components along the X and Y axes. Based on time-dependent perturbation theory, it can be shown that if the populations

Transcription

1 Block quations to th vctor paradigm Assum that thr ar N N α +N β idntical, isolatd spins alignd along an trnal magntic fild in th Z dirction with th population ratio of N α and N β givn b a Boltmann distribution (Eqn 1). N α g β ΔE / kt N β N N (1) Whr g N is th magntogric ratio of nuclus N, β Ν is th nuclar magnton for nuclus N, ΔE is th nrg diffrnc btwn th α and β stats, k is Boltmann s constant, and T is th absolut tmpratur. At thrmal quilibrium N α > N β, and thr will b a nt (classical) macroscopic magntiation alignd along th Z ais with no magntiation componnts along th X and Y as. Basd on tim-dpndnt prturbation thor, it can b shown that if th populations of N α and N β ar prturbd awa from thir thrmal quilibrium valus, th sstm will dca ponntiall back to th quilibrium valus. Eqn. whr Z N α -N β Z Z T1 Th paramtr T 1 is th longitudinal or spin-lattic rlaation tim and is a charactristic of th spin sstm. Th () macroscopic magntiation can b rotatd to li in an dirction through th application of an oscillating magntic fild. Th componnts of th magntiation along th transvrs as dca to ro (Eqns. 3). X T X (3) Y Y T Th paramtr T is th transvrs or spin-spin rlaation tim. n gnral, this rlaation tim is diffrnt than T 1. To complt th dscription of th motion of th macroscopic magntiation, th spin angular momntum, a strictl quantum mchanical quantit, must b includd. Although this is a quantum ffct, it can b thought of as th ffct of a groscopic motion of a bar magnt placd in an trnal magntic fild. Th angular momntum du to th groscopic motion causs th magntiation to prcss around th trnal magntic fild, just as a groscop prcsss in a gravitational fild. Th quation of motion, tmporaril in th absnc of rlaation, is givn as th cross product of th magntiation and th magntic fild (Eqn 4).

2 (4) γ N ( H ) Hr rprsnts th macroscopic magntiation containing angular momntum du to th groscopic motion and H is a uniform magntic fild; γ N is th magntogric ratio of th nuclus. Th cross product of Eqn. 4 is givn in dtrminant form in Eqn 5. i j k H H H N X Y Z X Y Z For an trnal magntic fild along th Z ais, on obtains Eqn. 6. γ (5) i j k γ N X Y Z H Z Th Larmor prcssion frqunc of th rotation is givn b γ N H; for protons (γ Η 4.55 H/T) in a 14.1 Tsla (6) magntic fild this frqunc is 6 H. Th chmical shifts of common nucli ar tpicall in th rang of th audio frquncis (- kh). Th larg diffrnc btwn th Larmor prcssion frqunc and th chmical shift frquncis suggsts that th motion can b simplifid b th introduction of a rotating coordinat sstm. Essntiall, this is a trick to rmov th larg not-vr-intrsting Larmor frqunc and lav onl th important frquncis du to th chmical shifts. t can b thought of as appling a fictitious magntic fild opposing that of th 14.1 T trnal magntic fild. Th frqunc for a nuclus prcssing actl at th Larmor frqunc (on-rsonanc) undr ths conditions bcoms ro. For rsonancs that ar not on-rsonanc, th frqunc is non-ro and will prcss in th rotating fram. athmaticall, th H in Eqn. 4-6 is rplacd with an ffctiv fild H H-ω L /γ. This is quivalnt to th as rotating at an angular vlocit of th Larmor frqunc. Upon solving th dtrminant in Eqn. 6, Eqns. 7 ar obtaind. X Y Ω Ω Z Ω is th chmical shift frqunc in th rotating fram, -γ N H Z. Th particular algbraic sign for Ω ariss from th X Y (7) convntion that a positiv chmical shift frqunc is assumd to b gratr than th rotating fram frqunc. Th dirction of motion is givn b a right handd coordinat sstm, ->->-->-. Equations 7 indicat that th magntiation vctor will prcss (rotat) around th Z ais indfinitl. Combining Eqns. 7 with th rlaation quations (Eqns and 3) ilds th Block quations for prcssion of th magntiation in a static magntic fild (Eqns. 8).

