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The Pauli exclusion principle and electron degeneracy pressure

Seeing as electron degeneracy pressure arises from the Pauli exclusion principal, we'll start by addressing what that principal means, and since it only requires a quick explanation, we'll be able to talk more about the effects of degeneracy pressure on some of the most fascinating objects in our universe, namely white dwarfs and neutron stars.

So without wasting any more time, let's get into it!

The Pauli exclusion principal states that:

No two electrons in the same atom can have identical values for all 4 of their quantum numbers, or in other words, electrons in the same atom can't share the same quantum state.

Now, I know I said that it would be a quick explanation, but that's probably only true if you already have an understanding of some basic QM (Quantum Mechanics).


It's not that difficult to understand if you have limited/no knowledge of QM, but it will probably require a little more explanation than the statement above, so let's dive into that now.

Hopefully, you're already familiar with atomic structure though (e.g the arrangement of electrons, protons and neutrons within an atom), otherwise I'd suggest taking a quick look at that elsewhere before continuing on with this article.

There's tons of free resources on the internet, although I do talk about it in this article here, so feel free to check that out!

Electrons are "point particles" of energy that can be ascribed 4 values based on their energy level, spin, angular momentum, and orbital path.

Although the more scientific terms for these are as follows....

The principal quantum number (n), where n = 1, 2, 3, 4, ....

Specifies the energy level of an electron, and can only be integer values that ascend from 1 (an electron can't have n = 1.5).

The reason for this is because electrons cannot exist between energy levels, and electrons that exist within the same energy level (they share the same n value) are said to be in the same "shell" as each other.

Each energy level has a specific amount of electrons that can exist in that level, and while the principal quantum number can be the same from atom to atom, the actual value of this will change, as electrons in different atoms have different levels of energy.

For example, an electron of a hydrogen atom may given n = 1, depending on where in the atom it exists, and the electron of a copper atom may also be given n = 1 for the same reason, but it's important to know that these electrons do not have the same amount of energy - you can't compare the principal quantum numbers of electrons that exist within different atoms.

n= 1 is the closest shell to the nucleus of an atom, and is the lowest energy state possible for an electron in that specific atom.

Electrons can move from a lower shell to a higher shell if they gain enough energy, in which case we describe the electron as being in an "excited" state.

If you've read some of my other articles before, then you'll probably be familiar with this term already.

Similarly, they can drop from a higher shell to a lower shell if they're able to release enough energy, which usually occurs via the emission of a photon of light.

The orbital angular momentum quantum number, or secondary/Aziumnthal quantum number (l)

As mentioned previously, there are different shells of energy levels in an atom, and each shell can be broken up into even smaller groups, commonly referred to as "sub-shells".

These sub-shells will have different shapes of orbital paths around the nucleus, and a maximum of two electrons can share the same sub-shell.

In essence, the secondary quantum number describes the shape of a sub-shell.

The range of values for the secondary quantum number are as follows, and they're often denoted with a letter to avoid confusion with the principal quantum number.

l 0, 1, 2, 3, ....

Letter s, p, d, f, ...

For example, an electron with n = 1 and l = 0, exists in the 1s sub-shell.

The electron spin quantum number (ms), where ms = +1/2 or -1/2

Specifies the direction in which an electron spins, of which +1/2 or -1/2 are the only possible options, which correspond to "up" or "down".

As the Pauli exclusion principal sets out, which we'll talk more about in a minute, no two electrons in the same orbital can spin in the same direction, and when there are two electrons in the same orbital that are spinning in different directions, it's said that these electrons are "paired".

For example, if two electrons both have n= 2 and l=3, then they both exist in the same 2f orbital, or sub-shell, but they must be spinning in opposite directions (one must have ms = +1/2 and the other -1/2).

Remember, they cannot share the same value for all for 4 quantum numbers.

The magnetic quantum number (ml), where ml = -l, ..., 0, ..., l

Specifies an orbital's orientation in space.

