2NMRandenergylevels
Thepicturethatweusetounderstandmostkindsofspectroscopyisthat moleculeshaveasetof energylevels andthatthelinesweseeinspectra aredueto transitions betweentheseenergylevels.Suchatransitioncanbe causedbyaphotonoflightwhosefrequency, ν ,isrelatedtotheenergygap, E ,betweenthetwolevelsaccordingto:
= h ν
where h isauniversalconstantknownasPlanck’sconstant.Forthecase showninFig.2.1, E = E 2 E 1 .
InNMRspectroscopywetendnottousethisapproachofthinkingabout energylevelsandthetransitionsbetweenthem.Rather,weusedifferentrules forworkingouttheappearanceofmultipletsandsoon.However,itisuseful,especiallyforunderstandingmorecomplexexperiments,tothinkabout howthefamiliarNMRspectraweseearerelatedtoenergylevels.Tostart withwewilllookattheenergylevelsofjustonespinandthemmoveon quicklytolookattwoandthreecoupledspins.Insuchspinsystems,asthey areknown,wewillseethatinprinciplethereareothertransitions,called multiplequantumtransitions,whichcantakeplace.Suchtransitionsarenot observedinsimpleNMRspectra,butwecandetectthemindirectlyusingtwodimensionalexperiments;thereare,asweshallsee,importantapplicationsof suchmultiplequantumtransitions.
Finally,wewilllookat stronglycoupledspectra.Thesearespectrain whichthesimplerulesusedtoconstructmultipletsnolongerapplybecause theshiftdifferencesbetweenthespinshavebecomesmallcomparedtothe couplings.Themostfamiliareffectofstrongcouplingisthe“roofing”or “tilting”ofmultiplets.Wewillseehowsuchspectracanbeanalysedinsome simplecases.
2.1Frequencyandenergy:sortingouttheunits
NMRspectroscopiststendtousesomeratherunusualunits,andsoweneed toknowabouttheseandhowtoconvertfromonetoanotherifwearenotto getintoamuddle.
Chemicalshifts
ItisfoundtoaverygoodapproximationthatthefrequenciesatwhichNMR absorptions(lines)occurscalelinearlywiththemagneticfieldstrength.So, ifthelinefromTMScomesoutononespectrometerat400MHz,doubling themagneticfieldwillresultinitcomingoutat800MHz.Ifwewanted toquotetheNMRfrequencyitwouldbeinconvenienttohavetospecifythe exactmagneticfieldstrengthaswell.Inaddition,thenumberswewouldhave Chapter2“NMRandenergylevels” c JamesKeeler,2002
ν = E2–E1
energy levels spectrum
Fig.2.1 Alineinthespectrum isassociatedwithatransition betweentwoenergylevels.
NMRandenergylevels
toquotewouldnotbeverymemorable.Forexample,wouldyouliketoquote theshiftoftheprotonsinbenzeneas400.001234MHz?
Weneatlyside-stepbothoftheseproblemsbyquotingthe chemicalshift relativetoanagreed referencecompound.Forexample,inthecaseofproton NMRthereferencecompoundisTMS.Ifthefrequencyofthelineweare interestedinis ν (inHz)andthefrequencyofthelinefromTMSis νTMS (alsoinHz),thechemicalshiftofthelineiscomputedas:
Fig.2.2 AnNMRspectrumcan beplottedasafunctionof frequency,butitismore convenienttousethechemical shiftscaleinwhichfrequencies areexpressedrelativetothatof anagreedreferencecompound, suchasTMSinthecaseof protonspectra.
Asallthefrequenciesscalewiththemagneticfield,thisratioisindependent ofthemagneticfieldstrength.Typically,thechemicalshiftisrathersmall soitiscommontomultiplythevaluefor δ by106 andthenquoteitsvalue in partspermillion,orppm.Withthisdefinitionthechemicalshiftofthe referencecompoundis0ppm.
ppm = 106 × ν
Sometimeswewanttoconvertfromshiftsinppmtofrequencies.Suppose thattherearetwopeaksinthespectrumatshifts δ1 and δ2 inppm.Whatis thefrequencyseparationbetweenthetwopeaks?Itiseasyenoughtowork outwhatitisinppm,itisjust (δ2 δ2 ).Writingthisdifferenceoutinterms ofthedefinitionofchemicalshiftgiveninEq.2.2wehave:
Multiplyingbothsidesby νTMS nowgivesuswhatwewant: (ν2 ν1 ) = 10 6 × νTMS × (δ2 δ1 ).
Itisoftensufficientlyaccuratetoreplace νTMS withthespectrometer reference frequency,aboutwhichwewillexplainlater.
Ifwewanttochangetheppmscaleofaspectrumintoafrequencyscalewe needtodecidewherezeroisgoingtobe.Onechoiceforthezerofrequency pointisthelinefromthereferencecompound.However,thereareplentyof otherpossibilitiessoitisaswelltoregardthezeropointonthefrequency scaleasarbitrary.
Angularfrequency
FrequenciesaremostcommonlyquotedinHz,whichisthesameas“per second”ors 1 .Thinkaboutapointontheedgeofadiscwhichisrotating aboutitscentre.Ifthediscismovingataconstantspeed,thepointreturns tothesamepositionatregularintervalseachtimeithascompeted360 ◦ of rotation.Thetimetakenforthepointtoreturntoitsoriginalpositioniscalled the period, τ .
2.2Nuclearspinandspinstates2–
Thefrequency, ν ,issimplytheinverseoftheperiod:
Forexample,iftheperiodis0.001s,thefrequencyis1/0.001=1000Hz. Thereisanotherwayofexpressingthefrequency,whichisin angular units.Recallthat360◦ is2π radians.So,ifthepointcompletesarotationin τ seconds,wecansaythatithasrotatedthough2π radiansin τ seconds.The angularfrequency, ω ,isgivenby
Theunitsofthisfrequencyare“radianspersecond”orrads 1 ν and ω are relatedvia
= 2πν.
Wewillfindthatangularfrequenciesareoftenthemostnaturalunitstousein NMRcalculations.Angularfrequencieswillbedenotedbythesymbols ω or whereasfrequenciesinHzwillbedenoted ν
Energies
Aphotonoffrequency ν hasenergy E givenby
where h isPlanck’sconstant.InSIunitsthefrequencyisinHzand h isin Js 1 .Ifwewanttoexpressthefrequencyinangularunitsthentherelationshipwiththeenergyis
Fig.2.3 Apointattheedgeofa circlewhichismovingata constantspeedreturnstoits originalpositionafteratime calledthe period.Duringeach periodthepointmovesthrough 2π radiansor360◦
where h (pronounced“hbar”or“hcross”)isPlanck’sconstantdividedby 2π
Thepointtonoticehereisthatfrequency,ineitherHzorrads 1 ,isdirectlyproportionaltoenergy.So,thereisreallynothingwrongwithquoting energiesinfrequencyunits.Allwehavetorememberisthatthereisafactor of h or h neededtoconverttoJoulesifweneedto.Itturnsouttobemuch moreconvenienttoworkinfrequencyunitsthroughout,andsothisiswhatwe willdo.So,donotbeconcernedtoseeanenergyexpressedinHzorrads 1
2.2Nuclearspinandspinstates
NMRspectroscopyarisesfromthefactthatnucleihaveapropertyknownas spin;wewillnotconcernourselveswithwherethiscomesfrom,butjusttake itasafact.Quantummechanicstellsusthatthisnuclearspinischaracterised byanuclearspinquantumnumber, I .Forallthenucleithatwearegoing
Strictly, α isthelowenergystate fornucleiwithapositive gyromagneticratio,moreof whichbelow.
NMRandenergylevels
tobeconcernedwith, I = 1 2 ,althoughothervaluesarepossible.Aspin-half nucleushasaninteractionwithamagneticfieldwhichgivesrisetotwoenergy levels;thesearecharacterisedbyanotherquantumnumber m whichquantum mechanicstellsusisrestrictedtothevalues I to I inintegersteps.So,in thecaseofaspin-half,thereareonlytwovaluesof m , 1 2 and + 1 2 .
