New approaches to promote the use of magnets-free NMR for analysis in chemistry, biology, and medicine

The diverse work of Mainz-based physicists in the field of nuclear magnetic resonance is being boosted by a new highly application-oriented approach: In October 2020, Dr. Danila Barskiy will join Johannes Gutenberg University Mainz (JGU) to set up a group focusing on nuclear magnetic resonance (NMR) spectroscopy, the objective being to explore approaches that do not require magnetic fields for chemical, biological, and medical applications. To do so, he has been awarded a Sofja Kovalevskaja Award worth EUR 1.6 million by the Alexander von Humboldt Foundation. “We are very pleased that Danila Barskiy and his research have been honored in this way. His work not only covers new and previously unexplored aspects, but also excellently complements research we are already undertaking here in Mainz,” said Professor Dmitry Budker, Barskiy’s host at the Institute of Physics at JGU and at the Helmholtz Institute Mainz (HIM).

Nuclear magnetic resonance spectroscopy is a standard analytical technique used to study the structure and dynamics of materials and living objects. With its sister technology Magnetic Resonance Imaging (MRI), NMR spectroscopy is used in organic chemistry, biochemistry, and medicine with fluids being positioned particularly well to this type of analysis. However, NMR spectroscopy is reaching its limits: Due to the weak interaction between atomic nuclei and the applied magnetic field, signals produced by NMR-active nuclei are typically extremely low and, therefore, require high magnetic fields for detection. This rules out the possibility of developing portable point-of-care devices, among other things.

Researchers aim to develop compact and portable NMR devices

Sofja Kovalevskaja Award winner Dr. Danila Barskiy has been looking at ways of improving NMR spectroscopy for around ten years, most recently at the University of California, Berkeley, from where he will be making the transition to Mainz. He is pursuing various approaches with the aim of designing compact and portable NMR devices that will ultimately be as small as a chip and affordable for wide analytical markets. According to Barskiy, the problem is the following: “Despite improvements being made, most NMR systems are still not compact because they need field strengths of several Tesla in order to distinguish between the chemical signatures in an NMR spectrum.”

Barskiy’s new interdisciplinary group will focus on developing miniaturized, portable NMR sensors. These sensors would employ the principle of zero to ultra-low field magnetic resonance, or ZULF NMR for short, using optically pumped magnetometers which would not require any strong magnetic fields. In addition to applications in chemical and biomedical research, such sensors could find use for detecting metabolic disorders at an early stage.

By heading up a work group at Mainz, Barskiy also wants to develop hyperpolarizers for benchtop NMR spectrometers. Hyperpolarization improves the alignment of nuclear spins in a sample, thereby amplifying their NMR signals. The scientist predicts that application-specific hyperpolarizers for tabletop NMR devices may be soon available and that they will be about the size of a coffee machine. And with a tabletop NMR device, it will be possible to perform highly sensitive analyses of fuels, biofluids such as blood or urine, and food extracts. “This will democratize NMR spectroscopy by providing access to wider audiences and will accelerate technological progress in developing countries,” Barskiy emphasized.

Long-standing collaboration with experts from UC Berkeley paves the way for new research

This represents another positive result of the collaboration between UC Berkeley, in particular the laboratory of Professor Alexander Pines, and the Mainz group led by Professor Dmitry Budker. “We have been working very productively with Professor Pines and his team, including Dr. Barskiy, for many years and developed ZULF NMR together,” said Budker. “Danila Barskiy was one of the first to recognize the importance of this research in the biological and medical context.” Budker points out that Barskiy’s plans fit perfectly with the work being done at Mainz, which is also being pursued as part of the European Union’s ZULF NMR Marie Curie Innovative Training Network together with other European partners.

Danila Barskiy studied at Novosibirsk State University and earned a doctorate in physical chemistry for his research at the International Tomography Center (ITC SB RAS). In 2015, he began working as a postdoc at Vanderbilt University in Nashville, Tennessee, and subsequently joined Professor Alexander Pines’ team at the University of California, Berkeley in 2017. “The conditions in Mainz are unique for me. The planned collaborations and the resources available fit perfectly with the projects I am pursuing. Thanks to the Sofja Kovalevskaja Award, I can not only begin an independent research career, the multidisciplinary research in Germany is promoted as a whole,” emphasized Barskiy.

The Alexander von Humboldt Foundation gave the 2020 Sofja Kovalevskaja Award to eight international research talents aged between 29 and 36 years. It is one of the most richly endowed research awards in Germany and provides young researchers with venture capital for innovative projects at an early stage in their careers. They undertake research for up to five years at German universities and research institutions and build up their own work groups at their host institutes. The award is funded by the German Federal Ministry of Education and Research (BMBF).





