Triple-Product: Decoding the Most Important Metric in Fusion Reactors

12 min readNov 25, 2023

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https://usfusionenergy.org/science-fusion#:~:text=Plasmas%20must%20meet%20three%20conditions,criterion%2C%20or%20the%20triple%20product.

Nowadays, going on a plane is a breeze (minus going through security 🫣). Picture it: you’re flying to Hawaii — you sit and relax over a smooth flight, are looked after by your flight attendants, and fed pretty good food for a designated period of time before you reach your destination — what a great system, right?

The aerodynamic efficiency + smooth airflow over the aircraft’s surfaces are primarily influenced by many parameters; such as the design of the airframe, the shape of the wings, and the surface finish. Small details like a smooth and glossy finish helps in the transition of air across the surface of the wing, reducing skin friction drag and ultimately contributing to the overall aerodynamic performance of the aircraft (1).

Okay so what am I getting at?

All of these different factors in the process of making a plane have a direct influence on the efficiency of the flight, and to make sure that you get to Hawaii in one piece.

I want to shift gears a little bit; I’m really interested in nuclear fusion, and creating more efficient fusion reactors. But to do that, we need to understand how different factors affect the efficiency of your machine. Hopefully you get it now that you’ve flown to Hawaii 🧘‍♀️

But you can’t just increase the efficiency of a fusion reactor with no understanding of the input parameters needed to do so. Let’s take a look at how different parameters in a nuclear fusion reactor directly impact the reactors ignition — a concept called Triple-product.

TL:DR

My video on this topic — feel free to watch for a simpler version ;)

The Building Blocks of a Fusion Reactor: The Lawson Criterion

The concept of nuclear fusion is simple- all we really want is clean energy on earth. But to do so, nuclear reactors (tokamaks) need to produce significantly more energy than they consume, so of course it’s important to understand which inputs we need to achieve that goal- but what are they?

Well, John D. Lawson already thought of that, back in 1955! He wondered what conditions needed to be met in order to get enough energy from a fusion reaction to use in larger scaled projects.

The Lawson criterion is expressed in terms of the triple product (nTτ), where n is the plasma density, T is the plasma temperature, and τ (pronounced “tau”) is the energy confinement time. (3 terms = triple product)

The Lawson criterion is given by the following inequality:
nTτ ≥ Lawson criterion

To simplify, the Lawson criterion is essentially a criteria/threshold that fusion plasma must surpass for the fusion reactor itself to become self-sustaining. If the triple product falls below the Lawson criterion: the rate of energy being lost from the plasma exceeds the rate of energy production, and the reaction cannot be sustained.

Big Q (Q)

The ratio Q is a measure of how well a fusion reactor is performing in terms of energy output compared to the energy input. For sustained fusion reactions, Q needs to be greater than 1. Achieving Q>1 means that the fusion reactions are producing more energy than is being externally supplied to maintain the plasma conditions.

The Lawson criterion, expressed in terms of the triple product, sets the foundation for achieving self-sustained fusion. Meeting, or even surpassing the Lawson criterion ensures that the conditions for sustained fusion are met. Ultimately, achieving a reaction of Q>1 shows a net positive energy output, signifying a successful and potentially practical fusion reactor (2)!

The relationship between the Lawson Criterion, the triple product, and Q is intertwined, which were put in place to help us understand when fusion reactions can happen effectively!

The Triple-Product x Lawson’s Criteria

So, the triple product is a broader concept, like saying “we need these three things to make fusion work,” and the Lawson criterion is a specific measurement or rule within that concept, specifying exactly how much of those three things we really need.

I mainly want to focus on triple-product, because it consists of the factors (T, n,τ) that make a fusion reaction possible.

Essentially, the triple product, in the context of nuclear fusion, quantifies the performance of a fusion grade plasma, measuring how good the overall fusion reactor is (3).

It reflects the combination of three very important measurements in a fusion reactor I mentioned before: temperature (T), density (n), and time (τ). By maximizing these three conditions in the plasma made of the right fuels, the fusion reactor has a great efficiency, and ultimately leads to the release of more energy.

Therefore, the higher the triple product, the greater the energy released!

Triple-product also allows for the comparison of different fusion concepts + tracks the progress of fusion research towards energy gain and commercial uses.

Retrieved from; https://www.youtube.com/watch?v=MCs86Ous1f4

Something really cool about triple product is that it’s independent of the specific machine used to create the fusion plasma, so it can be used to compare performance across different kinds of approaches to fusion (4). For example, it can be applied to quantities in magnetic confinement machines (tokamaks, stellarators), or inertial confinement machines (laser fusion).