BytraditioninNMRtheenergylevel(or state,asitissometimescalled) with m = 1 2 isdenoted α andissometimesdescribedas“spinup”.Thestate with m =− 1 2 isdenoted β andissometimesdescribedas“spindown”.For thenucleiweareinterestedin,the α stateistheonewiththelowestenergy. Ifwehavetwospinsinourmolecule,theneachspincanbeinthe α or β state,andsotherearefourpossibilities: α1 α2 , α1 β2 ,β1 α2 and β1 β2 .These fourpossibilitiescorrespondtofourenergylevels.Notethatwehaveaddeda subscript1or2todifferentiatethetwonuclei,althoughoftenwewilldispense withtheseandsimplytakeitthatthefirstspinstateisforspin1andthesecond forspin2.So αβ implies α1 β2 etc.
Wecancontinuethesameprocessforthreeormorespins,andaseachspin isaddedthenumberofpossiblecombinationsincreases.So,forexample,for threespinsthereare8combinationsleadingto8energylevels.Itshouldbe notedherethatthereisonlyaone-to-onecorrespondencebetweenthesespin statecombinationsandtheenergylevelsinthecaseofweakcoupling,which wewillassumefromnowon.Furtherdetailsaretobefoundinsection2.6.
2.3Onespin
Therearejusttwoenergylevelsforasinglespin,andthesecanbelabelled witheitherthe m valueofthelabels α and β .Fromquantummechanicswe canshowthattheenergiesofthesetwolevels, E α and E β ,are:
where ν0,1 isthe Larmorfrequency ofspin1(wewillneedthe1lateron,but itisabitsuperfluoushereasweonlyhaveonespin).Infactitiseasytosee thattheenergiesjustdependon m :
Youwillnoteherethat,asexplainedinsection2.1wehavewrittenthe energiesinfrequencyunits.TheLarmorfrequencydependsonaquantity knownasthe gyromagneticratio, γ ,thechemicalshift δ ,andthestrengthof theappliedmagneticfield, B0 : ThisnegativeLarmorfrequency fornucleiwithapositive γ sometimesseemsabit unnaturalandawkward,butto beconsistentweneedtostick withthisconvention.Wewill seeinalaterchapterthatall thisnegativefrequencyreally meansisthatthespin precessesinaparticularsense.
(2.3) whereagainwehaveusedthesubscript1todistinguishtheofnucleus.The magneticfieldisnormallygiveninunitsofTesla(symbolT).Thegyromagneticratioischaracteristicofaparticularkindofnucleus,suchasprotonor carbon-13;itsunitsarenormallyrads 1 T 1 .Infact, γ canbepositiveor negative;forthecommonestnuclei(protonsandcarbon-13)itispositive.For suchnucleiwethereforeseethattheLarmorfrequencyisnegative.
Totakeaspecificexample,forprotons γ =+2.67 × 108 rads 1 T 1 , soinamagneticfieldof4.7TtheLarmorfrequencyofaspinwithchemical shiftzerois
=−
1 +
Inotherwords,theLarmorfrequencyis 200MHz.
WecanalsocalculatetheLarmorfrequencyinangularunits, ω0 ,inwhich casethefactorof1/2π isnotneeded: ω0 =−γ(1 + δ) B0
whichgivesavalueof1.255 × 109 rads 1
Spectrum
Asyoumayknowfromotherkindsofspectroscopyyouhavemet,onlycertain transitionsare allowed i.e.onlycertainonesactuallytakeplace.Thereare usuallyrules–called selectionrules –aboutwhichtransitionscantakeplace; theserulesnormallyrelatetothequantumnumberswhicharecharacteristic ofeachstateorenergylevel.
InthecaseofNMR,theselectionrulereferstothequantumnumber m : onlytransitionsinwhich m changesbyone(upordown)areallowed.Thisis sometimesexpressedas
m = m (initialstate) m (finalstate) =±1
Anotherwayassayingthisisthatonespincanflipbetween“up”and“down” orviceversa.
Inthecaseofasinglespin-half,thechangein m betweenthetwostatesis (+ 1 2 ( 1 2 )) = 1sothetransitionisallowed.Wecannowsimplyworkout thefrequencyoftheallowedtransition:
Fig.2.4 Thetransitionbetween thetwoenergylevelsofa spin-halfisallowed,andresults inasinglelineattheLarmor frequencyofthespin.
Notethatwehavetakentheenergyoftheupperstateminusthatofthelower state.Inwords,therefore,weseeonetransitionattheminustheLarmor frequency, ν0,1 .
Youwouldbeforgivenforthinkingthatthisisallanenormousamountof efforttocomeupwithsomethingverysimple!However,thetechniquesand ideasdevelopedinthissectionwillenableustomakefasterprogresswiththe caseoftwoandthreecoupledspins,whichweconsidernext.
2.4Twospins
J12 J12
Fig.2.5 Schematicspectrumof twocoupledspinsshowingtwo doubletswithequalsplittings. Asindicatedbythedashed lines,theseparationofthe Larmorfrequenciesismuch largerthanthecoupling betweenthespins.
Weknowthatthespectrumoftwocoupledspinsconsistsoftwodoublets, eachsplitbythesameamount,onecentredatthechemicalshiftofthefirst spinandoneattheshiftofthesecond.Thesplittingofthedoubletsisthe scalarcoupling, J12 ,quotedinHz;thesubscriptsindicatewhichspinsare involved.Wewillwritetheshiftsofthetwospinsas δ1 and δ2 ,andgivethe correspondingLarmorfrequencies, ν0
1 and
2 as:
Ifthetwonucleiareofthesametype,suchaproton,thenthetwogyromagneticratiosareequal;suchatwospinsystemwouldbedescribedas homonuclear.Theoppositecaseiswherethetwonucleiareofdifferenttypes,such asprotonandcarbon-13;suchaspinsystemisdescribedas heteronuclear
Energylevels
Aswasalreadydescribedinsection2.2,therearefourpossiblecombinations ofthespinstatesoftwospinsandthesecombinationscorrespondtofour energylevels.Theirenergiesaregiveninthefollowingtable:
numberspinstatesenergy
Thesecondcolumngivesthespinstatesofspins1and2,inthatorder.Itis easytoseethattheseenergieshavethegeneralform:
where m 1 and m 2 arethe m valuesforspins1and2,respectively.
Forahomonuclearsystem ν0,1 ≈ ν0,2 ;alsobothLarmorfrequenciesare muchgreaterinmagnitudethanthecoupling(theLarmorfrequenciesareof theorderofhundredsofMHz,whilecouplingsareatmostafewtensofHz). Therefore,underthesecircumstances,theenergiesofthe αβ and βα statesare rathersimilar,butverydifferentfromtheothertwostates.Foraheteronuclear system,inwhichtheLarmorfrequenciesdiffersignificantly,thefourlevels areallatmarkedlydifferentenergies.ThesepointsareillustratedinFig.2.6.
Spectrum
Theselectionruleisthesameasbefore,butthistimeitappliestothequantum number M whichisfoundbyaddingupthe m valuesforeachofthespins.In thiscase:
M = m 1 + m 2 .
Theresulting M valuesforthefourlevelsare:
Fig.2.6 Energylevels,drawnapproximatelytoscale,fortwospinsystems.Ontheleftisshowna homonuclearsystem(twoprotons);onthisscalethe αβ and βα stateshavethesameenergy.Onthe rightisthecaseforacarbon-13 – protonpair.TheLarmorfrequencyofprotonisaboutfourtimesthat ofcarbon-13,andthisisclearreflectedinthediagram.The αβ and βα statesnowhavesubstantially differentenergies.
numberspinstates M
Theselectionruleisthat M =±1,i.e.thevalueof M canchangeupor downbyoneunit.Thismeansthattheallowedtransitionsarebetweenlevels 1&2,3&4,1&3and2&4.Theresultingfrequenciesareeasilyworked out;forexample,the1–2transition:
Thecompletesetoftransitionsare:
Theenergylevelsandcorrespondingschematicspectrumareshownin Fig.2.7.Thereisalotwecansayaboutthisspectrum.Firstly,eachallowed transitioncorrespondstooneofthespinsflippingfromonespinstatetothe other,whilethespinstateoftheotherspinremainsfixed.Forexample,transition1–2involvesaspin2goingfrom α to β whilstspin1remainsinthe α state.Inthistransitionwesaythatspin2is active andspin1is passive.As
Throughoutwewillusethe conventionthatwhencomputing thetransitionfrequencywewill taketheenergyoftheupper stateminustheenergyofthe lower: E = E
Fig.2.7 Ontheleft,theenergylevelsofatwo-spinsystem;thearrowsshowtheallowedtransitions: solidlinesfortransitionsinwhichspin1 flipsanddottedforthoseinwhichspin2 flips.Ontheright,the correspondingspectrum;itisassumedthattheLarmorfrequencyofspin2isgreaterinmagnitudethan thatofspin1andthatthecoupling J12 ispositive.
spin2flipsinthistransition,itisnotsurprisingthatthetransitionformsone partofthedoubletforspin2.
