“The everyday fact that one’s body does not collapse spontaneously into a black hole, therefore, depends on the spin-1/2 of the electron.” - Malcolm Levitt
quantum spin - a property of quantum particles that has many mathematical similarities to macroscopic spinning objects but also has unique quantum properties not observed in the macroscopic realm elementary particles - subatomic particles that make up all known matter and cannot be divided any further into constituent parts
Magnetic resonance experiments depend on the interactions between nuclear quantum spins and magnetic fields. elementary particles have spin, some basic properties of quantum spin, and which particles and nuclei are used commonly in magnetic resonance experiments.
Check out this PhET simulation to explore some of the different isotopes in the first few rows of the periodic table.
Image source (2)
The most relevant elementary particles in our work will be the main components of the atom: electrons, protons, and neutrons. All three have \(s = 1/2\) and are called spin-1/2 particles.
For spin-1/2 particles (\(s = 1/2\)), there would be two distinct spin states, typically called spin-up (\(m_s = +1/2\)) and spin-down (\(m_s = -1/2\)). We will be representing these quantum states as kets in this text.
note: do we need bra ket now?
As fermions, spin-1/2 particles are actually excluded from sharing identical quantum states (Pauli exclusion principle.) When fermions are put together (Aufbau principle). This even happens inside the protons and neutrons, which each are made up of of different combinations of three spin-1/2 quarks.
isotope - an isotope of a chemical element has the same atomic number (i.e. number of protons) but a different atomic mass (i.e. different number of neutrons); commonly written in the form “carbon-14” or \(^{14}\)C) where the number is the atomic mass of the isotope
For understanding NMR, we are interested in the total nuclear spin of the atomic nuclei, usually denoted by \(I\). Since both protons and neutrons contribute to the nuclear spin, we have to be explicit about what atomic isotope we are observing. If you need a refresher on how to determine the number of proton and neutrons in the nucleus for a given isotope, check out the helpful figure in the margin! NMR only works for isotopes that have non-zero nuclear spin, so to determine whether a given isotope may be a good candidate for NMR, it is helpful to follow some rules to determine the nuclear spin of the isotope.
Rules for finding the nuclear spin of a given isotope
- If the number of neutrons and the number of protons are both even, then the nucleus has NO spin (\(I = 0\)).
- If the number of neutrons plus the number of protons is odd, then the nucleus has a half-integer spin (i.e. \(I = 1/2, 3/2, 5/2\))
- If the number of neutrons and the number of protons are both odd, then the nucleus has an integer spin (i.e. \(I = 1, 2, 3\))
By far the most NMR research is done on spin-1/2 nuclei. The isotope we choose to observe with NMR should be stable.
leave this as question. so that the nuclei in our sample will not radioactively decay to different nuclei during the time we are making our measurements.
Which hydrogen isotope do you think is referenced in the periodic table above? Any advantages or drawbacks to choosing to use this particular isotope?
Which carbon isotope do you think is referenced in the periodic table above? Any advantages or drawbacks to choosing this particular isotope?
Which fluorine isotope do you think is referenced in the periodic table above? Any advantages or drawbacks to choosing to use this particular isotope?
Why might learning more about quantum spins be useful?
A lot of the conventions and naming schemes may understandably seem fairly arbitrary to you. Do you have any alternative names that you think would provide more useful information? Can you see any logic behind the conventions or are they truly arbitrary?
For each of the following nuclear isotopes, provide your assessment on whether they may be useful for NMR or not? (Look for non-zero nuclear spin, stability, relative abundance, etc.)
\(^{31}\)P:
\(^{15}\)C:
\(^{3}\)He:
\(^{29}\)Si:
Can you remeber how the aufbau principle is used in chemistry - the electron configuration of C atom?
Electrons in atomic orbitals as well as protons and neutrons inside the atomic nucleus all obey the Aufbau principle. Protons and neutrons are each made up of three quarks which each carry spin-1/2. Can you use the Aufbau principle to understand why we say the protons and neutrons each are spin-1/2 particles?
Why should we use stable nuclei?
Commonly denoted by \(s\), which is a positive, dimensionless number. Particles can either have an integer spin (i.e. \(s = 0, 1, 2, ...\)) or a half-integer spin (i.e. \(s = 1/2, 3/2, 5/2, ...\))
bosons - particles with integer spin; like to be buddies (i.e. bosons are happy to all crowd into the same quantum state together) fermions - particles with half-integer spin; like to be frenemies (i.e. two fermions cannot share the same quantum state, but tend to pair up with a fermion with opposite spin )
The quantum spin of a particle, commonly denoted by the spin quantum number \(s\), is one of the few physical characteristics that uniquely identify an elementary particle - along with other information like mass and electric charge. A particle that has different spin - even if all other physical characteristics are the same - can have very different quantum mechanical behavior. Every particle has an associated spin quantum number that is either an integer or half-integer.
Commonly denoted by \(m_s\), which is a dimensionless number that goes from \(-s\) to \(s\) in increments of 1.
