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A Physical Constants and Units A Physical Constants and Units A.1 Physical Constants The following table contains the values in SI units for some physical constants that are particularly important in solid-state physics. Name Symbol Value speed of light in vacuum c 299 792 458 m s−1 magnetic constant −7 −2 (permeability of free space) μ0 4π × 10 NA electric constant 2 −12 −1 (permittivity of free space) 0 =1/μ0c 8.854 188 × 10 Fm elementary charge e 1.602 176 × 10−19 C Planck constant h 6.626 069 × 10−34 Js in eV h/{e} 4.135 667 × 10−15 eV s reduced Planck constant = h/2π 1.054 572 × 10−34 Js in eV /{e} 6.582 119 × 10−16 eV s 2 −3 fine-structure constant α = e /4π 0c 7.297 353 × 10 inverse of αα−1 137.035 999 −15 magnetic flux quantum Φ0 = h/2e 2.067 834 × 10 Wb 2 −5 conductance quantum G0 =2e /h 7.748 092 × 10 S −1 2 inverse of G0 G0 = h/2e 12 906.404 Ω 9 −1 Josephson constant KJ =2e/h 483 597.9 × 10 Hz V 2 von Klitzing constant RK = h/e 25 812.807 Ω −23 −1 Boltzmann constant kB 1.380 650 × 10 JK 23 −1 Avogadro constant NA 6.022 142 × 10 mol Continued on the next page 588 A Physical Constants and Units Name Symbol Value −1 −1 molar gas constant R = NAkB 8.314 472 J mol K −24 −1 Bohr magneton μB = e/2me 9.274 009 × 10 JT −27 −1 nuclear magneton μN = e/2mp 5.050 783 × 10 JT 2 2 −10 Bohr radius a0 =4π 0 /mee 0.529 177 × 10 m −31 electron mass me 9.109 382 × 10 kg −24 −1 electron magnetic μe −9.284 764 × 10 JT moment −1.001 160 μB electron g-factor ge =2μe/μB −2.002 319 11 −1 −1 electron gyromagnetic γe =2|μe|/ 1.760 860 × 10 s T −1 ratio γe/2π 28 024.9532 MHzT −27 neutron mass mn 1.674 927 × 10 kg −26 −1 neutron magnetic μn −0.966 236 × 10 JT moment −1.913 043 μN neutron g-factor gn =2μn/μN −3.826 085 −27 proton mass mp 1.672 622 × 10 kg proton–electron mass ratio mp/me 1836.152 667 electron–proton −4 mass ratio me/mp 5.446 170 10 −26 −1 proton magnetic μp 1.410 607 × 10 JT moment 2.792 847 μN proton g-factor gp =2μp/μN 5.585 695 8 −1 −1 proton gyromagnetic γp =2μp/ 2.675 222 × 10 s T −1 ratio γp/2π 42.577 481 MHzT 2 −1 Rydberg constant R∞ = α mec/2h 10 973 731.569 m −18 Rydberg energy Ry = R∞hc 2.179 872 × 10 J in electronvolts 13.605 692 eV 2 −18 Hartree energy Eh = e /4π 0a0 4.359 744 × 10 J in electronvolts 27.211 384 eV A.2 Relationships Among Units The fundamental units of the SI system are meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (Cd). In the next table derived units are expressed in terms of these fundamental ones – and other derived units. A.2 Relationships Among Units 589 1 coulomb (C) = 1 A s 1 pascal (Pa) = 1 N m−2 1 farad (F) = 1 C V−1 =1A2 s2 J−1 1 siemens (S) = 1 Ω−1 =1AV−1 1 henry (H) = 1 V s A−1 =1JA−2 1 tesla (T) = 1 Wb m−2 1 joule (J)=1Nm=1kgm2 s−2 1 volt (V) = 1 W A−1 1 newton (N)=1kgms−2 1 watt (W) = 1 J s−1 1 ohm (Ω) = 1 V A−1 1 weber (Wb) = 1 V s = 1 J A−1 It is immediately established that the unit of magnetic moment is 1Am2 = 1JT−1. The following SI prefixes are used to denote the multiples and fractions of the units: Name Symbol Corresponding power of 10 yotta- Y 1024 zetta- Z 1021 exa- E 1018 peta- P 1015 tera- T 1012 giga- G 109 mega- M 106 kilo- k 103 milli- m 10−3 micro- μ 10−6 nano- n 10−9 pico- p 10−12 femto- f 10−15 atto- a 10−18 zepto- z 10−21 yocto- y 10−24 Non-SI Units Much of the solid-state physics literature continues to use CGS units. Some of the most frequently used units are: 1 Å= 10−10 m =0.1 nm , 1 erg = 10−7 J , 1 atm = 101 325 Pa , 1 bar = 105 Pa . The CGS unit of magnetic field strength is the oersted (Oe), while that of magnetic induction (flux density) and magnetization is the gauss (G). In SI, magnetic induction is given in teslas, while magnetic field strength and magnetization in A/m. Therefore 1 G =10 −4 T 590 A Physical Constants and Units when specifying the magnetic induction, however 1 G =10 3 A/m when it comes to magnetization. The relationship between the units of mag- netic flux is 1 Gcm2 =10 −8 Wb . The units of magnetic field strength are related by 103 1 Oe = A/m =79.58 A/m . 