Transition3–4issimilarto1–2exceptthatthepassivespin(spin1)is inthe β state;thistransitionformsthesecondlineofthedoubletforspin2. Thisdiscussionillustratesaveryimportantpoint,whichisthatthelinesofa multipletcanbeassociatedwithdifferentspinstatesofthecoupled(passive) spins.Wewillusethiskindofinterpretationveryoften,especiallywhen consideringtwo-dimensionalspectra.
Thetwotransitionsinwhichspin1flipsare1–3and2–4,andtheseare associatedwithspin2beinginthe α and β spinstates,respectively.Which spinflipsandthespinsstatesofthepassivespinsareshowninFig.2.7.
Whathappensisthecouplingisnegative?Ifyouworkthroughthetable youwillseethattherearestillfourlinesatthesamefrequenciesasbefore.All thatchangesisthelabelsofthelines.So,forexample,transition1–2isnow therightlineofthedoublet,ratherthantheleftline.Fromthepointofviewof thespectrum,whatswapsoveristhespinstateofthepassivespinassociated witheachlineofthemultiplet.Theoverallappearanceofthespectrumis thereforeindependentofthesignofthecouplingconstant.
Multiplequantumtransitions
Therearetwomoretransitionsinourtwo-spinsystemwhicharenotallowed bytheusualselectionrule.Thefirstisbetweenstates1and4(αα → ββ )in whichbothspinsflip.The M valueis2,sothisiscalleda double-quantum transition.Usingthesameterminology,alloftheallowedtransitionsdescribedabove,whichhave M = 1,are single-quantum transitions.Fromthe tableofenergylevelsitiseasytoworkoutthatitsfrequencyis ( ν0,1 ν0,2 ) i.e.thesumoftheLarmorfrequencies.Notethatthecouplinghasnoeffect onthefrequencyofthisline.
Thesecondtransitionisbetweenstates2and3(αβ → βα );again,both
spinsflip.The M valueis0,sothisiscalleda zero-quantum transition, anditsfrequencyis ( ν0,1 + ν0,2 ) i.e.thedifferenceoftheLarmorfrequencies.Aswiththedouble-quantumtransition,thecouplinghasnoeffectonthe frequencyofthisline.
Inatwospinsystemthedouble-andzero-quantumspectraarenotespeciallyinteresting,butwewillseeinathree-spinsystemthatthesituationis ratherdifferent.Wewillalsoseelateronthatintwo-dimensionalspectrawe canexploittoouradvantagethespecialpropertiesofthesemultiple-quantum spectra.
2.5Threespins
Ifwehavethreespins,eachofwhichiscoupledtotheothertwo,thenthe spectrumconsistsofthreedoubletsofdoublets,onecentredattheshiftof eachofthethreespins;the spintopology isshowninFig.2.9.Theappearance ofthesemultipletswilldependontherelativesizesofthecouplings.For example,if J12 = J13 thedoubletofdoubletsfromspin1willcollapsetoa 1:2:1triplet.Ontheotherhand,if J12 = 0,onlydoubletswillbeseenforspin 1andspin2,butspin3willstillshowadoubletofdoublets.
Energylevels
Eachofthethreespinscanbeinthe α or β spinstate,sothereareatotalof 8possiblecombinationscorrespondingto8energylevels.Theenergiesare givenby:
where m i isthevalueofthequantumnumber m forthe i thspin.Theenergies andcorresponding M values(= m 1 + m 2
)areshowninthetable: numberspinstates
Fig.2.8 Inatwo-spinsystem thereisonedoublequantum transition(1–4)andone zero-quantumtransition(2–3); thefrequencyofneitherof thesetransitionsareaffectedby thesizeofthecouplingbetween thetwospins.
Wehavegroupedtheenergylevelsintotwogroupsoffour;thefirstgroup allhavespin3inthe α stateandthesecondhavespin3inthe β state.Theenergylevels(forahomonuclearsystem)areshownschematicallyinFig.2.10.
Spectrum
Theselectionruleisasbefore,thatis M canonlychangeby1.However,in thecaseofmorethantwospins,thereistheadditionalconstraintthat only
Fig.2.9 The topology – thatis thenumberofspinsandthe couplingsbetweenthem – fora three-spinsysteminwhicheach spiniscoupledtobothofthe others.Theresultingspectrum consistsofthreedoubletsof doublets,atypicalexampleof whichisshownforspin1with theassumptionthat J12 is greaterthan J13
Fig.2.10 Energylevelsforahomonuclearthree-spinsystem.Thelevelscanbegroupedintotwosetsof four:thosewithspin3inthe α state(shownontheleftwithsolidlines)andthosewithspin3inthe β state, shownontheright(dashedlines).
one spincanflip.Applyingtheserulesweseethattherearefourallowed transitionsinwhichspin1slips:1–3,2–4,5–7and6–8.Thefrequenciesof theselinescaneasilybeworkedoutfromthetableofenergylevelsonpage 2–9.Theresultsareshowninthetable,alongwiththespinstatesofthepassive spins(2and3inthiscase).
transitionstateofspin2stateofspin3frequency
Fig.2.11 Energylevelsforathree-spinsystemshowingbythearrowsthefourallowedtransitionswhich resultinthedoubletofdoubletsattheshiftofspin1.Theschematicmultipletisshownontheright,where ithasbeenassumingthat ν0,1 =−100Hz, J12 = 10Hzand J13 = 2Hz.Themultipletislabelledwiththe spinstatesofthepassivespins.
Thesefourtransitionsformthefourlinesofthemultiplet(adoubletof doublets)attheshiftofspin1.Theschematicspectrumisillustratedin Fig.2.11.Asinthecaseofatwo-spinsystem,wecanlabeleachlineofthe
multipletwiththespinstatesofthepassivespins–inthecaseofthemultiplet fromspin1,thismeansthespinstatesofspins2and3.Inthesameway,we canidentifythefourtransitionswhichcontributetothemultipletfromspin 2(1–2,3–4,5–6and7–8)andthefourwhichcontributetothatfromspin3 (1–5,3–7,2–6and4–8).
Subspectra
Fig.2.12 Illustrationofthedivisionofthetwomultipletsfromspins1and2intosubspectraaccordingto thespinstateofspin3.Thetransitionsassociatedwithspin3inthe α state(indicatedbythefulllines ontheenergyleveldiagram)giverisetoapairofdoublets,butwiththeircentresshiftedfromtheLarmor frequenciesbyhalfthecouplingtospin3.Thesameistrueofthosetransitionsassociatedwithspin3 beinginthe β state(dashedlines),exceptthattheshiftisintheoppositedirection.
Onewasofthinkingaboutthespectrumfromthethree-spinsystemis todivideupthelinesinthemultipletsforspins1and2intotwogroupsor subspectra.Thefirstgroupconsistsofthelineswhichhavespin3inthe α stateandthesecondgroupconsistsofthelineswhichhavespin3inthe α state.ThisseparationisillustratedinFig.2.12.
Therearefourlineswhichhavespin-3inthe α state,andascanbeseen fromthespectrumtheseformtwodoubletswithacommonseparationof J12 However,thetwodoubletsarenotcentredat ν0,1 and ν0,2 ,butat ( ν0,1 1 2 J13 ) and ( ν0,2 1 2 J23 ).Wecandefinean effective Larmorfrequencyfor spin1withspin3inthe α spinstate, ν α3 0,1 ,as ν
andlikewiseforspin2:
Thetwodoubletsinthesub-spectrumcorrespondingtospin3beinginthe α statearethuscentredat ν α3 0,1 and ν α3 0,2 .Similarly,wecandefineeffective Larmorfrequenciesforspin3beinginthe β state:
NMRandenergylevels
Thetwodoubletsinthe β sub-spectrumarecentredat ν β3 0,1 and ν β3 0,2
Wecanthinkofthespectrumofspin1and2asbeingcomposedoftwo subspectra,eachfromatwospinsystembutinwhichtheLarmorfrequencies areeffectivelyshiftedonewayoftheotherbyhalfthecouplingtothethird spin.Thiskindofapproachisparticularlyusefulwhenitcomestodealing withstronglycoupledspinsystems,section2.6
Notethattheseparationofthespectraaccordingtothespinstateofspin 3isarbitrary.Wecouldjustaswellseparatethetwomultipletsfromspins1 and3accordingtothespinstateofspin2.