Between the 1950s and the early 2000s, the maximum triple product achieved by fusion energy experiments grew dramatically. During post WWII times in the 1950s, the US, the Soviet Union, and the UK received well funded efforts to design and build fusion energy systems (See Figure 1). A few early experiments initially reported promising results of high temperature plasmas and fusion neutrons, results of the D-T fusion reaction.

…..But it was soon realized that this was a misinterpretation of the data. The source of the fusion reactions were areas of plasma instabilities, not from the hot plasma alone. The actual achieved triple products were very low, lower than what was required for a full nuclear power plant. By the end of the 1950s, the research was declassified and the field entered into a brief period of defeat 😢.

***Figure 1: Photograph of the ZETA pinch experiment in the UK from the mid 1950s.*** *Description + image r*etrieved from; https://www.fusionenergybase.com/article/measuring-progress-in-fusion-energy-the-triple-products

Fast forward to today, a rush of tokamak construction followed, and the achieved triple products increased dramatically throughout the rest of the 20th century. The ability to drive higher triple products in tokamaks was the result of so many advancements in the understanding of plasma physics, improved technologies, and an increase in tokamak size (See Figure 2). The increase in size might not seem important, but larger tokamaks actually hold on to heat for a longer time, increasing their energy confinement times and therefore also their triple products (5).

Figure 2: *Photograph of the JET tokamak in 1991.* *Description + image r*etrieved from; https://www.fusionenergybase.com/article/measuring-progress-in-fusion-energy-the-triple-products

I mentioned the specific parameters that triple product measures- let’s take a look at how exactly each one is measured, why, and the specifics of it all!

Temperature 🌡️

Fusion reactor plasmas need to be hot 🔥 . While fusion in the sun’s core occurs at a temperature of 15 million degrees Celsius, fusion on Earth has to exceed this temperature, by several orders of magnitude to account for our low atmospheric pressure (6).

Fusion reactors (specifically tokamaks) consistently heat plasmas to temperatures of 100 million degrees Celsius or more. And these temperatures are necessary to provide the nuclei with enough energy to overcome repulsion, so that they fuse (hence nuclear fusion!)

Retrieved from; https://usfusionenergy.org/science-fusion#:~:text=Plasmas%20must%20meet%20three%20conditions,criterion%2C%20or%20the%20triple%20product.

Achieving + maintaining high plasma temperatures is essential for sustaining fusion reactions. In fusion, particles need to overcome the Coulomb barrier- essentially just a repulsive force due to the positive charges of atomic nuclei- to collide and fuse. The triple-product here is that higher temperatures = increased particle velocities, allowing for more successful collisions between nuclei, and ultimately an increased chance of fusion.

Temperatures in fusion research are written in units of keV (thousand electron volts), instead of degrees Celsius. 1keV ≈ 11,600,000°C.

Due to these large temperature requirements, it’s evident that effective plasma heating is such a big and important milestone in any fusion model. This whole idea explains why the 1keV temperature result of the 1968 tokamak was so exciting- although 1keV is too low for any sort of fusion mode, it’s a factor of ten of what’s required (7), which is still pretty impressive. Even in the present a temperature of 1keV is very exciting, when combined with reasonable parameters for plasma confinement (See Figure 3).

Figure 3: *Achieved triple products and temperatures for selected experiments.* *Full description + image retrieved from;* https://www.fusionenergybase.com/article/measuring-progress-in-fusion-energy-the-triple-products

It’s also important to know that many important fusion experiments don’t actually rely solely on high temperatures as a goal. Rather, their goals are set in place to be small achievements to eventually reach those levels; for example studying plasma instabilities, creating new confinement approaches, or exploring the limits of other emerging technologies. But each of these goals are connected to enabling high triple products down the line (8).

Density 🏋️

In a fusion reaction, there has to be enough atoms actually fusing… meaning that there are a lot of atoms needed for a fully sustained reaction.

The density (n) revolves around the number of particles per unit volume in the plasma, influencing the probability of particle collisions required for making fusion possible.

The more dense a plasma is, there is an increased chance for particle collisions, the most important requirement for fusion reactions. The distance of the particles within the plasma also determines the frequency of collisions. These distances heavily rely on the density of the atoms, influencing the overall efficiency of fusion.

To put it all into perspective, a plasma density of about 10¹⁴ particles per cubic centimetre is 250,000 times thinner than the earth’s air mantle 😮 (9). This extremely low density means that plasma in fusion has an overall power density only a bit bigger than an ordinary light bulb, despite its high temperature!

Keeping the plasma density high enough makes sure that the atoms do fuse, and to do so, the plasma is surrounded by very big electromagnets. These create magnetic fields are 10,000 x stronger than the earth’s magnetic field 🌎.

However a large increase in density for the plasma can create different types of collisions between nuclei and electrons, which can create large amounts of radiation (10). The radiation (called bremsstrahlung) takes energy from the plasma and prevents fusion from happening 😣.

