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INTERNATIONAL SYSTEM OF UNITS (SI)

1 SIbaseunits

Table1givesthesevenbasequantities,assumedtobemutuallyindependent,onwhichtheSIis founded;andthenamesandsymbolsoftheirrespectiveunits,called``SIbaseunits.' ' Definitionsof theSIbaseunitsaregiveninAppendixA.ThekelvinanditssymbolKarealsousedtoexpressthe valueofatemperatureintervaloratemperaturedifference.

Table1.SIbaseunits

SIbaseunit

Basequantity Name Symbol meter m kg s electriccurrent A thermodynamictemperature K amountofsubstance mol luminousintensity cd

2 SIderiived units Derivedunitsareexpressedalgebraicallyintermsofbaseunitsorotherderivedunits(including theradianandsteradianwhicharethetwosupplementaryunits– seeSec.3).Thesymbolsfor derivedunitsareobtainedbymeansofthemathematicaloperationsofmultiplicationanddivision. Forexample,thederivedunitforthederivedquantitymolarmass(massdividedbyamountofsub- stance)isthekilogrampermole,symbolkg/mol.Additionalexamplesofderivedunitsexpressedin termsofSIbaseunitsaregiveninTable2.

Table2.ExamplesofSIderivedunitsexpressedintermsofSIbaseunits

SIderivedunit Derivedquantity Name Symbol squaremeter m2 cubicmeter m3 , meterpersecond m/s meterpersecondsquared m/s2 reciprocalmeter mϪ1 massdensity() kilogrampercubicmeter kg/m3 specificvolume cubicmeterperkilogram m3/kg currentdensity amperepersquaremeter A/m2 magneticfieldstrength amperepermeter A/m amount-of-substanceconcentration (concentration) molepercubicmeter mol/m3 candelapersquaremeter cd/m2

2.1 SIde rived units withspecial namesandsymbols

CertainSIderivedunitshavespecialnamesandsymbols;thesearegiveninTables3aand3b. AsdiscussedinSec.3,theradianandsteradian,whicharethetwosupplementaryunits,are includedinTable3a. INTERNATIONAL SYSTEM OF UNITS (SI) (continued)

Table3a.SIderivedunitswithspecialnamesandsymbols,includingtheradianandsteradian

SIderivedunit

Expression Expression Derivedquantity Specialname Specialsymbol interms interms ofother ofSIbase SIunits units planeangle mиmϪ1 =1 solidangle sr m2 иmϪ2 =1 Hz sϪ1 N mиkgиsϪ2 ,stress Pa N/m2 mϪ1 иkgиsϪ2 ,, ofheat J Nиmm2 иkgиsϪ2 ,radiantflux W J/s m2 иkgиsϪ3 electriccharge, quantityofelectricity C sиA electricpotential, potentialdifference, electromotiveforce V W/A m2 иkgиsϪ3 иAϪ1 capacitance F C/V mϪ2 иkgϪ1 иs4 иA2 electricresistance ⍀ V/A m2 иkgиsϪ3 иAϪ2 electricconductance S A/V mϪ2 иkgϪ1 иs3 иA2 magneticflux Wb Vиsm2 иkgиsϪ2 иAϪ1 magneticfluxdensity T Wb/m2 kgиsϪ2 иAϪ1 H Wb/A m2 иkgиsϪ2 иAϪ2 Celsiustemperature(a) degreeCelsius ЊCK luminousflux lm cdиsr cdиsr(b) lx lm/m2 mϪ2 иcdиsr(b)

(a) SeeSec.2.1.1. (b) Thesteradian(sr)isnotanSIbaseunit.However,inphotometrythesteradian(sr)ismaintainedinexpressions forunits(seeSec.3).

Table3b.SIderivedunitswithspecialnamesandsymbolsadmittedforreasonsofsafeguardinghumanhealth(a)

SIderivedunit Derivedquantity Special Special Expressioninterms Expressioninterms name symbol ofotherSIunits ofSIbaseunits activity(ofa radionuclide) Bq sϪ1 absorbeddose, specificenergy (imparted),kerma Gy J/kg m2 иsϪ2 doseequivalent,ambientdose equivalent,directionaldose equivalent,personaldose equivalent,equivalentdose Sv J/kg m2 иsϪ2

(a) Thederivedquantitiestobeexpressedinthegrayandthesieverthavebeenrevisedinaccordancewiththe recommendationsoftheInternationalCommissiononRadiationUnitsandMeasurements(ICRU).