Multiplequantumtransitions
Therearesixtransitionsinwhich M changesby2.Theirfrequenciesare giveninthetable.
transitioninitialstatefinalstatefrequency
Thesetransitionscomeinthreepairs.Transitions1–4and5–8arecentred atthesumoftheLarmorfrequenciesofspins1and2;thisisnotsurprisingas wenotethatinthesetransitionsitisthespinstatesofbothspins1and2which flip.Thetwotransitionsareseparatedbythe sum ofthecouplingstospin3 ( J13 + J23 ),buttheyareunaffectedbythecoupling J12 whichisbetweenthe twospinswhichflip.
Fig.2.13 Therearetwodoublequantumtransitionsinwhichspins1and2both flip(1–4and5–8).The tworesultinglinesformadoubletwhichiscentredatthesumoftheLarmorfrequenciesofspins1and 2andwhichissplitbythesumofthecouplingstospin3.Aswiththesingle-quantumspectra,wecan associatethetwolinesofthedoubletwithdifferentspinstatesofthethirdspin.Ithasbeenassumedthat bothcouplingsarepositive.
Wecandescribethesetransitionsasakindofdoublequantumdoublet. Spins1and2arebothactiveinthesetransitions,andspin3ispassive.Justas
wedidbefore,wecanassociateonelinewithspin3beinginthe α state(1–4) andonewithitbeinginthe β state(5–8).Aschematicrepresentationofthe spectrumisshowninFig.2.13.
Therearealsosixzero-quantumtransitionsinwhich M doesnotchange. Likethedoublequantumtransitionsthesegroupinthreepairs,butthistime centredaroundthe difference intheLarmorfrequenciesoftwoofthespins. Thesezero-quantumdoubletsaresplitbythe difference ofthecouplingstothe spinwhichdoesnotflipinthetransitions.Therearethusmanysimilarities betweenthedouble-andzero-quantumspectra.
Inathreespinsystemthereisonetriple-quantumtransition,inwhich M changesby3,betweenlevels1(ααα )and8(βββ ).Inthistransitionallof thespinsflip,andfromthetableofenergieswecaneasilyworkoutthatits frequencyis ν0,1 ν0,2 ν0,3 ,i.e.thesumoftheLarmorfrequencies.
Weseethatthesingle-quantumspectrumconsistsofthreedoubletsofdoublets,thedouble-quantumspectrumofthreedoubletsandthetriple-quantum spectrumofasingleline.Thisillustratestheideathataswemovetohigher ordersofmultiplequantum,thecorrespondingspectrabecomesimpler.This featurehasbeenusedintheanalysisofsomecomplexspinsystems.
Combinationlines
Therearethreemoretransitionswhichwehavenotyetdescribed.Forthese, M changesby1butallthreespinsflip;theyarecalled combinationlines. Suchlinesarenotseeninnormalspectrabut,likemultiplequantumtransitions,theycanbedetectedindirectlyusingtwo-dimensionalspectra.Wewill alsoseeinsection2.6thattheselinesmaybeobservableinstronglycoupled spectra.Thetablegivesthefrequenciesofthesethreelines: transitioninitialstatefinalstatefrequency
Noticethatthefrequenciesoftheselinesarenotaffectedbyanyofthe couplings.
2.6Strongcoupling
Sofarallwehavesaidaboutenergylevelsandspectraappliestowhatare called weaklycoupled spinsystems.Thesearespinsystemsinwhichthe differencesbetweentheLarmorfrequencies(inHz)ofthespinsaremuch greaterinmagnitudethanthemagnitudeofthecouplingsbetweenthespins.
Underthesecircumstancestherulesfrompredictingspectraareverysimple–theyaretheonesyouarealreadyfamiliarwithwhichyouuseforconstructingmultiplets.Inaddition,intheweakcouplinglimititispossibleto workouttheenergiesofthelevelspresentsimplybymakingallpossiblecombinationsofspinstates,justaswehavedoneabove.Finally,inthislimitall ofthelinesinamultiplethavethesameintensity.
Weneedtobecarefulhereas Larmorfrequencies,the differencesbetweenLarmor frequenciesandthevaluesof couplingscanbepositiveor negative!Indecidingwhetheror notaspectrumwillbestrongly coupledweneedtocompare the magnitude ofthedifference intheLarmorfrequencieswith the magnitude ofthecoupling.
Youwilloften findthatpeople talkoftwospinsbeing strongly coupled whenwhattheyreally meanisthecouplingbetween thetwospinsis large.Thisis sloppyusage;wewillalways usethetermstrongcouplingin thesensedescribedinthis section.
NMRandenergylevels
IftheseparationoftheLarmorfrequenciesisnotsufficienttosatisfythe weakcouplingcriterion,thesystemissaidtobe stronglycoupled.Inthislimit none oftherulesoutlinedinthepreviousparagraphapply.Thismakespredictingoranalysingthespectramuchmoredifficultthaninthecaseofweak coupling,andreallytheonlypracticalapproachistousecomputersimulation. However,itisusefultolookatthespectrumfromtwostronglycoupledspins asthespectrumissimpleenoughtoanalysebyhandandwillrevealmostof theofthecrucialfeaturesofstrongcoupling.
Itisarelativelysimpleexerciseinquantummechanicstoworkoutthe energylevelsandhencefrequenciesandintensitiesofthelinesfromastrongly coupledtwo-spinsystem1 .Thesearegiveninthefollowingtable.
Inthistable isthesumoftheLarmorfrequencies:
ν0,1 + ν0,2 and D isthe positive quantitydefinedas
Theangle θ iscalledthe strongcouplingparameter andisdefinedvia sin2θ = J12 D
Thefirstthingtodoistoverifythattheseformulaegiveustheexpected resultwhenweimposetheconditionthattheseparationoftheLarmorfrequenciesislargecomparedtothecoupling.Inthislimititisclearthat
andso D = (ν0,1 ν0,2 ).Puttingthisvalueintothetableabovegivesus exactlythefrequencieswehadbeforeonpage 2–7.
When D isverymuchlargerthan J12 thefraction J12 / D becomessmall, andsosin2θ ≈ 0(sin φ goestozeroas φ goestozero).Underthesecircumstancesallofthelineshaveunitintensity.So,theweakcouplinglimitis regained.
Figure2.14showsaseriesofspectracomputedusingtheaboveformulae inwhichtheLarmorfrequencyofspin1isheldconstantwhiletheLarmor frequencyofspin2isprogressivelymovedtowardsthatofspin1.Thismakes thespectrummoreandmorestronglycoupled.Thespectrumatthebottom isalmostweaklycoupled;thepeaksarejustaboutallthesameintensityand whereweexpectthemtobe.
1 See,forexample,Chapter10of NMR:TheToolkit,byPJHore,JAJonesandSWimperis(OxfordUniversityPress,2000)
Fig.2.14 AseriesofspectraofatwospinsysteminwhichtheLarmorfrequencyofspin1ishelpconstant andthatofspin2ismovedinclosertospin1.Thespectrabecomemoreandmorestronglycoupled showingapronouncedroofeffectuntilinthelimitthatthetwoLarmorfrequenciesareequalonlyoneline isobserved.Notethatasthe “outer” linesgetweakerthe “inner” linesgetproportionatelystronger.The parametersusedforthesespectrawere ν0,1 =−10Hzand J12 = 5Hz;thepeakinthetopmostspectrum hasbeentruncated.
However,astheLarmorfrequenciesofthetwospinsgetcloserandcloser togetherwenoticetwothings:(1)the“outer”twolinesgetweakerandthe “inner”twolinesgetstronger;(2)thetwolineswhichoriginallyformedthe doubletarenolongersymmetricallyspacedabouttheLarmorfrequency;in factthestrongerofthetwolinesmovesprogressivelyclosertotheLarmor frequency.Thereisonemorethingtonoticewhichisnotsoclearfromthe spectrabutisclearifonelooksatthefrequenciesinthetable.Thisisthatthe twolinesthatoriginallyformedthespin1doubletarealwaysseparatedby J12 ;thesameistruefortheotherdoublet.