4π When CGS units are used, equations for electromagnetic quantities are usu- ally written in their nonrationalized form. The table below shows the con- version factors used in the transformation of the rationalized equations into nonrationalized ones that apply to the Gaussian quantities (denoted by ∗); 2 c =1/ 0μ0. Physical quantity Conversion formula (CGS–SI) ∗ −1/2 electric charge q =(4π 0) q ∗ −1/2 electric current j =(4π 0) j ∗ 1/2 electric field E =(4π 0) E ∗ 1/2 electric displacement D =(4π/ 0) D ∗ 1/2 magnetic field H =(4πμ0) H ∗ 1/2 magnetic induction B =(4π/μ0) B ∗ 1/2 scalar potential ϕ =(4π 0) ϕ ∗ 1/2 vector potential A =(4π/μ0) A ∗ resistivity ! =4π 0! ∗ −1 conductivity σ =(4π 0) σ ∗ permittivity = / 0 ∗ permeability μ = μ/μ0 magnetic susceptibility χ∗ = χ/4π ∗ 1/2 magnetization M =(μ0/4π) M ∗ 1/2 magnetic flux Φ =(4π/μ0) Φ For example, the expression ωc = eB/me for cyclotron frequency goes over into ∗ ∗ √ ∗ ∗ e B ωc = 0μ0e B /me = . mec To convert electromagnetic SI units into CGS units, the next relationship (of mixed units) has to be used: / √ μ0 dyn =0.1 . 4π A The nonrationalized form and CGS value of some electromagnetic con- stants are listed below. A.2 Relationships Among Units 591 Name of physical constant Symbol Value 1 2 elementary charge e∗ 4.803 24 × 10−10 dyn / cm 2 ∗2 −9 Bohr radius a0 = /mee 5.291 772 × 10 cm ∗ ∗ × −21 −1 Bohr magneton μB = e /2mec 9.274 009 10 erg G 2 fine-structure constant α = e∗ /c 1/137.036 ∗ ∗ −7 2 flux quantum Φ0 = hc/2e 2.067 834 × 10 Gcm Conversion Factors of Energy Equivalents In solid-state physics energies are very often given in electronvolts: 1 eV = (e/C)J. Thermal energies are usually specified using the relation E = kBT , with the temperature given in kelvins, while magnetic field energies are com- monly converted to field strengths given in teslas or gausses through E = μBB. In spectroscopy energy is often expressed in hertz units via E = hν or in in- verse (centi)meters, via E = hc/λ. The energies corresponding to these units are −23 E(1 K) = (1 K)kB =1.380 650 × 10 J , −24 E(1 T) = (1 T)μB =9.274 009 × 10 J , E(1 cm−1)=(1cm−1)hc =1.986 445 × 10−23 J , E(1 Hz) = (1 Hz)h =6.626 069 × 10−34 J . The rydberg (1Ry = 1R∞hc) and hartree (1 hartree =2Ry) units are widely used, too. The conversion factors relating energies given in the previous units are listed in the following table. 1eV=1 .602 176 × 10−19 J1K=1 .380 650 × 10−23 J 1.160 451 × 104 K8.617 343 × 10−5 eV 1.727 599 × 104 T1.488 731 T 7.349 865 × 10−2 Ry 6.333 631 × 10−6 Ry 8.065 544 × 103 cm−1 6.950 356 × 10−1 cm−1 2.417 989 × 1014 Hz 2.083 664 × 1010 Hz 1T=9 .274 009 × 10−24 J1Ry=2 .179 872 × 10−18 J 5.788 382 × 10−5 eV 13.605 692 eV 0.671 713 K 1.578 873 × 105 K 4.254 383 × 10−6 Ry 2.350 517 × 105 T 4.668 645 × 10−1 cm−1 1.097 373 × 105 cm−1 1.399 625 × 1010 Hz 3.289 842 × 1015 Hz 592 A Physical Constants and Units 1cm−1 =1 .986 445 × 10−23 J1Hz=6 .626 069 × 10−34 J 1.239 842 × 10−4 eV 4.135 667 × 10−15 eV 1.438 775 K 4.799 237 × 10−11 K 2.141 949 T 7.144 773 × 10−11 T 9.112 670 × 10−6 Ry 3.039 660 × 10−16 Ry 2.997 925 × 1010 Hz 3.335 641 × 10−11 cm−1 Binding and cohesive energies in solids are commonly given per atom or per mole. In addition to eV, Ry, and J, older tables also contain data given in calories. Its usual value is 1 cal =4.1868 J, however that of the thermochemical calory is 1cal=4.184 J.Consequently eV mRy kJ kcal 1 =73.499 =96.4853 =23.05 . atom atom mol mol Reference 1.
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