Mathematics, Symbols, and Physical Constants

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1 Mathematics, Symbols, and Physical Constants Greek Alphabet International System of Units (SI) The International System of units (SI) was adopted by the 11th General Conference on Weights and Measures (CGPM) in It is a coherent system of units built form seven SI base units, one for each of the seven dimensionally independent base quantities: they are the meter, kilogram, second, ampere, kelvin, mole, and candela, for the dimensions length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity, respectively. The definitions of the SI base units are given below. The SI derived units are expressed as products of powers of the base units, analogous to the corresponding relations between physical quantities but with numerical factors equal to unity. In the International System there is only one SI unit for each physical quantity. This is either the appropriate SI base unit itself or the appropriate SI derived unit. However, any of the approved decimal prefixes, called SI prefixes, may be used to construct decimal multiples or submultiples of SI units. It is recommended that only SI units be used in science and technology (with SI prefixes where appropriate). Where there are special reasons for making an exception to this rule, it is recommended always to define the units used in terms of SI units. This section is based on information supplied by IUPAC. Definitions of SI Base Units Greek Greek English Greek Greek English letter name equivalent letter name equivalent A a Alpha a N n Nu n B b Beta b X x Xi x G g Gamma g O o Omicron ŏ D d Delta d P p Pi p E e Epsilon ĕ R r Rho r Z z Zeta z S s Sigma s H h Eta e T t Tau t Q q J Theta th U u Upsilon u I i Iota i F f j Phi ph K k Kappa k C c Chi ch L l Lambda l Y y Psi ps M m Mu m W w Omega o Meter The meter is the length of path traveled by light in vacuum during a time interval of 1/ of a second (17th CGPM, 1983). Kilogram The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram (3rd CGPM, 1901). Second The second is the duration of periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom (13th CGPM, 1967). Ampere The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to newton per meter of length (9th CGPM, 1948).

2 Kelvin The kelvin, unit of thermodynamic temperature, is the fraction 1/ of the thermodynamic temperature of the triple point of water (13th CGPM, 1967). Mole The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in kilogram of carbon-12. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, or other particles, or specified groups of such particles (14th CGPM, 1971). Examples of the use of the mole: 1 mol of H 2 contains about H 2 molecules, or H atoms 1 mol of HgCl has a mass of g 1 mol of Hg 2 Cl 2 has a mass of g 2+ 1 mol of Hg 2 has a mass of g and a charge of kc 1 mol of Fe 0.91 S has a mass of g 1 mol of e has a mass of mg and a charge of kc 1 mol of photons whose frequency is Hz has energy of about kj Candela The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency hertz and that has a radiant intensity in that direction of (1/683) watt per steradian (16th CGPM, 1979). Names and Symbols for the SI Base Units Physical quantity Name of SI unit Symbol for SI unit length meter m mass kilogram kg time second s electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd SI Derived Units with Special Names and Symbols Name of Symbol for Expression in Physical quantity SI unit SI unit terms of SI base units frequency 1 hertz Hz s 1 force newton N m kg s 2 pressure, stress pascal Pa N m 2 =m 1 kg s 2 energy, work, heat joule J N m = m 2 kg s 2 power, radiant flux watt W J s 1 = m 2 kg s 3 electric charge coulomb C A s electric potential, volt V J C 1 = m 2 kg s 3 A 1 electromotive force electric resistance ohm W V A 1 = m 2 kg s 3 A 2 electric conductance siemens S W 1 = m 2 kg 1 s 3 A 2 electric capacitance farad F C V 1 = m 2 kg 1 s 4 A 2 magnetic flux density tesla T V s m 2 = kg s 2 A 1 magnetic flux weber Wb V s = m 2 kg s 2 A 1 inductance henry H V A 1 s = m 2 kg s 2 A 2 Celsius temperature 2 degree Celsius C K luminous flux lumen lm cd sr illuminance lux lx cd sr m 2 activity (radioactive) becquerel Bq s 1 absorbed dose (of radiation) gray Gy J kg 1 = m 2 s 2 dose equivalent sievert Sv J kg 1 = m 2 s 2 (dose equivalent index)

