NMR Specroscop: Principles nd Applicions Ngrjn Murli Advnced Tools Lecure 4
Advnced Tools Qunum Approch We know now h NMR is rnch of Specroscop nd he MNR specrum is n oucome of nucler spin inercion wih eernl mgneic field nd elecromgneic rdiion. This is sujec es ckled qunum mechnics. We lso know h NMR is ulk effec nd i is indeed ver hrd o descrie NMR of rel smple qunum heor s i ecomes comple mn od prolem. Bu, surprisingl, we cn focus on single spin or few coupled spins nd hen predic he course of NMR eperimen o represen our rel ssems. The sisicl ensemle of nucler spins ehve ecl like he isoled model spin ssem.
nercion of Spin wih Eernl Mgneic Field We lern in Lecure, he inercion of nucler mgneic momen m wih eernl mgneic field B 0 is known s Zeemn inercion nd he inercion energ known s Zeemn energ is given s: E μ B0 B0 E B m m where m 0 B 0 0 0 0,,..., B 0 NMR is rnch of specroscop nd so i descries he nure of he energ levels of he meril ssem nd rnsiions induced eween hem hrough sorpion or emission of elecromgneic rdiion.
Resonnce Resonnce Mehod implies h he frequenc of irrdiion is sme s h of he seprion of energ levels in frequenc uni. Deecion of Resonnce is reduced o he mesuremen of frequenc which here is deecle chnge in he re of rnsiion of spin se.
Hmilonin n qunum mechnics he energ of ssem is n imporn quni nd is descried n operor wih specil nme Hmilonin. Operors re imporn in h h we oserve heir mgniude when suile eperimen is performed o mesure hem. Such mesurle quniies re clled oservles nd he operor corresponding o such quniies re clled oservle operors. For single spin he Hmilonin is given s H μ B 0 B 0 The erms h re operors hve h over hem. We knew his inercion is clled Zeemn energ nd his erm is clled Zeemn erm.
Hmilonin The sic field is pplied long he -is of he loror frme. Then he Hmilonin is reduced o B B 0 H 0 Onl he operor is relevn s he sclr produc is non-vnishing for his erm. The possile eigen vlues m of he operor is,-+,, i.e. + vlues.
Zeemn Energ Thus he inercion energ of single spin in sic field pplied long he -is of he loror frme is E B0 m m kes + vlues from o + in seps of. Energ H =/ 5 N =/ 7 Al =5/ Mgneic Field Mgneic Field Mgneic Field
Resonnce of single spin =/ For =/, m cn e +/ or -/. Usull he / se is referred s nd he -/ se s. 0 0 0 0 0 0 0 0 m h B E B B E E E B E B E m B E
Resonnce of single spin =/ Qunum mechnics ss h rnsiion is llowed if he chnge of qunum numer m is + or -. So he rnsiion eween o se is llowed nd is clled one qunum or single qunum rnsiion. E m B 0 0 h 0
Hmilonin in Frequenc Unis Since energ cn e epressed in erms of frequenc uni, we cn lso wrie he Hmilonin in frequenc uni. Since, E m B We cn wrie he Hmilonin s, 0 0 h 0 H or H 0 0 in rd s in H -
Spin =/ Hmilonin Summr m se energ unis Frequenc rd / s Frequenc H +/ -/ħb 0 / 0 / 0 -/ /ħb 0 -/ 0 -/ 0
Two Spins =/ Hmilonin Le us eend his ide of wriing Hmilonin o wo spins ech wih spin =/ H 0 0 in H m m se Frequenc H +/ +/ / 0 +/ 0 +/ -/ / 0 -/ 0 -/ +/ -/ 0 +/ 0 -/ -/ -/ 0 -/ 0
Two Spins =/ Energ Levels The energ level digrm could e presened s elow Two proon spins nd 3 - H pir. E m m, m 0 0 m in H
Hmilonin Two Spins =/ nd J oupling Le us eend his ide of wriing Hmilonin o wo spins ech wih spin =/ nd J oupling eween hem. H H 0 H 0 0 J J in H The J coupling is lso known s sclr coupling s he operors involved re epressed s sclr produc. Noe h here is no pplied field dependen erm in his inercion mens he coupling vlue is independen of field srengh. The nucler mgneic momens sense he presence of oher spins rough he chemicl ond eween hem. in H 0 0 0 J in H for he cse 0 0 J
Energ Levels Two Spins =/ nd J oupling The energ levels of he wo spins ech wih spin =/ nd J oupling eween hem is hen wrien s E m, m 0 m 0 m J m m in H for he cse 0 0 J m m se Frequenc H +/ +/ / 0 +/ 0 +/4J +/ -/ / 0 -/ 0 -/4J -/ +/ -/ 0 +/ 0 -/4J -/ -/ -/ 0 -/ 0 +/4J
Energ Levels Two Spins =/ nd J oupling The energ level digrm direcl predics he NMR specrum. Lef: Energ level digrm; drk rrows show spin flip nd is rnsiions nd ligh rrow show flip of he spin nd is rnsiions. Righ: The NMR specrum resuling from he four rnsiions.
