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
Time Dependent NMR Experiments

Now that you have been introduced to some general concepts of spin-dynamics, as rotations in a
given coordinate system due to some term in the Hamiltonian operator, lets apply these to more
complex NMR experiments.

Most NMR experiments can be separated into 3 events - excitation, evolution, and detection.
i) Excitation or Preparation Period :
Usually a 900 or a 1800 pulse which is phase cycled. Often before excitation the system is
allowed to come to equilibrium, thus a delay of 3 to 7 times T1 is often used.
ii) Evolution Period:
The spin system is allowed to evolve in time under the influence of the Hamiltonian. In some
cases additional r.f radiation is applied to remove certain terms from the Hamiltonian such as is
during heteronuclear decoupling. For 2-Dimensional experiments this period of evolution is
varied and represent the indirect time dimension.
iii) Detection Period:
The signal is detected in quadrature, where often several scans are collected representing a whole
number multiple of the minimum all-over phase cycle required for the experiment. For 2dimenasional experiments this is referred to as the direct time domain. The detection period is
often initiated by a pulse to convert terms in the density matrix to those that the detected is
sensitive to. Often this pulse phase is phase cycled to suppress artifacts due to quadrature images
ant the emergence of unwanted terns from the final density matrix.
Figure 8
The General Time Dependent NMR Experiment.
Relaxation
Delay
Preparation
Evolution of under
the influence of H
Evolution
Aquisition
Time
Detection
The Spin Echo for One-Spin-1/2 System
90x
Relaxation
delay
180y


Acquire
 The system is allowed to come to equilibrium before the excitation pulse. Considering one spin, the
equilibrium density matrix is Iz.
Equilibrium Magnetization
z
x'
 The 90x pulse rotates Iz along the x-axis towards the y-axis giving rise to -Iy.
Magnetization after a 90x pulse
z
y'
x'

Once in the transverse plane as -Iy the magnetization will evolve according to:
 I y  I y cos t   I x sin  t  , where is the chemical shift.

After an evolution period  the density matrix is:
 ( )  I y cos    I x sin  
Magnetization precessing in the XY plane
at a rate of  for a period 


y'
x'

The 180y pulse converts Ix to -Ix and leaves -Iy unchanged giving:  I y cos    I x sin  
Application of a 180y pulse flips the magnetization
to the other side of the Y-axis
B1
y'
x'


y'
x'
The x and y magnetization evolve according to the expressions:
I y  I y cos t   I x sin  t  and I x  I x cos t   I y sin  t  ,
thus the expression for the density matrix becomes:
 (t   )  (I y cos(t )  I x sin t ) cos    (I x cos(t )  I y sin t ) sin  
which after another evolution period of  becomes:
 (2 )  (I y cos( )  I x sin  ) cos    (I x cos( )  I y sin  ) sin  


 I y  cos 2 ( )  sin 2    I x sin   cos    cos( ) sin  
 I y

Therefore after the second evolution period after the 180y pulse the density matrix becomes purely
-Iy, which means that the magnetization has refocused along the -y- axis.
After a second evolution period , at a rate , the
magnetization refocuses along the –Y-axis


y'
y'
x'
x'
The Spin Echo for a Two-Spin-1/2 System

Starting at equilibrium with the density matrix: I z  E  E  I z .
The 90x rotates the Z magnetization to the –Y axis
(Keeping track of just one spin which has two z-magnetizations
corresponding to the state of the other spin)

z
z


y'

x'
y'
x'

The 90x pulse converts the equilibrium density matrix to:  I y  E  E  I y

The Hamiltonian for during free evolution is the weakly coupled two spin Hamiltonian given by the
expression:
H  1I z  E  2E  I z  JI z  I z
the corresponding propagator is:
U  exp i1I z  E  i2E  I z  i2JI z  I z 
which can be separated into three propagators applied in succession:
U  U1U 2 U3  exp  i 2JI z  I z exp  i 2E  I z exp  i1I z  E

