Matter:AVeryShortIntroduction

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ABOLITIONISM RichardS.NewmanACCOUNTING ChristopherNobesADAMSMITH ChristopherJ.BerryADOLESCENCE PeterK.SmithADVERTISING WinstonFletcherAFRICANAMERICANRELIGION EddieS.GlaudeJrAFRICANHISTORY JohnParkerandRichardRathboneAFRICANPOLITICS IanTaylorAFRICANRELIGIONS JacobK.OluponaAGEING NancyA.PachanaAGNOSTICISM RobinLePoidevinAGRICULTURE PaulBrassleyandRichardSoffeALEXANDERTHEGREAT HughBowdenALGEBRA PeterM.HigginsAMERICANCULTURALHISTORY EricAvilaAMERICANHISTORY PaulS.BoyerAMERICANIMMIGRATION DavidA.GerberAMERICANLEGALHISTORY G.EdwardWhiteAMERICANNAVALHISTORY CraigL.SymondsAMERICANPOLITICALHISTORY DonaldCritchlowAMERICANPOLITICALPARTIESANDELECTIONS L.SandyMaiselAMERICANPOLITICS RichardM.ValellyTHEAMERICANPRESIDENCY CharlesO.JonesTHEAMERICANREVOLUTION RobertJ.AllisonAMERICANSLAVERY HeatherAndreaWilliamsTHEAMERICANWEST StephenAronAMERICANWOMEN’SHISTORY SusanWareANAESTHESIA AidanO’DonnellANALYTICPHILOSOPHY MichaelBeaneyANARCHISM ColinWardANCIENTASSYRIA KarenRadnerANCIENTEGYPT IanShaw

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DRUIDS BarryCunliffeEARLYMUSIC ThomasForrestKellyTHEEARTH MartinRedfernEARTHSYSTEMSCIENCE TimLentonECONOMICS ParthaDasguptaEDUCATION GaryThomasEGYPTIANMYTH GeraldinePinchEIGHTEENTH-CENTURYBRITAIN PaulLangfordTHEELEMENTS PhilipBallEMOTION DylanEvansEMPIRE StephenHoweENGELS TerrellCarverENGINEERING DavidBlockleyTHEENGLISHLANGUAGE SimonHorobinENGLISHLITERATURE JonathanBateTHEENLIGHTENMENT JohnRobertsonENTREPRENEURSHIP PaulWestheadandMikeWrightENVIRONMENTALECONOMICS StephenSmithENVIRONMENTALETHICS RobinAttfieldENVIRONMENTALLAW ElizabethFisherENVIRONMENTALPOLITICS AndrewDobsonEPICUREANISM CatherineWilsonEPIDEMIOLOGY RodolfoSaracciETHICS SimonBlackburnETHNOMUSICOLOGY TimothyRiceTHEETRUSCANS ChristopherSmithEUGENICS PhilippaLevineTHEEUROPEANUNION SimonUsherwoodandJohnPinderEUROPEANUNIONLAW AnthonyArnullEVOLUTION BrianandDeborahCharlesworthEXISTENTIALISM ThomasFlynnEXPLORATION StewartA.WeaverTHEEYE MichaelLandFAIRYTALE MarinaWarnerFAMILYLAW JonathanHerringFASCISM KevinPassmoreFASHION RebeccaArnoldFEMINISM MargaretWaltersFILM MichaelWoodFILMMUSIC KathrynKalinakTHEFIRSTWORLDWAR MichaelHowardFOLKMUSIC MarkSlobinFILMNOIR JamesNaremoreFOOD JohnKrebsFORENSICPSYCHOLOGY DavidCanterFORENSICSCIENCE JimFraserFORESTS JabouryGhazoulFOSSILS KeithThomsonFOUCAULT GaryGutting

THEFOUNDINGFATHERS R.B.BernsteinFRACTALS KennethFalconerFREESPEECH NigelWarburtonFREEWILL ThomasPinkFREEMASONRY AndreasÖnnerforsFRENCHLITERATURE JohnD.LyonsTHEFRENCHREVOLUTION WilliamDoyleFREUD AnthonyStorrFUNDAMENTALISM MaliseRuthvenFUNGI NicholasP.MoneyTHEFUTURE JenniferM.GidleyGALAXIES JohnGribbinGALILEO StillmanDrakeGAMETHEORY KenBinmoreGANDHI BhikhuParekhGARDENHISTORY GordonCampbellGENES JonathanSlackGENIUS AndrewRobinsonGENOMICS JohnArchibaldGEOGRAPHY JohnMatthewsandDavidHerbertGEOLOGY JanZalasiewiczGEOPHYSICS WilliamLowrieGEOPOLITICS KlausDoddsGERMANLITERATURE NicholasBoyleGERMANPHILOSOPHY AndrewBowieGLACIATION DavidJ.A.EvansGLOBALCATASTROPHES BillMcGuireGLOBALECONOMICHISTORY RobertC.AllenGLOBALIZATION ManfredStegerGOD JohnBowkerGOETHE RitchieRobertsonTHEGOTHIC NickGroomGOVERNANCE MarkBevirGRAVITY TimothyCliftonTHEGREATDEPRESSIONANDTHENEWDEAL EricRauchwayHABERMAS JamesGordonFinlaysonTHEHABSBURGEMPIRE MartynRadyHAPPINESS DanielM.HaybronTHEHARLEMRENAISSANCE CherylA.WallTHEHEBREWBIBLEASLITERATURE TodLinafeltHEGEL PeterSingerHEIDEGGER MichaelInwoodTHEHELLENISTICAGE PeterThonemannHEREDITY JohnWallerHERMENEUTICS JensZimmermannHERODOTUS JenniferT.RobertsHIEROGLYPHS PenelopeWilsonHINDUISM KimKnottHISTORY JohnH.Arnold

THEHISTORYOFASTRONOMY MichaelHoskinTHEHISTORYOFCHEMISTRY WilliamH.BrockTHEHISTORYOFCHILDHOOD JamesMartenTHEHISTORYOFCINEMA GeoffreyNowell-SmithTHEHISTORYOFLIFE MichaelBentonTHEHISTORYOFMATHEMATICS JacquelineStedallTHEHISTORYOFMEDICINE WilliamBynumTHEHISTORYOFPHYSICS J.L.HeilbronTHEHISTORYOFTIME LeofrancHolford‑StrevensHIVANDAIDS AlanWhitesideHOBBES RichardTuckHOLLYWOOD PeterDecherneyTHEHOLYROMANEMPIRE JoachimWhaleyHOME MichaelAllenFoxHOMER BarbaraGraziosiHORMONES MartinLuckHUMANANATOMY LeslieKlenermanHUMANEVOLUTION BernardWoodHUMANRIGHTS AndrewClaphamHUMANISM StephenLawHUME A.J.AyerHUMOUR NoëlCarrollTHEICEAGE JamieWoodwardIDENTITY FlorianCoulmasIDEOLOGY MichaelFreedenTHEIMMUNESYSTEM PaulKlenermanINDIANCINEMA AshishRajadhyakshaINDIANPHILOSOPHY SueHamiltonTHEINDUSTRIALREVOLUTION RobertC.AllenINFECTIOUSDISEASE MartaL.WayneandBenjaminM.BolkerINFINITY IanStewartINFORMATION LucianoFloridiINNOVATION MarkDodgsonandDavidGannINTELLIGENCE IanJ.DearyINTELLECTUALPROPERTY SivaVaidhyanathanINTERNATIONALLAW VaughanLoweINTERNATIONALMIGRATION KhalidKoserINTERNATIONALRELATIONS PaulWilkinsonINTERNATIONALSECURITY ChristopherS.BrowningIRAN AliM.AnsariISLAM MaliseRuthvenISLAMICHISTORY AdamSilversteinISOTOPES RobEllamITALIANLITERATURE PeterHainsworthandDavidRobeyJESUS RichardBauckhamJEWISHHISTORY DavidN.MyersJOURNALISM IanHargreavesJUDAISM NormanSolomonJUNG AnthonyStevens

KABBALAH JosephDanKAFKA RitchieRobertsonKANT RogerScrutonKEYNES RobertSkidelskyKIERKEGAARD PatrickGardinerKNOWLEDGE JenniferNagelTHEKORAN MichaelCookLAKES WarwickF.VincentLANDSCAPEARCHITECTURE IanH.ThompsonLANDSCAPESANDGEOMORPHOLOGY AndrewGoudieandHeatherVilesLANGUAGES StephenR.AndersonLATEANTIQUITY GillianClarkLAW RaymondWacksTHELAWSOFTHERMODYNAMICS PeterAtkinsLEADERSHIP KeithGrintLEARNING MarkHaselgroveLEIBNIZ MariaRosaAntognazzaLIBERALISM MichaelFreedenLIGHT IanWalmsleyLINCOLN AllenC.GuelzoLINGUISTICS PeterMatthewsLITERARYTHEORY JonathanCullerLOCKE JohnDunnLOGIC GrahamPriestLOVE RonalddeSousaMACHIAVELLI QuentinSkinnerMADNESS AndrewScullMAGIC OwenDaviesMAGNACARTA NicholasVincentMAGNETISM StephenBlundellMALTHUS DonaldWinchMAMMALS T.S.KempMANAGEMENT JohnHendryMAO DeliaDavinMARINEBIOLOGY PhilipV.MladenovTHEMARQUISDESADE JohnPhillipsMARTINLUTHER ScottH.HendrixMARTYRDOM JolyonMitchellMARX PeterSingerMATERIALS ChristopherHallMATHEMATICS TimothyGowersMATHEMATICALFINANCE MarkH.A.DavisMATTER GeoffCottrellTHEMEANINGOFLIFE TerryEagletonMEASUREMENT DavidHandMEDICALETHICS MichaelDunnandTonyHopeMEDICALLAW CharlesFosterMEDIEVALBRITAIN JohnGillinghamandRalphA.GriffithsMEDIEVALLITERATURE ElaineTreharne

MEDIEVALPHILOSOPHY JohnMarenbonMEMORY JonathanK.FosterMETAPHYSICS StephenMumfordTHEMEXICANREVOLUTION AlanKnightMICHAELFARADAY FrankA.J.L.JamesMICROBIOLOGY NicholasP.MoneyMICROECONOMICS AvinashDixitMICROSCOPY TerenceAllenTHEMIDDLEAGES MiriRubinMILITARYJUSTICE EugeneR.FidellMILITARYSTRATEGY AntulioJ.EchevarriaIIMINERALS DavidVaughanMIRACLES YujinNagasawaMODERNARCHITECTURE AdamSharrMODERNART DavidCottingtonMODERNCHINA RanaMitterMODERNDRAMA KirstenE.Shepherd-BarrMODERNFRANCE VanessaR.SchwartzMODERNINDIA CraigJeffreyMODERNIRELAND SeniaPašetaMODERNITALY AnnaCentoBullMODERNJAPAN ChristopherGoto-JonesMODERNLATINAMERICANLITERATURE RobertoGonzálezEchevarríaMODERNWAR RichardEnglishMODERNISM ChristopherButlerMOLECULARBIOLOGY AyshaDivanandJaniceA.RoydsMOLECULES PhilipBallMONASTICISM StephenJ.DavisTHEMONGOLS MorrisRossabiMOONS DavidA.RotheryMORMONISM RichardLymanBushmanMOUNTAINS MartinF.PriceMUHAMMAD JonathanA.C.BrownMULTICULTURALISM AliRattansiMULTILINGUALISM JohnC.MaherMUSIC NicholasCookMYTH RobertA.SegalNAPOLEON DavidBellTHENAPOLEONICWARS MikeRapportNATIONALISM StevenGrosbyNATIVEAMERICANLITERATURE SeanTeutonNAVIGATION JimBennettNELSONMANDELA EllekeBoehmerNEOLIBERALISM ManfredStegerandRaviRoyNETWORKS GuidoCaldarelliandMicheleCatanzaroTHENEWTESTAMENT LukeTimothyJohnsonTHENEWTESTAMENTASLITERATURE KyleKeeferNEWTON RobertIliffeNIETZSCHE MichaelTanner

NINETEENTH-CENTURYBRITAIN ChristopherHarvieandH.C.G.MatthewTHENORMANCONQUEST GeorgeGarnettNORTHAMERICANINDIANS ThedaPerdueandMichaelD.GreenNORTHERNIRELAND MarcMulhollandNOTHING FrankCloseNUCLEARPHYSICS FrankCloseNUCLEARPOWER MaxwellIrvineNUCLEARWEAPONS JosephM.SiracusaNUMBERS PeterM.HigginsNUTRITION DavidA.BenderOBJECTIVITY StephenGaukrogerOCEANS DorrikStowTHEOLDTESTAMENT MichaelD.CooganTHEORCHESTRA D.KernHolomanORGANICCHEMISTRY GrahamPatrickORGANIZEDCRIME GeorgiosA.AntonopoulosandGeorgiosPapanicolaouORGANIZATIONS MaryJoHatchPAGANISM OwenDaviesPAIN RobBoddiceTHEPALESTINIAN-ISRAELICONFLICT MartinBuntonPANDEMICS ChristianW.McMillenPARTICLEPHYSICS FrankClosePAUL E.P.SandersPEACE OliverP.RichmondPENTECOSTALISM WilliamK.KayPERCEPTION BrianRogersTHEPERIODICTABLE EricR.ScerriPHILOSOPHY EdwardCraigPHILOSOPHYINTHEISLAMICWORLD PeterAdamsonPHILOSOPHYOFLAW RaymondWacksPHILOSOPHYOFSCIENCE SamirOkashaPHILOSOPHYOFRELIGION TimBaynePHOTOGRAPHY SteveEdwardsPHYSICALCHEMISTRY PeterAtkinsPILGRIMAGE IanReaderPLAGUE PaulSlackPLANETS DavidA.RotheryPLANTS TimothyWalkerPLATETECTONICS PeterMolnarPLATO JuliaAnnasPOLITICALPHILOSOPHY DavidMillerPOLITICS KennethMinoguePOPULISM CasMuddeandCristóbalRoviraKaltwasserPOSTCOLONIALISM RobertYoungPOSTMODERNISM ChristopherButlerPOSTSTRUCTURALISM CatherineBelseyPOVERTY PhilipN.JeffersonPREHISTORY ChrisGosdenPRESOCRATICPHILOSOPHY CatherineOsborne

PRIVACY RaymondWacksPROBABILITY JohnHaighPROGRESSIVISM WalterNugentPROJECTS AndrewDaviesPROTESTANTISM MarkA.NollPSYCHIATRY TomBurnsPSYCHOANALYSIS DanielPickPSYCHOLOGY GillianButlerandFredaMcManusPSYCHOLOGYOFMUSIC ElizabethHellmuthMargulisPSYCHOTHERAPY TomBurnsandEvaBurns-LundgrenPUBLICADMINISTRATION StellaZ.TheodoulouandRaviK.RoyPUBLICHEALTH VirginiaBerridgePURITANISM FrancisJ.BremerTHEQUAKERS PinkDandelionQUANTUMTHEORY JohnPolkinghorneRACISM AliRattansiRADIOACTIVITY ClaudioTunizRASTAFARI EnnisB.EdmondsTHEREAGANREVOLUTION GilTroyREALITY JanWesterhoffTHEREFORMATION PeterMarshallRELATIVITY RussellStannardRELIGIONINAMERICA TimothyBealTHERENAISSANCE JerryBrottonRENAISSANCEART GeraldineA.JohnsonREPTILES T.S.KempREVOLUTIONS JackA.GoldstoneRHETORIC RichardToyeRISK BaruchFischhoffandJohnKadvanyRITUAL BarryStephensonRIVERS NickMiddletonROBOTICS AlanWinfieldROCKS JanZalasiewiczROMANBRITAIN PeterSalwayTHEROMANEMPIRE ChristopherKellyTHEROMANREPUBLIC DavidM.GwynnROMANTICISM MichaelFerberROUSSEAU RobertWoklerRUSSELL A.C.GraylingRUSSIANHISTORY GeoffreyHoskingRUSSIANLITERATURE CatrionaKellyTHERUSSIANREVOLUTION S.A.SmithSAINTS SimonYarrowSAVANNAS PeterA.FurleySCHIZOPHRENIA ChrisFrithandEveJohnstoneSCHOPENHAUER ChristopherJanawaySCIENCEANDRELIGION ThomasDixonSCIENCEFICTION DavidSeedTHESCIENTIFICREVOLUTION LawrenceM.Principe

SCOTLAND RabHoustonSEXUALSELECTION MarleneZukandLeighW.SimmonsSEXUALITY VéroniqueMottierSHAKESPEARE’SCOMEDIES BartvanEsSHAKESPEARE’SSONNETSANDPOEMS JonathanF.S.PostSHAKESPEARE’STRAGEDIES StanleyWellsSIKHISM EleanorNesbittTHESILKROAD JamesA.MillwardSLANG JonathonGreenSLEEP StevenW.LockleyandRussellG.FosterSOCIALANDCULTURALANTHROPOLOGY JohnMonaghanandPeterJustSOCIALPSYCHOLOGY RichardJ.CrispSOCIALWORK SallyHollandandJonathanScourfieldSOCIALISM MichaelNewmanSOCIOLINGUISTICS JohnEdwardsSOCIOLOGY SteveBruceSOCRATES C.C.W.TaylorSOUND MikeGoldsmithSOUTHEASTASIA JamesR.RushTHESOVIETUNION StephenLovellTHESPANISHCIVILWAR HelenGrahamSPANISHLITERATURE JoLabanyiSPINOZA RogerScrutonSPIRITUALITY PhilipSheldrakeSPORT MikeCroninSTARS AndrewKingSTATISTICS DavidJ.HandSTEMCELLS JonathanSlackSTOICISM BradInwoodSTRUCTURALENGINEERING DavidBlockleySTUARTBRITAIN JohnMorrillSUPERCONDUCTIVITY StephenBlundellSYMMETRY IanStewartSYNTHETICBIOLOGY JamieA.DaviesTAXATION StephenSmithTEETH PeterS.UngarTELESCOPES GeoffCottrellTERRORISM CharlesTownshendTHEATRE MarvinCarlsonTHEOLOGY DavidF.FordTHINKINGANDREASONING JonathanStB.T.EvansTHOMASAQUINAS FergusKerrTHOUGHT TimBayneTIBETANBUDDHISM MatthewT.KapsteinTOCQUEVILLE HarveyC.MansfieldTRAGEDY AdrianPooleTRANSLATION MatthewReynoldsTHETREATYOFVERSAILLES MichaelS.NeibergTHETROJANWAR EricH.Cline

TRUST KatherineHawleyTHETUDORS JohnGuyTWENTIETH‑CENTURYBRITAIN KennethO.MorganTYPOGRAPHY PaulLunaTHEUNITEDNATIONS JussiM.HanhimäkiUNIVERSITIESANDCOLLEGES DavidPalfreymanandPaulTempleTHEU.S.CONGRESS DonaldA.RitchieTHEU.S.CONSTITUTION DavidJ.BodenhamerTHEU.S.SUPREMECOURT LindaGreenhouseUTILITARIANISM KatarzynadeLazari-RadekandPeterSingerUTOPIANISM LymanTowerSargentVETERINARYSCIENCE JamesYeatesTHEVIKINGS JulianD.RichardsVIRUSES DorothyH.CrawfordVOLTAIRE NicholasCronkWARANDTECHNOLOGY AlexRolandWATER JohnFinneyWAVES MikeGoldsmithWEATHER StormDunlopTHEWELFARESTATE DavidGarlandWILLIAMSHAKESPEARE StanleyWellsWITCHCRAFT MalcolmGaskillWITTGENSTEIN A.C.GraylingWORK StephenFinemanWORLDMUSIC PhilipBohlmanTHEWORLDTRADEORGANIZATION AmritaNarlikarWORLDWARII GerhardL.WeinbergWRITINGANDSCRIPT AndrewRobinsonZIONISM MichaelStanislawski

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GeoffCottrell

MATTERAVeryShortIntroduction

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Contents

Acknowledgements

Listofillustrations

Whatismatter?

Atoms

Formsofmatter

Energy,mass,andlight

Thequantumworldoftheatom

Quantummatter

Fundamentalparticles

Wheredotheelementscomefrom?

Darkmatteranddarkenergy

Furtherreading

Index

Acknowledgements

Itisapleasuretothankmyeditor,LathaMenon,forsuggestingthisstimulatingtopicforaVSI.IwouldalsoliketothankPhilHopkins,LauraLauro,andJoMarks,fortheircommentsonthefirstdraft.

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Listofillustrations

Theperiodictableoftheelements©OxfordUniversityPress.

Thedifferentlengthscalesofmatter(inmetres)arrangedonacircle©GeoffCottrell.

Brownianmotion:thetrackofapollengrainparticleinwater©GeoffCottrell.

ScanningTunnellingMicroscopy(STM)imageoftheelectroncloudsofindividualcarbonatomsonagraphitesurfaceWithpermission:P.HopkinsandW.S.Steer,ImperialCollege,London.

Transformationsofthestatesofmatterwithincreasingtemperaturefromsolidtoliquid,togas,andtoplasma©GeoffCottrell.

Illustrationofanunstablesymmetryandabrokensymmetry©GeoffCottrell.

Thechainstructureofapolymer(plastic)molecule:polyethylene©GeoffCottrell.

Theatomic-scale‘chicken-wire’structureofgraphene,a2Dsheetofcarbon,oneatomthickMathier/Shutterstock.com.

Thehexagonalbeautyofasnowflake.Itappearstobesymmetrical,butithaslesssymmetrythanthewatervapourfromwhichitcondensedAlexeyKljatov/Shutterstock.com.

MichaelFaraday’ssketchofthemagneticlinesofforceRoyalInstitutionofGreatBritain/SciencePhotoLibrary.

ThecurvatureofspaceroundthemassivebodyoftheEarth

Rutherford’sexperimenttoscatteralphaparticlesfromgoldatomsledtothediscoveryoftheatomicnucleus

Representationofanatomshowingtheneutronsandprotonsclusteredtogetherintheatomicnucleus,surroundedbyacloudofelectrons

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©GeoffCottrell.

Theladderofquantumenergylevelsofthehydrogenatomshownasaseriesofhorizontallines

ThomasYoung’sdoubleslitexperiment

IllustrationofHeisenberg’suncertaintyprinciple,inwhichthequantumofactionhasbeen‘squeezed’indifferentways©GeoffCottrell.

Allparticlesintheworldareeitherfermionsorbosons©GeoffCottrell.

Sphericalanddumbbellshapedelectrondensitydistributionsforthehydrogengroundstateandsomeexcitedstates

Atomsinagasatdifferenttemperatures

Howelectronstatesinasolidareproduced©GeoffCottrell.

Howstatesarefilledbyelectrons©GeoffCottrell.

ThecreepofsuperfluidheliumoverthesidesofitscontainertodripoutunderneathAlfredLeitner/WikimediaCommons/PublicDomain.

ThefountaineffectJohnF.Allen.

TheintegerQuantumHallEffectReprintedfigurewithpermissionfromM.A.Paalanen,D.C.Tsui,andA.C.Gossard,QuantizedHalleffectatlowtemperatures,1982.Copyright2018bytheAmericanPhysicalSociety.<https://doi.org/10.1103/PhysRevB.25.5566>.

Acoffeemugistopologicallyequivalenttoabagel,becausetheybothhaveonehole;buttheybothbelongtoadifferenttopologicalclassfromthepretzelwithitsthreeholes(a)AfricaStudio/Shutterstock.com(b)BinhThanhBui/Shutterstock.com(c)diamant24/Shutterstock.com.

TheInternationalPrototypeKilogramisstoredinsidethreebelljarsinasafeinabasementintheParissuburbofSèvresCourtesyoftheBIPM.

Pairproduction:matterandantimatterarecreateddirectlyfromtheenergyofaphotonScienceSource/SciencePhotoLibrary.

Thelightestbaryon,theproton,ismadefromthreequarks,two‘up’andone‘down’

TheStandardModelofparticlephysics©GeoffCottrell.

Thefullskymapofthecosmicmicrowavebackgroundradiation,theafterglowoftheBigBang,showingthe

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infantuniverse©ESAandthePlanckCollaboration.

Thecurveofbindingenergypernucleonforvariousnuclei©GeoffCottrell.

TheCrabNebulaistheremnantofastarthatexplodedinAD1054NASA,ESAandAllisonLoll/JeffHester(ArizonaStateUniversity).

Illustrationofthegravitationallensingeffect,withwaterripplesinaswimmingpool©GeoffCottrell.

TheSmileyFace;thegravitationallensingofdistantgalaxies(curvedarcs)bythemassofanearerclusterofgalaxiesNASA/ESA/JPL-Caltech.NASAphoto,referencehttps://www.nasa.gov/content/hubble-sees-a-smiling-lens

Chapter1

Whatismatter?

Matteristhestufffromwhichyouandallthethingsintheworldaroundyouaremade.Ifyouhadthemostpowerfulmicroscopeimaginableyoucouldlookinsideyourbodyandseethatyouaremadeofatoms.Insideeveryatomisatinynucleus,andorbitingthenucleusisacloudofelectrons.Thenucleusismadeoutofprotonsandneutrons,andbyzoominginonanuclearparticleyouwouldfindthatinsideitthereareevensmallerparticles—quarks.Quarksarethesmallestparticlesthatwehaveseen,andlieatthelimitofresolutionofthemostpowerfulmicroscopesofmatter.Asfarasweknow,electronsandquarksarenotmadefromanythingsmallerandsotheyarecalledfundamentalparticles.Allmatterismadefromjusttheseparticles.

Atomsaresosmallthatamillionofthemcanfitacrossthebreadthofahumanhair.IfanappleweremagnifieduptothesizeoftheEarth,itsatomswouldbethesizeofapples.Thediameterofanatomisaround10–10m.(Whenquantitiesaregivenas10tosomepower(106say)thisissimply1followedby6zeros,inthiscase1,000,000oronemillion;thoseexpressedas10tosomenegativepower(10–6say)have1inthe6thplaceafterthedecimalpoint,thatis,0.000001oronemillionth.)Yourbodycontainsroughly1029atoms;asizeofabouttwometresdefinesthehumanscale.

Matterexistsinformsofimmensevarietyandcomplexity.Thefamiliarthingsaroundus—books,atable,water,acat—allhaveintricatestructuresandcompositions.Theyaremadeofvastnumbersofatoms,stickingtogetherinclumpsofdifferentshapesandsizes.The‘glue’thatholdsclumpsofatomstogetherandtheelectronstothenucleiofatomsistheelectricalforceofattractionbetweenoppositeelectricalcharges.Allthedifferentstructuresofmatterresultfromthemanypossiblewaysinwhichparticlesinteracttomakedifferentphysicalformsandarrangementsinspace.Theelectricalforce,invariousguises,alsoproducesthevarioustypesofinteratomic

bondsbetweenatoms,joiningthemtogethertomakemolecules,andsoitunderpinschemistry.Moleculescanbeassimpleaswater(twoatomsofhydrogenandoneofoxygen,orH2O),ortheycanbeascomplexasthemillionsofatomsinaproteinmacromoleculeinyourbody.

Asubstanceisanelementifitcannotbedecomposedintotwoormoredifferentsubstancesbycommonphysicalorchemicalprocesses.Thereareninety-twonaturallyoccurringdifferenttypesofchemicalelements,andeachtypehasitsownuniqueproperties.Intheyear1867onlysixty-threeoftheelementshadbeendiscovered.Thedifferentatomswerethenknowntohavedifferentatomicweightsrangingfromthelightest,hydrogen,withanatomicweightof1,uptotheheaviestknownatthetime,lead,withaweightof207.(Thebasicchemicalunitistheweightofahydrogenatom,1.67×10–27kg,whichdefinestheatomicmassunit.)Atthattimethechemistsweresearchingforpatternsintheirpropertiesthatmightrevealadeeperstructure.ThepropertiesoftheelementswerewellknowntotheRussianchemistDmitriMendeleev,whowrotedowntheirnamesandpropertiesoncardsandarrangedtheminorderoftheiratomicweights.Henoticedthatthechemicalpropertieshadapattern:theyrepeatedatregularintervals,aperiodiclaw.Mendeleevdescribedhisdiscovery:‘Isawinadreamwhereallelementsfellintoplaceasrequired.Awakening,Iimmediatelywroteitdownonapieceofpaper,onlyinoneplacedidacorrectionlaterseemnecessary.’Thepatternshowedthatwerealsosomeelementsmissingfromthetable.Heleftgapsforthese,confidentlypredictingtheelementsgermanium,gallium,andscandium,whichweresoondiscovered.

Themodernperiodictable(Figure1)isarrangednotbyatomicweight,butbyatomicnumber,thenumberofprotonsinanucleus,rangingfrom1(hydrogen)upto92(uranium).Theatomicnumberisequaltothenumberofelectronsintheatom.Elementsheavierthanuraniumareproducedartificially.HenryMoseleydevelopedanX-raytechniquetomeasurethenumberofprotonsinthenucleus,andwehavehimtothankforessentiallythemodernversionofthetable.Astheatomicnumberincreasesalongsuccessivehorizontalrowsthechemicalpropertiesrepeatinperiodsoftwo(hydrogenandhelium),thentwoperiodsofeight(lithiumtoneon;sodiumtoargon),andthenthreeofeighteen.Thetableismorethanaclassificationscheme;itrevealsapatternthatisdeeplyembeddedinnatureandthestructureofatoms.

1.Theperiodictableoftheelements.

Theperiodictableisgovernedbyquantumlaws.Theelectronsinatomsspreadoutaroundthenucleusinwhatareknownasatomicorbitals,whichformshell-likestructuresaroundthenucleus.Atomsseektominimizetheirenergies,whichiswhathappenswhentheirelectronshellsarecompletelyfull.Forexample,thefirstatomwithafullshellishelium(atomicnumber2),thesecondisneon(atomicnumber2+8=10),andthethirdisargon(atomicnumber2+8+8=18),andsoon.Thesestablefilled-shellatomsarethechemicallyinertnoblegases,whichsitinthelastcolumnofthetable.

Thedifferentatomsarelikelettersofanalphabet,whichcombinetomakemolecules,akintothewordsofalanguage.Howmanytypesofmoleculesarethere?TheEnglishlanguagehasaroundaquarterofamillionwordsincurrentuse,allbasedonatwenty-six-letteralphabet.Theletteraisalwaysana,whetheritappearsintheword‘cat’orinotherwordswithcompletelydifferentmeanings,like‘bat’.Similarly,eachhydrogenatominawatermoleculeisidenticaltothosethatcombinewithcarbontoformmethane(CH4),amoleculewithcompletelydifferentproperties.Intheoryitispossibletoformbillionsofdifferentstablechemicalcompoundsbycombiningthe

elementsoftheatomicalphabetindifferentways.

ThescaleofmatterTofixideasitisimportanttoappreciatethevastrangeofthelengthscalesofmatter,indicatedinFigure2.Fromtheverysmalleststructuresthatwebelievemayexist,tothelargest(thevisibleuniverse),wecoveramind-bendingrangeofsixty-twoordersofmagnitudeinsize,or1062.Thistakesusfromthequantumworldofthesmallestentities,tothatofthelargeststructures,whicharedominatedbytheforceofgravity.Thehumanlengthscaleisatthebottomofthecircle,lyingroughlyinthemiddlebetweenthetwoarmsofthediagram.ThelargeststructuresrangefromobjectsthesizeoftheEarth,uptotheclustersandsuperclustersofgalaxies.AlbertEinstein’sgeneraltheoryofrelativitydescribesthesehugestructuresandthespacethattheyoccupy.Attheverysmallendofthesizespectrum,thelawsofquantummechanicsdescribematter.Atpresentthereisnocompletetheoryconnectingthequantumworldandthegravitationallydominatedworld,andsothevarioustheoriesofquantumgravitythathavebeenproposed,whichmightbridgethegap,arenotdescribedhere.

2.Thedifferentlengthscalesofmatter(inmetres)arrangedonacircle.

Let’stakeanimaginaryzoomlensand,startingfromthehumanscale,zoomintoprogressivelysmallerscales.Toseethecellsinyourbody,youwouldneedtoincreasethemagnification100,000timesfromthehumanscale.Thisispossibleusingthewavelengthsofvisiblelight.However,lightcannotbeusedtoresolvestructuressmallerthanitswavelength.Thewavelengthsoflightarearound500nanometres(onenanometre,ornm,is10–9metres,andthereare20,000wavelengthsinacentimetre),andtoseesmallerthingswemustuseshorterwavelengths.Electronsacceleratedinelectronmicroscopeshavewavelengthssmallenoughtoenableustoseestructuresassmallasanatom.Inzoominginfromcellstoatoms,themagnificationhastobeincreasedby100,000times.Afurtherincreaseofmagnificationof100,000timesfillsourframewiththeatomicnucleus.Withatotalmagnificationnowofatrilliontimes,wehavearrivedatthescaleofthequantumworld,wherethewavenatureofmattermakesthingsappearveryfuzzy.Tozoominbeyondthis,andlookinsidetheproton,wemustaccelerateelectronstohigh-energiesandspeedsofover99.9percentofthespeedoflightandcrashthemintonucleitoseethesubstructuresinsidetheproton,thequarks.(Thespeedoflightinvacuumisc=300,000kilometrespersecond.)Therearenomicroscopestotakeusanyfurtheronthisjourney,andfromnowononemustrelyontheory.Thefinestdivisionsofspacethatarebelievedtoexistandhaveanymeaningintermsofthelawsofphysicsaretinyquantumfluctuationsonascaleof10–35metres.Anenormousleapinmagnificationof1017wouldbeneededtoseeanystructuresonthisscale.

Ifwestartagainonthehumanscaleandzoomouttenmilliontimes,thefieldofviewisfilledbytheEarth.Fromnowon,bigstructuresareshapedbygravity.Afurtherzoomof100timesfillstheframewithourstar,theSun.Thenextlargeststructure,thesolarsystem,hasadiameterof300trillionmetres,and,tobringthatintoview,wemustzoomout10,000timesfurtherstill.IttakeslightaroundfivehourstoreachPlutointheoutersolarsystem.Wemustwait4.2yearsforthelightwhichisnowleavingthesurfaceofthenextneareststartotheSun,ProximaCentauri,toreachus—thestaris4.2lightyearsaway.Aswecontinuetozoomoutanawe-inspiringsightcomesintoview:ourhomegalaxytheMilkyWay.TheMilkyWayisadisc-shapedspiralgalaxycontaining100billionstarsandhasadiameterof100,000lightyears.Tofitthisintheframeofthelenswewouldhavetozoomout10,000timesmore.