The number of spin states can be calculated using the spin quantum number of the particle, \(s\):
\[2s + 1\] Distinct spin states are indexed by the spin magnetic quantum number, \(m_s\), and this quantum number can be negative or positive. For example, if \(s = 1\) then there would be 3 possible spin states, with \(m_s = -1, 0, \textrm{ or } 1\).
vector - a mathematical quantity that has both a magnitude and direction and is usually visualized using an arrow; vector quantities will be denoted with little arrows on top, like \(\vec{A}\) angular momentum - a physical quantity related to how much stuff is spinning (or rotating) about some axis of rotation and how fast it is spinning (or rotating) axis of rotation - a straight line through all points in a rotating object that remain stationary; often where the axle of a rotating object is placed (e.g. through the center of a bike wheel)
Commonly denoted by \(\vec{S}\), which is a vector that has the same dimensions as angular momentum, but typically will be written in terms of \(\hbar = h/2\pi\) where h is Planck’s constant (If you see an \(h\) of \(\hbar\) anywhere, it is a sure sign you are dealing with quantum behavior!)
The name ‘spin’ comes about due to some of the mathematical similarities of quantum spin behavior and macroscopic spinning objects. Most notably both types of ‘spin’ seem to have some form of angular momentum. In fact, quantum spin is often referred to as ‘intrinsic angular momentum’. In classical physics, angular momentum is a vector typically denoted by \(\vec{L}\) and represented by an arrow that points along the axis of rotation. Clearly, the gods of convention gave up on trying to find sensible variable names for physical parameters by the time angular momentum came around. The direction the angular momentum arrow points is determined by whether the object is rotating clockwise or counterclockwise and can be found by using the pretty much arbitrary right-hand rule, shown in the figure below. We can deduce from this that the gods of convention have an obvious right-hand bias but were not in power around the time clocks were invented.
right-hand rule - make a thumbs up with your right hand and rotate your hand so that your fingers curl in the direction of rotation of the spinning object (e.g. as if the tips of your fingers were the head of the rotation arrow); your thumb now points in the direction of the angular momentum \(L\); Image adapted from (1)
For a quantum spins, this angular momentum vector is replaced with the spin angular momentum vector, \(\vec{S}\). As a helpful visualization of quantum spins, we will use a rotating sphere, and have an arrow with the spin angular momentum vector that depicts the spin direction. It is important to point out that despite this helpful visualization, there is not an actual particle actually spinning at the quantum level. Unfortunately, there is no perfect visualizations that encapsulate all the full weirdness of quantum particles!
magnetic moment - also known as magnetic dipole moment or magnetic dipole; the magnetic strength and orientation of a magnet or other object that produces a magnetic field dipole - two poles (e.g. the north and south pole of a magnet); often contrasted with monopole - one pole - (e.g. a positive electric charge would be considered an electric monopole) FUN FACT! No matter how you cut up a magnet, you always get two poles in the remaining pieces, and the intrinsic magnetic moment of fundamental particles is a magnetic dipole. Magnetic monopoles have never been found in nature, though scientists have searched for them because they would bring a nice symmetry to the laws of physics and have some pretty nifty physical properties. You can read more about magnetic monopoles here.
Physical objects can also have magnetic properties, which is encapsulated in the magnetic moment of the object. This is sometimes also called a magnetic dipole moment or magnetic dipole. The magnetic dipole moment can be visualized as an arrow that points from the south pole to the north pole of a tiny, little bar magnet. Essentially, the arrow representing the magnetic moment is aligned with the magnetic field it produces. The convention is that the magnetic field lines point away from the north pole and loop back to point towards the south pole, as shown in the figure in the margin.
Elementary particles can also have magnetic moments, including an intrinsic magnetic moment caused by the particle’s spin. This is called the spin magnetic moment of the particle and denoted by \(\mu_\textrm{S}\). There is a very simple and direct relationship between the spin magnetic moment and the spin angular momentum \(S\) of the particle:
\[\vec{\mu}_\textrm{S} = \gamma \vec{S}\] where \(\gamma\) is a constant called the gyromagnetic ratio and has particular values for each type of particle. We will talk more about \(\gamma\) later, but I think you have enough information to digest for the moment. This simple expression shows that if you know the spin angular momentum and the gyromagnetic ratio of the particle, you can easily calculate its spin magnetic moment. But even more importantly for our purposes, this equation tells us that the spin magnetic moment is always aligned (pointing in the same direction) or anti-aligned (pointing in exactly the opposite direction) with the spin angular momentum, depending on the sign of the gyromagnetic ratio.
Thus we can complete our full visualization of a quantum spin which contains both the spin magnetic moment and the spin angular momentum.
If I were to tell you the spin of a particular electron is \(\hbar/2\), which type of spin am I really talking about?
How many different allowed spin states does a spin-2 particle have? How about a spin-3/2 particle?
Quantum spins with \(s = 0\) are sometimes said to occupy a singlet state and spins with \(s = 1\) are sometimes said to occupy a triplet state. What do you think the reasoning is behind those names?
In the spin figure above, is the spin magnetic moment vector aligned or anti-aligned with the spin angular momentum vector? Would this spin have a positive or negative gyromagnetic ratio?
We will see soon that quantum spins like to align with external magnetic fields, from what you learned above, why may that information not surprise you?