The vector model of NMR Flashcards
Nuclear magnetic moments in the absence of an applied field
Randomly orientated
No net magnetisation
Nuclear magnetic moments in an applied field, B0
The individual spin states, ml, lose their energetic degeneracy and populate new energy levels according to a Boltzmann distribution
Consequently there is an excess of nuclear magnetic moments aligned with the applied field, leading to a net magnetisation, M0, which is represented as a vector
M0
Net/bulk magnetisation of the sample
Can be represented as a vector
Points along the direction of the applied field B0
The rotating frame in NMR experiments
In a magnetic field B0, the net magnetisation vector M precesses about the axis of the applied field at the Larmor frequency and is inclined at a precession cone angle
To simplify the magnetisation in an NMR experiment, we move from the static laboratory frame x, y, z to the rotating frame x’, y’, z’. which rotates about the z axis in the same direction and at the same rate as B1
Frame rotation matches the Larmor frequency
Z’ is inclined to the z-axis at the precession cone angle
Advantage of using the rotating frame method
Allows us to analyse what happens to M/to individual spins when the B1 magnetic field is applied at a frequency matching the Larmor frequency
The vector model of NMR
Considers what happens to M during the experiment as viewed in the rotating frame - basically we sit on the vector and observe the stationary origin
SW
Sweep/spectral width/window about the offset, in ppm
Refers to the chemical shift range over which the data is recorded
e.g. a sweep width of 2400 Hz covers 12 ppm of chemical shift if the 1H Larmor frequency is 200 MHz (2400Hz/200MHz = 12ppm)
What determines the centre of a spectrum?
The offset frequency i.e. the precise spectrometer frequency
Chemical shift range observed in the final spectrum
Extends from -0.5SW to +0.5SW
e.g. an offset frequency of 5 ppm with a spectral window of 10 ppm will yield a spectrum that starts at 0 ppm and ends at 10 ppm
Rf pulse
Radiofrequency pulse
Has a characteristic frequency (= spectrometer frequency) that depends on the nucleus you wish to observe and the magnetic field strength of the spectrometer
Excites the nuclei, which then emit Rf during the acquisition time, giving rise to an NMR signal in the form of an exponentially decaying sine wave (=FID)
How are NMR spectrometers generally named?
After the frequency at which hydrogen atoms resonate
e.g. a 500 will cause H atoms to resonate at approx. 500 MHz
Why is an Rf pulse that excites spins at only one frequency not desirable?
Because the NMR frequencies are spread out over a range of frequencies, called the range of chemical shifts
Why are short Rf pulse lengths used in FT-NMR?
Short pulse widths irradiate a wide range of different frequencies due to the Heisenberg Uncertainty Principle
A pulse length of 10 us causes the Rf power to be distributed over 1/pw = 1/0.00001 = 100,000 Hz of frequencies
Heisenberg Uncertainty Principle
The time it takes for a system to change significantly multiplied by the uncertainty in energy is always greater than h/2pi
The Heisenberg Uncertainty Principle in relation to NMR
As the pulse length is shortened, the uncertainty in the frequency results in a larger field of excitation
See green equation in notes
When would a longer, lower energy pulse be useful in NMR?
A longer, lower energy pulse will have less frequency spread so can be used for frequency-selective excitation or saturation
Pulse width
Refers to TIME
i.e. units are us
= the amount of time the pulse of Rf energy is applied to the particular sample in order to flip the bulk magnetisation from the z-axis into the xy-plane
Pulse angle
The angle by which the magnetisation (M) is displaced away from the z’-axis towards the y’ axis
Determined by the strength (i.e. time length) of B1
B1
Excitation pulse
What happens to M after the Rf pulse has been applied?
M precesses about the z’ axis as it relaxes back to its pre-pulse, equilibrium state - i.e. the excited nuclear spins lose energy and drop from the excited state back to the ground state (Boltzmann distribution)
What is the observable NMR signal?
The projection of the precession of M onto the x’y’ plane
The receiver coil is orientated along the y’ axis, meaning the intensity of the signal observed is dependent upon the amount of magnetisation in the y’ direction
How can maximum signal intensity be achieved?
By using a 90 degree pulse angle
When can we expect a 0 intensity signal?
When the pulse angle is 180 degrees
How long does relaxation of nuclei take?
Can vary from seconds to hours depending on the relaxation mechanisms available
Nuclei must relax fully before applying the next pulse in order to obtain a true and maximised signal
How does relaxation generally occur?
By transfer of magnetisation to other dipoles
Dipole-dipole relaxation
Occurs when the sample contains other magnetic dipoles (e.g. magnetic moments of other nuclei/unpaired electrons) that can interact with the nuclear magnetic moment
Occurs by two mechanisms:
T1 relaxation = spin-lattice relaxation
T2 relaxation = spin-spin relaxation
Factors affecting the effectiveness of relaxation
Relaxation depends on, and is proportional to, gamma(A)^2 multiplied by gamma(B)^2, where A is the starting nucleus and B is the receiving nucleus
i.e. nuclei with larger gyromagnetic ratios will have faster relaxation times than those with smaller gamma
Why is relaxation in paramagnetic complexes dominated by the presence of the unpaired electron(s)?