ButtheMilkyWayisnot,aswasbelievedonlyacenturyago,theentireuniverse.Ourgalaxyisoneofaboutthirtymembersofthelocalgroupofgalaxies;tofitthelocalgroupinourframewewouldneedtozoomoutaroundtentimesmore.Beyondthis,

thereareevenlargerstructures,suchastheComaclusterofgalaxies,whichhasadiameterofover300millionlightyearsandcontains1,000galaxies,gravitationallyboundtogetherintoaroughlysphericalclump.Toseethiscluster,wewouldneedtozoomoutafurther100times.Thelargestknownstructuresintheuniversearethegiantsuperclustersofgalaxies,andthehugefilamentsofgalaxieswhichsurroundvastvoidsinspace,butbeyondthat,thefurthestdistancethatwecanseeouttowithtelescopesistheboundaryofthevisibleuniverse,whichhasadiameterofaround100billionlightyears.Allobservablematteriscontainedinthissphere,consistingof100billiongalaxies,withatotalmattercontentof1080hydrogenatoms.WewilllookattheseverylargescalesinChapter9,butit’sworthnotingthattheaveragedensityofmatterinthevisibleuniverseisaboutafewhydrogenatomspercubicmetre.Forcomparison,theEarthhasadensityalmost1030timeslargerthanthat,makingourplanetahighlyatypicalregionoftheuniverse.Mostoftheuniverseisemptyspace.

Inthisbookwewill,inChapter3,seehowthefamiliarstatesofmatterofsolid,liquid,andgasarise,andlookatsomeotherstatesofmatter.If,likeIsaacNewton,weconsidermattertobedefinedasmass,thentheequivalenceofmassandenergy,describedinChapter4,takesourunderstandingofmattertoadeeperlevel,andrevealstheoriginoftheawesomereleaseofnuclearenergyfromtheatomicnucleus.InChapter5wewillentertheweirdandfuzzyquantumworldandseehowitexplainsthestructureofatomsandtheperiodictable.Whenlargenumbersofparticlesaggregatetogether,theycan,undercertainconditions,manifestdramaticandcoherentquantumbehaviouronmacroscopicscales.ThisisdescribedinChapter6,andhasledtothedevelopmentofquantummeasurementdevicesthatallowthebasicunitofmass,thekilogram,tobedefinedintermsofthefundamentalconstantsofnaturetoanunprecedenteddegreeofprecision.Theultimatebuildingblocksofmatter,whichincludeantimatter,areintroducedinChapter7,whichdescribeshowtheworldcanbeunderstoodintermsofaroundtwentydifferentquantumfields.Mostofthemassofnormalmattercanbeexplainedbytheenergyinthesequantumfields.Tounderstandwheretheelementscomefrom,inChapter8welookatthehistoryoftheuniverse,fromitsearliestmoments,andtracehowtheelementsarebuiltinstarsaswellasinthemostviolenteventsintheuniverse.Matterisenergyandenergycurvesspace.Thispropertyisbeingputtousebyastronomerstomapouthowmuchmattertheuniversecontains,andwhereitislocated.Finally,inChapter9,wearriveatthehumblingrealizationthatthenormalmatter,theatomsandmoleculesoffamiliareverydaylife,representonly5percentofallthetypesofmatterthatis‘outthere’.Theremainderofthematterintheuniverseseemstoconsistoftwocompletelymysterioussubstances:darkenergy,anddarkmatter.First,however,weneedtoassureourselvesthatthebasicconstituentsofmatter,atoms,reallydoexist.ThisisthesubjectofChapter2.

Chapter2

Atoms

ThebirthofWesternsciencecanbetracedbacktotheGreekphilosophersofantiquity.Inaround500BCThalesofMiletusfoundedaschoolthatsoughttoexplaintheworldbyapplyinglogicandreasontotheobservationofnature.Heproposedthatallmatterwascomposedofasingleprimarysubstance,whichhebelievedtobewater.Whatwasimportantwasnotthathewasmistakenaboutthis,butthathesetinmotionawayofthinkingthatwasbasedinlookingattheworldasitisinitself.Thisapproachledtotheprescientideathatthebuildingblocksofmatterareatoms,aconjecturethatisattributedtoLeucippusandDemocritusinabout450BC.

Takeanappleandchopitupintosmallerandsmallerpieces.Eventuallyapointisreachedwherenomorechoppingispossible,revealingtheultimategraininessofmatter,atoms.ThewordatomcomesfromtheGreekwordatomos,meaningindivisible.Theseultimateparticleswereconsideredtobeindestructible,differingonlyinshape,size,position,andarrangement.TheGreeksimaginedaspaceinwhichtheatomsmovearoundceaselessly,calledvoid.Theatomist’sbeautifullysimpleviewwasthattheuniverseiscomposedofjusttwoelements:atomsandvoid.

Inabout350BC,Aristotleadoptedthebeliefthatvariouscombinationsofjustfourelements,earth,fire,air,andwater,canexplaineverydaymatter.SogreatwasAristotle’sphilosophicalstandingthatthisviewpersistedunchallengedintothemedievalperiod,andinspiredthealchemists’fruitlesssearchforwaysofturningbasemattersuchasleadintogold.Intheirunquestioningadherencetothebeliefinaworldmadeoffourelements,theclosestthealchemistscametoaneffectivescientificmethodwasabasicformofempiricalchemistry.Thisinvolvedgrinding,mixing,heating,anddistillingcommonsubstancessuchaswater,oil,mercury,earth,sulphur,salt,andair.ThedarkagesofEuropelastedfromthe6thtothe14thcentury,markingaperiodwhenthedevelopmentofsciencelargelygroundtoahalt.

Writingatthestartofthe17thcentury,theFrenchphilosopherRenéDescartesconsideredtheprimaryqualityofmattertobeextension,namelythatwhichoccupieslength,breadth,andheightinspace,oraswewouldsay,volume.Thisperiodmarkedthebirthoftheageofenlightenmentwhenquantitativesciencereallybegantotakeoff.Fromaboutthattimetheatomichypothesisbegantore-emergeasneededbytheoriesoftheday,buttherewasstillnoproofoftheexistenceofatoms.

Thepropertyofmassiscentraltotheconceptofmatter.MassfirstappearedinNewton’slawsofmotion,whichformthebasisofmodernclassicalmechanics.Newtonaskedifthereisanysimplerulebywhichthemovementsoftheplanetscanbecalculatedgiventheirstatesofmotion.Inthisquesthedevelopedthemathematicalcalculusnecessarytosolveequationsdescribingthechangeinthestateofmotionofamasspointinaninfinitesimaltime,undertheinfluenceofanexternalforce.Heconnectedtheconceptofaforce,alreadywellknownfromthestudyofstatics,withacceleration,andindoingsointroducedtheconceptofmass.ThemeaningofmassinNewton’ssecondlaw(force=mass×acceleration)isstrictlyinertialmass,theresistanceofabodytoachangeinitsmotion.

Newton’sgreatbreakthroughwastolinkthelawsofmotionwiththelawofgravitationalattraction.Theforceonamassisdeterminedbythepositionsandmassesofnearbybodies.Newton’slawofgravitationstatesthatthegravitationalforcebetweentwomassesisequaltoaconstant(Newton’sgravitationalconstant,G)multipliedbyeachmassandtheinversesquareoftheirseparation.Gravityisauniversallong-rangeforce(namely,allmassesattracteachother).Thelawofmotioncombinedwiththelawofattractionenabledhimtocalculatethepastandfuturestatesofmassesactingundertheforceofgravity.WiththishesucceededinexplainingthemotionoftheMoon,theplanets,comets,andeventhetideswithgreatprecision.

WhatwasNewton’sviewofmatter?InhisOptiksof1704,Newtonwritesofmatterasbeingformedof‘solid,massy,impenetrable,moveableparticles…soveryhard,asnevertowearorbreakinpieces…’,aconceptionofatomsnotverydifferentfromthatofDemocritus.Whatwasnewwasthedeterministicmannerinwhichthe‘massyparticles’moveundertheactionofforces.

Weightandmassareeasilyconfused,butthereisanimportantdifferencebetweenthem.Whenweweighanobjectwhatwearereallydoingismeasuringhowstronglyitispulledbygravity.Ifweloadaspringbalancewithakilogramofapples,abigmass(theEarth)pullstheapples(asmallmass)downwardsagainstthepullofthespring.Bysymmetry,thesmallmassalsopullsthebigmass;theforceattractingthemtogetherisequalandopposite,whetherwethinkoftheEarthpullingtheapples,orthe

applespullingtheEarth.ThisexpressesNewton’sthirdlaw.However,theSunexertsagravitationalpullontheEarththatisanenormous1022timesstrongerthanthatoftheapples,andsotheappleshaveanegligibleeffectontheEarth.ThisiswhytheEarth’sorbitiscontrolledbytheSunandnotbyapples.(Bysymmetry,theEarthalsoexertsaforceontheSun.However,becausethemassoftheEarthisonlythreemillionthsofthemassoftheSun,theeffectissmall.Thetwobodiesorbitaroundtheircommoncentreofmass,whichlieswellwithinthebodyoftheSun.)IfwetakeourkilogramofapplestotheMoon,whichhasonlyone-sixthofthemassoftheEarth,thebalanceregistersonlyasixthofakilogram.Takethemoutinspace,farfromanygravitatingmatterorintoorbitaroundtheEarthonaspacestationandtheyexistinastateoffree-fallandareweightless.Massisthereforeanintrinsicpropertyofanobject;whereastheweightofthesameobjectwouldbedifferentiftheobjectwereontheEarth,ortheMoon,oronMars.

AfterNewton’sbrilliantachievements,thestoryofmatterandatomsshiftstotheworkoftheearlychemists.Bytheendofthe18thcentury,importantadvanceshadbeenmadeinquantitativechemistry,largelyattributabletothedevelopmentofaccuratebalancestomeasuretheweightsofreactingsubstances.Thelanguagechemistsusedalsomademorefrequentuseoftheword‘element’,ofwhichtherewerethenknowntobeaboutthirty.ThefirstquantitativechemicalexperimentsofJosephPriestley,thediscovererofoxygen,andAntoineLavoisiershowedthatfamiliarsubstanceswereoftencombinationsofelementssuchashydrogen,oxygen,carbon,iron,andsulphur.In1789Lavoisierdiscoveredthelawofconservationofmass;namely,whensubstancescombined,themassofthereactingchemicalswasalwaysequaltothemassoftheproducts.Forexample,themassofthehydrogenplusthemassoftheoxygenisalwaysequaltothemassofthewaterproduced.AtragicfatebefellLavoisier.HewasataxcollectorandanaristocratwholivedatthetimeoftheFrenchRevolution;hewasbrandedatraitorandguillotinedin1794.ThemathematicianJosephLagrangepaidtributetohim:‘ittookthemonlyaninstanttocutoffhishead,andonehundredyearsmightnotsufficetoreproduceitslike.’

Atomictheorygainedastrongerfootholdintheearly19thcenturywiththeexperimentsoftheEnglishchemistJohnDalton.Daltonproposedthateachchemicalelementisauniquetypeofatom,differingfromothersbyitsweight.Daltondiscoveredsimplenumericalratiosintheproportionsoftheweightsofelementswhentheycombinedchemically.Dalton’slawofconstantproportionsstatesthatwhenelementsformcompounds,theirweightsalwayscombineinsimplewhole-numberratios.Forexample,theweightofoxygenrequiredtocombinewithagivenweightofhydrogentoformwaterisalwayseighttimesaslarge.Daltonmadeanorderedtableoftherelativeweightsoftheelements:hydrogen,nitrogen,carbon,oxygen,

phosphorus,sulphur,copper,lead,silver,gold,platinum,andmercury.ThistablewasaninspirationtoMendeleevandothers,andwasaprecursortothemodernperiodictable.

Thesimplenumberratioscomingoutofchemistryledmanytobelieveintheexistenceofatoms,butothersstillharboureddoubts.Chemistshadrevealedmanyfactsaboutatomsandmolecules,butwereunabletomeasuretheirsizesandabsoluteweights.Thekeyobservationthatwouldultimatelyleadtothecertaintyoftheexistenceofatomswas,itseems,firstmadeinantiquity.Inthe1stcenturyBC,theRomanpoetLucretiushadfollowedinthefootstepsoftheatomistDemocrituswhenhewrotehisgreatepicpoemDeRerumNatura(OntheNatureofThings).ItisfortunatethatLucretiuswasawareoftheearlierwork,becausemostofDemocritus’originalbooksandmanuscriptsweredestroyedinthegreatfirethatdestroyedthelibraryatAlexandriainEgyptinabout48BC.Inhispoem,Lucretiusdescribesadarkenedroom,piercedbyashaftofsunlightenteringthroughaholeinashutter.Thebrilliantlightilluminatesmyriadsofminutedustparticlescaughtinthebeam.Theytrembleagitatedly,seeminglybeingjostledatrandom.Lucretiussuggeststhatthismotionresultsfromhordesofunseenatoms,rainingrepeatedtinyblowsuponthedustparticles.

Thesamephenomenonwasseenin1827bytheEnglishbotanistRobertBrownwhohadbeenstudyingpollengrainsinwater.Thetinyparticlesdancedabout(Figure3),asiftheyhadalifeoftheirown.Atthetimetherewasmuchinterestinvitalism—thesearchforahypotheticallifeforce,whichwasthoughttoinfuselivingmatter.Brownwonderedifhehaddiscoveredthevitalforce,andperformedthecrucialexperiment.Hereplacedthepollengrainsbyparticlesoffinelypowderedsilica,buttheparticleskeptonjiggling.Brownianmotion,asithascometobecalled,wasnotthevitalforce.

3.Brownianmotion:thetrackofapollengrainparticleinwater.Thegrainisgraduallynudgedawayfromitsstartingpositionasitisjostledbymoleculesandperformsa‘randomwalk’.

IttookthegeniusofEinsteintoseewhatBrownianmotionwassaying,andheusedthelawsofchancetodeducetheexistenceofmolecules,andtheirsizes.Evenifatomsarethemselvestoosmalltobeseen,theyareabletoproduceasensiblemotioninotherverysmallparticlesthatarevisible.Amicron-sized(10–6metres)dustparticleinaircanbethoughtofasasuper-largeairmoleculethatreceivesaround10,000molecularblowsononesurfaceatanyinstant.Asimilarnumberofblowsbattertheoppositeside,sothatonaveragethereisnonetforceontheparticle.However,thenumberofblowsissubjecttostatisticalvariations.Thelawsofprobabilitytellusthatthenumberofmolecularblowsfluctuatesasthesquarerootofthenumberitself.Thenumberofblowsreceivedbyadustparticlevariesbetween9,900and10,100,makingtheforceonitfluctuatebyabout1percent,whichaccountedforthe‘randomwalk’ofBrownianmotion.Einstein’spaperwaspublishedinhisannusmirabilisyearof1905anditcontainedthefirsteverestimateofthesizeofamolecule.

Itiswellknownthatoilandwaterdon’tmix.Oilspreadsoutintoathinlayertwomoleculesthickonthewaterwiththeoilmoleculeslinedupsidebyside,andbacktoback.ThepolymathBenjaminFranklinwasinterestedintheeffectofpouringoilon‘troubled’watersandtheeffectitisreputedtohaveincalmingthemdown.In1774hepouredateaspoonofoilonthesurfaceofapondinClaphamCommoninLondonand,asitspreadoveragreatarea,henotedthatthesurfacebecame‘assmoothasalookingglass’.TheEnglishphysicistLordRayleighlaterusedthiseffectinanelegantandsimpleexperimenttomeasurethesizeofmolecules.Theareacoveredbytheoil

iseasiertoseeifthewaterisfirstlightlydustedwithfinepowder,whichtheoilwillpushawaywhenitspreadsout.Asingledropofoilwithavolumeofonecubicmillimetrecoversanareaofaboutasquaremetre.Themolecularsize(abouttwonanometres)issimplythedropletvolumedividedbytheareaofthefilm.Sincethereareapproximately12atomsinanoilmolecule,thesizeofanatomworksouttobeabout1.7×10–10metres.

Sofar,wehaveseenhowtheprescientideasoftheGreekatomists2,500yearsagowereignoredforcenturiesuntilre-emergingatthestartoftheageoftheenlightenment.BythetimeofNewton,therewerethreeideasforadefinitionofmatter:allmatterismadeofatoms,atomstakeupspace,andatomshavemass.TheconceptofatomsbecomesmorefirmlyrootedbythetimeofDalton,andhisdiscoveryofthesimpleratiosofatomicmassesinchemicalreactions.ButitwasnotuntilEinstein’sinterpretationofBrownianmotionin1905thatanyobjectionstotheatomichypothesiswerefinallyquashed.

Itisnowpossibletoimageindividualatoms.InFigure4,atechniquecalledScanningTunnellingMicroscopy(STM)wasusedtoimageindividualcarbonatomssittingonthesurfaceofasampleofgraphite.Themethodreliesontheabilityofelectronsto‘tunnel’,quantummechanically,acrossthegapbetweenthesampleandafine-tippedmetalprobe.QuantummechanicaltunnellingisdescribedinChapter5.

4.ScanningTunnellingMicroscopy(STM)imageoftheelectroncloudsofindividualcarbonatomsona

graphitesurface.

Theatomichypothesisissofundamentalthatin1970itpromptedAmericanphysicistRichardFeynmantowriteinVolume1ofhisfamousLecturesonPhysics:

If,insomecataclysm,allofscientificknowledgeweretobedestroyed,andonlyonesentencepassedontothenextgenerationsofcreatures,whatstatementwouldcontainthemostinformationinthefewestwords?Ibelieveitistheatomichypothesis(ortheatomicfact,orwhateveryouwishtocallit)thatallthingsaremadeofatoms—littleparticlesthatmovearoundinperpetualmotion,attractingeachotherwhentheyarealittledistanceapart,butrepellinguponbeingsqueezedintooneanother.

InChapter3Itakeupthestoryofhowattractiveandrepulsiveforcesbetweenatomsproducethefamiliarstatesofmatter.

Chapter3

Formsofmatter

WaterisoneofthefeweverydaysubstancesthatcanexistnaturallyontheEarthassolid,liquid,andgas.Coolitdown,anditturnsintoiceashardasrock.OnSaturn’smoonTitan,thetemperatureisachilly180°Cbelowzeroandtherearemountainsofice,3kilometrestall.Onourmoretemperateplanetthenormalstateofwaterisliquid.Whenyouboilakettleajetofinvisiblegas,steam,isproduced.Thewhitecloudsthatcomeoutofthekettlecontainminutedropsofwaterthatcondenseintheairandscatterlight.Placeacoldsurfaceinthejetandthesteamcondensesbacktodropsofwaterthatrundownandcoalesce.Thesedifferentstatesorphasesofmatterarisebecauseofacompetitionbetweenopposites:thethermalmotiondrivingparticlesapartandattractiveinteratomicforcespullingthemtogether,repulsionandattraction.The‘glue’thatholdselectronstoatoms,bringsatomstogethertoformmolecules,anddrawsmoleculestogethertomakesolidsandliquids,iselectricity.Electricalforcesliebehindchemistry,biology,andlifeitself.Statesofmatterthatcanflow,theliquidsandgases,arecalledfluids.Solids,liquids,andgasesaretheso-calledgreatstatesofmatter.Asolidhasashapeandavolume,aliquidhasavolumebutnoshape,andagashasneithershapenorvolume.Liquidsandsolidsarecalledthecondensedstatesofmatter.

Thesimpleststateofmatterisgas.InthenineteenthcenturyJamesClerkMaxwellandLudwigBoltzmanndevelopedthekinetictheoryofgasesthatforgedalinkbetweenthestatisticalmicroscopicworldofmoleculesandthemacroscopicpropertiesofgas.An‘ideal’gasconsistsofalargenumberofatomsormoleculesflyingaboutrandomlyandcollidingasiftheyareperfectlyelasticminiaturebilliardballs.Betweentheirbriefcollisionstheymoveinstraightlines,whichiswhygasesfillcontainersofanyshapeandsize.Thereareavastnumber(oftheorderof1022)ofmoleculesinalitreofgasatstandardconditions,anumbersolargethatwecancalculatetheirstatisticalbehaviourwithgreatcertainty.Whenamoleculestrikesthewallofitscontaineritimpartsatinyimpulsetoitandbouncesback.Therelentless

batteringoftheLilliputianblowsofthemoleculesaveragesouttoproduceasizeablemacroscopicforce,thepressure,whichpushesequallyonallthewalls.

Eventhoughtherearesomanymoleculesinagas,theyareverysmall,andthereisagreatdealofspacebetweenthem.Thispropertyofgasesmakesthemcompressible.Ifyouliquefyairbystrongcooling,itcontractsto1/2000thofitsvolume.Thewallsofagascontainercanbesqueezedtocompressthegas,likethepistoninabicyclepump.The17th-centuryEnglishscientistRobertBoyleperformedexperimentsongases,whichhedescribedelegantlyas‘touchingthespringoftheair’.Boylediscoveredthatthegaspressureincreasesinproportiontotheinverseofitsvolume.Ifonehalvesthevolumeofthecontainer,themolecules,nowsquashedintohalfthespace,bombardthewallstwiceasintensely,sodoublingthepressure.

Themoleculesinagasdonotalltravelatthesamespeed.Thesloweronesgainkineticenergybecausefasteronesstrikethemmoreoften,andthefasteronesloseenergymorefrequentlybycollidingwithslowerones.CollisionsleadtoastatisticaldistributionofparticlespeedscalledtheMaxwell–Boltzmanndistribution,inwhichtheaveragespeedisrelatedtothetemperatureofthegas.Anairmoleculeatroomconditionshasanaveragespeedofaround350m/s,whichisroughlythespeedofsound,andsoundpropagatesaswavesofcompressionsandrarefactionsoftheair.Thelinkbetweentemperatureandmolecularspeedimpliesthattheremustbealowestpossibletemperature,absolutezero,atwhichmotionstops.Absolutezerois−273°C,orzerodegreesontheabsolutetemperaturescalemeasuredinKelvin(0K).

Thereisanimportantdistinctionbetweenheatandtemperature.Somebodieshavehightemperaturesbutcontainlittleheat;othersarecoolerandcontainagreatdealofheat.Heatdependsonboththetemperatureandhowmanyparticlesareinvolved.Twoequalpansofboilingwatercontaintwiceasmuchheatasone,eventhoughtheybothhavethesametemperature(100°C).

ThermodynamicsThepowerhouseoftheindustrialrevolutionwasthesteamengine.Intryingtoimprovetheperformanceofsteamengines,thetransformationsbetweenheat,work,andenergywerestudiedintensivelyandtheknowledgegainedgrewintothescienceofthermodynamics.Therearetwobasicformsofenergy:theenergyofmotion(orkineticenergy),andtheenergythatthebodyhasbyvirtueofitspositioninaforcefield(orpotentialenergy).Liftinga1kgmassthroughaheightofametreintheEarth’sgravitationalfieldincreasesitspotentialenergybyalmost10Joules.TheJoule

(J)istheSIunitofenergy,aconvenientlysizedunitformacroscopicbodies.Ittakesabout400,000Joulestoboilakilogramofwater,whichisenoughenergytoliftapersonaheightof500metres.

Anobjectcangainkineticenergybyincreasingitsspeed.Butwhentheobjectistakentothetopofabuilding,itgainspotentialenergyfromthegravitationalfieldoftheEarth.Ifitisnowdropped,theobjectpicksupspeedasitfalls,convertingitspotentialenergytokineticenergy.Energyisausefulconceptbecauseoftheprincipleoftheconservationofenergy:thesumofthepotentialandkineticenergyisconstantthroughoutthemotionoftheobject.Whentheobjecthitstheground,itsenergyisconvertedintotherandommotionsofmolecules,orheat.

Thefactthatheatisaformofenergymeantthatithadtobeincludedinthelawoftheconservationofenergy,whichbecametheFirstLawofThermodynamics.Whenabody(calleda‘system’inthermodynamics)isinthermodynamicequilibrium,thesystemcanbethoughtofasbeinginasealedboxfromwhichnoheatenergycanescapeorbeadded.Theenergyofathermodynamicsystemisconserved.

Itisimportanttodistinguishbetweenusefulformsofenergy,suchastheinterchangeablekineticorpotentialenergies,andenergythathasbecomesodegradedthatitcan’tbeusedtoperformwork.It’seasytodegradeenergyintoauselessform,andmuchhardertoreversetheprocess.Thekineticenergyofamovingcar,thedirectedenergiesofallitscomponents,will,onbraking,largelyendupasheat.Theorderedmotionofthecarhasnowbecomedisorganizedmotion,therandommotionoftheatomsandmoleculesinthebrakedrums.Thatheatenergyisnolongerusefulinmakingthecarmoveagain.TheirreversiblenatureofheatisembodiedintheSecondLawofThermodynamics.Thefirstlawsaysthatwecannotgetsomethingfornothing,andthesecondlawsaysthatwecan’tevenbreakeven!Thequalityofenergy,oritsabilitytodousefulwork,isrelatedtotheamountofdisorderinasystem,andismeasuredbythermodynamicquantitycalledtheentropy.EntropywasfirstdefinedbyRudolfClausiusandputonastatisticalmolecularbasisbyBoltzmann.Theconceptofdisorderinasystemofparticlesplaysafundamentalroleindetermininghowvariousarrangementsofatomsgiverisetothedifferentstatesofmatter.

Eachparticleinasystemhasanumberofdegreesoffreedom,orindependentwaysinwhichitcanmoveorabsorbenergy.Foragasinequilibrium,thetotalenergyisdistributedequallybetweenallthedegreesoffreedomoftheparticles,accordingtotheprincipleoftheequipartitionofenergy.Asimplemonatomicgas,likeneon,hasthreedegreesoffreedom,whichcorrespondtothethreedimensionsofspace.

Moleculeshaveextradegreesoffreedom.ThechemicalbondinadiatomicH2moleculeforexamplebehaveslikeaspringconnectingtwoatomicmassesthatcanvibrateorrotate,anditisthesethatbringinextradegreesoffreedom.Whenastandardamountofmatterinagivenstateincreasesitstemperaturebyonedegree,itabsorbsaquantityofheatcalledthespecificheat.

SolidsandliquidsWhathappenswhenwecoolacontainerofgas?Thegasmoleculesreboundfromthewallsofthecontainerwithreducedenergiesandthetemperatureofthegasstartstofall.Movingmoresluggishly,themoleculesarenowlessindependent,andspendmoretimeneartheirneighbourswhereinteratomicforcesslowthemdown.Interatomicforcesresultfromtheelectrostaticattractionofatomsormoleculesatfairlyshortrange(ofafewatomicdiameters)butbecomerepulsiveatveryclosespacing.Thinkoftheforcebetweentwoatoms.Atlargedistances(saymorethan10atomicdiameters),thereisvirtuallynoforcebecausethepositiveandnegativechargesineachatomcanceloutalmostcompletely.Butwhentheatomsapproacheachother,theyrevealtheirgranularnatureandstarttofeeleachother’sinternalstructures,eachbeingclosertosomeoftheother’satomicchargesthanothers.Thenegativeelectroncloudsofone,andthepositivenucleioftheother,attract.However,iftheatomsgettooclosetheirouterelectroncloudsoverlapandrepulsionsetsin.

Asthetemperatureofthegasfalls,thebalancebetweenthedisorderedthermalmotionandtheattractiveinteratomicforcesnowswingsinfavourofthelatterasthegascondensesintoaliquid(Figure5).Themoleculesarenowcloseenoughtoresistbulkcompression,andthisisthereasonwhyliquidsarelargelyincompressibleandareusedtotransmitforcesthroughpipesinhydraulicsystems.Atthemolecularlevel,acompromiseisstruckbetweenattractionandrepulsion,whichresultsinthemoleculeshavingtypicalequilibriumspacingsofabout3×10–10metres.Themoleculesofaliquidhavejustenoughthermalenergytoenablethemtoswapplaceswiththeirneighboursbyslidingaroundthem,givingaliquidthefluidpropertyofbeingabletoadapttotheshapeofitscontainer.

5.Transformationsofthestatesofmatterwithincreasingtemperaturefromsolidtoliquid,togas,andtoplasma.

Whentheliquidiscooledstillfurther,thermalmotionbecomesmorefeeble.Theliquidfreezestoasolid,andallfluiditydisappears.Themolecules,whilestillcontinuingtovibrateweakly,become‘locked’intospecificlocationsandthematerialbecomeshardandtakesonadefiniteshape,witheachmoleculehavingadefinitepositioninthesolid.Themoleculesinacrystallinesolidoccupyspecificpositionsonaregularperiodic3Dlattice.

Afundamentaldifferencebetweenasolidandaliquidisthedegreetowhichthemoleculesmaintaintheirregularorderingpatternsoverlargedistances.Thehallmarkofacrystallinesolidisthepresenceoflong-rangeorder,wheretheregularperiodicityofthemoleculararrangementextendsovermanylatticespacings.Aliquidontheotherhandisisotropicandhomogeneous.Ithasthedisorganizedstructureofagasbutitsmoleculesclumptogetherunderinternalforces,withoutexternalpressureneedingtobeapplied.Thisispossiblebecausethereisacriticaltemperature,abovewhichsomeexternalpressureisneededtohelpthemoleculesstayclosetogetherandopposethermalmotion.Asthecriticaltemperatureisapproached,thegasandliquidphasesmerge,withasmoothtransitionbetweenthem.Theliquidhasthedisorderedstructureofagas,butdiffersfromitbybeingabletomaintainastablevolumewithouttheneedofanexternalcontainer.Inpassingthroughaphasechange,asystemofmoleculesmustbreaktheirbondsandreformtheminnewways.Aquantityofenergy,thelatentheat,hastobeprovidedtodothis,whichdoesnotcontributetothekineticenergyoftheparticles.Whilethesemicroscopicstructuralchangesaretakingplace,thetemperatureofthematerialstaysconstant,despitetheadditionorremovalofheatenergy.

SymmetryTheregulararrangementsofatomsandmoleculesoncrystallatticesrevealadeep

aspectofmatter:symmetry.Weareveryfamiliarwiththeideasofsymmetry,forexamplethesymmetriesingeometricpatternssuchasrepeatingwallpaperortilingpatternsin2D;alsothoseofaperfect3Dsphere.Aspherecanberotatedthroughanyangle,aroundaninfinitenumberofpossibleaxesorbemirror-reflectedaboutaninfinitenumberofplanespassingthroughitscentre,anditstilllooksthesame.

Asymmetryoperationisdefinedasanactionthatcanbeperformedonanobjectthatleavestheobjectunchanged.Forexample,acrystallatticecanbeshiftedbyawholenumberoflatticespacingsalongoneofitslatticedirectionsanditlooksidentical.Acrystalhasdiscretesymmetriessuchastranslationalsymmetry,andalsoasetofrotationalandreflectionalsymmetries.However,inthedisorderedliquidandgasphasesthereareaninfinitenumberofcontinuoussymmetryoperations.Matterinthesephasescanbetranslated,reflected,androtatedinaninfinitenumberofwaysanditstilllooksthesame.Whenmatteriscondensedfromthehigh-energydisorderedgaseousorliquidphasesintoasolidcrystal,thedegreeofsymmetryisreduced,givingrisetowhatiscalledbrokensymmetry.

Aclassicexampleofsymmetrybreakingoccursinmagnets.Apermanentmagnet,suchasafridgemagnet,iscomposedofanarrayofmicroscopicmagnets.Whentheinternalmagnetsalllineupandpointinthesamedirectiontheirfieldsaddtoproduceastrongoverallmagneticfield,andthebodyasawholebehaveslikeonesinglemagnet.Thisconfigurationhasalowerenergythantheverymanypossiblestatesinwhichtheinternalmagnetsallpointinrandomdirections.However,ifthemagnetisheated,thereisacriticaltemperature,theCurietemperature,abovewhichallmagnetismislost.Abovethistemperature,therandomthermalmotionbecomessostrongthattheinternalmagnetscannolongerstaylinedup,andinsteadpointinallpossibledirections.Thishightemperaturestateisoneofmaximumdisorder,orentropy,andcausesthemagnettoloseitsmagnetism.Oncooling,however,thewholeassemblyspontaneously‘flips’back,toitslowerenergystate,withtheinternalmagnetsagainpointinginonepreferreddirection.Thefinaldirectioncanbeanywhere.However,ifthematerialisalreadythreadedbyaweak‘seed’magneticfield,suchastheEarth’sfield,theycanaligntothat.Thissuddenonsetofmagnetismisaclassicexampleofspontaneoussymmetrybreaking,andmarksthetransitionfromahightemperature,highentropystate,toalowtemperature,lowentropyone.TheenergylandscapeofthisprocessisillustratedinFigure6intheexampleoftheunstableequilibriumofaballbalancedonthetopofaperfectlysymmetrichill.Whentheballrollsdownavalleyitlowersitspotentialenergyandthesymmetryisbroken.

6.Illustrationofanunstablesymmetry(left)andabrokensymmetry(right).

Symmetryisbuiltintothelawsofnatureatthemostfundamentallevel.Everycontinuoussymmetryfoundinnatureisassociatedwithaconservedorinvariantquantity,aswasprovedinafamoustheorembyEmmyNötherin1915.Thepropertiesofspaceandtimetellusthatthelawsofnaturearethesameeverywhereintheuniverse.Forexample,thelawoftheconservationoflinearmomentumisindependentofthechoiceoftheoriginofthecoordinatesysteminwhichthemotionofbodiesismeasured.Spacehasasymmetrycalledtranslationalinvariance.Also,thefactthatthelawsofphysicsarethesameatalltimesturnsouttobeequivalenttothelawoftheconservationofenergy.Theconservationofenergydoesnotdependonwhattimeaclockissetto.Spaceisalsoisotropic;thelawoftheconservationofangularmomentumdoesnotdependonwheretheaxisofaspinningbodypointsinspace.

StickingtogetherInteratomicforcesarealldifferentmanifestationsoftheelectricalforce;in1785,Charles-AugustindeCoulombdiscoveredtheunderlyinglaw.The‘Coulomb’forcebetweentwochargedbodiesisproportionaltotheproductoftheirchargesand,likethegravitationalforce,isalong-rangeforce,beinginverselyproportionaltothesquareoftheirseparation.However,theelectricalforceisenormouslystrongerthangravity;therepulsiveelectricalforcebetweentwoelectronsisafactorof1042strongerthanthegravitationalforcewhichattractsthem.Giventhishugedifference,whydowemainlyexperiencegravityandnotelectricityonthehumanscale?Macroscopicobjectsaremadefromenormousnumbersofpositiveandnegative

charges,whicharesointimatelymixedthatthecombinedforcesofattractionofunlikechargesandrepulsionoflikechargescanceleachotheroutalmostperfectly.Lumpsofmatterarethereforealmostcompletelyelectricallyneutral.

Occasionallyasmallamountofchargecanbetransferredbyfrictionfromonebodytoanother,creatinganimbalance.Thisisstaticelectricity,wellknownsinceantiquity,whereactionslikepullingoffasweaterorcombinghairmakesparksjumporhairstandonend.In585BCThalesdescribeshowapieceofamber(whichisfossilizedtreeresin),whenrubbedonfur,canpickuplightweightobjectslikefeathers.TheGreekwordforamberiselektron,fromwhichwegetourwordforelectricity.

