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 more populated than μ opposed to B_ext. This is only significant 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 ^[1][2].

Chemical shielding

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:

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

The chemical shielding itself cannot be measured in experiment, instead, it must be taken relative to a standard reference ^[3], which results in the chemical shift δ_ij:

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

where i and j are Cartesian axes.

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

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

The chemical shielding tensor may be calculated by means of linear response:

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

LCHIMAG calculates the chemical shieldings. These are for individual atoms and exclude the contribution due to the augmentation currents in other PAW spheres. If two-center terms are important, e.g. for H, then they may be included using LLRAUG.

Quadrupolar nuclei - electric field gradient

Nuclei with I > ± ½ have an electronic quadrupolar moment. This means that, at the nucleus, there is a non-zero electric field gradient (EFG), i.e. the rate of change of the electric field with respect to position:

${\displaystyle V_{ij} = \frac{\partial^2 V}{\partial x_i \partial x_j} }$

This comes from the quadrupolar nuclei being non-spherical an so having a non-uniform electric charge distribution. The electric quadrupolar moment couples with the EFG and so the chemical enviornment of the nucleus may be probed using nuclear quadrupole resonance (NQR) ^[4]. The EFG is not directly measurable but the nuclear quadrupolar coupling constant C_q is, defined as:

${\displaystyle C_q = \frac{eQV_{zz}}{h} }$

where e is the charge of an electron, Q is the isotope-specific quadrupole moment, and h is the Planck constant.

The EFG can be calculated using LEFG, which also calculates C_q so long as Q are defined using QUAD_EFG.

Hyperfine coupling

As well as the nuclei, electrons also have spin. Analogously to the nuclei, this may couple with B_ext to provide information about its environment. The interaction between internally generated magnetic fields and the magnetic dipole moment of the nucleus split otherwise degenerate energy levels. This splitting is known as hyperfine splitting.

In most stable systems, all electrons are paired together, spin-up and spin-down, resulting in overall no spin. If a system has unpaired electrons, e.g. radicals, metal oxides, defects, then these systems can be investigated, e.g. using electron paramagnetic resonance (EPR) ^[5].

The hyperfine tensor A^I describes the interaction between a nuclear spin S^I and the electronic spin distribution S^e (in most cases associated with a paramagnetic defect state):

${\displaystyle E=\sum_{ij} S^e_i A^I_{ij} S^I_j }$

The hyperfine tensor can be calculated using LHYPERFINE.

How to

• Chemical shift tensors: LCHIMAG.
• 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.
• Hyperfine tensors: LHYPERFINE.

References