So far, we've talk about how atoms can have different shells that each have a unique energy level, and that these shells can be broken down further into sub-shells.

Well, these sub-shells can also be broken down further into "orbitals", which define the regions of space where there is a high-probability of finding an electron.

The orbital of an electron is essentially it's path around the nucleus of the atom, although they don't follow this path in the same way that the planets orbit around the Sun - we never really know where electrons are along this path, but we know that there's a high probability of finding one here.

That probably sounds a bit strange, but as we'll elaborate on later, electrons are quantum particles that we don't truly understand.

Briefly, the Schrödinger equation sets out the likelihood of finding an electron within an atom, which is referred to as the wave function of an electron, but we can never predict the exact properties of an electron with 100% certainty, as laid out by the Heisenberg uncertainty principal.

Here's a picture of the wave function for a hydrogen atom. As you can see, the formula for determining the actual probability of finding an electron is very complicated, but viewed as robust by quantum physicists (for the time being at least...)

Hydrogen wave function

Strangely, the Heisenberg principal is one of the reasons why absolute zero (the lowest theoretical temperature) can never be achieved, although you can read more about that here.

Only two electrons can occupy the same orbital of an atom, and the number of orbitals that can be found in a sub-shell will depend on which sub-shell is at question.

An expression for finding how many orbitals can exist in a sub-shell is 2l+1, where l is the secondary quantum number.

For example, the s sub-shell can only contain a maximum of 1 orbital, and the d sub-shell can only contain a maximum of 5.

Something that's also worth remembering is that electrons will individually fill out the maximum number of orbitals before pairing together in the same orbital - they would rather be by themselves than paired with another electron, since they're both negatively charged and hence repel each other with some force.

I'd advise reading through this several times, as it's necessary to be comfortable with these principals if you'd like to learn more about QM.

So with the basics out of the way with, let's refer back to Pauli's exclusion principal to understand what it means exactly for the structure of an atom.

A recap of the definition....

No two electrons in the same atom can have identical values for all 4 of their quantum numbers, or in other words, electrons in the same atom can't share the same quantum state.

You can look at this from a variety of different angles, in terms of its implications on how electrons can be structured around their relative nuclei, but the direction that's most relevant to astronomy, in my opinion, would be this:

If no two electrons can have identical values for all 4 quantum numbers, and they fill orbitals from the lowest energy level first, then electrons must be forced into higher and higher energy levels as more electrons exist within that atom.

The reason why that's significant is because it means that electrons don't really have much of a choice about how far out from the nucleus they are, since they'll occupy the lowest energy level that's available to them, considering the fact that they can't share the same quantum state as another electron, and that's it.... They're bound to either that orbital, or one that's even higher in energy (if they absorb some energy, for example by the absorption of infrared radiation).

There's nothing that an electron can do to occupy a region of space that's closer to the nucleus and hence lower in energy.

Every electron within a gravitational field such as the Earth's would experience pressure due to particles above them essentially weighing down on top of them, attempting to push the electrons towards the nucleus.

So, what happens?

Well, have you ever wondered why electrons don't just "fall" into their relative nuclei?

I mean electrons are negatively charged, and protons are positively charged, so why don't they just combine to produce neutrons?

Using classical mechanics, such as Isaac Newton's laws of motion, electrons would do exactly that - eventually fall into the nucleus due to losses in energy, as a result of being a charged particle that's accelerating, although that's a bit off topic for this article.

However, quantum mechanics redefines our understanding of the subatomic world, and it's becoming ever so increasingly clear to scientists that we live in a quantum world, despite its (sort of) rivalry with Einstein's relativistic world.

Today, we accept that electrons are quantum particles, and do not obey the rules of classical mechanics.

It's because electrons cannot exist between energy levels, and that n = 1 is the "ground" state for an electron, that they cannot fall into the nucleus.

The key takeaway is that electrons don't obey the same rules that we're used to on the macro scale of this universe...