U3 involves simple chemical shift evolution for spin 1 according to:
I y  I y cos t   I x sin  t 
where  is the chemical shift, giving:


 I y cos 1t  I x sin 1t  E  E  I y

U2 involves simple chemical shift evolution for spin 2 as for spin 1, giving:



 I y cos 1t  I x sin 1t  E  E  I y cos 1t  I x sin 1t

which can be simplified to:
 I y  E cos 1t  I x  E sin 1t  E  I y cos  2t  E  I x sin  2t
Free presession of the Z-Magnetizations of spin 1 under the
influence of chemical shift and scalar coupling
J

J

y'
x'

Lets follow the calculation from this point on first two terms describing spin 1 only. The effect of
evolution under spin-spin coupling is:




 I y  E cos Jt  2I x  I z sin Jt cos 1t  I x  E cos Jt  2I y  I z sin Jt sin 1t

The 180y pulse converts Ix to -Ix,, Iz to -Iz and leaves Iy unchanged giving:




 I y  E cos Jt  2I x  I z sin Jt cos 1t   I x  E cos Jt  2I y  I z sin Jt sin 1t
The 180y pulse flips both magnetization across the Y axis
and changes the state of the second nucleus to which the
transition can be attributed.
J

J

B1
J
y'
y'


x'

J
x'
Which is allowed to evolve for another period t under the chemical shift Hamiltonian as:

 
 


 I y cos 1t  I x sin 1t  E cos Jt


 cos 1t
 2 I x cos 1t  I y sin 1t  I z sin Jt 
 I x cos 1t  I y sin 1t  E cos Jt 

 sin 1t
 2 I y cos 1t  I x sin 1t  I z sin Jt 

 
 
which is simplifies to:



  I y  E cos 2 1t  I x  E sin 1t cos 1t cos Jt 


 2I  I cos 2  t  2I  I sin  t cos  t sin Jt 
z
1
y
z
1
1
 x





  I x  E cos 1t sin 1t  I y  E sin 2 1t cos Jt



  2I  I cos  t sin  t  2I  I sin 2  t sin Jt 
y
z
1
1
x
z
1



collecting terms and further simplifying:
 I  E cos  t  I  E sin  t cos  t  I  E cos  t sin  t  I  E sin  t cos Jt
 2I  I cos  t  2I  I sin  t cos  t  2I  I cos  t sin  t  2I  I sin  t sin Jt
2
y
x
z
1
2
x
1
1
y
z
1
1
x
1
y
1
1
z
1
2
y
1
x
1
z
2
1
which finally gives
 I y  E cos Jt  2I x  I z sin Jt
this means that the chemical shift terms have disappeared which implied that they have been refocused.
Free precession for a second period  under the influence of chemical
shift and scalar coupling evolution result in the chemical shift
terms in refocusing but the coupling terms does not.

J
J
y'

y'
J
J


J
J

x'
x'
Now consider the evolution under the J coupling Hamiltonian:




 I y  E cos Jt  2I x  I z sin Jt cos Jt  2 I x  I z cos Jt  0.5I y  E sin Jt sin Jt
which simplifies to:
 I y  E cos 2 Jt  2I x  I z sin Jt cos Jt  2I x  I z cos Jt sin Jt  I y  E sin 2 Jt
giving the final expression:
 I y  E cos 2Jt  2I x  I z sin 2Jt
which implies that under the scalar coupling evolution the system does not refocus unless the delay
time is set to 1/J.

In some experiment the delay time is set to 1/4J which gives the pure antiphase term, 2I x  I z .