Magnetic resonance without magnets: explained

Have you ever done MRI? If yes, you probably remember, it is not the most pleasant experience. MRI is claustrophobic, loud, and takes a long time. The main reason for these problems — high magnetic fields. Would it be cool to avoid them? Why can’t we simply do MRI without magnetic fields?

It turns out, we can do it though it is not that simple…

To understand why high magnetic fields are needed to obtain magnetic resonance images, we need to understand the principles of signal detection and the concept of polarization.

Some atomic nuclei possess a property called spin and, therefore, produce magnetic fields. These nuclei can simply be visualized as tiny magnets (magnetic dipoles) swirling in space as their hosts — molecules — move, rotate or vibrate depending on the physical state of the chemical system they comprise (solid, liquid or gas).

In the absence of external fields, nuclear spin orientations are random:

By placing the spins in high magnetic fields, one can align them in the same direction, creating “polarization”:

The level to which spins are polarized in the picture above is extremely exaggerated. In reality, even in the very high magnetic fields of modern magnetic resonance spectrometers and imaging scanners (thousands of times higher than the Earth’s magnetic field), nuclear spin polarization is only a fraction of percent. Complete (100%) polarization is also possible. However, instead of placing spins in high fields, it is practically more feasible to use a different approach. Physicists and chemists came up with several creative techniques to create extremely high polarization or hyperpolarization. Techniques based on chemical reactions (such as parahydrogen-induced polarization) are my favorite ones since they don’t require magnetic fields at all (in reality, polarization in such techniques can still be magnetic-field-dependent but the requirements are very modest).

Now about detection. Nuclear magnetism can be detected in several ways. I will talk about two ways and both of them are named after a brilliant physicist Michael Faraday.

The first detection technique is the so-called inductive detection. If spins are polarized and brought to the magnetic field perpendicular to the polarization axis, collective nuclear magnetization will start oscillating about the field axis. This phenomenon is called Larmor precession and it is a general result of classical physics (no quantum mechanics is necessary to explain it but quantum mechanics brings more fun). Oscillating magnetization can be picked up by a coil (solenoid) and a voltage across the coil terminals (electromotive force) will be generated according to Faraday’s law of induction:

Collective nuclear spin magnetization (M) is rotating about the magnetic field (B0). When a pre-polarized (or, simply speaking, pre-magnetized) sample is placed in such a field, M starts oscillating and its z-component generates an oscillating voltage across the coil’s terminals according to the Faraday’s law.

The problem is that it is really hard to construct magnets that produce high magnetic fields. Moreover, these fields need to be very uniform to be suitable for magnetic resonance purposes. This is why bores of MRI scanners are usually small – this way engineers achieve higher magnetic field homogeneity over the active volume.

The conventional logic of MRI engineers is straightforward: by using higher magnetic fields (1) we achieve higher nuclear spin polarization (indeed, polarization scales linearly with the applied field B0) and (2) we increase sensitivity of the detection (the voltage generated in the coil by Faraday’s law is linearly proportional to the oscillation frequency which, in turn, proportional to B0). Therefore, conventional MRI signal scales as a square of the magnetic field. Higher the field is — bigger the magnet should be, and we already reached the limits of our engineering capabilities to make the highest magnetic fields that are stable and uniform.

Now imagine that we already have high polarization produced without magnetic fields, for example, via chemical reactions mentioned above. Therefore, the signal is only linearly proportional to the magnetic field strength and even low magnetic fields can be used to produce high-quality images.

Recently my students and I played with MRI at the Earth’s magnetic field (compare 50 micro-Tesla vs. several Tesla of conventional MRI scanners). In our experiments, we simply bubbled a gas through a solution containing the required ingredients and picked up the signal by a coil:

Bubbling para-H2 gas through a solution containing a “polarizable” substrate and a SABRE catalyst.

The gas that we used is parahydrogen (para-H2) and the enhanced nuclear polarization in solution is the result of Signal Amplification By Reversible Exchange (SABRE) effect. This effect is a remarkable consequence of the interplay between chemical kinetics and nuclear spin dynamics. Unfortunately, explaining it here, in this post, would require many more words and even some math 🙂 But the main point is simple: bubbling the gas through the solution results in high nuclear polarization and, therefore, high MRI contrast. Without bubbling the gas, magnetic nuclei are oriented randomly and the signal is too low to be detected.

MRI of a SABRE solution at the Earth’s magnetic field (left) when para-H2 gas is bubbled and (middle) when the gas supply is ceased. (Right) Proton Nuclear Magnetic Resonance (NMR) spectrum of the solution at the Earth’s magnetic field.

This idea of avoiding magnetic fields can be extended even further. Indeed, if we don’t need a magnetic field to generate polarization, can we detect nuclear spin signals completely without magnets? This somewhat weird idea (magnetic resonance without magnets) is, in fact, one of the most outstanding achievements in magnetic resonance of the last decade. It was pioneered by chemists and physicists from UC Berkeley (laboratories of Alex Pines and Dmitry Budker).