3 Name of Symbol for Expression in Physical quantity SI unit SI unit terms of SI base units plane angle radian rad 1 = m m 1 solid angle steradian sr 1 = m 2 m 2 1 For radial (circular) frequency and for angular velocity the unit rad s 1, or simply s 1, should be used, and this may not be simplified to Hz. The unit Hz should be used only for frequency in the sense of cycles per second. 2 The Celsius temperature q is defined by the equation: q/ C = T/K The SI unit of Celsius temperature interval is the degree Celsius, C, which is equal to the kelvin, K. C should be treated as a single symbol, with no space between the sign and the letter C. (The symbol K, and the symbol, should no longer be used.) Units in Use Together with the SI These units are not part of the SI, but it is recognized that they will continue to be used in appropriate contexts. SI prefixes may be attached to some of these units, such as milliliter, ml; millibar, mbar; megaelectronvolt, MeV; kilotonne, ktonne. Physical Symbol quantity Name of unit for unit Value in SI units time minute min 60 s time hour h 3600 s time day d s plane angle degree (p/180) rad plane angle minute (p/10 800) rad plane angle second ² (p/ ) rad length ångstrom 1 Å m area barn b m 2 volume litre l, L dm 3 = 10 3 m 3 mass tonne t Mg = 10 3 kg pressure bar 1 bar 10 5 Pa = 10 5 N m 2 energy electronvolt 2 ev (= e V)» J mass unified atomic mass unit 2,3 u (=m a ( 12 C)/12)» kg 1 The ångstrom and the bar are approved by CIPM for temporary use with SI units, until CIPM makes a further recommendation. However, they should not be introduced where they are not used at present. 2 The values of these units in terms of the corresponding SI units are not exact, since they depend on the values of the physical constants e (for the electronvolt) and N a (for the unified atomic mass unit), which are determined by experiment. 3 The unified atomic mass unit is also sometimes called the dalton, with symbol Da, although the name and symbol have not been approved by CGPM. Conversion Constants and Multipliers Recommended Decimal Multiples and Submultiples Multiples and Multiples and submultiples Prefixes Symbols submultiples Prefixes Symbols exa E 10 1 deci d peta P 10 2 centi c tera T 10 3 milli m 10 9 giga G 10 6 micro m (Greek mu)

4 Multiples and Multiples and submultiples Prefixes Symbols submultiples Prefixes Symbols 10 6 mega M 10 9 nano n 10 3 kilo k pico p 10 2 hecto h femto f 10 deca da atto a Conversion Factors Metric to English To obtain Multiply By Inches Centimeters Feet Meters Yards Meters Miles Kilometers Ounces Grams Pounds Kilogram Gallons (U.S. Liquid) Liters Fluid ounces Milliliters (cc) Square inches Square centimeters Square feet Square meters Square yards Square meters Cubic inches Milliliters (cc) Cubic feet Cubic meters Cubic yards Cubic meters Conversion Factors English to Metric* Conversion Factors General* To obtain Multiply By Microns Mils 25.4 Centimeters Inches 2.54 Meters Feet Meters Yards Kilometers Miles Grams Ounces Kilograms Pounds Liters Gallons (U.S. Liquid) Millimeters (cc) Fluid ounces Square centimeters Square inches Square meters Square feet Square meters Square yards Milliliters (cc) Cubic inches Cubic meters Cubic feet Cubic meters Cubic yards To obtain Multiply By Atmospheres Feet of 4 C Atmospheres Inches of 0 C Atmospheres Pounds per square inch BTU Foot-pounds BTU Joules Cubic feet Cords 128 Degree (angle) Radians Ergs Foot-pounds *Boldface numbers are exact; others are given to ten significant figures where so indicated by the multiplier factor.

