Densy Marx Descrpon of NMR BCMBCHEM 89
Operaors n Marx Noaon Alernae approach o second order specra: ask abou x magnezaon nsead of energes and ranson probables. If we say wh one bass se, properes vary only because of changes n he coeffcens of each bass se funcon x = h< Ix > - hs s how we calculae observables = c + c + c3 + c4 = j c j j AB example: =. +. +. +. =. +.6 +.8 +. 3 =. +.8 -.6 +. 4 =. +. +. +. < Ix > = j,k c j * c k < j Ix k > We need o calculae < j Ix k > only once f we say wh hs bass se hese can be pu n a n x n marx. Marx equvalen: < Ix > = c, c, * c Ix c
Specal Case: Paul Spn Marces Ix = ½ ½ Noe: < Ix > = ½ < > = < Ix > = ½ < > = ½ Iy = -½ ½ Iz = ½ ½ How do hey work? Try somehng we know: Ix > = ½ ½ ½ = = ½ = ½ ½ Operaors are a marx of numbers, Spn funcons a vecor of numbers
Larger Collecons of Spn ½ Nucle =??????? I Ax =???????? < I Ax > = < ½ > = ½? I Xx = Easer way: drec producs: E I Ax wh X marces
Densy Marces and Observables for an Ensemble of Spns Expecaon values peran o sngle spn properes; we observe ne behavor of an ensemble of spns. For a parcular sysem wo spn ½ nucle operaor represenaons are he same; all varaons are n bass se coeffcens. We need o average over hese coeffcens. < Ix > = c, c, * c Ix c Producs of averaged coeffcens can also be colleced and used n marx form. Ths s called a densy marx,. < Ix > = Tr c j c k * Ix = Tr Ix = j,k c j * c k Ix jk For a spn sysem: c*,c* Ix Ix c = c*, c* Ix c + Ix c Ix Ix c Ix c + Ix c = c*c Ix + c*c Ix + c*c Ix + c*c Ix For a spn sysem: < Ix > = Ix + Ix + Ix + Ix
Solvng for he Tme Dependence of Our observables are me dependen magnezaon precesses. All me dependence can also be pu n bass se coeffcens, or n coeffcen producs of he densy marx. Schrodnger s me dependen equaon H = - h dd Allows us o solve for dc j d, dd c j c k * = c j dc k *d + dc j d c k * Resul n erms of a densy marx s Lovlle von Neuman eq. dd = h { H - H } = h [, H] Implcaons: If we know a any me equlbrum a me and know he Hamlonan H, we can solve for a any me, and calculae any observable as he Tr O O s any operaor.
Tes: Does X Magnezaon Precess n B? Wha elemens of dcae x magnezaon? <I x > = Tr [] [I x ] - examne spn ½ case = Therefore, consder = Wha s H?, H = -hb I z = -hb d d B Tr
Consder evoluon of one elemen dd = B -½ ½ = Soluon: = exp- or: Same happens for = cos sn hence elemens correspondng o x magnezaon precess = =
Densy Marx a Equlbrum Have a roue o an observable: < Ix > = Tr Ix Have a roue o : dd = h [, H ] Need a place o sar: Dagonal elemens: consder c n c n * a probably c n c n * = exp-e n ktz Z - E n ktz; small E n nn = Z - nn ; Z # of saes Noe: ex uses for densy marx really devaon marx
Off-dagonal elemens a equlbrum mus sasfy me dependen Schrodnger Eq. H = h = E; n lm of H = H Soluon s: n = n exphe n Bu, he members of he ensemble do no have a common orgn n me. Hence add a random phase facor: nm = n exphe n + m = n c mn c mn = cosn,m, + snn,m, Densy marx elemen = c c j * = cos cos j + sn cos j +cos I sn j +sn I sn j f = j, Pcos + sn P If I j, cos cos j ec =, and all off dagonals =
Examples for a spn ½ nucleus H = -hb I Z, E = +- h4b eq exp B kt exp B kt eq B kt No me varaon as expeced: dd = h[, H ] = [ H] [H ] = produc of dagonal marces B kt
Wha abou M Z and M X? Cures law for suscepbly kt B kt B kt B Tr Tr M Z Z 4 4 4 ] ][ [ ] ];[ ][ [ Tr M I Tr M X X X X X Expec no X magnezaon a equlbrum
Wha elemens do gve M X? M X Tr We have argued ha hese are relaed o ranson probables We have shown ha hese precess n a magnec feld
Roaon operaors a more general descrpon of roaon dd = h [, H] General soluon: = exp-hhexphh = RR - If bass se elemens are egen funcons of H, R s dagonal and easy o evaluae Chemcal shf evoluon: R Z = exp-b cs I Z R Z exp exp ;
M X and M X Precess a sn sn cos cos exp exp exp exp exp exp exp exp exp exp M X M Y
Roaon operaors for an RF pulse along X axs n he roang frame Hamlonan n he roang frame: H = -B hi X Formal soluon a end of pulse me s gven as: = exp-hh exphh = R X R X - Canno smply nser I X n exponenal operaor o evaluae marx elemens; I X mxes spn saes Elemens of do no jus oscllae, bu conver one elemen no anoher for example - o + somehng we assocae wh z magnezaon o wha we assocae wh y magnezaon n a snb dependen way.
X pulse wh =, ; ; ; ;,,,, R R R R X X X X cos sn sn cos R X For one spn Smlar argumens lead o Y pulse operaons
The One Pulse FID = R Z R X, R - X, R Z - 9 X eq M p p Z
Evoluon of he FID Frs pon n he FID: = R Z d w p R Z - d w ; d w = dwell me; = exp exp exp exp W W W W d d d d = R Z dw R Z - dw; second pon n FID Calculang he FID: FID = Tr{[ ] [M X + M Y ]} Programs lke GAMMA, SPINEVOLUTION work hs way
Densy Marx Smulaon of nd Order Specra For wo spn bass se s:,,, 3... 3....... + 34 + 43 3 + 3 4 + 4,, are assocaed wh lnes n an AX specrum. : for frs order specrum Don have o have a : assocaon f s no an egen funcon of H. c I + c I may evolve coherenly as one lne e, 3, are mxed Calculaon of Mx sll works for a second order sysem