## Introduction

In the previous blog post in this series, we introduced the 4 states of matter and concluded with stars being primarily composed of plasma (the 4^{th} state) and the fact that they require a process called nuclear fusion in order to sustain themselves. In part 2 of this series, we are going to pivot and talk about why Fusion is so powerful such that it powers stars. Because there is a lot of physics involved, this particular post will be more involved compared to the next one. So strap in!

## The Perfect Balance – Fighting Against Gravity

Stars like our Sun are in a delicate balance between two opposing forces; gravity attempting to crush the mass of the star inwards, and the pressure force from the trillions upon trillions of nuclear fusion reactions in the star’s core pushing outwards. Anything with mass (measurable weight) has gravity acting on it to squash it inwards. The more mass an object has, the stronger its overall force of gravity acting inwards. That is why most celestial objects (planets, moons and stars) are all spherical - because they have enough mass that gravity has acted to attempt to squash it perfectly inwards. The reason why most celestial objects don’t get crushed down to an infinitely small point, is because they have outward opposing forces to fight back against gravity. For planets and other similar objects, its varying degrees of degeneracy – basically a quantum mechanical principle that prevents too many particles from occupying the same space or energy state. For stars, its the explosive force from all of the fusion reactions. Given that stars are massive (our Sun is considered a very small star) and their gravity must be immense, nuclear fusion must be extremely powerful in order to fight back against gravity right? Let’s have a look at why.

### Einstein's Famous Equation

$$ E = mc^{2} $$

*"Everything is energy and that's all there is to it"*

- Albert Einstein

Energy equals mass multiplied by the square of lightspeed. This very simple looking equation is extremely powerful. It equates mass and energy; energy can “crystallize” into mass and mass can be transformed into energy - The Manhatten Project was the first real application of this famous equation and principle.

This equation (or forms of it) can be used to describe atoms when they are formed. When a new nucleus is formed, you’d expect that the total mass of the nucleus is equal to the combined mass of all of the protons and neutrons (nucleons). Measurements say that this is not so. All of the nucleons weighed and combined as individuals are heavier than if they were stuck together and weighed. That is like saying that the total mass of two 80kg humans (160kg) is actually heavier than if those humans were stuck together – it should still be 160kg!

How can this be? Well, there is an additional factor called Binding Energy \( \Delta E_{BE} \). In most textbooks, Binding Energy is the amount of energy you need to break a nucleus into its separate nucleons. In my understanding of nuclear physics, its a measure of stability and how much energy was released when the nucleus was formed in the first place - of course, this energy does need to be returned if you want to break up the nucleus! Let's look at what makes up a nucleus:

$$ m_{A} = Nm_{n} + Zm_{p} - \frac {\Delta E_{BE}} {c^{2}} $$

Where \( m_A \) is the total mass of the nucleus, \( N \) is the number of neutrons, \( Z \) is the number of protons, and \( m_{n} \) and \( m_{p} \) are the mass of a single neutron and proton respectively. The Binding Energy has been divided by \( c^{2} \) in order to convert it into a mass - by using Einstein's equation.

This equation tells us that when a nucleus is formed from nucleons, energy is lost/radiated away in the form of Binding Energy. If you want to break up the nucleus, you need to put that Binding Energy back. The more stable a nucleus is when it is formed, the more Binding Energy that is lost and therefore the harder you have to work to use a nuclear process to break it. This is visualized in the Binding Energy curve:

Iron (Fe) is the most stable element in the universe and cannot undergo fusion or fission. It doesn't need to! Ignoring isotopes, elements on the left of Iron are small and less stable and will undergo fusion to stick together in order to become more stable. Elements on the right of Iron are large and less stable and will undergo fission to split apart in order to become more stable.

You can already see why fusion is so powerful. The large differences in Binding Energy are what we would receive as pure energy and what we can use for energy production and in the case of armaments - atomic bombs. Fission is something that we already know how to do and although it is powerful, we can can clearly see that the differences in Binding Energy between the elements on the right are a lot smaller. So compared to fusion, we'd obtain less energy.

### The Semi-Empirical Mass Formula

To finish off, we're going to look at where Binding Energy comes from. What controls how much or how little Binding Energy is lost when a nucleus is created. Remember, we wouldn't be chasing after fusion if we couldn't get a lot of energy from it. That energy is the Binding Energy. The amount of Binding Energy released depends upon how stable the nucleus is when it is first formed. This is expressed via the Semi-Empirical Mass formula:

$$ \Delta E_{BE} = a_{V} A - a_{S} A^{\frac {2}{3}} - a_{C} \frac {Z(Z - 1)}{A^{\frac {1}{3}}} - a_{A} \frac {(N - Z)^{2}}{A} + \delta (N, Z) $$

This section is probably going to be the toughest to get your head around, but let's try! The first term ( \( a_{V} \) ) in this equation is called the volume term. It describes the fact that every nucleon has a “glue” called the Strong Nuclear Force that will indiscriminately attempt to bind them all together as closely as possible. Most of the Binding Energy comes from this and the fact that the more nucleons (the bigger the volume) you have, the more "glue" you have and the closer they stick together and remain stable.

The next term ( \( a_{S} \) ) is the surface term and is the first term that corrects the Binding Energy by reducing it. It says that the nucleons towards the outside of the nucleus are more weakly bound than those on the inside. The further out a nucleon is, the less surrounded it is by other nucleons and so the less Strong Nuclear "glue" that is available to keep it stuck. So stability weakens and Binding Energy is lowered.

The next term ( \( a_{C} \) ) is the coulomb term which also corrects the Binding Energy by reducing it. This term says that there are differences between the nucleons. Protons are positively charged whilst neutrons are neutrals. Same charges repel and so the protons actively repel each other over any distance using the Electromagnetic Force. There comes a point where the “glue” of the Strong Nuclear Force begins to not do so well against the many protons pushing each other away with the Electromagnetic Force. The bigger the nucleus, the more protons and therefore the higher the imbalance is with all of the protons pushing away from each other. So stability weakens.

The next term ( \( a_{A} \) ) is the Asymmetry Term and is probably the hardest to understand. Protons and Neutrons can exist in pairs in identical energy levels. If either is paired up and another neutron/proton comes along, a completely new energy level for BOTH neutrons and protons needs to be created – increasing the overall energy of the atom and reducing Binding Energy. You have to understand that stability is synonymous with low energy. If you even provide a means to accomodate higher energies, the nucleus will automatically be at a high energy and so stability is reduced. This term is lowered if there are equal numbers of protons and neutrons, because then it implies that the atom is at its lowest possible energy:

Both arrangements have the same amount of nucleons but different amount of pairs. The arrangement with asymmetrical pairs has to exist at a higher energy. The arrangment with symmetrical pairs can exist at a lower energy - the extra energy that would have been accessed will go towards the Binding Energy.

The last term ( \( \delta (N, Z) \) ) is the pairing term which can sometimes increase binding energy instead of reducing it. Very simply, the pairing term accounts for the tendency of protons and neutrons to form pairs within the nucleus - a phenomenon akin to the pairing of electrons in atomic orbitals. This pairing arises from the strong nuclear force, which binds nucleons together in pairs with opposite spins, resulting in enhanced stability for nuclei with an even number of protons and neutrons.

## Conclusion

In part 2 of this blog series, we've did a deep dive into why Fusion is so powerful. Parts 3 onwards will be less rigorous and go back to a more top level description about the fusion in stars and how we plan to make it reality here on Earth. See you next week!