It turns out that very weak magnetic signals can be detected by atomic magnetometers. These magnetometers use vapor cells, glass containers filled with the vapor of alkali metal. Atoms of alkali metals have unpaired valence electrons. Since electrons are tiny magnets in the same way as their nuclear brothers, they can also be polarized, this time by lasers. Properties of a laser beam passing through such polarized “electronic gas” can be modulated if electrons are subjected to additional magnetic fields. This is called the Faraday effect and it is a second way of detecting magnetic resonance signals. In a nutshell, electrons “feel” the magnetic fields around them in a very sensitive way, even if these fields are generated by nuclear spins from a sample placed on top of a vapor cell:

Magnetization (M) generated in a sample can be detected by atomic magnetometers. The most sensitive magnetometers operate at a near-zero magnetic field. Signal can be detected by lasers due to the Faraday effect using a variety of polarimetry schemes.

The end result of this signal detection scheme is the same as before – the voltage is generated and picked up, this time by a photodetector. If magnetic fields coming from the sample are time-dependent, the voltage is time-dependent as well and it can be converted into a spectrum containing useful chemical information about the sample.

You might wonder where this all can be applied? Well, it is worth mentioning Faraday here again:

I personally feel that MRI without magnets is not only possible but necessary. However, it will obviously not replace the conventional approach but will rather complement it. One may imagine sensitive magnetometers and Earth’s-field MRI applied to study chemicals at a large scale, to investigate processes taking place inside big industrial chemical reactors. Indeed, the Earth is the best magnet for these purposes – it is extremely homogeneous on the scale that we need. Nullifying magnetic fields (to produce zero field) is another option – remarkably, the sensitivity of atomic magnetometers is maximized at the near-zero-field conditions. It is hard to predict the direction that future will take but one thing is clear – magnetic resonance is entering a new era. So you better stay tuned! 😉

Further reading

Dipole-dipole interactions in NMR: explained

Interactions between spins are fundamental for understanding magnetic resonance. One of the most important ones is the magnetic dipole-dipole interaction. Spins act as tiny magnets, thus, they can interact with each other directly through space, pretty much the same way as classical magnetic dipoles (Figure 1). In NMR, dipole-dipole interactions very often determine lineshapes of solid-state samples and relaxation rates of nuclei in the liquid state.

Figure 1. Nuclear spin 1 produces the magnetic field that is sensed by a spin 2. Similarly, nuclear spin 2 produces the magnetic field that is sensed by a spin 1.

In many NMR textbooks, you can find the following expression for dipole-dipole (DD) Hamiltonian between two spins 1 and 2:

\hat{{H}}_{\rm DD} = d_{12} \left( 3 \hat{I}_{1z} \hat{I}_{2z} - \hat{\mathbf{I}}_1 \cdot \hat{\mathbf{I}}_2 \right)

where d_{12} = - \displaystyle\frac{\mu_0}{4 \pi} \displaystyle\frac{\gamma_1 \gamma_2 \hbar}{r^3} \frac{\left(  3 \cos^2{\theta} - 1 \right)}{2} , \gamma_1 and \gamma_2 are gyromagnetic ratios of the spins, r is the distance between them.  Meaning of the angle \theta can be seen from the Figure 2. But how was this expression derived?

Figure 2. Two nuclear spins, 1 and 2, are separated by the distance r\theta is defined as an angle between z-direction of the coordinate system and the vector connecting the two spins. If there is an external magnetic field B0, it is usually defined along the z-axis.

In physics, I rarely struggled with imagining abstract things and concepts, but rather, I was often lazy to do math thoroughly and derive equations from the beginning to the end. That’s why I decided to write here a full derivation, from the beginning to the end, for the Hamiltonian of interacting nuclear spins. We will start from a classical expression of the magnetic field produced by the dipole and finish by the analysis of a truncated dipole-dipole Hamiltonian. Usually, you don’t see such a full derivation in textbooks. Maybe, this is because textbooks have limited space… Luckily, here we do not have such limitations, thus, we can have some fun here! Let’s go then! 🙂

Classical equation describing the magnetic field \vec{\boldsymbol{B}}_{\rm dip} (\vec{\boldsymbol{r}}) produced by a magnetic dipole moment \vec{\boldsymbol{\mu}} is

\vec{\boldsymbol{B}}_{\rm dip} (\vec{\boldsymbol{r}}) = \displaystyle\frac{\mu_0}{4 \pi r^3} \left( 3 \left( \vec{\boldsymbol{\mu}} \cdot \hat{\boldsymbol{r}} \right) \hat{\boldsymbol{r}} - \vec{\boldsymbol{\mu}} \right)

where \hat{\boldsymbol{r}} = \displaystyle\frac{\vec{\boldsymbol{r}}}{|\vec{\boldsymbol{r}}|} is a unit vector. It is easy to show that \left( \vec{\boldsymbol{\mu}} \cdot \vec{\boldsymbol{r}} \right) \vec{\boldsymbol{r}} = \left( \vec{\boldsymbol{r}} \cdot \vec{\boldsymbol{r}}^{\: \intercal} \right) \vec{\boldsymbol{\mu}} .