3 X Y Ω Y T ΩX T X Y ( ) Z Z T1 Ths quations dscrib th volution of a magntiation vctor of an isolatd spin during a fr prcssion priod. Whn an RF fild is applid th Block quations ar transformd to thos in Eqns 9. Whr Ω is th chmical shift offst; ω is th rotating fram frqunc; B 1, ω, and φ ar th magnitud, frqunc, and phas of th applid RF fild, rspctivl. (8) *1/ T * Ω+ * ω * Ω *1/ T * ω * ω+ * ω ( )* ω γb1 cos( φ) ω γb1 sin( φ) Ω ω ω RF 1/ T (9) 1 n pulsd NR spctroscop, th applid radio frqunc is usuall much gratr than th offst frqunc du to chmical shifts and th duration of th RF puls is much shortr than 1/T 1 and 1/T. Undr ths conditions th trms arising from th applid RF fild dominat and th Block quations ar givn b Eqn 1. * ω * ω * ω+ * ω (1) From th solution of th st of diffrntial quations in Eqns. 8, th linshap for th NR rsonanc can b dtrmind. Thr ar two componnts of th linshap: on is in-phas with th citing RF fild and on is 9 out-of- phas. Th Lorntian absorption linshap (in-phas) is givn and th disprsiv (out-of-phas) linshap ar givn in Eqn. 11. A plot of ths linshaps ar plottd in Figur 1. On usful paramtr is th linwih of th absorption at half-hight (Δν 1/ ). This is givn as 1/πT H.

4 T A k* 1 + T Δ ω (11) ΔωT D k* 1 + T Δ Eqns. 9 can b also writtn in matri form (Eqn. 1). ω 1/ T Ω ω d T Ω 1/ ω + ω ω 1/ T 1 *1/ T 1 (1) Ths quations form a st of inhomognous diffrntial quations. This ariss sinc th rats of th X and Y componnts dpnd onl th magnitud of th Y and X magntiation, rspctivl, whras th Z magntiation dpnds on th diffrnc from th quilibrium stat. Th inhomognous charactr of ths quations maks th calculation of th propagation of th magntiation in a puls squnc difficult if th Hamiltonian is tim dpndnt, as it is in all usful puls squncs. olutions to ths quations provid a dtaild mthod to follow spin dnamics in th prsnc of rlaation and RF filds. A mthod to convrt th st of inhomognous quations to homognous is dscribd blow. Ω ω d ω Ω ω ω (13) Thr ar two simplifications that ar commonl applid to liminat th inhomognous charactr: olution 1: f rlaation is ignord, a homognous st of diffrntial quations is obtaind (Eqn. 13). This simplification givs ris to th simpl vctor dscription of th volution of th magntiation vctor. Th magnitud of th vctor rmains constant in this simplification. olution : rlaation is introducd during fr prcssion priods, but sinc th Hamiltonian during a puls dos not commut with th dnsit oprator, rlaation is ignord during all Figur 1. Lorntian absorption and disprsion linshaps. Not that th disprsion lin has finit amplitud ovr a much widr rang than th absorption lin. This is important to rmmbr in stting spctral wihs.