Thefirstfundamentalparticleofmattertobeidentifiedwasdiscoveredin1897byJ.J.Thompsonwhenhechippedelectronsoffatomsinelectricaldischarges.Hisapparatus,a‘cathoderaytube’,wasanevacuatedglasstubewithtwosealed-inelectrodes.Whenahighvoltageisappliedtotheelectrodes,streamsofelectrons(‘cathoderays’)areprojectedinstraightlinesfromthenegativeelectrode(thecathode),throughthetubetowardsthepositiveelectrode(theanode).Whereverthe‘rays’striketheglass,itglowswithamysteriousyellowish-greenfluorescence.Bybringingupelectricallychargedplatesandmagnets,thecathoderayscanbedeflectedinasystematicway,andThompsonusedthesefieldstomeasuretheelectriccharge(e)andmassoftheelectron.Hisdiscoverythatatomscontainsmallerparticles,electrons,endedforevertheconceptoftheindivisibleGreekatom.

ChemicalbondsWepictureatomsashavingapositivenucleus,surroundedbyacloudofelectronsflyingabout.Allatomsareattractedtoeachother,byweakforcesarisingfromthemotionofelectronsaroundthenuclei.ThisisthevanderWaalsforce,andcancausechemicallyinertgases,likeargon,toliquefyandsolidifyatlowtemperatures.Buttounderstandhowstrongchemicalforcesbondatomsintomolecules,weneedtoconsiderthewayinwhichthemorereactiveatomsinteractwhentheirelectroncloudsoverlap.ThestructureoftheperiodictableofFigure1ishelpful.Atomsandmoleculesindividuallyandinaggregatesseektheirlowestenergyandmoststableconfigurations,whichcorrespondtotheclosed,orsaturated,atomicshellsofelectrons.Theeightelectronsintheclosedshellsofatomslikeneonandargonformasphericalballofcharge.Alleightelectronsoccupythesameshellandenjoyanequallystrongelectrostaticforce.Theelectroncloudsaresymmetricalandcomplete;thisgivesthenoblegasesnoincentivetojoinupwithotheratomstoformmolecules.

Inmovingalongtheperiodictableonepositionfromneontosodiumanextraelectronisadded,calledavalenceelectron,whichmustgointoanewouteratomicshell,makingsodiumhighlyreactive.Ifwemoveonespacebackfromargon,wegettochlorine,whichisoneelectronshortofafilledshellandsoalsoreactive.Whensodiumandchlorineatomscometogethertheirouterorbitalsoverlap,enablingthemtostrikeamutuallyadvantageousdeal.Thesodiumcandonateitsoutermostvalenceelectron,whichiseasilyionized,tochlorinetocompleteitsoutershell,andthewholestructurebecomesastablemoleculeofcommonsalt(sodiumchlorideorNaCl).Eachatombenefitsbyachievingthestableclosed-shellstructureofanoblegas,andeachisachargedion,Na+andCl−;theionsareheldtogetherbyanionicbond.Saltformsacubicioniccrystalwithalatticeofalternatingpositiveandnegativecharges.Theelectropositiveelementsthatdonatetheirsurplusouterelectrons,likesodium,aremainlymetals.Theelectronegative,orelectronacceptor,elementsarefoundinsubstancescontainingoxygen,sulphur,chlorine,andfluorine.

Hydrogenisdifferent.Ithasoneelectroninashellmadefortwo(helium),andsoitcangoeitherway,H+,orH−,inmakinganioniccompound.Hydrogencanacceptanelectronfromsay,lithium,tomakelithiumhydride,orshareitselectrontosatisfyanelectron-hungryelementsuchasfluorineoroxygen.Theseweakerbondsarecalledhydrogenbondsandarecommoninmanyorganicandbiologicalmolecules,forexamplebetweenthebasepairsthatformthetwisteddoublehelixDNAmolecule.

Twoprotonscanjoinupbysharingtwoelectronstoformaneutralhydrogenmolecule,H2.Thetwoelectronsformastrongcovalentbond;the‘twoness’ofthepairedelectronscomesfromaquantumrulecalledthePauliexclusionprinciple(seeChapter5),whichallowstwoandonlytwoelectronstotakepart.Covalentbondscanthereforebecomesaturated,forifathirdhydrogenatomshouldapproachthecovalentlybondedpair,itwouldbeexcludedandsocannotformastabletriatomicmolecule.

Theversatileelementcarbonhasfourelectronsinitsoutershell,andfourvacancies.Thisstructureenablescarbontoformorganiccompoundswithoxygen,hydrogen,andmanyotherssuchasthebiologicalmoleculesoflife.Weare‘carbon-basedlifeforms’,andourbodiescontainpolymermolecules.Apolymerisalong-chainmolecule,inwhichthousandsofatomsormoleculeslineup,likebeadsonanecklace,withotheratomsstucktothesidestofillupthesparebonds.Manyorganicpolymershavethischain-likeformincludingcommonplasticssuchaspolyethylene,alongchainofcarbonatoms,withhydrogenatomsattachedtothesides(Figure7).

7.Thechainstructureofapolymer(plastic)molecule:polyethylene.

Someelementscanexistinseveraldifferentphysicalforms,calledallotropes.Graphite,charcoal,anddiamondareallallotropesofcarbon.Theextremehardnessanddurabilityofdiamondformarisesfromits3Dcubicarrangementoffourstrongcovalentbonds,whichcontrastsstronglywiththepropertiesofitsotherallotropessuchasgraphite,whichinvolvesonlythreecovalentbonds.Graphitehasalayeredstructurewithweakforcesinbetweenthetoughlayersmakingitslipperyandabletowriteonpaper;the‘lead’inapencilisinfactgraphite.Thereareothercarbonallotropestoo,andalargeclassofthesearethefullerenes,whichcanformsingle-atomthicknesscarbonnanotubes,andhollow60-atom-strongspheresofBuckminsterfullerene(‘buckyballs’).Carboncontinuestosurpriseus,andin2004,anewphaseofcarbonwasextracted,graphene,whichhasremarkablemechanicalandelectricalproperties.Itisthethinnestpossiblelayerofgraphite,asheetofcarbonatoms,oneatomthick,arrangedina2Dhoneycomblattice(Figure8).

8.Theatomic-scale‘chicken-wire’structureofgraphene,a2Dsheetofcarbon,oneatomthick.

Crystals

Thebeautifulformsofgemstonesorasnowflake(Figure9)revealtheunderlyingsymmetryofthearrangementsofmoleculesincrystals.Whenlargenumbersofparticlesaggregate,theyalwaystrytominimizetheirpotentialenergy,whichdrawstheminto‘close-packed’configurations.Theminimumenergyconfigurationofthreeatomsisatriangle.Ifmoreatomsareaddedinthesameplane,theysettleintoacompact2Dhexagonalpattern,suchasthegraphitesurfaceshowninFigure4.Thisiseasilydemonstratedbyfillingashallowtraywithequalballbearings.Onshakingthetray,theywillformalayerwithacompacthexagonalpattern.Anewlayerofatomscanbeaddedontop,fittingsnuglyintothehollowsofthelowerones.Withfouratoms,wemoveintothreedimensions,wheretheminimumenergyconfigurationisatetrahedron.Asmoreatomsattachthemselvestotheseedcrystal,theypreferentiallyoccupythetriangularfacesandgrowouttoformlargerhexagonalclusters.TheefficientarrangementsoftheatomsandmoleculesintheHexagonalClose-Packed(HCP)crystalstructureiscommontomanycrystals.

9.Thehexagonalbeautyofasnowflake.Itappearstobesymmetrical,butithaslesssymmetrythanthewatervapourfromwhichitcondensed.

Thehexagonalstructureofthesnowflakereflectsthearrangementofitsmolecules.In

awatermolecule,twohydrogenatomsbondwithtwohalf-emptyelectroncloudsthatemergeatrightanglesfromtheoxygenatom.Thehydrogenatomspriseapartthebondsslightly,toanangleof105°sothatthemoleculeisshapedlikeashallowletter‘V’withtheoxygenatomattheapex.Thisgeometrygiveswaterauniquesetofproperties.Whenwatermoleculesformice,theirminimumenergyconfigurationisahexagonalring,withaholeinthecentre.Thesolidexpandsslightlyonfreezingwhichexplainswhywaterpipesburstandrockssplit.Theslightlylowerdensityoficealsoexplainswhyicebergsfloat(90percentoftheirmassliesbelowthesurface).

Intermediatestates:glassesandliquidcrystalsItwouldbeeasytothinkthatmatterneatlydividesupintojustthesolid,liquid,andgasphases.Butnatureisfarmorecomplexthanthat.MaterialshaveaverywiderangeofdifferentformsandIwillhighlighttwoexamples:glass,aformthatisneithersolidnorliquidbuta‘frozenliquid’,andaliquidcrystal,whichhaspropertiesinbetweenaliquidandacrystal.

Whenaliquidiscooleddowntoformacrystallinesolid,themoleculesmustmovefromtheirclose-spaced,disorganizedconfigurationoftheliquidphase,tothelong-rangeorderedconfigurationandregularityofacrystallattice.Ittakestimeforthemoleculestomakethesepositionaladjustmentsand,ifthecoolingistoorapid,theydonotalwayshaveenoughtimetore-formintothelong-rangeorderedconfigurationofthecrystal.Themoleculesareineffect‘caughtunawares’bytheirlossofmobilityandgetstuckbeforetheycangettotheircrystallinepositions.Thisisthevitreous,orglassystateofmatterandonewherethereisaliquid-likedisorganizedmolecularorder,butthematerialhastherigidityofasolid.Solids,likeglasses,thatdonothavedefinitegeometricorcrystallinestructuresarecalledamorphoussolids,andintheirdisorderedstatethereisnotenoughenergyforthemoleculestoflowpasteachother.Aglassisa‘frozenliquid’,neitheraregularsolidnoraliquid,andisinahighlyviscousmetastablestate,whichmeansthatoverlongperiodsoftimeglasswillgraduallytendtowardscrystallinity.

Whenacooledmonatomicliquidlikeargonfreezes,theatomsformaclose-packedregularlattice.Eachatomcanbeimaginedtobeahardsphere,whichmeansthatithasnopreferreddirectioninspacewhensettlingintoitsminimum-energyconfiguration.Buttherearecertainmaterials,calledliquidcrystals,whichcontainhighlyanisotropicrigidrod-likemoleculesthatbehavedifferently.Thinkofemptyingaboxofmatchesonatable.Thematchesspreadoutandcanenduppointinghaphazardlyinanydirection.Ifthematchesaregatheredtogether,theymustallalign

inthesamedirection.

Similarly,thelongmoleculesinaliquidcrystalphasedonotoccupyregularpositionsonalattice,buttheycanallbemadetopointinasingledirection.Aliquidcrystalhastheorderandpositionalcharacteristicsofaliquid,butitalsohaswhatiscalledorientationalorder.Iftheliquidcrystalisheated,therod-likemoleculesgobackintothestandardliquidphase,withthemoleculespointinginalldirectionsatrandom.Aliquidcrystalisthereforeastateofmatterthatisintermediatebetweenaliquidandanorderedcrystal.Thesematerialsarethebasisoftheliquidcrystaldisplays(LCDs)familiarinsmartphones,computerscreens,andTVs.Liquidcrystalscanbeorientedbyapplyingelectricforces,movingeasilyfromadisorderedstatetoonewherethemoleculesarelinedup,aconfigurationthatchangesthewayinwhichthematerialtransmitslight.InanLCD,thisisachievedbyapplyingvoltagestothepixelsofthescreen.

ThefourthstateOver99percentofthenormalmatterintheuniverseisplasma,sometimescalledthefourthstateofmatter.Aplasma(Figure5)isagasthatissohotthatittransformstoanewstate.Tounderstandhowplasmadiffersfromagas,wehavetothinkabouttheatomsthatmakeupthegas.Whenweheatagastoveryhightemperatures,thespeedoftheatomsbecomessolargethattheircollisionsknockoffsomeoftheorbitingatomicelectrons,whicharenowfreetomovearoundontheirown.Theatomsthatloseelectronsareionized,andarenowpositivelychargedions.Plasmaisahigh-energygasthatiscomposedoftwocommingledpopulationsofchargedparticles;thelightnegativeelectrons,andtheheavypositiveions.

ThenameplasmacomesfromtheGreekwordmeaning‘somethingmoulded’,andwascoinedbyapioneerofthesubject,IrvingLangmuir,inthe1920s.Twowell-knownnaturallyoccurringterrestrialplasmasarelightningandtheauroraornorthernorsouthernlights.Theauroraisproducedwhenhigh-energychargedparticlesemittedbytheSunstriketheEarthandionizegasesthatglowhighupintheatmosphere.Otherfamiliarplasmasincludecandleflames,andneonandsodiumlighting.Industrialandscientificapplicationsofplasmasrangefromthemanufactureofmicrocircuitstorealizingthepotentialforunlimitedcleanenergyfromthermonuclearfusionpower.OutsidetheatmospherewearesurroundedbyalayercalledthemagnetospherewhichistheplasmasystemformedbytheinteractionofthesolarwindwiththeEarth’smagneticfield.TheSunandallthestarsareballsofhotplasmaheldtogetherbygravity,andthesolarwindisastreamofturbulentplasmathatis

blownawayfromtheSun’ssurface.Plasmasalsoformnearexoticastrophysicalobjectslikeblackholes.Whenmatterfallstowardsablackhole,itsettlesfirstintoaspinningdiscsurroundingtheblackhole.Thefrictioninthegasheatsittoveryhightemperatures,formingplasmasohotthatitemitsX-rays.

Plasmascanalsobeproducedwhenatomsareionizedbyhigh-energyparticles,suchascosmicrays,orthoseproducedbynaturalradioactivity.Asmallamountofnaturalionizationispresentinmostgases.Plasmacanalsoformwhenelectricitypassesthroughgas.Airisnormallyagoodinsulator.However,ifaverystrongelectricfieldisappliedtoit,asoccursduringanelectricalstorm,itcan‘breakdown’andbecomeanelectricalconductor.Inalightningstrike,freechargesareacceleratedtohighspeedsbeforecollidingwithmoleculesofgasandknockingoffchargedfragments.Thiscangenerateacascadeofchargedparticles,amplifyingandconcentratingtheion–electronpairstoformaplasmaarc,andcarvingoutaconductingpaththroughthegasbysheerbruteforce.

Themostnotablepropertyofplasmaisitselectricalconduction,whichresultsfromthemobilechargesitcontains.Thecurrentsthatflowinplasmasgeneratemagneticfieldswhichexertforcesthatcausetheplasmato‘pinch’andformnarrowfilaments.SuchfilamentarystructureshavebeenobservedintheprominencesandflaresonthesurfaceoftheSun.Theelectromagneticpropertiesofplasmasinterrestrialthermonuclearfusionreactorsareexploitedinvariousways,bothtoheatthem,andtoconfinethem,ofteninsidetoroidalmagnetic‘bottles’.(Atorusistheshapeofaring-doughnut.)Iftheveryhotplasmainsuchadeviceweretotouchmaterialwalls,theimpactsoftheenergeticplasmaparticleswoulderodethem,byknockingatomsout.

Oneoftheuniquepropertiesofplasmaisthatitexhibitscollectivebehaviour.Examplesofthisarewavesthatcanpropagatethroughplasmawithoutanyparticlecollisions.Inair,normalsoundwavesspreadoutwhenmoleculescollide,passingontheirenergyascompressionsandrarefactions.Inplasmathereareequalnumbersofpositiveandnegativecharges,makingitelectricallyneutraltoaveryhighlevelofprecision.Ifthiswerenotthecase,theelectricforcessetupbyevenasmallchargeimbalancewouldmovechargesarounduntilneutralitywasobtained.Plasmascansupportarangeofdifferentwaves.Supposeforexamplethatasuddendisturbancecauseselectronstobunchupinsomeregionoftheplasma.Theions,beingheavyandsluggish,cannotrespondfastenoughtorestoretheimbalance,andelectricforcesdevelopandpushtheelectronsback.Butthemomentumoftheelectronsmakesthemovershoottheiroriginalpositions,whichtheydountiltheelectricforcepullsthembackagain.Again,theyovershoot.Thecyclerepeats,creatingadisturbancethatpropagatesthroughtheplasmaasLangmuirwaves.TheoutermostpartoftheSun’s

atmosphere,thesolarcorona,isveryhotplasma,whichsupportstheseLangmuirwaves.Plasmashostawiderangeofothermodesofoscillation,involvingthecollectivemotionofchargedparticlesmovinginelectromagneticforces.

Inthischapter,wehaveseentheforcesatworkwhenatomsgettogether.Thecentralideaisthataggregationsofatomswilltendtodropintothelowestenergystatethatisavailabletothem.Thereisacompetitionbetweeninteratomicelectricforces,whichtendtoattractatomsandmoleculestogetherintoclumps,andthedispersiveeffectofthethermalmotion.Ingases,thethermalmotionwinsandinsolidsthevictorsaretheinteratomicforces.Arangeofintermediatestatescanalsoexist,suchasthe‘frozenliquids’ofglasses,andtheorientationalpropertiesofmoleculesinliquidcrystals.Mostofthenormalmatterintheuniverseisinthefourthstateofmatter,high-energyplasmawhereparticleenergiesaresohighthatelectronsareknockedofftheatomsandthematterexistsasanintimatelymixedgasofionsandelectrons.

InChapter4,wemoveawayfromtheNewtonianworldandseehowourunderstandingofmatterwascompletelytransformedbythegreatdiscoveriesofelectromagnetismandrelativity.

Chapter4

Energy,mass,andlight

Atthebeginningofthe20thcentury,physicswasturnedonitsheadbytwogreatrevolutions:relativityandquantummechanics.Thesechangedforeverourunderstandingofmatter.InthischapterIwilloutlineEinstein’sspecialtheoryofrelativityof1905,whichdescribeswhathappenswhenobjectsmoveatspeedsclosetothespeedoflight.Thetheorytransformedourunderstandingofthenatureofspaceandtime,andmatterthroughtheequivalenceofmassandenergy.In1916Einsteinextendedthetheorytoincludegravityinthegeneraltheoryofrelativity,whichrevealedthatmatteraffectsspacebycurvingspacearoundit.

Toputthingsinperspective,weshouldfirsttouchontheclassicalNewtonianpicture.Newtonhad,by1700,establishedlawsofmotionandthetheoryofgravitation.Providedspeedsarenottoohigh,andmassesnottoolarge,Newton’slawsprovideaverygoodframeworkforunderstandingtheworld,enablingus,forexample,toputamanontheMoon.Newton’suniverserestedontwoassumptions.Firstistheideaofanabsolutetime;hislawsseemtocontainthenotionthatthereisacosmicclocktickingaway,whicheveryoneintheuniversewouldagreewith,wherevertheyare.Secondistheconceptofanabsoluteandimmutablespace.

Newtonwasawarethat,apartfromgravity,thereareotherforcesinnature,suchastheelectricforce.Electricalchargesattractorrepeleachotheratlargedistances,along-rangepropertysharedalsobymagnets.Whenyouholdapairofmagnetsinyourhands,youcanfeeltherepulsionoflikepolesandtheattractionofoppositepoles.Itisnothardtoimaginethatthemagnetsareimmersedinsomesortofinvisible‘forcefield’.Electricityandmagnetismaredeeplyconnected,afactthatwasdiscoveredbyDanishphysicistHansChristianØrstedin1820whenheobservedthatamagneticcompassneedlewasdeflectedbyanearbywirecarryingelectriccurrent.André-MarieAmpèrewentontodeterminetheforcelawbetweencurrent-carryingwires.

Ourunderstandingoftheconnectionbetweenelectricityandmagnetismtookagiantleapforwardinthe1830swithMichaelFaraday’sexperimentsoncoils,batteries,andcircuits.Faradaydiscoveredthatachangingmagneticfieldproduceselectricforces,aneffectknownasinduction,whichgovernsallpracticalelectricgenerators.Faraday’sexperimentsledhimtoabrilliantinsight.WhatFaraday‘saw’inhismind’seyewastheelectromagneticfield,aninvisibletensionorstressthatspreadsoutthroughemptyspace.Thefieldmakesitspresencefeltbyproducingforcesthatactonnearbysusceptiblebodies.Faradayimaginedthatchargedormagneticbodiesproduceabundleoflinesofforce,sproutingfromtheirsurfaces(Figure10).Thelinestransmittheirforcestothebodiesasifconnectedbyinvisiblecables,pushingorpullingonthem.Thelinescanberenderedvisiblewhenironfilingsarescatteredonacardplacedoverabarmagnet.Aspeckofironisitselflikealittlemagnet,liningupwiththemagneticfield,justasacompassneedlealignstothatoftheEarth.

10.MichaelFaraday’ssketchofthemagneticlinesofforce,revealedbyironfilingsscatteredoverabarmagnet.

Faraday’sbrilliantexperimentsshowedthatthelinesofforcespreadoutthroughspacebetweenthebodiesoncurvedpaths.ThisideaclashedwiththewaythatNewtonimaginedhowthegravitationalforcewastransmittedinstantaneouslybetweentwoseparatedmasspointsdirectlyalongthelinejoiningthem.WhileNewton’stheorieshadbeenimmenselysuccessfulinexplainingthemotionoftheplanets,theconceptofinstantaneous‘actionatadistance’seemedoutofcharacterwithmostoftheprocessesfamiliartousfromeverydaylife.

Inaphysicalfieldeverypointinspacecanbelabelledwithanumberrepresentingthefieldstrength,whichvariesfrompointtopoint.Onweathermaps,forexample,

temperaturesorpressuresarerepresentedbyagridofordinarynumbers.Mapsliketheserepresentso-calledscalarfields,wherethefieldquantityisrepresentedbyanumberassociatedwitheverypointinspace.Therearealsomorecomplicatedvectorfields,suchasamapofthewindvelocityforwhichtwonumbers,speedanddirection,areneededateachpoint.Weathermapsareshownwitharrowsindicatingthewindspeed(thelengthofthearrow),andthecompassbearing(itsdirection).

In1864Faraday’sintuitionaboutfieldswasputonamathematicalbasisbyJamesClerkMaxwellinhisfamoussetofequationsthatdescribedelectricandmagneticfields,unifyingthemintoasingleentity:theelectromagneticfield.Maxwellusedvectorfieldstodescribehowthemagnitudesanddirectionsofelectricandmagneticforcesvaryinspaceandtime.Healsorealizedthatfieldsspreadoutintoemptyspace,disconnectedfromanymatter.Hisequationsshowedthatfieldsspreadoutatthevelocityoflight,andheguessedthatlightisanelectromagneticwave.

Whatisanelectromagneticwave?Imagineanelectron,withitsattachedfieldlinessproutingout,beingshakenrapidlytoandfro.Whathappenstotheelectricfield?Closetotheelectron,thefieldlinesadjustrapidlytoitschangingpositions.Butittakeslongerfortheinformationaboutthevaryingpositionoftheelectrontoreachpointsfurtheraway.Theinformationspreadsoutinspace,somethinglikethewaythatbucketsofwaterarepassedfromonepersontoanotherina‘bucketbrigade’;ittakestimeforabuckettobepasseddownalineofpeople.Maxwell’sequationspredictthatwhenanelectronisshaken,theoscillatingelectricfieldgeneratesacomplementaryoscillatingmagneticfieldwhichinturngeneratesanoscillatingelectricfield,andsoon.Thetwointerlinkedfieldsmoveoutthroughemptyspaceasasingleundulatingentity,transportingenergywiththem.Theequationscontaintwoeasilymeasuredphysicalconstants,andwhenMaxwellputtheseintohistheoryhediscoveredthatitpredictedthatthewavestravelatafixedspeedinavacuum,thespeedoflight,c.Infact,lightisanelectromagneticwave.Maxwell’stheorywasthemostimportantscientificdiscoveryofthe19thcentury;thegreattriumphwasthatinonestrokeMaxwellhadunifiedthreebranchesofphysics:electricity,magnetism,andoptics.

Therewasmore.TheMaxwellequationsalsopredictedthatthefieldshouldvibratewithamuchwiderrangeofwavelengthsthanjustthoseofvisiblelight.Oureyeshaveevolvedtosensethenarrowbandofwavelengthsinsunlight.However,therearelongerwavelengthsbeyondtheredendofthespectrumandshorteronesbeyondthebluethatwecannotsee.Thereisavastelectromagneticspectrumfromtheshortwavelengthofgammarays(equaltothediameterofaproton)toradiowavesmanythousandsofkilometreslong.Maxwell’spredictionoftheelectricalnatureoflightwasconfirmedsoonafterwardsbyHeinrichHertz’sbrilliantexperimentsonthe

generationanddetectionofradiowaves.

ThespeedlimitWhileMaxwell’stheorywastremendouslysuccessful,itclashedinasubtlewaywithNewton’sideas.Hereisathoughtexperiment.FirsttheNewtonianview.Supposeyouaretravellingonabusatasteady70km/handyouthrowaballforwardsat10km/h.Fromyourpointofviewintheframeofthebus,theballtravelsat10km/h.Butifaroadsideobservermeasurestheball’svelocity,theywouldfindittobe70+10=80km/h.InNewton’sworld,theconceptofavelocityhasameaningonlywhenitismeasuredwithrespecttoanothervelocity.Avelocityaddswhenyouaremovingtowardsabody,andsubtractswhenyouaremovingawayfromit;Newtonianvelocitiesarerelative.

Nextconsiderwhathappenswhenthebusdriverswitchesontheheadlights.Ifyoumeasurethespeedofthelightonthebus,you’dfindittobec.Whatwouldourroadsideobserverfind?Theywouldnotmeasureittobe70+c,butstillmeasurethespeedoflighttobec,thesameasyoumeasureittobe.Thisraisesaquestion:isthereavelocitythatlightismovingrelativeto?

Onewayoutoftheproblemoftheadditionofvelocitieswasthepossibleexistenceoftheaether,ahypotheticalall-pervasivelight-bearingmediumthroughwhichlightpropagates,butasubstancethatdoesnotinteractwithmatter.Allcommonwavesneedamediumforpropagation:forexample,soundwavesaretransmittedthroughtheair,andripplesmoveacrossthesurfaceofwater.Whenwegazeupatthestars,theirlighthastravelledgreatdistancesthroughthevacuumofspace.Iftheaetherexists,thenthespeedoflightshoulddependonthemotionoftheEarththroughit.ThefactthatsoundwavespropagatefasterdownwindthanupwindpromptedafamousexperimentbyAlbertMichelsonandEdwardMorleyinwhichtheyattemptedtomeasurethespeedoflightalongandacrosstheEarth’spatharoundtheSun.Theexperimentsalwaysyieldedthesameanswer:cisaconstant.Experimentstodetectthedependenceofthevelocityoflightonthemotionoftheobserverallfailed.

Intryingtomakesenseoutofthebafflingnon-existenceoftheaether,DutchphysicistHendrikLorentzexaminedtheMaxwellequations,andinparticularthewaytheychangewhenexpressedindifferentinertialframes.Aninertialframeisagridofcoordinatesthatismovingatsomeconstantspeedinastraightlinewithrespecttosomeotherinertialframe.Lorentzfoundthattheequationstakeondifferentformswhenformulatedindifferentframes,whichwouldimplythatthespeedoflightshould

changeingoingfromonetoanother,apredictionthatclearlycontradictedthenegativeMichelson–Morleyexperiment.Lorentzdefinedamathematicaltransformation(theLorentztransformation),whichenabledtheMaxwellequationstohavethesameformindifferentinertialframes.

ThiswastheproblemthatEinstein,a26-year-oldclerkinapatentofficeinBern,attackedwithironlogicin1905.EinsteininterpretedtheLorentztransformationasexpressingaprofoundphysicalrelationshipbetweenspaceandtimeforobserversindifferentinertialframes.Hisspecialtheoryofrelativitywasunderpinnedbytwoideas:namelythatthelawsofphysicsshouldbethesame,andthatthespeedoflightinavacuumisalwaysthesameforobserversindifferentinertialframes.Einsteinwasfamousforhisthoughtexperiments,andonethatheperformedwhenconstructinghisspecialtheorywastoaskwhattheworldwouldlooklikeifhecouldrideonalightbeam;hewouldnotbeabletoseehisownimageinamirror,becauselightwouldneverleavehisface.

Ifthespeedoflightisthesameforallinertialobservers,thenournotionsofspaceandtimehavetobeadaptedandmergedtoallowforit.EinsteinreplacedNewton’sconceptsofabsolutespaceandabsolutetimewithasinglemergedentity:thefabricoffour-dimensionalspacetime(threespacedimensionsplustime)inwhichspaceandtimebecomeelasticandchangefordifferentobservers.Forexample,ifyouhavearulertravellingatgreatspeedpastyou,youwouldseetherulershortenedalongitsdirectionofmotion;thisiscalledtheLorentz–Fitzgeraldcontraction.Also,iftherewereaclockmovinguniformlyathighspeedpastyou,youwouldseeitrunningslow;thisiscalledtimedilation.Timedilationhasbeenverywelltestedinlaboratoryexperimentsandisaconsequenceofthefactthatallbodiesaremovingthroughspacetimeatlightspeed.ThespecialtheoryofrelativityineffectcompletedMaxwell’stheory.EinsteinsaidthatMaxwellwastheonlyoneofhispredecessorswhowasonalevelwithNewton.

Butthemostfar-reachingpredictionofthespecialtheorycamefromconsideringhowNewton’slawsofmotionhadtobemodifiedforbodiesmovingathighspeeds.Thespeedoflightisnature’sultimatespeedlimit,andnomaterialbodycanmovefasterthanthespeedoflight.Newton’ssecondlawtellsushowabody’svelocitychangesinresponsetoaforce,whichistherateofchangeofmomentum.So,ifaconstantforcewereappliedtoamass,therewouldbenolimittothevelocitythatitcouldattain,evenexceedingthespeedoflight,whichwouldviolaterelativity.Newton’sformulaforthemomentumofaparticlethereforehadtobemodified,witharelativistictransformationthatledtothemostfamousequationinphysics,thelawoftheequivalenceofmassandenergy:

TheequationtellsusthatthemassmandenergyEarereallythesamething;theyarejustmeasuredindifferentunits(thec2factorsimplyconvertsmassunitstoenergyunits).Ifanobjectistravellingveryfast,itcarriesalotofenergywithit,butthereisalimittohowmuchitcanhave.Evenwhentheobjectisstationaryithasa‘latent’amountofenergycalledtherest-massenergy,andthisistherealmeaningofthe‘m’intheequation.ThisisdifferentfromNewtonianmechanicswherethekineticenergyofastationaryparticleiszero,notafinitequantity.Therest-massenergyisenormousformatterbecausetheconversionfactorc2issolarge.Evenasmallmasscontainsahugeamountofenergy.Theenergyequivalentofonegramofmatter(theweightofabusinesscard)is25millionkilowatt-hours,enoughtoheatandlightalargecityforaday.Whydon’twenoticetheenergycontentofordinarymatterineverydaylife?Ifnoneofitisgivenoffexternally,nonecanbeobserved.Einsteinaskedustoimagineaverywealthybutparsimoniousman.Ifheneverspendsasinglecoinhowwouldweknowthatheisfabulouslyrich?

Matterasmass-energyEinstein’sfamousresultwasannouncedinoneofhisfourannusmirabiluspapersof1905.Ithasthetitle‘Doestheinertiaofabodydependuponitsenergy-content?’andinitheputthemassontheleft-handsideoftheequation:

Whileitismathematicallyidenticaltothebetter-knownform,putthiswaytheformulaanswersthequestioninthepaper’stitlebydefiningmassinenergyunits.Forexample,therestmassesofanelectronandaprotonarerespectively0.511MeV/c2and938.25MeV/c2.(Theelectronvoltunitofenergy,eV,isdefinedonp.55.)

Itiseasytoaddsomeenergytomattertoincreaseitsmass,forexamplebyheatingit.Ifyouboilakettleofcoldwatertherelativisticmassincreaseistiny(about5×10–13kilograms).Eveninenergy-liberatingchemicalreactions,asinburningfuel,againonlyanegligiblefraction(1partinabillion)ofthemassisreleasedasenergy.Lavoisier’slawofmassconservationisthereforeaverygoodapproximationforchemicalprocesses.ButinnuclearreactionsmuchmoreenergyisreleasedandEinstein’sequationisseenasthemostawesomeoneinsciencebecauseitisthebasisofnuclearpower.

ThenuclearreactionstowhichweoweourexistencearethefusionreactionsthattakeplacedeepinsidetheSun.There,thetemperatureishighenoughtoenablefourprotonstofusetogethertoproduceanalphaparticle(thenucleusofaheliumatom),whichhasarestmassof3727.3MeV/c2.Informingthealphaparticle,theprotonsloseasignificantamount(0.7percent)oftheirmass,andthedeficit,26.7MeV,goesintoheatenergy.Analphaparticleisnearly1percentlighterthanthesumofitsparts.

Evenhigherrest-massenergyfractionsarereleasedinthemostspectaculareventsintheuniverse,forexamplewhenblackholesmerge.Severalmergersofblackholepairshavenowbeenobservedviatheirgravitationalradiationsignals,whichwerefirstobservedin2015.Thesemergerscaninvolveblackholeswithmassesoftensofsolarmasses(asolarmassisthemassoftheSun).Thefinalstagesofthemergersoccurinafewtenthsofasecondandcreatesuchviolentrupturesinthefabricofspacetimethattheequivalentofafewsolarmassesisconvertedintogravitationalwaveenergy.Thereisanevenmoreextremeprocessinvolvingantimatter,whichwewilllookatinChapter6.Whenmatterencountersantimatter,itannihilatescompletely.Alloftherest-massenergyisconvertedtoradiationwith100percentefficiency;insciencefictionthisprocesshasbeenenvisagedasthepropulsionsystemfortheStarshipEnterprise.

CurvedspacetimeNewton’sgreatdiscoveryofthelawsgoverningthemotionsoftheplanetshidaprofoundlysubtleandremarkablefactaboutthenatureofmass.Newtoncalculatedtheorbitofaplanetbycombininghissecondlawofmotion(force=mass×acceleration)withthelawofgravity(theforcebetweentwomassesisproportionaltothemassofone×themassoftheother).Buttherearetwodifferenttypesofmassinvolvedhere.Themassthatappearsinthelawofmotionisconnectedwithabody’sresistancetoachangeinitsmotion,itsinertia.Thisisitsinertialmass,whichhasnothingtodowithweight.TheinertialmassofabodyisthesamewhetheritisonthesurfaceoftheEarthorisindeepspacefarawayfromanyplanet.Butthemassthatappearsinthelawofgravityisconnectedwithgravityandweight.Anobjectwithmoremassisattractedmorestronglytoothermassesbygravity.Inthiscasethemassiscalledgravitationalmass,andcanbethoughtaboutassomethinglikethe‘charge’ofgravity.

