There are multiple excellent guides to the fundamental principles of dilution refrigeration, including “Low-Temperature Physics” by C. Enss and S. Hunklinger, “Matter and Methods at Low Temperatures” by F. Pobell, and our favorite “Experimental Principles and Methods Below 1 K” by O. Lounasmaa. These are handy reference guides to have around the lab; nonetheless, here we present the basic principles of dilution refrigeration as a “quick start” guide.
The first consideration is to understand the evaporative cooling of the helium isotopes. Helium-4 (4He), the common isotope with two protons and two neutrons, liquefies at 4.2 K at standard pressure (sea level). We can cool liquid helium further by pumping on it; extracting the latent heat of vaporization, unfortunately, the vapor pressure of helium decreases exponentially. Since the cooling power [dQ/dt] is proportional to the number of particles per unit time moved into the vapor phase [dn/dt] the cooling power is limited by the exponentially decreasing vapor pressure. This limits the ultimate temperature achieved by pumping on liquid helium to around 0.8 K, but more realistically 1.2 K in the laboratory.
To get even colder, one must switch to the rare isotope of helium, helium-3 (3He) with its two protons and one neutron. Because of the missing neutron 3He has a smaller molar mass and hence a higher vapor pressure than 4He at the same temperature, this will be important for the operation of the dilution refrigerator later. Because of the higher vapor pressure the practical base temperature achieved by pumping on liquid 3He is decreased to 0.3 K. This is an excellent temperature, but due to the scarcity of 3He there are no practical continuously operating 3He refrigerators. Instead, a small reservoir (commonly called a pot) of 3He is usually condensed by a pumped 4He pot, which is then pumped by a closed charcoal sorption pump in cyclic operation.
Figure 1: Phase Diagram Frank Pobell (1995)
How then can we achieve continuous operation of a refrigerator below 1 K? The solution to this problem came from the mind of Heinz London; the key comes from the phase diagram of 3He in 4He, shown in Figure 1. The addition of 3He suppresses the superfluid transition of 4He and below approximately 0.87 K there is a phase separation resulting in two fluids. The first is a nearly pure liquid of 3He (at least at the lowest temperatures) which we will call the pure phase, and the second is a dilute solution of 3He in 4He which we call the dilute phase (This is a remarkable phenomenon, two isotopes of the same atomic species phase separating!) Even as the temperature approaches zero, the solubility remains at approximately 6.7%. This finite solubility is key to the operation of a dilution refrigerator.
Let us now look at a cartoon of a dilution refrigerator, Figure 2. If we imagine that we have already condensed a mixture of 3He and 4He (roughly 30% 3He), which we call the mix, into the fridge, the liquid will completely fill the mixing chamber, heat exchangers, and partially fill the still. If we now pump on the still we will perform evaporative cooling of both the 4He and the 3He. As the temperature is lowered, especially below 1.2 K, the evaporation will be predominantly 3He because of its higher vapor pressure. This allows cooling of the entire liquid into the region of phase separation. Now we expect to have two distinct liquids with the pure (3He rich) phase floating on top of the dilute phase. The boundary between these two phases should lie in the mixing chamber.
Figure 2: Dilution Refrigerator Frank Pobell (1995)
We now have the right conditions to operate a dilution refrigerator. We are pumping on the still and preferentially extracting 3He atoms. As a result; the dilute phase in the still now has less than its desired 3He concentration from the phase diagram and so there is an osmotic pressure driving 3He up the dilute phase from the mixing chamber to the still. In order to replenish the 3He in the dilute phase, 3He passes across the phase boundary from the pure side into the dilute, undergoing dilution. Hence the name of the refrigerator. One can imagine that this process of going from the pure phase to the dilute is very similar to the process of going from the liquid phase to the gas phase, except the exponentially decreasing vapor pressure of the gas phase is now replaced with a finite solubility (think vapor pressure) of the 3He in the dilute phase. This is the key principle of operation of the dilution refrigerator, and the reason the cooling power should be proportional to T2, instead of e-1/T as it is for evaporative cooling.
The final step of the dilution refrigerator is to purify the 3He that was pumped from the still and return it to the pure side of the mixing chamber. This may sound simple, but in practice, it is the most difficult part of the dilution refrigerator. The returning pure 3He must be adequately pre-cooled by the dilute phase at every point along the fridge to ensure that it is properly pre-cooled before entering the mixing chamber.
Figure 3: Zero Point Cryogenics' Model I Dilution Refrigerator
One final note on the principle of operation is that the cooling power of the dilution refrigerator is proportional to the number of 3He atoms being extracted per unit of time. In practice, when left on its own the still will cool too low, such that the vapor pressure of the 3He drops and the number of moles of 3He being cycled also drops. In this case, we will add some heat to the still to raise the temperature and hence the vapor pressure of 3He, allowing for faster circulation and higher cooling powers.
Zero Point Cryogenics – Colder for longer
Zero Point Cryogenics is an Edmonton-based company that manufactures cryogenic equipment called dilution refrigerators, which are the primary low-temperature platform for quantum computers. The company is focused on designing robust and reliable dilution refrigerators for quantum computing industry partners that will satisfy their space, size, and operation requirements. By creating the world’s most reliable dilution refrigerator, ZPC customers can focus on developing their quantum technologies, instead of operating their refrigerators.
For more information, visit zpcryo.com.
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