\left( \vec{\boldsymbol{\mu}} \cdot \vec{\boldsymbol{r}} \right) \vec{\boldsymbol{r}} = \left( \mu_x r_x + \mu_y r_y + \mu_z r_z \right) \cdot \begin{pmatrix} r_x \\ r_y \\ r_z \end{pmatrix} = \begin{pmatrix} \mu_{x} r_{x} r_{x} + \mu_{y} r_{y} r_{x} + \mu_{z} r_{z} r_{x} \\ \mu_{x} r_{x} r_{y} + \mu_{y} r_{y} r_{y} + \mu_{z} r_{z} r_{y} \\ \mu_{x} r_{x} r_{z} + \mu_{y} r_{y} r_{z} + \mu_{z} r_{z} r_{z} \end{pmatrix}

which is the same as

\left( \vec{\boldsymbol{r}} \cdot \vec{\boldsymbol{r}}^{\: \intercal} \right) \vec{\boldsymbol{\mu}} = \begin{pmatrix} r_{x} r_{x} & r_{x} r_{y} & r_{x} r_{z} \\ r_{y} r_{x} & r_{y} r_{y} & r_{y} r_{z} \\ r_{z} r_{x} & r_{z} r_{y} & r_{z} r_{z} \end{pmatrix} \cdot \begin{pmatrix} \mu_x \\ \mu_y \\ \mu_z \end{pmatrix} = \begin{pmatrix} r_{x} r_{x} \mu_{x} + r_{x} r_{y} \mu_{y} + r_{x} r_{z} \mu_{z} \\ r_{y} r_{x} \mu_{x} + r_{y} r_{y} \mu_{y} + r_{y} r_{z} \mu_{z} \\ r_{z} r_{x} \mu_{x} + r_{z} r_{y} \mu_{y} + r_{z} r_{z} \mu_{z} \end{pmatrix}

So, we write the magnetic field produced by a magnetic dipole \vec{\boldsymbol{\mu}}_2 as

\vec{\boldsymbol{B}}_{\mu_2} = \displaystyle\frac{\mu_0}{4 \pi r^3} \left( 3 \left( \hat{\boldsymbol{r}} \cdot \hat{\boldsymbol{r}}^{\: \intercal} \right) \vec{\boldsymbol{\mu}}_2 - \vec{\boldsymbol{\mu}}_2 \right)

The energy of a magnetic dipole \vec{\boldsymbol{\mu}}_1 interacting with the magnetic field \vec{\boldsymbol{B}}_{\mu_2} produced by a magnetic dipole \vec{\boldsymbol{\mu}}_2 (dipole-dipole interaction) is therefore

E_{\rm DD} = - \left( \vec{\boldsymbol{\mu}}_1 \cdot \vec{\boldsymbol{B}}_{\mu_2} \right) = -\displaystyle\frac{\mu_0}{4 \pi r^3} \left( 3 \cdot \vec{\boldsymbol{\mu}}_1 \left( \hat{\boldsymbol{r}} \cdot \hat{\boldsymbol{r}}^{\: \intercal} \right) \vec{\boldsymbol{\mu}}_2 - \left( \vec{\boldsymbol{\mu}}_1 \cdot \vec{\boldsymbol{\mu}}_2 \right) \right)

The transition from classical to quantum mechanics is realized by substituting the measurable quantities by corresponding quantum mechanical operators:

E_{\rm DD} \rightarrow \hat{H}_{\rm DD} \quad \vec{\boldsymbol{\mu}}_1 \rightarrow \gamma_1 \hbar \hat{\mathbf{I}}_1\quad \vec{\boldsymbol{\mu}}_2 \rightarrow \gamma_2 \hbar \hat{\mathbf{I}}_2

\label{Eq_Hdd} \hat{H}_{\rm DD} = - \displaystyle\frac{\mu_0}{4 \pi } \displaystyle\frac{\gamma_1 \gamma_1 \hbar}{r^3} \left( 3 \cdot \hat{\mathbf{I}}_1 \left( \hat{\boldsymbol{r}} \cdot \hat{\boldsymbol{r}}^{\: \intercal} \right) \hat{\mathbf{I}}_2 - \left( \hat{\mathbf{I}}_1 \cdot \hat{\mathbf{I}}_2 \right) \right) = b_{12} \hat{\mathbf{I}}_1 \hat{\mathbf{D}} \hat{\mathbf{I}}_2

here b_{12} is a factor which depends only on the types of the nuclear spins and the distance between them, and a tensor of dipole-dipole interactions contains information about the mutual orientation of two spins:

