Category:NMR: Difference between revisions

Many nuclei have an inherent, non-zero spin I and therefore a magnetic dipole moment μ, conventionally along the z-axis:

${\displaystyle \mu_z = \gamma \textbf{I}_z }$

In the absence of a magnetic field, these are degenerate states. When an external magnetic field B_ext is applied, the energy difference between two states is given by the following equation:

${\displaystyle \Delta E = \gamma \hbar \textbf{B}_{ext} }$

Conventionally, the z-axis is chosen for the direction of B_ext. Along this axis, μ aligned with B_ext will be slightly more energetically favorable and so populated than μ opposed to B_ext. This is only signficant in the presence of strong magnetic fields.

In the presence of B_ext, μ precesses at its Larmor frequency ω_L, determined by the strength of the magnetic field and the nucleus' gyromagnetic ratio γ:

${\displaystyle \omega_L = \gamma \textbf{B} }$

A weak, oscillating magnetic field applied perpendicular (i.e. in the transverse frame) to B_ext (reference frame), e.g. using a radio-frequency (RF) pulse at frequency ω_rf, can cause μ to oscillate with the RF. If ω_rf is similar to ω_L, then resonance occurs, hence nuclear magnetic resonance (NMR). μ flips from the reference to the transverse frame and the relaxation of μ back to the reference frame creates a signal that is measured in NMR.

Each nuclear isotope has a different gyromagnetic ratio. Even with the same isotope, the frequency can subtly differ based on the chemical environment. Electrons are also charged and so their movement in atoms, i.e. the electronic current, generates a magnetic field opposed to B_ext. This induced magnetic field B_ind reduces the magnetic field at the nucleus, decreasing the frequency measured in NMR. In this way, the electrons shield the nucleus from B_ext. Since the electron density of a molecule or crystal is determined by its molecular orbitals, its chemical environment can be probed by these subtle differences in frequency. This shielding relation between B_ext and B_ind is described by the chemical shielding tensor σ_ij:

${\displaystyle \textbf{B}_{ind}(\textbf{R}) = -\sigma(\textbf{R}) \textbf{B}_{ext}}$

where i and j are Cartesian axes.

It may be calculated by means of linear response:

${\displaystyle \sigma_{ij}(\textbf{R}) = - \frac{\partial B^{\mathrm{ind}}_i(\mathbf{R})}{\partial B^{\mathrm{ext}}_j} }$

The chemical shielding itself cannot be measured in experiment, instead it must be taken relative to a standard reference ^[1], which results in the chemical shift δ_ij:

${\displaystyle \delta_{ij}(\mathbf{R}) = \sigma_{ij}^{\mathrm{ref}} - \sigma_{ij}(\mathbf{R}) }$

The chemical shift (in ppm) is measurable and is related to the measured frequency ω_sample via the following equation:

${\displaystyle \delta = \frac{\omega_{ref} - \omega_{sample}}{\omega_{sample}} \times 10^6 }$

How to

• Chemical shift tensors: LCHIMAG.
• Hyperfine tensors: LHYPERFINE.
• Electric field gradient tensors: Electric Field Gradient. The main tags are:
  • LEFG to switch on the field gradient tensor calculation,
  • QUAD_EFG to specify the input nuclear quadrupole moments.