Relaxation in NMR Source of relaxation in NMR is fluctuation of nuclear interaction with respect to the external magnetic field. Dipolar interaction: N H B 0 CSA interaction: ! 11 ! 22 ! 33 B 0 Other interactions: Quadrupolar, Paramagnetic, etc. Relaxation in NMR Where does the fluctuation come from? •Random events: Rotational diffusion Translational diffusion Vibrational/Librational motions Conformational sampling/variability •Non-random events: Magic angle spinning B 0 quenching Correlation function The behavior of the random fluctuation can be described by a correlation function. What is a correlation function: 1. Correlation function of 2 variables is the expected value of their product. This could be evaluated as a function of space or time. In liquid NMR random processes are typically defined as a function of time. 2. It is a measure of how quickly the two variables change as a function of time or space. Correlation function How does it relate to NMR relaxation? Remember the master equation: Taking the ensemble average, first term vanishes (H 1 (t) is random): d ˜ "(t) dt = #i ˜ H i (t), ˜ "(0) [ ] # ˜ H 1 (t ), ˜ H 1 (t'), ˜ " (t ') [ ] [ ] 0 t $ dt' d ˜ " (t) dt = # ˜ H 1 (t), ˜ H 1 (t'), ˜ "(t') [ ] [ ] 0 t $ dt ' Expand H 1 (t) in spin and time dependent operators: H 1 (t) = V " F a " # (t) = V " + F " * " # (t) spin random function of time
12
Embed
Relaxation in NMRnmr/workshops/ws_nmr_09/tjandra/TIFR_corr... · Relaxation in NMR Source of relaxation in NMR is fluctuation of nuclear interaction with respect to the external
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Relaxation in NMR
Source of relaxation in NMR is fluctuation of nuclear
interaction with respect to the external magnetic field.
Dipolar interaction:!
N
HB0
CSA interaction:!
!11
!22
!33
B0
Other interactions: Quadrupolar, Paramagnetic, etc.
Relaxation in NMR
Where does the fluctuation come from?
•Random events:
! Rotational diffusion
! Translational diffusion
! Vibrational/Librational motions
! Conformational sampling/variability
•Non-random events:
! Magic angle spinning
! B0 quenching
!
Correlation function
The behavior of the random fluctuation can be described by a
correlation function.
What is a correlation function:
1. Correlation function of 2 variables is the expected value of
their product. This could be evaluated as a function of space or
time.
In liquid NMR random processes are typically defined as a
function of time.
2. It is a measure of how quickly the two variables change as a
function of time or space.
Correlation function
How does it relate to NMR relaxation?
Remember the master equation:
Taking the ensemble average, first term vanishes (H1(t) is random):
!
d ˜ " (t)
dt= #i ˜ H
i(t), ˜ " (0)[ ] # ˜ H
1(t), ˜ H
1(t'), ˜ " (t ')[ ][ ]
0
t
$ dt'
!
d ˜ " (t)
dt= # ˜ H
1(t), ˜ H
1(t'), ˜ " (t')[ ][ ]
0
t
$ dt '
Expand H1(t) in spin and time dependent operators:
!
H1(t) = V"Fa
"
# (t) = V"+F"*
"
# (t)
spin random function of time
Correlation function
How does it relate to NMR relaxation?
Substituting back:
!
d ˜ " (t)
dt= # ˜ V $ (t), ˜ V %
+(t'), ˜ " (t ')[ ][ ]F$ (t)F%
*(t ')dt '
0
t
&$,%
'
!
F" (t)F#*(t') =G"# ( t $ t' ) Correlation function
It describes the random function of time that contributes to changes in the spin density
Correlation function
How quickly the NH dipole changes as a function of time with
respect to the external magnetic field.
!
C(t) = Dq02( )*
"LF 0( )( )•Dq02( )"LF t( )( )
LF=Laboratory frame, B0 is static
In liquid it reduces to:
!
C(t) =1
5P2
r µ LF0( )•
r µ LF
t( )( )N
H
!
r µ
!
C(t) = d" d# p($,#,")P2
0
%
&0
2%
&0
2%
& (cos$0t )d$
p(",#,$) is the normalized distribution function of µ
Relaxation in NMR
In NMR observed quantities: T1, T2, and NOE are determined
by the Fourier transform of an appropriate time correlation function (the spectral density) evaluated at certain frequencies, thus certain magnetic fields.
C(t)
t
J(%)
%
F.T.
Relaxation in NMR
Motional model
Define correlationfunction
Calculate spectraldensity
Fit experimentaldata
Goodness of fit
Not accepta
ble
acceptable
Done
Approximate acorrelation function
Calculate spectraldensity
Fit experimentaldata
Pick models that can describe the fitted parameters
Lipari-Szabo Model
Free Approach
General approach
Lipari-Szabo Correlation functionConditions:
• Overall and internal motions are independent
• C(t) = Co(t) CI(t)
• CI(t) = &P2(µ(0) • µ(t))'! in the molecular frame.
• CI(0) = &P2(µ(0) • µ(0))' = 1
• CI(() = S2
• Area under the approximated correlation function is the same as the real one: