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.
· Pulse along the x-axis through :
· Pulse along the y-axis through :
· Free precession about the z-axis at a frequency of :
· Free precession due to J-coupling at a frequency of :
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 2-dimenasional 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.
The Spin Echo for One-Spin-1/2 System
· The system is allowed to come to equilibrium before the excitation pulse. Considering one spin, the equilibrium density matrix is Iz.
Equilibrium Magnetization
· The 90x pulse rotates Iz along the x-axis towards the y-axis giving rise to -Iy.
Magnetization after a 90x pulse
· Once in the transverse plane as -Iy the magnetization will evolve according to:
, where w is the chemical shift.
· After an evolution period t the density matrix is:
Magnetization precessing in the XY plane
at a rate of w for a period t.
· The 180y pulse converts Ix to -Ix and leaves -Iy unchanged giving:
Application of a 180y pulse flips the magnetization
to the other side of the Y-axis
· The x and y magnetization evolve according to the expressions:
and ,
thus the expression for the density matrix becomes:
which after another evolution period of t becomes:
· 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 t, at a rate w, the
magnetization refocuses along the –Y-axis
The Spin Echo for a Two-Spin-1/2 System
· Starting at equilibrium with the density matrix: .
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)
· The 90x pulse converts the equilibrium density matrix to:
· The Hamiltonian for during free evolution is the weakly coupled two spin Hamiltonian given by the expression:
the corresponding propagator is:
which can be separated into three propagators applied in succession:
· U3 involves simple chemical shift evolution for spin 1 according to: where w is the chemical shift, giving:
· U2 involves simple chemical shift evolution for spin 2 as for spin 1, giving:
which can be simplified to:
Free presession of the Z-Magnetizations of spin 1 under the
influence of chemical shift and scalar coupling
· Lets follow the calculation from this point on first two terms describing spin 1 only. The effect of evolution under spin-spin coupling is:
· The 180y pulse converts Ix to -Ix,, Iz to -Iz and leaves Iy unchanged giving:
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.
· Which is allowed to evolve for another period t under the chemical shift Hamiltonian as:
which is simplifies to:
collecting terms and further simplifying:
which finally gives
this means that the chemical shift terms have disappeared which implied that they have been refocused.
Free precession for a second period t under the influence of chemical
shift and scalar coupling evolution result in the chemical shift
terms in refocusing but the coupling terms does not.
· Now consider the evolution under the J coupling Hamiltonian:
which simplifies to:
giving the final expression:
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, .
· A similar expression exists for spin 2. The complete expression is therefore:
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 g is:
Thus the operator corresponding to Mz should be scaled accordingly thus strictly speaking the equilibrium density matrix should be written as:
· The equilibrium density matrix for a heteronuclear two-spin system is therefore:
which can be rewritten as:
The scalar in front is often neglected thus spin dynamics calculation often assume an initial density matrix of the form:
Vector diagrams corresponding to the equilibrium density matrix
of an IS spin system, where S is a weak nucleus.
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.
· 13C 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.
The INEPT pulse Sequence
· 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:
· After a 900x pulse on the proton channel the density matrix becomes:
The vector diagram for product operators after the initial 900x pulse on proton
· 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:
which when t=1/4J becomes:
The vector diagrams corresponding to the density after the spin echo part of the pulse sequence
· At this stage a 900y pulse is applied to the proton channel and a 900x pulse to the carbon channel giving:
The vector diagram for the density matrix after the final 900 pulses
· Evolving under the influence of the scalar coupling interaction this becomes:
· The first and last terms are not observable however the last two are.
Vector representation of the density matrix during the detection period
· The term leads to an anitphase doublet at the carbon frequency separated by the coupling constant. The second term leads to in-phase doublet opposite in sign. 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
· 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
· 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:
The vector diagrams corresponding to the density just at the first echo of the refocused INEPT pulse sequence
· After the 90opulses along the y-axis in both the carbon and proton channel the density matrix becomes:
The vector diagrams corresponding to the density after the 90oy pulses in the C and H channel.
· This is allowed to evolve under J coupling, the chemical shift evolution is ignore since it is refocused:
· After a 180ox pulse in both channels it becomes:
· Another period of J evolution gives:
which can be simplified to:
which at t = 1/4J becomes:
The vector diagrams corresponding to the density at the start of the detection period of the refocused INEPT pulse sequence.
· 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
· 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 sinqcos2q.
· 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.
· Another pulse sequence that achieves carbon signal enhancement by polarization transfer is the Distortionless Enhancement by Polarization transfer, DEPT sequence.