Energ Levels Two Spins =/ nd J oupling The NMR specrum hs ll he informion of he energ levels or he Hmilonin of he ssem. Trnsiion Spin ses Frequenc H - 0 -/J 34-0 +/J 3-0 -/J 4-0 +/J
A Bi More of Qunum Mechnics So fr we sw h he energ of inercion of nucler spins wih mgneic field nd wih ech oher cn e epressed Hmilonin h in urn epressed in erms of he vrious componens of he spin ngulr momenum. Wih his descripion we could clcule he rnsiions. We did no descrie how we ecie he rnsiion nd wh hppens during he NMR eperimen such s one pulse eperimen or spin echo eperimen. To undersnd he ime course of NMR eperimen we hve o follow he ssem s i evolves during he eperimen. We hve o undersnd how he spin ssem ses re defined in qunum mechnics nd how he chnge in n eperimen nd how we cn oserve he spin ssem se.
Wve Funcion: The se of he Spin n qunum heor, we don no know priori wheher spin s =/ is in he se +/ or in he se -/. So we wrie in generl he se o e superposiion of he wo possile se The coefficiens nd re numers comple numers in generl nd heir phse cn vr in ime nd heir mgniudes give rise o he vlue of he oservle quniies in NMR. This funcion is clled wve funcion nd s i is comple funcion we cn lso wrie is comple conjuge s Then numer jus is
Properies of Wve Funcion We cn hen compue numer s r ke produc n his clculion we hve used he fc h he spin ses re orhogonl. numer jus is 0 0
Epecion Vlues Once we define he se of he ssem wve funcion, we cn now sk wh is he chnce h we hve componen of he spin ngulr momenum long -is nd is given he epecion vlue of he spin ngulr momenum operor long -is. Since he denominor is numer we cn se i o equl o sing he wve funcion is normlied, ll we hve o do is compue And we will use he fc h he se nd re eigen ses of ˆ ˆ
Epecion Vlues Then he epecion vlue of is We hve used he orhogonl proper of he spin ses nd in evluing he ove vlue. The ove equion mens h he verge vlue of he componen long -is is given he difference in he proili of finding he spin in he +/ se or -/ se when mn mesuremens re mde.
Epecion Vlues We cn now see wh hppens o he spin ngulr momenum componens in he rnsverse plne nd using he fc h Agin he epecion vlue for hese componens lso depend on he coefficiens h re proili funcions. i i i
Bulk Mgneiion We re now red o rrive he ulk mgneiion h is induced when collecion such individul spins re eposed o mgneic field. The ulk mgneiion long -is hen sum of he epecion vlue of he -componen of he individul spins. The r ove he funcions on he righ hnd side indices sum over he ensemle of spins. N i i i i i N i i M N M N i N N M where
Populions The proili difference h gives componen long - is now cn e inerpreed s populion difference in he wo ses h give rise o he -mgneiion n N nd n M N n n
Bulk Mgneiion Trnsverse Plne We cn in he sme w compue mgneiion long -is nd - is. A equilirium here is onl -mgneiion nd no mgneiion in he rnsverse plne. This mens h he ensemle verge on he righ goes o ero. This is clled he rndom phse pproimion or h here is no coherence eween he spin ses. N N M N i N M 0 0
Time Evoluion in Qunum Mechnics We discussed in he vecor model how mgneiion roes in he presence of pplied fields oh D nd RF. We wroe he equion of moion s he ime derivive of he mgneic momen equl o he orque on he momen. n he sme w, he moion of he spins cn e epressed in erms of is se undergoing chnge effeced he inercion Hmilonin. Suppose if we consider single spin =/ in he roing frme hen he Hmilonin is And d ih d H
Time Evoluion in Qunum Mechnics Then, B mulipling on eiher side < nd < nd using < > = < > =, nd < > = < > = 0, we cn simpl sepre he ove equion ino wo equions on he coefficiens s i d d H i d d i i d d d d nd i d d i d d
Time Evoluion in Qunum Mechnics The soluion of hese wo equions is srighforwrd, Since we know now he vlue of he coefficiens s n ime we cn evlue he epecion vlues of he spin ngulr momenum componens in he,, nd is nd i d d i d d i i e e 0 nd 0 i
Time Evoluion in Qunum Mechnics Le us jus evlue one of hese in deil i i e 0 0 0 0 e 0 0 cos isin 0 0 cos isin cos 0 0 0 0 sin i 0 0 i 0 0 cos 0 sin 0
Time Evoluion in Qunum Mechnics Similr clculions cn e done for he oher componens nd in summr we hve cos 0 sin 0 cos 0 sin 0 0 Free evoluion does no ffec he -componen. The nd componens roe in he plne. These resuls re sme s we go from he vecor model.
Time Evoluion of Bulk Mgneiion in Qunum Mechnics We know h he vlue of he ulk mgneiions re given heir respecive epecion vlues nd hus we cn compue he se of he mgneiion componens n ime. 0 sin 0 cos 0 sin 0 cos M M M N N N M 0 sin 0 cos 0 sin 0 cos M M M N N N M 0 0 M M N N M
Time Evoluion Due o RF Pulse in Qunum Mechnics So fr we worked wih roing frme Hmilonin corresponding o jus he pplied sic mgneic field. H Now le us s we hve n RF field lso long -is nd for simplici le us lso ssume h we re on-resonnce =0. So he new Hmilonin in he roing frme in he presence of RF long -is is H Where is he mpliude of he RF in unis of rdins/sec. We cn repe he clculions in he sme line s efore nd he resuls for he mgneiion componens cn e given nlog.
Time Evoluion Due o RF Pulse in Qunum Mechnics The effec of RF long is hen, Under -pulse he mgneiion precess in he plne s we sw in he vecor model. For emple if we se =/ nd =0 wih mgneiion onl long + is, we will end up long is. 0 0 M M N N M 0 sin 0 cos 0 sin 0 cos M M M N N N M 0 sin 0 cos 0 sin 0 cos M M M N N N M
Operor Formlism We hve colleced so much knowledge in qunum frme work o descrie he spins inercing wih he mgneic field, RF, nd mong hemselves nd now we see h if we follow he evoluion of he componens of he spin ngulr momenum operor, we cn predic he course of he mgneiion of spin in n NMR eperimen. To fcilie h, we will descrie he se of he spin ssem direcl hese spin ngulr momenum operors. is hen sid h we hve descried he ssem densi operor s opposed o descriing he se wve funcion. For single isoled spin ssem, he rirr se cn e given he operor, ρ
Operor Formlism Now we cn wrie he ime evoluion of his densi operor Liouville-von Neumnn equion o descrie he spin ssem. ρ d d ρ i Hρ ρ H ih ρ ρ e 0 e ih Noe h in he ove he order in which he erms re wrien is imporn since Hρ ρ H
Operor Formlism Roions Le us s =0 = = =0, hen ρ0 Also Le us s he Hmilonin is Then ρ ih i i The whole clculion cn e wrien in noion s elow H ih e ρ0 e e ρ cos e sin cos H ρ 0 ρ sin
Operor Formlism Roions Le us s he Hmilonin is Then Using he noion we descried efore i i i i e e e e H H ρ ρ ρ 0 0 H sin cos sin cos
Roions -Summr Le us now summrie he roions under pulses ou differen es nd flip ngles. - - - - - -
Roions -Summr Posiive roions re couner-clockwise s shown nd negive roion is clockwise.