Time ⏰

Of course the more time a reaction runs, the closer we get to harnessing fusion power. Most recently, a nuclear fusion reactor research facility successfully harnessed plasma at 70 million degrees Celsius for as long as 17 minutes, 36 seconds! (11)

When adding time into density + temperature, there has to be enough time to let the fusion reaction to actually happen, and to cool down. The more and more the plasma is heated (T), the number of reactions increase overall, and so does the time (τ)!

Time is represented by the energy confinement time. The energy confinement time measures the average time particles spend within the plasma before losing their energy. It also measures how effectively the plasma can confine and maintain the high-temperature conditions required for a full fusion reaction.

In a fusion reaction, 80% of the energy is carried by the neutrons, while the other 20% energy is carried by the helium nuclei in the plasma. The newly formed helium (20%) bounces around the vessel, colliding with unburnt nuclei used for fuel, and heats them up, reducing the need for the external heating systems. But it takes a pretty long time for this to happen depending on the density + temperature of the plasma (12).

Overall, after a certain time, a specific value of the fusion product called ignition becomes self-sustaining. This means that the heat generated by the reaction is enough to keep the plasma hot, and the external heating systems can be turned off 🥵. This self-sustaining state is a key milestone in the development of nuclear fusion, and it’s all done using temperature, density, and time.

We can see that all three products live in harmony, and have to work together if we want to see fusion energy becoming a reality. Now that the groundwork for the Lawsons Criterion has been laid and understanding the role of all three products required for a fusion reaction, I really want to show my linear regression model which predicts the relationship between different parameters, and how they affect the overall plasma power of a reactor!

Linear Regression Analysis of Lawson’s Criterion + Impact on Ignition

The data set I used to predict values was based off of a Plasma_ML_Dataset.csv, which included parameters like plasma power, particle velocity, particle temperature, e.t.c. And, since the whole basis of this project is on triple product, I chose this particular data set because it included temperature (t).

Since this was a dataset based off of some pretty hard analysis’ in fusion, creating the predictions was not very easy.

So how exactly did I build the ML prediction model?

Using Python through Jupyter Notebook, I was able to import the dataset, which included all the original factors in a fusion plasma.

I defined “Plasma Power” as what I wanted the ML model to predict, and dropped all unnecessary parameters. I made three different scatterplots using all the parameters I wanted to analyze: Particle Temperature, Particle Velocity, and Primary Gas Flow Rate, to see what effect each parameters had on Plasma Power. I got the following results:

What I found the most interesting was how much each parameter was so different, very inconsistent, and hard to predict for humans. For example, you would think that the hotter the particle, the more plasma power — right? Wrong: in the first scatterplot, we can see that the highest temperature is 2900K, but it has the second lowest plasma power. So maybe the ML can make better + more accurate predictions than us to solve these uncertainties!

With the machine learning p.o.v, you have some inputs, and you can make a linear model that predicts continuous values as well. So in this case, our input variables were continuous values to predict the output of how much plasma power is actually being produced 🔥, an essential component in a fusion reactor.

After defining all important variables + cleaning up the data, the model spit out the “test predictions” as the following:

What this means is that the model predicted values based off of other values, and came up with those outputs. Obviously, the Sklearn Plasma Predictions and Test Predictions are not very close, so what did I do?

I asked the model for the “best predictions” to ensure that the model was as accurate as can be, where the best predictions + Sklearn Plasma Predictions were identical, meaning these predictions are the most accurate.

Visualizing the Data 🔍

If you want to see a more washed down + simplified version of Particle Temperature (remember we chose this specific variable because of the triple-product), VS. Plasma Power to visualize the data and to see how accurate the model was:

We can see that the highest point of plasma power actually does not have the hottest particle temperature, instead, a plasma with less power has the highest temperature. Our model predicted something very close to that, how the highest particle temperature did not produce the largest amount of plasma power.

Hopefully as you read this, you started to make the connection between the triple-product and how it’s essentially a performance indicator for a fusion reactor (or in this case — plasma power)! We have such unpredictable values for humans to predict, based off of triple product, so our ML algorithm helps us to make the best predictions, enhancing the triple product!

You landed in Hawaii just in time for checking into your hotel! You think about your overall flight; there were no errors, no delays, a little turbulence 😅, and no mechanical malfunctions within the plane itself.

One day, hopefully fusion reactors can be the same — perform perfectly, with little to none turbulence (haha get it?). We want to measure this performance using all the factors of the Triple Product, which make up the Lawson’s Criterion. And one day, we will get there!

…but for now, I could really use a trip to Hawaii

Thank you so much for reading! If you liked this, check out my other stories on Medium that I’ve published :)

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