A similar expression exists for spin 2. The complete expression is therefore:


 I y  E  E  I y cos 2Jt  2I x  I z  I z  I x sin 2Jt
Heteronuclear Two-spin Systems

When considering multiple spin systems that include nuclei of different type one has to consider
difference in the population differences. For example other words the population difference for
hydrogen nucleus in a magnetic field is four times greater than that of a carbon. This population
difference is determined by the Boltzmann distribution which related it to the corresponding energy
difference. The z-magnetization of a nucleus with  is:
Mz 
E Bo

kT
kT
Thus the operator corresponding to Mz should be scaled accordingly thus strictly speaking the
equilibrium density matrix should be written as:
Bo
 eq 
Iz
kT

The equilibrium density matrix for a heteronuclear two-spin system is therefore:
 eq 
 1Bo
 B
Iz E  2 o EIz
kT
kT
which can be rewritten as:
 eq 
 1Bo
kT


2
EIz 
I z  E 
1


The scalar in front is often neglected thus spin dynamics calculation often assume an initial density
matrix of the form:
 eq  I z  E 
2
EIz
1
Vector diagrams corresponding to the equilibrium density matrix
of an IS spin system, where S is a weak nucleus.
Iz X E
H
z
E X Sz
C
y
x
z
H
C
y
y
x
z
x
z
y
x
Carbon Spectroscopy and Spectral Editing.

Next to proton the carbon nucleus is probably the most popular nucleus for investigation by NMR.

It main limitation is that its signal is very weak in natural abundance samples due to the fact that it is
both a rare and a weak nucleus at the same time.

C is only 1.1 % natural abundant and its geomagnetic ratio is about 25% that of proton which
means that it takes approximately 6400 scans for carbon to be equivalent to one scan on proton.

To make matters even worse is that 13C nuclei also have long T1’s meaning that the repetition time
are very long thus limiting the number of scans that can be collected over a given period.

Coupling to the normally abundant protons it its environment distributed the signal over a large
number of line in the complicated coupling patterns. Thus most often proton decoupling is
employed such that only on line is present in the sample for each carbon. In this case the carbon
signal is also enhanced due to the nuclear Overhauser effect, which arises when irradiating the
proton with the decoupling r.f field. This will be discussed further later in the course.

Another way to enhance the signal on carbon and to shorten its relaxation time is to transfer
polarization from the protons to the carbon. IN this case the carbon signal is enhanced four times
and attains the proton relaxation times which are often shorter than that of carbon, thus more scans
can be collected over a given period, further increasing the ultimate signal to noise ratio.

A pulse sequence that uses polarization transfer from H to C is known as Insensitive Nuclei
Enhancement by Polarization Transfer, INEPT, shown below.
13
The INEPT pulse Sequence
x
2
H
y
x
2


x
x
C


2

Acquire
Notice that up to the final 900pulses the inept pulse sequence is the same as the spin echo pulse
sequence in the proton channel.

Since this is a heteronuclear system the calculations will have to start with the density matrix:
eq  I z  E 

C
E  Sz
H
After a 900x pulse on the proton channel the density matrix becomes:

 I y  E  C E  Sz
H
The vector diagram for product operators after the initial 900x pulse on proton
-Iy X E
z
H
E X Sz
z
C
y
x

z
H
C
z
y
y
x
x
y
x
The first term in the density matrix evolves just as in the spin echo pulse sequence the second term
in just inverted by the 1800x pulse in the carbon channel giving the density matrix:

 I y  E cos 2Jt  2I x  S z sin 2Jt  c E  S z
h
which when t=1/4J becomes:

2I x  S z  c E  S z
h
The vector diagrams corresponding to the density after the spin echo part of the
pulse sequence
Ix
H
X
z
C
y
x
-E X Sz
Sz
z
H
C
y
y
x
z
x
z
y
x

At this stage a 900y pulse is applied to the proton channel and a 900x pulse to the carbon channel
giving:

2I z  S y  c E  S y
h
The vector diagram for the density matrix after the final 900 pulses
Iz
H
Sy
X
z
C
E
z
y
x

H
Sy
X
z
C
z
y
y
y
x
x
x
Evolving under the influence of the scalar coupling interaction this becomes:




2 I z  S y cos 2Jt  0.5E  S x sin 2Jt  c E  S y cos 2Jt  2I x  S z sin 2Jt
h



 2I z  S y cos 2Jt  E  S x sin 2Jt  c E  S y cos 2Jt  2 c I x  S z sin 2Jt
h
h
The first and last terms are not observable however the last two are.

