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Exciton-polariton Bose-Einstein condensate at room temperature and h_{eff} hierarchy

Ulla gave in my blog a link to a very interesting work about Bose-Einstein condensation of quasi-particles known as exciton-polaritons. The popular article tells about a research article published in Nature by IBM scientists.

Bose-Einstein condensation happens for exciton-polaritons at room temperature, this temperature is four orders of magnitude higher than the corresponding temperature for crystals. This puts bells ringing. Could h_{eff}/h=n be involved?

One learns from Wikipedia that exciton-polaritons are electron hole pairs- photons kick electron to higher energy state and exciton is created.These quasiparticles would form a Bose-Einstein condensate with large number of particles in ground state. The critical temperature corresponds to the divergence of Boltzmann factor given by Bose-Einstein statistics.

- The energy of excitons must be of order thermal energy at room temperature: IR photons are in question. Membrane potential happens to corresponds to this energy. That the material is organic, might be of relevance. Living matter involves various Bose-Einstein condensate and one can consider also excitons.

As noticed the critical temperature is surprisingly high. For crystal BECs it is of order .01 K. Now by a factor 30,000 times higher!

- Does the large value of h
_{eff}=n×h visible make the critical temperature so high?

Here I must look at Wikipedia for BEC of quasiparticles. Unfortunately the formula for n

^{1/3}is copied from source and contains several errors. Dimensions are completely wrong.

It should read n

^{1/3}= (ℏ)^{-1}(m_{eff}kT_{cr})^{x}, x= 1/2.

[not x=-1/2 and 1/ℏ rather than ℏ as in Wikipedia formula. This is usual: it would important to have Wikipedia contributors who understand at least something about what they are copying from various sources].

- The correct formula for critical temperature T
_{cr}reads as

T

_{cr}= (dn/dV)^{y}ℏ^{2}/m_{eff}, y=2/3.

[T

_{cr}replaces T_{c}and y=2/3 replaces y=2 in Wikipedia formula. Note that in Wikipedia formula dn/dV is denoted by n reserved now for h_{eff}=n×h].

- In TGD one can generalize by replacing ℏ with ℏ
_{eff}=n ×ℏ so that one has

T

_{cr}→ n^{2}T_{cr}.

Critical temperature would behave like n

^{2}and the high critical temperature (room temperature) could be understood. In crystals the critical temperature is very low but in organic matter a large value of n≈ 100 could change the situation. n≈ 100 would scale up the atomic scale of 1 Angstrom as a coherence length of valence electron orbitals to cell membrane thickness about 10 nm. There would be one dark electron-hole pair per volume taken by dark valence electron: this would look reasonable.

_{cr}is also proportional to (dn/dV)

^{2}, where dn/dV is the density of excitons and to the inverse of the effective mass m

_{eff}. m

_{eff}must be of order electron mass so that the density dn/dV or n is the critical parameter. In standard physics so high a critical temperature would require either large density dn/dV about factor 10

^{6}higher than in crystals.

Is this possible?

- Fermi energy E
_{}is given by almost identical formula but with factor 1/2 appearing on the right hand side. Using the density dn_{e}/dV for electrons instead of dn/dV gives an upper bound for T_{cr}≤ 2E_{F}. E_{F}varies in the range 2-10 eV. The actual values of T_{cr}in crystals is of order 10^{-6}eV so that the density of quasi particles must be very small for crystals: dn_{cryst}/dV≈ 10^{-9}dn_{e}/dV .

- For crystal the size scale L
_{cryst}of the volume taken by quasiparticle would be 10^{-3}times larger than that taken by electron, which varies in the range 10^{1/3}-10^{2/3}Angstroms giving the range (220-460) nm for L_{cryst}.

- On the other hand, the thickness of the plastic layer is L
_{layer}= 35 nm, roughly 10 times smaller than L_{cryst}. One can argue that L_{plast}≈ L_{layer}is a natural order of magnitude for L_{cryst}for quasiparticle in plastic layer. If so, the density of quasiparticles is roughly 10^{3}times higher than for crystals. The (dn/dV)^{2}-proportionality of T_{cr}would give the factor T_{cr,plast}≈ 10^{6}T_{cr,cryst}so that there would be no need for non-standard value of h_{eff}!

But is the assumption L

_{plast}≈ L_{layer}really justified in standard physics framework? Why this would be the case? What would make the dirty plastic different from super pure crystal?

For background see the chapter Criticality and dark matter of "Hyper-finite factors, p-adic length scale hypothesis, and dark matter hierarchy".

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

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