Summary of Transformation on the density matrix


Rotations about a given axis is represented by the exponential operator with a phase corresponding
to the axis.
x  axis :
y  axis :
R x ( )  exp( iI x )
R y ( )  exp( iI y )
z  axis :
zz  axis :
R z ( )  exp( iI z )
R zz ( )  exp( iI z  I z )
Pulse along the x-axis through   B1t :
I x  R x ( )I x R x1 ( )  I x
I y  R x ( )I y R x1 ( )  I y cos  I z sin 
I z  R x ( )I z R x1 ( )  I z cos  I y sin 

Pulse along the y-axis through   B1t :
I x  R y ( )I x R y1 ( )  I x cos  I z sin 
I y  R y ( )I y R y1 ( )  I y
I z  R y ( )I z R y1 ( )  I z cos  I x sin 

Free precession about the z-axis at a frequency of   B0 :
I x  R z (t )I x R z 1 (t )  I x cos t  I y sin t
I y  R z (t )I y R z 1 (t )  I y cos t  I x sin t
I z  R z (t )I z R z 1 (t )  I z

Free precession due to J-coupling at a frequency of   2J :
Ix E 
R zz (t )I x  ER zz1 (t )  I x  E cos t  2I y  I z sin t
I x  Iz 
R zz (t )I x  I z R zz1 (t )  I x  I z cos t  0.5I y  E sin t
Iy E 
R zz (t )I y  ER zz1 (t )  I y  E cos t  2I x  I z sin t
Iy Iz 
R zz (t )I y  I z R zz1 (t )  I y  I z cos t  0.5I x  E sin t
Iz 
R zz (t )I z R zz1 (t )  I z
Spin Dynamics for a Two-spin system during the COSY Experiment.

Starting with the equilibrium density matrix of a two-spin system, reflecting the z-magnetization of
the two nuclei as follows:
Iz  E  E  Iz

After the initial 900x excitation pulse the density matrix becomes:
 Iy E  EIy

Allowing the system to evolve under the chemical shift of spin 1 we get:


 I y cos1t1  I x sin 1t1  E  E  I y

Now letting the system evolve under the chemical shift of the second nucleus:



 I y cos1t1  I x sin 1t1  E  E  I y cos2t1  I x sin 2t1

which can be simplified as:
 I y  E cos1t1  I x  E sin 1t1  E  I y cos2t1  E  I x sin 2t1

These are all single quantum coherences that will evolve between detectable and undetectable terms
under the influence of the J coupling term.

At this stage, in the interest of simplicity, the terms involving spin 2 will be dropped since the
treatment is identical. Thus the density matrix under consideration is:
 I y  E cos1t1  I x  E sin 1t1

Now applying scalar coupling evolution the density matrix becomes:




 I y  E cosJt1  2I x  I z sin Jt1 cos1t1  I x  E cosJt1  2I y  I z sin Jt sin 1t1
which simplifies to:
 I y  E cosJt1 cos1t1  2I x  I z sin Jt1 cos1t1  I x  E cosJt1 sin 1t1  2I y  I z sin Jt sin 1t

The amplitude factors can be repressed as periodic functions of sums and differences of the angular
frequencies as:
1
 I y  Ecos1  J t1  cos1  J t1   I x  I z sin 1  J t1  sin 1  J t1 
2
1
 I x  Esin 1  J t1  sin 1  J t1   I y  I z cos1  J t1  cos1  J t1 
2

Factoring out 2, thus working in frequency units we get:


1
J
J 
J
J 




 I y  Ecos 2 1  t1  cos 2 1  t1   I x  I z sin 2 1  t1  sin 2 1  t1 
2
2
2 
2
2 








1
J
J 
J
J 




 I x  Esin 2 1  t1  sin 2 1  t1   I y  I z cos 2 1  t1  cos 2 1  t1 
2
2
2 
2
2 







The above expression says that the observable term,
Iy E
has amplitude factor
J
J
J
J




cos 2 1   + cos 2 1   and I x  E has factor sin 2 1   + sin 2 1   . The first
2
2
2
2




term, if detected about the y-axis would give a signal composed of two in-phase doublets, split by J,
centered at +1 and -1. The second term if detected about the x-axis would give a signal composed
of an anti-phase doublet centered about +1 and -1, split by J.
Splitting patterns in the frequency domain due to amplitude factors in the indirect
time dimension
cos(Jcos(J


sin(Jsin(J


cos(Jcos(J


sin(Jsin(J



Now lets apply the final 900x pulse:


1
J
J 
J
J 




 I z  Ecos 2 1  t1  cos 2 1  t1   I x  I y sin 2 1  t1  sin 2 1  t1 
2
2
2 
2
2 








1
J
J 
J
J 




 I x  Esin 2 1  t1  sin 2 1  t1   I z  I y cos 2 1  t1  cos 2 1  t1 
2
2
2 
2
2 






i)
ii)
iii)
iv)
The first term is not observable and does not evolve into an observable terms and thus will be
ignored.
The second terms is a double quantum coherence and does not evolve into an observable
term.
The third term is observable.
The last term is not observable but evolves into an observable term under the scalar coupling
term.
In other formulations of the COSY experiment additional pulses can be employed to make the other
terms observable. This is done in the double quantum filtered COSY where the fourth term is
converted to an observable term. This you will be doing in your assignment.