2.1.1DegreeCelsiusInadditiontothequantitythermodynamictemperature(symbolT), expressedintheunitkelvin,useisalsomadeofthequantityCelsiustemperature (symbolt)definedbytheequation

t=TϪT0 , whereT0 =273.15Kbydefinition.ToexpressCelsiustemperature,theunitdegreeCelsius,symbol ЊC,whichisequalinmagnitudetotheunitkelvin,isused;inthiscase,``degreeCelsius' ' isaspecial nameusedinplaceof``kelvin.' ' AnintervalordifferenceofCelsiustemperaturecan,however,be expressed in the unit kelvin as well as in the unit . (Note that the thermodynamic temperatureT0 is exactly 0.01 K below the thermodynamic of the of .) INTERNATIONAL SYSTEM OF UNITS (SI) (continued)

2.2 UseofSIderivedunitswithspecialnamesandsymbols ExamplesofSIderivedunitsthatcanbeexpressedwiththeaidofSIderivedunitshaving specialnamesandsymbols(includingtheradianandsteradian)aregiveninTable4.

Table4.ExamplesofSIderivedunitsexpressedwiththeaidofSIderivedunitshavingspecialnamesandsymbols

SIderivedunit

Expression Derivedquantity Name Symbol intermsof SIbaseunits angularvelocity radianpersecond rad/s mиmϪ1 иsϪ1 =sϪ1 angularacceleration radianpersecondsquared rad/s2 mиmϪ1 иsϪ2 =sϪ2 dynamicviscosity pascalsecond PaиsmϪ1 иkgиsϪ1 momentofforce newtonmeter Nиmm2 иkgиsϪ2 surfacetension newtonpermeter N/m kgиsϪ2 heatfluxdensity, wattpersquaremeter W/m2 kgиsϪ3 radiantintensity wattpersteradian W/sr m2 иkgиsϪ3 иsrϪ1(a) wattpersquare metersteradian W/(m2 иsr) kgиsϪ3 иsrϪ1(a) heatcapacity, jouleperkelvin J/K m2 иkgиsϪ2 иKϪ1 specificheatcapacity, jouleperkilogram specificentropy kelvin J/(kgиK) m2 иsϪ2 иKϪ1 specificenergy jouleperkilogram J/kg m2 иsϪ2 thermalconductivity wattpermeterkelvin W/(mиK) mиkgиsϪ3 иKϪ1 energydensity joulepercubicmeter J/m3 mϪ1 иkgиsϪ2 electricfieldstrength voltpermeter V/m mиkgиsϪ3 иAϪ1 electricchargedensity coulombpercubicmeter C/m3 mϪ3 иsиA electricfluxdensity coulombpersquaremeter C/m2 mϪ2 иsиA faradpermeter F/m mϪ3 иkgϪ1 иs4 иA2 permeability henrypermeter H/m mиkgиsϪ2 иAϪ2 molarenergy joulepermole J/mol m2 иkgиsϪ2 иmolϪ1 molarentropy,molar heatcapacity joulepermolekelvin J/(molиK) m2 иkgиsϪ2 иKϪ1 иmolϪ1 (xand␥rays) coulombperkilogram C/kg kgϪ1 иsиA absorbeddoserate graypersecond Gy/s m2 иsϪ3

(a) Thesteradian(sr)isnotanSIbaseunit.However,inradiometrythesteradian(sr)ismaintainedinexpressions forunits(seeSec.3).