5 To obtain Multiply By Feet Miles 5280 Feet of 4 C Atmospheres Foot-pounds Horsepower-hours Foot-pounds Kilowatt-hours Foot-pounds per min Horsepower Horsepower Foot-pounds per sec Inches of 0 C Pounds per square inch Joules BTU Joules Foot-pounds Kilowatts BTU per min Kilowatts Foot-pounds per min Kilowatts Horsepower Knots Miles per hour Miles Feet Nautical miles Miles Radians Degrees Square feet Acres Watts BTU per min Temperature Factors F = 9/5 ( C) + 32 Fahrenheit temperature = 1.8 (temperature in kelvins) C = 5/9 [( F) 32)] Celsius temperature = temperature in kelvins Fahrenheit temperature = 1.8 (Celsius temperature) + 32 Conversion of Temperatures From To Celsius Fahrenheit t F = (t C 1.8) + 32 Kelvin T K = t C Rankine T R = (t C ) 18 Fahrenheit Celsius t t C = F Kelvin t T k = F Rankine T R = t F Kelvin Celsius t C = T K Rankine T R = T K 1.8 Rankine Kelvin T T K = R 1.8 Farenheit t F = T R Physical Constants General Equatorial radius of the earth = km = miles (statute). Polar radius of the earth, km = miles (statute). 1 degree of latitude at 40 = 69 miles. 1 international nautical mile = miles (statute) = 1852 m = ft. Mean density of the earth = g/cm 3 = lb/ft 3 Constant of gravitation (6.673 ± 0.003) 10 8 cm 3 gm 1 s 2.

6 Acceleration due to gravity at sea level, latitude 45 = cm/s 2 = ft/s 2. Length of seconds pendulum at sea level, latitude 45 = cm = in. 1 knot (international) = ft/min = ft/s = miles (statute)/h. 1 micron = 10 4 cm. 1 ångstrom = 10 8 cm. Mass of hydrogen atom = ( ± ) g. Density of mercury at 0 C = g/ml. Density of water at 3.98 C = g/ml. Density, maximum, of water, at 3.98 C = g/cm 3. Density of dry air at 0 C, 760 mm = g/l. Velocity of sound in dry air at 0 C = m/s ft/s. Velocity of light in vacuum = ( ± ) cm/s. Heat of fusion of water 0 C = cal/g. Heat of vaporization of water 100 C = cal/g. Electrochemical equivalent of silver g/s international amp. Absolute wavelength of red cadmium light in air at 15 C, 760 mm pressure = Å. Wavelength of orange-red line of krypton 86 = Å. p Constants p = /p = p 2 = log e p = log 10 p = log 10 2p = Constants Involving e e = /e = e 2 = M = log 10 e = /M =log e 10 = log 10 M = Numerical Constants 2 = = log e 2 = log 10 2 = = = log e 3 = log 10 3 = Symbols and Terminology for Physical and Chemical Quantities Name Symbol Definition SI unit Classical Mechanics mass m kg reduced mass m m = m 1 m 2 /(m 1 + m 2 ) kg density, mass density r r = M/V kg m 3 relative density d d = r/r q l surface density r A, r S r A = m/a kg m 2