ItwasimplicitinNewton’sformulationofplanetarymotionthatinertialmassandgravitationalmassareoneandthesamething,anassumptionthatisbynomeansobvious.Einsteinelevatedtheequalityofthetwotypesofmasstoaprinciple:the

principleofequivalence,whichhetooktobethefoundationofhisgeneraltheoryofrelativityof1915.Itisoneofthegreatachievementsofscience.Inthis,herecognizedthesignificanceofGalileoGalilei’sfamousexperimentofdroppingalightballandaheavyball,whichareobservedtofallwiththesameaccelerationinthegravityoftheEarth.Butimaginerepeatingthesameexperiment,thistimeinsidearocketshipoutinspace,farawayfromanygravitatingmatter.Iftheshipfiresitsengineandstartsaccelerating,thedroppedmasseswouldappeartoacceleratetowardsthefloor.Wecanimaginethattheobjectsareatfirstjustfloatinginspaceand,whentherocketsarefired,thefloorapproachesthematanever-increasingspeed.Toanon-boardobserver,thismotionmimicstheaccelerationofgravity.Theobserverthereforecannottelliftheshipisbeingaccelerated,orifalargegravitationalmasshassuddenlyappearedbehindthevessel.

IttookEinsteintenyearstoincorporatetheeffectsofgravityandaccelerationinthemorecomplexandmathematicallydemandinggeneraltheory.TodothishehadtoabandonthenotionthatspacetimecanbedescribedbyregularEuclideangeometryand,inthepresenceofgravitatingmasses,mustinsteadbebasedonthegeometryofcurvedsurfaces.Whatdoesitmeantosaythatspacetimeiscurved?IntheplanegeometryofflatEuclidianspace,theanglesofatriangleaddupto180°.ButifyoudrawabigtriangleonthesurfaceoftheEarth,youfindthattheanglesdon’taddupto180°,becauseofthecurvatureoftheEarth.Ifyouimaginedrawingatriangleinspace,nearanobjectwithagravitationalfield,thegeometryofspaceitselfiscurvedandhereyouwouldagainfindthattheanglesofthetriangledon’taddupto180°.

Theprincipleofequivalenceledtoaprofoundviewofgravity,asthemanifestationofthegeometryofspaceandtime.Thefactthatdifferentmassbodiesfallinginagravitationalfieldallacceleratebythesameamountandsofollowthesamepathsrevealsadeeptruth:thepathsthatthebodiesfollowarethemselvesinherentfeaturesofspacetime.Mattercurvesspacearounditself.Imaginearubbersheetwithanobjectwithlargemass-energyliketheEarthsittinginthemiddlemakinga‘dent’init(Figure11),whichrepresentsthecurvatureofspacearoundamassivebody.TheMoonfeelsthecurvatureofspaceandmovesinitsorbitaroundtheEarth,asifitwereamarblerollingaroundinsideabowl.TheconceptofNewton’sforceofgravitywasreplacedbythecurvatureofspacetime.

11.ThecurvatureofspaceroundthemassivebodyoftheEarth.

Thecurvatureofspacearoundamassnotonlyaffectsthemotionofmasses,butitalsodeflectslight.In1919ArthurEddingtonfamouslyputEinstein’stheorytoakeytestduringatotaleclipseoftheSun.TheapparentpositionsofstarsobservedneartheedgeoftheSunwereseentobedisplacedfromtheirnormalpositionsontheskybypreciselytheamountthatwaspredictedbyEinstein’stheory.

Generalrelativityisthecorrecttheorytouseindescribingwhathappensnearmatterthathassufferedextremegravitationalcollapse.Theultimatelight-bendinglimitoccursintheimmenselystronggravitationalfieldofablackhole,themostcompactformofmatterweknowof.IfallthematterintheSun,whichhasadiameterof1.4millionkilometres,werecompressedintoanobjectjust3kilometresacross,thegravitationalfieldatthesurfacewouldbesoenormousthatspacetimewouldwrapitselfaroundtheobjectandablackholewouldform.Lightwouldbebentsostronglyatthesurfacethatitwouldbecompletelytrapped,aswouldanymaterialobjectfallingintoit.WecanrestassuredthatotherprocesseswillpreventtheSunfromcollapsingintoablackhole,butthisexampleservestoshowhowextremethephysicalconditionswouldbe,aswellastheinherentweaknessofgravity.Spacetimeissomassivelydistortedaroundablackholethattimeitselfappearstobefrozenatthesurface.TheeffectofgravityonthetimeindicatedbyclocksisevenimportantforsatellitesorbitingintheweakergravityoftheEarth.Theglobalpositioningsystem,orGPS,onwhichwedependforournavigation,wouldnotworkifgeneralrelativitywerenotconsidered.

Tosummarize;inthe19thcentury,electromagneticforcescametobeunderstoodaselectricandmagneticfieldsfillingemptyspacewhichcoupletogetherandoscillateaswaves.Infact,lightisanelectromagneticwave.Einsteinbuiltthespecialtheoryofrelativityontwopostulates:thatthespeedoflightisthesameforallobservers,andthelawsofphysicsarethesameforobserversmovingatuniformspeeds.Fromthese,hededucedthatspaceandtimemergeintoasinglefabric:spacetime,inwhichspace

andtimearedistortedwhenseenbydifferentobservers.Newton’slawsofmechanicsalsohadtobemodified,whichrevealedthatacentralpropertyofmatter,itsmass,isequivalenttoenergy.WhenEinsteinincludedacceleratedmotionandgravityasthebasisofhisgeneraltheoryofrelativity,accelerationandgravitywerefoundtobeequivalent.Thegeometryofspacetimeisdeterminedbythedistributionofmass-energyinspace.ThephysicistJohnWheelerputitsuccinctly:mattercurvesspace,andspacetellsmatterhowtomove.

Einstein’sgeneraltheoryofrelativityrevolutionizedphysics.InChapter5,Iwillturntotheworldofverysmallparticlesofmatterandthesecondgreatrevolutionin20th-centuryphysics:quantummechanics.

Chapter5

Thequantumworldoftheatom

Bytheendofthe19thcentury,thephysicallawsdescribingthebehaviourofmacroscopicchunksofmatterhadlargelybeenestablished.TheseincludedNewton’slawsofmotionandgravity,whichcouldpredictphenomenarangingfromthecollisionsofbilliardballs,tothemotionsoftheplanets.Therewerealsothelawsofthermodynamics,Boltzmann’sstatisticalmechanics,andtheunificationofelectricity,magnetism,andopticsinMaxwell’slawsofelectromagnetism.Butthestructureandtheinnerworldofatomsandmoleculesremainedamystery.Therewereunresolvedpuzzles,suchastheoriginofsharpatomicspectrallines,seeninthelightfromthestars.Thewavelengthsoftheselinescoincidedtellinglywiththoseemittedbychemicalelementsheatedinlaboratoryflames.Thischapterchartsthestoryofhow,inparallelwithprofounddiscoveriesofthepropertiesofatoms,therevolutionarytheoryofquantummechanicswasborn.

Severalimportantcluesaboutthestructureofatomshadalreadybeenuncovered,particularlyduringthefinaldecadeofthe19thcentury.TheseincludedthehighlypenetratingX-rays(byWilhelmRöntgen)andradioactivity(byHenriBecquerel).Radioactiveatoms,suchasuranium,fireoffpartsofthemselvesatgreatspeeds,disintegratingspontaneouslyandemittingdifferenttypesofionizingradiationorparticles.Onekindofradioactivity,gammarays,ishigh-energyelectromagneticradiation.Anotherkindconsistsofpositivelychargedalphaparticles,whichareverystableentities,andwereshownbyErnestRutherfordtobeheliumnucleiejectedatspeedsof10,000km/s(about30,000timesthespeedofsound),andthethird(betaparticles)arehigh-speedelectrons.

TheheartofmatterToprobeinsidetheatom,Rutherfordhitontheideaofusingtheheavyweightalpha

particlesasatomic‘bullets’.In1912,withErnestMarsdenandHansGeiger,hefiredalphaparticlesatathingoldfoil(Figure12).Mostofthefastparticlesploughedstraightthroughthemillionsofgoldatomstheyencountered,brutallyshovingasidethelightweightelectrons.Butoccasionallythealphaparticleswerescatteredthroughsmallangles,fewerweredeflectedbylargerangles,andasmallfractionwereevenscatteredbackwards.InRutherford’sownwords:‘itwasasifyouhadfireda15-inchshellatapieceoftissuepaperanditcamebackandhityou’.Theexperimentshowedthatinthecentreoftheatomisatiny,positivelycharged,andheavynucleus.Occasionallyanalphaparticlemightstrikeagoldnucleushead-on,feelitsstrongelectrostaticrepulsion,andrebound.Thesingleprotonnucleusofahydrogenatomhasadiameterof10–15m,about100,000timessmallerthantheatomitself.Atomsaremostlyempty.IfahydrogenatomwereenlargedfourtrilliontimestothesizeofLondon’sWembleystadium,itsnucleuswouldbethesizeofapea.

12.Rutherford’sexperimenttoscatteralphaparticlesfromgoldatomsledtothediscoveryoftheatomicnucleus.

Thenucleusissmall,massive,andpositivelycharged,butofwhatwasitmade?Ahundredyearsearlier,Dalton’schemicalexperimentshadrevealedthesimplenumberratiosoftheatomicweightsoftheelements,inunitsoftheweightofthehydrogenatom.Itwasguessedthatnucleiweremadefromthebuildingblocksofprotons,thenucleiofhydrogenatoms.In1919Rutherfordconfirmedthisbyknockingprotonsoutofnitrogen,againusingalphaparticles.

Thediscoverythatthenucleuscontainsprotonsraisedadeeperquestion.Inmovingalongtheperiodictable,theatomicweightofanelementincreasesmuchfasterthan

itsatomicnumber.Forhydrogentheatomicnumberandweightareboth1.Inhelium(atomicnumber2),theweightisfourtimeslarger.Bythetimewegettothe92ndelement,uranium,theatomweighsabout238timesasmuchashydrogen.Somethingwasneededtomakeupthedeficitbetweenthechargeandthemassinthenucleus,andRutherfordguessedthat,inadditiontoprotons,otherheavyparticlesmustbepresent.Hecalledtheseneutronssincetheycarrynocharge.JamesChadwickdiscoveredtheneutronin1932whenhefiredalphaparticlesatberylliumatomsandknockedneutronsoutofthem.Aneutronweighsatenthofapercentmorethanaproton;butapartfromitselectricalneutrality,thetwoaresimilar.Theprotonsandneutronsinanucleusarecallednucleons(Figure13).Elementswiththesamenumberofprotonsintheirnucleiallhavethesamechemicalproperties,buttheycanhavedifferentnumbersofneutrons,whichmakesthemisotopesofthatelement.Forexample,helium-4hastwoprotonsandtwoneutronswhereashelium-3hastwoprotonsandonlyoneneutron.

13.Representationofanatomshowingtheneutronsandprotonsclusteredtogetherintheatomicnucleus,surroundedbyorbitingelectrons.(Caution:theimageofthenucleushasbeenmagnifiedbyover100,000timesrelativetotheelectronorbits;ifitweretoscale,itwouldbeinvisible.)

Howisthenucleusheldtogether?Withuptoaround100protonscrammedintoitstinyvolume,themutualelectrostaticrepulsioninthenucleusisenormousandadifferentforce,thestrongnuclearforce,bindsthewholeclusterofparticlestogether.Thestrongforceismuchmorepowerfulthantheelectricalforce,andoperatesoveramuchshorterrangeofaround10–15metres.Wecanappreciatehowstrongthisforceisbycomparingitwithhowtightlyanelectronisboundtoanatom.Theenergyunitthatisconventionallyusedonthemicroscopicscaleofmatteristheelectronvolt,oreV.Theworkneededtomoveanelectronthroughanelectricpotential,saybetweentheterminalsofa1.5-volt‘AA’battery,is1.5eVofenergy.(Thereare6.25×1018eVinoneJoule.)Toremoveanelectronfromanatom,ittakesabout15eVofenergy,whichcouldbesuppliedby10AA-sizebatteries.Buttoremoveaprotonfromanucleusitwouldtakeabout10MeVofenergy,whichisequivalenttosevenmillionbatteries.Ifyoucouldreachintoanatomicnucleuswithastrongpairoftweezersandpluckoutaneutron,theforceneededwouldbeaboutthesameasliftingaheavysuitcase.

QuantizingtheatomThediscoveryoftheatomicnucleusledtoapictureoftheatomasaminiaturesolarsystem,notunlikeFigure13,inwhichthenucleussitsinthecentreliketheSun,withtheelectronsorbitingarounditlikeplanets.However,itwasquicklyrealizedthatiftheelectronsbehavedclassically,atomswouldrapidlyradiateawayalltheirenergyandcollapseundertheattractiveforcebetweenpositiveandnegativecharges.Wheredoesthestabilityofatomscomefrom?

Thekeyconceptthatwasneededtounderstandatomicstabilityhadalreadybeenproposedin1900,butinadifferentareaofphysics—thestudyoftheradiationemittedbyhotbodies.Instudyingthermalradiation,GermanphysicistMaxPlanckhadaradicalidea:radiationenergymustbequantized.Planckproposedthattheenergyoftheelectromagneticfieldisbundledupindiscretepackets,orquanta,ratherthanformingwaveswithacontinuousrangeofenergiesrightdowntozero,aspredictedbyMaxwell’stheory.TheenergyofPlanck’squantumisproportionaltothefrequencyoftheradiation,sothataquantumofhigh-frequencybluelightcarries,forexample,roughlytwiceasmuchenergyasoneofredlight.Theconstantofproportionality,h,isafundamentalconstantofnature,calledPlanck’sconstant.Ithasthesameunitsasthemomentumofaspinningbody,orangularmomentum.Planck’sconstantisaverysmallnumberanddefinesthesmalleststepstakenbynature,thequantumofaction.

SoonafterPlanck’srevolutionaryproposal,Einsteinextendedthequantizationideabyshowingthatlighttravellingthroughspaceconsistsofparticles,calledphotons,carryingdiscretepacketsofenergy.Thisideaexplainedawell-knownphenomenoncalledthephotoelectriceffectinwhichmetalssuchaszinccanbemadetoejectelectronsbyshininghigh-frequencyultravioletlightonthem.Topriseanelectronoutofametal,aminimumamountofenergyhastobedeliveredtoit,andforthisthehighfrequencyofthephotoniscrucial.Ifthefrequencyofthelightisbelowacriticalvaluenoelectronsareemittedhoweverintensethelightsource.

TheDanishphysicistNielsBohrappliedtheideaofthequantizationofenergytotheelectronicstructureofatoms.Hetheorizedthatanelectronmovinginacircularorbitaroundthenucleuscouldonlyoccupydiscreteorbitswithwholenumbervalues(1,2,3…)oftheangularmomentum,inunitsofh/2πThewholenumbervaluesarecalledprincipalquantumnumbers.Theenergiesofthedifferentorbitsformadiscreteladderofallowedstateswithunequallyspacedrungs(Figure14).Thequantumruleisthattheelectroninanatomisallowedtositonlyontherungsoftheladder,butneverinbetweenthem.Itcanjumpuptoahigherrung(anexcitedstate)byabsorbingaquantumofenergyequaltothedifferenceinenergybetweentherungs,orfalldowntoalowerenergyrungbyemittingaphotonwithaspecificenergyandthereforefrequency.Thelowestrungontheladderiscalledthegroundstate,and,becausetherearenostatesbelowit,electronsarepreventedfromcrashingintothenucleus;itisthisthatmakesatomsstable.Inadditiontoexplainingthestabilityofatoms,Bohr’smodelalsoresolvedthelong-standingquestionoftheoriginofthesharpatomiclinesinatomicspectra.Whenheusedthetheorytopredicttheobservedwavelengthsofthehydrogenlinespectrum,hismodeloftheatomwasimmediatelyhailedasamajortriumph.

14.Theladderofquantumenergylevelsofthehydrogenatomshownasaseriesofhorizontallines.Theenergiesarenegativenumbersbecausetheyrepresenttheenergyneededtoremoveanelectronfromagivenleveltoinfinity.Theprincipalquantumnumbersarelabelledbyn.

ParticlesandwavesNatureimposesstrictquantumrulesonmatter.Anelectroncannothaveanyenergyintheatom,butisconstrainedtoaladderofdiscretequantumenergies.Howisitpossibletomakesenseoutofthisstrangebehaviour?Thekeyistherealizationthatmicroscopicparticlespossesswavelikecharacteristics.Wavesofalltypeshavebasicfeaturessuchaswavelength(thedistancebetweensuccessivewavecrests)andamplitude(theheightofthewave).Theycanpropagatefreelythroughspaceastravellingwaves,orasstationaryorstandingwaves,forexamplethevibrationsofaguitarstring.Electronsinatomsbehavelikestandingwaves.

Thequintessentialpropertyofwavesisinterference.Ifyouthrowtwostonessimultaneouslyintoastillpond,thecircularripplesfromeachspreadoutandcross.Wherethewavesmeet,eachaddstotheother(calledlinearsuperposition).Wheretwowavecrestsmeet,theyaddtomakealargedoubleheighthump(constructiveinterference);whentwotroughsmeet,theymakeatroughofdoubledepth.Ifacrestandatroughmeet,theycanceleachotheroutsothatthewatersurfaceislevel

(destructiveinterference).Lightshowstheseinterferencepatterns,andin1801ThomasYoungperformedhisfamousdoubleslitexperiment(Figure15),whichestablisheddecisivelythewavenatureoflight.Whenlightisshoneonapairofcloselyspacedparallelslitscutinascreen,thewavesspreadoutonthefarsidetoproduceapatternofbrightanddarkinterferencebandsofconstructiveanddestructiveinterference.

15.ThomasYoung’sdoubleslitexperimentshowingbrightanddarkinterferencebandsorfringeswhichareproducedbytheconstructiveanddestructiveinterferenceofwavesemergingfromtheslits.

AfterEinstein’sexplanationofthephotoelectriceffectintermsofphotonsithadbecomeclearthatlighthasbothaparticlelikeandwavelikecharacter.Wave–particledualitywas(andstillis)ahardconceptforustodigest;buttheevidencefortheexistenceofphotonsandYoung’swavelikeinterferencepatternsisirrefutable.In1924anideaoccurredtoayoungFrencharistocrat,PrinceLouisdeBroglie.Iflighthasadualnature,thenwhyshouldn’tthesmallestparticlesofmatteralsohaveawavelikecharacter?DeBrogliedefinedthewavelengthofaquantumparticle,thequantumdeBrogliewavelengthλdB,intermsofitsmomentummv(mass×velocity):

ThesizescaleonwhichwavelikequantumeffectsareimportantinmatterisfixedbythemagnitudeofPlanck’sconstant,inthenumerator.Theparticle’smomentumappearsinthedenominator,whichpredictsthatthemoremassiveapieceofmatterisand/orthehigheritsspeed,thesmalleristhequantumwavelength.Thismeansthatmacroscopicobjects,likebilliardballs,havedeBrogliewavelengthsthatareunnoticeablysmall.Butonthescaleofatomsthewavenatureofmatteriscentral.

DeBroglieconceivedofwavesthatcorrespondtofreelymovingparticles.But,whenviewedasstandingwaves,theygiveinsightintohowBohr’selectronorbitsarequantized.Ifoneimaginesbendinganelectronwaveintoacirclearoundanucleus,thereareonlycertainwaysthatthiscanbedonesothatitsendsjoinupsmoothly.Themathematicalconditionisthatthecircumferenceoftheorbitmustcontainanexactnumberofwavelengths.Ifeitherthewavelengthorthecircumferencedeviatesevenslightlyfromthiscondition,theelectronwavewouldnotjoinupsmoothlyandwouldrapidlygetoutofstepwithitselfandcancelout.Thisisthereasonwhyelectronscannotsitinthegapsbetweentherungsonthequantumenergyladder.

TheuncertaintyprincipleInclassicalmechanics,Newton’slawsofmotiondescribecompletelythemotionofamacroscopicparticlemovinginaforcefield.Boththepositionandthemomentumofapoint-massaremathematicallywell-definedandindependentquantitiesanditstrajectoryissharplydefinedeverywhereinspacetime.Butthisisnottrueatthequantumlevelwhereparticlestravelfromonelocationtoanotheralongeverypossiblepaththroughspacetime.Itmakesnosensetothinkaboutpinpointinganextendedquantumobjecttoaregionofspacesmallerthanitsownwavelength.Furthermore,thewavelengthofaquantumobjectvariesinverselywithitsmomentumaccordingtodeBroglie’sformula,andthepositionandmomentumvariablesarenotindependent.Alargematterwavelengthimpliesasmallmomentumandviceversa.Positionandmomentumvariablesformspecialpairs,calledcomplementaryvariables.

In1926GermanphysicistWernerHeisenbergconstructedthefirstcompletetheoryofquantummechanics(calledmatrixmechanics,involvingmathematicalobjectscalledmatrices).Inthishefocusedonwhatitmeanstomakeameasurementofaquantumsystemandconcludedthattheknowledgewecanobtainaboutitisfundamentallylimitedbynature,withaprecisionthatisnotrelatedtoanyimperfectionsorlimitationsinthemeasuringapparatus.Aquantumentitydoesn’titselfknowwhereitisandhowfastorinwhatdirectionitisgoing—thequantumworldisinherentlyindeterminate.Heisenberghituponthekeyidea,theuncertaintyprinciple,which

assertsthatwecanknoweitherwhereaquantumobjectisorwhereitisgoing,butwecan’tknowbothatthesametime.Theprincipleunderpinsthewholeofquantummechanicsandtellsusthatthequantumentitiesinitareneitherpureparticlesnorpurewaves.

Theuncertaintyprinciplelinksthespreadoruncertainty(indicatedbytheΔsymbol)inaparticle’sposition(Δx)withthatofitsmomentum(Δp),sothattheproductofthetwoisgreaterthanorequaltoPlanck’sconstant/4π:

Howdoesthiswork?Supposewewanttomeasurethepositionandthespeedofanelectron.Wecanshinealightontheelectronandfindoutwhereitisandhowitmovesbythelightthatitscattersbacktous.However,theelectronhasaverysmallmassand,toavoiddisturbingittoomuch,weneedtoshineonlyadimlightonit.However,theelectronmustscatteratleastonephotonforittobeseenatall;whenitdoes,thereisanexchangeofenergyandmomentum.Thephotonthereforeinevitablyimpartsmomentumtotheelectron,aninteractionthatmakesitfundamentallyimpossibletoobserveaparticlewithoutdisturbance.Whataretheoptions?Wecanchoosetodisturbtheelectronaslittleaspossibleandsomakeamoreprecisemeasurementoftheelectron’smomentumbyusingalow-energylong-wavelengthphoton,illustratedontheleftofFigure16.Butwhenwedothat,sincewecannotpinpointanobjectinspacewithaprecisionthatissmallerthanthewavelengthofthelightused(inthiscaselarge),ourmeasurementofthepositionbecomesuncertain.Alternatively,wecanchoosetoreducetheuncertaintyinmeasuringtheelectron’spositionbyusingahigh-energyshort-wavelengthphoton(suchasagammarayphoton),asshownontherightofthefigure.But,withahigh-energygammarayphoton,the‘kick’receivedbytheelectronisnowsolargethatthemomentummeasurementbecomesveryuncertain.Theuncertaintyprinciplemakesitimpossibletorefinebothpositionandmomentummeasurementssimultaneously,toarbitrarilyhighdegreesofprecision.

16.IllustrationofHeisenberg’suncertaintyprinciple,inwhichthequantumofactionhasbeen‘squeezed’indifferentways.Theuncertaintyinmeasuringthemomentumofaparticle,Δp,isinverselyproportionaltotheuncertaintyofitsposition,Δx,sothattheareainmomentum-positionspace(indicatedbythegreyed-outrectangles)isproportionaltoPlanck’sconstant.Thelong,medium,andshortwavelengthsoflightusedtomeasureaquantumparticleareindicatedaboveeachrectangle,anddescribedinthetext.

Theuncertaintyprincipleisverypowerful.Amongitsmanyimportantconsequencesisthepropertythatwhenaquantumparticleisstronglylocalizedinspaceitexperienceslargefluctuationsinmomentumandthereforekineticenergy,knownaszero-pointenergy.Quantumentitiesconstantlyfluctuateintheirlowestenergystate.Theuncertaintyprincipledeniesmattereverquitereachingtheabsolutezerooftemperature,because,ifitdid,theparticleswouldhavepreciselocationsinspace,whichisforbidden.

Theconceptofquantumlocalizationenergyalsohelpsusunderstandthestabilityofatoms.Imaginebuildingahydrogenatombybringinganelectronandaprotontogetherfrominfinity.Thestartingpointistwowidelyseparatedparticles,andtheendpointisthequantum-restrictedmotionofanelectronmovingintheelectrostaticforcefieldofitsprotonnucleus.Astheelectronapproachestheproton,thekineticenergyitacquiresfromtheelectrostaticfieldmakesitjitteraboutincreasinglywildly,surroundingtheprotonasavibratingfuzzyball.Atfirst,theballislarge,butastheelectronradiatesawayitsenergyitshrinksdown,anditbecomesincreasinglyconfinedtothevicinityoftheproton,occupyingarestrictedvolumedefinedbysizeΔx.AsΔxgetssmaller,thespreadinitsmomentumΔpmustincreaseinlinewiththeuncertaintyprinciple.Inturn,thekineticenergyoftheelectronincreases,whichopposesanyfurthershrinkage.Eventuallythecompetingeffectsofelectrostaticattractionandthequantumlocalizationenergybalanceeachotherastheelectronrelaxesintoitsminimumenergygroundstate.Nature’scompromiseistoproduce

stableatomswithquantumladdersofelectronicenergylevels.

InterpretingthequantumworldShortlyafterHeisenbergpublishedhismatrixtheory,AustrianphysicistErwinSchrödingerpublishedanalternativeapproach,whichwasbasedonanequationdescribingthedynamicsofwavelikequantumparticlesmovinginaforcefield.TheSchrödingerequationissimilartoclassicalwaveequationsgoverningthewayripplesmoveoverapondorsoundwavespropagatethroughtheair.ItisthefundamentalequationofquantumtheoryandtheanaloguetoNewtoniandynamicsinthemicroworld.TheHeisenbergandSchrödingerformulations,althoughoutwardlydifferent,arephysicallyequivalent.

SolutionstoSchrödinger’sequationarecalledwavefunctions,anddescribewavesthatarelabelledwithasymbolΨ.Thisimmediatelyraisedthequestionofhowthewavefunctionshouldbeinterpreted.Allwaveshavepositiveandnegativevalues(forexampleseawavescanswellaboveordipbelowmeansealevel).So,ifaquantumparticleisrepresentedbyawave,howshouldoneinterpretnegativevalues?MaxBornassertedthatthesquareofthewavefunctionΨ2(whichisnevernegative)representstheprobability(andnotthecertainty)offindingtheparticleinaparticularregionofspace.

AsecondessentialfeatureofSchrödinger’sequationisthatitislinear.Inalinearequation,thesumoftwoormoresolutionsisitselfasolution.Thismeansthatthesumofwavefunctionsisalsoawavefunctionrepresentingamixedstatethatpermitsaquantumsystemtobeinmorethanonestateatatime.Eachquantumstatecanevolveindependently,asiftheotherswerenotthere.Thisfeaturegivesrisetoabizarreanduniquefeatureofthequantumworld,quantumsuperposition.Whenaquantumsystemisinasuperpositionofstates,itisnotpossibletospecifyitsphysicalcharacteristics.

Quantumsuperpositioncanhelpusmakesenseoutoftheoddbehaviourofhowanatomicelectroncanjumpbetweenrungsonthequantumladder,seeminglyvanishingfromonelevel(stateA)andpoppinguponanother(stateB),withoutpassingthroughanyintermediatestates.WhentheatommakesaquantumjumpfromstateAtostateB,theelectroninteractsbrieflywithaphoton.Duringthattime,thewavefunctionsoftheinitialandfinalstatesaresuperposed,andtheelectronissaidtobeinasuperpositionofstates.Thesuperpositionisamixtureoftwostates:theelectronisinstateA,andtheelectronisinstateB.Astheatomictransitionunfolds,the

wavefunctionforstateAgetsweaker,whileB’sgetsstrongeruntilfinallyitaloneremains.Aroughanalogytothisisthetransient‘squeak’thatisproducedbyaclumsilyblownwindinstrument,whenitflipsbetweentwomodesofoscillation.

Thereisakeyexperimentwhichgetstotheheartofthesuperpositionprinciple:Young’sdouble-slitexperiment,performedwithparticlesinsteadoflight.ThisexperimentwasafavouriteofRichardFeynman’s,aboutwhichhesaid:‘inrealityitcontainstheonlymystery…ofallquantummechanics’.Let’slookagainattheexperimentofFigure15,usingabeamofelectronsfiredatametalscreencontainingtwoparallelslits.Onthefarsideisaglassscreen.Ifelectronspassthroughtheslits,theywillstriketheglassscreen,andshowupastinyflashesoflight.Ifoneoftheslitsiscoveredup,thepatternofflashesontheglassscreenisfoundtobeuniform,withtheelectronspassingthroughtheopenslitandhittingtargetpointsonthefarscreenasiftheyweretinybullets.Iftheblockedslitisnowuncovered,theelectronscanpassthroughbothslitsandthepatternofflashesontheglassscreenchangesradically.Theuniformpatternnowswitchestoawavelikepattern,withbandsofconstructiveanddestructiveinterference.Significantly,electronsarenowbeingdeniedaccesstocertainregionsofthescreenthattheycouldreadilyhitwhenonlyoneslitwasopen.Thisdramaticresultcompletelycontradictsourmentalpicturethatelectronsbehaveliketinybullets;why,ifelectronsarelikebullets,shouldopeningasecondslitinfluencetheelectronspassingthroughthefirst?

Thisexperimentcanbetakenonestagefurther.Thefiringrateoftheelectrongunisnowturneddownsothatthereisonlyoneelectronflyingthroughtheapparatusatanytime.Withbothslitsopen,thepositionsoftheelectronhitsontheglassscreenarerecordedand,overtime,graduallybuildupawavelikeinterferencepattern,identicaltotheoneoftheearlierexperiment,whenbothslitswereopen.Buthowcanthisbeifthereisonlyoneelectronintheapparatus?Doesanelectronspreadoutandsomehowpassthroughbothslitsatthesametimeinordertointerferewithitself?Ifwetrytoidentifythroughwhichslitanelectronpasses,saybyscatteringaphotonfromit,theactofmeasurementdisturbsthestateoftheelectronandthepatternontheglassscreenimmediatelyflipsbacktotheuniformone.

Let’ssummarizethisweirdbehaviour.Ifnobodyisobservingtheelectron,itcanapparentlygothroughbothslitsatonce.Inthatcasetheelectronwavefunctionisinasuperpositionofstates:namely,theelectrongoesthroughthefirstslit,andthesameelectrongoesthroughthesecondslit.Butiftheelectronisobserved,itswavefunctionissaidtocollapseintoasinglestatecorrespondingtoitgoingthroughonlyoneslit.Ineffect,lookingandnotlookingattheelectroncreatestwodifferentexperiments,eachofwhichproducesdifferentresults.

Theconceptofthecollapseofthewavefunctioniscentraltothe‘Copenhageninterpretation’ofquantummechanics,asetofideasputforwardbyNielsBohrin1927.ThenamecomesfromthelocationofBohr’sinstitute.Bohrrecognizedthatourknowledgeofthequantumworldcomesonlyfromthemeasurementswemakeatthemacroscopiclevel,usingtypicallaboratoryapparatus.Becauseaquantumsystemisdisturbedwhenmeasurementsaremadeonit,itismeaninglesstoaskwhataquantumparticleisdoingwhennooneislookingatit.Theonlythingonecandoistocalculatetheprobabilitiesthataquantumsystemoccupiescertainstates.Theinstantthatanobservermakesameasurement,thesystemisforcedtocollapseintoauniquestate,whereuponthesystem‘decides’whatstateitisin.EinsteinwasneverhappywiththeprobabilisticbasisoftheCopenhageninterpretationand,inafamousseriesofexchangeswithBohr,famouslyannouncedthat‘Goddoesnotplaydice’,towhichBohrreplied:‘Einstein,stoptellingGodwhattodo.’Itturnsoutthathoweverwechoosetointerpretthewavefunction,theSchrödingerequationhaspassedexactingexperimentaltests.

ButevenSchrödingerwasnothappywitheitherhistheoryortheCopenhageninterpretation.Toillustratetheabsurditiesthatarise,heconcoctedahypotheticalthoughtexperimentcalledSchrödinger’scat.Heimaginedplacingacatinasealedbox,alongwitharadioactiveatomandadevicetoreleaseadeadlypoisongastheinstantthattheatomdisintegrates,anactionthatwouldkillthecat.Afteracertaintime,thechancethattheatomhasdisintegratedis50percent,andtheatomisinasuperpositionoftwostates:notdisintegratedanddisintegrated.Iftheatomhasdisintegrated,thepoisonhasbeenreleasedandthecatisdead.Otherwisetheatomisstillintact,andthecatisalive.Sincetheboxisclosed,wehavenowayofknowingwhetherthecatisaliveordead,andtheexperimentinvitesustoconsiderthatthesuperpositionofthetwostatesoftheatomhas‘leaked’outandaffectedthecontentsofthebox,includingthecat,whichmustalsobeinasuperpositionofbeingbothdeadandalive.IftheCopenhageninterpretationiscorrect,allisinlimbountilanobserverlooksinsidethebox.Atthispointthesuperpositioncollapsesandthecatbecomeseitherdeadoralive.Theabsurdityliesinthefactthatweexperiencerealworldcatsasbeingeitherdeadoralive,butneverboth.

Thisweirdstateofaffairshasencouragedseveralotherinterpretationsofquantummechanics.Oneofthesewasputforwardinthe1950sbyHughEverettandiscalledthe‘many-worldsinterpretation’.Inthis,thereisnocollapseofthewavefunction,andinsteadEverettpositsthateverythingthatcanhappendoeshappen.Thismeansthatwhentheradioactiveatomandthecatenterasuperposedstate,physicalrealitysplitsintotwoseparateandparallelversions:oneinwhichthecatisaliveandoneinwhichitisdead.Ifthisisextendedtoencompassallpossibleactsofmeasurementinthe

universe,physicalrealitysplitsintoamultiplicityofseparateuniverses,aconceptthathasencouragedcosmologiststoproposethemultiverse.Themultiverseisalargenumberofparalleluniverseswhichcompriseeverythingthatexists,includingallmatter,allphysicallaws,andthefundamentalconstantsofnature.Forafullerdiscussion,thereaderisdirectedtoMartinRees’sbookintheBibliography.

