Thank you all for reading, following, and recommending. It encourages me.

1. Introduction to 3He isotope, where and how to find it

Short summary base in the popularization book series Kvant for high school students - USSR "Вблизи абсолютного нуля" 2001, by Prof. V. S. Edelman:

"...There are several isotopes with the nuclear charge Z = 2. Of these, 2 are important for the physics of low & ultra-low temperatures, both isotopes are stable:

1. Helium with mass number 3, designated by 3He

2. Helium with mass number 4, that is, 4He, which is a superfluid and we will not talk about it. (the mass number is the sum of protons and neutrons)

• In nature, there is a small amount of 4He, and in the atmosphere, constitutes 5x10-4 %. It is obtained from natural gas. According to geologists' estimates, the total amount of this isotope in all deposits is 7x1010 m3. It is formed as a result of radioactive decay of heavy elements, for example, uranium. Precisely for this reason, it accumulates in underground deposits.
• 3He is also found in nature, but its concentration in the entrails of the earth constitutes 10-9%. Found in uranium deposits. It means that it is impossible to extract any quantity of it from natural sources.

However, there is a solution to obtain it inside nuclear power plants, the nuclear reactions that occur, an isotope of hydrogen, tritium, is obtained 3H or T. Tritium is an uncommon product of the nuclear fission of uranium235 & plutonium239, with a production of about 1 atom per 10,000 fissions. Tritium's decay product helium-3 has a very large cross-section (5330 barns) for reacting with thermal neutrons, expelling a proton; hence, it is rapidly converted back to tritium in nuclear reactors.

• Tritium serves as "fuel" in thermonuclear installations. In addition, is used for the manufacture of nuclear weapons. But accumulating it is a tedious job since the time of its radioactive semi-disintegration constitutes 12.3 years. It disintegrates by radiating a particle and transforming itself into 3He. Separating the latter from 3H is not difficult. In this way, the 3He reaches the physics laboratories..."

3H → 3He + 2 leptons( e− + νe (neutrino))

see cc Wiki:

https://en.wikipedia.org/wiki/Tritium

A slide with the summary:

2. How to cold down 3He atoms to such extremely very low temperatures? *

The answer is the Pomeranchuk effect, based on an anomalous melting (or anomalous solidification) of 3He: at a temperature below 0.3 K, the entropy of liquid 3He is less than that of the solid 3He, and heat we know that heat is absorbed during the formation of a solid phase.

According to the Clausius-Clapeyron equation, the dependence of the melting point - Tm on pressure - P, is also anomalous: Tm decreases with increasing P.

This thermal effect was theoretically predicted by I. Ya. Pomeranchuk in 1950, experimentally discovered by American physicists W. Fairbank & G. Walters in 1957.

The Pomeranchuk effect is used to obtain ultralow temperatures (up to 1-1.5 mK): during adiabatic compression of 3He, a solid phase is formed along the melting curve and T decreases accordingly.

Below 1–1.5 mK, the ordering of the 3He nuclear spins in the solid phase leads to a sharp drop in the entropy of solid 3He, which rules out the possibility of the effect accordingly to the Nernst theorem.

But there is more to say for the solid phase, lowering adiabatically the temperature of a mixture of solid and liquid 3He.

The compression below the temperature of 0.32 K due to the fact that the entropy S of the system of disordered nuclear spins of solid 3He remains constant up to the Neel temperature TN (solid 3He) = 1 mK, and the entropy of liquid 3He decreases linearly, characteristic of a Fermi liquid.

As a result, below 0.32 K, the entropy of liquid 3He becomes less than the entropy of solid 3He, and the heat of fusion of 3He becomes negative.

According to the Clausius-Clapeyron equation, a change in compression of an equilibrium mixture of liquid and solid 3He leads to a decrease in P and is used to obtain at ultra-low temperatures the Fermi liquid 3He isotope that can be used to obtain the phase diagram of figure 1 in this thread for T from 10 mK to 1.5 mK.