Thesespectraillustratetheso-called roof effectinwhichtheintensities ofthelinesinastronglycoupledmultiplettiltupwardstowardsthemultiplet fromthecoupledspin,makingakindofroof;Fig.2.15illustratestheidea. ThespectrainFig2.14alsoillustratethepointthatwhenthetwoLarmor frequenciesareidenticalthereisonlyonelineseeninthespectrumandthis isatthisLarmorfrequency.Inthislimitlines1–2and2–4bothappearatthe Larmorfrequencyandwithintensity2;lines1–3and3–4appearelsewhere buthaveintensityzero.
The“takehomemessage”isthatfromsuchstronglycoupledspectrawe caneasilymeasurethecoupling,buttheLarmorfrequencies(theshifts)are nolongermid-waybetweenthetwolinesofthedoublet.Infactitiseasy
Fig.2.15 Theintensity distributionsinmultipletsfrom strongly-coupledspectraare suchthatthemultiplets “tilt” towardsoneanother;thisis calledthe “roof ” effect.
NMRandenergylevels
enoughtoworkouttheLarmorfrequenciesusingthefollowingmethod;the ideaisillustratedinFig.2.16.
Ifwedenotethefrequencyoftransition1–2as ν12 andsoon,itisclear fromthetablethatthefrequencyseparationoftheleft-handlinesofthetwo multiplets(3–4and2–4)is D
Fig.2.16 Thequantities J12 and D arereadilymeasurablefrom thespectrumoftwostrongly coupledspins.
Theseparationoftheothertwolinesisalso D .Rememberwecaneasily measure J12 directlyfromthesplitting,andsoonceweknow D itiseasyto compute (ν0,1 ν0,2 ) fromitsdefinition,Eqn.2.4.
Nowwenoticefromthetableon 2–14thatthesumofthefrequenciesof thetwostrongerlines(1–2and2–4)orthetwoweakerlines(3–4and2–4) givesus :
Nowwehaveavaluesfor = (ν
,
+
,2 ) andavaluefor (ν0,1 ν0,2 ) we canfind ν0,1 and ν0,2 separately:
InthiswaywecanextracttheLarmorfrequenciesofthetwospins(theshifts) andthecouplingfromthestronglycoupledspectrum.
Notationforspinsystems
Thereisatraditionalnotationforspinsystemswhichitissometimesusefulto use.Eachspinisgivenaletter(ratherthananumber),withadifferentletter forspinswhichhavedifferentLarmorfrequencies(chemicalshifts).Ifthe Larmorfrequenciesoftwospinsarewidelyseparatedtheyaregivenletters whicharewidelyseparatedinthealphabet.So,twoweaklycoupledspinsare usuallydenotedasAandX;whereasthreeweaklycoupledspinswouldbe denotedAMX.
Ifspinsarestronglycoupled,thentheconventionistousedletterswhich arecloseinthealphabet.So,astronglycoupledtwo-spinsystemwouldbedenotedABandastronglycoupledthree-spinsystemABC.ThenotationABX impliesthattwoofthespins(AandB)arestronglycoupledbuttheLarmor frequencyofthethirdspiniswidelyseparatedfromthatofAandB.
TheABXspinsystem
Wenotedinsection2.5thatwecouldthinkaboutthespectrumofthreecoupledspinsintermsofsub-spectrainwhichtheLarmorfrequencieswerereplacedbyeffectiveLarmorfrequencies.Thiskindofapproachisveryuseful forunderstandingtheABpartoftheABXspectrum.
β sub-spectrum
α sub-spectrum
Fig.2.17 ABpartsofanABXspectrumillustratingthedecompositionintotwosub-spectrawithdifferent effectiveLarmorfrequencies(indicatedbythearrows).Theparametersusedinthesimulationwere ν0,A =−20Hz, ν0,B =−30Hz, JAB = 5Hz, JAX = 15Hzand JBX = 3Hz.
AsspinXisweaklycoupledtotheotherswecanthinkoftheABpartof thespectrumastwosuperimposedABsub-spectra;oneisassociatedwiththe Xspinbeinginthe α stateandtheotherwiththespinbeinginthe β state.If spinsAandBhaveLarmorfrequencies ν0, A and ν0, B ,respectively,thenone sub-spectrumhaseffectiveLarmorfrequencies ν0, A + 1 2 JAX and ν0, B + 1 2 JBX TheotherhaseffectiveLarmorfrequencies ν0, A 1 2 JAX and ν0, B 1 2 JBX
TheseparationbetweenthetwoeffectiveLarmorfrequenciesinthetwo subspectracaneasilybedifferent,andsothedegreeofstrongcoupling(and hencetheintensitypatterns)inthetwosubspectrawillbedifferent.Allwe canmeasureisthecompletespectrum(thesumofthetwosub-spectra)but onceweknowthatitisinfactthesumoftwoAB-typespectraitisusually possibletodisentanglethesetwocontributions.Oncewehavedonethis,the twosub-spectracanbeanalysedinexactlythewaydescribedaboveforan ABsystem.Figure2.17illustratesthisdecomposition.
TheformoftheXpartoftheABXspectrumcannotbededucedfrom thissimpleanalysis.Ingeneralitcontains6lines,ratherthanthefourwhich wouldbeexpectedintheweakcouplinglimit.Thetwoextralinesarecombinationlineswhichbecomeobservablewhenstrongcouplingispresent.
OfcourseinrealitytheLarmor frequenciesouttobetensor hundredsofMHz,not100Hz! However,itmakesthenumbers easiertohandleifweusethese unrealisticsmallvalues;the principlesremainthesame, however.
2.7Exercises
E2–1
InaprotonspectrumthepeakfromTMSisfoundtobeat400.135705MHz. Whatistheshift,inppm,ofapeakwhichhasafrequencyof 400.136305MHz?Recalculatetheshiftusingthespectrometerfrequency, νspec quotedbythemanufactureras400.13MHzratherthan νTMS inthedenominatorofEq.2.2:
ppm = 106 × ν νTMS νspec . Doesthismakeasignificantdifferencetothevalueoftheshift?
E2–2
Twopeaksinaprotonspectrumarefoundat1.54and5.34ppm.Thespectrometerfrequencyisquotedas400.13MHz.Whatistheseparationofthese twolinesinHzandinrads 1 ?
E2–3
CalculatetheLarmorfrequency(inHzandinrads 1 )ofacarbon-13resonancewithchemicalshift48ppmwhenrecordedinaspectrometerwith amagneticfieldstrengthof9.4T.Thegyromagneticratioofcarbon-13is +6.7283 × 107 rads 1 T 1 .
E2–4
Considerasystemoftwoweaklycoupledspins.LettheLarmorfrequency
ofthefirstspinbe 100Hzandthatofthesecondspinbe 200Hz,andlet thecouplingbetweenthetwospinsbe 5Hz.Computethefrequenciesof thelinesinthenormal(singlequantum)spectrum.
Makeasketchofthespectrum,roughlytoscale,andlabeleachlinewith theenergylevelsinvolved(i.e.1–2etc.).Alsoindicateforeachlinewhich spinflipsandthespinstateofthepassivespin.Compareyoursketchwith Fig.2.7andcommentonanydifferences.
E2–5
Forathreespinsystem,drawupatablesimilartothatonpage 2–10showing thefrequenciesofthefourlinesofthemultipletfromspin2.Then,taking ν0,2 =−200Hz, J23 = 4HzandtherestoftheparametersasinFig.2.11, computethefrequenciesofthelineswhichcomprisethespin2multiplet. Makeasketchofthemultiplet(roughlytoscale)andlabelthelinesinthe samewayasisdoneinFig.2.11.Howwouldtheselabelschangeif J23 = 4Hz?
Onanenergyleveldiagram,indicatethefourtransitionswhichcomprise thespin2multiplet,andwhichfourcomprisethespin3multiplet.
E2–6
Forathreespinsystem,computethefrequenciesofthesixzero-quantum
transitionsandalsomarktheseonanenergyleveldiagram.Dothesesix transitionsfallintonaturalgroups?Howwouldyoudescribethespectrum?
E2–7
Calculatethelinefrequenciesandintensitiesofthespectrumforasystemof twospinswiththefollowingparameters: ν0,1 =−10Hz, ν0,2 =−20Hz, J12 = 5Hz.Makeasketchofthespectrum(roughlytoscale)indicating whichtransitioniswhichandthepositionoftheLarmorfrequencies.
E2–8
Thespectrumfromastrongly-coupledtwospinsystemshowedlinesatthe
Makesurethatyouhaveyour calculatorsetto “radians” when youcompute sin2θ followingfrequencies,inHz,(intensitiesaregiveninbrackets):32.0(1.3), 39.0(0.7),6.0(0.7),13.0(1.3).Determinethevaluesofthecouplingconstant andthetwoLarmorfrequencies.Showthatthevaluesyoufindareconsistent withtheobservedintensities.