Ithasbeensuggestedthatquantummechanicsisnotthefinaltheory,butjustaverygoodapproximationtophysicalrealityonverysmallscalesandwithverydifferentlawsthatdescribemacroscopicbodies.Themicroscopicandmacroscopicworldsappeartobehermeticallysealedfromeachother.Yetitisalsotruethatmacroscopicbodiesaremadeoutoflargenumbersofquantumentities,anditisreasonabletoaskwhereisthedividingline?Thedouble-slitexperimenthasrecentlybeenperformedusingquitelargemolecules,buckyballs,inlieuofelectrons.BeamsoftheseC60molecules(eachofwhichweighsoveramilliontimesthatofanelectron)displayquantumsuperpositionandinterferenceeffects.

Oneaspectofthequantumworldisparticularlyspooky.Quantumentitiescanpassthroughseeminglyimpenetrablebarriersasiftheyareghosts,aphenomenoncalledquantumtunnelling.TheSTMimageofFigure4wasproducedbythequantumtunnellingofelectronsbetweenafine-tippedneedleprobeandthecarbonatomslyingunderthetip.Classically,theelectronsdonothaveenoughenergytoclimbovertheenergybarrieratthetipoftheprobe.However,thereisanalternativeformoftheuncertaintyprinciple(ΔEΔt≥h/4π)thatconnectstheuncertaintiesofanotherpairofcomplementaryvariables,energy(ΔE)andtime(Δt).ThisallowsaquantumparticletoborrowtheenergyΔEitneedstosurmountanenergybarrier,provideditpaysbacktheloaninatimeΔtgovernedbytheuncertaintyprinciple.Inthequantumworldtheconservationofenergycanbeviolated,provideditisatemporaryinfringementandtheloanispaidbackinfull.Thistransactionallowsquantumparticlestospreadoutintoclassicallyforbiddenregionsandpassthroughbarriers.

Quantumtunnellingoccurswhenalphaparticlesareemittedbyradioactivenuclei,andbetweenprotonsinthehotcoresofstars.Whentwoprotonsapproachclosetoeachothertheyencounteralargeelectrostaticrepulsion,whichtriestopushthemapart.However,theirwavelikenatureallowsthemtospreadoutandtunnelacrossthegap,bringingthemcloseenoughforthestrongnuclearforcetofusethemtogetherandliberatefusionenergy.Withoutquantumtunnellingthestarswouldnotshine,andwewouldnotexist.

MatterandforceIntheeverydayworldwetakeforgrantedcertainobjectsbeingidentical.Examplesarebilliardballswhichhavethesamemass,size,andcomposition.Billiardballscaneasilybelabelled,saybypaintingthemdifferentcolours,sothatastheymovearoundthebilliardtableduringagame,wekeeptrackofwheretheygo.Inthequantumworld,theconceptofidenticalorindistinguishableparticleshasanaltogetherdifferentandstrictermeaningwhichforbidsthembeinglabelled.Twoquantumparticlesareconsideredtobeidenticalifthecoordinatesoftheirwavefunctionscanbeswappedroundwithoutchanginganyoftheproperties.

Inthetwo-slitexperiment,forexample,ifwetrytodiscoverthroughwhichslittheelectronpasses,theobservationcausesitsquantummatterwavetogetoutofstepwithitself(alossofcoherence)andsothewavelikeinterferencepatternontheglassscreenisdestroyed.Theactofobservationistantamounttolabellingaquantumentity,whichnatureforbids.Thetrajectoriesofidenticalquantumparticlesareunobservable.Unlikebilliardballs,electronscan’tbepainteddifferentcolours.

Indistinguishableandidenticalparticlesrelatetohowmatterandforcearedifferentiatedatafundamentallevel.Atthehumanlevel,wenormallyperceivematterandforcetobedifferentkindsofthings.Matterisclearlysomethingtangible,andforcesseemtobenature’swayforchunksofmattertopushorpullotherchunksofmatteraround.However,inthemicroworld,forcesarethemselvescarriedbyparticles.Alltheparticlesoftheworldbelongtooneorotheroftwogreatclasses:fermionsandbosons.Matterparticlesarefermionsandforce-carryingparticlesarebosons,whichmediateforces.Thetwofundamentaltypesofparticlesgettheirnamesfromthedifferentstatisticallawsthattheyobeywhenlargenumbersofidenticalparticlescometogether.Fermions,namedaftertheItalianphysicistEnricoFermi,obeyFermi-Diracstatisticsandbosons,namedaftertheIndianphysicistSatyendraNathBose,obeyBose-Einsteinstatistics.

Symmetryplaysacentralroleindifferentiatingfermionsfrombosons.Inasystemofidenticalparticles,theprobabilityΨ2cannotchangeifanytwoofthemareexchanged.Onswappingapairofparticles,thisleadstotwopossibilities:eitherthesignofthewavefunctionchanges(Ψ→−Ψ),inwhichcasetheparticlesarefermionsandhaveantisymmetricwavefunctions,oritisunchanged(Ψ→Ψ),andtheparticlesarebosonswithsymmetricwavefunctions.

Fermionsandbosonseachhaveverydifferentproperties.Whenfermionsclump

together,theyavoidsharingeachother’squantumstatesandinsteadprefertospreadthemselvesacrosstheladderofquantumenergylevels,fillinguptheavailablestatesfromthegroundstateupwards(Figure17).Ifanextrafermionjoinsthecrowd,itmustoccupyahigherrungoftheenergyladderandthecombinedpieceofmatteroccupiesmorevolume.Theinabilityoffermionstosharethesamequantumstatepreventstheelectronsinanatomfromgettingtooclosetothoseofitsneighbours,andgivesrisetomanypropertiesofmatter,suchasitssolidity.Bosons,ontheotherhand,don’tcareaboutthestatesoccupiedbytheotherbosons,andevenprefertooccupythesamegroundstatelevel.Photonsarebosons.Insideeverycompactdiscplayerisalaser,inwhichhugenumbersofbosonsformasinglecoherentwavelikestatewithallthequantumwavesoscillatinginperfectsynchrony,likeasquadofwell-drilledsoldiersmarchingperfectlyinstep.Laserbeamscanbemadeasintenseasrequiredsimplybyaddingmorephotons.

17.Allparticlesintheworldareeitherfermionsorbosons.Matterparticlesarefermionsandforce-carryingparticlesarebosons.Identicalquantumparticlesclumptogetherontheladderofenergylevelsaccordingtotheirtype;fermionsaggregatewithoneparticleperstate,whereasbosonspreferentiallycollapseintothelowestenergygroundstate.

Whatdetermineswhetheraparticleisabosonorafermionisconnectedtooneofthemostmysteriousandotherworldlyofquantumproperties:spin.Quantumparticlescanhaveanintrinsicangularmomentumasiftheyarespinningaboutanaxis.Thereisnoexactcounterparttoquantumspininclassicalphysics.Theclosestclassicalanalogytospinisrotation,forexampletherotationoftheEarthspinningonitsaxis.Buthereisthecatch.TheEarthrotatesthroughanangleof360°onitsaxisintwenty-four

hoursandontwosuccessivedaystheSunrisesatvirtuallythesametime.But,byhowmuchmustaquantumparticleberotatedforittolookthesame?Likeanyfamiliarobject,abosoncanberotated360°forthis.Afermion,ontheotherhand,hastorotatetwice(720°)forthingstolookthesame.ItisasiftheEarthhadtorotatetwice,justtoarriveatthenextsunrise.Howweirdisthat!

Afermionisanyparticlewithoddhalf-integerspin(like1/2,3/2,etc.),whileabosonhasanintegerspin(0,1,2,etc.).ThespinsofsomebosonsandfermionsaregiveninTable1.Fermionnumbersinordinarymatterareconserved(exceptwhenannihilatedbyantimatterparticles,whichwewillcomeacrossinChapter6).Butbosonshavenosuchlimitation;theyarecreatedanddestroyedinvastnumbersbyanactionassimpleasswitchingonandoffalight.

Table1. Propertiesofsomebosonsandfermions

Whatdoesitmeantosaythatbosonsmediatetheforcesbetweenmatterparticles?Imaginetwoskaters,onfrictionlessice.Astheyglidetowardseachother,onethrowsaheavyballtotheother,whocatchesit.Theybothmoveoffinnewdirections,eachconservingmomentumandenergyintheexchange.Adistantobserveralsoobservestheskatersbutistoofarawaytoseetheballandseesonlythechangeintheirmotions.Theobserverwouldconcludethattheskatershaveinteractedviaaforce.Inthisanalogytheskatersrepresentfermionicmatterparticlesandtheballaforce-carryingboson.Theexchangeofaparticleillustrateshowforcesoperateonamicroscopicscale;however,macroscopicanalogiesofthequantumworldshouldalwaysbetreatedwithcaution.

Bosonsmediatetheforcesbetweenmatterparticlesbyflittinginandoutofexistenceasvirtualparticles;thesearethe‘heavyballs’ofthemicroscopicworld.WewillseethecentralroletheyplayformatterinChapter7.Howdoesavirtualphotonwork?Duringitsbriefexistence,avirtualphotonborrowsitsenergyfromthevacuum,asallowedbyHeisenberg’suncertaintyprinciple.Thefleetingexistenceofvirtualparticlesmeansthatevenazero-energysystemcanspontaneouslyproduceenergeticparticles.Whentwoelectronsrepeleachother,theelectromagneticforcebetweenthemiscarriedbyavirtualphoton,aspin-1vectorboson.Theparticleiscalledavectorbosonbecauseitisthequantumofavectorfield,andtherearethreepossiblespacedirectionsinwhichtheparticle’sspinaxiscanpoint.Quantummechanicsforbidsavirtualparticleeverbeingobserved.Eitherthevirtualquantumhastobeabsorbedinitsentiretybythereceivingparticleornotatall.

ThestructureoftheperiodictableOneofthegreatesttriumphsofquantummechanicsisthatitexplainsthestructureoftheperiodictable.WhenSchrödinger’sequationisappliedtoatoms,itpredictselectronwavefunctionsthatrepresentvariouskindsofstandingwaves.Thesearecalledorbitalsanddefinetheregionsofspacewhereanelectronismostlikelytobefound.Anatomicorbitalisspecifiedbyfourquantumnumbers,whichdefineanumberofdifferentorbitalshapesinthree-dimensionalspace.Thewavefunctionforthegroundstateofthehydrogenatomisasphericalball,withamaximumnearthenucleus,graduallydecreasingwithradius(Figure18).Butinanexcitedstate,theelectronspendsmoretimefurtherawayfromthenucleus.Someofthesehigher-energywavepatternsarelikehollowsphericalballs,withacentralcore,andothersaredumbbellshapedwithtwolobescentredonthenucleus.Therearethreepossibledumbbellorientations,oneforeachdirectioninspace,whichhavethesameenergy.Stateswithevenhigherenergieshavemorecomplexelectrondensitydistributions.

18.Sphericalanddumbbellshapedelectrondensitydistributionsforthehydrogengroundstate(left)andsomeexcitedstates.Thedensityrepresentsthewavepattern,andtheamountoftimeanelectronspendsatagivenpointnearthecentralnucleus.

Electronsfilluptheatomiclevelsfromthegroundstateupwards.In1925WolfgangPaulidiscoveredtheprinciplethatgovernsthewayelectronsfillthestates.Iftwoelectronsthatareinthesamestateareexchanged,Ψcannotchange.Sinceelectronsarefermions,thisimpliesthatΨ=0,inotherwordsthereiszeroprobabilitythattwoelectronsoccupythesamestate.ThisisthePauliexclusionprinciplewhichplaysakeyroleinatomicstructureandgivesrisetotheshell-likestructureoftheelementsintheperiodictable,forminggroupswith2,8,8,18,…,members.

Inbuildingupatomsbeyondhydrogen,theelectronclouddistributionsaresimilartotheonesshowninFigure18,but,owingtothelargernuclearcharges,theelectroncloudsarepulledinmoretightly.Thenextatomintheperiodictableishelium,withtwoprotonsinthenucleus.ThePauliprincipleallowstwospin-pairedelectrons,withopposed‘up’and‘down’spins,togointothegroundstatetoformthefirststableshellintheatom.Inhelium,theelectroncloudsformamorecompactballthaninhydrogen,andthisisreflectedinitslargerionizationenergyof24.6eV,whichisalmosttwiceasmuchasforhydrogen.

Thenextelementinthetable,lithium,hasthreeprotons.Theextrathirdelectronisexcludedfromjoiningthetwointhefirstshellandinsteadhastogointothenextavailablehigher-energystate,andstartsanewshellasavalenceelectron.There,itisfurtherfromthenucleusandpartiallyshieldedfromthenuclearchargebytheinnerelectroncloud.Becausetheouterelectronisonlyweaklyboundtotheatomitisfairlyeasytoremove.Thismakeslithiumchemicallyreactiveandeasilyionized,thehallmarkofelementsthatformmetals.Mostoftheelementsoftheperiodictablearemetals.Non-metalsaremainlyfoundintheupperrightofadiagonallinerunningfromborontoastatine.

Aprominentfeatureoftheperiodictableisits‘eightfold’periodicity.Thisarisesfromtherebeingfourbasicshapesofatomicorbitals:onesphericalwavefunctionandthreedumbbells,oneforeachspacedirection.Eachofthesecantakeapairofspin-opposedelectrons,makingeightstatesinacompleteshell.So,inmovinghorizontallythroughthenextrowofthetable,fromlithiumtoneon,eightelectronsareaddedonebyonetobalancethenuclearcharges.Neonhasaclosedshellofeightelectrons,whichmakesitachemicallyinertnoblegas,likehelium.Beyondneonisthereactiveelementsodium;theaddedvalenceelectronmuststartanewshell,andsotheprocessrepeats.

Inthischapterwehavelookedattheprofounddiscoveriesoftheearly20thcentury

whichexposedtheinnerstructureoftheatom,andtherevolutionarynewphysicswhichgrewupalongside:quantummechanics.Theatomicnucleusisverysmall,makingtheatommostlyemptyspace,inhabitedbytheelectrostaticfieldthatgripstheelectronsintheirquantumshells.Thenucleuscontainstwotypesofparticles,protonsandneutrons,whichareheldtogetherbythestrongshort-rangenuclearforce.ThesmoothandcontinuouselectromagneticfieldenvisagedbyMaxwellisreallymadefromswarmsofnumerousquanta.Thedividinglinebetweenthequantumandthemacroscopicworldsismarkedbythesizeofafundamentalconstantofnature—Planck’sconstant.Heisenberg’suncertaintyprinciplegovernsthebehaviourofthemicroscopicworld,andshowshowparticlescantunnelthroughbarriersthatareclassicallyinsurmountable.Theworldcontainstwotypesofparticles,fermionsandbosons.Fermionsarematterparticles,andbosonsaretheforce-carriers.Onlyonefermioncanoccupyagivenquantumstate,whichisthereasonwhyfermionicmatteroccupiesspace.Bosons,ontheotherhand,prefertocrowdintothesamequantumstatewheretheycanformcoherentfields.Thegreatsuccessofquantumtheoryisthatitexplainsthestructureoftheperiodictable,thepropertiesofatoms,molecules,materials,andlifeitself.

Althoughthequantumpropertiesofmatterarerestrictedmainlytomicroscopicscales,theycansometimesrevealthemselvesmacroscopically.InChapter6wewilllookathowaggregationsofverymanyparticlescanreveallarge-scalequantumeffects.

Chapter6

Quantummatter

Allmatterisquantummatter.Thelawsofquantummechanicsunderpinthebehaviourofmolecules,atoms,andsubatomicparticles.Undernormalconditionsthequantumwavelengthsofmatteraretoosmalltobediscernibleonmacroscopicscales.But,underspecialcircumstances,largepiecesofmattercontainingtypically1023particlescanmanifestlarge-scalemacroscopicquantumbehaviour.Theseformthefocusofthischapter.

Whenparticlescondensetoformamacroscopicpieceofmatter,twoconditionshavetobemetforittodisplaylarge-scalequantumeffects.First,theindividualentitiesthatmakeuptheaggregatemustlosetheir‘identities’,sothattheirquantuminteractionscanextendoverlargedistances,enablingparticlestointeractwithotherparticles.Second,theparticlesmustbefreelyexchangeable,sothatthesystemcan‘recognize’thatitcontainsidenticalparticlesandobeytheappropriatestatistics:Bose-EinsteinorFermi-Dirac.Thesecondconditionmeansthatparticlesneedtobeabletomovearoundeasily,astheycaninafluid,andthesystemswearediscussingareknownasquantumfluids.

Thereareimportantdifferencesbetweensystemsmadeofbosonsorfermions.Athightemperaturesthedifferenceisminimal,becausetheparticlesaresoenergeticthattheprobabilitythattwoareinthesamequantumstateisverysmall.Butatabsolutezero,thefermionsmusteachoccupydifferentstates,inaccordwiththeexclusionprinciple,whereasbosonscanalldropintothelowest-energystate.Fermionsfilltheladderofavailablequantumstatesuptoamaximumenergy,calledtheFermilevel,abovewhichthestatesareempty.

TheonsetofquantumbehaviourinagasisillustratedinFigure19.Athightemperatures,particlesmoveclassicallyinstraightlines,andonlychangetheir

directionsofmotionasaresultoftheirbriefbilliard-ball-likecollisions.Asthetemperaturefalls,themomentaoftheparticlesarereducedandtheirdeBrogliewavelengthsincrease.Atacertaincharacteristictemperature,thewavelengthsofindividualparticlesbecomecomparablewiththeiraverageseparation,andtheycaninteractwiththeothersinawavelikemanner.Thisinvolvestheentitiesbeingabletodiffractthroughgaps,andinterfereconstructivelyanddestructivelywitheachother.Thecharacteristictemperatureforonsetofquantumbehaviourdependsinverselyonthemassesoftheparticlesinquestion,andismuchlargerforelectronsthanitisforheliumatomsatthesamedensity.Iftheparticlesarebosons,afractionofthemcancondenseintothesamelow-energygroundstateandformwhatiscalledaBose–EinsteinCondensate(orBEC).Asthetemperatureapproachesabsolutezero,essentiallyallthebosonsavalancheintothisstate.ThetransitiontoaBECmarksaquantumphasetransitionfromclassicaltoquantumbehaviour,involvingcoherentmatterwaves.Aquantumphasetransitionthereforecontrastssharplywiththeclassicaltransitions,discussedinChapter3,inwhichthesolid,liquid,andgasphasesarisefromthecompetitionofthermalandinteratomicforces.ExperimentswithtwooverlappingBECs,eachcontaininglargenumbersofparticles,haveshownthattheycaninterfereinawavelikemanner.

19.Atomsinagasatdifferenttemperatures.Top:athightemperaturestheatomsbouncearoundlikelittlebilliardballs.Asthetemperaturedecreases(centre),thedeBrogliewavelengthsoftheatomsincreaseuntilthewavefunctionsstarttooverlap.Inthisregime(bottom)theatomslosetheirindividualidentitiesandquantumeffectsbecomeimportant.Iftheparticlesarebosons,theatomsallhavethesamewavefunctionandcansettleintoaBosecondensate.

Althoughtheexoticstatesofmatterdescribedinthischapterinvolveatomsatverylowtemperatures,thereisanexampleofaquantumfluidthatoccursundermore

temperateconditions.Itinvolveselectronsandexplainswhysomematerialsconductelectricityandothersdon’t.

ConductorsandinsulatorsAnelectriccurrentisthemovementorflowofchargesbetweenpointsinspace,orthroughconductingmaterials.Materialsthatdon’tconductelectricityareinsulators.Commoninsulatorsincludeglass,amber,manyceramics,plastics,andvariouscrystallinesolidssuchasdiamond.Themoleculesinthesematerialsareheldtogetherbystrongchemicalbonds,wheretheelectronsaregrippedtightlytotheirparentmoleculesandsoarepreventedfrommovingaroundinthematerialtoformacurrent.Metals,ontheotherhand,containmobileelectronsandarethemostcommonconductors.AswesawinChapter5,mostoftheelementsintheperiodictablearemetals.

Metallicconductionarisesnaturallyandspontaneouslyfromtheeasewithwhichavalenceelectroncanbeliberatedfromametalatom,resultinginapositiveionandanelectronwhichisfreetowanderoffintothematerial.Ametalwireiseffectivelya‘highway’forfreeelectrons.Whenawirejoinstheterminalsofabatteryandsocompletesanelectricalcircuit,theelectronsfeeltheelectricalforceofthebatteryand,beingmobile,respondbyflowingthroughthewiretowardsthepositiveterminal.

Theconductionelectronsinametalformacollectivefluid,whichisakindofplasma.Thepositiveionsinametalareimmersedintheelectronfluid,andthewholeassemblyisheldtogetherbytheattractionofpositiveandnegativecharges.Thisismetallicbonding,whichgovernsmanypropertiesofmetals,notonlytheirhighelectricalconductivities.Thefluidityoftheelectron‘glue’inametalenablestheionstoassembleintoclose-packedcrystalstructures.Also,theshininessorlustreofmetalscomesfromtheinabilityofelectromagneticwavestopassthrougharegionofmobilecharges.Thisexplainswhylightisreflectedfromametalsurface.

Apieceofcopperandadiamondareeachproducedbytheaggregationofahugenumberofcopperorcarbonatoms,yettheirelectricalconductivitiesdifferbyanenormousfactorof1020.Tounderstandwhythereissuchabigdifference,weneedtoconsiderwhathappenstothequantumladderofstateswhenalargenumberofatomscometogether.Let’sbuildupasolid,atombyatom.TworungsofthequantumenergyladderareshowninFigure20.Supposewebringtwoatomsclosetoeachother,sothattheirwavefunctionsoverlap.BecausethePauliexclusionprincipleforbidselectronsfromoccupyingthesameenergystate,theenergylevelssplit(knownas

hybridization)toformextraclose-spacedlevels.Asmoreandmoreatomsareaddedtothesolid,furthersplittingtakesplaceuntilthelevelsformbands,eachcontainingmanyfinelyseparatedstatesinenergy.Thereisonestateineachbandforeachatominthesolid.Crucially,thereisalsoabandgapseparatingthebands,inwhichtherearenostates.

20.Howelectronstatesinasolidareproduced.Left:twoenergylevelsforasingleatomaresharplydefined;centre:fortwointeractingatomstheenergylevelssplitintotwo;right:withmanyatomsthelevelssplitfurtherandformenergybands.

Thenextstepinbuildingupourmaterialistoseehowelectronspopulatethebands.Beingfermions,theelectronsobeyFermi-Diracstatisticsandsoexcludeeachotherfromoccupyingthesamequantumstate.Imaginepouringelectronsintothesolid,withoneelectrontoastatesoastofillupthebands,fromthebottomupwards.Therearetwopossibilities.Eitherabandiscompletelyfilledwithelectrons,oritispartlyfull.Ifitisfull,theelectronseffectivelyforma‘logjam’,andthesolidisaninsulator(Figure21).Inthiscase,theonlywaythematerialcanconductelectricityisforelectronstojumpacrossthebandgap,intothenexthigherbandwhereemptystatesareavailable.However,aconsiderableamountofenergyisrequiredtomakesuchabigjump,andinagoodinsulatorlikediamondthebandgapis5.47eV,whichisseveralhundredtimeslargerthantheenergyofroomtemperatureelectrons.Theelectronssimplycan’tacquireenoughenergyfromthermalfluctuationstojumpthatfar.

21.Howstatesarefilledbyelectrons.Theenergybandsarerepresentedbyboxes.Left:thepartlyfilledbandofametal;right:thefullyfilledbandofaninsulator.

ThingsareverydifferentinthemetalofFigure21,wheretheconductionelectronsonlypartiallyfillaband.AnelectronneartheFermilevelneedsonlytomakeverysmallenergyjumpstofindnearbyemptystateswhereitisfreetomovearound.Ametallicconductoristhereforecharacterizedbyapartlyfilledband.Sinceeachatomdonatesroughlyoneelectrontothemetal,theamountofspaceavailableforeachelectronisrestrictedtoavolumedefinedapproximatelybythespacingofthecrystallattice.Bytheuncertaintyprinciple,thisspatiallocalizationforcestheconductionelectronsintohigher-energyquantumstates,withanequivalenttemperaturethatisoftheorderof100,000K.Thisiswellabovetheboilingtemperatureofmetals,andsotheconductionelectronsinmetalsformaquantumfluid.MatterwhichhasahighenoughdensityforitspressuretoarisefromthePauliexclusionprinciple,andnotfromthermalmotion,iscalleddegeneratematter.Thefreeelectronsinametalcanberegardedasadegenerategas,whiletheremainderoftheelectronsinitareinboundquantumstates.

Althoughconductionelectronswanderfreelythroughthebodyofametal,thereisanenergycostthatcomeswithelectricalconduction.Electronsscatterfromimperfectionssuchasimpurityions,varioustypesofcrystallatticedefects,andfromthermalvibrations.Whentheyscatter,theelectronslosekineticenergythatendsupasheat.Scatteringisaformoffrictionthatresultsinelectricalresistance,whichiswhyelectricfireskeepyouwarm,andwhycomputermicroprocessorchipsneedtobecooled.

SuperfluidsandsuperconductorsSuperfluidsandsuperconductorsdisplaysomeofthemostdramaticandexoticmacroscopicquantumphenomenaofallthestatesofmatter.However,onlyveryfewtypesofatomsformquantumfluidsbecausemostmaterialsfreezesolidattemperatureswellabovethelevelwherequantumeffectsareimportant,andtheyfailtomeettherequirementforparticleexchange.Thebest-knownexamplesofatomsthatdoformquantumfluidsarethenoblegasisotopes,helium-4andhelium-3,whichremainliquiddowntoabsolutezeroatnormalpressures.

Forhelium-4thecharacteristictemperaturefortheonsetofquantumbehaviouris3K,whereasforthelighterhelium-3isotopeitisabout5K.Thequantumspinruleforacompositeparticle,suchasanatom,isthatifitcontainsanevennumberoffermionsitisaboson,otherwiseitisafermion.Theprotons,neutrons,andelectronsthatmakeupatomsareallspin-½fermions,andthespinsalwaysalignparallelorantiparalleltoeachother.Itfollowsthatasystemwithanoddnumberoffermionsmustitselfhavehalf-integralspin.Ahelium-4atomiscomposedofsixfermionsandisthereforeaboson,andoneofhelium-3withitsfivefermionsisafermion.

Thestudyofmatterattemperaturesclosetoabsolutezerobeganin1908whenDutchphysicistKamerlinghOnnessucceededinliquefyingheliumatatemperatureof4K.Bypumpingontheliquidtolowerthepressure,evenlowertemperaturescouldbeattained,makingitsomeofthecoldestmatterintheuniverse,withtemperaturesbeloweventhatofthecosmicmicrowavebackgroundtemperatureof2.7K.

Normalfluidsareviscous.Viscosityistheinternalfrictionor‘stickiness’betweenthemoleculesinafluidthattendstoresistuniformflow.Ittakesenergytoovercomeviscosity;thinkofswimminginapool.In1937,RussianphysicistPyotrKapitsadiscoveredthatwhenhelium-4iscooledbelowatemperatureofabout2K,theliquidspontaneouslymakesaphasechangetoanexoticsuperfluidphaseinwhichtheviscositydroppedprecipitouslybyafactorofmorethan1011.Superfluidhelium-4hasarangeofdramaticquantummechanicalproperties,suchastheabilitytoflowthoughverynarrowcapillarytubeswithoutanyfriction,andformathinfilm(calledaRollinfilm),whichcancreepupthesidesofacontainingvesselandclimbout(Figure22).Whenitisheated,asuperfluidproducesaspectacularspoutingfountain(Figure23).Belowthequantumcharacteristictemperature,allthebosonicatomsofhelium-4tendtocondenseintolowestenergyquantumstate,andformaBoseCondensate,inwhichtheoverlappingmatterwavesformasinglecoherentquantumentitywherebillionsofatomsactasiftheywereonegiant‘atom’movingcollectively.Thisisanalogoustoabattalionofperfectlydrilledsoldiersmarchinginstep.Ifonemoves,theothersmust

follow.

22.Thecreepofsuperfluidheliumoverthesidesofitscontainertodripoutunderneath.

23.Thefountaineffect.Mildlyheatedsuperfluidheliumproducesaspoutingfountain.

Thequantumcoherenceinsuperfluidsexplainswhysuperfluidheliumcanflowfrictionlesslythroughfinecapillarytubes.Normalviscousliquidscannotflowthroughsuchnarrowchannels,sinceanyslightirregularitiesorroughnessonthewall

scattersfluid,creatingviscousforceslargeenoughtoblocktheflow.Butinasuperfluid,theviscousforcesarerenderedineffectivebecauseeveryatommustobedientlyfollowalltheothers,andmacroscopicflowbecomespossible.Collectivebehaviouronamacroscopicscaleisalsofoundinalaser,inwhichaBosecondensateofspin-1photonsoccupiesasinglequantumstate,inwhichthereisnolimittothenumberofparticles.AlaserbeamisacoherentlightwaveandaBosecondensateofhelium-4isacoherentmatterwave.

AfterOnneshaddiscoveredhowtoliquefyhelium,heusedthetechniquetocoolmetalsdowntoverylowtemperaturetostudytheirelectricalconductivities.In1911hewasastoundedtodiscoverthattheresistanceofmercurysuddenlyfellprecipitouslytozerobelowatransitiontemperatureofabout4K.Thiswasthediscoveryofsuperconductivity.Thecurrent(orsupercurrent)thatflowsinasuperconductordoesnotdecaymeasurablyovertimescalesofyears.Othermetallicsuperconductorsweresoondiscovered.Akeydefiningpropertyofasuperconductoristhatitcompletelyexpelsallmagneticfieldsfromitsinterior,exceptinaverythinsurfacelayer.Asuperconductordoesthisbyproducinginternalcurrentsthatgeneratemagneticfieldstocanceltheexternallyappliedones.ThisisknownastheMeissnereffect,anditcanhavespectacularconsequences.Ifabarmagnetisloweredintoasuperconductingbowl,theworkdoneinexpellingthemagneticfluxcansupporttheweightofthemagnet,whichislevitatedandfloatsabovethebowl.

Superconductivitywaseventuallyunderstoodtobearathersubtleeffect.In1957JohnBardeen,LeonCooper,andRobertSchrieffer(collectivelyknownasBCS)deducedthatpaired-upelectrons(calledCooperpairs)carrythesupercurrent.Inthenormalnon-superconductingstatetwoelectronsrepeleachother.Butinlow-temperaturesuperconductors,theCooperpairsresemblebosonsthatinteractdynamicallywithsmalldistortionsintheionlattice,toformaweakcollectiveboundstate.ACooperpairhasbeenpicturedasbeingsomethinglikeatwo-electron‘molecule’withinthesuperconductor,anentitythatcanexistoverdistancesofmanymillionsofatomicspacings.ThetheorydescribesCooperpairscondensingintoasinglemacroscopicquantumcondensate,asuperfluid,andmovingasasingleunit.ThisisreminiscentoftheBECphaseinhelium-4;butinthiscasethefluidcontainschargedparticlesandthereforecanbemanipulatedwithelectricfields.ProvidedtheCooperpairsremainintact,thesuperfluidflowsthroughinthesuperconductorwithnoelectricaldissipationorresistance.

TheweakbindingofCooperpairsmakestheconventionallow-temperaturesuperconductingstateafragileonethatcaneasilybedestroyedbythermalfluctuationsandsuchmaterialsgenerallyhavetransitiontemperaturesbelowabout20

K.Butin1986,GeorgBednorzandAlexMuellermadeasurprisediscoveryofaremarkablenewclassofceramicsuperconductors,withmuchhighersuperconductingtransitiontemperatures.Thesehigh-temperaturesuperconductorsarecurrentlynotunderstoodand,todate,thehighesttemperaturesuperconductorknownisarelativelycommoncompound,hydrogensulphide(H2S),which,underveryhighpressures,isasuperconductorat203K(−70°C).Theultimategoalistodiscoveraroomtemperaturesuperconductor.Suchamaterialcould,forexample,openthepossibilityoftransmittingelectricalpowerveryefficientlyoverlargedistances.

TheJosephsonjunctionIftwometalsareseparatedbyaverythininsulatingbarrier,nomorethanafewhundredatomsthick,theycanformwhatisknownasatunneljunction,whichcansupportasmallelectroncurrentflowingacrossbyquantummechanicaltunnelling.In1962,EnglishphysicistBrianJosephsonrealizedthatifthemetalswerereplacedbysuperconductors,andformedaquantumsystem,alargersupercurrentcanflowbetweenthem.InChapter5wesawhowthesquareoftheamplitudeofthewavefunctionisinterpretedasaprobabilityand,forasingleparticle,thephaseofthewaveislargelyirrelevant.Butthisisnolongertruewhenthereisaphasedifferencebetweentwointeractingquantumobjects.InaJosephsonjunctionthephasesofthewavefunctionsdifferacrossthebarrier,andtheresultingsupercurrentflowingbetweenthemturnsouttoberelatedtothegradientofthephaseofthewavefunction.

TheJosephsonjunctionhasinterestingproperties.First,sincetheCooperelectronpairsinthesuperconductorscarrycharge,thejunctioncanbemanipulatedbyapplyingexternalelectromagneticfields.Ifavoltageisappliedtothejunction,itburstsintooscillation(typicallyatmicrowavefrequenciesforappliedvoltagesinthemillivoltrange).Thisisbecausethephasesofthewavefunctionsacrossthejunction‘beat’witheachotherratherliketwoout-of-tunepianostrings,whichdrivesanalternatingcurrent,theJosephsoncurrent.Thereverseisalsopossible.Ifthejunctionisexposedtoanoscillatingmicrowavefield,thesupercurrentshowsquantized‘steps’thatoccuratparticularvoltages.Thesequantizedstepsprovideaveryprecisewayofmeasuringvoltages(to1partin108),oralternativelyofmeasuringafundamentalconstantofnature,e/h,withgreatprecision.ThedegreeofquantizationissoprecisethatJosephsonjunctionsareusedinmetrologytodefinethevolt.

Thereareotherpossibilities.TwoJosephsonjunctionscanbeconnectedinparallel,toformwhatisknownasaSQUID(orSuperconductingQuantumInterferenceDevice).IntheSQUIDcircuit,whenamagneticfieldisapplied,thephaserelationship

betweenthetwojunctionsisaltered,andtheSQUIDcanbeusedasamagnetometertomeasureverysmallmagneticfields,suchasthoseproducedbytinycurrentsinthebrain.SQUIDscanalsobeusedashigh-speedswitchesbeingdevelopedforquantumcomputers.