* see for example: "Вблизи абсолютного нуля" 2001, Series Kvant by Prof. V. S. Edelman

Therefore this way, it was possible to obtain very low temperatures in 3He isotope and find the phase diagram shown at the beginning of this thread where there are normal state and superconducting phases depending upon the value of pressure and temperature.

Some of these phases resulted to be superconducting phases, even in the liquid state.

3He isotope became the first Fermi liquid (conducting) at ultralow temperatures with different order parameters in its superconducting phases.

If to the phase diagram figure adds another dependence, an external magnetic field H, then new phenomena related to a kind of order parameter can be studied.

For a figure look at:

Article On a possible component of the magnetic field of neutron stars

Will liquid 3He isotope be able to transmit electric current? *

In general, liquid He is an excellent dielectric

A. Shalnikov introduced a β-radioactive source into liquid helium and began to study how the electrons leaving the source moved under the action of the electric field.

He used a softshell of molybdenum oxide, the pores filled with tritium.

Tritium reaction: 31H → 32He (helium three) +  e− (β particle) + νe (neutrino) disintegrates by emitting β particles with an energy of several keV, becoming 3He.

Those β particles collide with helium atoms. Since their energy greatly exceeds the ionization energy of the helium atom, they, when colliding with it, split it into an electron and an He+ ion.

Naturally, the separated electrons and ions attract each other and again form atoms. But since the β particles are constantly emitted, close to the target, there will nonetheless be a cloud of electrons and ions.

This makes the mobility of electrons and ions of liquid He more favorable at very low temperatures, increasing their mobility - μ and electric current - I considerably.

I = N e μ E

* see for example: "Вблизи абсолютного нуля" 2001, Series Kvant by Prof. V. S. Edelman

And here comes Prof. Anthony Leggett in the 70s, at that time at the University of Sussex, UK who was awarded the 2003 Nobel Prize in Physics (with V. Ginzburg and A. Abrikosov) for pioneering contributions to the theory of superconductors & superfluids

Article A theoretical description of the new phases of liquid ³He

The different types of symmetry breaking:

• The global phase of BCS superconductors is due to the breaking of the gauge.
• The preferred direction is due to the breaking of the 3D rotation symmetry.
• The spontaneous fields are due to the breaking of the spin symmetry.

Each quasiparticle in 3He carries 2 vectors, the L (3D space) and the S vectors (Pauli matrix complex space), each has 3 components, therefore the direct tensor product gives a total of 3 times 3 = 9 degrees of freedom, each with 3 components and we have 18 in total since those for the A phase it could be a real and an imaginary part.

3He atoms in the liquid state are:

• Fermions & obey Fermi Dirac statistics, therefore there should not be Bose Einstein condensation.
• Similar to the BCS superconductivity with Cooper pairs.

In addition, they follow the Landau Fermi liquid theory FL (the 50s) starting with a noninteracting Fermi gas and turning on interactions slowly, to get a Fermi liquid.

FL describes quasiparticles which can be thought of as dressed helium atoms with an effective mass m* and a mean time between collisions τc is defined.

Zero sound for frequencies ω τc >>1 similar to plasmon as in a collisionless plasma and without self-consistency are defined.

To have an idea of how the Oder parameter of the Cooper pairs looks in a liquid 3He isotope and in an HTSC, I have done this slide transformation from a talk given in 2003 to the Canadian Physics Students association.

Another slide involving this new thread, which was the families of unconventional superconductors around the 2000s

Soon there will be a preprint on this subject from our group.

An interesting science journalism blog for this thread:

https://www.bbvaopenmind.com/en/science/physics/helium-3-lunar-gold-fever/

Before continuing into this talk, we have to understand why in nature, symmetry is broken by some physical laws such as the presence of an internal magnetic vector field Hint as in the case of the heavy-fermion UPt3or the application of an external elastic tensor field Sij as in the case of the triplet Sr2RuO4.

There are basic symmetries for superconductivity which are simplified in the slide I attach to this post:

• Rotational symmetry
• Translational symmetry
• Time reversal symmetry
• Gauge symmetry

The book: Helium Three. Editors L. Pitaevskii & W. Halperin in the series: Modern Problems in Condensed Matter Sciences. 1st Edition, 1990, North-Holland ELSEVIER

has an extensive review on the liquid 3He isotope superconducting properties including the physics and the group theory, very instructive even though it is complicated.

I refer the lector to that book but I advise that the 3He isotope is a liquid and that the angular momentum L for liquids is not conserved as in the symmetric solids.

"Therefore the analogy that implies that liquid crystals are like unconventional superconducting crystals is one idea that does not convince me".

Nature describes liquids with classical or quantum hydrodynamics but describes solids with classical or quantum crystal theory. Dot!

https://www.elsevier.com/books/helium-three/pitaevskii/978-0-444-87476-4

Well now I am prepared for the crystal ternary compound strontium ruthenate, a triplet low-temperature superconductor which is represented as Sr2RuO4

• It was discovered by Prof. Yoshiteru Maeno & a group of collaborators in 1994
• Tc ~ 1.5 K but strongly depends on non-magnetic pair breakink, i.e, is an unconventional superconductor.

Yoshiteru Maeno; H. Hashimoto; et al. (1994). "Superconductivity in a layered perovskite without copper". Nature. 372 (6506): 532–534.

Article Superconductivity in a layered perovskite without copper

Please see cc Wiki commons license:

https://en.wikipedia.org/wiki/Strontium_ruthenate

There are several important references from the experimental point of view about triplet pairing in Sr2RuO4, they are old, but those were the ones I studied in detail, the experiments were made with quire pure crystal samples:

I attach 2 slides to this post:

• We can see that from the value of Tc ~ 1.5 K, the ruthenate follows the line of heavy fermions and also has several signatures of a 3He isotope, the jumps in one macroscopic quantity observed for the first time by Prof. Lupien.
• Upt3 also has a jump in the elastic constant c11, but as we noticed in our 2002 article, it is the H internal antiferromagnetic field that causes TRB (time-reversal breaking).
• In Sr2RuO4 the breaking of the field, in this case, elastic field, namely c66 comes from the experimental sep ut, and my guess is that the stochiometric Sr impurities appearance of DOS impurity levels and dressed quasiparticles play the main role.

P. Contreras.

I will follow from now in this thread on the phenomenology of the MN model available at:

Article Model for Unconventional Superconductivity of Sr2RuO4: Effec...

The attached slide presents the topic.

In BCS superconductors we can have two kinds of phases:

• The intermediate phase in the Ginzburg Landau Theory.
• The mixed-phase was predicted by Abrikosov's theory.

I will dedicate several posts to them before I continue.

Londons equations in superconductors, first phenomenological theory for the physical phenomenon of superconductivity:

• It establishes a relationship between current j and magnetic field H in conventional superconductors.
• It was first obtained in 1935 by the brothers' Fritz & Heinz London in their work "The Electromagnetic Equations of the Supraconductor. "

London, F., and London, H. (1935) Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 149(866), 71–88. http://www.jstor.org/stable/96265

• It provided the first satisfactory explanation for the Meissner effect, the decay of the magnetic field in superconductors. Afterward in 1953, the Pippard equation for pure superconductors was obtained.
• The London equation provides the key to understanding the nature of superconducting ordering due to the condensation of current carriers in the state with the smallest possible momentum p = 0
• Therefore it tells us about the phase of the carrier's condensation, we call them here supercarriers since there is no current dissipation (The Cooper Pair)
• There are 2 equations, one for the effect of an external electric field and one for the magnetic field, the gauge is div A = 0.

The second important phase in superconductors was discovered experimentally by Meissner. According to a demagnetization factor, one may obtain an intermediate state where the external field H penetrates a superconductor.