E2–9
Figure2.18showstheABpartofanABXspectrum.Disentanglethetwo subspectra,markintheroughpositionsoftheeffectiveLarmorfrequencies andhenceestimatethesizeoftheAXandBXcouplings.Also,givethevalue oftheABcoupling.
Fig.2.18 TheABpartofanABXspectrum
3Thevectormodel
Formostkindsofspectroscopyitissufficienttothinkaboutenergylevels andselectionrules;thisisnottrueforNMR.Forexample,usingthisenergy levelapproachwecannotevendescribehowthemostbasicpulsedNMRexperimentworks,letalonethelargenumberofsubtletwo-dimensionalexperimentswhichhavebeendeveloped.Tomakeanyprogressinunderstanding NMRexperimentsweneedsomemoretools,andthefirstoftheseweare goingtoexploreisthe vectormodel
ThismodelhasbeenaroundaslongasNMRitself,andnotsurprisingly thelanguageandideaswhichflowfromthemodelhavebecomethelanguage ofNMRtoalargeextent.Infact,inthestrictestsense,thevectormodelcan onlybeappliedtoasurprisinglysmallnumberofsituations.However,the ideasthatflowfromeventhisratherrestrictedareainwhichthemodelcan beappliedarecarriedoverintomoresophisticatedtreatments.Itistherefore essentialtohaveagoodgraspofthevectormodelandhowtoapplyit.
3.1Bulkmagnetization
Wecommentedbeforethatthenuclearspinhasaninteractionwithanapplied magneticfield,andthatitisthiswhichgivesrisetheenergylevelsandultimatelyanNMRspectrum.Inmanyways,itispermissibletothinkofthe nucleusasbehavinglikeasmallbarmagnetor,tobemoreprecise,a magneticmoment.Wewillnotgointothedetailshere,butnotethatthequantum mechanicstellsusthatthemagneticmomentcanbealignedinanydirection1 .
InanNMRexperiment,wedonotobservejustonenucleusbutavery largenumberofthem(say1020 ),sowhatweneedtobeconcernedwithisthe neteffectofallthesenuclei,asthisiswhatwewillobserve.
Ifthemagneticmomentswerealltopointinrandomdirections,thenthe smallmagneticfieldthateachgenerateswillcanceloneanotheroutandthere willbenoneteffect.However,itturnsoutthatatequilibriumthemagnetic momentsarenotalignedrandomlybutinsuchawaythatwhentheircontributionsarealladdedupthereisanetmagneticfieldalongthedirectionofthe appliedfield( B0 ).Thisiscalledthe bulkmagnetization ofthesample.
Themagnetizationcanberepresentedbyavector–calledthe magnetizationvector –pointingalongthedirectionoftheappliedfield( z ),asshownin Fig.3.1.Fromnowonwewillonlybeconcernedwithwhathappenstothis vector.
Thiswouldbeagoodpointtocommentontheaxissystemwearegoingto use–itiscalleda right-handedset,andsuchasetofaxesisshowinFig.3.1. Thenameright-handedcomesaboutfromthefactthatifyouimaginegrasping
1 Itisacommonmisconceptiontostatethatthemagneticmomentmusteitherbealigned withoragainstthemagneticfield.Infact,quantummechanicssaysnosuchthing(seeLevitt Chapter9foraveryluciddiscussionofthispoint).
Chapter3“Thevectormodel” c JamesKeeler,2002
magnetization vector magnetic field z x y
Fig.3.1 Atequilibrium,a samplehasanetmagnetization alongthemagnetic field direction(the z axis)whichcan berepresentedbya magnetizationvector.Theaxis setinthisdiagramisa right-handedone,whichiswhat wewillusethroughoutthese lectures.
z x y
Fig.3.2 Ifthemagnetization vectoristiltedawayfromthe z axisitexecutesaprecessional motioninwhichthevector sweepsoutaconeofconstant angletothemagnetic field direction.Thedirectionof precessionshownisfora nucleuswithapositive gyromagneticratioandhencea negativeLarmorfrequency.
Thevectormodel
the z axiswithyourrighthand,yourfingerscurlfromthe x tothe y axes.
3.2Larmorprecession
Supposethatwehavemanaged,somehow,totipthemagnetizationvector awayfromthe z axis,suchthatitmakesanangle β tothataxis.Wewillsee lateronthatsuchatiltcanbebroughtaboutbyaradiofrequencypulse.Once tiltedawayfromthe z axiswefindisthatthemagnetizationvectorrotates aboutthedirectionofthemagneticfieldsweepingoutaconewithaconstant angle;seeFig.3.2.Thevectorissaidto precesses aboutthefieldandthis particularmotioniscalled Larmorprecession
Ifthemagneticfieldstrengthis B0 ,thenthefrequencyoftheLarmor precessionis ω0 (inrads 1 )
orifwewantthefrequencyinHz,itisgivenby
Fig.3.3 Theprecessing magnetizationwillcutacoil woundroundthe x axis,thereby inducingacurrentinthecoil. Thiscurrentcanbeamplified anddetected;itisthisthatforms thefreeinductionsignal.For clarity,thecoilhasonlybeen shownononesideofthe x axis.
where γ isthegyromagneticratio.Theseareofcourseexactlythesame frequenciesthatweencounteredinsection2.3.Inwords,thefrequencyat whichthemagnetizationprecessesaroundthe B0 fieldisexactlythesameas thefrequencyofthelineweseefromthespectrumononespin;thisisno accident.
Aswasdiscussedinsection2.3,theLarmorfrequencyisasignedquantity andisnegativefornucleiwithapositivegyromagneticratio.Thismeansthat forsuchspinstheprecessionfrequencyisnegative,whichispreciselywhatis showninFig.3.2.
Wecansortoutpositiveandnegativefrequenciesinthefollowingway. Imaginegraspingthe z axiswithyourrighthand,withthethumbpointing alongthe + z direction.Thefingersthencurlinthesenseofapositiveprecession.InspectionofFig.3.2willshowthatthemagnetizationvectorisrotating intheoppositesensetoyourfingers,andthiscorrespondstoanegativeLarmorfrequency.
3.3Detection
Theprecessionofthemagnetizationvectoriswhatweactuallydetectinan NMRexperiment.Allwehavetodoistomountasmallcoilofwireround thesample,withtheaxisofthecoilalignedinthe xy -plane;thisisillustrated inFig.3.3.Asthemagnetizationvector“cuts”thecoilacurrentisinduced whichwecanamplifyandthenrecord–thisistheso-called freeinduction signalwhichisdetectedinapulseNMRexperiment.Thewholeprocessis analogoustothewayinwhichelectriccurrentcanbegeneratedbyamagnet rotatinginsideacoil.
Essentially,thecoildetectsthe x -componentofthemagnetization.Wecan easilyworkoutwhatthiswillbe.Supposethattheequilibriummagnetization vectorisofsize M0 ;ifthishasbeentiltedthroughanangle β towardsthe x axis,the x -componentis M0 sin β ;Fig.3.4illustratesthegeometry. Althoughthemagnetizationvectorprecessesonacone,wecanvisualize whathappenstothe x -and y -componentsmuchmoresimplybyjustthinking abouttheprojectionontothe xy -plane.ThisisshowninFig.3.5. Attimezero,wewillassumethatthereisonlyan x -component.Aftera time τ1 thevectorhasrotatedthroughacertainangle,whichwewillcall 1 . Asthevectorisrotatingat ω0 radianspersecond,intime τ1 thevectorhas movedthrough (ω0 × τ1 ) radians;so 1 = ω0 τ1 .Atalatertime,say τ2 ,the vectorhashadlongertoprecessandtheangle 2 willbe (ω0 τ2 ).Ingeneral, wecanseethataftertime t theangleis = ω0 t .
Fig.3.4 Tiltingthe magnetizationthroughanangle θ givesan x-componentofsize M0 sin β
Wecannoweasilyworkoutthe x -and y -componentsofthemagnetizationusingsimplegeometry;thisisillustratedinFig.3.5.The x -component isproportionaltocos andthe y -componentisnegative(along y )andproportionaltosin .Recallingthattheinitialsizeofthevectoris M0 sin β ,we candeducethatthe x -and y -components, M x and M y respectively,are:
M x = M0 sin β cos(ω0 t )
M y =− M0 sin β sin(ω0 t ).