TheadvantagesofusingthespecialnamesandsymbolsofSIderivedunitsareapparentinTable 4.Consider,forexample,thequantitymolarentropy:theunitJ/(molиK)isobviouslymoreeasily understoodthanitsSIbase-unitequivalent,m2 иkgиsϪ2 иKϪ1 иmolϪ1.Nevertheless,itshould alwaysberecognizedthatthespecialnamesandsymbolsexistforconvenience;eithertheformin whichspecialnamesorsymbolsareusedforcertaincombinationsofunitsortheforminwhichthey arenotusediscorrect.Forexample,becauseofthedescriptivevalueimplicitinthecompound-unit form,communicationissometimesfacilitatedifmagneticflux(seeTable3a)isexpressedinterms ofthevoltsecond(Vиs)insteadoftheweber(Wb). Tables3a,3b,and4alsoshowthatthevaluesofseveraldifferentquantitiesareexpressedinthe sameSIunit.Forexample,thejouleperkelvin(J/K)istheSIunitforheatcapacityaswellasfor entropy.Thusthenameoftheunitisnotsufficienttodefinethequantitymeasured. Aderivedunitcanoftenbeexpressedinseveraldifferentwaysthroughtheuseofbaseunitsand derivedunitswithspecialnames.Inpractice,withcertainquantities,preferenceisgiventousing certainunitswithspecialnames,orcombinationsofunits,tofacilitatethedistinctionbetweenquan- titieswhosevalueshaveidenticalexpressionsintermsofSIbaseunits.Forexample,theSIunitof frequencyisspecifiedasthehertz(Hz)ratherthanthereciprocalsecond(sϪ1),andtheSIunitof momentofforceisspecifiedasthenewtonmeter(Nиm)ratherthanthejoule(J). INTERNATIONAL SYSTEM OF UNITS (SI) (continued)

Similarly,inthefieldofionizingradiation,theSIunitofactivityisdesignatedasthebecquerel (Bq)ratherthanthereciprocalsecond(sϪ1),andtheSIunitsofabsorbeddoseanddoseequivalent aredesignatedasthegray(Gy)andthesievert(Sv),respectively,ratherthanthejouleperkilogram (J/kg).

3 SIsupplementaryunits Aspreviouslystated,therearetwounitsinthisclass:theradian,symbolrad,theSIunitofthe quantityplaneangle;andthesteradian,symbolsr,theSIunitofthequantitysolidangle.Definitions oftheseunitsaregiveninAppendixA. The SI supplementary units are now interpreted as so-called dimensionless derived units for which the CGPM allows the freedom of using or not using them in expressions for SI derived units.3 Thus the radian and steradian are not given in a separate table but have been included in Table 3a together with other derived units with special names and symbols (seeSec.2.1). This interpretation of the supplementary units implies that plane and are considered derived of one (so-called dimensionless quantities), each of which has the whichhastheunitone,symbol1,asitscoherentSIunit.However,inpractice,whenoneexpresses thevaluesofderivedquantitiesinvolvingplaneangleorsolidangle,itoftenaidsunderstandingifthe specialnames(orsymbols)``radian' ' (rad)or``steradian' ' (sr)areusedinplaceofthenumber1.For example,althoughvaluesofthederivedquantityangularvelocity(planeangledividedbytime)may beexpressedintheunitsϪ1,suchvaluesareusuallyexpressedintheunitrad/s. Becausetheradianandsteradianarenowviewedasso-calleddimensionlessderivedunits,the ConsultativeCommitteeforUnits(CCU, Comité Consultatif des Unités) of the CIPM as result of a 1993 request it received from ISO/TC12, recommended to the CIPM that it request the CGPM to abolish the class of supplementary units as a separate class in the SI. The CIPM accepted the CCU recommendation, and if the abolishment is approved by the CGPM as is likely (the question will be on the agenda of the 20th CGPM, October 1995), the SI will consist of only two classes of units: base units and derived units, with the radian and steradian subsumed into the class of derived units of the SI. (The option of using or not using them in expressions for SI derived units, as is convenient,wouldremainunchanged.)

4 DecimalmultiplesandsubmultiplesofSIunits:SIprefixes Table 5 gives the SI prefixes that are used to form decimal multiples and submultiples of SI units. They allow very large or very small numerical values to be avoided. A prefix attaches directly to the name of a unit, and a prefix symbol attaches directly to the symbol for a unit. Forexample,onekilometer,symbol1km,isequaltoonethousandmeters,symbol1000mor103 m. WhenprefixesareattachedtoSIunits,theunitssoformedarecalled``multiplesandsubmultiples ofSIunits' ' inordertodistinguishthemfromthecoherentsystemofSIunits.

Note: AlternativedefinitionsoftheSIprefixesandtheirsymbolsarenotpermitted.Forexample, itisunacceptabletousekilo(k)torepresent210 =1024,mega(M)torepresent 220 =1048576,orgiga(G)torepresent230 =1073741824.