• This is a macroscopic phase defined by a phenomenological non-linear wave function very slowly varying inside an interval called coherence length ξ(T) which depends on temperature.
• κ = λ/ξ(T), where κ is the GL parameter and λ is the GL penetration depth

https://es.wikipedia.org/wiki/Teor%C3%ADa_Ginzburg-Landau

• The Intermedium phase in a superconductor was first described by Ginzburg & Landau, as a phase separation into macroscopic non-superconducting & superconducting domains forming a Husimi Q representation according to a wiki source * in 1950

V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950)

Husimi Q-repr was introduced by K. Husimi in 1940 as a quasiprobability dist to be used in QM to classically represent the phase-space distr of a quantum state such as light in a phase space formulation & is also applied in the study of quantum effects in superconductors.

* https://en.wikipedia.org/wiki/Husimi_Q_representation

The third phase was discovered by A. Abrikosov, triangular vortex lattices of normal states inside superconductors II type:

Abrikosov vortex is a superconducting current (supercurrent) vortex circulating around a normal (non-superconducting) core (vortex filament), inducing a magnetic field with a magnetic flux equivalent to a magnetic quantum flux

A. Abrikosov in 1957 in his work “On the magnetic properties of superconductors of the second group”, theoretically showed that the penetration of a magnetic field into a type 2 superconductor occurs in the form of quantized vortex filaments (such a system is energetically “favorable”).

A. A. Abrikosov (1957). Journal of Physics and Chemistry of Solids. 2 (3): 199–208. DOI:10.1016/0022-3697(57)90083-5

Each such filament (vortex) has a normal (non-superconducting) core with a radius on the order of the coherence length of the superconductor ξ.

Around this normal cylinder, in a region with a radius of the order of the penetration depth of the magnetic field λ, an eddy current of Cooper pairs (supercurrent) flows, oriented so that the Hint it creates is directed along with the normal core, that is, it coincides with the direction of Hext.

Each vortex carries one flux quantum Φ0= 0.5 ħ/e

Image cc commons license from:

https://www.doitpoms.ac.uk/tlplib/superconductivity/type.php

The following pictures are adapted for the science popularization book:

Электроны шагают в ногу, или история сверхпроводимости by Prof. В.И. Рыдник 1986, Серия: Наука и прогресс Издательство: Знание / Electrons keep pace or the history of superconductivity by Prof. V. I. Rydnik, 1986, Series: Science and progress Publisher: Znanie

• They show nicely and clearly, the 2 types of phases and how they can coexist together in a superconductor, theory, and original articles mentioned in the 3 previous posts.

Prof. Ph.D. Vitaly Rydnik has several popularization physics books in the Series: Science and progress Publisher: Znanie.

This research article is related to vortices in the triplet Sr2RuO4 and therefore follows the previous posts:

Article Superconducting rings show hints of half-quantum vortices

We have added a new publication related to this thread:

Article Tight-Binding Superconducting Phases in the Unconventional C...

In Dec 2002, we firstly proposed under the guidance & idea of my Ph.D. supervisor, Prof. MB Walker at the University of Toronto, that in the triplet SC compound strontium ruthenate (Sr2RuO4), there were 2 phases depending on an external elastic tensor field Sij.

For that, we built & calculated the Ginzburg-Landau Theory using the Gibbs free energy G which is adequate for external fields.

• If the external elastic stress σij is small then a BCS with one component GL (1,0) phase appears at a second-order phase transition Tc, +. If T is decreased more, then a second triplet phase with 2 GL components (1, i) appears at a second-order phase transition Tc, -. In those cases, σij is under the control of the experimentalist.

Article Theory of elastic properties of Sr2RuO4 at the superconducti...

In Dec 2021, (19 years later) we proposed that in Sr2RuO4 there could be 3 phases depending on non-magnetic disorder % under the influence of in-situ Sr-non-magnetic atoms, this work has been accepted for publication and we are waiting for the galley proofs.

Preprint Quasi-point versus point nodes in Sr2RuO4, the case of a fla...