Vector representation of the density matrix during the detection period
-E
H
X
z
Sx
C
y
x

E
z
H
z
Sy
C
y
y
x
X
x
z
y
x
The  E  S x sin 2Jt term leads to an anitphase doublet at the carbon frequency separated by the

coupling constant. The second term c E  S y cos 2Jt leads to in-phase doublet opposite in sign.
h
The first term arises from what was initially proton magnetization while the second term arises from
carbon and is approximately 25% of the intensity. The final resulting signal is an antiphase doublet
that has a ratio of 3 to –5.
The signal from the anti phase doublet and the in-phase doublet add to give an anti
phase doublet at a 3 to –5 ratio
4

3
4

5

1
1

Notice that is this sequence decoupling is not used since the antiphase doublet will collapse and
destructively interfere with itself resulting in the loss of signal. Only signal arising from the second
terms remains which is just the same intensity as a normal decoupled carbon spectrum and thus
nothing is gained.

Before one can decouple the anti-phase doublet must be refocused as an in-phase doublet, this is
done by appending the INEPT sequence with yet another spin echo sequence. This is known as thr
refocused INEPT.
The refocused INEPT pulse sequence
x
2
H
y
x
y
x
C




x
2
Decouple

Acquire
x
2




Just before the 90opulses along the y-axis in both the carbon and proton channel the density
matrix can be taken from the INEPT pulse sequence just before the final pulses as:

2I x  S z  c E  S z
h
The vector diagrams corresponding to the density just at the first echo of the
refocused INEPT pulse sequence
Ix
H
X
z
-E X Sz
Sz
C
z
y
x

H
z
C
y
y
x
x
z
y
x
After the 90opulses along the y-axis in both the carbon and proton channel the density matrix
becomes:

 2I z  S x  c E  S x
h
The vector diagrams corresponding to the density after the 90oy pulses in the C
and H channel.
-E X Sx
-Iz X Sx
H
z
C
y
x
z
H
C
y
y
x
z
x
z
y
x

This is allowed to evolve under J coupling, the chemical shift evolution is ignore since it is
refocused:




 2 I z  S x cos Jt  0.5E  S y sin Jt  c E  S x cos Jt  2I z  S y sin Jt
h
o
After a 180 x pulse in both channels it becomes:


2I z  S x cos Jt  E  S y sin Jt    c E  S x cos Jt  2I z  S y sin Jt 
h

Another period of J evolution gives:
2I z  S x cos Jt  0.5E  S y sin Jt cos Jt  E  S y cos Jt  2I z  S x sin Jt sin Jt 

 c E  S x cos Jt  2I z  S y sin Jt cos Jt  2I z  S y cos Jt  0.5I z  S y sin Jt sin Jt 

h
which can be simplified to:


2I z  S x cos 2 Jt  sin 2 Jt  E  S y 2 sin Jt cos Jt  




c
E  S x cos 2 Jt  sin 2 Jt  I z  S y 2 sin Jt cos Jt 
h


2I z  S x cos 2Jt  E  S y sin 2Jt  c E  S x cos 2Jt  c I z  S y sin 2Jt
h
h

which at t = 1/4J becomes:

ESy  c Iz Sy
h
The vector diagrams corresponding to the density at the start of the detection
period of the refocused INEPT pulse sequence.
I z X Sy
E X Sy
H
z
C
y
x
z
H
C
y
y
x
z
x
z
y
x

The first term is detectable and experiences and is cosine modulation during the detection period,
and thus will give rise to an in-phase doublet.

The second term is initially undetectable but evolves into a the detectable like the first terms but
experiences sine modulation in the detection period and thus gives rise to an anti-phase doublet.

