5 - The Spin Quantum Number
Abstract (TL;DR):
The Spin Quantum Number begins with an experiment by Otto Stern and Walther Gerlach, which demonstrated that a beam of atoms split evenly into two. This is due to a property of the electrons that causes them to generate their own magnetic field and, consequently, their own internal momentum. This momentum causes them to behave in two, specific ways, called “spins”.
—
Originally posted by geometricanimations
This is it! We have finally reached the conclusion of orbitals. I can already hear the cheers from the Chemistry 101 students reading this. Let’s march on and finish up.
Be sure to check out Part 4 before this one. You can’t understand where this one begins without seeing where the other one ends.
Now, it is time to explain the last quantum number.
The Quantum Dive - Internal Energy
Stern-Gerlach Experiment
Electrons are trying to get rid of their energy in any way possible. Orienting themselves in a way that lowers energy is a good starting point, but they still have plenty of internal energy. In fact, electrons have their own internal angular momentum, separate from the orbital angular momentum (l).
The Stern-Gerlach Experiment Setup
(1) The emitter for the silver atom beam. (2) The silver atom beam. (3) The nonuniform magnets (north and south poles are different shapes, distorting the magnetic field). (4) What they expected the result to be. (5) What the result actually was.
This was determined after a 1922 experiment by German physicists Otto Stern and Walther Gerlach. The aptly named Stern-Gerlach Experiment involved firing a beam of silver atoms through a nonuniform magnetic field and observing the results on a detecting wall at the end. The result, surprisingly, was that the beam was deflected in only two directions, landing flush against a plate in two spots.
But how could that be? There’s no way that there should be two, specific places in which the atoms ended up.
If you think about it, if you threw twenty bar magnets at a magnetic wall past two huge magnets and looked at where they ended up, they would all be stuck on the wall in specific places according to their poles. After all, you have no idea what direction those magnets were when you threw them or how much attraction or repulsion one magnet had than another one, so you should have no idea exactly how they will come out. Shouldn’t it have been the same for these atoms?
In order to explain the significance of this inconsistency, I’ll need to diverge for a second into the physical quantity of momentum.
Types of Momentum
We’ve spent quite a bit of time talking through orbital angular momentum. But there are several types. First, of course, is linear momentum, the momentum all physics students are likely familiar with. This momentum is acquired by multiplying a mass by its velocity, or its displacement over some time in some direction. This tells you how much, in numerical form, an object moves in a particular direction. In fact, you likely already knew the definition of “momentum” without knowing how to put it in words. Classical angular momentum is almost completely the same. The only difference is that an axis is involved, around which your mass rotates.
For classical angular momentum, instead “mass” and “velocity,” the two relative quantities are the rotational inertia (also known as the moment of inertia or angular mass) and angular velocity. The latter is simply the rate at which rotation occurs about some axis. On the other hand, the rotational inertia, is used to depict how much torque, or rotational force, you need to move an object a certain distance around an axis with respect to a certain position. The classic example of this is the tightrope walker.
If it was super easy to make an acrobat flip around the rope (for example, if gravity didn’t exist), then you would say that their moment of inertia is low. But when one walks across a rope you usually see them stick their arms out or hold a long rod. That single act increases the moment of inertia because you have changed the center of the mass, where the force acts. That is, more rotational force would be needed to rotate the acrobat.
With this foundation in momentum, we have one tool to explain the final quantum number.
Magnetism
The next tool comes through an explanation of the electron’s behavior and its innate properties.
First, before I discuss their behavior, we need to review a few things about electrons. Electrons are not individual points or balls of negative charge that orbit around a nucleus as our Earth orbits the Sun. They’re objects with wave-like and particle-like qualities (that we call and treat as particles for the sake of discussion or math). Its location at any given moment is best shown with a cloud of probabilities (the electron cloud model shown back in Part 1). This is why we call the patterns of their motion “orbitals”, which represent that cloud, and not “orbits”.
With that in mind, I mentioned, from the now legendary Part 2, that electrons, as charged particles, can generate a magnetic field just by moving. We also know that electrons are “moving” around the nucleus, and, therefore, they must be exhibiting some sort of magnetic field.
This is known as orbital magnetization, which causes the electron to behave like magnetic dipoles, which are objects with two opposing poles that generate magnetic fields (like a bar magnet or the north and south pole of the Earth).
Crucially, though, the creation of this kind of magnetic field requires circulation around a nucleus, like a loop. If there is no net movement around the nucleus, there wouldn’t be any kind of magnetic field created from the electron’s movement. There is one such orbital that fits this situation, and, believe it or not, you’ve already read about it - the “s” orbital.
In Part 4, we explained that the “s” orbital corresponds to an orbital quantum number (l) of 0, which means the electron itself has an orbital angular momentum of 0. From our discussion of classical angular momentum, this would mean that there is no rotation around some central axis. But the orbital angular momentum describes the probability of finding an electron around the nucleus. An orbital angular momentum of 0 means that this probability is the same in every direction - the orbital is symmetrically spherical. The electron could be at any given place in its spherical cloud of possibilities at any given time without any directional bias. That cloud does not flow around the nucleus, it remains completely fixed. Without that flow, there can’t be a magnetic field from orbital.