• In 2022, we compared with the strontium doped HTCS LaSrCu0, and the same methodology allowed us to explain experimental data in this compound, with simple and elegant physical principles:

Preprint Coalescence of Impurity Strontium Atoms in Lightly-Substitut...

so we can finally have concluded the comparison in the open-access article:

Article Tight-Binding Superconducting Phases in the Unconventional C...

• Depending on the non-magnetic disorder, there can be three phases in Sr2RuO4: 1 phase with the Miyake Narikiyo tiny gap microscopic model, which is a BCS-like phase with only Cooper pairs/supercurrent carriers. Another phase in the same MN model with triplet bosons and dressed by the impurity potential normal state quasiparticles. On the other hand, our methodology allows us to go to the GS and there we find a 3rd point nodes phase, theoretically. That means three phases.

A popularization article will follow this thread, that was the goal to create it if God allows me to do so.

Best Regards and thank you so much to all readers.

In the binary crystal alloy, Uranium platinum three (UPt3) also were observed different superconducting phases as in the 3He isotope and in Sr2RuO4

• In ultrasound attenuation below Tc, researchers observed two phases.
• In specific heat below Tc, researchers observed 2 phases.
• However, in the H vs T diagram experimentalists observed 3 superconducting phases.

To summarize, UPt3 is an unconventional superconductor that was discovered by Prof. Stewart & collaborators in 1984 and also has different superconducting phases with some symmetries broken states.

UPt3 is a heavy-fermion metal alloy, the superconductivity is related to intrinsic magnetism inside the compound. It is a crystal with a hexagonal symmetry point group. I will add another post when electricity will allow me to do so.

The Order Parameters (OP) for the heavy-fermion UPt3

The fitting of the exp. data in UPte3 was made using 3D OP that is superconducting ultrasound and electronic specific heat, therefore it was considered that the nature of superconductivity in uranium platinum three is a 3D nature.

The problem is that I worked a lot on strontium ruthenate for 20 years and I also needed a 3D OP to fit the ultrasound data, the specific heat data, and the electronic thermal conductivity data, that experimental data came from different groups around the world, therefore they had different amounts of non-magnetic dirt, but I considered for the beginning that the quantum nature of strontium ruthenate is 2D and it is given by the Miyake-Narikiyo model, despite we need to consider the z-axis hopping to fit experimental data, the question is still open.

Why? But because the predominant and most intriguing FS sheet is the gamma (with a 2D nature), the other 2, the FS alpha & beta sheets have a (1D) dominant behavior for the {xy0} to {00z} doping, but nothing quantum special on them, we fitted the data with a triplet model, the same kind as the 2D model, only one more hopping term that extends to 3D the IR of the OP.

For Uranium Platinum three the OP a 3D hexagonal system, theorists established years ago that there were combinations of four 1-dimensional Irreducible representations, i.e., the A1, A2, B1 & B2; and two 2-dimensional irreducible representations, namely the E1 &E2.

Experimentally, the study of UPt3 has been guided by the discovery of its complex superconducting phase diagram, which displays several different superconducting phases, separated by well-marked second-order phase transitions as in strontium ruthenate, with the difference that Tc is bigger for strontium ruthenate and disorder could be seen marvelously in several 3D experiments.

In heavy-fermion UPt3, an antiferromagnetic (AF) order often coexist with superconductivity, a short-range magnetic order is observed by neutron scattering below TN ~ 5K.

The magnetic moment is very small with ~ 0.01 μB but it does coexist with the superconducting state.

For a review check on UPt3, please follow:

R. Joynt & L. Taillefer, 2002. The superconducting phases of UPt3. Reviews of modern physics, Vol. 74, pp. 235- 294.

Article The superconducting phases of UPt3

Superconductivity according to Prof. Jean-Pascal Lectures in 2011, extracts adapted from his excellent lectures.

Let us return to the 3He isotope for a couple of posts:

It was mentioned the anomalous behavior of the liquid-solid 3He transition, without explaining what it consists of. Now let us explain it *:

If it is cooled the isotope 3He down to T = 0 K ---> that the thermal motion is banned at all, as a consequence, there is no transition heat & the melting curve is horizontal.

