Spin Hamiltonians

Free evolution of nuclear spins is governed by a Hamiltonian (${\displaystyle {\hat {H}}}$), which in the case of liquid-state nuclear magnetic resonance may be split into two major terms. The first term (${\displaystyle {\hat {H}}_{z}}$) corresponds to the Zeeman interaction between spins and the external magnetic field, which includes chemical shift (${\displaystyle \sigma }$). The second term (${\displaystyle {\hat {H}}_{J}}$) corresponds to the indirect spin-spin, or J-coupling, interaction.

${\displaystyle {\hat {H}}={\hat {H}}_{z}+{\hat {H}}_{J}}$, where:

${\displaystyle {\hat {H}}_{z}=-\hbar \sum _{a}\gamma _{a}(1-\sigma _{a}){\hat {I}}_{a}\cdot B_{0}}$, and

${\displaystyle {\hat {H}}_{J}=-\hbar 2\pi \sum _{a>b}J_{ab}{\hat {I}}_{a}\cdot {\hat {I}}_{b}}$.

Here the summation is taken over the whole system of coupled spins; ${\displaystyle \hbar }$ denotes the reduced Planck constant; ${\displaystyle \gamma _{a}}$ denotes the gyromagnetic ratio of spin a; ${\displaystyle \sigma _{a}}$ denotes the isotropic part of the chemical shift for the a-th spin; ${\displaystyle I_{a}}$ denotes the spin operator of the a-th spin; ${\displaystyle B_{0}}$ is the external magnetic field experienced by all considered spins, and; ${\displaystyle J_{ab}}$ is the J-coupling constant between spins a and b.

Importantly, the relative strength of ${\displaystyle {\hat {H}}_{z}}$ and ${\displaystyle {\hat {H}}_{J}}$ (and therefore the spin dynamics behavior of such a system) depends on the magnetic field. For example, in conventional NMR, ${\displaystyle |B_{0}|}$ is typically larger than 1 T, so the Larmor frequency ${\displaystyle \nu _{0}=-\gamma B_{0}/2\pi }$ of ¹H exceeds tens of MHz. This is much larger than ${\displaystyle J}$-coupling values which are typically Hz to hundreds of Hz. In this limit, ${\displaystyle {\hat {H}}_{J}}$ is a perturbation to ${\displaystyle {\hat {H}}_{z}}$. In contrast, at nanotesla fields, Larmor frequencies can be much smaller than ${\displaystyle J}$-couplings, and ${\displaystyle {\hat {H}}_{J}}$ dominates.

Polarization

Before signals can be detected in a ZULF NMR experiment, it is first necessary to polarize the nuclear spin ensemble, since the signal is proportional to the nuclear spin magnetization. There are a number of methods to generate nuclear spin polarization. The most common is to allow the spins to thermally equilibrate in a magnetic field, and the nuclear spin alignment with the magnetic field due to the Zeeman interaction leads to weak spin polarization. The polarization generated in this way is on the order of 10⁻⁶ for tesla field-strengths.

An alternative approach is to use hyperpolarization techniques, which are chemical and physical methods to generate nuclear spin polarization. Examples include parahydrogen-induced polarization, spin-exchange optical pumping of noble gas atoms, dissolution dynamic nuclear polarization, and chemically-induced dynamic nuclear polarization.

Excitation and spin manipulation

NMR experiments require creating a transient non-stationary state of the spin system. In conventional high-field experiments, radio frequency pulses tilt the magnetization from along the main magnetic field direction to the transverse plan. Once in the transverse plan, the magnetization is no longer in a stationary state (or eigenstate) and so it begins to precess about the main magnetic field creating a detectable oscillating magnetic field.

The thermal equilibrium state of a ¹H-¹³C pair in high-field corresponds to a state in which both spins are polarized along the B₀ field, with ¹H polarization about 4 times higher than that of ¹³C spins. This is a stationary state at high field. If the field is non-adiabatically (rapidly) switched off, the state starts to evolve. The polarization oscillates between the ¹H and ¹³C spins at the J-coupling frequency (210 Hz in this example), and this gives rise to J-spectra in ZULF NMR.

In ZULF experiments, constant magnetic field pulses are used to induce non-stationary states of the spin system. The two main strategies consist of (1) switching of the magnetic field from pseudo-high field to zero (or ultra-low) field, or (2) of ramping down the magnetic field experienced by the spins to zero field in order to convert the Zeeman populations into zero-field eigenstates adiabatically and subsequently in applying a constant magnetic field pulse to induce a coherence between the zero-field eigenstates. In the simple case of a heteronuclear pair of J-coupled spins, both these excitation schemes induce a transition between the singlet and triplet-0 states, which generates a detectable oscillatory magnetic field. More sophisticated pulse sequences have been reported including selective pulses,^[5] two-dimensional experiments and decoupling schemes.^[6]

Signal detection

NMR signals are usually detected inductively, but the low frequencies of the electromagnetic radiation emitted by samples in a ZULF experiment makes inductive detection impractical at low fields. Hence, the earliest approach for measuring zero-field NMR in solid samples was via field-cycling techniques.^[7] The field cycling involves three steps: preparation, evolution and detection. In the preparation stage, a field is applied in order to magnetize the nuclear spins. Then the field is suddenly switched to zero to initiate the evolution interval and the magnetization evolves under the zero-field Hamiltonian. After a time period, the field is again switched on and the signal is detected inductively at high field. In a single field cycle, the magnetization observed corresponds only to a single value of the zero-field evolution time. The time-varying magnetization can be detected by repeating the field cycle with incremented lengths of the zero-field interval, and hence the evolution and decay of the magnetization is measured point by point. The Fourier transform of this magnetization will result to the zero-field absorption spectrum.

The emergence of highly sensitive magnetometry techniques has allowed for the detection of zero-field NMR signals in situ. Examples include superconducting quantum interference devices (SQUIDs), magnetoresistive sensors, and SERF atomic magnetometers. SQUIDs have high sensitivity, but require cryogenic conditions to operate, which makes them practically somewhat difficult to employ for the detection of chemical or biological samples. Magnetoresistive sensors are less sensitive, but are much easier to handle and to bring close to the NMR sample which is advantageous since proximity improves sensitivity. The most common sensors employed in ZULF NMR experiments are optically-pumped magnetometers, which have high sensitivity and can be placed in close proximity to an NMR sample.