Relaxation and Dynamics

The classical equations describing the precession of the induced magnetization in the magnetic field,
is given by the expression:
M
 B  M
t
which can be expressed as the set of equations:




M x
  B y M z  Bz M y
t
M x
  B x M z  B z M x 
t
M x
  Bx M y  ByM x
t
where the magnetic field defines the z-axis, implying that:
B  (0, 0, B o )

The system of equation reduces to:
M x
 B o M y ;
t

and
M z
0
t
M   M x  iM y
The system of equation becomes:
M 
 Bo M  ;
t

t
 B o M x ;
Changing the representation where:
M   M x  iM y

M y
M 
 Bo M - ;
t
M z
0
t
These are now three independent equation. Assuming that Bo = the solution are the familiar
equations:
M  (t )  M o exp( i o t);
M  (t )  M o- exp( i o t);
M z (t )  M oz
these equations illustrate the M+ is the positive rotating component of the magnetization, M- in the
negative rotating component and that Mz remains unchanged with time.

We know that both the transverse and longitudinal component of the magnetization relax at rates
determined by the spin-spin and spin-lattice relaxation times, T2 and T1, respectively. As a
consequence the system of equations above has to be modified to include relaxation as follows:
M x
1
 B o M y 
Mx
t
T2
M y
t
 B o M x 
M 
1
 B o M  
M
t
T2
1
My
T2
M 
1
 B o M - 
M
t
T2


M z
1
  Mz
t
T1
M z
1
  Mz
t
T1
which have the familiar solutions:

1 
t
M  (t )  M o exp  i o T2 


1 
 t
M  (t )  M o- exp   i o T
2




 t 
M z (t )  M eq  M oz  M eq exp  - 
 T1 




which show that the transverse magnetization decay as 1/T2, and the z-magnetization as 1/T1. The
corresponding Fourier transform gives the characteristic absorption and dispersion line shape at the
resonance frequency .
Relaxation need not only occur to the lattice or other spins in an incoherent way. It is possible to
have relaxation between nuclei that is coherent, referred to a cross-relaxation, which must be treated
like kinetics between in the magnetizations.
When the x-populations of a spin system are not at equilibrium, either through population inversion,
selective inversion or by selective irradiation, (i.e. decoupling), the populations will attempt to reestablish a equilibrium trough the single, double and zero quantum transitions.
In the presence of persistent radiation a new equilibrium is achieved. In the case of a heteronuclear
spin system, such as carbon-proton, when the proton transitions are stimulated constantly by a
decoupler pulse, the equilibrium population are perturbed to such an extent that population
differences corresponding to carbon transition is greatly increase thus enhancing the carbon signal.
This is known as the Nuclear Overhauser Effect.
This effect was widely exploited before the onset of 2D NMR, in making conformational
determination, since the NOE enhancement depends on the internuclear distance between the nuclei
involved. Thus by irradiating the signal of a nucleus in question and by observing an enhancement
of another nucleus it was possible to identify proximal partners, and rough estimates of the
internuclear distances.
Steady State NOE Difference Spectra

Cross relaxation effects can be included in modeling the system by including the appropriate kinetic
terms, i.e. a decaying term from the source nucleus and a build up term on the target nucleus. In this
case the transverse magnetizations are not considered and an equations for at least two nuclei are
required, as follows:
M1z
1
  12 M1z   21 M 2 z  M1z
t
T1
M 2 z
1
  12 M1z   21 M 2 z  M 2 z
t
T1

Which can be expressed in matrix form as:
1

  12 

M
  1z 
T1



t M 2 z   
12



  M 1z 


1 
  21   M 2 z 
T1 
 21
In general this matrix must be diagonalized to solve for the new z-magnetizations, the solution will
be of the form:
M1z  A1a (t)M 1z  A1c (t)M 2 z
M 2 z  A2c (t)M 1z  A2a (t)M 2 z


The coefficients thus depend on the cross relaxation, and spin-lattice relaxation rates of each nucleus
involved. The coefficients indicate whether they are auto-correlation or cross-correlations.
The analogous quantum mechanical expressions are:
I1z  A1a (t) I1z  A1c (t) I 2 z
I 2 z  A2c (t) I1z  A2a (t) I 2 z
in this case we are considering cross correlation between two one-spin systems, thus there is no
direct product required in this case. In most cases, over most mixing times, the cross-correlation
coefficient is negative. The auto-correlation coefficients are always positive.

The NOESY pulse sequence has correlation based on cross-relaxation relaxation. Thus it will start
by preparing a frequency labeled z-magnetization, which will be allowed to cross relax. They a
subsequently made into transverse magnetization and observed.
The NOESY Pulse Sequence
90
90x
Evolution
90
Mixing
mix
t1
Acquire
t1
1) Start by considering two one-spin systems at equilibrium as:
I1z  I 2 z
2) Applying the 900x pulse gives:
 I1y  I 2 y
3) Allowing the systems to evolve during t1 as:
 I1y cos(2v1t1 )  I1x sin( 2v1t1 )  I 2 y cos(2v2t1 )  I 2 x sin( 2v2t1 )
4) The second pulse is typically a 90-x pulse:
I1z cos(2v1t1 )  I1x sin( 2v1t1 )  I 2 z cos(2v2t1 )  I 2 x sin( 2v2t1 )
since the second pulse phase is designed to remove any transverse signal from this part of the
sequence the second and fourth terms are dropped.
I1z cos(2v1t1 )  I 2 z cos(2v2t1 )
5) during the mixing time cross relaxation and spin-lattice relaxation occurs according to the
treatment above giving solutions:
A1a ( )I1z  A1c ( )I 2 z cos(2v1t1 )  A2c ( )I1z  A2a (I 2 z cos(2v2t1 )
which can be rearranged as:
A1a ( ) cos(2v1t1 )  A2c ( ) cos(2v2t1 )I1z  A1c ( ) cos(2v1t1 )  A2a ( ) cos(2v2t1 ))I 2 z
6) The final 900 pulse rotates these z-magnetization back to the transverse plane, lets consider a
900y pulse:
A1a ( ) cos(2v1t1 )  A2c ( ) cos(2v2t1 )I1x  A1c ( ) cos(2v1t1 )  A2a ( ) cos(2v2t1 ))I 2 x
7) The density matrix is allowed to evolve under the chemical shift tersm whilst being detected:
A1a ( ) cos(2v1t1 )  A2c ( ) cos(2v2t1 )I1x cos(2v1t 2 )  I1y sin( 2v1t 2 )

 A1c ( ) cos( 2v1t1 )  A2a ( ) cos( 2v 2 t1 ))  I 2 x cos( 2v 2 t 2 )  I1 y sin( 2v 2 t 2 )

assuming a receiver phase along the x-axis the signal becomes:
A1a ( ) cos(2v1t1 )  A2c ( ) cos(2v2t1 )exp i 2v1t 2 
 A1c ( ) cos( 2v1t1 )  A2a ( ) cos( 2v2 t1 )) exp i 2v2 t 2 
upon collection of a sin modulated data set, the resulting 2D spectrum contain both absorption
diagonal and cross-paks that can be phase and the sign is known.