Everything happens exactly the same as for 4He, only the “horizontality” ends already at 1 - 2 mK for 3He, meanwhile, for the superfluid isotope 4He, it ends at about 1 K.

The difference in that interval is huge & immediately becomes clear that the cause, must be a qualitative difference between the atom of 3He and that of 4He.

What does that difference consist of?

We know that the nucleus of the isotope 3He has spin & magnetic moment of the atoms in solid 3He at a temperature of 2 mK.

As a result of their interaction, they are arranged in a regular structure: the spins of each pair of neighboring atoms are oriented in opposite directions.

Such a state, called antiferromagnetic, is found in nature no less often than the characteristic ferromagnetic state of iron, in which all magnetic moments of atoms are parallel to each other.

The interaction energy - E corresponds to a particle, and the phase transition temperature Tc in the ordered state, whether it is the formation of atoms based on ions and electrons, of a solid body, based on vapor, or the magnetic ordering, always have the same order of magnitude E ~ kB Tc

When kB Tc >> E, the spin disorder is observed, since, according to the law of

Boltzmann, all admissible states remain equally filled, and the difference in their energies is within the limits of E.

* see "Near absolute zero/Вблизи абсолютного нуля" 2001, Series Kvant by Prof. V. S. Edelman

Continuation of the previous post:

* The difference is that the nucleus of 3He has spin and magnetic moment linked to it, and the magnetic moments of the atoms in solid 3He at a temperature of 2 mK, as a result of their interaction, are arranged in a regular structure: the spins of each pair of neighboring atoms are oriented in opposite directions (see the figure attached to this post)

This state is called antiferromagnetic and is found in nature no less often than the characteristic ferromagnetic state of iron, in which all the magnetic moments of the atoms are parallel to each other.

To destroy the magnetic order, solid 3He must be heated. It is not difficult to estimate approximately the amount of such heat:

"If the energy of interaction is ~kB Tc, to completely destroy the order it is necessary, at the rate of one atom, to supply approximately the same energy. And since 1 mol of solid 3He, which occupies a volume of 40 cm3, contains NA ~ 6·1023 atoms, an energy of 3·10-26 x 6·1023 ~ 2·10-2 J is required to heat this vol."

According to the absolute temperature measured in mK, the result obtained is a huge amount of energy. This physics happens when it comes to solid 3He.

In the liquid, when the atoms transit from one place to another and change neighbors all the time, the maximum order is equivalent to what for solid 3He is called an absolute disorder, that is, it is impossible to tell where a neighbor is looking, but on average half of them are looking one way and the other half another.

Therefore, at T > 0.002°K with thermal motion, that is, with the alteration of the order in the spin system, more energy is bound in the solid 3He than in the 3He gas at the same temperature. See attached Fig.

Because of this, during melting, heat is given off, and during crystallization, it is absorbed.

This thermal effect (called the Pomeranchuk effect after the Soviet scientist who predicted it) is observed by the fact that the substance has to undergo a phase transition at a much lower temperature than the thermostat possesses.

will continue...

* see "Near absolute zero/Вблизи абсолютного нуля" 2001, Series Kvant by Prof. V. S. Edelman

The Pomeranchuk effect

The Pomeranchuk effect is the basis for one method of cooling, and is also magnetic, although its realization does not require a magnetic field.

I translate & unquote from "Near absolute zero/Вблизи абсолютного нуля" 2001, Series Kvant by Prof. V. S. Edelman

"...The figure attached to this thread shows a cryostat can be structured by three volumes separated from each other by elastic siphons of bronze; the innermost volume is filled with liquid 3He, and the other two with 4He.

Initially, this apparatus is cooled to ~ 0.3°K, and the pressure of 4He in the chambers rises up to 25 atm, that is, it acquires a somewhat lower value than the solidification pressure of 4He. Then the outer volume is closed, and the pressure of 3He rises to 29 atm, which is also somewhat lower than the pressure of the transition of 3He to the solid phase previously discussed in other posts.

