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N = < n ( E, ( T ) , T ) > D ( E ) dE 0 This expression - PowerPoint PPT Presentation

May 09, 2023 •199 likes •705 views

N = < n ( E, ( T ) , T ) > D ( E ) dE 0 This expression implicitly determines = ( T ). U = < n ( E, ( T ) , T ) > E D ( E ) dE 0 To determine C V from this expression one must take into account the temperature

  1. � ∞ N = < n ( E, μ ( T ) , T ) > D ( E ) dE 0 This expression implicitly determines μ = μ ( T ). � ∞ U = < n ( E, μ ( T ) , T ) > E D ( E ) dE 0 To determine C V from this expression one must take into account the temperature dependence of μ in addition to the explicit dependence of < n > on T . 8.044 L19B15

  2. ��� � � � � ε−µ 8.044 L22B1

  3. � �� ε � ε=µ ��� ε � 8.044 L22B2

  4. � ��� ε ε=µ 8.044 L22B3

  5. ��� ε=µ � ε 8.044 L22B4

  6. ��� ε=µ ε � 8.044 L22B5

  7. � � �� µ ε � 8.044 L22B6

  8. � � ∞ N = < n > D ( E ) dE ǫ 0 ⎡ ⎤ 3 / 2 √ 1 (2 S + 1) V 2 m � � ∞ = E dE ⎣ ⎦ E/k B T − 1 4 π 2 2 0 e n √ 3 / 2 (2 S + 1) V 2 m E � ∞ � = dE 4 π 2 n 2 E/k B T − 1 0 e √ 3 / 2 (2 S + 1) V 2 mk B T x � ∞ � = dx 4 π 2 n 2 0 e x − 1 ' - � √ ( π/ 2) ζ (3 / 2) ⎛ ⎞ V = (2 S + 1) ζ (3 / 2) λ 3 ( T ) ⎝ ⎠ 8.044 L22B7

  9. √ ζ (3 / 2) = 2 . 612 . . . and λ ( T ) = h/ 2 πmk B T . For a fixed number density n = N/V the critical temperature for S = 0 Bosons is given by ⎞ 2 / 3 ⎛ h 2 n 2 / 3 T c = ∝ n ⎝ ⎠ 2 πmk B ζ (3 / 2) At a fixed temperature the critical number density is given by n c = ζ (3 / 2) λ − 3 ( T ) ∝ T 3 / 2 8.044 L22B8

  10. For a fixed number of atoms N , once T falls below T c those atoms that can not be accommodated in states with finite ǫ begin accumulating in the ground state where ǫ = 0. � ∞ < n > D ( ǫ ) dǫ = N (1 − ( T/T c ) 3 / 2 ) N 0 = N − � 0 �� � ∝ T 3 / 2 8.044 L22B9

  11. 1900 1920 1940 1960 1980 2000 300 K 30 K 3 K 300 mK BOSE­EINSTEIN CONDENSATION 30 mK IS A QUANUM MECHANICAL EFFECT 3 mK 300 µ K 30 µ K 3 µ K 300 nK 30 nK

  12. 1900 1920 1940 1960 1980 2000 300 K CLASSICAL MODEL QUANTUM REALITY 30 K λ 3 K 300 mK 30 mK 3 mK 300 µ K 30 µ K POINT­LIKE WAVES 3 µ K PARTICLES PROPAGATING FOLLOWING THROUGH 300 nK TRAJECTORIES SPACE 30 nK

  13. 1900 1920 1940 1960 1980 2000 30 00 0 K K 1 30 3 0 K K λ ∝ m × V 3 3 K K 30 00 0 m mK K FO FOR ATOMS MOV NG AT THERMAL VELOC TY OR R ATOMS MOVI ATOMS MOV NG AT THERMAL VELOC TY ING AT THERMAL VELOCI IT 30 3 0 m mK K AT A T R T ROOM TEMPERATURE ROOM TEMPERATURE ( OOM TEMPERATURE (300K) 300K 300K ), , 3 3 m mK K λ < THE R PHYS CAL S ZE. < T THEI HE R PHYSI IR PHYS CAL S ZE ICAL SI IZE. 0 μ K 3 30 00 F FO OR OR THE ELECTRONS MOV NG AROUND THE NUCLE R THE ELECTRONS MOV NG AROUND THE NUCLE THE ELECTRONS MOVI ING AROUND THE NUCLEI I 0 μ K 30 λ ≈ 1 ANGSTROM. I IN N THOSE ATOMS N T THOSE ATOMS HOSE ATOMS, , 1 ANGSTROM. ANGSTROM 3 μ K 3 30 00 0 n nK K 3 30 0 n nK K

  14. 1900 1920 1940 1960 1980 2000 300 K 30 K 3 K 300 mK 30 mK 3 mK 300 µ K 30 µ K 3 µ K THE WAVE NATURE OF THE ELECTRONS 300 nK STABILIZES THEM AGAINST LOSING ENERGY AND 30 nK FALLING INTO THE NUCLEUS.

