 1.054 572 66(63) × 10−34 Js Planck mass (c/G)1/2 = 2.176 71(14) × 10−8 kg Planck length (G/c3)1/2 = 1.616 05(10) × 10−35 m Planck time (G/c5)1/2 = 5.390 56(34) × 10−44 s Elementary charge e = 4.803 206 8(15) × 10−19 C e = 1.602 177 33(49) × 10−20 emu e2 = 23.070 796 × 10−20 in esu e4 = 5.322 616 1 × 10−38 in esu −31 Mass of electron me = 9.109 389 7(54) × 10 kg = 5.485 799 03(13) × 10−4 u Sp.-V/AQuan/1999/10/08:06:45 Page 9 2.2 PHYSICAL CONSTANTS /9 Mass of unit atomic weight u = 1.660 540 2(10) × 10−27 kg (12C = 12 scale) Boltzmann constant k = 1.380 658(12) × 10−23 JK−1 = 8.617 385(73) × 10−5 eV K−1 k1/2 = 1.175 014 × 10−8 erg1/2 K−1/2 Gas constant (12C scale) R = 8.314 510(70) J K−1 mol−1 = 1.987 216 cal K−1 mol−1 = 82.057 83(70) cm3 atm K−1 mol−1 Joule equivalent (chemical, exact) [4] = 4.184 J cal−1 23 −1 Avogadro constant NA = 6.022 136 7(36) × 10 mol 25 −3 Loschmidt constant n0 = 2.686 763(23) × 10 m −3 Volume of gram-molecule at STP NA/n0 = V0 = 22.414 10(19) × 10 (T = 273.15 K, P = 101 325 Pa) m3 mol−1 −2 Standard atmosphere pressure (exact) P0 = 1 013 250 dyn cm = 760 mmHg Ice point 0◦ C = 273.150 K Triple point (H2O) = 273.160 K −1 Faraday NAe/c = 96 485.309(29) C mol Atomic constants 1 −1 Rydberg constant for H RH = 10 967 758.306(13) m 1/RH = 911.763 345 0 A˚ − Rydberg constant for infinite nuclear mass R∞ = 10 973 731.534(13) m 1 2 4 3 2π mee /ch 1/R∞ = 911.267 053 4 A˚ − cR∞ = 3.289 841 950 × 1015 s 1 Fine structure constant α = 7.297 353 08(33) × 10−3 2πe2/hc 1/α = 137.035 989 5(61) α2 = 5.325 136 20 × 10−5 −10 Radius for first Bohr orbit a0 = 0.529 177 249(24) × 10 m 2 2 2 (infinite nuclear mass) h /4π mee −1 −17 Time for (2π) revolutions in first τ0 = 2.418 884 4 × 10 s 1/2 3/2 −1 3 3 4 Bohr orbit me a e = h /8π mee Frequency of first Bohr orbit = 6.579 683 7 × 1015 s−1 π 2 = . × −21 2 Area of first Bohr orbit a0 8 797 356 70 10 m τ −1 = . × 6 −1 Electron speed in first Bohr orbit a0 0 2 187 691 4 10 ms Atomic unit of energy = 4.359 748 2(26) × 10−18 J 2 (Hartree = 2 Rydbergs) e /a0 = 2chR∞ = 27.211 396 1(81) eV Energy of Rydberg ryd = 2.179 874 1(13) × 10−18 J (often adopted as atomic unit) = 13.605 698 1(40) eV Atomic unit of angular momentum h/2π = 1.054 572 66(63) × 10−34 kg m2s−1 2 2 −15 Classical electron radius e /m0c l = 2.817 940 92(38) × 10 m 2 −2 27 −1 −2 Schrodinger¨ constant for fixed nucleus 8π meh = 1.638 197 48 × 10 erg cm Schrodinger¨ constant for 1H atom = 1.637 305 78 × 1027 erg−1 cm−2 Sp.-V/AQuan/1999/10/08:06:45 Page 10 10 / 2 GENERAL CONSTANTS AND UNITS 1 6 −1 Hyperfine structure splitting of H νH = 1 420.405 751 768 × 10 s ground state Doublet separation in 1H atom = 0.365 866 231 cm−1 2 2 2 10 −1 (1/16)RHα [1 + α/π + (5/8 − 5.946/π )α ] = 1.096 839 36×10 s 1 −31 Reduced mass of electron in H atom me(mp/mH) = 9.104 431 3 × 10 kg Mass of 1H atom = 1.673 534 4 × 10−27 kg = 1.007 825 050(12) u Mass of proton = 1.672 623 1(10) × 10−27 kg = 1.007 276 470(12) u Mass of neutron = 1.674 928 6(10) × 10−27 kg = 1.008 664 904(14) u Mass of deuteron = 3.343 586 0(20) × 10−27 kg = 2.013 553 214(24) u Mass energy of unit atomic mass uc2 = 1.492 419 1 × 10−10 J = 931.494 2(28) MeV 2 −14 Rest mass energy of electron mec = 8.187 111 1 × 10 J = 0.510 999 06(15) MeV Mass ratio proton/electron = 1 836.152 701(37) 7 −1 Specific electron charge e/me = 1.758 819 62 × 10 emu g 17 −1 e/me = 5.272 808 6 × 10 esu g Quantum of magnetic flux h/e = 1.379 510 77 × 10−17 erg s esu−1 hc/e = 4.135 669 2 × 10−7 gauss cm2 −1 Quantum of circulation h/me = 7.273 896 2 erg s g −12 Compton wavelength h/mec = 2.426 310 58(22) × 10 m −13 h/2πmec = 3.861 593 23(35) × 10 m Band spectrum constant h/8π 2c = 27.992 774 × 10−40 gcm (moment of inertia/wave number) Atomic specific heat constant = 4.799 216 × 10−11 sK c2/c = h/k −21 Magnetic moment of 1 Bohr magneton µB = 9.274 015 4(31) × 10 µ 1 α 1/2 5/2τ −1 = / π −1 B= 2 me a0 0 he 4 mec erg gauss Electron magnetic moment µe = 1.001 159 652 193(10)µB −3 Proton magnetic moment µp = 1.521 032 202(15) × 10 µB 4 Gyromagnetic ratio of proton γp = 2.675 221 28(81) × 10 −1 −1 corrected for diamagnetism of H2O rad s gauss −24 Magnetic moment of 1 nuclear magneton µn = 5.050 786 6(17) × 10 −1 he/4πmpc erg gauss Atomic unit of magnetic moment = 2.541 747 8 × 10−18 erg gauss−1 2µB/α Magnetic moment per mole of 1 Bohr = 5 584.938 8 erg gauss−1 mol−1 magneton per molecule Zeeman displacement = 4.668 643 7(14) × 10−5 −1 −1 3/4πmec (e in emu) cm gauss in frequency = 1.399 624 18(42) × 106 s−1 gauss−1 Sp.-V/AQuan/1999/10/08:06:45 Page 11 2.2 PHYSICAL CONSTANTS /11 The electron–volt and photons [5] −10 Wavelength associated with 1 eV λ0 = 12 398.428 2 × 10 m −1 Wave number associated with 1 eV s0 = 8 065.538 51 cm = 8.065 538 51 kilo–kayser 14 −1 Frequency associated with 1 eV ν0 = 2.417 988 36(72) × 10 s −19 Energy of 1 eV E0 = 1.602 177 33(49) × 10 J = 0.073 498 617 6 ryd Photon energy associated with unit hc = 1.986 448 0 × 10−23 J wave number Photon energy associated with = 1.986 448 0 × 10−8/λ erg (λ in A)˚ wavelength λ Speed of 1 eV electron = 5.930 968 92 × 105 ms−1 8 1/2 (2 × 10 (e/mec)) Speed2 = 3.517 639 23 × 1011 m2 s−2 Wavelength of electron of energy V in eV = V −1/2(12.264 263 × 10−8) cm −1/2 −1/2 h(2me E0) V Temperature associated with 1 eV = 11 604.45 K E0/k Temperature associated with 1 eV = 5039.75 K in common logs = (E0/k) log e Temperature associated with 1 kilo-kayser = 624.849 3 K in common logs = 103(hc/k) log e Energy of 1 eV per molecule = 23 060.054 2 cal mol−1 Radiation constants Radiation density constant a = 7.565 91(25) × 10−15 8π 5k4/15c3h3 erg cm−3 K−4 Stefan–Boltzmann constant = ac/4 σ = 5.670 51(19) × 10−5 erg cm−2 K−4 s−1 −5 First radiation constant c1 = 3.741 774 9(22) × 10 (emittance) = 2πhc2 erg cm2 s−1 First radiation constant (radiation density) 8πhc = 4.992 487 0 × 10−15 erg cm Second radiation constant = hc/kc2 = 1.438 769(12) cm K Wien displacement law constant = 0.289 775 5 c2/4.965 114 23 Some general constants [1, 5] Density of mercury (0◦ C, 760 mmHg) = 13.395080 g cm−3 Ratio, grating to Siegbahn scale of X-ray wavelengths [5] λg/λs = 1.002 077 89(70) [λs (Cu Kα1) = 1.537 400 kXu] Lattice spacing of Si (in vacuum, 22.5◦ C) = 0.543 101 96(11) × 10−9 m Molar volume of Si = 12.058 817 9(89) cm3 mol−1 Maximum density of water = 0.999 972 g cm−3 Cesium resonance frequency = 9 192 631 770 Hz (defining the SI second) [6] Sp.-V/AQuan/1999/10/08:06:45 Page 12 12 / 2 GENERAL CONSTANTS AND UNITS 2.3 GENERAL ASTRONOMICAL CONSTANTS by Alan D.