\hat{\mathbf{D}} = 3 \cdot \left( \hat{\boldsymbol{r}} \cdot \hat{\boldsymbol{r}}^{\: \intercal} \right) - \hat{1}

here \hat{1} is a unit matrix. Note that we write Hamiltonian \hat{H}_{\rm DD} in units of [rad/s], that is why one \hbar is missing. In spherical coordinates:

\hat{\boldsymbol{r}} = \begin{pmatrix} \sin{\theta} \cos{\phi} \\ \sin{\theta} \sin{\phi} \\ \cos{\theta} \end{pmatrix}


\hat{\mathbf{D}} = \begin{pmatrix} 3 \sin^2{\theta} \cos^2{\phi} - 1 & 3 \sin^2{\theta} \cos{\phi} \sin{\phi} & 3 \sin{\theta} \cos{\theta} \cos{\phi} \\ 3 \sin^2{\theta} \cos{\phi} \sin{\phi} & 3 \sin^2{\theta} \sin^2{\phi} - 1 & 3 \sin{\theta} \cos{\theta} \sin{\phi} \\ 3 \sin{\theta} \cos{\theta} \cos{\phi} & 3 \sin{\theta} \cos{\theta} \sin{\phi} & 3 \cos^2{\theta} - 1 \end{pmatrix}

Looks good, doesn’t it? Now, let’s evaluate the product \hat{\mathbf{I}}_1 \hat{\mathbf{D}} \hat{\mathbf{I}}_2:

\hat{\mathbf{I}}_1 \hat{\mathbf{D}} \hat{\mathbf{I}}_2 = \\ \begin{pmatrix} \hat{I}_{1x} & \hat{I}_{1y} & \hat{I}_{1z} \end{pmatrix} \begin{pmatrix} 3 \sin^2{\theta} \cos^2{\phi} - 1 & 3 \sin^2{\theta} \cos{\phi} \sin{\phi} & 3 \sin{\theta} \cos{\theta} \cos{\phi} \\ 3 \sin^2{\theta} \cos{\phi} \sin{\phi} & 3 \sin^2{\theta} \sin^2{\phi} - 1 & 3 \sin{\theta} \cos{\theta} \sin{\phi} \\ 3 \sin{\theta} \cos{\theta} \cos{\phi} & 3 \sin{\theta} \cos{\theta} \sin{\phi} & 3 \cos^2{\theta} - 1 \end{pmatrix} \begin{pmatrix} \hat{I}_{2x} \\ \hat{I}_{2y} \\ \hat{I}_{2z} \end{pmatrix} =

\begin{pmatrix} \hat{I}_{1x} & \hat{I}_{1y} & \hat{I}_{1z} \end{pmatrix} \begin{pmatrix} \hat{I}_{2x} \left( 3 \sin^2{\theta} \cos^2{\phi} - 1 \right) + \hat{I}_{2y} \left( 3 \sin^2{\theta} \cos{\phi} \sin{\phi} \right) + \hat{I}_{2z} \left( 3 \sin{\theta} \cos{\theta} \cos{\phi} \right) \\ \hat{I}_{2x} \left( 3 \sin^2{\theta} \cos{\phi} \sin{\phi} \right) + \hat{I}_{2y} \left( 3 \sin^2{\theta} \sin^2{\phi} - 1 \right) + \hat{I}_{2z} \left( 3 \sin{\theta} \cos{\theta} \sin{\phi} \right) \\ \hat{I}_{2x} \left( 3 \sin{\theta} \cos{\theta} \cos{\phi} \right) + \hat{I}_{2y} \left( 3 \sin{\theta} \cos{\theta} \sin{\phi} \right) + \hat{I}_{2z} \left( 3 \cos^2{\theta} - 1 \right) \end{pmatrix} =

= \hat{I}_{1x} \hat{I}_{2x} \left( 3 \sin^2{\theta} \cos^2{\phi} - 1 \right) + \hat{I}_{1x} \hat{I}_{2y} \left( 3 \sin^2{\theta} \cos{\phi} \sin{\phi} \right) + \hat{I}_{1x} \hat{I}_{2z} \left( 3 \sin{\theta} \cos{\theta} \cos{\phi} \right) + + \hat{I}_{1y} \hat{I}_{2x} \left( 3 \sin^2{\theta} \cos{\phi} \sin{\phi} \right) + \hat{I}_{1y} \hat{I}_{2y} \left( 3 \sin^2{\theta} \sin^2{\phi} - 1 \right) + \hat{I}_{1y} \hat{I}_{2z} \left( 3 \sin{\theta} \cos{\theta} \sin{\phi} \right) + + \hat{I}_{1z} \hat{I}_{2x} \left( 3 \sin{\theta} \cos{\theta} \cos{\phi} \right) + \hat{I}_{1z} \hat{I}_{2y} \left( 3 \sin{\theta} \cos{\theta} \sin{\phi} \right) + \hat{I}_{1z} \hat{I}_{2z} \left( 3 \cos^2{\theta} - 1 \right)

