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FIXING by THINKING: The power of dimensional analysis

FIXING by THINKING: The power of dimensional analysis. E.N. Economou Dept. of Physics, U of C FORTH March, 200 6. In the atomic idea “there is an enormous amount of information about the world, if just a little imagination and thinking are applied”. R.P.Feynman. 1. The atomic idea.

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FIXING by THINKING: The power of dimensional analysis

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  1. FIXING by THINKING:The power of dimensional analysis E.N. Economou Dept. of Physics, U of C FORTH March, 2006 In the atomic idea “there is an enormous amount of information about the world, if just a little imagination and thinking are applied”. R.P.Feynman FORTH, E.N. Economou

  2. 1. The atomic idea • Everything is made of indestructible, indivisible (α – τομα) microscopic particles which attract each other • This attraction must be counterbalanced for equilibrium to be established HOW?

  3. 2. The wave – particle duality comes to rescue, i.e., QM, which can be distilled in three basic principles: (a) Heisenberg Principle: (b) Pauli Principle: (c) “Schrödinger” Principle: (a)+(b)  Equilibrium when Pforces = Pkinetic energy (c)  Stability against small perturbations; changes still possible

  4. 3. Equilibrium  minimization of Et Under constant Pext and Text Equilibrium  minimization of G

  5. Limits setting universal constants Forces Particles (spin ½)

  6. For condensed matter only EM forces matter which are characterized by: e • The kinetic energy is characterized by • The three quantities define a system of units: E.g.,the unit of length • More convenient to use (instead of ) determines the atomic radius or

  7. Examples: General Formula for any quantity A (a) so that (b) Replace by ; (c) A may depend on mα, T, P, c, etc. Example: Ionization Potential ,

  8. Shell & Subshell structure of the atomic orbitals based on their angular dependence • The latter is the same as that of (same degree) polynomials satisfying Laplace equation. Zero degree  const  no angular dependence  s orbitals 1st degree  x, y, z  sinθcosφ, sinθsinφ, cosθ  3 p orbitals 2nd degree  xy, yz, zx, x2-y2, y2-x2  …  5 d orbitals 3rd degree   7 f orbitals etc.. xyz, z(x2-y2), x(y2-z2),y(z2-x2) y(x2 - 1/3 y2), z(y2 - 1/3 z2), x(z2 - 1/3 x2)

  9. Molecules (diatomic) exception: noble gases , ,

  10. Solids , , ,

  11. Solids , ,

  12. Comparison with experimental data Fe Al Cu Si Exp. Theory Exp. Theory Exp. Theory Exp. Theory 2,67 2,99 2,67 3,18 (gr/cm3) 7,86 7,92 2,79 2,73 8,96 9,01 2,33 2,36 (eV/atom) 4,28 3,82 3,39 3,04 3,49 3,82 4,63 2,69 B () 1,68 1,29 0,722 0,73 1,37 1,29 0,998 0,54 (Κm/s) 4,63 4,11 5,68 5,28 3,93 3,85 6,48 4,87 ΘD(K) 470 422 428 495 343 406 645 430

  13. Melting temperature

  14. DC electrical resistivity ρe [dimensions of time]

  15. DC electrical resistivity ρe (2) At T=295 K at T2K ρ10-3 μΩ∙cm10-5μΩ∙cm for pure Cu ρ2x1023μΩ∙cm for yellow sulfur What went wrong?

  16. Formula for ρe , Depends on ; If WAVE ?

  17. “The fact that periodicity of a crystal would be essential was somehow suggested to me by remembering a demonstration in elementary physics where many equal and equally coupled pendulums were hanging at constant spacing from a rod and the motion of one of them was seen to “migrate” along the rod from pendulum to pendulum. Returning to my rented room one evening in early January, it was with such vague ideas in mind that I began to use pencil and paper and to treat the easiest case of a single electron in a one–dimensional periodic potential..” F. Bloch WAVE + PERIODICITY SYSTEMATIC CONSTRUCTIVE INTERFERENCE, CANCELS SCATTERING FREE-LIKE

  18. Metals Destructive interference  Gaps Eg  Semiconductor

  19. Specific Heat Classically , QM:

  20. FLUIDS, Ι Velocity of sea waves [g, λ (ork=2π/λ), d, ρΜ]

  21. 102 101 100 0.232 10-1 0.84 km/h 10-3 10-2 10-1 100 101 102 103 104 105 TSUNAMI WINDINDUCED λ(m) 1.7 cm

  22. FLUIDS • Drag force , , LARGE BODIES, HIGH SPEED , , SMALL BODIES, LOW SPEED Reynolds number

  23. Black Body Radiation

  24. Why Earth is round? • Why are there mountains? • What is the largest possible height of a mountain in a planet? When the shear stress exceeds the critical value ,

  25. STARS • MINIMUM NUMBER OF NUCLEONS: • MAXIMUM NUMBER OF NUCLEONS: • NUMBER OF NUCLEONS IN OUR SUN: • MAXIMUM NUMBER OF NUCLEONS IN A WHITE DWARF: ,

  26. Universe • Homogeneous & Isotropic • Expanding according to Hubble’s law: • Hubble’ s constant (indipendent of R): • , dimensionally;  Euclidean geometry

  27. the equivalent of 5.51 protons per m3 • This density equals to the critical one with an uncertainty of 2% • protons per m3, i.e. about 4.2% of ε The Rest? • DARK MATTER • DARK ENERGY

  28.  Einstein’ s cosmological constant

  29. Cosmic Epochs • Inflation • Early t  10-4 s • Photon domination 1ms  t  50 ky • Matter domination 200 ky  t  5 ky • Dark energy domination 20 Gy  t

  30. Photon domination Matter/ Dark Energy Domination in Gy for

  31. BB q→B P,n→nuclei εph→εnucl decoupling Protostars Galaxies Today 10-4s 1s 70 ky 380 ky 180 My 500 My 8 Gy 13.7 Gy t

  32. Acoustic waves in the infant universe of a basic frequency and its overtones • The seeds of the future galaxies • This cosmic “music” must have been inprinted in the cosmic photons (cosmic background radiation) • Is it there? • Can we listen to it?

  33. OBSERVATIONS WMAP Wilkinson Microwave Anisotropy Probe

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