ShapesofquantummatterUnderstandingtheworkingsofacomplexphysicalsystemin3Dpresentsaformidablechallenge,andtomakeprogressit’softenusefultostudyasimplifiedsystemthatretainsenoughofthefeaturesoftheoriginalonetoberelevant.Anexampleofthisisthereductionofa3Dsystemto2D,amotivationthatin1980ledKlausvonKlitzingtotheNobel-Prize-winningdiscoveryoftheQuantumHallEffect(QHE).

TheclassicalHalleffecthadbeendiscoveredacenturyearlierbytheAmericanphysicistEdwinHallwhenhewasstudyingelectriccurrentsflowinginathinmetalsheet.Hediscoveredthatwhenamagneticfieldisappliedperpendiculartothesheet,avoltagecalledtheHallvoltageappearsacrossthesample.Whenanychargedparticle,suchasanelectron,triestomoveinastraightlinethroughamagneticfield,thefieldtendstobenditintoacircularorbit.InHall’sexperiment,thepresenceoftheHallvoltagecouldbeexplainedbytheeffectofthemagneticfieldbendingelectronssideways,towardstheedgesofthestripwhereelectricalchargebuildsup.TheHallvoltage,anditsassociatedelectricalresistance,changesmoothlyasthemagneticfieldstrengthisvaried.

VonKlitzingwasinterestedinthequantumanalogueoftheHalleffect,andhadsetupanexperimenttostudycurrentsflowinginathin2Dlayerofelectrons,cooledtoverylowtemperaturessothattheelectronsformedacoherentquantumsystem.Thelayerwasimmersedinaverystrongperpendicularmagneticfield,sothattheelectronscouldcompletefullycircularorbits,lyingintheplaneofthesample.ThiscircularmotionisreminiscentofelectronmotionintheBohrmodeloftheatom;andwecanthinkofthe2Delectronlayerasbeingdividedupintoanumberofregionsdefinedbythesizesoftheseclosedorbits.Whenthiscircularmotionisquantized,aladderofdiscretequantumenergystatesforms,whichproducesanenergygapseparatingoccupiedandemptybands,justasinanordinaryinsulator.

WhenvonKlitzingvariedthestrengthofthemagneticfield,hediscoveredadramaticeffect:theHallresistancedidnotchangesmoothly,butjumpeddiscontinuouslybetweenstepsona‘staircase’ofbroadplateaux,revealingquantizationona

macroscopicscale(Figure24).ThisishowtheQuantumHallEffectwasdiscovered.AsecondbigsurprisewasthediscoverythattheHallconductance(theinverseofresistance)isquantizedinexactintegermultiplesofafundamentalconstantofnature:e2/h.Subsequentmeasurementshaveshownthatthelevelofquantizationholdstoanextremelyhighprecisionofatleast1partinabillion.Thisisfoundtoberobustlyinsensitivetothedetailsofthesamplegeometryandimperfectionsarisingfromitspreparation,insharpcontrastwiththeclassicalHalleffect,where,dependingonconditions,thereisconsiderablevariability.

24.TheintegerQuantumHallEffect:themeasuredHallresistanceisplottedagainstthestrengthoftheappliedmagneticfield.Theintegersnrefertodifferenttopologiesoftheelectronwavefunctions.

Thisunprecedentedbehaviourcouldnotbeunderstoodonthebasisofthestandardtheoryofelectricalconduction.Whilethebulkofthesamplebehaveslikeastandardbandgapinsulator,somethingunusualhappensattheedge.There,anelectroncannotcompleteacircularorbitwithouthittingthe‘hard’outeredge,bouncingback,andskippingalongtheedgeinaseriesofjumps.Theseunidirectionalorbitsareknownas‘skippingorbits’whichformanedgecurrent.AQHEmaterialhastheinterestingandunusualpropertythatitisaninsulatorinitsbulk,andaconductoronitssurface.

TheQuantumHallsystemhascometobeunderstoodbyrecognizingthattheelectronlayermustbeconsideredasamacroscopicquantumsysteminitsownright.Thespecialpropertiesofthesystemrelatetothevariousshapes,ortopologies,oftheelectronwavefunctions.Topologyisabranchofmathematicsdealingwiththe

geometricalpropertiesofobjectswhichareunaffectedbycontinuousdeformationssuchasbending,squashing,orstretching.Objectsbelongtodifferenttopologicalclassesiftheyarepierced,orpartsofthemaregluedtogether,tomakeholes.Considertheexamplesofamug,abagel,andapretzelshowninFigure25.

25.Acoffeemugistopologicallyequivalenttoabagel,becausetheybothhaveonehole;buttheybothbelongtoadifferenttopologicalclassfromthepretzelwithitsthreeholes.

Amug(let’ssaymadeoutofsoftclay)canbedeformedintoabagelshapeandbackagainsimplybyacontinuoussqueezingandpullingprocesswithouttheneedtocutanyholes.Themugandthebagelarethereforetopologicallyequivalent;theyhaveoneholeeachandbelongtothesametopologicalclass.However,neitheroftheseshapescanbedeformedtothepretzelshapewithoutpiercingtwoextraholes;thepretzelthereforebelongstoadifferenttopologicalclass.

IntheQuantumHallsystem,eachplateauinthemeasuredHallconductance(Figure24)representsadifferenttopologicalclass,eachofwhichislabelledbyaninteger,n.Ataverycrudelevelthedifferentclassescanbethoughtaboutasdifferentwaysof‘knotting’theelectronwavefunction.Topologyhasaddedanewmethodtoclassifymatter.Forexample,itispossibletothinkofdeformingonetypeofsolidinsulatorintoadifferentone,byvariousactions.Thesemightincludephysicaldeformationssuchasstretching,compressing,orbendingthematerial.Theseactionsaltertheenergylevelsandbandsinthesolid.However,providedtheinsulatoralwaysremainsaninsulatorduringaparticulardeformationprocess(inotherwordsthereisalwaysaforbiddenenergygappresent),thefinalinsulatormustbelongtothesametopologicalclassastheinitialone.Butifatanypointtheenergygapshouldcloseup,causingthematerialtobecome,evenbriefly,ametal,theinitialandfinalinsulatorsbelongtodifferenttopologicalclasses.

SincethediscoveryoftheQuantumHallEffect,ourunderstandingofwhatarenowcalledtopologicalinsulatorshasadvancedconsiderably.Movingawayfromtwodimensions,totheworldofpracticalmaterials,topologicalinsulatorsin3Dhavenowbeendiscovered.Itturnsoutthatmanycommoninsulatingchemicalcompounds,forexamplethosecontainingelementssuchasbismuth,selenium,andtellurium,aretopologicalinsulators,allcharacterizedbyconductingsurfacesandinsulatinginteriors.Thesematerialsdisplayremarkableproperties;ifatopologicalinsulatoriscutintopieces,newconductingsurfacesappearwherecutshavebeenmade.In2016,DavidThouless,DuncanHaldane,andMichaelKosterlitzwereawardedtheNobelPrizefortheircontributionstounderstandingthesenewexoticstatesofmatter.

ThekilogramAccordingtoIsaacNewton,akeypropertyofmatterismass,andthekilogramisthefundamentalunitofmassintheSIsystem.Itisalsotheoneremainingunittobedefinedbyaphysicalobject,alumpofplatinum–iridiumalloycastin1879andkeptinasecurevaultintheInternationalBureauofWeightsandMeasures(BIPM)inParis.Theobjectisaffectionatelyknownas‘LeGrandK’(Figure26).Onceevery

fortyyears,itistakenoutsothatreplicakilosfromaroundtheworldcanbecomparedwithit.

26.TheInternationalPrototypeKilogramisstoredinsidethreebelljarsinasafeinabasementintheParissuburbofSèvres.

Theproblemisthat‘LeGrandK’isnotstable:itsmasshasdivergedfromthoseofitsclones.Thedifferenceissmallandamountstoamasslossofjust50microgramsacentury,whichisabouttheweightofagrainofsand.Aweightlossoftwopartsin107percenturymayappeartobesmall,butitisneverthelesssignificant.In2018itwasthereforedecidedthatthephysicalkilogramwillbereplacedbyan‘electrical’kilogramthatreliesnotontheintegrityofaphysicallumpofmatter,butonthemeasurementoffundamentalphysicalconstants.Matterwillbeweighedagainsttheelectromagneticforce.

ThenewandmoreprecisedefinitionoftheelectricalkilogramreliesontheJosephson

andQuantumHalleffects,whichprovidethehighprecisionoftheelectricalvoltageandcurrentmeasurementsneeded.TheJosephsonjunctionstandardwillbeusedasavoltagestandardandaQuantumHalleffectdeviceforthecurrent/resistancestandard.Withtheproposednewelectricalstandardkilogram,laboratoriesanywhereintheworldwillbeabletodefinethekilograminsitu,withouttheneedtotransportamaterialobject.Thenetuncertaintyintheweightoftheelectricalkilogramis1partin108.

Thischapterhasexploredsomeexamplesofquantumfluids,matterthatdisplaysquantumeffectsonmacroscopicscales.Quantumfluidbehaviourexplainsthedifferencebetweenconductorsandinsulators,aswellasthedramaticpropertiesofsuperfluidsandsuperconductors.Theseexoticmacroscopicquantumsystemsalsohavepracticaluses,andhaveledtonewandveryprecisewaystomakeelectricalmeasurements.Thesetechniqueswillbeusedtomakeagreatlyimproveddefinitionofthefundamentalunitofmass,thekilogram,onewhichislinkedtofundamentalphysicalconstants,andnottoaphysicalpieceofmatter.

InChapter7wewillzoominonsubatomicparticlesandlookatthesmallestandmostfundamentalparticlesofmatter.

Chapter7

Fundamentalparticles

Allthecompositeformsofmatter—nucleons,atoms,molecules,livingcreatures,planets,stars,andsoon—arebuiltoutofasmallnumberofdifferenttypesofparticleswhichinteractindifferentwaysviatheforcesofnature.Itistheforcesthatgiverisetotheenormousvarietyandmanyformsofmatter.Whendiscussingmatter,wemustthereforeincludetheforces.Therearefourfundamentalforces:thetwolong-rangeforcesofgravityandelectromagnetism,andthetwoshort-rangeforcesthatoperateinthenucleus,thestrongnuclearforce,andtheweaknuclearforce.Thestrongforcebindsnucleitogether,andtheweakforceisconnectedwithcertaintypesofradioactivity,andtheenergyproducingprocessesinthestars.

Justoveracenturyago,itwasreasonabletohaveimaginedthattheultimatebuildingblocksoftheuniversewereatoms.By1932,theatomsofnormalmattercouldbeexplainedintermsofjustthreetypesofsubatomicparticles,protons,neutrons,andelectrons.Thisbeautifullysimplepictureofthefundamentalcomponentsofmatterdidnotlastlong.Bythen,matterwasstartingtobeprobedathigherenergies,andtheoreticaldevelopmentsinquantummechanicshadtakenplacewhichwouldradicallychangeourunderstandingofthemicroscopicworld.

WhiletheSchrödingerequationsuccessfullydescribesmuchofthesubatomicworld,itdoesnotdescribethepropertiesofparticlesmovingatspeedsclosetothespeedoflight.Thisshortcomingwassignificantbecause,forexample,whilethespeedofanelectroninthegroundstateofahydrogenatomisonlyabout1/137thofthespeedoflightandthereforenon-relativistic,thisisnottruefortheinnermostelectronsinheavyatomslikegold,inwhichspeedsapproachhalfthespeedoflight.(Thenumber1/137.036…isaconstantofnatureknownasthefinestructureconstant.)So,in1928EnglishphysicistPaulDiracsetouttomakequantumtheoryconsistentwithspecialrelativity.

Thefirstattemptstounifyquantumtheorywithspecialrelativityhadfounderedonamathematicalproblem.Inclassicalwaveequations,spaceandtimearetreatedseparately,butinspecialrelativity,theyarewovenintoasinglefabric:4Dspacetime.Diracdiscoveredthathecouldincorporatespecialrelativitybyusingmatricestodividetheequationintofourparts,resultinginthefamousDiracequation.Thepropertyofelectronspincameoutnaturallyfromthetheory,directlyfromspecialrelativity.Theelectronalsoemergedasafundamentalparticleofzerosize.Twoofthefourpartsoftheequationwereeasytointerpret:thesecorrespondtotheelectronspinningclockwise(up)oranticlockwise(down).Butthepuzzlewas:towhatdidtheothertwocorrespond?

AntimatterInspecialrelativity,theenergyEofasystemappearsintheequationsasasquaredquantity,i.e.asE2.Sinceasquarerootcanhavepositiveandnegativevalues,twiceasmanysolutionsresult.Thenegativevalueofthesquarerootimpliedthatanelectronhasnegativeenergy—somethingthatmadenosense.Diracrealizedthatanegativelychargedelectronwithnegativeenergycouldbeinterpretedasapositivelychargedelectronwithpositiveenergy.Ineffecthetransferredthenegativesignoftheenergytotheelectricalchargewhereitconvertedanegativeelectronintoapositiveelectron,orpositron,aparticleofantimatter.Matterandantimatteraremirrorimagesofoneanother;whateveronedoes,theotherdoestheopposite.Forexample,ifanelectronspinsclockwise,apositronspinsanticlockwise.

Didthepositronreallyexist,orwasitjustaquirkofthemathematics?AfewyearsafterDirac’sprediction,thephysicistCarlAndersonwasstudyingcosmicrayelectrons.CosmicraysareenergeticparticlesthatoriginateintheSunandouterspaceandbombardtheEarth.Andersonhadsetupacloudchamberdetectorinsideastrongmagnettorecordthetracksofthecosmicrays.Whenachargedparticlepassesthroughacloudchamber,itproducesacondensationtrailoftinyliquiddropletsbehindit,likethecontrailofajetaircraftflyinginthestratosphere.Whenanegativelychargedelectronpassesthroughamagneticfielditisdeflectedsideways,alwaysinthesamedirection.Andersonobservedthatinadditiontotheelectrons,therewereparticleswiththesamemassastheelectron,butwithpositivecharges,whichweredeflectedintheoppositedirection.Dirac’spositronshadbeenfound.

Whenanelectronandapositronmeet,theyannihilateeachother,andconvertalloftheirrest-massenergy,atotalof1.022MeV,intoaburstofgammaradiation.Theprocessissymmetric.Ifanenergeticgammarayphotoncollideswithanatom,the

strongelectricfieldaroundtheatomicnucleuscausesmattertobeconverteddirectlyfromenergyinaprocesscalledpairproduction(Figure27).Theelectron–positronpairsAndersonobservedwerepairproductionevents.

27.Pairproduction:matterandantimatterarecreateddirectlyfromtheenergyofaphoton.Agammarayphotonentersfromthetop,passesclosetoanatomicnucleus,andproducesanelectron(left-handedspiral),andapositron(right-handedspiral).Arecoilingatomicelectronemergesdownwards.

Allparticles,notjustelectrons,haveantiparticlepartners,anditwasnotlongbeforeothertypeswerediscovered,alwaysoneofapair,whereelectricalchargeisstrictlyconservedinequalandoppositeamounts.Ifanegativelychargedantiprotonpassesnearaproton,thechargescancelsoastoleavebehindtwoneutralparticles,aneutron,andanantineutron.Wholeanti-atomshavebeenmadeinwhichapositroniscombinedwithanantiprotontomakeantihydrogen.Thelightfromaquantumtransitioninantihydrogenwasobservedforthefirsttimein2017andwasfoundtohaveawavelengththatisidenticaltothatofthecorrespondingtransitioninnormalhydrogen.

QuantumfieldsandforcesInearlyquantumtheory,theparticletookcentrestage,andthetheoryprovidedthescaffoldingneededtocalculatetheparticle’squantumstatesanditsassociatedenergylevels.Thisapproachhowevercouldnotexplainhowparticleswerecreatedanddestroyed.Thenextstepwastodevelopquantummechanicsintoamoregeneralframework:QuantumFieldTheory(QFT).QFTplacestheemphasisonthefield,andassumesthatspaceisfilledbyanumberofinteractingparticleandforcefields,eachofwhichischaracterizedbyitsquantumstates.Theprimaryconcepthereisthatofanall-pervasivequantumfield,inwhichtheparticles(thefieldquanta)arefluctuations.Justasphotonsarethefieldquantaofaquantizedelectromagneticfield,electronsarethefieldquantaofaquantizedelectronquantumfield.

Thefirstsuccessfulquantumfieldtheorytobeconstructedwasquantumelectrodynamics(QED),whichdescribestheinteractionoflightandmatter.Atthestart,itwasbesetbytheoreticaldifficulties.Oneofthemostglaringwastheinteractionofanelectronwithitsownelectromagneticfield,theso-calledself-energyoftheelectron.TheCoulombelectricforcefromanelectron(thesourceofthefield)variesastheinverseofthedistancesquared.So,iftheelectronhaszerosize,theforceoughttobeinfinitelylargeatthepositionoftheparticle(onedividedbyzerodistanceisinfinity).Theenergyintheelectromagneticfieldsurroundinganelectronshouldthereforebeinfiniteand,sinceenergyandmassareequivalent,themassoftheelectronshouldalsobeinfinite.

Theirksomeinfinitieswererelatedtotheelectron’sself-energyandthepolarizationofthevacuum.Tovisualizevacuumpolarization,imaginetryingtocreatethemostperfectofallvacuums,byremovingallparticlesfromsomeregionofspace.Whenyouhavedonethat,Heisenberg’suncertaintyprincipletellsusthatsomethingstillremains.Quantummechanicsallowsvirtualparticlestopopinandoutofexistenceconstantly;thevacuumhasafiniteenergydensity.So-calledemptyspaceisreallyaseethingcauldronofactivity,filledwithanenergycalledthevacuumenergy.Theelectricfieldsurroundinganelectroncontainsabubblingseaofvirtualparticles.Theself-energyoftheelectronarisesfromtheinteractionofitsintrinsic,or‘naked’,chargewiththeseparticlesandsoproducesanextraenergydensity.Thisinteractionconfersupontheelectronsomethingcalleditselectromagneticmass.

ThetheoristHansBethecircumventedtheproblemofinfinitieswithamathematicaltrick,calledrenormalization,whichinvolvessubtractingoneinfinitequantityfromanothertoleaveafiniteanswer.Seenbysometobeunsatisfactory,renormalizationwasfoundtoworkwellinpracticeandenabledQEDtobedevelopedintothemost

accuratephysicaltheoryofwhichweknow.OneofthefirsttestsitfacedwastoexplainsomethingcalledtheLambshift.WillisLambhadobservedthataparticularquantumstateofhydrogenwassplitintotwoclose-spacedenergylevels.TheLambshiftiscausedbytheself-interactionoftheelectronwithitsfieldandcouldnotbeexplainedbyDirac’stheory.ThesplittingispredictedextremelypreciselyinthemoderntheoryofQED,developedin1947byRichardFeynman,JulianSchwinger,andShin’ichiroTomonaga.

Physicistshadalsobeenstudyingthestrongandweaknuclearforces.Nuclearforcesdifferfromtheelectromagneticandgravitationalforcesinthattheyoperateoververyshortranges,belowaround10–15metres.Onscalesofthesizeofanatom(10–10metres)nuclearforcesessentiallydon’texist.NuclearforcesarelikeVelcro:whentwopiecesofVelcroareincontacttheyarestronglyattached,butwhentheyarepulledaparttheyfeelnoforcewhatsoever.Thestrongforceisalsoindifferenttoelectricalcharge;itbindsthepositivelychargedprotonsandneutronsinthenucleustogetherequallystrongly.RecallfromChapter5theanalogyofthetwoiceskatersexchangingaheavyballtoillustratethemicroscopicnatureofforces.Inanyshort-rangeforce,themediatingforceparticlemusthavealargemassandashortlifetime,asrequiredbytheenergy-timeformoftheuncertaintyprinciple.In1935,HidekiYukawaproposedthatthestrongforceinvolvestheexchangeofavirtualparticlecalledapion.Themassofapionliesinbetweenthoseofanelectronandaproton,andbelongstoaclassofparticlecalledameson(fromtheGreekwordmeaning‘intermediate’).Pionsareunstableandlivebrieflivesinsidenuclei;butwhenanucleusisstruckveryhardbyanotherparticle,pionscanbedrivenoutintotheworldasrealparticleswithshortlifetimes.Theseshort-livedpionswereobservedin1947incosmic-rayparticleinteractions.Thepionmodelforthestrongforcewasanapproximatetheory,andhascometobereplacedbythequarktheory,whichisdescribedbelow.

Theweaknuclearforce,ormoreproperlytheweakinteraction,hasamoresubtlecharacter.ItisresponsibleforbetaradioactivityandalsoplaysakeyroleinhowtheSunburns.Aloneneutronisunstableandwilldecayintoaprotonandanelectroninaroundtenminutes.Insidethenucleushowevertheneutronismorestable,butiftheatomisradioactiveaneutroncandecayintoaproton,changingthechemicalidentityofanatomtothenexthigherelementintheperiodictable.Thisbeta-decayprocessisaccompaniedbytheemissionofanenergeticelectron,thebetaparticleofradioactivity.Whenbeta-decaywasfirststudied,itwasfoundthatelectronswere‘spatout’ofnucleiwithacontinuousrangeofenergies,fromzerouptoamaximumvalue.Thiswasproblematic,becauseitappearedthatthelawsofenergyandmomentumconservationwerebeingviolated.

WolfgangPaulihitontherightsolutionwhenhesuggestedthatanunseenandmysteriousparticle,aneutrino(or‘littleneutron’)wasemittedalongwiththeelectron,soastobalanceenergyandmomentum.Inbeta-decaytheemittedparticleisanantineutrino;intheinverseprocess,inversebeta-decay,protonsareconvertedintoneutronsbyemittingpositronsandneutrinos.Neutrinoshaveverylowmass,noelectricalcharge,andcanpenetrateverydeeplyintomatter.TheSunandthestarsproducetrillionsofthemeverysecond,buttheyinteractsoweaklywithordinarymatterthataneutrinocanpassthroughalightyearoflead(tentrillionkilometres),andstillhaveonlya50:50chanceofhittinganucleus.Weareblissfullyunawareofthevastnumbersofneutrinosthatpassunfeltthroughourbodies.Eachsecondabillionneutrinoszipthroughyourthumbnail,dayandnight.Theelusivenatureoftheneutrinomeantthatitwasnotdetecteduntil1957,inreactionsinanuclearreactor.

Fromthedescriptionoftheweakinteractionjustgivenitisreasonabletoask:whyisitregardedasoneofthefourfundamentalforcesofnature?Intheirquesttounderstandadeeperphysicalreality,physicistsoftenseekapparentlydifferentaspectsofnaturethathaveacommoncauseandsoareamenabletounification.OneexamplewasMaxwell’sunificationoftheelectricandmagneticfields,whichaswehaveseenrevealedthemorefundamentalelectromagneticfield.AnotherwasDirac’sunificationofspecialrelativitywithquantummechanics,whichresultedinthepredictionofantimatter.Athirdexampleistheunificationoftheelectromagneticandweaknuclearforces(theelectroweakinteraction),whichwaspostulatedin1967bySheldonGlashow,StevenWeinberg,andAbdusSalam.Inessence,theirtheoryproposedthattheparticlescarryingforcesinelectromagnetismandtheweaknuclearforcearereallythesame.Theyonlyappeartobedifferentbecausetheforce-carryingbosons(theWandZparticles)havemassintheweakinteraction,butinelectromagnetismthecorrespondingparticle,thephoton,ismassless.

Theweaknuclearforceisonlyabout1/1000thasstrongastheelectromagneticforce,yetthetwohavemuchincommon.Tounderstandhowtheyrelatetoeachother,recallthatthelong-rangeelectromagneticforceiscarriedbythephoton,whichismasslessandcarriesnoelectricalcharge.ThiscanbecomparedwiththeWandZcarriersoftheweakforce,whichhavemassesabout100timestheprotonmass,aboutthesamemassasasilveratom.Wparticlescanbepositivelyornegativelycharged,andtheZisneutral.Whenaneutronundergoesradioactivebeta-decay,itemitsachargedboson,thenegativeWparticle.Intheinteraction,theneutronrecoilstoconserveenergyandmomentum,andsotheWhasthekeypropertyofaforcecarrier.ThepropertiesofthefourfundamentalforcesaresummarizedinTable2.

Table2. Propertiesofthefourfundamentalforces

Unliketheotherforcesofnature,theWchangestheidentityofaparticle.Whenaneutronconvertsintoaproton,oneunitofnegativechargeisremoved,andtheemittedWsubsequentlydecaysintoanelectronandaneutrino.TheshortlifetimeoftheWisrelatedtoitslargemass;itisaheavybosonwithalimitedrange.Inelectromagneticinteractions,avirtualphotoncarriesnocharge,andstrictlyspeakingshouldberegardedasforminganeutralcurrent.Theanalogousneutralcurrentintheweakinteractioniscarriedbyaneutralboson,theZparticle.Thethreebosons(W+,W–,andZ0)arecalledintermediatevectorbosons.In1983,experimentsattheSuperProtonSynchrotronacceleratoratCERNinGenevadirectlyobservedtheWsandtheZ0.

Buttherewasstillsomethingmissing—anunderstandingoftheoriginofmass.QuantumfieldtheorypredictedthattheWandZparticlesshould,likethephoton,bemassless.SohowdotheWsandZsgettheirlargemasses?Amajorcluecamefromanideainthetheoryofcondensedmatter—thebehaviourofphotonsinsuperconductors.InChapter6wesawthatasuperconductorishighlyintolerantofexternalmagneticfields(theMeissnereffect),andtriestoexpelthemcompletelyfromitsbulkbygeneratingsupercurrentsthatcancelthemout.Thispropertyofasuperconductorappliestoallmagneticfields,includingthoseassociatedwiththequantumfluctuationsofvirtualphotons.Whenavirtualphotonflitsintoexistenceinsideasuperconductor,thesuperconductorrespondsbygeneratingsupercurrentsthattrytocanceloutthefluctuatingmagneticfieldofthephoton.Asaresult,thefieldofthephotonisweakened,andmoreenergyisneededtosustainthefluctuations.Thisextraenergyconfersmassonthevirtualphoton,calledtheeffectivemass.Photonsacquiremassbybeinginsideasuperconductor.

ThisprofoundideawasseizedupontoexplaintheoriginofthemassesoftheWandZbosons.Coulditbethatallofspaceisfilledwithanunidentifiedquantumfield,which,byanalogywithsuperconductivity,interactswithfundamentalparticlestogivethemtheirmasses?In1964physicistsPeterHiggs,RobertBrout,andFrançoisEnglertproposedsuchaquantumfield,theHiggsfield,whichfillstheuniverse.If

true,itimpliedthatwewould,ineffect,belivinginsideacosmicsuperconductor.Inthistheory,allfundamentalparticlesaremasslessuntiltheyinteractwiththeHiggsfieldbywhatiscalledtheHiggsmechanism.Acrudephysicalpictureofthisisthatparticlesacquiremassby‘sticking’tothefieldasiftheyaretryingtomovethroughthicktreacle.TheHiggsfieldcontributesafiniteenergydensitytothevacuumand,becauseithasnopreferreddirectioninspace,isascalarfield.

TheHiggsmechanismrelatestotheunificationofelectromagnetismandtheweakinteractionviaspontaneoussymmetrybreaking.RecallfromChapter3howsymmetryisbrokenwhenaliquidfreezes,orwhenthetemperatureofapieceofironfallsbelowtheCurietemperatureandbecomesamagnet.Atveryhightemperaturesandenergies,theWandZparticlesdon’tinteractwiththeHiggsfield,renderingthemmassless,likethephoton.Butatlowtemperaturesthesymmetrybreaks,andtheWsandZsinteractwiththeHiggsfield,fromwhichtheyacquiretheirmasses.Intheelectroweakinteraction,theterms‘high’and‘low’energyrelatetoacriticalenergyofabout100GeV,whichdefinesthescaleoftheelectroweakforce.

ThequantumparticleoftheHiggsfieldistheHiggsboson.Thesearchforthismuchsought-afterparticlewasakeyobjectiveofexperimentsthatstartedin2013inCERN’sLargeHadronCollider(theLHC).Theexperimentsinvolvedcollidingbeamsofprotonswitheachotherwithacombinedenergyof8trillionelectronvolts(8TeV,or8×1012eV).ThesecollisionshittheHiggsfieldveryhard,andmadeitoscillateandproduceaquantumoffieldexcitation:theHiggsboson.Themeasuredmassoftheparticle(125GeV/c2)wasconsistentwithitsveryshortlifetimeofabout10–22sec.TheHiggsparticlewasnotobserveddirectly,butitsexistencewasinferredunambiguouslyfromtheparticlesintowhichitsubsequentlydecayed.

QuarksTostudythesmallestparticlesofmatteraverypowerfulmicroscopeisneeded.Iftheobjectsofstudyaresmallerthananucleustheymustbeprobedwithveryshort-wavelength,high-energyparticles.High-energyparticleacceleratorsarethemicroscopesofthesubatomicworld;thebiggerandmorepowerfultheaccelerator,thestrongerthemicroscope.

Fromthe1940stothe1960s,manyhundredsofnewtypesofsubatomicparticlesturnedupinacceleratorexperiments.Inthese,high-energyprotonsweresmashedintotargets,producingaplethoraofunstableparticles.Thesituationwasnotunlikethatofthepreviouscentury,whenchemistshadbeenconfrontedwithabewilderinglylarge

numberofchemicalelementswithdifferentproperties.Tryingtomakesenseoutofalltheparticleswas,asRichardFeynmanputit,‘liketryingtofigureoutapocketwatchbysmashingtwoofthemtogetherandwatchingthepiecesflyout’.By1954,somanyparticleshadbeendiscoveredthatFermicomplained,‘IfIcouldrememberthenamesofalltheseparticles,Iwouldhavebeenabotanist.’

Subatomicparticlesbelongtotwobroadfamilies:hadronsandleptons.Hadrons(fromtheGreekwordmeaning‘thick’)haveafamilyresemblancetoprotonsandneutronsinthattheyfeelthestrongnuclearforce,theweakinteraction,andelectromagneticforces.Thelightestmembersareprotonsandneutrons,butmosthadronsaremuchmoremassive,andareinexcitedstateswithhighinternalenergiesandshortlifetimes.Thehadronfamilyissubdividedintotwofurthertypes,baryonsandmesons.Thewordbaryonmeans‘heavy’.Leptons(fromtheGreekfor‘thin’,or‘small’)areparticleslikeelectronsandneutrinos,andfeelonlytheweakandelectromagneticforces.

Physicistsscrutinizedthemassofdatainthehadronzoo,searchingforpatternsthatmightrevealaclassificationthatwouldbeforthesubatomicworldasprofoundasMendeleev’speriodictablehadbeenfortheelements.In1964MurrayGell-MannandGeorgeZweigdidindeedfindpatternsamongtheplethoraofhadrons,andconcludedthatalltheparticlescouldbebuiltoutofquarks,elementaryparticleswithfractionalcharges.Gell-Mannchosethename‘quark’fromalineinJamesJoyce’sbookFinnegansWake:‘ThreequarksformusterMark.’

Normalmatterismadefromtwotypesofquarks:an‘up’quark(u,mass2.3MeV/c2),anda‘down’quark(d,mass4.8MeV/c2).Eachtypecarriesadifferentfractionalelectricalcharge;theup-quarkcarries+⅔oftheelectroncharge,andthedownquark–⅓ofit.Baryonsaremadefromthreequarks,andmesonsfromaquark–antiquarkpair.Aproton(Figure28)canbeimaginedasatinysphere,about10–15ofametreacross,containingthreequarks:(u,u,d)givingitanetchargeof+1unit.Theneutronissimilar,buthas(u,d,d)quarksgivingitzerocharge.Thequarksmoveroundinsideanucleonatalmostthespeedoflight,bouncingoffeachotherandoffthewalls.Allthehadronsintheparticlezoocouldseeminglybeexplainedasvariousboundstatesofquarks.Wasthisjustacleverwayoforganizingthehadronzoo,ordidquarksreallyexist?

28.Thelightestbaryon,theproton,ismadefromthreequarks,two‘up’(‘blue’and‘red’colourcharges)andone‘down’(‘green’colourcharge).Allthreecolours(shownherebydifferencesinshading)mustbepresent,makingtheparticlecolourless.

Decisiveevidencethatquarksreallydoexistcamefromaseriesof‘deepelectronscattering’experimentsmadeinthe1960sand1970sbyJeromeFriedman,HenryKendall,andRichardTaylor.Theyusedthe3-kilometre-longStanfordLinearAccelerator(SLAC)tobombardprotonswithelectronshavingspeedsclosetothespeedoflight.Theelectronswereseentoscatterfromtinygranularobjectsmovingaboutveryfastinsidetheproton.Thesetinyobjectswerethesought-afterquarks.

Soon,othermoremassivevarietiesofquarkswerediscovered,comingindifferent‘flavours’thatsignifytheirsymmetryproperties.Thereisacharmquark(c)withamassof1.3GeV/c2,thestrangequark(s)withmass0.95GeV/c2,andthebottomquark(b)withmass4.2GeV/c2.In1995anevenmoremassive180GeV/c2topquark(t)wasdiscoveredusingthehigh-energycollideratFermilab.Theunstableheavierquarksdecayrapidlytothelowermassstates.Therearesixtypesofquarks,andallarespin-½fermions.Asfarasweknow,quarksaretheultimatefundamentalparticlesofmatter.

Quantumchromodynamics(QCD)Averystrongforceisrequiredtobindthreeveryenergeticquarkstogetherinahadron.Theforceparticlesthattiequarkstogetherarecalledgluons.Buthowisitthatthreespin-½fermionsareabletocoexistinanucleon,apparentlyinviolationofthePauliexclusionprinciple?Itturnsoutthatquarksandgluonscarryadifferentkindofchargecalledcolourcharge,whichisameasureofthestrengthwithwhichtheparticlesinteractviathestrongforce.Unlikeelectromagneticcharge,whichcomesintwovarieties,positiveandnegative,colourchargecomesinthreevarieties:red,green,andblue.Theassociatedcolourforceisthebasicforceofthestronginteraction.Asinelectromagnetism,colourchargeisastrictlyconservedquantity.Itisimportantto

appreciatethatthecolournamesarejustlabels,anddonotsignifyanythingthatwewouldrecognizeinthefamiliarworldascolours.Thelabelsmerelyhelpusdistinguishbetweendifferentquantumstatesandrecognizethat,say,ablueup-quarkisdifferentfromagreenup-quark.Thismaysoundalittlecontorted,butthekeyquantumruleisthatquarksonlyformcombinationsthatarecolourless.Inahadron,allthreecoloursmustbepresent.Mesonsareautomaticallycolourlessbecausecolourandanticolourchargescanceloutinquark–antiquarkpairs.Toaddanotherlayerofcomplication,itturnsoutthatquarkscanalsochangecolour.Gluonsarethemselvescolouredandsocantransfercolourchargefromonequarktoanother.