Let’s color terms to make it easier grouping them:

\hat{I}_{1x} \hat{I}_{2x} \left( 3 \sin^2{\theta} \cos^2{\phi} - 1 \right) + \hat{I}_{1x} \hat{I}_{2y} \left( 3 \sin^2{\theta} \cos{\phi} \sin{\phi} \right) + \hat{I}_{1x} \hat{I}_{2z} \left( 3 \sin{\theta} \cos{\theta} \cos{\phi} \right) + \hat{I}_{1y} \hat{I}_{2x} \left( 3 \sin^2{\theta} \cos{\phi} \sin{\phi} \right) + \hat{I}_{1y} \hat{I}_{2y} \left( 3 \sin^2{\theta} \sin^2{\phi} - 1 \right) + \hat{I}_{1y} \hat{I}_{2z} \left( 3 \sin{\theta} \cos{\theta} \sin{\phi} \right) + \hat{I}_{1z} \hat{I}_{2x} \left( 3 \sin{\theta} \cos{\theta} \cos{\phi} \right) + \hat{I}_{1z} \hat{I}_{2y} \left( 3 \sin{\theta} \cos{\theta} \sin{\phi} \right) + \hat{I}_{1z} \hat{I}_{2z} \left( 3 \cos^2{\theta} - 1 \right)

Groupling the red terms gives

\left( \hat{I}_{1x} \hat{I}_{2x} \cos^2{\phi} + \hat{I}_{1y} \hat{I}_{2y} \sin^2{\phi} \right) 3 \sin^2{\theta} - \left( \hat{I}_{1x} \hat{I}_{2x} + \hat{I}_{1y} \hat{I}_{2y} \right)

Let’s not forget about intrinsic connections of spin angular momentum with raising and lowering operators:

\hat{I}_{1x} \hat{I}_{2x} = \displaystyle\frac{\left( \hat{I}_{1+} + \hat{I}_{1-} \right)}{2}\displaystyle\frac{\left( \hat{I}_{2+} + \hat{I}_{2-} \right)}{2} = \displaystyle\frac{1}{4} \left( \hat{I}_{1+} \hat{I}_{2+} + \hat{I}_{1+} \hat{I}_{2-} + \hat{I}_{1-} \hat{I}_{2+} + \hat{I}_{1-} \hat{I}_{2-} \right)

\hat{I}_{1y} \hat{I}_{2y} = \displaystyle\frac{\left( \hat{I}_{1+} - \hat{I}_{1-} \right)}{2 i}\displaystyle\frac{\left( \hat{I}_{2+} - \hat{I}_{2-} \right)}{2 i} = -\displaystyle\frac{1}{4} \left( \hat{I}_{1+} \hat{I}_{2+} - \hat{I}_{1+} \hat{I}_{2-} - \hat{I}_{1-} \hat{I}_{2+} + \hat{I}_{1-} \hat{I}_{2-} \right)

Therefore, grouping the red terms gives

\left( \hat{I}_{1x} \hat{I}_{2x} \cos^2{\phi} + \hat{I}_{1y} \hat{I}_{2y} \sin^2{\phi} \right) 3 \sin^2{\theta} - \left( \hat{I}_{1x} \hat{I}_{2x} + \hat{I}_{1y} \hat{I}_{2y} \right) = \left( \hat{I}_{1+} \hat{I}_{2-} + \hat{I}_{1-} \hat{I}_{2+} \right) \frac{3}{4} \sin^2{\theta} + \left( \hat{I}_{1+} \hat{I}_{2+} + \hat{I}_{1-} \hat{I}_{2-} \right) \frac{3}{4} \sin^2{\theta} \cdot \left( \cos{ 2 \phi} \right) - \frac{1}{2} \left( \hat{I}_{1+} \hat{I}_{2-} + \hat{I}_{1-} \hat{I}_{2+} \right) = \left( \hat{I}_{1+} \hat{I}_{2-} + \hat{I}_{1-} \hat{I}_{2+} \right) \frac{1}{4}  \left(1 - 3 \cos^2{\theta} \right) + \left( \hat{I}_{1+} \hat{I}_{2+} + \hat{I}_{1-} \hat{I}_{2-} \right) \frac{3}{4} \sin^2{\theta} \cdot \left( \cos{ 2 \phi} \right)