7 Symbols and Terminology for Physical and Chemical Quantities (continued) Name Symbol Definition SI unit Classical Mechanics (continued) momentum p p = mv kg m s 1 angular momentum, action L l = r p J s moment of inertia I, J I = Sm i r 2 i kg m 2 force F F = dp/dt = ma N torque, moment of a force T, (M) T = r F N m energy E J potential energy E p, V, F E p = òf ds J kinetic energy E k, T, K e k = (1/2)mv 2 J work W, w w = òf ds J Hamilton function H H(q, p) J = T(q, p) + V(q) Lagrange function L L(q, q) J T(q, q) V(q) pressure p, P p = F/A Pa, N m 2 surface tension g, s g = dw/da N m 1, J m 2 weight G, (W, P) G = mg N gravitational constant G F= Gm 1 m 2 /r 2 N m 2 kg 2 normal stress s s = F/A Pa shear stress t t = F/A Pa linear strain, e, e e = Dl/l l relative elongation modulus of elasticity, E E = s/e Pa Young s modulus shear strain g g = Dx/d l shear modulus G G = t/g Pa volume strain, bulk strain q q = DV/V 0 l bulk modulus, K K = V 0 (dp/dv) Pa compression modulus h, m t x,z = h(dv x /dz) Pa s viscosity, dynamic viscosity fluidity f f = 1/h m kg 1 s kinematic viscosity n n = h/r m 2 s 1 friction coefficient m, ( f ) F frict = mf norm l power P P = dw/dt W sound energy flux P, P a P = de/dt W acoustic factors reflection factor r r = P t /P 0 l acoustic absorption factor a a, (a) a a = 1 r l transmission factor t t = P tr /P 0 l dissipation factor d d = a a t l Electricity and Magnetism quantity of electricity, electric charge Q C charge density r r = Q/V C m 3 surface charge density s s = Q/A C m 2 electric potential V, f V = dw/dq V, J C 1 electric potential difference U, DV, Df U = V 2 V 1 V electromotive force E E = ò(f/q) ds V electric field strength E E = F/Q = grad V V m 1 electric flux Y Y = òd da C electric displacement D D = ee C m 2 capacitance C C = Q/U F, C V 1 permittivity e D = ee F m 1 permittivity of vacuum e e 0 = m 0 c 0 F m 1 relative permittivity e r e r = e/e 0 l dielectric polarization P P = D e 0 E C m 2

8 Symbols and Terminology for Physical and Chemical Quantities (continued) Name Symbol Definition SI unit Electricity and Magnetism (continued) (dipole moment per volume) electric susceptibility c e c e = e r 1 l electric dipole moment p, m p = Qr C m electric current I I = dq/dt A electric current density j, J I = òj da A m 2 magnetic flux density, magnetic B F = Qv B T induction magnetic flux F F = òb da Wb magnetic field strength H B = mh A M 1 permeability m B = mh N A 2, H m 1 permeability of vacuum m 0 H m 1 relative permeability m r m r = m/m 0 l magnetization (magnetic dipole M M = B/m 0 H A m 1 moment per volume) magnetic susceptibility c, k, (c m ) c = m r 1 l molar magnetic susceptibility c m c m = V m c m 3 mol 1 magnetic dipole moment m, m E p = m B A m 2, J T 1 electrical resistance R R = U/I W conductance G G = 1/R S loss angle d d = (p/2) + f I f U 1, rad reactance X X = (U/I)sin d W impedance (complex impedance) Z Z = R + ix W admittance (complex admittance) Y Y = 1/Z S susceptance B Y = G + ib S resistivity r r = E/j W m conductivity k, g, s k = 1/r S m 1 self-inductance L E = L(dI/dt) H mutual inductance M, L 12 E 1 = L 12 (Di 2 /dt) H magnetic vector potential A B = Ñ A Wb m 1 Poynting vector S S = E H W m 2 Electromagnetic Radiation wavelength l m speed of light in vacuum c 0 m s 1 in a medium c c = c 0 /n wavenumber in vacuum n ~ n ~ =n/c 0 = 1/nl m 1 wavenumber (in a medium) s s = 1/l m 1 frequency n n = c/l Hz circular frequency, pulsatance w w = 2pn s 1, rad s 1 refractive index n n = c 0 /c l Planck constant h J s Planck constant/2p = h/2p J s radiant energy Q, W J radiant energy density r, w r = Q/V J m 3 spectral radiant energy density in terms of frequency r n, w n r n = dr /dn J m 3 Hz 1 in terms of wavenumber r, n w n r n = dr/dn J m 2 in terms of wavelength r l, w l r l = dr /dl J m 4 Einstein transition probabilities spontaneous emission A nm dn n /dt = A nm N n s 1 stimulated emission B nm dn n /dt = r n (n nm) B nm N n s kg 1 radiant power, F, P F = dq/dt W radiant energy per time radiant intensity I I = df/dw W sr 1 radiant exitance (emitted radiant flux) M M = df/da source W m 2