Oneofthemoststrikingpropertiesofquarksisthattheyareneverseenalone.The‘principleofconfinement’relatestothewayquarksinteractwiththegluonfieldsinsideahadronandkeepsthemtightlyboundinside.Theconceptof‘tying’thingstogetherisparticularlyaptforquarksbecausetheforcelawbetweenquarksbehavesasiftheyareconnectedbyelasticstrings.Ifyoustretchanelasticband,theforceneededincreasesthemoreitisstretched.Butifnoforceisapplied,theelasticisslack.Theforcelawbetweenquarksworksinpreciselytheoppositewaytotheinversesquarelawofgravityorelectromagnetism:theforcebetweenquarksincreasesthefurtheraparttheyget!

Theinteractionsbetweencolourchargesaredescribedinaquantumfieldtheory,quantumchromodynamics,orQCD,constructedbyDavidGross,FrankWilczek,andDavidPolitzer.QCDdiffersfromQEDinakeyrespect.Inelectromagnetism,twounchargedphotonscanpassfreelythrougheachother,oblivioustoeachother’sexistence.TheairwavesarefullofelectromagneticsignalsfrommultipleradioandTVstationsandmobilephonesignals,allofwhichcrossandsuperimposewithoutaffectingeachother.Butgluonscarrycolourcharges,whichmeanstheyinteractstronglywitheachother,apropertythathashugeconsequences.Ifyouhitaquarkinsideanucleonhardenoughitcanemergeasmalldistanceintotheouterworld,butitisstillconnectedtotheothertwoquarksviathecolourforce,whichaswehavesaidbehaveslikeapieceofstrongelastic.Thefurtheroutthequarkemerges,thestrongertheforceitfeelspullingitback,and,topullitoutevenmore,moreenergyisneeded.Eventuallysomuchenergyhastobeprovidedthattheelasticsnaps.Whenthathappensalltheenergystoredinthe‘stretchedelastic’ofthecolourfieldconvertsintomatterandantimatter,andaquark–antiquarkpairforms.Stringsofquark–antiquarkpairsarecommonlyproducedinparticleacceleratorsexperiments,andareknownasparticlejets.

TheStandardModel

TheStandardModelofParticlePhysics(Figure29)isoneoftheverygreatachievementsofscience,andisthedistillationofmanydecadesofworkbyenormousnumbersofscientists.Themodeliswellvalidatedbyexperiment,andbringstogetherallthefundamentalmatterandforceparticlesweknowof.ThefoundationsonwhichthestandardmodelisbuiltareQED,theelectroweakinteraction,andQCD.Allthefundamentalparticlesneededtobuildnormalmatterarehere:fromquarks,tonuclei,toatoms,tochemistry,tolife,allthewayuptothevisibleuniverse.Gravityisnotincludedinthemodelbecausethereisatpresentnoviabletheoryofquantumgravity.

29.TheStandardModelofparticlephysics,thesumofourknowledgeofthefundamentalmatterandforceparticlesfromwhichtheuniverseismade.

Thematterparticles,thefermions,areshownontheleft,andtheforceparticles,thebosons,areontheright,withtheHiggsbosoninthecentre.Eachfermionhasanantimattertwin,whicharenotshownhere.Allofthematteroftheeverydayworldisbuiltfromjustthreematterparticleswhichoccupytheleft-handcolumn:theupanddownquarks,andtheelectrontogetherwithitsneutrino.Inmovingonecolumntotherightthebasicpatternrepeats,butthistimeitcontainsthecharmandstrangequarksandthemuon(μ)anditsneutrino;thiscolumniscalledthesecondgeneration.Secondgenerationparticlesaremoremassivethanthoseinthefirst,andarelessstable.Thepatternisrepeatedonemoretime,makingathirdgenerationofevenmoremassive

andunstableparticles,containingthetopandbottomquarks,thetaulepton(τ),anditsneutrino.Inall,thestandardmodelcontainstwelveparticlesofmatter,governedbythreeforcesthatresultfromtheexchangeoffourforce-carryingparticles:thephoton(γ),thegluon(g),andtheWandZbosons.

Inourstoryofmatter,wehave,inasense,arrivedatourdestination:atrulyDemocritanviewoftheworldintermsofitsbasicbuildingblocks:fundamentalparticlesofzerosize.Theknifethatcutsapieceofmatterupineversmallerpiecescan,asfaraswecansee,cutnomore.Ifthefundamentalparticlesarereallydimensionlesspoints,interactingviatheforcesofnature,wearriveataremarkablepictureinwhichmatterisempty,anditisthefourinteractionsthatgiveitshapeandform.Butifhistoryisanythingtogoby,weshouldnotbeconfidentthatwehavealltheanswers.Onmanyoccasions,scientistshavebelievedthattheyhavefinallyreachedtheultimatepicture,onlytofindthatnatureissubtlerthantheyhadthoughtandthatunderneaththereisadeeperreality.

EventhoughtheStandardModeliscurrentlythebest-testedandmostcomprehensivetheoryofthebasiccomponentsofmatter,therearestillplentyofmysteriesandglaringomissions.Wehavealreadynotedtheabsenceofgravityfromthepicture.Ifyouletgoofanapple,itfallsintheEarth’sgravitationalfield.Matterthereforedoesinteractwithgravity,yetthereisnothingintheStandardModeltoexplainthis.Weknowthatthegravitationalfieldsupportsoscillations,becausegravitationalwaveshavebeendetectedfromviolenteventsinthecosmos.Thepredictedforce-carrierofthegravitationalfieldisthegraviton,butthisparticlehassofarnotbeendetected.Also,wedonotknowwhytherearetwelvematterparticles,groupedintothreegenerations.Nordoweunderstandwhyquarksandleptonsaredifferent,andwhyfermionsandbosonsdiffer.Inthe1970s,atheory,calledsupersymmetry,orSUSYforshort,wasputforwardthatintimatelyconnectsmatterandforcebyproposingthateveryparticleintheStandardModelhasasupersymmetricpartner,orsuperpartner.Foreveryfermion,thereisaSUSYbosonandviceversa,effectivelydoublingthenumberofparticles.Althoughsupersymmetryexplainstheoreticallythedistinctionbetweenmatterandforceparticles,theissueisthatifsupersymmetricparticlesdoexist,theyshouldalreadyhavebeenobservedintheLHC;sofar,therehasbeennosignofthem.

Therearefurtherproblems.ThemassesofthefundamentalparticlesarebelievedtocomefromthewaythatthequarksandleptonsandtheWandZparticlesinteractwiththeHiggsfield.Wedonotunderstandwhytheseparticlesinteractinsuchawaytogivethemtheirdifferentmasses.Whenitcomestotheoriginsofmassitisclearthatthereisstillmuchtodiscover.Inaddition,thereisevidencethattheamountofmatter

intheuniverseissignificantlygreaterthancanbeaccountedforinthestandardmodel.Thisistheso-calleddarkmatteranddarkenergy,whichwillbediscussedinChapter9.

TheoriginofmassOneofthemostbasicquestionsaboutordinarymatteris:wheredoesitsmasscomefrom?Thisisadifferentquestionfromwheredothemassesofthefundamentalparticlescomefrom,whichwehavejustlookedat.Weknowthat99.9percentofthemassofatomsresidesintheirnuclei.Nucleonsaremadeoutofquarks,whichinteractviathestrongcolourforcesthroughthegluonfields.QuarksandotherparticlesderivetheirmassviainteractionswiththeHiggsfield.Wecanask:howmuchmassdothethreequarkscontributeindividuallytothemassofaproton?

Aprotonhasarestmassof938.28MeV/c2.Anup-quarkisestimatedtohaveamassofabout2.3MeV/c2,andadown-quarkamassofabout4.8MeV/c2.Forthethreequarksthetotalis9.4MeV/c2.Butthisisonlyabout1percentofthemassoftheproton!Whereistheremaining99percentofthemass?TheanswercomesfromapplyingEinstein’sformula:m=E/c2,whichtellsusthatwhereverthereisenergy,thereismass.Itturnsoutthattheremaining99percentofthemassofaprotonoraneutronisinthemassequivalentoftheenergyofthegluoncolourfieldsandthekineticenergyofthequarks.Thisistheoriginofmostofthemassofordinarymatter—itispureenergy.

Inthischapterwehavelookedattheintensivepictureofmatter—peelingoffitsonionskinlayerstorevealthesmallestmostfundamentalparticlesofwhichitismade.Onlyahandfulofelementaryparticlesmakeuptheworld:quarks,leptons,andtheforceparticles,whichappearintheStandardModelofParticlePhysics.Theseparticlesarethefieldquantaofafewquantumfields.Theelementaryparticlesgettheirmassesbyinteractingwiththeall-pervasiveHiggsfield,butthedominantsourceofthemassofordinarymattercomesfromtheenergyofthequarkandgluonfieldsinsidenucleons.

TheStandardModelisatoweringachievementofscience,butitisnotcomplete.InChapter9,wewilllookattwomissingandmysteriouspiecesofthejigsawpuzzle:darkmatteranddarkenergy.Butbeforethat,let’ssetthescenefortheverylargescalesofmatter,bytracingthechemicalhistoryoftheuniverseinthe13.8billionyearssincetheBigBang.

Chapter8

Wheredotheelementscomefrom?

ThemedievalistsbelievedtheEarthtobesurroundedbya‘crystalsphere’containingAristotle’sunearthlyfifthelement,quintessence.However,when19th-centuryastronomersattachedprismstotelescopesandsplitthelightfromtheSunandotherstarsintodifferentcolours,theysawspectrapepperedwithnarrowatomiclineshavingwavelengthsmatchingthoseemittedbyatomsinlaboratoryexperiments.TheatomsinthestarsarethesametypesasthosefoundonEarthandthatmakeupourbodies.CarlSaganwasfamousforsayingthatweare‘starstuff’.Hewasright.

Tounderstandwherethechemicalelementscomefrom,wehavetogobackafractionofasecondafterthebirthoftheuniverseinthe‘BigBang’.Thepowerfulallianceoftelescopeandspectrometerwhichhadfirstrevealedchemicalelementsinstars,struckagaininthe1920swhenEdwinHubbleandMiltonHumasonusedthethenbiggesttelescopeintheworld,the100-inchHookertelescopeatMountWilsoninCalifornia,tomeasurethedistancesandtheradialvelocitiesofgalaxies.Justasthesirenonamovingvehicledropsinpitchasitspeedsawayfromyou,thewavelengthsofthespectrallinesfromthestarsinrecedinggalaxiesaredisplaced(orredshiftedtolongerwavelengthsattheredendofthespectrum).Thesizeofthisredshiftallowstheradialrecessionvelocitiesofthegalaxiestobedetermined.Hubble’slawtellsusthatthegalaxiesaremovingawayfromusatspeedsthatincreaseinproportiontotheirdistances.Theuniverseisexpanding,andifweimagineamovieoftheexpansionrunbackwards,weinferthattherewasamomentofcreation,atimewhenallthegalaxiescametogetheratonepoint,13.8billionyearsago.TheBigBangwasinfinitesimallysmall,infinitelyhot,andinfinitelydense.

TheoriginofmatterAllthematterandenergyintheuniverseeruptedoutoftheBigBang.Atveryearly

timesoflessthanamicrosecond,theenergieswerefargreaterthanthosethatwecanproduceinourbiggestparticleaccelerators,andwecanonlyextrapolateourcurrenttheoriesbacktotheseearlytimes,anextrapolationwhichinevitablyincursuncertainties.Theshortestlengthscalethathasanymeaninginphysicsistheso-calledPlancklengthscaleof10–35metres,adistancebelowwhichourideasaboutgravityandspacetimearenolongervalid,andquantumeffectsdominate.ThetimetakenforlighttotravelonePlancklengthiscalledthePlancktime(10–43seconds).Thisisthesmallestunitintowhichwecansubdivideourunitoftimeandsowecansaynothingaboutthephysicalconditionsoftheuniverseearlierthanthistime.

Astheuniverseexpanded,itcooled.Thiscoolingisanaturalconsequenceofthelawoftheconservationofenergy.Bythetimeitwasamicrosecondold,theuniversehadcooledtoatemperatureof1013K(or1GeV),whichisequivalenttothemass-energyofaproton.Itwasthereforetoohotforthequarksandgluonstobindtogethertomakenucleons.Theuniversewouldhaveconsistedofexoticquark–gluonplasmaorquarksoup,asearinglyhotcauldronofquarks,gluons,andphotons,tooenergetictosticktogether.Thisisthoughttohavebeentheonlytimeinthehistoryoftheuniversewhenfreequarkscouldhaveexisted.

Atthistime,conditionsweresoextremethatmatterandenergywerefreelyinterchangeable.Thelawsofphysicsdonotdistinguishbetweenmatterandantimatter,soifpreciselyequalamountsofitwerecreatedandannihilatedwithcompletesymmetry,wewouldhavehadauniversefilledonlywithradiation,andnomatter.Yet,13.8billionyearslater,weobservethatthereareabout10billionphotonsforeveryprotonorneutronintheuniverse.Thispuzzleisknownasthebaryogenesisproblemandraisesthequestion:whereisalltheantimatter?Itisofcoursepossiblethatthereareentiregalaxiesmadefromantimatter.Anantigalaxywouldhavetobe‘cordonedoff’fromnormalgalaxies;otherwisegalaxy–antigalaxypairswouldannihilateeachotherinenormousexplosionsofgammarays.Suchhugeexplosionshavenotbeenobserved.Webelievethatthe1080protonsofmatterthatconstitutethevisibleuniverseareprotonsandnotantiprotons,andthattheyweremadefromquarksleftoverfromthevastnumberofquark–antiquarkannihilationsthatareinferredtohavetakenplaceatearlytimes.TheRussianphysicistAndreiSakharovsuggestedasolutiontothebaryogenesisproblem:forevery10billionantiquarksformedintheearlyuniverse,therewere10billionandonequarks,leavinganetbalanceofmatteroverantimatter.Thismodelrequiresaverysmallbiasinthelawsofphysicstofavourtheformationofmatter.

Wherecouldsuchabiashavecomefrom?Theslightasymmetryinthelawscouldbe

relatedtotheviolationofatypeofsymmetrycalledparity,orP-symmetry,theinabilityofnaturetodistinguishbetweentheworldanditsmirrorimage.ThisisincludedintheStandardModel,andrelatestotheweaknuclearinteraction.In1956Chinese-AmericanphysicistChien-ShiungWu,atthesuggestionofT.D.LeeandC.N.Yang,setupanexperimenttomeasurethemagneticspinofradioactivelybeta-decayingcobalt-60nuclei.Byaligningthespinningnucleiinamagneticfield,shefoundthatthenucleishotmoreelectronsoutfromtheirsouthpolesthanthenorth.Thiswasagreatsurprisebecauseitshowedthatthereisanabsolutedifferencebetweenthetwopolesofanucleus.Theasymmetryiscalledthe‘failureofconservationofparity’.

Anothersymmetryofthephysicallawsischargeconjugationsymmetry,orC-symmetry.Itisatransformationsuchthat,ifparticlesareswitchedwiththeirantiparticles,thesignsofallchargesarechanged.Thisistrueforalltheforcesexceptfortheweakinteraction,forwhichC-symmetryisviolated.Experimentshaveshownthatatypeofmesoncalledakaon,firstseenincosmicrays,decaysintopairsofpionsforwhichthecombinedCPsymmetrycanalsobebrokenduringtheweakinteraction.Thereisasmalltendencyfordecayingkaonstoproducemorepositronsthanelectrons,andtheseexperimentsshowedthatnaturehasaninherent‘handedness’.

Theimmensenumbersofquark–antiquarkannihilationsintheearlyuniverseproducedelectromagneticradiationthatcooleddownastheuniverseexpanded.Thewavelengthsofthisafterglowradiationwerestretchedoutbytheexpansionofspaceitself,anditisnowobservableasthe2.7Kcosmicmicrowavebackground(CMB)radiationthatbathestheuniverse(Figure30).

30.Thefullskymapofthecosmicmicrowavebackgroundradiation,theafterglowoftheBigBang,showingtheinfantuniverse.Themottledappearanceisduetotemperaturefluctuations,andtheseprovideinformationonthelarge-scaledistributionofmatteratearlytimes.

Astheuniverseexpanded,temperaturesfellbelow1013K(1GeV),anditbecameenergeticallyfavourableforquarkstocondenseintoprotonsandneutrons.Bythetimethattheuniversewasthreeminutesold,thetemperaturehaddroppedto109K(100keV),andthefirstnuclearreactionscouldtakeplacetosynthesizethelightelementsofmatter.Nucleiofdeuterium,helium-3,helium-4,andlithium-7wereformedinacomplexofreactionscalledBigBangnucleosynthesis.Duringthisepoch,itwashotenoughfornuclearreactionstofuseasignificantproportionoftheprotonsintoheliumbuttherewasnotenoughtimetosynthesizemanyheaviernuclei.Theprimordialrawmaterialforfurtherprocessinginstarswastherefore‘lockedin’atthistime,resultinginacompositionof75percenthydrogen,25percenthelium,andtracesofotherlightelements.

Itwashoweverstilltoohotforneutralatomstoexist,andtheuniversecontinuedtoexpandasplasma,inwhichthephotonsscatteredcopiouslyfromfreeelectrons.Toanembeddedimaginaryobserver,theuniversewouldhavelookedlikeadensefog.Byanageof380,000yearsthetemperaturehadfallentosome3,000K,whenelectronscouldcombinewithnucleitoformthefirststableatoms,atwhichpointthefogcleared(‘recombination’).Fromthenon,thephotonswerenolongerstronglycoupledtomatterandcouldtravelfreelythroughspace.Thephotonsthatemergedfromtheedgeofthefogbank,the‘surfaceoflastscattering’,aretheoldestwecannowdetectinourtelescopesasthecosmicmicrowavebackgroundradiation.Theseancientphotonscarrywiththeminformationaboutthephysicalconditionsintheuniverseatthisstageofitsevolution,andshowthatthefirstlarge-scaleaggregationsofmatterhadalreadystartedtoform.TheCMBtemperaturefluctuationsshowninFigure30areverysmall,representing1partin100,000oftheaverage2.7Ktemperaturethatweobserve.Inthesubsequentexpansion,thedensityfluctuationscontinuedtogrowandconcentrateprimordialmatterintothestars,galaxies,clusters,andsuperclustersofgalaxies.

FurnacesofmatterByabout500millionyears,thedensityfluctuationsinthegashadgrownintodistinctcloudsthatbecamedetachedfromthegeneralexpansionoftheuniverse,andthesestartedtocollapseinonthemselvesundertheirself-gravity.ThecloudsfragmentedintosmallerclumpsduringastarlessperiodcalledtheDarkAge.Thegasaccretedonthecoresofwhatwouldbecomethefirststars,andwascompressedandheated,justastheairinabicyclepumpgetshotwhenitissqueezed.Atfirstthehotatomsradiatedtheirheatintospace,butthenasthedensityincreased,theenergywastrappedinside,creatingathermalpressuretocountertheinwardpullofgravity.

Atthisstageanascentstarhasastablemassandsize,andiscalledaprotostar.Asthecoreoftheprotostarreachesthetemperatureatwhichfusionreactionsbetweenprotonscanoccur(around15milliondegrees),thehydrogenstartstoburnliberatingenergy,whichistransportedouttothesurfacebyphotonsandradiatedintospace.Thetemperatureofastarincreasesstronglywithitsmass.Thefirststarsweremassiveandhotandshonewithanintenseultravioletlight,whichionizedthesurroundinggas.Thisepochiscalledtheeraofreionization.TheionizationofinterstellargasaroundnewlybornstarscontinuestodayinactivestarformingregionssuchastheOrionnebulaintheMilkyWay.

Withtheappearanceofthefirststars,thesecondphaseoftheproductionofthechemicalelements,stellarnucleosynthesis,couldthengetgoing.Toappreciatehowstellarnucleosynthesisworks,weneedfirsttounderstandabasicconceptrelatingtothestabilityofnuclei.Considerthestabilityofdifferent-sizednuclei.Inasmallnucleuswithonlyafewnucleons,alargefractionoftheparticlesliesonthesurface,wheretheyenjoytheshort-range‘Velcro-like’strongforcebindingthemtotheirneighbouringnucleonswithwhichtheyareinclosecontact.However,sincethenucleonsonthesurfacefeelthebindingforcesonlytowardsthecentreofthenucleus,theycanbe‘evaporated’relativelyeasilyfromit.Thisisthereasonwhyverysmallnucleiarenotthemoststable.Attheotherextreme,alargenucleuswithmanyprotonsproducesastrongrepulsiveelectrostaticforce,whichtriestobreakthenucleusapart.Theheaviestelementstendtobespontaneouslyradioactive.Alargenucleussuchasuranium,whichisonthevergeofinstability,canpushoutanalphaparticle(theoriginofalpharadioactivity),removingtwounitsofpositivecharge,andsobecomesmorestable.

Thestabilityofnucleicanusefullybepicturedonadiagramofthebindingenergypernucleon(Figure31).Thebindingenergycurvehasa‘U’shape,withavalleywherethemoststableintermediatemassnuclei(suchasironwith56protonsand30neutrons)arefound.Lessenergyisneededtoremoveanucleonfromanelementhighupinthediagramthanonelowerdowninthevalley.Thenucleihighupinthediagramarethereforelessstablethantheoneslowerdown.Thelightelements:deuterium,tritium(thehydrogenisotopewithoneprotonandtwoneutrons),helium-3,helium-4,etc.areatahighlevelontheleft,andtheheavyunstablenucleisuchasuraniumareabovethevalleyontheright.Energy-producingnuclearreactionscanproceedintwopossibledirections,alwaysmovingtowardsthemorestableelementsnearthevalleyfloor.Fusionreactionscombinelightnucleiintoheavierones,andfissionreactionsinvolvethesplittingofheavynucleiintosmallerfragments.

31.Thecurveofbindingenergypernucleonforvariousnuclei.Energy-producingnuclearreactionsmoveinthedirectionofthemoststablenucleus,iron,atthebottomofthecurve.

Thenuclearreactionsinstarsarethermonuclearfusionreactions,wherelightelementsfusetogethertomakeheavieronesandliberateenergy.Therearetwomainpartstoastar,thehotcore,wherefusiontakesplace,andthesurroundinglow-densityenvelope,whichformsablanketandisthepartofthestarthatyoucansee.Whentwonucleifuseinthecoreofastar,theymustapproacheachothercloselyenoughtobegrippedbythestrongnuclearforce.But,beingpositivelycharged,theyhavefirsttoovercometheirmutualelectrostaticrepulsion,andthismeanstheymusthavehighenergiesandhightemperatures.Theeasiestreactionisbetweentwoprotons.Twoprotonscombinetoproducedeuterium,aninitialstepintheso-calledp-pchainoffusionreactionsthatleadstotheproductionofhelium.Thep-pchainproducestheenergythatpowersstarsofcomparativelylowmassliketheSun.

Theheavierelementsarebuiltupina‘buildingblock’approach,inwhichthestableunitsofheliumnuclei(alphaparticles)fusetogethertomakebiggernuclei.Whenthetheoryofthisprocesswasfirstbeingworkedout,thereappearedtobeasnag.Itseemedthatelementbuildingcouldnevergetoverthefirststep,thefusionoftwoalphaparticles.Twoalphaparticlesproduceanunstablenucleus,beryllium-8,whichdecaysrapidlyawaybeforeathirdalphaparticlecanjoinit.Thereisnostablenucleusofmass8innature.In1953,EnglishastronomerFredHoylepredictedthatthe

nucleusofthenextelementintheseries,carbon-12,musthavewhatiscalledanexcitedstateresonance.Theconceptofresonanceisfamiliartoanyonewhohasseenawineglassshatteredbyanoperasinger.Atitsresonantfrequency,theglassabsorbsenergybysympatheticoscillationsthatgrowinamplitudeuntilthematerialfractures.Instars,theresonanceofthecarbon-12nucleushastheeffectofgreatlyenhancingtheprobabilitythatcarbon-12willformbeforetheberyllium-8candecay.Hoyleurgedexperimentaliststosearchfortheresonance,whichtheydulydiscoveredattheenergyhehadpredicted.Carboniscreatedinstarsatatemperatureof100milliondegreesbythreealphaparticlesfusingtogetherintothecarbon-12excitedstate,theso-called‘triplealpha’process.

Forbiggernucleitofusetogether,highertemperaturesareneededtoovercometheirlargerelectrostaticrepulsions.Forexample,instellarcarbonburning,twocarbonnuclei,eachwithsixprotons,mustfuse,whileforoxygenburningthenucleihaveeightprotons.Thecoretemperatureofastarofaround20solarmassescanreachseveralbilliondegrees,whichishighenoughforelementsuptonickeltobesynthesized.Thehotcoreofsuchastarissurroundedbyanenvelopeofhydrogenandhelium.Workinginwardsfromtherelativelycooloutersurfacetothehotcore,therearenestedonionskin-likeshellsofincreasingtemperatureinwhichdifferentnuclearreactionsoccur.Thereactionsproceedfromheliumburninginthecooleroutershellproducingcarbonandoxygen;theseelementsburninthenexthottershelldownproducingneonandmagnesium,thensulphur,and,inthehightemperaturecore,siliconburningproducesnickel.Thefusionproductsofeachnuclearreactionfeedintotheinnershellsandthestarevolvesasitconsumesitsfuels.

Insummary:thestarsshinebyliberatingthefusionenergyofthelightelements,whichsithighabovethevalleyinthebindingenergycurve,toproducethemiddleweightelements,whichsitlowerdowninthevalley.Bythetimethestarhasconsumedallofitseasilyavailablefusionenergyfuel,itcangetverylittleenergyfromburningsilicontomakenickel,sincebothelementslienearthefloorofthevalley.Thefinalsilicon-burningstageisthereforerapid,lastingonlyafewdays,andthenickeldecaystoformastellarcoreofiron.Theirongroupofelementshavethemosttightlyboundnucleiintheperiodictable,andtomakeelementsbeyondironrequiresaninputofenergy,whichcannotoccurbythermonuclearprocesses.Asfarassynthesizingtheelementsinsidestars,ironistheendoftheline.Thequestionis:howareelementsheavierthanironcreated?

Thedeathofstars

Whenastarrunsoutoffuel,thethermalpressurethatoncekeptitinflatedvanishes,andtheinexorableinwardpullofgravitycausesittocollapse.Starshaveessentiallythreepossiblefates:alowmassstarliketheSunwillcollapseintoawhite-dwarfstar,amiddle-rangemassstarcanformaneutronstar,orifthemassishighenough,matteriscompressedintothemostcompactformpossibleinablackhole.

Inawhitedwarf,thegravitationalcollapseishaltedbyanewsourceofpressure:electrondegeneracypressure.Thisquantummechanicalpressurecomesfromthehighenergiesoftheelectrons,whentheyarecompressedintoasmallvolume,inlinewiththeuncertaintyprinciple.Thedensityofmatterinawhitedwarfisaboutamilliontimesthedensityofwater.Thereishoweveralimittohowmuchelectrondegeneracypressurecanbeprovided.IfthemassofthecollapsingstarislargerthanacriticalmasscalledtheChandrasekharmass(about1.4solarmasses),theelectrondegeneracypressureisinsufficienttopreventfurthercollapse.

Whena20solarmassiron-coredstarrunsoutoffuel,theChandrasekharlimitisexceeded,andelectrondegeneracypressurecannotsupportthestar.Thecentralpressureplummets,andthestarcannolongerfightagainsttheinexorablepullofgravity.Theironcorebeginstocollapsereleasingahugeamountofgravitationalpotentialenergy,andimplodes,reachingone-thirdofthespeedoflightinasecond.Thetemperaturerisesdramaticallyandtheironnucleidisintegrateintotheirconstituentnucleons.Theelectronsandprotonsmergeviatheweakinteractiontoformaneutron-richformofmatter.Ifmorethanasmallfractionofitsnucleonsremainedasprotons,theelectricalrepulsionwouldbeoverwhelming.

Suddenlysomethingextraordinaryhappens.Anothernewpressuresource,thequantumdegeneracypressureoftheneutrons,setsinandhaltsthecollapsedeadinitstracks.Unawareofwhathasjusthappenedinthecore,therestofthestarcontinuesitsheadlonginfall.Whentheouterlayersofthestarhitthenow-stablecore,theybouncebackviolently,sendingoutpowerfulshockwaves.Thestarexplodesasoneofthemostviolenteventsthatweknowofintheuniverse,asupernova,flingingtheouterlayersofthestarintospacecarryingtheirrichcargoofelements.

Asupernovaisanuclearexplosionwithayieldthatisamassive1027timesbiggerthanaman-madeH-bomb.ThesupernovashineswithaluminosityabilliontimesthatoftheSun,andforafewmonthscanoutshineitshostgalaxy.Theexpandingblastwavecontainsmanysolarmassesofmaterial,consistingofmiddleweightelementsanddustgrains,immersedinafluxofhigh-speedneutrons.Theexpellednucleirapidlyabsorbtheneutrons,whichareconvertedtoprotonsviabeta-decay,transmutingthenucleiintoelementsfromleaduptouranium.

Theneutron-richstellarcoreisdestinedtobecomeaneutronstar,onethemostexoticformsofmatterofwhichweknow.Ineffectitisagiantatomicnucleusheldtogetherbygravity,anobjectthesizeofanaveragecity,weighingabouttwiceasmuchastheSun.Ateaspoonfulofneutronstarmatterweighsabilliontonnes,aboutthesameasMountEverest.

Thereisasymbiosisbetweenstarsandthegascloudsofthetenuousinterstellarmediumoutofwhichtheyform.Intheirdeaththroes,starsinjectheavyelementsintothemediumtoenrichtheprimordialgas,andthemixturegetsrecycledbacktomakenewstars.TheSunandthesolarsystemare4.5billionyearsold,andweremadefrommatterfromatleastonegenerationofearlierstars,whichhadburnedoutbillionsofyearsbeforetheSunandoursolarsystemformed.Theelementsofourfamiliarworld,thecarbononwhichlifeisbased,theoxygenthatwebreathe,andtheironinourcarsallcomefromthestars.

Thereisroughlyonesupernovaexplosioninourgalaxyeachcentury.On4JulyAD1054Chineseastronomersrecordedthepositionofa‘gueststar’inthesky.Modernastronomerslookedinthesamedirection,anddiscoveredtheCrabnebula(Figure32).Thefilamentsofgasinthenebulaarerushingoutwardsintospace,andiftheirmotionisbacktracked,theymusthavecomefromasinglepoint,thesupernovaexplosionof1054.Thisiswhatasupernovalookslikeafter1,000years.

32.TheCrabNebulaistheremnantofastarthatexplodedinAD1054.

Thealchemists’dreamrealizedThecollapsedneutronstarremnantofasupernovarotatesrapidly,draggingroundwithitanimmenselystrongmagneticfield.Suchobjectscanbepulsars,objectsthatemitbeamsofelectromagneticradiationsweepinground,likethebeamsoflighthouses.Pulsarsproduceasequenceofpreciselytimedpulses,likeclocks,astheirbeamssweeppastourdirection.TheCrabnebulacontainsapulsarthatpulsesthirtytimesasecond.

Thefirstbinarypulsarwasdiscoveredin1974,andconsistsoftwoneutronstarscloselyorbitingaroundtheircommoncentreofmass.Thissystemisoftremendousinterestbecauseitallowsgeneralrelativitytobetestedinthestronggravitationalfieldsandcurvedspacetimenearhighlycollapsedobjects.Thetworotatingmassesdistortspacearoundthem,whichswirlsaround,generatingripplesinspacetime,gravitationalwavescarryingawayenergyatlightspeed.Thetwostarsaresteadilylosingenergy,whirlingeverfasteraroundtheircommoncentreofmassastheycome

evercloser.Theorbitalperiodofthebinarysystemhasshortenedbyaboutaminuteinthetimesinceitsdiscovery,astheirrotationalenergyisconvertedintoradiation.(TheEarthorbitingtheSunalsolosesenergybygravitationalradiation,butfortunatelyforusatanimmeasurablysmallrate.)Afterafinalrapid‘inspiral’phase,thebinaryneutronstarswilleventuallymergeinacataclysmicmergerthatwillwrenchthefabricofspacetimeviolentlyandreleaseastrongpulseofgravitationalwaveenergy.

Gravitationalwavesweredetectedforthefirsttimein2015,frompairsofmergingblackholes.Astronomershadalsobeenanticipatingthedetectionofgravitationalwavesemittedwhentwoneutronstarscollideandmerge.On17August2017theirpatiencewasfinallyrewardedwhenthreegravitationalwaveobservatories,twointheUSA(LIGO)andonenearPisainItaly(VIRGO),detectedaburstofgravitationalradiationfromthemergeroftwoneutronstars(aneventcalledGW170817).Thetimingofthecosmicsignalatthethreesitesenabledthephysiciststotriangulateitspositionwithenoughprecisiontocorrelatetheeventwithabrightgammarayflashrecorded1.7secondslaterbyNASA’sFermispacetelescope.TheobservationofGW170817triggeredanalertthatgalvanized100teamsofastronomerstosearchforanopticalcounterpart.Thistheyfoundinagalaxy130millionlightyearsaway.Overtheensuingweeks,observatoriesdetectedelectromagneticradiationatX-ray,ultraviolet,optical,infrared,andradiowavelengths.

Mergingneutronstarsdonotjustproducegravitationalwaves;theyalsospewouthotdensechunksofmatteratspeedsofuptohalfthespeedoflight.Theexpandingcloudofdebrisformsafireballinwhichtheprotonsandneutronscombinerapidlytoformheavynuclei.Thesenucleicapturemoreneutrons,makingthenunstableandhighlyradioactive.Theneutronsinthenucleiareconvertedintoprotonsviaslowerbeta-decayprocesses,andreleaseenergywhichlightsupthefireball.Spreadingoutintospace,thefireballcontainsarichcocktailofsomeoftheheaviestelementsintheperiodictable.Theaftermathofthecollisionisknownasakilonova,abrighttransientevent,lessluminousthanasupernova,but10milliontimesmorethantheSun.

Observationsofthespectrallinesemittedbytheseelements,usingtheVeryLargeTelescopeinChile,revealedsignaturesofheavyrareearthelements(lanthanides).Thisprovidedevidencethatheavyelementsintheperiodictable(elementsfromniobiumtouranium)werecreatedinthemerger.IthasbeenestimatedthatGW170817hasejectedafewtimesthemassoftheEarthingoldandplatinumintospace,exceedingeventhewildestdreamsofthealchemists!