5 pulss. This is not a bad approimation whn th puls tim is much shortr than th rlaation rats. n this simplification th lngth of th vctor dos not rmain constant. Whil th introduction of rlaation during th fr prcssion priods maks a computation a bit mor ralistic, th simplst approach of totall ignoring rlaation lads to a vr intuitiv pictur of th motion of th magntiation. Th formal solution to Eqn. 13 givn in Eqn. 14. Ω ω * t Ω ω ω ω t (14) Or, b simplifing, on obtains Eqn. 15 with R bing th rat matri arising from th groscopic motion of th nuclar spin. A compact vrsion of this quation is givn in Eqn. 15. t R Th ponntial matri can b simplifid b various mthods (Appndi 1) to ild th simpl matri quation in Eqn 16, whr A is a 3 X 3 rotation matri in thr dimnsional spac. * t t A n th spcial cas whr onl chmical shift is prsnt (fr prcssion ilding rotation around th Z ais), matri A bcoms simpl a Z-ais rotation matri (Eqn. 17). Th transvrs componnts and volv. (15) (16) Ω Ω * t cos( Ωt) sin( Ωt) sin( t) cos( t) Ω Ω 1 (17) f th frqunc du to th applid RF fild B 1 (along th X ais) is much gratr than th chmical shift Ω, th lattr can b ignord and a simpl X-ais rotation matri is obtaind for an X ais rotation (Eqn. 18). ω * t ω 1 cos( wt) sin( wt) sin( wt ) cos( wt ) (18) Changing th phas of th applid RF fild to gnrat a Y ais rotation ilds Eqn. 19. ω * t ω cos( wt ) sin( wt) 1 sin( wt ) cos( wt ) (19)

6 Ths rotation matrics ar usd to calculat th (approimat) trajctor of th magntiation in th absnc of rlaation. f thr is mor than on non-commuting (Appndi ) oprator in th ponntial matri, thn th rsulting rotation matrics ar no longr along th X, Y, or Z ais. n othr words if th RF fild strngth along th X ais is comparabl to th chmical shift offst, thn th rotation occurs simultanousl around th Z and Y ais. This dos not lad to a simpl rotation matri around on of th X, Y, or Z ais, but a rotation about an ais that is tiltd awa from all of th as. As an ampl of two simultanous, non-commuting rotations, considr a RF puls causing a rotation about th X ais at a frqunc ω 1 for a tim t with a simultanous Z rotation du to a chmical shift offst Ω. Th ponntial rat matri and th corrsponding rotation matri is givn in Eqn. Not that th frqunc ω is th vctor sum of ω 1 and Ω, a consqunc of th introduction of a rotating coordinat sstm. This lads to a rotation ais that is tiltd out of th XY plan having a magnitud gratr than ithr ω 1 or Ω individuall. ω Ω Ω ω Ω + cos sin 1 cos ( ) ( ω t) ( ω t) ( ω t) 1 1 Ω ω ω ω ω Ω ω1 * t ω1 Ω ω1 sin ( ω ) cos( ω ) sin( ωt ) ω ω t t ω1ω ω 1 Ω ω 1 + ω ω ω ω ( 1 cos( ωt) ) sin( ωt) cos( ωt) () ω1 ω Ω + Th unwild natur of ths rotation matrics can b bpassd through th us of similarit transforms to mov th sstm into a tiltd fram of rfrnc, prform a rotation, and thn rturn th sstm back from th tiltd fram. Eqn. 1 givs th appropriat similarit transform for this situation. Acting on a column vctor at th right, th rightmost rotation R (tan -1 Ω/ω 1 ) rotats th tiltd rotation ais to along th X ais. Th nt R rotation rotats th magntiation vctor with considration that th frqunc of th rotation is gratr than ω 1. 1 Ω Ω + ω 1 1 Ω R tan R * ω1t R tan ω1 ω1 ω 1 (1) For situations in which th chmical shift frqunc is similar to th RF, th motion of th magntiation is no longr a simpl rotation about th X, Y or Z as. n th analsis of puls squncs, it is commonl assumd that th applid RF is infinitl gratr than th chmical shift frqunc lading to th simplification in Eqn. 1. n th prsnc of rlaation, th us of similarit transforms is not possibl (s blow).