Electrons cannot be forced closer to the nucleus than what quantum mechanics permits them.

Until that force becomes incredibly high...

This pressure of electrons refusing to move closer to the nucleus is known as electron degeneracy pressure, which you'll surely recognise from the title of this article!

The word "degenerate[te]" here, simply describes how any two electrons can't exist in the same quantum state as each other.

Something that's also worth noting is that the Pauli exclusion principal applies to all fermions, which include protons, neutrons and electrons, but not for example bosons, such as photons of light.

I know that it's taken a long time to get here, but QM is a very, very complicated world, and it's difficult to make statements assuming any level of intuition, as you might do if we were talking about the moon orbiting around the Earth.

Electron degeneracy pressure stops electrons from collapsing into lower energy levels, and exists because any two electrons of the same atom simply can't exist in the same quantum state as each other.

This pressure is exactly what stops stars like the sun, from collapsing beyond a white dwarf at the time of running out of fuel, which is held up purely because QM prohibits it from collapsing further under its own gravitational field.... within limits....

When you collapse matter, and increase its density, whether that be gases inside a balloon, or a solid block of gold (although obviously the latter would be quite hard to do on Earth), what you're essentially doing is decreasing the space between particles.

At an atomic level, nothing much is changing - just that these atoms are getting closer to each other.

Eventually though, the atoms will be unable to move any closer to each other due to electromagnetic forces, hence the object will be rendered incompressible.

Relating this back to astronomy though, at this point, you have a white dwarf.

An object that's being held up solely because its gravity can't collapse it any further, as the electrons refuse to exist any closer to their relative nuclei, and the atoms themselves can't get any closer to each other.

If you've read my article about the life cycle of a star, which you can check out here if not, then you'll know that white dwarfs do not undergo any nuclear fusion, unless they steal matter from a neighbouring star that's too close for their own good!

The takeaway from this?

Electron degeneracy pressure arises from electrons being quantum particles that obey Pauli's exclusion principal, which means that they're unable to exist in the same quantum state as another electron in the same atom, and will not be pushed any closer to the nucleus than what QM allows them.

This is the reason that stars similar in mass to the Sun do not collapse down further than a white dwarf at the time of "death", despite the absence of any nuclear fusion - electrons do the job themselves of resisting the pull of gravity.


As I've hinted at a few times in this article, this isn't a hard and fast rule for any electron in any condition.

When the force becomes so large, electrons will have no choice but to move, as they're under incredible amounts of pressure to do so.

They do this however, while still obeying the laws of quantum mechanics.

We've discussed how they can't exist in between energy levels, or "shells", so that rules out the possibility of them just sliding closer to the nucleus...

We've also discussed how the Pauli exclusion principal prohibits them from sharing identical values for all 4 quantum numbers with any other electron in the same atom, essentially meaning that they can't just join with the orbital of another electron....

Instead, the electrons combine with the protons within their relative nucleus to form an object that's made exclusively of neutrons!

Otherwise known as a neutron star.

It may come as a surprise to you to learn that all atoms are more than 99.9% empty space, meaning that you, me and everything that we see around us is almost exclusively empty space... a mere vacuum!

If you're currently debating your own existence in an attempt to understand that, then maybe give this article a read, as it proves it to be true, mathematically!

We're going a bit off-topic now, but this explains why neutron stars are so incredibly dense - they're no longer made of atoms that are mostly empty space, as electrons have combined with the protons to form an object that's exclusively neutrons.

There's little to no empty space, simply because there are no atoms!

Just a bunch of neutrons to form what's essentially one massive nucleus.

With that said, I think we'll leave it there for this article.

Quantum mechanics is an extremely complicated, deep area of physics that really tests your imagination more than anything else, but with patience, anyone can learn most of the concepts - just try not to get your hopes up about actually knowing anything with 100% certainty, seeing as uncertainty rules QM!

Thanks for reading!

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