  15. 1900 1920 1940 1960 1980 2000 300 K 30 K 3 K 300 mK 30 mK 3 mK 300 µ K 30 µ K 3 µ K THE WAVE NATURE OF PROTONS ALLOWS THEM 300 nK TO GET CLOSE ENOUGH DURING COLLISIONS IN THE SUN TO INITIATE FUSION. 30 nK

  16. 1900 1920 1940 1960 1980 2000 300 K 30 K 3 K 300 mK 30 mK 3 mK 300 µ K 30 µ K 3 µ K QM ALLOWS MOLECULES TO HAVE A STATISTICAL 300 nK CHANCE OF ADSORBING ON A SURFACE INSTEAD OF 30 nK REMAINING IN THE BULK GAS

  17. 1900 1920 1940 1960 1980 2000 IN 1924 AND 1925 SATYENDRA BOSE AND ALBERT 300 K EINSTEIN INVESTIGATED THE INFLUENCE OF QM ON 30 K THE COLLECTIVE BEHAVIOR OF PARTICLES. 3 K 300 mK 30 mK 3 mK 300 µ K 30 µ K 3 µ K 1921 300 nK "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect" 30 nK

  18. 1900 1920 1940 1960 1980 2000 30 00 0 K K 3 30 0 K K 3 3 K K 30 00 0 m mK K 30 3 0 m mK K 3 3 m mK K 0 μ K 30 3 00 0 μ K 30 3 μ K 3 30 00 0 n nK K THEY APPL ED THE R THEORY TO THE S MPLEST TH HEY APPLI EY APPL ED THEI IED THE R THEORY TO THE SI IR THEORY TO THE S MPLEST IMPLES POSS BLE CASE A GAS OF NON­ NTERACT NG ATOMS POSS BLE CASE: OSSI IBLE CASE A GAS OF NON­ NTERACTI : A GAS OF NON­I INTERACT NG IN ATOM TOMS 3 30 0 n nK K

  19. 1900 1920 1940 1960 1980 2000 30 00 0 K K 3 30 0 K K 3 3 K K 30 00 0 m mK K 30 3 0 m mK K 3 3 m mK K 0 μ K 30 3 00 0 μ K 30 3 μ K AS AS THE ATOMS GET COLDER THE R VELOC TY S THE ATOMS GET COLDER THEI THE ATOMS GET COLDER THE R VELOC TY IR VELOCI IT 3 30 00 0 n nK K DI D M N SHES AND THE R WAVELENGTH GROWS. IM NI MI IN SHES AND THE R WAVELENGTH GROWS ISHES AND THEI IR WAVELENGTH GROWS. 30 3 0 n nK K

  20. 1900 1920 1940 1960 1980 2000 30 00 0 K K 30 3 0 K K 3 K 3 K 30 00 0 m mK K 3 30 0 m mK K 3 m 3 mK K 0 μ K 3 30 00 0 μ K 30 3 μ K WH WHEN THE WAVELENGTH BECOMES COMPARABLE TO HEN THE WAVELENGTH BECOMES COMPARABLE T EN THE WAVELENGTH BECOMES COMPARABLE TO 30 3 00 0 n nK K TH THE SEPARAT ON A PHASE TRANS T ON OCCURS. HE SEPARATI E SEPARAT ON A PHASE TRANSI ION, , A PHASE TRANS TI IT ON OCCURS ION OCCURS. SOME OF THE ATOMS LOSE THE R DENT TY AND BECOME SO OM ME OF THE ATOMS LOSE THEI E OF THE ATOMS LOSE THE R I IR DENT TY AND BECOM IDENTI ITY AND BECOME 30 3 0 n nK K PA PART OF A S NGLE WAVE SPANN NG THE CONTA NER. AR RT OF A S NGLE WAVE SPANNI T OF A SI INGLE WAVE SPANN NG THE CONTA NER. ING THE CONTAI INER

  21. 1900 1920 1940 1960 1980 2000 HALF THE ATOMS IN THE WORLD FOLLOW THE RULES 300 K OF BOSE AND EINSTEIN AND ARE CALLED "BOSONS". THE OTHER HALF FOLLOW RULES SET OUT BY ENRICO 30 K FERMI AND PAUL DIRAC AND ARE CALLED "FERMIONS". 3 K 300 mK 30 mK 3 mK 300 µ K 30 µ K 3 µ K Enrico Fermi 1938 P.A.M. Dirac 1933 (with Erwin Schrodinger) "for his demonstrations of the existence "for the discovery of new productive forms 300 nK of new radioactive elements produced by of atomic theory" neutron irradiation, and for his related discovery of nuclear reactions brought 30 nK about by slow neutrons"

  22. 1900 1920 1980 2000 1940 1960 FI F N TE T IN NI IT TE E T T = T = 0 = 0 30 00 0 K K <n> <n> ~ kT 3 30 0 K K 1 1 ε F /k B T << FE ER ERM RM MI I 3 3 K K ε F ε ε F ε 30 00 0 m mK K 30 3 0 m mK K 3 3 m mK K 0 μ K 30 3 00 0 μ K 30 3 μ K 3 30 00 0 n nK K 3 30 0 n nK K

