Lecture 6: Diatomic gases (and others)

Transcription

1 Lecture 6: Datomc gases and others) General rule for calculatng n complex systems Ams: Deal wth a quantsed datomc molecule: Translatonal degrees of freedom last lecture); Rotaton and Vbraton. Partton functon as a product of ndependent factors. Law of equpartton of energy as the hghtemperature lmt of a quantum system. Black-body radaton Planck s formula for the spectrum of black-body radaton. Heat capacty of solds. March 04 Lecture 6 1

2 Datomc gas n a box Quantum states n a cube of sde a. n, J, r ma I + + tr translaton n tr Partton functon,, s n, J, r n, J, r. ) l + m + n + J J + 1) + r + 1/ ) aton β ) raton J+1) states wth wth the the same energy.e.dfferent m j j J + 1) β ) β ) β ) β ) J + 1) β ) β ) n, J, r tr. J tr Rotatonal degeneracy r A smple product of separate partton functons Average energy follows from. 1 d dln tr + d β d β + March 04 Lecture 6

3 General result The result apples generally.. If the energy s a sum of components + + A B C and.. A B C + + A B C p For a contrbuton Q), dependng only on Q, the probablty of fndng a value, Q, s Q ) g Q ) kt ) g Q ) kt ) ndependent of other parts of the system. March 04 Lecture 6

4 Datomc molecule agan Translatonal energy as before, tr kt Rotatonal energy J + 1 βj J + 1) J T Ik Hgh temperature lmt s a characterstc temp. T >> T. Occuped levels closely spaced compared wth T ). Change Σ to an ntegral over varable to x JJ+1), dx J+1)dJ. ) β x I d x I β 0 1 d 1 hgh T kt d β β Low temperature lmt T << T. Only J0 and J1 states are occuped. March 04 Lecture 6 4 ) ) I 1+ 1 ) β I d d β I ) β I Classcal result orthogonal atons) 0 T 0

5 Rotatonal contrbuton to specfc heat Overall pcture Quenched at at low temperatures Classcal lmt Typcally T s a few K. H has the smallest moment of nerta and T 88K. General concluson All quantum systems behave n a smlar way. In the low temperature lmt, kt<< E, the exctaton s quenched..e. the exctaton plays no part n the thermal propertes of the system. In the hgh temperature lmt kt>> E, the system behaves classcally. Quantum of of energy March 04 Lecture 6 5

6 Vbratonal energy Quantum ratons Energy levels r +1 r r r 0 β ) 1 d d β kt ) 1 ) 1 1 β ) neglect 1/ Geometrc seres Planck s formula for a sngle oscllator Hgh temperature lmt T >> k T kt ) 1+ kt kt Characterstc Temp Low temp. lmt kt ) 0 Typcally T >1000K Hence for typcal datomc at room temp < tot > 5/)kT. March 04 Lecture 6 6

7 Black body radaton Planck spectrum of black-body body radaton Radaton occupes standng wave states n a box of sde, a. Energy of the modes: k l, m, n a g k)d k g n ) l + m + n ck n a Planck formula gves us the average energy n mode, j. j Total energy comes from a sum over all the modes Frst get densty of states usng the result calculated n the last lecture vz. a g k)d k. k ) d g )d ) l + m + n 1 March 04 Lecture 6 7 kt ) 1 a c j j d d k polarsatons

8 Energy densty of black-body body radaton Energy densty U T ) u T ) u T, )d V u T, )d 0 1 c d kt ) 1 g)d/a Mean No. of of states per unt vol. energy of of sngle oscllator n n d dat at at.. u T, ) 1 kt ) 1 c Planck s radaton formula Total power n the radaton U T ) u T ) V c u T ) u T ) c kt c ) kt ) 4 0 x kt ) x) d x 1 d 1 4 /15 Stefan s Law March 04 Lecture 6 8

9 Black-body radaton Classcal formula Raylegh-Jeans) kt per mode ut,) kt g)d. A result know as the ultra-volet catastrophe. Cosmc mcrowave background. Cosmc Background Explorer, COBE) T o.74±0.06)k λ5mm ν60ghz λ0.5mm ν600ghz March 04 Lecture 6 9

10 Heat capacty of solds See later lectures for the full story. Vbratons n a sold Dulong and Pett: ratonal modes per atom <> kt Molar heat-capacty of all smple solds ~R. Works at hgh temperatures. Fals, for example, for damond, whch has a much lower value. Ensten: Realsed that the new quantum theory could lan the phenomena. He quantsed the nteratomc ratons. Assumed all oscllators have frequency E. Usng our prevous result multpled by, for the -modes per atom) Low temperature lmt, T), mproved by Debye T March 04 Lecture 6 10

11 Slde 10 Correcton Ensten bography corrected. March 04 Lecture 6 11