In addition, the siphons are distended, thanks to which the 4He, in the external volume, becomes partially solid. After this, by means of a cryostat of the solution, the apparatus is cooled to 25 mK, and the 4He in the intermediate volume is sent outside...".

Let us summarize once more time the effect of the magnetization inside a superconductor of a cylindrical from the Gibss thermodynamic potential that serves in an external field H, a perfect diamagnetic:

The free thermodynamic Gibss for external potential:

(0) G = U - T S - H . M

From the thermodynamics in a metal:

(1) M = Bins + 4 π H, if Bins = 0, then we have H = - 1/ (4 π) M

(2) χ (chi) = - 1/ (4 π) < 0 for a diamagnetic, i.e, diamagnetic suceptibility , in general is a tensor of second order x-1ij = ∂ Mi / ∂ Hj and specifices the magnetic flux density B(strengt is H)

The following blog discusses inverse problems in statistical mechanics.

One of those problems elaborated is how to establish ground zero in physics as a fundamental basis for studying the physical properties of a particular state:

https://torquato.princeton.edu/research/statistical-mechanics/

Worthly to read, I recommend it.

Well, the second protagonist of the saga is Prof. Y . Maeno from Kyoto U., who kindly sent his experimental samples on Sr2RuO4 & data 20 years ago to several LT Labs. I as a theoretician learned to train data fitting.

Thank you, Dear Prof.Y. Maeno, I learned a lot from your data and studies.

Best Regards.

I will start to talk about the Fermi Liquid Theory (FLT) in this and the thread on quasiparticles as well, and the relation to isotope 3He:

Some comparative values to start: the idea of quasiparticles is valid in several approximations, please look at the attached adapted screenshot.

Why in Sr2RuO4, the Δγ0? seems to be ~ 1 meV, even with the disorder?

Soon an answer.

Here there is a possible answer, Dear readers, follow the DOI link and I apologize for the late reply:

Article The Effect of Nonmagnetic Disorder in the Energy Gap at Zero...

Dear Doctor

"ABSTRACT

The isotope effect in superconductors is usually summarized by giving the observed values of p in the equation MpTc=constant, where M is the isotopic mass and Tc the superconducting transition temperature. Fröhlich predicted the value p=1/2, but the measurements in some instances show deviations from this prediction. An explanation of the deviation of p from ½ is offered based on an analog of Wien's displacement law applicable to the vibration spectrum of real crystal lattices. The departure of p from the value ½ is attributed to the departure of the frequency spectrum from a simple power law. For many superconducting elements, p may be estimated from specific heat data, when such data are available to the desired degree of accuracy. A value of p is calculated for Sn which is in good agreement with some of the experiments. The large value 0.73 observed for Pb is shown to be reasonable. The values of p for the other superconducting elements are discussed. It is concluded that the observed deviations of p from ½ are not necessarily in conflict with the theories of Fröhlich and Bardeen."

"Distrontium ruthenate, also known as strontium ruthenate, is an oxide of strontium and ruthenium with the chemical formula Sr2RuO4. It was the first reported perovskite superconductor that did not contain copper.Superconductivity in SRO was first observed by Yoshiteru Maeno et al. Unlike the cuprate superconductors, SRO displays superconductivity in the absence of doping.[2] The superconducting order parameter in SRO exhibits signatures of time-reversal symmetry breaking,[3] and hence, it can be classified as an unconventional superconductor."

"The presence of two phases is a clear indication that 3He is an unconventional superfluid (superconductor), since the presence of two phases requires an additional symmetry, other than gauge symmetry, to be broken."

Dear Prof. Sundus F Hantoosh

Thank you so much for your detailed post.

Yes, it requires another symmetry to be broken.

In the case of strontium ruthenate is the time reversal.

In the case of the isotope 3He, in the small A phase it has also a broken symmetry.

Best Regards.