Young padawans mentoring, you seek: How I did research with 10 undergraduate students

Doing research is a lot of fun! 🙂 You can do things that no one has ever done before. It is like fulfilling your childish curiosities by playing around with nature, unraveling its mysterious principles, and sometimes finding useful applications for things that were previously considered to be purely fundamental. With time your experience grows – you finish your Ph.D., and at some point, you realize that you can hardly find time to check all of your ideas alone. Through this experience, I understood that I definitely need support. The support of undergraduate students! 🙂

In this post, I would like to share some thoughts and realizations I learned from working with undergrads. While reading, try to focus on what could be learned from my experience and not on my particular situation. Most likely, a lot of the things I describe cannot be applied to every postdoc or graduate student. However, I believe that experience I have obtained can be extremely valuable for other Jedi masters looking to mentor their padawans!

Take-home messages:

  • Undergraduate students can be extremely motivated and capable of solving complex problems
  • A mentor needs to take into account different personalities when working with undergraduates and assign tasks based on this
  • Working in groups can be fun but can diminish the sense of individual accomplishment
  • Undergraduate students are willing to learn (even if it takes more time to accomplish their goals)
  • Focus. Having one or two motivated students can be better than having a big group of unmotivated individuals
  • Undergrads are not small graduate students: they have a limited schedule and, usually, need step-by-step instructions to successfully complete tasks

My journey of mentoring 10 undergraduate students

“Always two there are, no more, no less. A master and apprentice.”

In October 2016, I was invited to work at the Pines Lab. It is hard to explain how excited I was! When I learned that I had the privilege of working under Alex Pines – an absolute legend, mentor, and leader in the field of magnetic resonance – I was over the moon. I was also thrilled by the atmosphere of UC Berkeley, where the concentration of IQ per square meter is mind-blowing. Although there were a lot of advantages related to my new position, I was also faced with some challenges in the new lab. One of which was the different dynamics amongst my lab associates. In previous labs, the associates consisted primarily of graduate students and postdocs. However, in Pines Lab, we have a different approach. As a well-accomplished scientist, Alex now believes that funding is best spent on postdocs and therefore, only a few of us are now working in the lab. However, we can’t accomplish everything alone and needed to reach out to others for help. Thankfully,  UC Berkeley provides a win-win solution for this challenge: postdocs are open to mentoring students and the campus is full of bright and motivated undergrads eager to do research!

I started a quest for my Jedi padawans by sending an E-mail to the chemical engineering undergraduate academic advisor requesting to advertise “the opening” for summer positions in the Pines Lab. Maybe my ad was just too good to refuse, or the UC Berkley students were absolutely eager for work, but in only two days I received more than thirty (thirty!) E-mails from students who were willing to work in the Lab! The hardest part was choosing only a small fraction of them.  Long story short, this is how I ended up having 10 students… Yes, believe me, I could not take any less as they were all extremely motivated, knowledgeable, and willing to learn more at the same time! This being said, be sure that you take the right amount of students required for the project that you have in mind.

Today, after more than a year of working with my students, teaching and mentoring them, finishing papers together and working on new ones, I feel happy and satisfied with my selection and decisions. My experience allowed me not only to learn a lot about how undergraduate students think and about what motivates them and what can boost their productivity, but it also allowed me to find the mentoring/teaching style that I feel most comfortable with.

The best students

“Truly wonderful, the mind of a child is.”

I found that, surprisingly, resumes of undergrads do not always correlate directly with their performance in the lab. However, it does not mean that you should completely disregard their scholarly performance when you are looking for them. The successful student should (1) have enough knowledge to be able to quickly grasp the ideas and concepts you work with but at the same time (2) have enough curiosity to learn more about things they don’t know about. Unfortunately, NMR as a research topic requires at least a basic understanding of quantum mechanics, therefore making it an overall challenging subject for freshmen and sophomore students (however, there could be exceptions!). Therefore, in my opinion, the best way to find a prospective student is to talk to them in person, ask about their interests/hobbies, what type of work they like and what they don’t like. From my conversations with potential undergrads, I realized that those who like a more structured way of working and need a lot of supervising would not be the best match for me.