Groupling the blue terms gives

\left( \hat{I}_{1x} \hat{I}_{2y} + \hat{I}_{1y} \hat{I}_{2x} \right) 3 \sin^2{\theta} \cos{\phi} \sin{\phi} =  \left( \hat{I}_{1+} \hat{I}_{2+} - \hat{I}_{1-} \hat{I}_{2-} \right)\displaystyle\frac{3}{4} \sin^2{\theta} \cdot \left( - i \sin{2 \phi} \right)

where we took into consideration that

\hat{I}_{1x} \hat{I}_{2y} =\displaystyle\frac{\left( \hat{I}_{1+} + \hat{I}_{1-} \right)}{2} \frac{\left( \hat{I}_{2+} - \hat{I}_{2-} \right)}{2 i} = \frac{1}{4 i} \left( \hat{I}_{1+} \hat{I}_{2+} - \hat{I}_{1+} \hat{I}_{2-} + \hat{I}_{1-} \hat{I}_{2+} - \hat{I}_{1-} \hat{I}_{2-} \right)
\hat{I}_{1y} \hat{I}_{2x} = \displaystyle\frac{\left( \hat{I}_{1+} - \hat{I}_{1-} \right)}{2 i} \frac{\left( \hat{I}_{2+} + \hat{I}_{2-} \right)}{2} = \frac{1}{4 i} \left( \hat{I}_{1+} \hat{I}_{2+} + \hat{I}_{1+} \hat{I}_{2-} - \hat{I}_{1-} \hat{I}_{2+} - \hat{I}_{1-} \hat{I}_{2-} \right)

Red and blue terms can be combined nicely to form

\left( \hat{I}_{1+} \hat{I}_{2-} + \hat{I}_{1-} \hat{I}_{2+} \right) \displaystyle\frac{1}{4} \left(1 - 3 \cos^2{\theta} \right) + \hat{I}_{1+} \hat{I}_{2+} \left( \frac{3}{4} \sin^2{\theta} \cdot e^{- 2 i \phi} \right) + \hat{I}_{1-} \hat{I}_{2-} \left( \frac{3}{4} \sin^2{\theta} \cdot e^{+ 2 i \phi} \right)

Now let’s focus on purple terms:

\left( \hat{I}_{1x} \hat{I}_{2z} + \hat{I}_{1z} \hat{I}_{2x} \right) \left( \hat{I}_{1x} \hat{I}_{2z} + \hat{I}_{1z} \hat{I}_{2x} \right)  \left( 3 \sin{\theta} \cos{\theta} \cos{\phi} \right) + \left( \hat{I}_{1y} \hat{I}_{2z} + \hat{I}_{1z} \hat{I}_{2y} \right)  \left( 3 \sin{\theta} \cos{\theta} \sin{\phi} \right) = \left( \left( \hat{I}_{1+} \hat{I}_{2z} + \hat{I}_{1-} \hat{I}_{2z} + \hat{I}_{1z} \hat{I}_{2+} + \hat{I}_{1z}\hat{I}_{2-} \right) \cos{\phi} \right) \displaystyle\frac{3}{4} \sin{2 \theta} + \left( \left( -i \hat{I}_{1+} \hat{I}_{2z} + i \hat{I}_{1-} \hat{I}_{2z} -i \hat{I}_{1z} \hat{I}_{2+} + i \hat{I}_{1z} \hat{I}_{2-} \right) \sin{\phi} \right) \frac{3}{4} \sin{2 \theta} = \left( \hat{I}_{1+} \hat{I}_{2z} + \hat{I}_{1z} \hat{I}_{2+} \right) \left( \frac{3}{4} \sin{2 \theta} \right) e^{-i \phi} + \left( \hat{I}_{1-} \hat{I}_{2z} + \hat{I}_{1z} \hat{I}_{2-} \right) \left( \frac{3}{4} \sin{2 \theta} \right) e^{+i \phi}

Overall, we have split our dipolar Hamiltonian into 6 term, so-called “Dipolar Alphabet”:

\hat{\mathbf{I}}_1 \hat{\mathbf{D}} \hat{\mathbf{I}}_2 = \hat{A} + \hat{B} + \hat{C} + \hat{D} + \hat{E} + \hat{F}


\hat{A} \quad = \quad \hat{I}_{1z} \hat{I}_{2z} \left( 3 \cos^2{\theta} - 1 \right)

\quad\quad \hat{B} \quad = \quad \left( \hat{I}_{1+} \hat{I}_{2-} + \hat{I}_{1-} \hat{I}_{2+} \right) \cdot \displaystyle\frac{\left(1 - 3 \cos^2{\theta} \right)}{4}

\quad\quad \hat{C} \quad = \quad \left( \hat{I}_{1+} \hat{I}_{2z} + \hat{I}_{1z} \hat{I}_{2+} \right) \left( \displaystyle\frac{3}{4} \sin{2 \theta} \right) e^{-i \phi}