Operors Two Spins We cn eend he sme represenion of he se of he ssem operors o wo spins nd. The wo uni operors E nd E do no represen n oservle. We need lso he producs of he spin operors of he wo spins o descrie he ssem. Noing h he producs wih he E operor is he sme s he 8 operors ove, we hve E E
We hve lred seen he evoluion of single spin operors under he Zeemnn erm nd RF pulses. For wo spins he respecive spin operors evolve seprel wih hese erms s if he re independen. The evoluion under he coupling pr of he Hmilonin is slighl more involved. Hmilonin for Two Spins Le us recll he Hmilonin for wo spins nd H 0 0 J in H for he cse 0 0 J Le us conver he ove in o he roing frme Hmilonin in ngulr frequenc unis. H J J in rd/secfor he cse The ime evoluion of he wo spin ssem is under he influence of he ove Hmilonin nd cn e clculed in sepre seps. J ρ 0 ρ
Evoluion under oupling-two Spins Le us now illusre he evoluion under he coupling pr of he Hmilonin H J J J cos sin This kind of evoluion is clled i-liner roion s he roion is ou he produc of wo operors oh long he -direcion. Noe he second erm on he righ, i is lso produc erm in which he se of spin is represened is -componen which lso senses he se of he -componen of he coupled second spin. J J
Evoluion under oupling-two Spins We cn now illusre ll of such evoluions: J J J J J J J sin cos sin cos J J J J J J J sin cos sin cos
Evoluion -Two Spins Le us now see he phsicl mening of he se erms h we hve colleced. Le us s =0 we hve nd nd see he evoluion. As one cn see if we follow one spin hrough he evoluion under vrious prs of he Hmilonin we cn wrie for he oher spin inducion cos sin sin cos cosj sinj J However, We cn deec onl he or operors nd no he produc operors sin cosj sinj J cos
Evoluion -Two Spins The oserved signl of he spin is hus come from he erm cos cos cosj sin J n he oserved signl S he -componen is he rel pr nd he -componen is he imginr pr. S cos cos cosj i sin cosj J epi ep R ep ij ep ij ep ep R ep R ep i J epi J ep R
Evoluion -Two Spins Fourier rnsform S will hen give rel nd imginr pr of he specrum wih lines he wo frequencies in he eponen. S R epi J epi J ep Noe he horionl is in unis of. The muliples re in sme phse wih respec o ech oher in oh he rel nd imginr prs.
Evoluion -Two Spins Thus he evoluion of under wo-spin Hmilonin gives in phse doule specrum J Thus he operor is clled he in-phse operor.
Evoluion -Two Spins Suppose =0 if we hve he operor, hen we cn clcule he resuling specrum in similr w. cos sin sin cos cosj sinj J sin cosj sinj J cos Noing gin h we cn onl deec or componens, he deeced signl is S sin sinj i cos sinjep R i sinj ep i ep R i i ep ij ep ij ep ep i J epi J ep R ep R
Evoluion -Two Spins Up on Fourier rnsform he specrum would e s elow, J The muliiples re in opposie phse in oh he rel nd imginr pr nd so he operor is clled ni-phse componen of he spin.
Summr of Phsicl Nure of Operors - Two Spins - is sorpion - is sorpion
Summr of Phsicl Nure of Operors - Two Spins - mgnei ion on spin in - phse - nd - mgnei ion on spin - mgnei ion on spin in - phse - nd - mgnei ion on spin,, ni - phse - nd - mgnei ion on spin ni - phse - nd - mgnei ion on spin muliple qunum coherences non - equlirium populion,,,,,
Muliple Qunum oherences ZQ Zero Qunum - ZQ Zero Qunum - DQ Doule Qunum - DQ Doule Qunum - / 0 +/ 0 +/4J / 0 +/ 0 +/4J / 0 -/ 0 -/4J -/ 0 +/ 0 -/4J 0 0 0 0 ZQ DQ
Summr of Phsicl Nure of Operors - Three Spins J =0H, J 3 =H, J =7H, J 3 =0H, c J =5H, J 3 =5H