Biophysical Chemistry: NMR Spectroscopy Spin Dynamics & Chemical - PowerPoint PPT Presentation

Pulse/Fourier Transform NMR Chemical Exchange Summary Biophysical Chemistry: NMR Spectroscopy Spin Dynamics & Chemical Exchange Lieven Buts Vrije Universiteit Brussel 25th November 2011 Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary Outline Pulse/Fourier Transform NMR 1 Thermal Equilibrium Effect of RF Pulses The Fourier Transform Chemical Exchange 2 Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Summary 3 Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Outline Pulse/Fourier Transform NMR 1 Thermal Equilibrium Effect of RF Pulses The Fourier Transform Chemical Exchange 2 Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Summary 3 Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Quantum Description of a Spin-1/2

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Larmor Precession (1) The interaction between an individual spin and a uniform external magnetic field leads to precession of the spin around the direction of the external field: The angle θ between the direction of the field and the direction of the spin remains constant throughout this motion. De frequency of the precession is the Larmor frequency ν = γ ( 1 − σ ) B 0 or ω = γ ( 1 − σ ) B 0 . 2 π Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Larmor Precession (2) The interaction between the spin and the external field is far stronger than all other interactions between the nucleus and other particles in its environment. Therefore, as a first approximation, the nucleus behaves like an isolated gyroscope which rotates independently, with no regard for its surroundings or the motions of the molecule which it is part of. Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Populations (1) The ratio between the populations of the two energy levels ( n α and n β ) is determined by the energy difference ∆ E and the temperature T : n β = e − ∆ E kT n α from which we find that n α − n β = ∆ E n α + n β 2 kT The Boltzmann constant ( k = k B = 1 . 38066 × 10 − 23 J K ) functions a conversion factor from temperature to thermal energy. Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Populations (2) At a temperature of T = 300K the average thermal energy is kT = 4 . 14 × 10 − 21 J . The energy difference between the two stationary states of a spin I = 1 / 2 is very small, even for 1 H (which has the largest gyromagnetic ratio of all practically available nuclei) in a strong external field: γ = 26 . 73 × 10 7 T − 1 s − 1 ; B 0 = 9 . 4T ; ∆ E = � γ B 0 = 2 . 65 × 10 − 25 J This implies that the difference between the two populations is very small: n α − n β = ∆ E 2 kT = 3 . 2 × 10 − 5 n α + n β In other words, about one low-energy spin out of every 10 5 has no counterpart in the high-energy orientation. Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Populations at Thermal Equilibrium Each individual spin contributes a certain fraction of " α character" (proportional to | c α | 2 ) and a complementary fraction " β character" (proportional to | c β | 2 = 1 − | c α | 2 ) to the ensemble (the collection of all spins). The populations n α and n β of the two energy levels are the avarage values | c α | 2 and | c β | 2 over all spins in the ensemble. Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Thermal Coupling Very infrequently, the nucleus does interact with a surrounding particle, which can lead to a change of its orientation with respect to the external field, as expressed by the angle θ . The energy that drives these interactions comes from the thermal energy of the atoms, that is associated with their random motions. The minuscule energy changes of the nuclear spins are associated with equally minuscule temperature changes of the system. Because of the energy difference between the α and β states there is a small preference for random flips that move the spin state towards the lower energy level. As a result, a thermal equilibrium between the α and β populations is slowly established. This equilibrium is described by the Boltzmann distribution. Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Bulk Magnetisation at Equilibrium (1) The individual dipole moments of all spins can be added together to find the total or bulk magnetisation of the sample.men. In the x and y directions, the spins are oriented completely randomly: which results in a net magnetisation of zero in these directions. Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Bulk Magnetisation at Equilibrium (2) In the z direction there is a small preference for the low energy state, as reflected by the slightly larger population n α : n α − n β = ∆ n eq = ∆ E n α + n β N 2 kT Because of this, a small net magnetisation remains in the direction of the positiove z axis. The magnitude of this remainder is proportional to the population difference ∆ n eq : M 0 = 1 2 γ � ∆ n eq Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Larmor Precession at Equilibrium At equilibrium, the x and y components of the spin dipoles remain randomly distributed throughout the precessional motion, and theirsum remains zero. The distribution of the α and β components is also unaffected by the precessional motion, and therefore the z component of the total magnetisation also remains constant. The bulk magnetisation vector therefore remains constant as the individual spins precess around the z axis. Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Outline Pulse/Fourier Transform NMR 1 Thermal Equilibrium Effect of RF Pulses The Fourier Transform Chemical Exchange 2 Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Summary 3 Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Design of a Modern NMR Spectrometer (1)

Design of a Modern NMR Spectrometer (2)

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform Bulk Magnetisation at Equilibrium (3) At thermal equilibrium the spins are almost equally distributed in all directions, with a small preference for the low-energy state: (For the purpose of the illustration, the population difference has been greatly exaggerated.) Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Design of a Modern NMR Spectrometer (3)

Effect of an RF Pulse A well-tuned RF pulse coherently rotates all spins about the x axis. The net effect is that the bulk magnetisation as a whole undergoes the same rotation:

Design of a Modern NMR Spectrometer (4)

Return to Equilibrium (Relaxation) When the excitation by the RF pulse ends, the system returns to its equilibirium state. The oscillating variation of the net magnetisation in the ( x , y ) plane is the source of the obervable signal:

The Magnetic Field of an RF Pulse Physics tells us that only the magnetic component of the RF radition coming from the excitation coil affects the spins. Because of the position of the coil with respect to the sample this magnetic component � B 1 rotates in the x , y plane, with a frequency ω RF and a phase φ RF determined by the operator: B 1 , x = B 1 cos ( ω RF t + φ RF ) B 1 , y = B 1 sin ( ω RF t + φ RF )

Pulse/Fourier Transform NMR Thermal Equilibrium Chemical Exchange Effect of RF Pulses Summary The Fourier Transform The Rotating Frame In order to simplify the description of the precession of spins around a field that is itself rotating, we introduce a new frame of reference that rotates around the z axis at the frequence of the RF wave ( ω RF ): � e ′ x = cos (Φ( t )) � e x + sin (Φ( t )) � e y � e ′ y = cos (Φ( t )) � e y − sin (Φ( t )) � e x � z = � e ′ e z Φ( t ) = ω RF t + φ RF Lieven Buts Biophysical Chemistry: NMR Spectroscopy