The total signal is thus a 5 to 3 in-phase doublet at the carbon frequency.
The signal from the in-phase doublet and the anti-phase doublet add to give an in
phase doublet at a 5 to 3 ratio
4
4

5

3

1
1

Upon decoupling this anti-phase doublet self cancels leaving only the contribution from the first
term which originated as a proton z-magnetization. The signal in this case is 4 times as intense as
with ordinary carbon spectroscopy.

Note the signal from this term has a sine dependence on the delay time, as a ratio of the coupling
constant. In other words at t=1/4J the signal is optimum while at t=1/2J it is zero.

Now lets investigate the behavior of the carbon signal from an CH2 group as a function of the echo
time. Recall that the signal from the carbon will be a 1:2:1 triplet centered at the carbon frequency
where the other lines are separated by 2J. The outer two line can thus the thought of as behaving like
a C-H system with a coupling that is twice as large and thus will give rise to carbon signal with a
sine dependency at twice the rate.

The 1:3:3:1 signal from a CH3 group can be though of as arising from two CH groups having an
effective coupling constant of J and 3J/2, the former being three times as intense as the latter. It
dependency on the delay time in units of the coupling constant can be shown to be sincos2

The signal from any quaternary carbons will be invariant with the echo time and will arise solely
from the direct signal from the carbon.

As a consequence of the unique dependency of the signal from different carbon types on the
echo-time for the INEPT pulse sequence one can perform spectral editing. One could select a
echo time in which the signal from the methylene carbons are opposite in phase to the methyl
and methane carbons. i.e methyls and methines, up and methylenes down.
The echo time dependency of the CH and CH2 groups
explained in terms of a vector treatment.
The variations of CH, CH2 and CH3 carbon signals with the echo
time in terms of evolution angle under the scalar coupling term.
The DEPT pulse sequence.
x
y
2
H
y


y

Decouple

Acquire
y
2
C



Another pulse sequence that achieves carbon signal enhancement by polarization transfer is the
Distortionless Enhancement by Polarization transfer, DEPT sequence.

One advantages of this sequence over refocused INEPT is that produces the same multiplet
patterns as with direct carbon detection when decoupling is not used (Hence, “Distortionless”).
Since most commonly proton decoupling is used this advantage is lost.

This pulse sequence is similar to INEPT however instead of an initial 180 oy pulse in the carbon
channel is a 90oy (not a 180oy), and the final pulse in the carbon channel is a 180oy pulse. The
sequence also ends in a refocusing delay just like the refocused INEPT sequence which allows
decoupling.

A complete density matrix treatment will not be given but a brief explanation will suffice:
1) After 90o H pulse the proton magnetization evolves under the scalar coupling term for 1/2J
giving rise to an anti-phase term, I x  S z , just like in the INEPT sequence.
2) The 90 pulse in the carbon channel gives rise to a situation where the proton and carbon
magnetization evolve coherently, which is known as a heteronuclear mutiple quantum
coherence (HMQC), I x  S y . This coherence evolves under both the carbon and proton
shifts, but is not influences by the J coupling term. To remove the effect of the proton
chemical shift evolution it is refocused by the 180ox pulse.
3) The final pulse in the proton channel transfers the HMQC to an antiphase carbon
magnetization, I z  S y which is allowed to refocus to an in-phase detectable term, E  S x ,
before the detection period.

The density matrix at the end of the pulse sequence at the detection time depends on the type of spin
system it is applied to and the angle of the final proton pulse:
1) For as CH group the final density matrix is  sin  E  S y where the optimum angle is 90o.
2) For as CH2 group the final density matrix is  sin 2 E  S y where the optimum angle is 45o.
3) For as CH3 group the final density matrix is  3 sin  cos 2  E  S y where the optimum angle is
3
.
2
The efficiency of DEPT is superior to refocused INEPT when range of J values is to be considered
since the selection is based on the pulse angle of the final proton pulse and not the value of J.
cos 1


By collecting a series of DEPT spectra at different final proton pulse angle and observing the change
in intensity of the carbon signal with the excitation pattern one can assign the type of carbon.
The DEPT Spectrum of a Terpene taken at 45, 90, and 135 degrees.