I found that for me, the most important students’ qualities are:

  • Ability to work independently
  • Responsibility
  • Enthusiasm about doing research
  • Personality matching (see below)

The last point, personality matching, is the main lesson I gained from my experience. Personality matters much more than anything else in research with undergrads, and I feel that this can be even truer for higher levels of graduate school in which professor-student interactions play a crucial role in a lab’s success!

Personality matters

“Good relations with the Wookiees, I have.”

Have you ever heard about the Myers-Briggs 16 personality types? If not, you should learn about it as soon as possible. Basically, this “theory” allows you to rationalize your own behavior and better understand the motivations of different types of people. It also explains why some individuals are naturally stronger in certain tasks and others are better at different ones. Therefore, when managing undergraduate research, you can always find a fulfilling and interesting problem for a student if you identify his/her personality type and look at things creatively.

Find the right problem for the right people. If you think the people are not right for the problems you post, chances are, it is not the right problem for the people you have.

For example, I suggested two of my students work on the project of using Earth-field MRI combined with SABRE hyperpolarization. It soon became very clear to me that one of the students was more interested in studying the basics of MRI and the way the Earth-field imaging instrument works than the other.  At one point I caught myself thinking that the second student was just lazier than the first one and much less motivated about going to the lab in general… However, purely by accident, I learned that the second student was interested in 3D modeling. We had another project in the lab that required machining a chemical reactor. Therefore, I asked him to make a 3D modeling of the reactor and found that he loved it. Ever since then, he was an absolutely different person – working hard and passionately while finishing the task much earlier than I expected. It was not the laziness of the student but rather my misunderstanding of his interests that did not allow him to be as motivated as he can be at the beginning of our work.

Take-home message – find the right problem for the right people. If you think the people are not right for the problems you post, chances are, it is not the right problem for the people you have. This is especially true for doing research with undergrads: they are volunteering their time to help you and it is your fault if are not motivated!

Finding the right project

“Already know you that which you need.”

I found that it is very important for students to work on problems that have clearly posted goals. For example, when choosing between two projects, one of which is very cool but relatively abstract and the second one which is less intellectually complicated but more practical, students often choose the second one. They want to see the immediate outcome of their research even though it is sometimes hard to get enough results given the limits of their residence in the lab. The most compelling project for undergrads is one that to them, makes an observable difference in the world!

The most compelling project for undergrads is one that to them, makes an observable difference in the world!

One of these application-oriented projects was the building of the para-H2-based polarizer. I explained to my students that I want to make a device that can automatically produce boluses of hyperpolarized liquids (for example, aqueous solutions of 13C-hyperpolarized metabolites such as pyruvate or lactate). Once produced, these boluses can be injected into animals and their metabolism can be monitored in vivo using NMR/MRI.

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Nick and Lucia conducting SABRE experiments.

This idea is not new and many researchers work on different aspects of this problem. However, one of the biggest issues preventing the widespread applicability of para-H2-based techniques is the presence of platinum-group metals in solutions with hyperpolarized molecules. Obviously, injecting even trace quantities of the metals in vivo should be avoided.

Therefore, we decided to focus on this part of the problem and at the same time started building para-H2-based polarizer. Two of my students were working on the testing of different scavengers’ performance by means of inductively coupled plasma atomic emission spectroscopy (ICP-AES) and three of the students were helping to build a so-called “hyper-cart”, a transportable cart containing all of the components necessary for producing hyperpolarized compounds.

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Elizabeth, Lucia, and Nevin are working on building the parahydrogen “hyper-cart”.

After performing a lot of tests, we were able to identify the nature of two commercially available metal scavengers that can efficiently and very quickly clear the hyperpolarized solutions from the trace metal quantities. Since the result was important, we decided to submit a paper describing the metal scavenging process to the Journal of Physical Chemistry Letters and it was successfully published there. It was a pleasure to see the students’ motivation when, even during their midterm exams, they were still able to find time to stay at work late to finish the figures and tables for the paper! Overall, despite challenges, this was a good research project for undergrads: a simple and clear idea resulting in a measurable advancement to the field.

Managing vs. Mentoring

“Always pass on what you have learned.”

