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Unformatted text preview: 1 5.61 Physical Chemistry Lecture #36 Page NUCLEAR MAGNETIC RESONANCE Just as IR spectroscopy is the simplest example of transitions being induced by light’s oscillating electric field, so NMR is the simplest example of transitions induced by the oscillating magnetic field. Because the strength of mattermagnetic field interactions are typically two orders of magnitude smaller than the corresponding electric field interactions, NMR is a much more delicate probe of molecular structure and properties. The NMR spin Hamiltonians and wavefunctions are particularly simple, and permit us to demonstrate several fundamental principles (about raising and lowering operators, energy levels, transition probabilities, etc.) with a minimal amount of algebra. The principles and procedures are applicable to other areas of spectroscopy electronic, vibrational, rotational, etc. – but for these cases the algebra is more extensive. Nuclear Spins in a Static Magnetic Field For a single isolated spin in a static magnetic field, the contribution to the energy is: H ˆ = − m B ˆ i ˆ i = − γ I B 0 0 where γ is called the gyromagnetic ratio. If we choose our z axis to point in the direction of Energy the magnetic field then: H ˆ = − m B ˆ ˆ = − γ I B 0 z 0 z 0 If we assume the nuclear spin is ½ (As it is for a proton) then we can easily work out the energy levels of this Hamiltonian: 1 1 ω E ± = ± 2 γ B 0 ≡ ± 2 0 where ω = γ B 0 is called the nuclear Larmor frequency (rad/sec). Now, nuclei are never isolated in chemistry – they are always surrounded by electrons. As we learned for the hydrogen atom, the electrons near the nucleus shield the outer electrons from the bare electric field produced by the nucleus. Similarly, the electrons shield the nucleus from the bare electric field we apply in the laboratory. More specifically, the electron circulation produces a field, B’ opposed to B 0 and of magnitude equal to σ B 0 where σ is a constant. bare nucleus with nucleus electrons Thus, the effective field, B , at the nucleus is B 0 B (1 σ ) B = (1 − σ ) B 0 Note that σ is different for each chemically different nuclear spin – this is the famous h ω h ω (1 −σ ) chemical shift – and permits resolution of lines in NMR spectra corresponding to chemically different sites. The Hamiltonian is modified accordingly ˆ ˆ H = − m B ˆ ( 1 − σ ) = − γ I B ( 1 − σ ) 0 z 0 z 0 Zero Field High Field Thus, instead of “seeing” a magnetic field of magnitude B , a proton in a molecule will see a 5.61 Physical Chemistry Lecture #36 Page 2 magnetic field of magnitude (1 σ ) B 0 and the associated Hamiltonian and spin state energies will become: 1 1 E ± = ± 2 γ B 0 ( 1 − σ ) ≡ ± 2 ω 0 ( 1 − σ ) This is illustrated in the figure above. Note the sign of the Hamiltonian is chosen so that the α state (spin parallel to B ) is lower in energy than the β state ( spin antiparallel to B )....
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 Spring '08
 greenwood
 Calculus, Magnetism, Magnetic Field, Nuclear magnetic resonance, High Field

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