At this stage the first and second terms are dropped and the calculation continues with just the third
and fourth terms, which are single quantum coherences, as:


1
J
J 
J
J 




 I x  Esin 2 1  t1  sin 2 1  t1   I z  I y cos 2 1  t1  cos 2 1  t1 
2
2
2 
2
2 







Notice that the second term will evolve into a term observable on spin 2, with frequency 2, but has a
phase factor with a time dependence of 1, thus it will give rise to a cross peak. The first term is
observable on spin 1 and has time of 1 in its phase factor thus it will give rise to an autocorrelation
peak on the main diagonal.

Lets consider evolution on spin 1:


1
I x  E cos 21t2  I y  E sin 21t2
2
sin 2 1  J2 t1  sin 2 1  J2 t1




 

J
J 


I z  I y cos 2 1  t1  cos 2 1  t1 
2
2 




Continuing with evolution on spin 2:


1
I x  E cos 21t2  I y  E sin 21t2
2

sin 2 1  J2 t1  sin 2 1  J2 t1






 

J
J 


 I z  I y cos 22t2  I z  I x sin 22t2 cos 2 1  t1  cos 2 1  t1 
2
2 



Now considering scalar evolution:


1  I x  E cos Jt2  2I y  I z sin Jt2 cos 21t2  
J
J 


 
 sin 2 1  t1  sin 2 1  t1 
2  I y  E cos Jt2  2I x  I z sin Jt2 sin 21t2  
2
2 






 I z  I y cos Jt2  0.5I x  E sin Jt2 cos 22t2  
J
J 



 cos 2 1  t1  cos 2 1  t1 
2
2 


 I z  I x cos Jt2  0.5I y  E sin Jt2 sin 22t2  


which simplifies to:
0.5I x  E cos 21t2 cos Jt2  I y  I z cos 21t2 sin Jt2  


 sin 2 1 

 0.5I y  E sin 21t2 cos Jt2  I x  I z sin 21t2 sin Jt2  
I z  I y cos 22t2 cos Jt2  0.5I x  E cos 22t2 sin Jt2  


 cos 2 1 

 I z  I x sin 22t2 cos Jt2  0.5I y  E sin 22t2 sin Jt  

J
J 

t1  sin 2 1  t1 
2
2 

J
J 

t1  cos 2 1  t1 
2
2 

In terms of periodic functions of sums and differences of the angular velocities this becomes:


J
J  


0.25I x  E cos 2 1  t2  cos 2 1  t2  
2
2  






J
J 


 0.5I y  I z sin 2 1  t2  sin 2 1  t2  
2
2   



J
J 




 sin 2 1  2 t1  sin 2 1  2 t1 



 
 0.25I  E sin 2   J t  sin 2   J t   
y
1
2
1
2



2
2  






J
J  



 0.5I x  I z cos 2 1  2 t2  cos 2 1  2 t2  



 




J
J  


0.5I z  I y cos 2  2  t2  cos 2  2  t2  
2
2  






J
J  


 0.25I x  E sin 2  2  t2  sin 2  2  t2  
2
2   



J
J 




 cos 2 1  2 t1  cos 2 1  2 t1 



 
 0.5I  I sin 2   J t  sin 2   J t   
z
x
2
2
2
2



2
2  






J
J  



 0.25I y  E cos 2  2  2 t2  cos 2  2  2 t2  



 



The first term is an auto-correlation peak, present on the main diagonal, and can be simplified to,
considering only the detectable terms:

J
J  
J
J 




0.25I x  Esin 2 1  t1  sin 2 1  t1  cos 2 1  t2  cos 2 1  t2 
2
2  
2
2 






J
J  
J
J 




 0.25I y  Esin 2 1  t1  sin 2 1  t1  sin 2 1  t2  sin 2 1  t2 
2
2  
2
2 





Assuming simultaneous detection in the direct frequency domain, t2, with receiver phase x. The
signal can be represented as:


J
J


cos 2 1  2 t2  i sin 2 1  2 t2 

J
J 







0.25sin 2 1  t1  sin 2 1  t1  

2
2
J
J 



 



 cos 2 1  t2  i sin 2 1  t2 
2
2 



which is:


J
J   
J 
J  




0.25sin 2 1  t1  sin 2 1  t1  exp  i 2 1  t2   exp  i 2 1  t2 
2
2   
2 
2  






Applying a Fourier transform in just the direct domain there is an in phase doublet centered at 1,
with a time dependence in the indirect dimension of sin 2 1  0.5J t1  sin 2 1  0.5J t1, which
with a non-phase sensitive Fourier transformation gives an anti-phase doublet of in-phase doublets.
Auto-correlation signal on the main diagonal from the basic COSY pulse sequence.