\quad\quad \hat{D} \quad = \quad \left( \hat{I}_{1-} \hat{I}_{2z} + \hat{I}_{1z} \hat{I}_{2-} \right) \left( \displaystyle\frac{3}{4} \sin{2 \theta} \right) e^{+i \phi}

\hat{E} \quad = \quad \hat{I}_{1+} \hat{I}_{2+} \left( \displaystyle\frac{3}{4} \sin^2{\theta} \right) e^{- 2 i \phi}

\quad\quad \hat{F} \quad = \quad \hat{I}_{1-} \hat{I}_{2-} \left( \displaystyle\frac{3}{4} \sin^2{\theta} \right) e^{+ 2 i \phi}

To summarize, the Hamiltonian of two interacting spins is a 4 \times 4 matrix composed of 6 operators. Each of the letters of the dipolar alphabet corresponds to certain matrix elements in the final Hamiltonian (Figure 3).

Figure 3. If written in Zeeman basis (αααβ, βα and ββ), dipole-dipole Hamiltonian can be split into the following 6 terms of the “Dipolar Alphabet”.

Without an externally imposed direction in space (for example, in the case of two equivalent spins in zero magnetic field), all of the terms of the dipole-dipole Hamiltonian need to be used for calculating an NMR spectrum. This is because all orientations in space are equivalent. However, in the presence of the external high magnetic field, the Hamiltonian can be simplified via the use of so-called “secular approximation”.

The secular approximation concerns the case where the Hamiltonian is the sum of two terms:

\hat{H} = \hat{A} + \hat{B}

where \displaystyle\hat{A} is a “large” operator and \hat{B} is a “small” operator. In our case, \hat{A} can be an operator describing the interaction with the magnetic field (Zeeman Hamiltonian) and \hat{B} is DD Hamiltonian. Eigenstates of the Zeeman Hamiltonian are familiar αααβ, βα, ββGenerally, \hat{B} does not commute with \hat{A} , therefore, if written in the eigenbasis of \hat{A}, it has finite elements everywhere.

The secular approximation for \hat{B} means that we leave only the blocks that correspond to the eigenvalue structure of the operator \hat{A} (Figure 4) and disregard all other elements.

Figure 4. Energy level structure of the Zeeman Hamiltonian: lowest energy level (E1) corresponds to the state αα, energy levels E2 and E3 correspond to the states αβ and βα. The highest energy level (E4) corresponds to the nuclear spin state ββ. The essence of secular approximation is to make the to-be-simplified Hamiltonian match the eigenvalue structure of the main Hamiltonian. One can see from Figure 3 that the first two terms of the DD Hamiltonian will match the structure of the Zeeman Hamiltonian.

In general, we can omit a matrix element b_{nm} that is much smaller than

|b_{mn}| \ll |E_m - E_n|

For homonuclear case (e.g., two interacting protons), this means that only the first two terms of the dipolar Alphabet will survive:

\hat{\mathbf{I}}_1 \hat{\mathbf{D}} \hat{\mathbf{I}}_2 = \hat{A} + \hat{B} =

= \hat{I}_{1z} \hat{I}_{2z} \left( 3 \cos^2{\theta} - 1 \right) + \left( \hat{I}_{1+} \hat{I}_{2-} + \hat{I}_{1-} \hat{I}_{2+} \right) \cdot \displaystyle\frac{\left(1 - 3 \cos^2{\theta} \right)}{4} =

= \hat{I}_{1z} \hat{I}_{2z} \left( 3 \cos^2{\theta} - 1 \right) - \left( \hat{I}_{1x} \hat{I}_{2x} + \hat{I}_{2y} \hat{I}_{2y} \right) \cdot \displaystyle\frac{\left(3 \cos^2{\theta} - 1\right)}{2} =

= \displaystyle\frac{\left( 3 \cos^2{\theta} - 1\right)}{2} \cdot \left( 2 \hat{I}_{1z} \hat{I}_{2z} + \hat{I}_{1z} \hat{I}_{2z} - \hat{I}_{1z} \hat{I}_{2z} - \left( \hat{I}_{1x} \hat{I}_{2x} + \hat{I}_{2y} \hat{I}_{2y} \right) \right) =

= \displaystyle\frac{\left(3 \cos^2{\theta} - 1\right)}{2} \cdot \left( 3 \hat{I}_{1z} \hat{I}_{2z} - \hat{\mathbf{I}}_1 \cdot \hat{\mathbf{I}}_2 \right)

Overall, this is how you go from the classical description of the magnetic field of the dipole to the truncated form of the Hamiltonian in the high nagnetic field. In the next post I will show how this Hamiltonian leads to the characteristic lineshape of the NMR line for solids.