Table of Contents

1. Introduction
2. Level 1: The Basics
3. Level 2: The Theory
   1. [N]uclear…
   2. …[M]agnetic…
   3. …[R]esonance
4. Level 3: The Acquisition
   1. The 90° Pulse
   2. Inter-pulse Evolution
      1. The Rotating Frame
      2. Chemical Shift Evolution
5. Level 4: The Spectrum
6. Level n: Other Concepts
   1. Relaxation Pathways
      1. The T1 Pathway
      2. The T2 Pathway
   2. Decoupling

Introduction

If you think about it—really hard—we are all just little tiny magnets, aren’t we?

Nuclear magnetic resonance (NMR) spectroscopy is an extremely powerful tool with many interesting applications. It can be used to image tissue, identify chemical structures, map chemical taxonomy, and more!

So, what exactly is this magic?
This guide will explain the what, the why, and the how of NMR spectroscopy!

Level 1: The Basics

The goal of NMR is to analyze a sample. Usually, this sample is dissolved in an NMR inactive solvent, and placed in an NMR tube.

Solid-state NMR, where a pure crystalline sample is analyzed without solvent is also possible. Also, an MRI machine in a hospital is just an NMR instrument—making the PATIENT the SAMPLE!

Level 2: The Theory

NMR relies on a few basic principles that are fundamental to physics and chemistry.

[N]uclear…

Every “thing” in the universe is made up of atoms. These atoms can have different properties, but they are all similar in their construction. They are all made of neutrons, protons, and electrons. An NMR active nucleus is one that has a property called spin. One such atom—and the most commonly measured—is the proton (¹H). Others include ¹³C, ¹⁵N, and ³¹P! The spin causes these atoms to act like tiny bar magnets.

The NMR inactive solvents mentioned before are those which have had their protons exchanged with deuterium (²H). Deuterium is inactive because it has an even number of protons (1p⁺) and neutrons (1n⁰)! Common solvents include CDCl₃, DMSO-d₆, MeOD, D₂O, and Acetone-d₆.

With that in mind, you can measure any organic molecule!

NMR measures the presence of NMR active nuclei which are those with an odd number of particles in its nucleus.

…[M]agnetic…

So, now you have a molecule with a bunch of NMR active atoms. Now what? Well, you put them into a really strong magnetic field, of course!

An NMR instrument uses a supercooled, superconductive coil to generate an intense magnetic field called B₀ in a ring around your sample. When you put NMR active nuclei into a very strong magnetic field, they begin to align themselves parallel to the magnetic field lines. The population of nuclei that align with the field is called P(α). The population that aligns against the field is called P(β).

Since the α population is aligned with the field, it is a slightly more favourable state. Thus, there is an extremely slight bias in the population in P(α). Considering that α and β populations cancel out 1-to-1 (they are aligned oppositely), there is a slight bulk magnetization with the magnetic field due to the slight bias overpopulation of P(α).

Figure 1. Left: total ensemble of spins in the sample; Right: vector of the bulk magnetization.

Conventionally, this “bulk magnetization” initially points towards the positive Z axis, or “+Z”. Note that this is different from typical 3D axes; the Z axis – not the Y axis – is the vertical one in NMR.

Putting NMR active nuclei in a magnetic field aligns them with (α) or against (β) the field lines. The population difference between the two states leads to bulk magnetization along the +Z.

…[R]esonance

A nucleus with spin also precesses. This YouTube video explains nuclear precession very well. If you don’t want to watch it, here is my best attempt at explaining it.

A nucleus with spin has angular momentum. When the bulk magnetization is aligned parallel to the magnetic field, this momentum is hard to observe. However, when the bulk magnetization is between +Z and -Z, the rotation of the nucleus is observed as precession.

— Trev

This precession occurs at an angular frequency (the frequency of one full rotation) which is characteristic for a specific nucleus at a specific magnetic field strength. This angular frequency of precession is called the Larmor frequency.