One of the biggest realizations that occurred to me while working with my students was understanding of the difference between managing and mentoring.

Managing is about organizing, making plans, setting up goals, splitting them into micro-goals, setting up the deadlines and so on. In other words, all the stuff that I hate… Mentoring, on the other hand, is giving padawans the opportunity to learn by doing their own work, providing resources and support, discussing the best ways to achieve the goals (as opposed to stating them), referring to the resources with useful information and troubleshooting when things don’t work. Through my experience, I found that mentoring is something I truly enjoy. You can clearly see the growth of students who are willing to learn!

Let’s take the previous example of the para-H2 cart. One of my chemical engineering students, Vincent, told me during his interview that he likes automation and programming. Since we needed automatically actuated valves to run gases and liquids, I suggested that he build an Arduino-based setup to control the valves. I did not have the background to teach him everything about Arduino, but I knew where he could find information about how to do it. Eventually, he created a very good setup that was extremely helpful. Check out this video created by him explaining all the components of his setup:

This is why personality matching is important. Some students would require more of a managing advisor while my approach leaned towards mentoring and allowing the student to figure out the details by himself. Not everyone would be able to do what Vincent has done, but his desire to learn, coupled with his desire to make a significant contribution to the project led to a successful outcome.

Journal Clubs

“Powerful you have become, the dark side I sense in you.”

I also found another valuable tool for mentoring students – Journal Clubs. I first learned about this type of meetings looking at Mark Does’ lab at the Vanderbilt Institute of Imaging Science. In Mark’s lab, students and postdocs meet once a week (in addition to weekly group meetings) and discuss a paper they want to learn more about. Mark does not even show up there – it is the students’ task to meet and study together.

My journal clubs were slightly different than the ones at Vanderbilt. Since my students are only undergrads, letting them learn completely by themselves would not be the most productive option. Therefore, a schedule of students was created before each semester. For each journal club, one of the students had to choose a paper of interest. Ideally, it should be a paper related to our research but this was not necessary. After choosing a topic of interest, a student would send his paper for the rest of the group in advance to read it and prepare for the discussion. When arriving at the journal club, the student would first make a short presentation about the paper and then discuss it with the rest of the group. I would then show up after 30-40 mins and work together to answer any questions the students had. We also discussed NMR-related and general scientific questions after I arrived.

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Joseph is explaining approaches to study chemical kinetics by NMR.

Based on the students’ reviews, they liked the journal clubs a lot. It helps them understand the research process and how to work with each other without a lot of guidance. I originally realized that students were not as eager to ask each other questions while I was in the room. This is why I decided to let them have 30-40 minutes on their own: this way I always entered a room full of discussion and exploration. They didn’t need my micro-management and ultimately worked together to facilitate ideas and find the answers to unknown questions.

Conclusions

“Do or do not. There is no try.”

Here is the biggest realization. Undergraduate students are not small graduate students. While graduate students have the time and ability to focus on projects, undergraduates have other responsibilities to classes, college activities, and simply navigating their busy university life. Therefore, they need to be treated differently than graduate students. Although optimism is great, do not be discouraged if you undergraduate students have other time commitments and responsibilities that interfere with their projects. They are not smaller, younger graduate students but rather have unique requirements and standards of learning all of their own.

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Hyun, Nick, James, Lucia, Nevin, and Dario making a birthday gift for me.

I want to express gratitude and say thanks to all undergraduate students I had a pleasure to mentor: Patricia Buenbrazo, Nevin Widarman, Hubert Situ, James (Xingyang) Li, Dario Gelevski, Vincent Stevenson, Lucia Ke, Elizabeth Chyn, Nick (Hao) Zhang, Hyun Park, Sean Littleton. Some of them have already graduated, some of them are still working on exciting projects and I hope to have a chance to mentor many more! 🙂 I also want to say a lot of thanks to Jessica Andrews who helped me to write this post and suggested the Master Yoda quotes idea. Check out her website: she writes about TV and movies and she is very passionate about it!

And of course, let the force be with you.

And with your undergraduate students! 🙂

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.

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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?

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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}} .

Indeed,

\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}

Therefore,

\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}

where

\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).

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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.

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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.