3 Thisinterpretationwasgivenin1980bytheCIPM.Itwasdeemednecessary becauseResolution12ofthe11thCGPM,whichestablishedtheSIin1960,didnotspecifythenatureofthesupplemen- taryunits.Theinterpretationisbasedontwoprincipalconsiderations:thatplaneangleisgenerallyexpressedastheratioof twolengthsandsolidangleastheratioofanareaandthesquareofalength,andarethusquantitiesofdimensionone(so-called dimensionlessquantities);andthattreatingtheradianandsteradianasSIbaseunits– apossibilitynotdisallowedbyReso- lution12– couldcompromisetheinternalcoherenceoftheSIbasedononlysevenbaseunits.(SeeISO31-0 foradiscussionoftheconceptofdimension.) INTERNATIONAL SYSTEM OF UNITS (SI) (continued)

Table5.SIprefixes

Factor Prefix Symbol Factor Prefix Symbol

1024 =(103)8 yotta Y 10Ϫ1 deci d 1021 =(103)7 zetta Z 10Ϫ2 centi c 1018 =(103)6 exa E 10Ϫ3 =(103)Ϫ1 milli m 1015 =(103)5 peta P 10Ϫ6 =(103)Ϫ2 micro ␮ 1012 =(103)4 tera T 10Ϫ9 =(103)Ϫ3 nano n 109 =(103)3 giga G 10Ϫ12 =(103)Ϫ4 pico p 106 =(103)2 mega M 10Ϫ15 =(103)Ϫ5 femto f 103 =(103)1 kilo k 10Ϫ18 =(103)Ϫ6 atto a 102 hecto h 10Ϫ21 =(103)Ϫ7 zepto z 101 deka da 10Ϫ24 =(103)Ϫ8 yocto y

5 UnitsOutsidetheSI UnitsthatareoutsidetheSImaybedividedintothreecategories: – thoseunitsthatareacceptedforusewiththeSI; – thoseunitsthataretemporarilyacceptedforusewiththeSI;and – thoseunitsthatarenotacceptedforusewiththeSIandthus muststrictlybeavoided.

5.1 UnitsacceptedforusewiththeSI Thefollowingsectionsdiscussindetailtheunitsthat are acceptable forusewiththeSI.

5.1.1 ,degree,liter,andthelike CertainunitsthatarenotpartoftheSIareessentialandusedsowidelythattheyareaccepted by the CIPM for use with the SI. These units are given in Table 6. The combination of units of this table with SI units to form derived units should be restricted to special cases in order not to lose theadvantagesofthecoherenceofSIunits.

Additionally, it is recognized that it may be necessary on occasion to use time-related units other than those given in Table 6; in particular, circumstances may require that intervals of time beexpressedinweeks,months,oryears.Insuchcases,ifastandardizedsymbolfortheunitisnot available,thenameoftheunitshouldbewrittenoutinfull.

Table6.UnitsacceptedforusewiththeSI

Name Symbol ValueinSIunits min 1min=60s hour · time h 1h =60min=3600s d 1d =24h=86400s degree Њ 1Њ =(␲/180)rad minute · planeangle ' 1' =(1/60)Њ=(␲/10800)rad second " 1" =(1/60)' =(␲/648000)rad liter l,L(b) 1L =1dm3 =10Ϫ3 m3 metricton(c) t1t=103 kg

(b) Thealternativesymbolfortheliter,L,wasadoptedbytheCGPMinordertoavoidtheriskofconfusionbetweentheletter landthenumber1.Thus,althoughbothlandLareinternationallyacceptedsymbolsfortheliter,to avoid this risk the symbol tobeusedintheUnitedStatesisL.Thescriptletterᐉisnotanapproved symbol for the liter.

