Lecture 3: Line broadening mechanisms in NMR

Transcription

1 Lecture 3: Line broadening mechanisms in NMR Lecture aims to explain: 1. Fourier transform limit (or energy-time uncertainty principle) 2. Resonance broadening due to interaction with neighbouring nuclei in a solid state lattice 3. Electron-nuclear interaction: Knight and chemical shifts

2 Uncertainty principle and Fourier transform limit

3 Energy-time uncertainty and line broadening 1/ +1/ 2 2 E = γ H 0 In an ensemble of identical nuclei magnetic resonance frequency is not a delta-function as follows from the energy-time uncertainty principle. Using quantum mechanical language: E t h Here t is a characteristic life-time of the nucleus in a spin state with a certain projection on the external field E is the natural (or homogeneous) linewidth of the resonance. Fast-decaying states have a broad linewidth, while slow decaying states have a narrow linewidth.

4 Fourier transform of exponential decay Using classical language based on Fourier transform: processes in the time domain transformed into the frequency domain Consider spin population decay: Use definition of Fourier transform: n( t) = n exp( t / τ ) 0 + F t [ n( t)] = n( t)exp( 2πift) dt For the exponential decay we ll obtain a Lorentzian: F t [ n( t)] = 1+ 2τ 2 (2πτ ) ( f f 0 ) 2 This gives full width at half maximum (FWHM) in the frequency domain: f FWHM = 1 πτ

5 Resonance broadening due to interaction with neighbouring nuclei

6 Rough estimation of the local magnetic field due to neighbouring nuclei Example 3.1 Estimate the local magnetic field experienced by an individual nucleus due to the magnetic dipole-dipole interaction with the neighbouring nuclei. For the estimate use 69 Ga in a GaAs crystal having a lattice constant of 0.56 nm. The gyromagnetic ratio for 69 Ga of rad s -1 T -1. Example 3.2 Estimate the NMR line broadening for 69 Ga in a GaAs crystal. Give your answer in terms of the frequency broadening measured in Hz. The gyromagnetic ratio for 69 Ga of rad s -1 T -1. Find the magnitude of a characteristic time associated with such line broadening. Assume T1 of 1 second.

7 Electron-nuclear interaction

8 Basic considerations Total Hamiltonian of the electrons and nuclei: H = H ( H ) + H (0) + H ( H ) + H total nz e ez en Nuclear Zeeman coupling H en Orbital and spin parts for electron at H=0 Electron Zeeman coupling corresponds to the extra magnetic fields the nuclei experience owing to the electrons Interaction between nuclear spins and the electron orbital and spin coordinates We may say that the nuclei experience a direct interaction with H via the Zeeman term and an indirect via the interplay of the electron Zeeman term and the electron-nuclear interaction

9 Chemical shift

10 Chemical shift Chemical shifts are due to the orbital motion of electrons In ethanol CH 3 CH 2 OH three types of protons are observed. Relative intensities of NMR peaks correspond to the relative number of different protons. Using classical theory, a moving charge q produces magnetic field H at the coordinate origin: H = q r v qµ r mv qµ 0 r m r m 0 µ 0 = = 3 3 Here L is the angular momentum. Note, electrons in s-states do not contribute to the chemical shifts as L=0 L 3 r

11 Chemical shifts: inhomogeneous broadening Resonance frequency modified due to the chemical shift obeys the equation: = γ ( H0 + H ) H = σh ω where 0 Example 3.3 In an InP/GaInP quantum dot 31 P resonance linewidth is 4 khz nearly independently of the external magnetic field for H < 1.5 T. Above ~1.5T the width gradually increases and reaches 12 khz for H = 7 T. Explain the observed phenomenon.

12 Knight shift

13 Electron-nuclear spin interaction For the electron s-state having a large density at the position of the nucleus the magnetic dipole-dipole interaction is not a good approximation. The following expression (written in Gauss system) should be used instead: H 8π = γ e γ n 3 2 I Sδ ( r) Here S and I are nuclear and electron spins, γ e and γ n corresponding gyromagnetic ratios This expression describes so-called hyperfine interaction between electrons and nuclei

14 Knight shifts: inhomogeneous broadening First discovered by W Knight as a resonance frequency shift of 0.23% for 63 Cu in metallic copper compared to that in CuCl: the shift 1 order of magnitude larger than chemical shifts. Related to paramagnetism of unpaired electron spins. Also present in semiconductors where metallic-like properties can be achieved by doping, and unpaired electron spins can be created optically.

15 SUMMARY Finite life-time leads to a finite linewidth in the frequency domain. Using Fourier transform limit the process with a characteristic time τ will be described with a full width at half maximum (FWHM) in the frequency domain : f FWHM = 1 πτ Broadening of the magnetic resonance may occur due to interaction with neighbouring nuclear spins: dipole-dipole coupling Electron-nuclear interaction influences the NMR linewidth via two contributions: orbital contribution (resulting in chemical shifts) and interaction with unpaired s-electron spins (Knight shifts) Hyperfine electron-nuclear interaction is given by (in Gauss system): H 8π = γ e γ n 3 2 I Sδ ( r)