UPDATE March 30, 2022: The time crystal model cannot explain the second sound of superfluid helium-4. Our plate model of superfluidity is better.
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Let us study of the transition of liquid helium-4 at 2.17 K, that is, the lambda point.
Helium-4 macroscopic physical properties close to the lambda point 2.17 K
W.H. Keesom et al. (1935) measured as the melting heat 4 J/g at 2.5 K, at an unknown pressure less than 135 atmospheres.
The vaporization heat at 2 K is 23 J/g.
We may conclude that the phase transition of liquid helium-4 at 2.17 K is "substantial", compared to its melting from a solid form.
The density of liquid helium-4 grows rapidly as it is cooled from 4.2 K to 2.17 K. When it is further cooled, the density starts falling. The density curve suggests that the phase transition actually happens gradually above 2.17 K. The specific heat, on the other hand, suggests that it happens mostly under 2.17 K.
Thermal conductivity of superfluid helium-4 below 2.17 K grows rapidly to a huge value as we cool the liquid. The conductivity according to Wikipedia is several hundred times that of copper. It is as if the liquid would contain a gas which transfers heat energy swiftly to distant locations. This may be one of the reasons why some people believe that superfluid helium contains a Bose-Einstein condensate which would behave like a gas.
Black body radiation at 2 K has a frequency ~ 2 * 10¹¹ Hz and a wavelength ~ 1.5 mm.
The speed of sound in superfluid helium-4 is 220 m/s at 2 K. The wavelength corresponding to 2 * 10¹¹ Hz would be 1 nanometer. The diameter of a helium atom is 0.28 nm.
Superfluid helium-4 cannot contain a Bose-Einstein gas of helium atoms
Let us have many photons in a box whose walls are perfect mirrors. There is no problem in putting many photons in the lowest energy state where the box length is 1/2 of the photon wavelength.
Low-energy photons do not interact. They do not collide and can live in a perfect harmony.
On the other hand, helium atoms do collide in a box. The assumed Bose-Einstein gas in superfluid helium-4 is so dense that it is essentially full of helium atoms. Why would the atoms not collide?
The atoms are quite heavy objects. They behave almost classically. Classically, the atoms will collide all the time. The hypothesis of a weakly interacting Bose-Einstein gas cannot hold.
^
/
o helium atom
/
/
__/ XXXXXX
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uneven capillary
wall
Also, the atoms of a gas would collide with uneven walls of a capillary and lose their momentum if the gas is moving relative to the capillary. There would be no superfluidity.
Nikolai Bogoliubov (1947) suggested that a superfluid contains a Bose-Einstein gas of quasiparticles. We will look at Bogoliubov's model in a subsequent blog post.
Why sound propagates fast in a material but heat propagates much slower?
Let us have a fluid. Audible sound waves have a very long wavelength and their scattering from individual molecules in the fluid is negligible.
Short wavelength vibrations of heat scatter from individual molecules.
How could we transfer heat quickly?
Maybe adding order to the fluid would help. If the coupling is stronger, heat propagates faster. Also, scattering may be less from an ordered lattice. Thermal conductivity of ice is four times the conductivity of water.
The obvious solution to very fast heat transfer is to move matter around. If there would exist a gas of heat-carrying electrons within the fluid, thermal conductivity could be much better. Metals are good conductors of heat for this reason.
Another option is to transform the fluid into a time crystal where molecules move large distances around without bumping into each other. They can carry thermal vibrations at a large speed to distant locations.
A pot of water boiling on the stove may have a fairly stable convection pattern. It is an approximate time crystal with a very efficient heat transfer rate.
Superfluid helium seems to have a third method of a very fast heat transfer: heat can propagate in waves which do not scatter much. The speed is 20 m/s at 1.8 K. We have to analyze this.
The time crystal model of superfluid helium-4
Huge thermal conductivity suggests that superfluid helium-4 is a time crystal where atoms can move in rings over large distances.
ring of atoms
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o o o o o
^ o o o o o o o o o o o o o |
| o o o o XXXXXX o o o o v
XXXXXXXXXXXXXXXXXXXXXXXXX
uneven capillary wall
The model may also explain why the liquid is superfluid. A ring can move over uneven walls like the continuous track of a bulldozer moves over an uneven terrain.
chain of helium atoms
o o o o o o o --->
o o o o o o o o o o o o o static helium atoms
XXXXX o o o XXXX o XX
XXXXXXXXXXXXXXXXX
uneven capillary wall
Besides the continuous track, there is an alternative explanation for superfluidity: static helium atoms form a layer over the uneven capillary wall and make the wall smooth. After that, chains of helium atoms can slide within the capillary without friction.
The time crystal model requires that chains of helium atoms can slide past each other with zero friction. Classically, the atoms will radiate electromagnetic radiation or phonons as they slip past each other. It might be that the highly ordered structure of the time crystal causes destructive interference in the created waves, such that the radiation to the outside is essentially zero.
Superfluid helium-4 at 0 K has to be a time crystal
Helium-4 can only become a solid crystal at pressures larger than 25 atmospheres. At the normal 1 atm pressure it stays as a superfluid even at 0 K.
Matter at 0 K must, in some sense, be highly ordered at 0 K. Otherwise, it would lose energy through radiation. But a superfluid does not form a static crystal lattice.
If a superfluid is highly ordered but does not form a static lattice, then it probably is a "moving crystal", that is, a time crystal.
Conclusions
The time crystal model explains superfluidity and huge thermal conductivity.
We need to check if the time crystal model fits all the facts that we know about superfluid helium.
Can we explain the second sound with the time crystal model?
Russell J. Donnelly (2009) has a nice paper about the second sound. Let us look at it.
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