(c) This is the name to be used for this unit in the United States; it is also used in some other English-speaking countries. However,``' ' isusedinmanycountries. INTERNATIONAL SYSTEM OF UNITS (SI) (continued)

5.1.2 ,bel,shannon,andthelike ThereareafewhighlyspecializedunitsnotlistedinTable6thataregivenbythe InternationalOrganizationforStandardization(ISO)ortheInternationalElectrotechnicalCommis- sion (IEC) and which are also acceptable for use with the SI. They include the neper (Np), bel (B), octave, phon, and sone, and units used in information technology, including the baud (Bd), bit (bit), erlang(E),hartley(Hart),andshannon(Sh).4 ItisthepositionofNIST that the only such additional units that may be used with the SI are those given in either the International Standards on quantities and unitsofISOorofIEC.

5.1.3 Electronvoltandunifiedatomicmassunit

The CIPM also finds it necessary to accept for use with the SI the two units given in Table 7. These units are used in specialized fields; their values in SI units must be obtained from experiment and,therefore,arenotknownexactly.

Note: Insomefieldstheunifiedatomicmassunitiscalledthedalton,symbolDa;however,this nameandsymbolarenotacceptedbytheCGPM,CIPM,ISO,orIECforusewiththeSI. Similarly,AMUisnotanacceptableunitsymbolfortheunifiedatomicmassunit.Theonly allowednameis``unifiedatomicmassunit' ' andtheonlyallowedsymbolisu.

Table7.UnitsacceptedforusewiththeSIwhosevaluesinSIunitsareobtainedexperimentally

Name Symbol Definition eV (a) unifiedatomicmassunit u (b)

(a) Theelectronvoltisthekineticenergyacquiredbyanelectroninpassingthroughapotentialdifferenceof1Vinvacuum; 1eV=1.60217733ϫ10Ϫ19 Jwithacombinedstandarduncertaintyof0.00000049ϫ10Ϫ19 J. (b) Theunifiedatomicmassunitisequalto1/12ofthemassofanatomofthenuclide 12C;1u=1.6605402ϫ 10Ϫ27 kgwithacombinedstandarduncertaintyof0.0000010ϫ10Ϫ27 kg.

5.1.4 Naturalandatomicunits In some cases, particularly in basic , the values of quantities are expressed in terms of fundamental constants of nature or so-called .The use of these units with the SI is permissible when it is necessary for the most effective communication of information. In such cases, the specific natural units that are used must be identified. This requirement applies even to the system of units customarily called ``atomicunits' ' used in theoretical atomic physics and , inasmuch as there are several different systems that have the appellation ``atomic units.'' ExamplesofphysicalquantitiesusedasnaturalunitsaregiveninTable8. NIST alsotakesthepositionthatwhiletheoreticalresultsintendedprimarilyforothertheorists may be left in natural units, if they are also intended for experimentalists, they must also be giveninacceptableunits.

4 Thesymbolinparenthesesfollowingthenameoftheunitisitsinternationallyacceptedunitsymbol,buttheoctave,phon, andsonehavenosuchunitsymbols.Foradditionalinformationontheneperandbel,seeSec.0.5ofISO31-2. Thequestionofthebyte(B)isunderinternationalconsideration. INTERNATIONAL SYSTEM OF UNITS (SI) (continued)

Table8.Examplesofphysicalquantitiessometimesusedasnaturalunits

Kindofquantity Physicalquantityusedasaunit Symbol Planckconstantdividedby2␲h electriccharge elementarycharge e

energy Hartreeenergy Eh length Bohrradius a0 length Comptonwavelength() ␭C magneticflux magneticfluxquantum ⌽0 magneticmoment Bohrmagneton ␮B magneticmoment nuclearmagneton ␮N mass electronrestmass me mass protonrestmass mp speed speedofelectromagneticwavesinvacuum c

5.2 UnitstemporarilyacceptedforusewiththeSI

Becauseofexistingpracticeincertainfieldsorcountries,in1978theCIPMconsideredthatit was permissible for the units given in Table 9 to continue to be used with the SI until the CIPM considers that their use is no longer necessary. However, these units must not be introduced where they are not presently used. Further, NIST strongly discourages the continued use of these units except for the nautical , knot, are, and ; and except for the , , rad, and rem until the 2000 (the cessation date suggested by the Committee for Ineragency ResearchandPolicyCoordinationorCIRRPC,aUnitedStatesGovernment interagencygroup).5

Table9.UnitstemporarilyacceptedforusewiththeSI (a)