The2017kilonovawasthefirsteverdetectionoftwocollidingneutronstars,andwasthefirstastronomicaleventinwhichgravitationalandelectromagneticwaveswere

observedtogether.Itheraldedthebirthofanewastronomy:multimessengerastronomy.Thenear-simultaneousarrivalofgravitationalandelectromagneticpulsesfromaneventwhichhappened130millionyearsagoisitselfremarkableandmeansthatpropagationspeedsofgravitationalandelectromagneticwavesdifferbynomorethan1partin1015,inlinewithpredictionsofEinstein’stheoryofrelativity.

Insummary,therearethreekeyprocessesbywhichthechemicalelementshaveformedinnature.ThelightelementsweresynthesizedintheBigBang,themiddleweightelementswere(andstillarebeing)forgedinsidestars,andtheheavyweightelementswere(andstillarebeing)producedinviolentstellarexplosionsandcataclysmicevents.TheelementsfromwhichyouandIaremadewereproducedinthelivesanddeathsofstarsthatexistedbillionsofyearsbeforethesolarsystemformed.ThehydrogenatomsinourbodiesgobacktotheBigBangitself.

InChapter9wewilllookattwomysteriousanddominantformsofmatterintheuniverse:darkmatteranddarkenergy.

Chapter9

Darkmatteranddarkenergy

Whenwelookintodeepspacewithourtelescopes,weseeacosmosfilledwithahundredbilliongalaxies.Eachgalaxycontainsaroundatrillionstars,andmanygalaxiesaresimilartoourown,thespinningdiscoftheMilkyWay.Onthelargestlengthscales,fromthesizeofasteroidsuptothatofthevisibleuniverse,matterisdominatedbyasingleforce:gravity.

Gravityistheweakestofthefourforcesofnature.Itistheforcethatkeepsourfeetfirmlyplantedontheground,andreachesoutintospacetoguidetheplanetsintheirorbitsaroundtheSun,andgripsthetrillionsofstarsastheyswirlaroundtheirhomegalaxies.ThemotionoftheplanetsisdescribedextremelyaccuratelybyNewton’slawofgravity.Indiscoveringthelaw,Newton’sbrilliantideawastoimaginethattheforceonafallingappleandtheMoonarereallythesame;bothbodiesmoveinEarth’sgravity,onlyfortheMoontheforceisdiminishedinstrengthbyitsgreaterdistancefromtheEarth.Heimaginedacannononahighmountain,firingshotstowardsthehorizonatever-higherspeeds,sothattheylandfurtherandfurtheraway.EventuallyonegoesfastenoughtocircletheEarthataconstantheightandsobecomesasatellite.

Newton’sthoughtexperimenttellsusthatthevelocityofasatellite,itsheight,andthemassofthebodyaroundwhichitorbitsareconnected.If,forexample,themassofthecentralbodyislarge,thespeedofthesatellitemustalsobelargetoremaininorbitandmaintainthebalancebetweenitsoutwardcentrifugalforceandtheincreasedinwardpullofgravity.Measuringofthesizeoftheorbitsofstarsingalaxiesandtheirorbitalspeedsisthebasisofthemethodthatastronomershaveusedtomeasurethemassesofgalaxies.

Thestarsinadiscgalaxyorbitaroundthecentrewithspeedsthatdependonhowfarouttheyare.Ifthebulkofthegravitatingmassofagalaxyisassumedtobe

concentratedinthemiddle,wherethereisoftenaprominentbulgecontainingahighdensityofstars,therotationalvelocitiesofthestarsintheouterdiscshoulddecreasethefurtheroutyougo.Thisdecreaseisseenfortheplanetsofthesolarsystem,wheretheSun’sgravityweakenswithdistance.Inthe1970s,AmericanastronomerVeraRubinmeasuredtherotationalvelocitiesofthediscsofnearbygalaxiesandfoundthat,contrarytoexpectation,therotationalspeedsdidnotdecreasewithdistance,butremainedconstant.Thegalaxieswererotatingtooquicklyforthevisiblemattertheycontain.Whensheworkedoutthemassesofthegalaxies,shediscoveredthattherewasaboutfivetimesasmuchmassascouldbeaccountedforbythecombinedmassofthestarsandgas.

TheSwissastronomerFritzZwickyhadreachedasimilarconclusioninthe1930swhenmeasuringthemotionsofgalaxiesthatswarmtogetherinclusters.Inthe1,000-galaxy-strongComaclusterhediscoveredthatthegalaxiesintheouterpartsoftheclusterweremovingmuchfasterthanexpectedgiventheamountofvisiblemassinthecluster.Thissuggestedthattheclusterwasbeingboundtogetherbyunseengravitationalmass,whichhedubbed‘dunkleMaterie’,ordarkmatter.Theinferencewasthatmostofthemassingalaxiesandclustersofgalaxiesisinvisiblematter,whichwenowbelievetobedistributedinlargehaloesenvelopingthevisiblepartsofthegalaxies.Thisso-called‘missingmass’revealsitselfbyitsgravitationalinteractionwithnormalmatter,butdoesnotgiveofforreflectlightinanyway.

ThemethodofmeasuringthemassesofgalaxiesusedbyRubinandZwickyreliedonNewton’slawofgravitation.But,aswesawinChapter4,Einsteininhisgeneraltheoryofrelativityshowedthatmasscurvesspaceandcurvedspacenotonlyinfluencesthemotionofmassivebodies,butitcanalsobendlight.Thislight-bendingaspectofgravityprovidesapowerfulwaytomeasurethemassoflargebodies.

ThelightthatwasonceemittedbytheatomsinthestarsofthemostdistantgalaxiesbillionsofyearsagoisonlynowreachingtheEarth.Initslongjourney,thelighthashadtotravelthroughvastdistancesofbillionsoflightyears.If,somewherealongitsroute,aphotonfromoneoftheseverydistantgalaxieshappenstopassneartoamassivebodyandencountersthecurvedspacethatsurroundsit,thelightisbentandendsuponadifferenttrajectory.Thisissimilartowhathappenswhenlightraysarebentorrefractedwhentheypassfromairintoatransparentmediumsuchaswateroraglasslens,ofthetypeyoumightfindinamagnifyingglass.Theexcitingideathatcomesoutofthisisthatitispossibletousethislight-bendingeffect,calledgravitationallensing,tomeasurehowmuchmassispresentinspaceandhowitisdistributed.Inagravitationallens,theonlythingthatmattersisthecurvatureofspace;lightcan’ttellthedifferenceifthecurvatureiscausedbydarkmatterorbythe

starsandothermatterinagalaxy.

Tohelpvisualizehowgravitationallensingworks,considerFigure33,whichshowsthetiledmosaiconthebottomofaswimmingpool,viewedlookingverticallydownwardsthroughthewater.Ripplesonthesurfaceofthewaterbehavelikerefractinglenses,anddistorttheimageofthebackgroundtiles,whichrepresentthedistantbackgroundgalaxies.Wecanthinkoftheripplesofthesurfaceofthewaterasbeinganalogoustothecurvatureofspaceassociatedwiththeconcentrationsofinterveningmassintheuniverse.Therearethreecases.Whenthewatersurfaceisstill,thereisnodistortionandthetilingpatternappearstoberegular;thiscorrespondstoanemptyuniversecontainingnointerveningmatter.Ifthewatersurfaceisgentlyrippled,thegridpatternappearsweaklydistorted,acasethatcorrespondstoweaklensing;thiswouldbeanalogoustothepresenceofsmallconcentrationsofmassforgravitationallensing.Largerripplescreatemoreextremedistortions,whichbreakupthepatternintomultipleimages.Thiscorrespondstostronglensing,andthepresenceoflargermasses.

33.Illustrationofthegravitationallensingeffect,withwaterripplesinaswimmingpool.Snapshotsofmosaictilepatternonthebottomofaswimmingpoolwhenthewatersurfaceis:(left)still;(centre)weaklyrippled,and(right)stronglyrippled.

Stronggravitationallensingcanbeproducedbythecurvingofspacearoundlargemassconcentrations,suchasclustersofgalaxies,andcanstretchouttheimagesofbackgroundgalaxiesintolongluminousarcs(Figure34).Weakgravitationallensingismorecommonlyobservedand,asthenamesuggests,isalessdramaticformwhichchangestheshapesofbackgroundgalaxiesinmoresubtleways.Inthiscase,informationonthemassofthelenscanstillbederivedthroughthestatisticalanalysisoftheshapedistortionsoftheverymanybackgroundgalaxieswhoselinesofsightpassclosetoit.Intheswimmingpoolanalogy,theamountofdistortiontothetilingpatternprovidesinformationonthesizeoftheripplesinthewater.

34.TheSmileyFace;thegravitationallensingofdistantgalaxies(curvedarcs)bythemassofanearerclusterofgalaxies.Twolargegalaxiesinthenearerclusterformthe‘eyes’oftheface.

Gravitationallensinghasrecentlyjoinedthearmouryofmodernastronomicaltechniquesthatwillinformonthequantityanddistributionofmatterintheuniverse,particularlythroughobservationsthatwillbemadewiththecominggenerationofdedicatednewtelescopes.GravitationallensinghasbeenobservedbytheHubbleSpaceTelescope,butonlyinsmallpatchesofskyofafewsquaredegrees.Toassesstheamountofdarkmatterthereisintheuniverse,itwillbenecessarytosurveyamuchlargerpartofthesky.OnetelescopethathasbeendesignedtodothisistheEuropeanSpaceAgency’svisibleandinfraredspacetelescope,EUCLID,dueforlaunchin2020.

Whatisdarkmatter?Thesimpleansweristhatnobodyknows.Atpresentweknowmoreaboutwhatdarkmatterisn’tthanwhatitis.Therearetwomainideasforwhatitmightbe.Oneisthatitissimplyordinarymatter,butinaformthatabsorbsoremitslittleornolight.This

possibilitycomesunderthebroadheadingofMassiveCompactHaloObjects(MACHOs),whichincludefailedlow-massstars(orbrowndwarfs),Jupiter-sizedplanets,whitedwarfs,andcompactobjectssuchasneutronstars.Theproblemwithmostofthesepossibilitiesisthat,beingmadeofnormalmatter,MACHOswillabsorbandemitelectromagneticradiation—namelytheyoughtto‘glow’atvariouswavelengths.Objectsthatglowinthiswayappeartoberuledoutbyobservations.

Thesecondandleadingpossibilityfordarkmatteristhatitisanexotictypeofsubatomicparticle,knownasaWeaklyInteractingMassiveParticle,orWIMP.WIMPscouldhavebeenmadeintheBigBangalongwithquarksandradiation.AWIMPwouldneedtobeheavy(havingamassofbetweenoneand1,000protonmasses),bestableforatleasttheageoftheuniverse,andinteractatmostonlyweaklywiththeotherparticlesintheStandardModelofParticlePhysics.ThereisnoparticlecurrentlyintheStandardModelwiththeseproperties.OtheroptionsforWIMPsincludethepossibilitythatnatureissupersymmetric.ThelightestsuperpartnertoaStandardModelparticle,thehypotheticalneutralino,isapossiblecandidatedarkmatterparticle.Neutrinosarenowbelievedtohaveverysmallmassesofamillionthofthemassoftheelectron,andhavethereforebeenconsideredaspossibleWIMPcandidates.Neutrinoscomeunderthecategoryof‘hot’darkmatter,whichinthiscontextmeansthattheymoveatspeedsclosetothespeedoflight.

Thefactthatdarkmatterconstitutesroughlysixtimestheamountofordinarymatterthatweknowofmeansthatitmusthaveplayedapivotalroleincontrollingthegrowthofthelarge-scalestructuresofgalaxiesandclustersofgalaxiesintheearlyuniverse.Onverylargelengthscales,theuniverseissmoothanduniform,butitis‘lumpy’onthescalesofgalaxiesandclustersofgalaxies.Theroleofdarkmatterintheformationofthesestructureshasbeeninvestigatedusingcomputersimulations,withtheobjectiveofexplainingthestructureswenowobserveinthelocaluniverse.Thesimulationshavebeenabletoreproducesuccessfullythepropertiesoftheobservedstructures,butonlyonthebasisthatthegravitationalpullissuppliedby‘cold’darkmatter(CDM).The‘cold’inCDMmeansthatthevelocityofthedarkmatterisassumedtobeverymuchsmallerthanthespeedoflight,anditcannotcoolbyemittingphotons,sinceitisdark.Hotdarkmattertendstosmoothoutthesmall-scalestructuretoomuch.This,andthesmallnessoftheirmasses,appearstoruleoutthecandidacyofneutrinosasWIMPs.

DarkmatterparticlesthatmayinhabitthehalooftheMilkyWaywouldbeexpectedtostreamcontinuallythoughthediscofthegalaxyandthereforeshouldpassthroughthesolarsystem.Iftheydo,thereisthepossibilityofdirectdetectionwhentheystriketheEarth.Ifthereisaweaknon-gravitationalinteractionbetweendarkandnormal

matterparticles,darkmatterparticlesmightberevealedinrarecollisionswiththenucleiofordinarymatter.Apossiblefingerprintwouldbetheproductionofaphotonfromadarkmatterparticleinteractingwithaheavynucleus.AnexperimentsetuptosearchforsucheventsistheLargeUndergroundXenonexperiment(LUX),whichisalargetankofliquidxenon,surroundedbybanksofhighlysensitivephotomultiplierdetectors.LUXissited1.5kmundergroundintheHomestakegoldmineinSouthDakota,deepenoughtoblockoutspuriousparticles.ThechanceofaWIMPstrikeonaxenonnucleusisexpectedtobeverylowandtherehavebeennosignificantdetectionstodate.

DarkenergyAfterpublishinghisgeneraltheoryofrelativityin1916Einsteinwentontoapplyhisequationstothecosmos.Atthetimetheuniversewasbelievedtobestatic.Therewasaproblem:ifyoufillamodeluniversewithanumberofmassesandletitgo,themassesfalltogetherundertheirgravitationalattraction—theuniversecouldn’tremainstatic.Einsteintriedtorectifythisbyaddingarepulsivetermtohisequations,calledthecosmologicalconstant,usuallydenotedbytheGreeksymbolΛ.WecanthinkoftheΛ-termasakindofantigravityforcewarpingspacetime,workinginoppositiontogravityandpushingobjectsapart.

However,withinafewyearsofEinstein’sproposedcosmologicalmodel,Hubblehadannouncedhisdiscoverythattheuniversewasexpandingandnotstatic.Einstein’soriginalequation,withouttheΛ-term,couldhaveexplainedanexpandinguniverse.OnhearingofHubble’sdiscovery,Einsteinfamouslydiscardedhiscosmologicalconstantsayingthatitwasthe‘biggestblunderofmylife’.But,aswewillsee,thereisfreshevidencetosuggestEinstein’sreactionmayhavebeenpremature.

Willtheuniversecontinuetoexpandforever,orwillitonedaystarttocontract,eventuallyendinginabig‘crunch’?ThefateoftheuniversedependsonacompetitionbetweenthekineticenergyoftheexpansionoftheBigBangandthegravityofallthematterintheuniversetryingtopulleverythingtogether.TherecessionvelocitiesofthegalaxiesthatledtothediscoveryoftheHubblelawarerelatedtothekineticenergyofexpansionandopposingitisthegravitationalpullofmatter.Ifthematterdensityistoosmall,itsgravityistooweaktostoptheuniverseexpandingforever.Ifthedensityistoolarge,theneventuallytheexpansionwillstop,theuniversewillcontract,andtherewillbeabigcrunch.Butifthedensityhasafinelytunedcriticalvalue,theuniversewillkeeponexpandingforever,andspacewillhavea‘flat’geometry.Thecriticaldensityisverysmall,equivalenttoaboutfive

hydrogenatomspercubicmetre(forcomparison,thereareabout1025hydrogenatomsinaglassofwater).Theevidenceseemstoindicatethattheaveragedensityoftheuniverseisclosetothecriticalone.

Inthe1990s,twogroupsofastronomersweretryingtomeasurethegeometryoftheuniversebypinningdowntheHubbleexpansionastightlyaspossibleoverextremelylargedistancescales.Theywereobservingaverybrightclassofstellarexplosionsindistantgalaxies,calledtype1asupernovae.Type1asupernovaeareimportantbecausetheygeneratenominallythesameamountofluminousenergy,andsocanbeusedas‘standardcandles’fordistancemeasurements.(Bymeasuringtheapparentbrightnessofaknownstandardcandle,itsdistancecanbeinferredfromtheinversesquarelaw.)

Theastronomersfoundthatthemostdistantsupernovaeareconsistentlymuchfainterthanhadbeenexpected.Thesurprisingconclusionwasthatthespacethroughwhichthelighthadtravelledhadexpandedmorethanexpected,andthatthesupernovaearefurtherawaythanhadpreviouslybeenthought.Thisimpliedthattheexpansionoftheuniverseisaccelerating,whichisnotwhatisexpectedfromauniversefilledwithgravitationallyattractingmasses.Itisasifyouthrowaballupintotheair,andjustasitisstartingtoslowdown,itacceleratesawayfromyouandkeepsongoing.Thatishowsurprisingtheresultwas.

AnaccelerationintheexpansionoftheuniverseishoweverexactlywhatEinstein’sΛ-termcanprovide.Thecosmicrepulsionitengendershasbecomeknownasdarkenergy,amysteriousunknownformofenergythatfillsspace.Thedarkenergydensityisconstant,whichmeansthatastheuniverseexpandsandcreatesmorevolume,thetotalamountofdarkenergyitcontainsincreasesinstepwiththeexpansion.Thisinitselfisanextraordinarynotion.Webelievethatdarkenergyisspreadsmoothlyacrosstheuniverseandthatitsmass-energydensityissmall.WithinthevolumeoftheEarthforexamplethemassequivalentofdarkenergyisonemillionthofagram.Thepuzzleiswhyitissosmall.Weknowthatemptyspacehasalatentenergydensitycalledvacuumenergythatisrelatedtothevirtualparticlesthatconstantlyflitinandoutofexistenceonthequantumscale.However,thequantumvacuumenergydensityfallsshortoftheinferreddarkenergydensitybyanenormousfactorof10120,andsothetruenatureofdarkenergyremainsagreatmystery.Intheveryearlyuniverse,theeffectsofdarkenergywouldhavebeenmaskedbythethenmuchhigherenergydensitiesofmatterandradiation.Butastheuniverseexpanded,thepresenceofthedarkenergycomponenthasbecomemoremarkedanditisonlyinthepastsixbillionyearsthattherateoftheexpansionoftheuniversehasbecomesignificantlyaffected.

BringingitalltogetherAlltheastronomicalobservationshavebeenbroughttogetherinthestandardmodelofcosmology,theΛ-CDMmodel,whichisfoundedonEinstein’sgeneralrelativityandincludesthecosmologicalconstantandacomponentofcolddarkmatter.ThemodelaccountsfortheprimordialabundancesofthelightelementsinthehotBigBang,theangularscalesoftheinitialdensityfluctuationsofthecosmicmicrowavebackgroundradiation,thelarge-scalestructureofthedistributionofgalaxies,andtheaccelerationoftheexpansionoftheuniverse.Oneofthestrongestconstraintsofthemodelisthespectrumoffluctuationsinthecosmicmicrowavebackground(Figure30).TheΛ-CDMmodelfitstheseobservationsandisconsistentwithauniversewithanageof13.8billionyears,aflatspacegeometrywithanaveragedensityclosetothecriticalone,andatightlyconstrainedmixofmass-energycomponentsofmatter.Themass-energyoftheuniverseismadeupof:70percentdarkenergy,25percentcolddarkmatter,and5percentnormalmatter(thefamiliaratoms,quarks,gluons,andleptons).Thecontributionoftheenergydensityofthecosmicmicrowavebackgroundradiationphotonsandneutrinosissmall.

Thestarkrealitythatemergesisthattheordinarymatteroftheatomsandmoleculesofourbodies,andthoseofalllivingcreatures,thematterthatisstudiedinbiology,chemistry,materialsscience,andengineeringandinmuchofastrophysics,constituteslessthanatwentiethofthematterwebelievetoexistintheuniverse.Normalmatterappearsthereforetobejustan‘impurity’inthematterthatisreally‘outthere’.Thestrikingandhumblingfactisthatwedonotknowwhatthebulkofthematterintheuniverseis.

Toroundoffthisstoryofmatter,wemightharkbacktoFeynman’sremark,quotedattheendofChapter3—thatthemostimportantsinglefacttopassontothenextgenerationsisthatallthingsaremadeofatoms.Since1970,whenhewrotethosewords,thegreatadvancesthathavebeenmadeinastronomyhaveuncoveredtheunforeseenandapparentlydominantformsofmatter:darkmatteranddarkenergy.Agreatchallengeforfuturescientistsistofindoutwhatthesemysteriousformsofmatterreallyare.

Furtherreading

PeterAtkins,Galileo’sFinger(OUP,2003).JimBaggott,Mass(OUP,2017).StephenBlundell,VSISuperconductivity(OUP,2009).BrianCathcart,TheFlyintheCathedral(Penguin,2004).FrankClose,VSIParticlePhysics(OUP,2012).RichardFeynman,QED(PrincetonUniversityPress,1988).JohnGribbin,Einstein’sMasterWork(IconBooks,2015).JohnPolkinghorne,VSIQuantumTheory(OUP,2002)(thisbookdescribesquantum

entanglement,whichisnotcoveredinChapter5).MartinRees,JustSixNumbers(Weidenfeld&Nicolson,1999).CarloRovelli,Realityisnotwhatitseems(AllenLane,2016).RussellStannard,VSIRelativity(OUP,2008).PaulStrathern,Mendeleev’sDream(Penguin,2001).StephenWeinberg,TheFirstThreeMinutes(BasicBooks,1993).StephenWeinberg,ToExplainTheWorld(AllenLane,2015).FrankWilczek,TheLightnessofBeing(AllenLane,2009).

Index

Aabsolutespaceandtime39,44–45absolutezerooftemperature21,63,78–79,84–85actionatadistance40–41aether43–44alchemy10–11,129allotrope30–31alphaparticle47,52–54,69,122–125amorphoussolid33–34antimatter8–9,47,72,98–100,111–113Aristotle10–11,117atomicmassunit2–3atomicnumber3,54atomicweight2–3,54

Bbandgap82–83,90–92baryogenesis118–119baryon108–110beta-decay103–105,119,127,130betaparticle(electron)52–53BigBang117–118,120–121,131–132,139–142blackhole35–36,47,50,129–130Bohr,Niels57–58,60,66–67Boltzmann,Ludwig20–22,52Bose–EinsteinCondensate(BEC)80,85–87,90boson70–74,78–80,85,88,104–105,107–108,112–115Boyle,Robert20brokensymmetry26–27,107,120Brownianmotion14–16Buckminsterfullerene30–31

CCERN105,107–108Chandrasekharmass126–127colddarkmatter(CDM)138collectivebehaviour37,81,85–88colourforce106,110–112,115–116complementaryvariables60–61,68–69conservationofenergy21–22,26–27,68–69,103,118conservationofmass13,46–47conservationofparity,failure119Cooperpair88–89cosmicmicrowavebackground(CMB)radiation85,120–121cosmicrays36,99,102–103,120cosmologicalconstant139,141–142Coulombelectricalforce27–28,101crystal24–26,29,32–33Curietemperature26,107curvatureofspacetime49–50

DDalton,John13–14,16,54darkage,ofuniverse122deBrogliewavelength59–61,79–80degeneratematter83–84degreeoffreedom22–23Democritus10,12,14doubleslitexperiment58–59,65–66,68,70

EEddington,Arthur50Einstein,Albert5–6,15–16,39,44–51electromagneticmass101–102electromagneticwave41–42,50–51,81,111–112,115–116,120,129–131electrondegeneracypressure126–127electronvolt54–56electroweakinteraction104,107,112energy,kineticandpotential20–22,26,32–33,45–46,63–64,126–127,139–140entropy22,26extension,apropertyofmatter11

FFaraday,Michael40–42fermienergy,orfermilevel78–79Fermilab110

fermion70–75,77–79,83,85,110–115Feynman,Richard17,65–66,102,108,142field8–9,21–22,26,39–43,49–51,60–61,74,101,105–108,111–112,114–115Franklin,Benjamin16fundamentalparticles1,28,97fusionenergy,controlled35–37fusionenergy,instars47,69,122–126

Ggalaxy7–8,118–119,127–130,132–134,138GalileoGalilei48gammaradiation42–43,52–53,61–63,99–100,120,129–130gas3,19–21glass33globalpositioningsystem50gluon73,106,110–113,115,118,141–142graphene30–31gravitationallensing134–137gravitationalwave47,114–115,129–131graviton106,114–115gravity,Newton’slawof11–12,40–41,47–48,132–134groundstate57–58,63–64,70–71,74–75,79,98

Hhadron105–111Halleffect90heat21–22Heisenberg’suncertaintyprinciple60,62,68–69,76–77,83–84,101–103,126Higgsmechanism105–108,115Higgs,Peter107Hoyle,Fred124–125Hubble,Edwin117–118hybridization,ofquantumlevels82hydrogenbond30

Iidenticalparticles69–70,78inertialframe44inertialmass11,47–48insulator81,92,94interference,ofwaves58–59ion28–29,35,37–38ionicbond29

J

Josephsoneffect89

KKapitsa,Pyotr85–87kilogram94kilonova130–131

LLambshift102LargeHadronCollider(LHC)107–108,114–115laser70–71,87latentheat24–25Lavoisier,Antoine13LeGrandK94–95Lepton108–109,112–113,141–142lightning30,36linesofforce40–41liquid19,23,25–26liquidcrystal33localgroup,ofgalaxies8long-rangeorder24–25Lorentz,Hendrik44Lucretius14

Mmass,gravitational47–48mass,inertial11,47–48mass,originof105–107,115massandweight12–13mass–energyequivalence46MassiveCompactHaloObjects(MACHOs)137Maxwell,JamesClerk20,41–45Meissnereffect87–88,105–107Mendeleev,Dmitri2–3,13–14meson102–103,108–111,120metallicbonding81MichelsonandMorleyexperiment43–44minimumenergyconfiguration32–33multimessengerastronomy131

Nneutrino73,103–104,108–109,112–113,137–138,141–142neutron1,54–56,73,85,97,99–100,102–105,108–109,115–116,127,130neutrondegeneracypressure126–127

neutronstar126Newton,Isaac8–9,11–13,40–41,43,45,47–48,60–61,94–95,132–133noblegas3,28–29,76,84Nöther’stheorem26–27nucleon54,97,109–112,118,122–124

OOnnes,Kamerlingh85,87–88orbital,atomic3,29,74–76orientationalorder34–35

PPauliexclusionprinciple30,75,82–84periodictableoftheelements3–4,29,54,74–76,125–126,130photoelectriceffect56–57,59–63photon56–57,59–60,65,70–71,73,87,101,104–107,111–113,118–119,121,134Plancklength118Planck’sconstant56,60plasma24,35,81,118,121polymer30–31potentialenergy21–22Priestley,Joseph13principleofconfinement,ofquarks111principleofequivalence,ingeneralrelativity48–49proton1,3,46–47,54–56,63–64,73,108–109pulsar129

Qquantum,ofenergy56–58quantumchromodynamics(QCD)110,112quantumelectrodynamics(QED)101–102,111–112QuantumHallEffect90–91,96quantumnumber57–58,74–75quantumphasetransition79quarks1,106,108–115,118–119,121quarksoup118quintessence117

RRayleigh,Lord16reionization,epochof122relativity,generaltheoryof5–6,47relativity,specialtheoryof44–47renormalization102

rest-massenergy45–46Rubin,Vera133

SSagan,Carl117Sakharov,Andrei118–119ScanningTunnellingMicroscopy16–17Schrödinger,Erwin64,67Schrödinger’scat67SLAC109–110solid19solidity,ofmatter70–71spacetime44–45specificheat22–23spin,quantum72spontaneoussymmetrybreaking26,107StarshipEnterprise47staticelectricity28steamengine21stellarnucleosynthesis122–123strongnuclearforce54–56,97,102–103,106,108–111superclustersofgalaxies5–6,8,121SuperconductingQuantumInterferenceDevice,SQUID89–90superconductor84,105–107superfluid84supernova,corecollapse127supernova,type1a140superpositionprinciple58–59,64–67supersymmetry(SUSY)114–115symmetry25

Ttemperature20–21thermodynamics,lawsof21Thompson,J.J.28timedilation44–45topologicalstateofmatter92–94tunnelling,quantum68–69

Uuncertaintyprinciple60,62

Vvacuumenergy74,101–102,107,140–141

valence29,76VanderWaalsforce28–29vectorboson74,105Velcro102–103,122–123VeryLargeTelescope130virtualparticles74,101–103,105–107,140–141viscosity85–87visibleuniverse5–6,8

Wwavefunction64WeaklyInteractingMassiveParticles(WIMPs)137weaknuclearforce,orinteraction97,103–105whitedwarfstar126

XX-ray3,52–53,129–130

YYoung,Thomas58–59

Zzero-pointenergy63Zwicky,Fritz133–134

CHAOSAVeryShortIntroduction

LeonardSmith

OurgrowingunderstandingofChaosTheoryishavingfascinatingapplicationsintherealworld-fromtechnologytoglobalwarming,politics,humanbehaviour,andevengamblingonthestockmarket.LeonardSmithshowsthatweallhaveanintuitiveunderstandingofchaoticsystems.Heusesaccessiblemathsandphysics(replacingcomplexequationswithsimpleexampleslikependulums,railwaylines,andtossingcoins)toexplainthetheory,andpointstonumerousexamplesinphilosophyandliterature(EdgarAllenPoe,Chang-Tzu,ArthurConanDoyle)thatilluminatetheproblems.Thebeautyoffractalpatternsandtheirrelationtochaos,aswellasthehistoryofchaos,anditsusesintherealworldandimplicationsforthephilosophyofsciencearealldiscussedinthisVeryShortIntroduction.

‘…Chaos…willgiveyoutheclearest(butnottoopainfulidea)ofthemathsinvolved…There’salotpackedintothislittlebook,andforsuchatechnicalexplorationit’ssurprisinglyreadableandenjoyable-Ireallywantedtokeepturningthepages.Smithalsohassomeexcellentwordsofwisdomaboutcommonmisunderstandingsofchaostheory…’

popularscience.co.uk

www.oup.com/vsi

EPIDEMIOLOGYAVeryShortIntroduction

RodolfoSaracci

Epidemiologyhashadanimpactonmanyareasofmedicine;andlungcancer,totheoriginandspreadofnewepidemics.andlungcancer,totheoriginandspreadofnewepidemics.However,itisoftenpoorlyunderstood,largelyduetomisrepresentationsinthemedia.InthisVeryShortIntroductionRodolfoSaraccidispelssomeofthemythssurroundingthestudyofepidemiology.Heprovidesageneralexplanationoftheprinciplesbehindclinicaltrials,andexplainsthenatureofbasicstatisticsconcerningdisease.Healsolooksattheethicalandpoliticalissuesrelatedtoobtainingandusinginformationconcerningpatients,andtrialsinvolvingplacebos.

www.oup.com/vsi

FORENSICPSYCHOLOGYAVeryShortIntroduction

DavidCanter

Liedetection,offenderprofiling,juryselection,insanityinthelaw,predictingtheriskofre-offending,themindsofserialkillersandmanyothertopicsthatfillnewsandfictionareallaspectsoftherapidlydevelopingareaofscientificpsychologybroadlyknownasForensicPsychology.ForensicPsychology:AVeryShortIntroductiondiscussesalltheaspectsofpsychologythatarerelevanttothelegalandcriminalprocessasawhole.Itincludesexplanationsofcriminalbehaviourandcriminality,includingtheroleofmentaldisorderincrime,anddiscusseshowforensicpsychologycontributestohelpinginvestigatethecrimeandcatchingtheperpetrators.

www.oup.com/vsi

GALAXIESAVeryShortIntroduction

JohnGribbin

GalaxiesarethebuildingblocksoftheUniverse:standinglikeislandsinspace,eachismadeupofmanyhundredsofmillionsofstarsinwhichthechemicalelementsaremade,aroundwhichplanetsform,andwhereonatleastoneofthoseplanetsintelligentlifehasemerged.InthisVeryShortIntroduction,renownedsciencewriterJohnGribbindescribestheextraordinarythingsthatastronomersarelearningaboutgalaxies,andexplainshowthiscanshedlightontheoriginsandstructureoftheUniverse.

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GLOBALWARMINGAVeryShortIntroduction

MarkMaslin

Globalwarmingisarguablythemostcriticalandcontroversialissuefacingtheworldinthetwenty-firstcentury.ThisVeryShortIntroductionprovidesaconciseandaccessibleexplanationofthekeytopicsinthedebate:lookingatthepredictedimpactofclimatechange,exploringthepoliticalcontroversiesofrecentyears,andexplainingtheproposedsolutions.Fullyupdatedfor2008,MarkMaslin’scompellingaccountbringsthereaderrightuptodate,describingrecentdevelopmentsfromUSpolicytotheUKClimateChangeBill,andwherewenowstandwiththeKyotoProtocol.Healsoincludesachapteronlocalsolutions,reflectingthenowwidelyheldviewthat,tomitigateanyimpendingdisaster,governmentsaswellasindividualsmusttoacttogether.

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NOTHINGAVeryShortIntroduction

FrankClose

Whatis‘nothing’?Whatremainswhenyoutakeallthematteraway?Canemptyspace-avoid-exist?ThisVeryShortIntroductionexploresthescienceandhistoryoftheelusivevoid:fromAristotle’stheoriestoblackholesandquantumparticles,andwhythelatestdiscoveriesaboutthevacuumtellusextraordinarythingsaboutthecosmos.FrankClosetellsthestoryofhowscientistshaveexploredtheelusivevoid,andtherichdiscoveriesthattheyhavemadethere.Hetakesthereaderonalivelyandaccessiblehistorythroughancientideasandculturalsuperstitionstothefrontiersofcurrentresearch.

‘Anaccessibleandentertainingreadforlaypersonandscientistalike.’PhysicsWorld

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PLANETSAVeryShortIntroduction

DavidA.Rothery

ThisVeryShortIntroductionlooksdeepintospaceanddescribestheworldsthatmakeupourSolarSystem:terrestrialplanets,giantplanets,dwarfplanetsandvariousotherobjectssuchassatellites(moons),asteroidsandTrans-Neptunianobjects.Itconsidershowourknowledgehasadvancedoverthecenturies,andhowithasexpandedatagrowingrateinrecentyears.DavidA.Rotherygivesanoverviewoftheorigin,nature,andevolutionofourSolarSystem,includingthecontroversialissuesofwhatqualifiesasaplanet,andwhatconditionsarerequiredforaplanetarybodytobehabitablebylife.HelooksatrockyplanetsandtheMoon,giantplanetsandtheirsatellites,andhowthesurfaceshavebeensculptedbygeology,weather,andimpacts.

“Thewritingstyleisexceptionallyclearandpricise”AstronomyNow

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