Because the gyromagnetic ratio of an electron is ~1000x larger than that of a proton, so relaxation is very efficient
Spectra of paramagnetic complexes often have broad line widths
For relaxation to occur…
…dipoles must set up a magnetic field that oscillates at approximately the NMR resonant frequency
Molecules can match their frequencies to better or worse extents depending on:
Their size/shape
The viscosity of the medium
The temperature
Long relaxation times (i.e. slower relaxation) occur with…
…small molecules, low viscosities and high temperatures
Long relaxation times (i.e. inefficient relaxation) give…
…narrower signals
Effect of concentration on relaxation
Higher concentration leads to more efficient (faster) relaxation, which gives broader signals
This is because dipole-dipole relaxation is a through-space effect and decreases with distance (proportional to r^-6)
T1 relaxation
= spin-lattice relaxation
= longitudinal relaxation
= relaxation in the z-direction
Corresponds to the process of (re-)establishing the Boltzmann population distribution of alpha and beta spin states in the magnetic field i.e. corresponds to the ‘growing back’ of magnetisation in the z’ direction
T2 relaxation
= spin-spin relaxation
= transverse relaxation
= relaxation in the xy plane
Corresponds to the loss of phase coherence among nuclei/corresponds to the ‘fading away’ of magnetisation in the x’y’ plane
Independent of the relaxation of the z’ component
What determines the line width of an NMR signal?
T2 Short T2 = broader lines (T2)^-1 = piW(1/2) OR W(1/2) = 1/piT2
W(1/2)
= width at half height
What determines the maximum repetition rate during the acquisition of an NMR signal?
T1
Short T1 = magnetisation recovers more rapidly so a spectrum can be acquired in less time
i.e. T1 determines how frequently we can pulse the nucleus
Sources of line broadening in NMR
Viscous medium
Molecule has a large molecular mass (e.g. protein
Low temperature (increases viscosity)
Nucleus has a quadrupole moment and is lower than cubic symmetry
Paramagnetic centres/impurities in the sample
Sample inhomogeneity (e.g. poor mixing, presence of solid particles)
Instrumental problems e.g. tuning, misshapen NMR tube
Why is T2 always shorter than T1?
Because return of magnetisation to the z-direction inherently causes loss of magnetisation in the xy plane
How is T1 measured?
By the inversion-recovery experiment
Steps in the inversion-recovery experiment
- Application of a 180degree pulse inverts the bulk magnetisation to the -z axis
- Immediately after this inversion, the spins begin to recover along the +z axis
- After a delay (tau), the spin are irradiated with a 90degree pulse
- this delay is varied, which produces multiple signals - Plotting the intensities of the signals vs the delay (tau) will produce a curve, which can be fit to an exponential function of the form M(tau) = M0 - 2M0e^-tauT1)
Value of tau to give a signal with zero intensity
tau = ln2T1
Short delay (i.e. short tau)
Gives signals with negative intensities
Delay (tau) = ln2T1
Gives signals with zero intensity
Long delay (i.e. long tau)
Gives signals with positive intensities
Delay time between scans
5 x T1
Quadrupolar relaxation
Depends on the size of the nuclear quadrupole and the electric field gradient of the nucleus
Electric field gradient
Measures the rate of change of the electric field at an atomic nucleus generated by the electronic charge distribution and the other nuclei
Very sensitive to the electronic density in the immediate vicinity of a nucleus
Couples with the quadrupole moment of quadrupole nuclei to generate an effect that can be measured by NMR
What does the electric field gradient depend on?
The symmetry of the molecule
It is smallest when the environment is cubic/highly symmetrical
It is largest when the molecule possesses no symmetry
When is quadrupole relaxation very efficient?
When the quadrupole moment is large and/or the nuclear environment is far from cubic symmetry
This speeds up T2 relaxation and gives the NMR excited states very short lifetimes, leading to Heisenberg broadening and therefore broad lines in the spectrum
When are quadrupolar nuclei NMR invisible?
When the lifetimes of the NMR excited states are so short that the resonances are 1000s of Hz wide
This gives lines so broad that they are essentially indistinguishable from the baseline noise
35Cl NMR of CDCl3 c.f. NaCl
In CDCl3, the environment around Cl is very asymmetrical so gives a broad, ill-defined peak
In NaCl, there is only one Cl- (i.e. the environment is as symmetrical as is possible) so the signal is much sharper
Spectroscopically useful nuclei
Have quadrupole moments < 0.15 x 10^-28 m^2
e.g. 2H, 6Li, 10/11B, 17O, 23Na
