Lecture #8 Redfield theory of NMR relaxation

Transcription

1 Lecure #8 Redfield heory of NMR relaxaion Topics The ineracion frame of reference Perurbaion heory The Maser Equaion Handous and Reading assignmens van de Ven, Chapers 6.2. Kowalewski, Chaper 4. Abragam Chaper VIII.C, pp ,

2 The Maser Equaion When he Hamilonian can be wrien as he sum of a large saic componen plus a small ime-varying perurbaion Ĥ = Ĥ + ( Ĥ1 ) our goal is o find an equaion of he form: σ = iĥ σ Γ σ ( σ B ) Relaxaion Relaxaion superoperaor from which we can calculae any direc and cross relaxaion erms of ineres. In general, Γ differs across differen relaxaion mechanisms. Reminder: Bloch s d M! equaions are: = γ M! B z M xx + M y y ( M M z ) z d T 2 Roaions Roaions Relaxaion erms T 1 2

3 Redfield Theory Redfield heory, also known as Wangsness, Bloch, and Redfield (WBR) heory, is more general han ha derived by Solomon. Boh Solomon and Redfield heory rely on 2 nd -order perurbaion heory (can be a limiaion, bu usually no for liquids) Like Solomon s approach, Redfield heory is semi-classical, using he same Bolzmann correcion for hermal equilibrium values. However, raher han direcly dealing wih energy level populaions, he heory is derived in erms of he densiy operaor. This allows for a more general descripion of relaxaion and allows he derivaion of relaxaion raes from muliple mechanisms, e.g. dipolar coupling, CSA, scalar relaxaion of he 1 s and 2 nd kind, ec. 3

4 Liouville-von Neumann equaion We ve seen where we can express he spin Hamilonian as he sum of a large saic componen plus a small ime-dependen spin laice ineracion erm. Ĥ() = Ĥ + Ĥ1() perurbaion Saring wih he Liouville-von Neumann equaion for he densiy operaor dσ d = iĥ σ I is helpful o swich o a roaing frame of reference where he spin-laice ineracion erm is isolaed from he saic par of Ĥ. 4

5 The Ineracion Frame Le σ! = e iĥ σ H! ( ) = e iĥ Ĥ Roaion abou he Ĥ axis in Liouville space H1! = e iĥ Ĥ 1, hen d σ d! = d d eiĥ σ which simplifies o: = iĥ e iĥ σ + e iĥ d σ d, d σ d! = i H1! σ! (see homework) Hence in his frame of reference, known as he ineracion frame, where he ime dependence of he densiy operaor depends only on H 1!. 5

6 Redfield heory Sep 1. Le s sar by formally inegraing d σ d! = i H1! σ!. σ! = σ! = σ! = σ! i H1 " ( " ) σ! ( " )d! i H1 " ( " ) σ! d " H1 " ( " ) H1! (!! ) σ! (!! )d!! d! i H1! (! ) σ!!!!! d! H1!! H1!! σ d!! d! + i!!! This is saring o ge ugly, so we ll jus keep he firs 3 erms (o be jusified laer). 6

7 Redfield heory (con.) Sep 2 is o ake he ensemble average (echnically his is an ensemble of ensembles), and noe.. H1! (! ) σ! d " = because H1! This leads o. σ! σ!! = = H1! (! ) H1! (!! ) σ! H1! and we ve assumed H1! and σ! are uncorrelaed. d!! d! If, we can always incorporae he non-zero par ino Ĥ. Technically, from here on we should be using σ! jus going o go wih he simpler noaion of σ!. bu we re 7

8 Redfield heory (con.) Sep 3. Choose = Δ very small, such ha σ! Δσ " = σ! ( Δ) σ" Δ! = H1 " ( " ) H1! (!! ) σ! Sep 4. Define a new variable, τ =!!! d!! d! σ! This assumpion is he one ha allows us o drop hose higher order erms. In essence, σ! is assume o vary slowly in ime as compared o H1!. Δσ " = σ! ( Δ) σ" Δ! = H1 " ( " ) H1! (! τ ) σ! dτd # 8

