NMR, the vector model and the relaxation

Transcription

1 NMR, the vector model and the relaxation Reading/Books: One and two dimensional NMR spectroscopy, VCH, Friebolin Spin Dynamics, Basics of NMR, Wiley, Levitt Molecular Quantum Mechanics, Oxford Univ. Press, Atkins. NMR: The Toolkit, Oxford Science Publications, Hore, Jones and Wimperis Understanding NMR Spectroscopy, Willey, Keeler CHI551 1

2 Effect of the Magnetic Field on Matter Interaction of Magnetic Field and Matter: Macroscopic Magnetism E = µ B Permanent magnetic moment, e.g. magnets: Induced magnetic moment µ χ induced = VB µ V: Volume χ: Magnetic susceptibility B: Applied field µ = 4π 1-7 H/m CHI551

3 Effect of the Magnetic Field on Matter Interaction of Magnetic Field and Matter: Macroscopic Magnetism Source of magnetism: 1 circulation of electron currents negative contribution to susceptibility magnetic moments of electrons positive contribution to susceptibility 3 magnetic moments of nuclei positive contribution to susceptibility with 1 and > 3 Spins and magnetism: ˆ µ = γŝ γ = gyromagnetic or magnetogyric ratio positive or negative and characteristic of the nuclei. The magnetic moment of the electron has been predicted by Dirac. However, today the magnetic moments of quarks and nucleons, and thereby nuclei, are not year understood. CHI551 3

4 Effect of the Magnetic Field on Nuclei Properties at the atomic level: Nuclear Spin Quantum Number, I For protons and neutrons: I = ½ For atoms: I depends on how spin are paired or unpaired: I =, µ N = I = ½, µ N, eq = I 1, µ N, eq > I 1, µ N, eq < No nuclear spin Nuclei with even numbers of both protons and neutrons A and Z are even e.g. 1 C, 16 O, 3 S Distribution of positive charge in the nucleus is spherical spinning sphere e.g. 1 H, 13 C, 19 F, 31 P as well as 195 Pt Non-spherical distribution of charges electric dipole, eq eq = in spherical environment e.g. D, 14 N, 17 O A = even, Z = odd, I = integer A = odd, I = half integer Unpaired nuclear spins I nuclear magnetic moment µ N. Spinning charges angular momentum I. The allowed orientation of m are indicated by nuclear spin angular momentum quantum number, m I with m I = I, I-1,, -I+1 and I, a total of m I +1 states!! This is associated with a magnetic moment µ, a characteristic of the nuclei: µ = γ" I CHI551 4 N

5 Effect of the Magnetic Field on one Nucleus Macroscopic Magnetism Microscopic Magnetism z In such an axis system, a positive rotation is defined by the curl of the fingers For γ > H! θ!! µ = γ" I N H exerts a force torque on µ N, causing a precession, perpendicular to µ N and H,with a frequency Larmor frequency:!! τ = µ H! = γh γ = gyromagnetic or magnetogyric ratio, associated with the angular momentum I and a characteristic of the nucleus 1 H, I =½; γ = rad/s/t -5 MHz at T H, I =1; γ = rad/s/t MHz at T 13 C, I =½; γ = rad/s/t -15 MHz at T 9 Si, I =½; γ = rad/s/t MHz at T CHI551 5

6 Effect of the Magnetic Field on one Nucleus Energies of spin states For I = ½, m I = ± ½ For I = 1, m I = -1,, 1 ENERGY No applied field ΔE Applied Magnetic field m I = - ½ m I = + ½ ENERGY No applied field ΔE ΔE Applied Magnetic field m I = - 1 m I = m I = + 1 µ N in the direction of the applied field for m I > µ N opposed to the applied field for m I < µ N perpendicular to the applied field for m I = ΔE: Energy difference between two states, it is associated with a radio frequency ν. This is what will be detected in the NMR experiment. ΔE = γ! H Δm =! I CHI551 6

7 Nuclear spins in B Energy E β E α B From quantum mechanics: Hˆ ΔE " " = µ H γ = γ! H Δm =! Boltzman Distribution: = N! HI z E = γ! mi H I ˆ For = 6 MHz B = 1.49 Tesla For = 5 MHz B = Tesla Note that B Earth = 5 µt N N β α = e E β E kt α = e γ! B kt γ B 1 kt! α At room temperature, the ratio = N N β CHI551 7

8 Nuclear spins in B = γb B N α most stable α spin N N α β = 1.66 N β less stable β spin M Net magnetization aligned with B not the spins! No net magnization perpendicular to B Note that: M << B and cannot be easily detected directly! Low sensitivity of NMR sensibility is proportional to B 3/, hence the development of high field NMR spectrometer. CHI551 8