Level 3: The Acquisition

This one gets a little technical… bear with me!

Now that everything is aligned, we have just one problem. Nothing is really happening. Things are spinning, but we can’t see it. This is because, now that all of the atoms are either spin up or spin down, they are precessing so that their bulk magnetization is pointing along the Z axis—straight up and down.

For visual reference, imagine looking at a spinning top from the top (and that it is spinning but locked in place). You cannot observe any change, as it all averages out pretty quick.

— Trev

To actually detect something, you have to change the direction of B₀ to have a non-zero component in the X-Y plane! That way, we can measure an oscillating current in a coil of wire. Though not entirely accurate, you can imagine this as touching the spinning top. The top will start to wobble, then try to return back to its original position – all while still spinning.

There is more information about pulse sequences in this post.
This post just covers the basic ¹H sequence for now.

The 90° Pulse

The bulk magnetization can be moved using a radio-frequency (RF) pulse to apply a sort of “torque” to the atoms, effectively yanking the spins in a different direction. This is predictable, as long as we know the strength (in Teslas) of the magnetic field. Thus, we can fine tune exactly how many degrees we turn the vector by calculating the amount of time needed to send the RF pulse. This is known as the pulse-width and is usually in the microseconds. Usually, the 180° pulse is calculated first, then divided by two to get the more common 90° pulse.

The NMR instrument that I use has a 9.4 T magnet, meaning that the pulse frequency needed to affect protons is near 400 MHz. We usually name the instruments after this frequency, so I would call it a 400 MHz NMR instrument!

When you rotate a magnet inside a coil of wire, the electrons flow in the coil with a sinusoidal period. The frequency of this wave is equal to the angular frequency I mentioned here. This is then measured as a wave in the time domain. In a molecule with many different protons in different chemical environments, this time domain signal contains the summation of all the signals. Imagine adding two sine waves together. This is important for later.

Figure 2. Two sine waves (blue & green) added together to make a compound waveform (red wave).

Inter-pulse Evolution

Evolution describes the change of atoms between events such as pulses. The time between pulses in an experiment can be the difference between having a signal and not! Without getting into TOO much detail, there are two major types of evolution. The first is chemical shift evolution and the second is called scalar evolution. Scalar evolution has a smaller effect overall, but it is important for building interesting pulse sequences!

The Rotating Frame

Evolution can be described using the rotating frame perspective. Imagine a merry-go-Round with three people on it. You are watching this merry-go-round spin with an angular frequency of ω from the outside perspective. The three people on this merry-go-round represent protons that are spinning counter-clockwise at their Larmor frequency – ω in this case. This is the observer’s frame.

Now, here’s the kicker. The rotating frame has the same merry-go-round, but YOU are sitting in the center, facing towards one of the three people. The person that you are facing is now the reference frequency – again, in this case, ω. Now you would observe that all three people are no longer moving, as they are now moving in relation to you.

The next mental exercise considers the fact that all of these “proton” people might not experience the same magnetic field. Perhaps this is a disturbance in the homogeneity of the magnetic field, but most commonly, it is because the proton is experiencing a local magnetic field. A variety of things can cause this, but an important contributor is the local chemical and magnetic environment of the proton.

Chemical Shift Evolution

Even though the resonant frequency of a single proton is ~400 MHz in a 400 MHz instrument, that doesn’t take into account the different chemical and magnetic environments in an organic molecule. Measured in Hz, chemical shift allows us to tell proton environments from each other, as each of them (usually) show a peak with a certain amount of shift. As you now know, a higher magnetic field increases the Larmor frequency, and vice versa.

All of these shifted frequencies are present in the time domain measurement after the 90° pulse. Using a Fourier Transform, we can get the individual frequencies of the proton environments, enabling the elucidation of much more information!