7 Th rotation matrics obtaind from th ponntiation of th rat matri of nuclar prcssion lad to th wll known volution of th product oprators for a singl spin. Th magntiation for a singl isolatd spin can b rprsntd b th column matri in Eqn 3. (3) At quilibrium, th magntiation vctor lis along th Z ais and is rprsntd b Eqn (4) Appling a θ rotation of th magntiation vctor around th X ais ilds Eqn cos( θ ) sin( θ) sin( θ) sin( θ ) cos( θ) 1 cos( θ) (5) Or in standard product oprator trms on obtains Eqn 6. θ cos( θ ) sin( θ ) Th tnsion to mor than on isolatd spin involvs th Bloch quations for ach spin individuall. For two spins in th absnc of rlaation, this lads to th 6X6 matri in Eqn. 7. Not that sinc th spins ar isolatd no matri lmnts connct th and spins. (6) ω Ω Ω ω d ω ω ω Ω ω Ω ω ω (7) Th solution is th sam as abov. Th sam simplifications for chmical shift and RF rotations appl as for th singl spin cas. That is, sinc diffrnt spins alwas commut simultanous and rotations.g. + ild simpl spin spcific rotation matrics around X, Y, or Z, but simultanous non-commuting rotations such as + or + do not

8 giv simpl rotation matrics. imultanous non-commuting rotations can b dalt with b similarit transforms (tiltd frams) or b solving th ponntial matri numricall. Rlaation can b rintroducd to this dscription b svral mthods. A vr clvr mthod is to transform th inhomognous quation into a homognous on b adding a row and column to th matri to introduc th diffrnc from quilibrium (Jnr,Lvitt, Allard, Ernst##). An ampl of this is shown in Eqn. 8. E/ E/ d 1/ T Ω ω (8) Ω 1/ T ω 1/ ( T1* ) ω ω 1/ T1 Th lmnt (1,4) of th matri introducs th quilibrium magntiation. This lads to Eqn. 9 for th tim dpndnc of in th prsnc of a RF fild, which is idntical to that givn in Eqn 14. Equation 8 can b solvd using th matri ponntial. Th rsulting matri is not a simpl rotation matri sinc th rlaation trms do not prsrv th lngth of th vctor. Howvr whn rlaation occurs during pulss, this mthod ilds a complt dscription. d d ( 1/ 1* ω ω 1/ 1 ) T + + T ( ω ω 1/ 1( )) + T (9)

9 For a htronuclar spin sstm a st of diffrntial quations can b st up similar to th following (no rlaation). d Ω + ω πj From ths diffrntial quations, th rat matri is obtaind Ω ω πj Ω ω πj ω ω J Ω ω π Ω ω π J ω ω πj Ω ω ω ω d π J Ω ω ω ω πj Ω ω ω ω πj Ω ω ω ω ω ω Ω Ω ω ω Ω Ω ω ω Ω Ω ω ω Ω Ω ω ω ω ω This is th matri rprsntation for th volution of th dnsit oprator in th Cartsian product oprator basis in th absnc of rlaation. t has a solution of th form t R * t Again if onl on non-commuting rotation is usd thn th rsulting matri is a simpl rotation matri, hr as 3 dimnsional subspacs of 15 dimnsional spac. n mab, mor familiar nomnclatur d σ ihσ i[ H, σ ]

10 Th solution to this quation in rgular oprators is iht σ ( t ) * σ () * iht

11 Appndi 1 Connction of ponntial rat matrics with rotation oprators. (dimnsional cas) Rat matri R, rat a, tim t 1 R 1 Rat Rat + ( Rat) + ( Rat) +! 3! ( Rat ) + ( Rat) + + Rat + ( Rat) + ( Rat) +! 4! 3! 5! at at ! 1 4! 1 at + at + at 1 3! 1 5! ( at) ( at) at ( at) ( at)! 1 + 4! ! 1 5! 1 cos( at)* + sin( at)* R cos( at) sin( at) sin( at) cos( at) D rotation matri Rat E 1 cos( at) sin( at) 1 cos( at) sin( at) cos( at) sin( at) +

12 Commuting oprators A* B B* A * * Non-commuting oprators A*B -B*A * * 1 1 1

13 Appndi 3. Eponntial commuting oprators. A + B + ( A+ B) + 1 ( A+ B) +! 1 A + A+ A +! B + B+ 1 B +! A B 1 + ( A+ B) + ( A+ AB+ B) +! comparing th scond ordr trms 1 ( A+ B ) A+ AB+ BA+ B! and 1 ( A+ AB+ B )! A+ B A B onl if ABBA