9 Redfield heory (con.) Sep 5. Inroduce a correlaion superoperaor: G τ = H1! (! ) H1! (! τ ) Sep 6. Make some more assumpions. Δσ" Δ Δ = Ĝ τ σ" dτ Assume Δ sufficienly small ha Δ σ" Δ d σ" d, bu ha Δ is sufficienly large ha we can exend he inegraion o. dσ! d = Ĝ τ σ! dτ Can we acually find such a Δ?!? 9

10 Redfield heory assumpions Le s look a some numbers Consider Δ = 1-6 s. Typically NMR relaxaion imes are on he order of milliseconds o seconds, hence Δ σ" d σ" Δ d. Tissue waer correlaion imes due o molecular umbling are on he order of 1-9 s. Hence G Δ for Δ = 1-6 s. In general, Redfield heory is valid for relaxaion imes several orders of magniude longer ha he correlaions imes driving he relaxaion processes, e.g. liquids. 1

11 Redfield heory (con.) Sep 7. We need o find an explici expression for Ĥ 1 H1! = e iĥ Ĥ 1. The bes choice is o express eigenoperaors of H. Ĥ 1 as a linear combinaion of eigenvalue Le  q be an eigenoperaors of Ĥ : Ĥ  q = e q  q Examples. Le Ĥ = ω I Î z ω S Ŝ z hen Î + = Î x + iî y H Î + = Ĥ ( Î x + iî ) y = ω I iî ω y IÎ x = ω IÎ+ Î + Ŝ + H Î + Ŝ + = ω I +ω S (See Problem Se 1 Î+Ŝ+ for more examples) 11

12 Redfield heory (con.) Hence, le Ĥ1 = F q q  q Eigenoperaor Random funcions of ime (ypically dependen on molecular orienaion) We firs noe, ha for Ĥ 1 o represen a physical process, Ĥ 1 mus be Hermiian. However, he F q Âqs can be complex. Thus, for every F q Âq he sum mus also conain a erm F * q Âq *, which, we will denoe as F q  q. 12

13 Redfield heory (con.) Noing Ĥ  q = e q  q e iĥ  q = e ie q  q We have H1! = e iĥ Ĥ 1 q = F q  q e ie q We can now wrie he full expression for he correlaion superoperaor as G τ = H1! (! ) H1! (! τ ) = F p! F q (! τ )  p  q e ie p! p q e ie q! τ Using a secular approximaion, one can show ha only he erms for which e p = -e q need o be kep. The oher erms average ou as hey oscillae fas as compared o he relaxaion rae of σ!. 13

14 Thus Ĝ τ Redfield heory (con.) = F q " q F q ( " τ )  q  q e ie qτ Defining a se of correlaion funcions: G q ( τ ) = F q ( " )F q " τ yields d σ! d = G q ( τ )e ie qτ  q  q σ! dτ. q Sep 9. Define a se of specral densiy funcions: J ( q e ) q = G q τ e ie qτ dτ, wih G q ( τ ) = G q e τ τ c Correlaion ime characerisic of he perurbaion. 14

15 Redfield heory (con.) The specral densiy funcions are: J ( q e ) q = G q τ e ie qτ dτ, As defined, J q s, are complex, however, in pracice he real par is much larger han he imaginary componen *. J q ( ω) = G q ( τ )e iωτ dτ G q ( τ )cosω q τ = G q wih G q ( τ ) = G q e τ τ c A more formal reamen shows he imaginary componens basically cancel due o he combinaion of he erms: A q  q and  q  q hus we ll use.. Correlaion ime characerisic of he perurbaion. τ c 1+ω 2 τ c 2. * If you re curious abou hese imaginary erms, look up dynamic frequency shifs in he NMR lieraure. 15

16 The Relaxaion Superoperaor Puing i all ogeher, we can now define he relaxaion superoperaor as: Γ = J ( q e ) q  q  q q wih J ( q e ) q = F q " F q ( " τ )e ie qτ dτ Subsiuing: dσ! d = Γ σ! We can now add he Bolzmann correcion and finally swich back o he laboraory frame * (see homework) o yield: dσ d = iĥ σ Γ σ ( σ B ) The Maser Equaion of NMR * Noe he relaxaion superoperaor is he same is boh frames of reference, as befis relaxaion imes. 16

17 Nex Lecure: Redfield heory II 17