So, what does it look like? Imagine that a “proton” person experiences a greater magnetic field. Thus, its Larmor frequency – which I will now call λ – increases, and now λ > ω. In the rotating frame, you would observe this person as walking counter-clockwise – the direction of the merry-go-round. The inverse is true for a lower magnetic field, where the person would be walking clockwise – against the direction of the merry-go-round. The speed that the person moves away from the reference person around the circle represents the chemical shift. In other terms, it is λ - ω = δ where δ is the chemical shift.

By subtracting the reference frequency, you essentially remove the contribution of the large, constant magnetic field (B0) and isolate the relative difference due to the specific chemical environment of the nucleus in the sample.

Level 4: The Spectrum

That was all pretty abstract—sorry about that. Allow me to put this all into visual terms!

Here is an FID, or Free Induction Decay. This is what is directly measured from the instrument, in the time domain. You can think of this as taking an audio recording of a bell that you just rung. There are many different frequencies present here, all added together.

Figure 2. An unprocessed free-induction-decay spectrum.

Then, if we apply a Fourier Transform, we get this.

Figure 3. An FID that has been processed with the Fourier Transform.

Each of the peaks corresponds to a frequency that was measured in the original FID, giving undeniable structural information about the molecule being measured. This spectrum is one I acquired for caffeic acid.

Figure 4. The structure of caffeic acid.

So, what do you get out of an NMR? Using the information in Figure 4 and more, you can resolve the structure, determine concentration, or even map how many different compounds are in your solution (including impurities!). NMR is extremely powerful!

I will soon write a guide on how to interpret an NMR spectrum. For now, though, this article will focus on the theory.

Level n: Other Concepts

Relaxation Pathways

There are two relaxation pathways that cause nuclei to accelerate their relaxation to the ground state.

The T1 Pathway

AKA the spin-lattice relaxation pathway – occurs when protons rotate in relation to a ¹³C atom as the molecule tumbles. If the tumbling occurs near the Larmor frequency, this induces unwanted α --> β or β --> α transitions and returning to the ground state quicker. This is actually a physical phenomenon!

This type of relaxation occurs mostly with medium-sized molecules (around 1000 Da).

The T2 Pathway

Random effects of tumbling molecules leads to randomized Larmor frequencies among spins, reducing coherence of the atoms. T2 relaxation occurs in the same way that Chemical Shift Evolution does.

Moreover, molecules with π-bonding have a magnetic field. If these molecules tumble slowly (over the course of acquisition window), the change in local magnetic field induces field heterogeneity within the sample. However, this doesn’t occur for quickly tumbling molecules, as the field averages out during the acquisition window.

Decoupling

The idea of decoupling is quite simple. However, I will first go over what coupling is so I can explain how to remove it!

Spin-spin J-coupling occurs because the local magnetic field produced by one proton can affect the Larmor frequency of a neighbouring proton. Since each proton can be either α or β, the following systems are possible: αα, αβ, βα, or ββ. For the sake of example, we know that proton A is spin α. On its own, it has a Larmor frequency of λ. If its neighbour (proton B) is spin α (meaning that it is aligned WITH B0) it has an additive effect on the local magnetic field. This means that the magnetic field at proton A is higher, increasing its Larmor frequency, and thus, its chemical shift. Moreover, the opposite is (almost) equally as likely, where proton B is spin β. This means that the field would be subtracted from, as it is aligned AGAINST B0 – equally subtracting from proton A’s chemical shift.

In the spectrum, we would see proton A’s singlet split into a doublet centered at the singlet’s ppm. Furthermore, proton B would exhibit the same behaviour.

In the context of a ¹³C spectrum, a detected carbon is able to couple with the protons that are attached to it. So, a methyl carbon would be a quartet, a methylene would be a triplet, and so on. Recall that a RF pulse at the Larmor frequency of a proton applies a torque to it. Now, imagine that you never turn that off! The proton would just keep going around and around, constantly flipping alphas to betas and betas to alphas. If we apply a carbon pulse while continually applying a proton pulse, we can average out the effect of coupling so that each carbon peak becomes a singlet! This is the concept of decoupling.