  23. 1900 1920 1980 2000 1940 1960 FI F N TE T IN NI IT TE E T T = T = 0 = 0 30 00 0 K K <n> <n> ~ kT 3 30 0 K K 1 1 ε F /k B T << FE ER ERM RM MI I 3 K 3 K ε F ε ε F ε 30 00 0 m mK K 30 3 0 m mK K <n> <n> N δ ( ε ) BOSE BOSE OS 3 m 3 mK K ~ kT (GOOD GUESS) 0 μ K 3 30 00 ε ε 0 μ K 30 3 μ K 30 3 00 0 n nK K 30 3 0 n nK K

  24. 1900 1920 1980 2000 1940 1960 FI F N TE T IN NI IT TE E T T = T = 0 = 0 30 00 0 K K <n> <n> ~ kT 30 3 0 K K 1 1 ε F /k B T << FE ERM ER RM MI I 3 3 K K ε F ε ε F ε 30 00 0 m mK K 3 30 0 m mK K <n> <n> N δ ( ε ) BOSE BOSE OS 3 m 3 mK K ~ kT (GOOD GUESS) 0 μ K 30 3 00 ε ε 0 μ K 30 N 0 /N <n> <n> 1 N δ ( ε ) 3 μ K BOSE BOSE OS N 0 (T) δ ( ε ) continuous Tc T part (ACTUAL) 3 30 00 0 n nK K ε ε 30 3 0 n nK K

  25. 1900 1920 1940 1960 1980 2000 O 2 LIQUEFIES AT 90K 300 K O 2 FREEZES AT 50K 30 K H 2 LIQUEFIES AT 20K 3 K H 2 FREEZES AT 14K 300 mK He LIQUEFIES AT 4K 30 mK 3 mK 300 µ K 30 µ K 3 µ K REAL ATOMS OR MOLECULES DO INTERACT WITH EACH OTHER AND UNDERGO LIQUEFICATION AND 300 nK FREEZING DUE TO THESE INTERACTIONS. 30 nK

  26. 1900 1920 1940 1960 1980 2000 TEMPERATURES BELOW 4.2 K CAN BE ACH EVED N TE EMPERATURES BELOW 4.2 K CAN BE ACH EVED MPERATURES BELOW 4.2 K CAN BE ACHI IEVED I IN 30 00 0 K K 4 He e B e BY BY PU Y PUMPI PUMP MP NG ING ON THE VAPOR ABOVE THE L QU NG ON ON TH THE V E VAP APOR OR AB ABOV OVE T E THE HE LI LIQ QUI UID D 3 30 0 K K 3 3 K K 30 00 0 m mK K 30 3 0 m mK K 3 3 m mK K 0 μ K 30 3 00 0 μ K 30 3 μ K 3 30 00 0 n nK 3 30 0 n nK K

  27. 1900 1920 1940 1960 1980 2000 SUPERCONDUCTIVITY WAS DISCOVERED BY 300 K KAMERLINGH ONNES IN 1911. 30 K 3 K RESISTANCE 300 mK 30 mK 3 mK 300 µ K 4K TEMPERATURE 30 µ K 3 µ K 1913 "for his investigations on the properties of matter at low temperatures which led, inter alia, to the production 300 nK of liquid helium" 30 nK

  28. 1900 1920 1940 1960 1980 2000 300 K The Nobel Prize in Physics 1972 30 K 3 K The Nobel Prize in Physics 1972 was awarded jointly to John Bardeen, Leon Neil Cooper 300 mK and John Robert Schrieffer " for their jointly developed theory of superconductivity, usually called the BCS-theory ". 30 mK 3 mK 300 µ K 30 µ K 3 µ K 300 nK 30 nK

  29. 1900 1920 1940 1960 1980 2000 300 K LIQUID HELIUM­4 FLOWS 30 K INSIDE THE TUBE 3 K 300 mK 30 mK 3 mK SOLID TUBE 300 µ K FINE POWDER 30 µ K 3 µ K SUPERFLUIDITY WAS DISCOVERED IN HELIUM­4 IN 300 nK THE 1930s AT A TEMPERATURE OF 2 KELVIN. 30 nK

  31. 1900 1920 1940 1960 1980 2000 DOUG OSHEROFF SEES STRANGE GL TCHES DOUG OSHEROFF SEES STRANGE GLI OUG OSHEROFF SEES STRANGE GL TCHE ITCHES 30 00 0 K K N THE MELT NG CURVE OF 3 He O ON N T THE MELTI HE MELT NG CURVE O ING CURVE OF H e 3 30 0 K K 3 3 K K 30 00 0 m mK K 3 30 0 m mK K 3 m 3 mK K 0 μ K 3 30 00 0 μ K 30 3 μ K 3 30 00 0 n nK 3 30 0 n nK K

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