We can explain this a second way using our old friend, the standing wave. Like a guitar string, held down at both ends and plucked, the electron is also “held” at a certain distance from the nucleus, expressing its energy in a stationary position around it. The pattern it forms around the nucleus is also spherical and symmetric, the pattern necessary to remain constant and stable - a guitar string that never stops vibrating. Since this wave doesn’t “travel” around the nucleus, there can’t be a magnetic field from orbital motion.
So, with all that said, what did Stern and Gerlach see in 1922? Well, let’s go back to their experiment and investigate a little deeper. They shot a beam of silver atoms through a nonuniform magnetic field. Silver…why don’t we take a look at its quantum numbers? Perhaps we find something familiar?
To do this, we will look at the last part of silver's electron configuration on our periodic table, since these subshells on non-noble gas elements have reactive valence electrons (Part 3 and 4). For silver, we are interested in the 5s1 orbital. 5, as we know, is the principle quantum number (n). But what do we have here in the orbital quantum number (l) section? An s-orbital, which we've just learned means there is no magnetic field from the orbit of an electron. So what gives?
A representation of the 5s-orbital, which shows the possible positions of an electron in either blue or white. The gaps between each circle represent the parts in the standing waves where the wave is flattest, between the crests (the high points) and troughs (the low points). There is no possibility of an electron being there. We’ll discuss this in more detail in the future.
Image via The Orbitron by Mark Winter
Spin Quantum Number (s)
Stern and Gerlach were smart men. They knew, from Bohr’s prior experimentation of the electron, that it must be quantized. In other words, the electron had to exist in specific, discrete states. So when they crafted this experiment, they expected a quantized result. That is exactly what they saw - atoms being redirected by the magnets to two, specific areas.
But just because they were correct in their hypothesis doesn’t mean their foundation was solid – a lesson for all of you upcoming scientists.
Although that’s a result that you could hypothesize based on an expectation gained after seeing the ice in a glass of water melt, the foundation is all wrong.
Image by Bill Watterson
The quantum mechanical theory of the time wasn’t correct. Their conclusion that the result was due to the quantization of the electron was not the reason for the quantized result. The result was due to the electron’s “spin”.
In the presence of an external magnetic field (B), like the one Stern and Gerlach fired silver atoms through, electrons have two half-spins.
Remember, we don’t believe the electrons actually spin. This is just a representation of the quantum mechanical concept.
Image by William Reusch via Michigan State University
Remember, the electrons became magnetic dipoles, meaning, according to how their magnetic field influenced them, they would act like a “north pole” or a “south pole”. What happens when you push the north pole of a bar magnet into the north pole of another magnet? They repel each other! What do you think happens when you push an electron with a north pole “spin” into a north pole magnet? Yes, they repel each other. That goes for an electron with a south pole “spin” and a south pole magnet as well.
It was because of the electron’s angular momentum was directed in two, specific directions that the result of the Stern-Gerlach experiment showed registered silver atoms at two, specific spots on the plate. We call each of the electrons two rotations “half-spins”, since one beam was split evenly into two.
The Higgs Boson
All elementary particles that hold these spins equal to ½ are called fermions, whereas if their spin value was 1 (what the value would be if the one beam went straight through without any division), it would be called a boson, a word you’ve likely heard before if you’re into physics. These are the two categories in which all particles in the universe reside.
- Image of the Higgs Boson, also known as the God Particle
Summary of The Four Quantum Numbers
So…how about another summary?
The Principle Quantum Number (n) determines the energy level, and thus, the electron’s shell. It is conventionally shown with a number from 1 to 5. The Orbital Quantum Number (l) determines where the electrons are with respect to the nucleus. It shows the sub-shell and is typically denoted with a letter (s, p, d or f) according to its number (0, 1, 2 and 3). The Magnetic Quantum Number (m) tells how many subshells there are. It is determined by looking at the subshell and taking the range from -l to l. Lastly, the complex Spin Quantum Number (s) determines whether the electron’s “spin” is +½ or -½.
Lastly, there’s the Pauli Exclusion Principle, watching over these Quantum Numbers to make sure they behave themselves. No two electrons can have the same quantum mechanical state within the same atom. It was the aforementioned spin quantum number that proved this; remember that the electrons in the Stern-Gerlach did not simply mix – there was a distinction between +½ and -½. Furthermore, given that the Principle only works with differing quantum mechanical states, only fermions follow this Principle.
Image via futurespaceprogram
This opens up a very important piece of orbital theory.
That spin quantum number shows a great deal, indeed, folks. Because that Pauli Exclusion Principle makes it so that each subshell can only have two electrons – one with the positive spin and one with the negative spin.
And that, my friends, will lead us into the Aufbau Principle, the ending of our journey…Which we will discuss next time.
Please ask questions if you are lost somewhere in these five parts. And share amongst all of your friends – not just the ones interested in science. We can all learn. We just need to light the fire.