Name Symbol ValueinSIunits nauticalmile 1nauticalmile=1852m knot 1nauticalmileperhour=(1852/3600)m/s ångström Å 1Å=0.1nm=10Ϫ10m are(b) a 1a=1dam2 =102 m2 hectare(b) ha 1ha=1hm2 =104 m2 barn b 1b=100fm2 =10Ϫ28 m2 bar bar 1bar=0.1MPa=100kPa=1000hPa=105 Pa Gal 1Gal=1cm/s2 =10Ϫ2 m/s2 curie Ci 1Ci=3.7ϫ1010 Bq roentgen 1R=2.58ϫ10Ϫ4 C/kg rad rad(c) 1rad=1cGy=10Ϫ2 Gy rem rem 1rem=1cSv=10Ϫ2 Sv

(a) SeeSec.5.2regardingthecontinueduseoftheseunits. (b) Thisunitanditssymbolareusedtoexpressagrarianareas. (c) Whenthereisriskofconfusionwiththesymbolfortheradian,rdmaybeusedasthesymbolforrad.

5 In1993theCCU(seeSec.3)wasrequestedbyISO/TC12toconsideraskingtheCIPMtodeprecate theuseoftheunitsofTable9exceptforthenauticalmileandknot,andpossiblytheareandhectare.TheCCUdiscussedthis requestatitsFebruary1995meeting. INTERNATIONAL SYSTEM OF UNITS (SI) (continued)

AppendixA. DefinitionsoftheSIBaseUnitsandtheRadianandSteradian A.1 Introduction ThefollowingdefinitionsoftheSIbaseunitsaretakenfromNIST SP 330;thedefinitionsoftheSI supplementaryunits,theradianandsteradian,whicharenowinterpretedasSIderivedunits(see Sec.3),arethosegenerallyacceptedandarethesameasthosegiveninANSI/IEEE Std 268-1992. SIderivedunitsareuniquelydefinedonlyintermsofSIbaseunits;forexample, 1V=1m2 иkgиsϪ3 иAϪ1. A.2 Meter(17thCGPM,1983) Themeteristhelengthofthepathtravelledbylightinvacuumduringatimeintervalof 1/299792458ofasecond.

A.3 Kilogram(3dCGPM,1901) Thekilogramistheunitofmass;itisequaltothemassoftheinternationalprototypeofthe kilogram.

A.4 Second(13thCGPM,1967) Thesecondisthedurationof9192631770periodsoftheradiationcorrespondingtothe transitionbetweenthetwohyperfinelevelsofthegroundstateofthecesium-133atom.

A.5 Ampere(9thCGPM,1948) Theampereisthatconstantcurrentwhich,ifmaintainedintwostraightparallelconductorsof infinitelength,ofnegligiblecircularcrosssection,andplaced1meterapartinvacuum,would producebetweentheseconductorsaforceequalto2ϫ10Ϫ7 newtonpermeteroflength.

A.6 Kelvin(13thCGPM,1967) Thekelvin,unitofthermodynamictemperature,isthefraction1/273.16ofthethermodynamic temperatureofthetriplepointofwater.

A.7 Mole(14thCGPM,1971) 1. Themoleistheamountofsubstanceofasystemwhichcontainsasmanyelementaryentities asthereareatomsin0.012kilogramofcarbon12. 2. Whenthemoleisused,theelementaryentitiesmustbespecifiedandmaybeatoms,molecules, ions,,otherparticles,orspecifiedgroupsofsuchparticles. Inthedefinitionofthemole,itisunderstoodthatunboundatomsofcarbon12,atrestandintheir groundstate,arereferredto. Notethatthisdefinitionspecifiesatthesametimethenatureofthequantitywhoseunitisthe mole.

A.8 Candela(16thCGPM,1979) Thecandelaistheluminousintensity,inagivendirection,ofasourcethatemitsmonochromatic radiationoffrequency540ϫ1012 hertzandthathasaradiantintensityinthatdirectionof(1/683) wattpersteradian.

A.9 Radian Theradianistheplaneanglebetweentworadiiofacirclethatcutoffonthecircumferencean arcequalinlengthtotheradius.

A.10 Steradian Thesteradianisthesolidanglethat,havingitsvertexinthecenterofasphere,cutsoffanarea ofthesurfaceofthesphereequaltothatofasquarewithsidesoflengthequaltotheradiusofthe .