Fig.3.5 Illustrationoftheprecessionofthemagnetizationvectorinthe xy-plane.Theanglethroughwhich thevectorhasprecessedisgivenby ω0 t.Ontheright-handdiagramweseethegeometryforworkingout the x and y componentsofthevector. time
PlotsofthesesignalsareshowninFig.3.6.Weseethattheyareboth simpleoscillationsattheLarmorfrequency.Fouriertransformationofthese signalsgivesusthefamiliarspectrum–inthiscaseasinglelineat ω0 ;the detailsofhowthisworkswillbecoveredinalaterchapter.Wewillalso seeinalatersectionthatinpracticewecaneasilydetect both the x -and y -componentsofthemagnetization.
3.4Pulses
Wenowturntotheimportantquestionastohowwecanrotatethemagnetizationawayfromitsequilibriumpositionalongthe z axis.Conceptuallyitis
M y M x
Fig.3.6 Plotsofthe x-and y-componentsofthe magnetizationpredictedusing theapproachofFig.3.5. Fouriertransformationofthese signalswillgiverisetotheusual spectrum.
Fig.3.7 Ifthemagnetic field alongthe z axisisreplaced quicklybyonealong x,the magnetizationwillthenprecess aboutthe x axisandsomove towardsthetransverseplane.
Thevectormodel easytoseewhatwehavetodo.Allthatisrequiredisto(suddenly)replace themagneticfieldalongthe z axiswithoneinthe xy -plane(sayalongthe x axis).Themagnetizationwouldthenprecessaboutthenewmagneticfield whichwouldbringthevectordownawayfromthe z axis,asillustratedin Fig.3.7.
Unfortunatelyitisallbutimpossibletoswitchthemagneticfieldsuddenly inthisway.Rememberthatthemainmagneticfieldissuppliedbyapowerful superconductingmagnet,andthereisnowaythatthiscanbeswitchedoff; wewillneedtofindanotherapproach,anditturnsoutthatthekeyistouse theideaof resonance
Theideaistoapplyaverysmallmagneticfieldalongthe x axisbutone whichisoscillatingatorneartotheLarmorfrequency–thatis resonant with theLarmorfrequency.Wewillshowthatthissmallmagneticfieldisableto rotatethemagnetizationawayfromthe z axis,eveninthepresenceofthevery strongappliedfield, B0
Conveniently,wecanusethesamecoiltogeneratethisoscillatingmagneticfieldastheoneweusedtodetectthemagnetization(Fig.3.3).Allwe doisfeedsomeradiofrequency(RF)powertothecoilandtheresultingoscillatingcurrentcreatesanoscillatingmagneticfieldalongthe x -direction.The resultingfieldiscalledthe radiofrequency or RFfield.Tounderstandhowthis weakRFfieldcanrotatethemagnetizationweneedtointroducetheideaof the rotatingframe
Rotatingframe
WhenRFpowerisappliedtothecoilwoundalongthe x axistheresultisa magneticfieldwhichoscillatesalongthe x axis.Themagneticfieldmoves backandforthfrom + x to x passingthroughzeroalongtheway.Wewill takethefrequencyofthisoscillationtobe ωRF (inrads 1 )andthesizeof themagneticfieldtobe2 B1 (inT);thereasonforthe2willbecomeapparent later.Thisfrequencyisalsocalledthe transmitterfrequency forthereason thataradiofrequencytransmitterisusedtoproducethepower.
Itturnsouttobealoteasiertoworkoutwhatisgoingonifwereplace, inourminds,this linearly oscillatingfieldwithtwocounter-rotatingfields; Fig.3.8illustratestheidea.Thetwocounterrotatingfieldshavethesame magnitude B1 .One,denoted B + 1 ,rotatesinthepositivesense(from x to y ) andtheother,denoted B1 ,rotatesinthenegativesense;botharerotatingat thetransmitterfrequency ωRF
Attimezero,theyarebothalignedalongthe x axisandsoadduptogive atotalfieldof2 B1 alongthe x axis.Astimeproceeds,thevectorsmoveaway from x ,inoppositedirections.Asthetwovectorshavethesamemagnitude andarerotatingatthesamefrequencythe y -components alwayscancel one anotherout.However,the x -componentsshrinktowardszeroastheangle throughwhichthevectorshaverotatedapproaches 1 2 π radiansor90◦ .Asthe angleincreasesbeyondthispointthe x -componentgrowsoncemore,butthis timealongthe x axis,reachingamaximumwhentheangleofrotationis π Thefieldscontinuetorotate,causingthe x -componenttodropbacktozero andriseagaintoavalue2 B1 alongthe + x axis.Thusweseethatthetwo
Fig.3.8 Illustrationofhowtwocounter-rotating fields(shownintheupperpartofthediagramandmarked B+ 1 and B1 )addtogethertogivea fieldwhichisoscillatingalongthe x axis(showninthelowerpart).The graphatthebottomshowshowthe fieldalong x varieswithtime.
counter-rotatingfieldsadduptothelinearlyoscillatingone.
Supposenowthatwethinkaboutanucleuswithapositivegyromagnetic ratio;recallthatthismeanstheLarmorfrequencyisnegativesothatthesense ofprecessionisfrom x towards y .Thisisthesamedirectionastherotation of B1 .Itturnsoutthattheotherfield,whichisrotatingintheoppositesense totheLarmorprecession,hasnosignificantinteractionwiththemagnetizationandsofromnowonwewillignoreit.
Fig.3.9 Thetoprowshowsa fieldrotatingat ωRF whenviewedina fixedaxissystem.Thesame field viewedinasetofaxesrotatingat ωRF appearstobestatic.
Wenowemployamathematicaltrickwhichistomovetoaco-ordinate systemwhich,ratherthanbeingstatic(calledthe laboratoryframe)isrotating aboutthe z axisinthesamedirectionandatthesamerateas B1 (i.e.at ωRF ).Inthisrotatingsetofaxes,or rotatingframe, B1 appearstobestatic
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a crowd most typical of Asia. Ten thousand students receive instruction in its schools. It contains:
The houses, which are set in small compounds approached by narrow alleys, are composed of clay with low roofs and without
THE MINAR KALAN, BOKHARA
THE ARK, BOKHARA
windows. A hole in the roof suffices for a chimney, and the open door affords light.
Samarkand, the administrative centre of the province of the same name and founded in 1871, is a close reproduction of a large Indian cantonment. The streets are wide, well paved, fringed with tall poplars and set with shops which are kept by Europeans. For the Russians, as the centre of the province and the location of army headquarters, it has special importance. Although without any architectural pretensions—the buildings are all one-storey structures on account of frequent visitations from earthquakes—its comparatively lofty position makes it an agreeable station and one of the most attractive gathering-places for Europeans in Asiatic Russia. The city is situated upon the south-western slopes of the Chupan Ata range, 7 versts from the Zerafshan river. The close proximity of the hills naturally influences its rainfall, which is greatest in March and April. The period from June to September is dry; and by February or March the trees are in bloom. By a happy choice in construction it has been planned upon exceptionally generous lines which, although imparting to the outskirts a desolate aspect, have been the cause of securing to the community a number of spacious squares, around which are placed the barracks and certain parks. The principal square, named after General Ivanoff, a former Governor of the province, is Ivanovski Square. Another interesting memento of the Russian conquest of Turkestan is situated between the military quarter and the green avenues of the Russian town, in a spot where the heroes who fell in the defence of the citadel in 1868 were buried. At the same place, too, a memorial has been erected to Colonel Sokovnin and Staff-Captain Konevski, who were killed in 1869.
The population of Samarkand at the census of 1897 was 54,900: Males. Females. 31,706 23,194
According to the statistics of 1901, which are the most recently available, these figures had increased by a few thousands; they were
then 58,194: Males.
In the town itself
and various medical, charitable and other institutions.
The native quarter, which is separated from the Russian town by the Abramovski Boulevard—so named in honour of General Abramoff, another military Governor of the province—covers an area of 4629 dessiatines. It was built by Timur the Lame. The streets with few exceptions are narrow, winding and unpaved; the houses are of baked mud, mean and cramped, with flat earthen roofs and no windows. In this division there are:
The value of Government property in the Russian and native areas of the city is estimated at 4,077,681 roubles. The city revenue approximates 147,616 roubles. The native quarter is the great commercial centre of the province and the trade returns for the city and its surrounding district amount to 17,858,900 roubles out of 24,951,320 roubles for the entire province. Of the squares the most celebrated is the Registan, with a length of 35 sagenes and width of 30 sagenes. It is bounded by three large mosques: the Tillah Kori— the Gold Covered; Ulug Beg; and Shir Dar—the Lion Bearing.
