** The strange observation of Brian Skinner about the effect**

After writing the above comments I learned from a popular article (see this) about and objection (see this) challenging the claimed discovery (see this). The claimed finding received a lot of attention and physicist Brian Skinner in MIT decided to test the claims. At first the findings look quite convincing to him. He however decided to look for the noise in the measured value of volume susceptibility χ_{V}. χ_{V} relates the magnetic field B in superconductor to the external magnetic field B_{ext} via the formulate B= (1+χ_{V})B_{ext} (in units with μ_{0}=1 one has B_{ext}=H, where H is used usually).

For diamagnetic materials χ_{V} is negative since they tend to repel external magnetic fields. For superconductors one has χ_{V}=-1 in the ideal situation. The situation is not however ideal and stepwise change of χ_{V} from χ_{V}=0 to χ_{V} to some negative value but satisfying |μ_{V}| <1 serves as a signature of high Tc superconductivity. Both superconducting and ordinary phase would be present in the sample.

Figure 3a of the article of authors gives χ_{V} as function of temperature for some values of B_{ext} with the color of the curve indicating the value of B_{ext}. Note that μ_{V} depends on B_{ext}, whereas in strictly linear situtation it would not do so. There is indeed transition at critical temperature T_{c}= 225 K reducing χ_{V}=0 to negative value in the range χ_{V} ∈ [-0.05 ,-.06 ] having no visible temperature dependence but decreasing somewhat with B_{ext}.

The problem is that the fluctuations of χ_{V} for green curve (B_{ext}=1 Tesla) and blue curve (B_{ext}=0.1 Tesla) have the same shape. With blue curve only only shifted downward relative to the green one (shifting corresponds to somewhat larger dia-magnetism for lower value of B_{ext}). If I have understood correctly, the finding applies only to these two curves and for one sample corresponding to Tc= 256 K. The article reports superconductivity with Tc varying in the range [145,400] K.

The pessimistic interpretation is that this part of data is fabricated. Second possibility is that human error is involved. The third interpretation would be that the random looking variation with temperature is not a fluctuation but represents genuine temperature dependence: this possibility looks infeasible but can be tested by repeating the measurements or simply looking whether it is present for the other measurements.

**TGD explanation of the effect found by Skinner**

One should understand why the effect found by Skinner occurs only for certain pairs of magnetic fields strengths B_{ext} and why the shape of pseudo fluctuations is the same in these situations.

Suppose that B_{ext} is realized as flux tubes of fixed radius. The magnetization is due to the penetration of magnetic field to the ordinary fraction of the sample as flux tubes. Suppose that the superconducting flux tubes assignable 2-D surfaces as in high Tc superconductivity. Could the fraction of super-conducting flux tubes with non-standard value of h_{eff} - depends on magnetic field and temperature in predictable manner?

The pseudo fluctuation should have same shape as a function temperature for the two values of magnetic fields involved but not for other pairs of magnetic field strengths.

- Concerning the selection of only preferred pairs of magnetic fields Haas-van Alphen effect gives a

clue. As the intensity of magnetic field is varied, one observes so called de Haas-van Alphen effect (see this) used to deduce the shape of the Fermi sphere: magnetization and some other observables vary periodically as function of 1/B. In particular, this is true for χ_{V}.

The value of P is

P

_{H-A}== 1/B_{H-A}= 2π e/hbar S_{e},

where S

_{e}is the extremum Fermi surface cross-sectional area in the plane perpendicular to the magnetic field and can be interpreted as area of electron orbit in momentum space (for illustration see this).

Haas-van Alphen effect can be understood in the following manner. As B increases, cyclotron orbits contract. For certain increments of 1/B n+1:th orbit is contracted to n:th orbit so that the sets of the orbits are identical for the values of 1/B, which appear periodically. This causes the periodic oscillation of say magnetization. From this one learns that the electrons rotating at magnetic flux tubes of B

_{ext}are responsible for magnetization.

- One can get a more detailed theoretical view about de Haas-van Alphen effect from the article of Lifschitz and Mosevich (see this). In a reasonable approximation one can write

P= e× ℏ/m

_{e}E_{F}= [4α/3^{2/3}π^{1/3}]× [1/B_{e}] , B_{e}== e/a_{e}^{2}=[x^{-2}16 Tesla ,

a

_{e}= (V/N)^{1/3}= =xa , a=10^{-10}m .

Here N/V corresponds to valence electron density assumed to form free Fermi gas with Fermi energy E

_{F}= ℏ^{2}(3pi^{2}N/V)^{2/3}/2m_{e}. a=10^{-10}m corresponds to atomic length scale. α≈ 1/137 is fine structure constant. For P one obtains the approximate expression

P≈ .15 x

^{2}Tesla^{-1}.

If the difference of Δ (1/B

_{ext}) for B_{ext}=1 Tesla and B_{ext}=.1 Tesla correspond to a k-multiple of P, one obtains the condition

kx

^{2}≈ 60 .

- Suppose that B
_{ext,1}=1 Tesla and B_{ext,1}=.1 Tesla differ by a period P of Haas-van Alphen effect. This would predict same value of χ_{V}for the two field strengths, which is not true. The formula used for χ_{V}however holds true only inside given flux tube: call this value χ_{V,H-A}.

The fraction f of flux tubes penetrating into the superconductor can depend on the value of B

_{ext}and this could explain the deviation. f can depend also on temperature. The simplest guess is that that two effects separate:

χ

_{V}= χ_{V,H-A}(B_{H-A}/B_{ext})× f(B_{ext},T) .

Here χ

_{V,H-A}has period P_{H-A}as function of 1/B_{ext}and f characterizes the fraction of penetrated flux tubes.

- What could one say about the function f(B
_{ext},T)? B_{H-A}=1/P_{H-A}has dimensions of magnetic field and depends on 1/B_{ext}periodically. The dimensionless ratio E_{c,H-A}/T of cyclotron energy E_{c,H-A}= hbar eB_{H-A}/m_{e}and thermal energy T and B_{ext}could serve as arguments of f(B_{ext},T) so that one would have

f(B

_{ext},T)=f_{1}(B_{ext})f_{2}(x) ,x=T/E

_{H-A}(B_{ext})) .

One can consider also the possibility that E

_{c,H-A}is cyclotron energy with hbar_{eff}=nh_{0}and larger than otherwise. For h_{eff}=h and B_{ext}= 1 Tesla one would have E_{c}= .8 K, which is same order of magnitude as variation length for the pseudo fluctuation. For instance, periodicity as a function of x might be considered.

If B

_{ext,1}=1 Tesla and B_{ext,1}=.1 Tesla differ by a period P one would have

χ

_{V}(B_{ext,1},T)/χ_{V}(B_{ext,2},T) =f_{1}(B_{ext,1})/f_{1}(B_{ext,2})

independently of T. For arbitrary pairs of magnetic fields this does not hold true. This property and also the predicted periodicity are testable.

For a summary of earlier postings see Latest progress in TGD.

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