9 Symbols and Terminology for Physical and Chemical Quantities (continued) Name Symbol Definition SI unit Electromagnetic Radiation (continued) irradiance (radiant flux received) E, (I) E = df/da W m 2 emittance e e = M/M bb l Stefan-Boltzmann constant s M bb = st 4 W m 2 K 4 first radiation constant c 1 2 c 1 = 2phc 0 W m 2 second radiation constant c 2 c 2 = hc 0 /k K m transmittance, transmission factor t, T t = F tr /F 0 l absorptance, absorption factor a a = f abs /f 0 l reflectance, reflection factor r r = f refl /F 0 l (decadic) absorbance A A = lg(1 a i ) l napierian absorbance B B = ln(1 a i ) l absorption coefficient (linear) decadic a, K a = A/l m 1 (linear) napierian a a = B/l m 1 molar (decadic) e e = a/c = A/cl m 2 mol 1 molar napierian k k = a/c = B/cl m 2 mol 1 absorption index k k = a/4pn l complex refractive index ˆn ˆn = n + ik l molar refraction R, R m ( n R = 1 ) V m m 3 mol 1 angle of optical rotation a ( n ) l, rad Solid State lattice vector R, R 0 m fundamental translation vectors for the a 1 ; a 2 ; a 3, R = n 1 a 1 + n 2 a 2 + n 3 a 3 m crystal lattice a; b; c (circular) reciprocal lattice vector G G R = 2pm m 1 (circular) fundamental translation b 1 ; b 2 ; b 3, a i b k = 2pd ik m 1 vectors for the reciprocal lattice a*; b*; c* lattice plane spacing d m Bragg angle q nl = 2d sin q l, rad order of reflection n l order parameters short range s l long range s 1 Burgers vector b m particle position vector r, R j m equilibrium position vector of an ion R o m displacement vector of an ion u u = R R 0 m Debye-Waller factor B, D l Debye circular wavenumber q D m 1 Debye circular frequency w D s 1 Grüneisen parameter g, G g = av/kc V l Madelung constant a, M a N A z + z e E 2 coul = 4pe 0 R 0 l density of states N E N E = dn(e)/de J 1 m 3 (spectral) density of vibrational modes N w, g N w = dn(w)/dw s m 3 resistivity tensor r ik E = r j W m conductivity tensor s ik s = r 1 S m 1 thermal conductivity tensor l ik J q = l grad T W m 1 K 1 residual resistivity r R W m relaxation time t t = l/v F s Lorenz coefficient L L = l/st V 2 K 2 Hall coefficient A H, R H E = r j + R H (B j) m 3 C 1 thermoelectric force E V Peltier coefficient P V Thomson coefficient m, (t ) V K 1

10 Symbols and Terminology for Physical and Chemical Quantities (continued) Name Symbol Definition SI unit Solid State (continued) work function F F = E E F J number density, number concentration n, (p) m 3 gap energy E g J donor ionization energy E d J acceptor ionization energy E a J Fermi energy E F, e F J circular wave vector, k, q k = 2p/l m 1 propagation vector Bloch function u k (r) y(r) = u k (r) exp(ik r) m 3/2 charge density of electrons r r(r) = ey*(r)y ( r) C m 3 effective mass m* kg mobility m m = n drift /E m 2 V 1 s 1 mobility ratio b b = m n /m p l diffusion coefficient D dn/dt = DA(dn/dx) m 2 s 1 diffusion length L L = D t m characteristic (Weiss) temperature f, f W K Curie temperature T C K Néel temperature T N K Credits Material in Section XII was reprinted from the following sources: D. R. Lide, Ed., CRC Handbook of Chemistry and Physics, 76th ed., Boca Raton, Fla.: CRC Press, 1992: International System of Units (SI), conversion constants and multipliers (conversion of temperatures), symbols and terminology for physical and chemical quantities, fundamental physical constants, classification of electromagnetic radiation. D. Zwillinger, Ed., CRC Standard Mathematical Tables and Formulae, 30th ed., Boca Raton, Fla.: CRC Press, 1996: Greek alphabet, conversion constants and multipliers (recommended decimal multiples and submultiples, metric to English, English to metric, general, temperature factors), physical constants, series expansion.