9 Nuclear spins in B A more real model: Effect of B on of sum of spins However, due to the isotropic distribution of spins, there is no contribution to the magnetism of the material. Origin of net magnetization: Molecular motions Induction of overall fluctuating fields time scale = nanosecond, and a biased magnetic moment aligned with B longitudinal magnetic moment. Stable anisotropic distribution of nuclear spin polarization, also named thermal equilibrium ΔE = γ! H =! CHI551 9

10 Nuclear spins in B A more real model: Build-up of Longitudinal magnetization sudden introduction of a sample in B M z t = M t t 1 exp[ T 1 on ] Loss of Longitudinal magnetization sudden removal of a sample from B M z t = M t exp[ t T 1 off ] Longitudinal relaxation see Section on relaxation CHI551 1

11 Observation of the NMR phenomenon Sequences RMN and magnetization What happens when M, aligned with the z axis, is moved away from its equilibrium position effect of B 1? It will vary under the control of two factors: 1 Torque!!!!! τ = µ B = γi B Relaxation physical phenomena, which will bring back the system to equilibrium The resulting motion will be a precession around the magnetic field at the Larmor frequency it is like a gyroscope in a gravitational field with a dissipation of the energy to return to the equilibrium position = πv = γb How does the magnetization is moved away from its equilibrium position? Application of a B 1 field perpendicular to B - Effect of B 1 field perpendicular to B : Magnetization will be precessing around B +B 1. However, because B 1 << B the effect will be only very small. - Effect of a B 1 field perpendicular to B but rotating in the O xy plane at close to Resonance effect, which corresponds to a adsorption of a continuous wave at and which is like having a B +B 1 field very effective to tip the magnetization in the xy plane. B 1 is the high frequency alternative field, which is generated by a solenoid oriented perpendicular to the z axis. The intensity and the duration of the radiofrequency wave are controlled. This corresponds to two rotating fields in opposite direction CHI551 11

12 Sequences RMN and magnetization B A time Influence of magnetization under B 1 in the rotating frame z M B 1 O x M α y t M xy -x B 1 O x B A γb A = 1 - and α = γb 1 τ After a π pulse After relaxation, back to equilibrium! M Consequence of π-pulse: inversion of population -M CHI551 1

13 Simple NMR experiment sequence and magnetization I II III IV V time z Time I Time II omitting relaxation Influence of magnetization under B 1 M B1 Ox B A B A y π/ -x B 1 O x γb A = Time III-V with T 1 relaxation M z M z M M xy M xy III IV V at full relaxation CHI551 13

14 Sequences RMN and magnetization M xy in the laboratory frame M y M x sin texp / { t } { t } t = M T t = M T cos texp / M xy magnetization transverse magnetization decays slowly, because of the inhomogeneous field leading to a return to equilibrium net magnetization along B. This is called transverse magnetization T. For small molecules, T T 1 i.e. nuclear spins precess millions of Larmor precession cycles before losing their synchrony. CHI551 14

15 Observation of the NMR phenomenon Spectrometer NMR probe: A coil oriented perpendicular to B is used to generate B 1 and to detect the signal. The signal of frequency close to 1 is amplified and compared to 1 giving - 1, which is the chemical shift. A second coil provide a second field B, generation of highly precise frequencies n using a clock Quartz, 1 MHz temperature regulated: Channel 1 with a frequency 1, used for the observation Channel with a frequency, typically used for decoupling or a second type of spins 13 C, 19 Sn. Channel 3 with a frequency 3, Lock, stabilization of the field by a reference nucleus usually H Clock 1 Mhz Decoupling Channel Sample Coil Receptor Spectrum Calculator 1 Signal: s t dφ = dt Synth. Gate Signal Mixer Synth. 1 Gate 1 D Correction to B Mixer Synth. Lock H D S N = CHI n

16 Observation of the NMR phenomenon Rf pulse and signal CHI551 16

17 NMR Signals Imaginary s B t sin t exp { λt} For a one-line spectrum: { λt} + i sin[ t] { λt} s t = cos[ t]exp exp s 1 1 {[ i λ] t} t = a exp 1 Real s A t cos t exp { λt} For a spectrum containing l lines: { i λ t} s t = al exp l l l For a each lines: One frequency l, One damping rates λ l, One amplitude a l. a l l /π Hz λ l s -1 e.g. vs. 4 lines of different frequencies CHI551 17

18 CHI { } { } exp cos λ λ λ + t t NMR Signals and Fourier Transformation dt t s e S t i + = Line shapes Lorenztian Adsorption line Dispersion line { } { } exp sin λ λ + t t 1 +T T 1 + T T λ = 1/T

19 NMR Signals, Fourier Transformation and Apodization Functions Apodization function A: FID*A before FT. Multiplying FID by an apodization function allows the improvement of signal to noise ratio or line width. Typically exponential decay, exp-at, is used in order to obtain an increased S/N at the expense of line broadening. Note that a truncated FID experimental/acquisition time shorter than FID corresponds to multiplying the FID by a step function, whose FT corresponds to sinax/x. This will induced wiggles at each peak of the spectrum. CHI551 19