sin(Jt1sin(Jt1
f1



f2

The second term is an cross-peak, present off the main diagonal, and can be simplified to,
considering only the detectable terms:

J
J  
J
J 




 0.25I x  Ecos 2 1  t1  cos 2 1  t1  sin 2  2  t2  sin 2  2  t2 
2
2  
2
2 






J
J  
J
J 




 0.25I y  Ecos 2 1  t1  cos 2 1  t1  cos 2  2  t2  cos 2  2  t2 
2
2  
2
2 





Assuming simultaneous detection, with receiver phase x, in the direct frequency domain, t2. The
signal can be represented as:

J
J  


 sin 2  2  t2  i cos 2  2  t2  
2
2  



J
J   


 0.25cos 2 1  t1  cos 2 1  t1  

2
2   


J
J 



 sin 2  2  t2  i cos 2  2  t2 
 
2
2   



which can be shown to be:


J
J   
J 
J  




 0.25i cos 2 1  t1  cos 2 1  t1 exp  i 2  2  t2   exp  i 2  2  t2 
2
2   
2 
2  






Applying a Fourier transform in just the direct domain there is an anti-phase doublet centered at 2,
with a time dependence in the indirect dimension of,  0.25cos 2 1  0.5J t1  cos 2 1  0.5J t1
which with a non-phase sensitive Fourier transformation gives an anti-phase doublet of anti-phase
doublets. (This signal is 900 out of phase with respect to the diagonal signal since it is premultiplied
by i.)
Cross-peak signal off the main diagonal from the basic COSY pulse sequence.

cos(Jt1cos(Jt1
f1



f2

Notice that the auto-correlation signal is purely sine modulated in the indirect domain and the crosspeak signal is purely cosine modulated. One way to remove the signals corresponding to the
negative frequencies is to find a modification of the pulse phases to give complementary data, being
cosine modulated in t1 for the auto-correlation peaks and sine modulated for the cross-peak, since
these. Thus there would be one set of data that is sine modulated which is added to another set that
is cosine modulated for both types of peaks giving signal just at the positive frequency, just like
Quadrature detection in the 1D case.

Changing the pulse phase of the first pulse from x to –y in the basic COSY pulse sequence this can
be achieved, as follows:
Relaxation
90x
90-y
Evolution


Acquire
t1
t2
The signal arises from the terms:
I x  E cos 21t1 cos J1t1  I z  I y cos 21t1 sin J1t1
which occur just after the second pulse, which evolve to giver the observable terms, namely the first
term gives the diagonal peak and the second term gives the cross-peak. Evolution in the second time
domain is neglected. The above expression can be simplified to give:
I x  Ecos 2 1  0.5J t1  cos 2 1  0.5J t1   I z  I y sin 2 1  0.5J t1  sin 2 1  0.5J t1 

The first term gives an in-phase pattern of in-phase doublets in the indirect domain and an in-phase
doublet in the direct domain and is on the main diagonal. The second term give an in-phase doublet
of anti-phase doublets in the indirect domain and an anti-phase doublet in the direct domain, and is
off-the main diagonal. The signal from the cross-peak is 900 out of phase with respect to the diagonal
peak.
Auto-correlation and cross-peak signal from the modified COSY sequence.

cos(Jt1cos(Jt1
f1

sin(Jt1sin(Jt1



f2
f1



f2
The Double Quantum Filtered COSY Pulse Sequence.

2
 
2 2 




t1
1
receiver
2
t2
3 4 5
The Phase Cycle for the DQF-COSY Pulse Sequence

Scan



reciever
1
x
x
x
x
2
x
x
y
-y
3
x
x
-x
-x
4
x
x
-y
y
Starting with the equilibrium density matrix of a two-spin system, reflecting the z-magnetization of
the two nuclei as follows:
Iz  E  E  Iz

After the initial 900x excitation pulse the density matrix becomes:
 Iy E  EIy

Allowing the system to evolve under the chemical shift of spin 1 we get:


 I y A(1 , t1 )  I x A(1 , t1 )  E  E  I y

Now letting the system evolve under the chemical shift of the second nucleus:



 I y A(1 , t1 )  I x A(1 , t1 )  E  E  I y A( 2 , t1 )  I x A( 2 , t1 )
which can be simplified as:

 I y  EA(1 , t1 )  I x  EA(1 , t1 )  E  I y A( 2 , t1 )  E  I x A( 2 , t1 )1

These are all single quantum coherences that will evolve between detectable and undetectable terms
under the influence of the J coupling term.

At this stage, in the interest of simplicity, the terms involving spin 2 will be dropped since the
treatment is identical. Thus the density matrix under consideration is:
 I y  EA(1 , t1 )  I x  EA(1 , t1 )

Now applying scalar coupling evolution the density matrix becomes:

 
 I y  EA(1 , J , t1 )  2I x  I z A(1 , J , t1 )  I x  EA(1 , J , t1 )  2I y  I z A(1 , J , t1 )