The Registan is the heart of ancient Samarkand. Prior to the advent of the Russians pardon and punishment were dealt from it to the people by their rulers, executions performed and wars declared, as the authorities pleased. Even up to the present day the Registan has preserved in some degree its importance as a popular tribune.
THE TOMB OF TAMERLANE, SAMARKAND
From it self-constituted orators, holy men and politicians, expound their doctrines before a people gathered together from the most distant corners of the Continent of Asia. The Registan is only one feature of this delightful city; for here, too, are the stately ruins of the Bibi Khanum, tomb of the wife of Timur, and the Gur Amir where Timur’s remains lie amid a scene eloquent in its simple grandeur. Although, unfortunately, this building has been spoilt by attempts at restoration, its encrusted tiles are as beautiful as when they were made, 400 years ago. Here, too, is the resting-place of the Shah Zindeh; and in its Urda or ancient citadel, now a weak, bastioned fort, is the Kok Tash—the coronation-stone of the descendants of Timur. The charm of the Gur Amir is supreme. Within its dome, before the horse-hair standard, the sheer force of association and the infinite suggestion of the spot make one feel the great presence of this renowned soldier. Beneath the cupola there is a nephrite cenotaph; perhaps, as Colonel C. E. de la Poer Beresford has said, the largest block of green jade in the world.[5] Close to it other tombs, lighter in colour, are erected to the memory of Ulug Beg and Mir Sayid, Timur’s grandson and tutor. Around these is a carved gypsum balustrade and in the crypt below, under a simple bricktomb, lies the vanquisher of Toktamish Khan, of Sultan Bayazid, of Persia, the Caucasus and India—Timur himself.
In its economic aspect Samarkand occupies a very important position. Although scarcely serving as a mart to the produce of British India and Afghanistan, it is nevertheless a great emporium of trade. The roads, leading to the town or from it, as the case may be, are an index of its wide-reaching commercial influence. They run from Samarkand to Karki on the Amu Daria; and to Tashkent viâ Jizak; while Khojand, Khokand, Namangan, Andijan, Margelan and Osh are all in direct communication with it. Caravans from the east and north, from Persia and from China, carts perched on two gigantic wheels or transport bullocks laden with skins, even sheep carrying small packages—all are impressed into service and seem to be revolving in a constant stream round Samarkand. There is a steady traffic and the numerous bazaars are the centre of a brisk trade in skins and pelts. Unlike the bazaars of Bokhara, along the sides of which the merchants have their stalls, the passage-ways are open to the heavens. After the wonderful picture of Asiatic life presented by Bokhara, there are those who complain of a feeling of disappointment at the more subdued current which flows through Samarkand. Nevertheless the town has a charming setting. The snow-peaks of the Hissar chain and the curtain of enchanting fields and spreading vineyards, which hides the hideous aspect of the Kara Kum, add to the pleasure which is derived from the delicate mingling of the colours of the street life. There is, indeed, a very special type found in the bazaars of Bokhara and Samarkand. Dressed in the choicest of silks, so soft that it suggests the rustle of the wind through the peach-trees and dyed in tones of yellow, green and brown, in shades of magenta and purple, in a note of blue reflecting the sky or touched with the blush of a red rose, are men of fine stature. They move with their long-skirted gowns clasped at the waist and their silken trousers tucked into brown, untanned boots, the seams of which are delicately embroidered. Every individual reserves to himself a most exclusive manner, representing the embodiment of dignity. There is such an air of contentment about the gaily-clad crowd as it passes from stall to stall; such perfect selfpossession, suggested humility and independence that the difference in size between Bokhara and Samarkand goes unnoticed; the
atmosphere being no less pleasing, the picture no less acceptable, in the smaller city than in the capital.
SAMARKAND A BIRD’S-EYE VIEW
As the administrative focus of the Syr Daria province Tashkent is the principal city of Russian Turkestan and the seat of the GovernorGeneral. The Russian quarter at once recalls memories of other spheres of Central Asia. The streets are wide and long. Dusty but much frequented, they are bordered by high, white poplars set in double rows, while upon each side there run the gurgling waters of the irrigation canals. The city is laid out in a sector of a circle, three great boulevards radiating from the cathedral, a handsome, octagonal building in freestone. Surmounted by the dome and golden cross, which mark in Russia all Orthodox places of worship, it occupies the centre of Konstantinovski Square. It contains the remains of General von Kauffman, Governor-General of Russian Turkestan between 1867 and 1882 and, incidentally, conqueror of Khiva in 1873 and of Samarkand in 1868. He died May 4, 1882.
Tashkent, situated upon the slopes of the Tian Shan 172 sagenes above sea-level, lies in the midst of an extensive oasis whose fertile acres are watered by the river Chirchik and its tributaries. January is the coldest month, while July is the hottest. The prevailing breezes are north and north-east; but the characteristic peculiarity of the climate is the absence of wind, which makes the high temperature in the summer particularly oppressive. Spring weather begins in March; the hot season, commencing in May, continues until the middle of August. Speaking generally the place possesses the attributes of the climate in the plains of Central Asia while distinguished by its greater yearly rain-fall—384 millimètres—in consequence of the proximity of the mountains. The drinking water question, an ever-attendant difficulty in Central Asia, is no less acute in Tashkent, constituting a serious drawback to conditions of life there. An ample supply of water is available for irrigation, the Chirchik river, as well as numerous wells and springs, being diverted for this purpose.
SOBORNAYA BOULEVARD IN TASHKENT
The Russian quarter, founded in 1865 after the capture of the native town from the Khan of Khokand upon June 15 by the Russian forces under General Chernaieff, is separated from the native by the Angar canal. It is divided into official and residential areas, and contains many large streets. The Sobornaya, in which are situated the best shops, is perhaps more animated than any other thoroughfare in the town, while the Romanovski Street, which crosses the official quarter, is devoted principally to the Government offices. Three wide streets—the Gospitalnaya, Dukhovskaya, and the Kailuski Prospekt—along which it is proposed to erect business premises, also run from this quarter to the station. The residential part is of much later construction; its population is more scattered, the houses are surrounded with dense gardens and the streets are wider. The houses in each section are, for the most part, single storeyed. The chief public works are the Alexandrovski Park, Konstantinovski Square, Gorodskoi Garden and the gardens surrounding the residence of the Governor-General. The Turkestan Public Library, founded by General von Kauffman with the object of furthering the education of the country, now contains more than 40,000 volumes.
The following table shows the existing statistics of the Russian quarter:
The town revenues for 1902 were as follows: Revenue. 427,572 roubles.
The permanent garrison is never less than 10,000 men. Barracks and store-house accommodation for military supplies abound in the place. Between the spacious station and the Russian city, a distance of one verst, there are very commodious infantry quarters. A long row of buildings, somewhat more remote and erected upon slightly rising ground, contains the lines of the Cossack establishment. The climate of Tashkent is too unhealthy to be endured in the hot weather. In summer the garrison moves to Chigman, a defile 671 sagenes above sea-level, situated 80 or 90 versts beyond the town on the river Chirchik, where there is a sanatorium for the troops. The families of the officers usually pass the season at the village of Troitzki, 25 versts from Tashkent. Five versts from the city is Nikolski, the first Russian settlement founded in the Syr Daria province. Lying between it and the Russian town is the native quarter. Recalling Andijan, Margelan, Khokand and Osh, it lacks the animation of the streets of Bokhara and is destitute of the architectural beauties of Samarkand. Surrounded on three sides by gardens, the fourth side touches the Russian town with which it is connected, as also with the station, by means of a horse tramway. It is divided into four parts called respectively Kukchinski, Sibzyarski, Shaikhantaurski and Bish Agatchski. Each is separated into districts, these sub-divisions totalling 206 in all.
The two quarters of Tashkent occupy to-day an area of 20 square versts. Forty years ago the site of the Russian settlement covered no more ground than that required by the village which contained the garrison. This original section has now disappeared, becoming merged as time passed and the colony expanded with the Fortress Esplanade, while the population has similarly increased. In May 1871 the combined figures of the native and Russian colony gave only 2701 inhabitants. In 1897 the census returns showed the population to be:
