Phys. 344 Ch 7 Lecture 8 Fri., April. 10 th,


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1 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, 009 Fri. 4/0 8. Ising Mdel f Ferrmagnets HW30 66, 74 Mn. 4/3 Review Sat. 4/8 3pm Exam 3 HW Mnday: Review fr est 3. See nline practice test lectureprep is t bring in questins 7.6 BseEinstein Cndensatin / Bsns with µ 0 his is kind f tricky stuff. Cnceptually: it appeals t the heart f the difference between distinguishable and indistinguishable particles and what we mean by temperature. Quantitatively: he math desn t let us cme up with a single, perfect mdel. S, we ll talk sme abut the cncepts and then we ll g abut building sme mathematical tls that d what we need. If yu feel like the mathematical mdel is kind f cbbledtgether, yu re abslutely right. It isn t perfect; ur gal is just t see that the qualitative behavir we expect is in there Why Des it Happen Okay, what s special abut Bsns is that a) their indistinguishable and b) they can ccupy the same singleparticle state as each ther. 0 S, bviusly, if yu take away all the energy yu can, every particle will happily chabitate in the singleparticle grund state they have cmpletely degenerated, cndensed int a single state. Lw As yu add energy / raise the temperature, sme f the particles will rise t lwlying energy states, but there will still be a large ppulatin in the grund state, a large ppulatin in the cndensate, fr higher temperatures than ne wuld classically expect. In pint f fact, there will always be sme particles sharing the grund state; hwever, as increases, it becmes an insignificant fractin f the ppulatin. First, why des the grundstate ppulatin eventually becme insignificant? hink f the distributin f particles in terms f energy: he average ccupancy f a particular state depends n the energy/k f that state: n. hat bviusly tells us that a given high energy state is ( µ ) e less ppulated than a given lw energy state. Meanwhile, the average number f particles with a given energy als depends n hw many states have the same energy, i.e., the density f states: nw / g( ) d g d. his f curse grws with energy at higher energies, there are mre states with the same energy. S the prduct f these tw determines hw many particles have a given energy. S, while the grund state will
2 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, 009 always be the mst ppular single state, the mst ppular energy level will g be where ever g( ) n is peaked, s the grund state quickly ( µ ) e becmes nt s ppular an energy level, and the cndensate becmes insignificant. dg( ) n d he peak shuld ccur at g g e ( µ ) 0 ( µ ) µ e ( ) ( e ) ( µ ) ( ) e ) ( µ ) 0 ( e ) ( µ ) ( e ) hat said, why des the grundstate remain significant fr lwish temperatures? Let s think abut the ther tw kinds f particles fr cmparisn. Fermins Okay, these can t have multiply ccupied states, s the lwest energy / zertemperature cnfiguratin fr a system f them is simply the first singleparticle states being full. Distinguishable Particles vs. Bsns Like indistinguishable bsns, the lwest energy cnfiguratin f the system has all f the particles being in the lwest energy level the difference is that each particle is distinguishable, s while they all have the same energy, they are in distinct states. Anther difference is that this situatin mre rapidly fades int bscurity as temperature rises. Here s why: their distinguishability means that there are a lt mre unique states available when yu add just a little energy: will particle Bb be, r will Alice, r will Carl, r will Dug, the mst ppular energy level (where ( µ ) g( ) n g e peaks) shifts up higher faster with increasing temperature it des fr is Bsns. get deeper int this, we need t recall just what temperature means. If yu re in the habit f directly assciating temperature with average energy, this may S be a little hard t swallw, but remember: S U the temperature f a system depends nt just n hw much energy yu put in it, but als hw much disrder it induces (quantified in the entrpy).
3 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, Distinguishable system. Imagine yu have an particle system f distinguishable particles and anther particle system f bsns. If yu add, say units f energy t the distinguishable system, then there are + ( + )! ( + ) ΩDist(, q ) ( )!. S, the temperature assciated with adding ne unit is rughly U q dist S k ln (( + ) / ) Als nte that f all these pssibilities, f them have  particles remaining in the grund state;the remaining pssibilities have nly  particles remaining in the grund state. Put anther way, the prbability f having  particles in the grund state is Ω Pr( ) Ω ( + ) / ( + ) ttal Bsn System. In cntrast, if yu add tw units f energy t a Bsn system, there are nly tw ways they can be distributed: all t ne particle, r ne t ne, and ne t anther. ΩBse (, q ). S the temperature assciated with adding ne unit is rughly U q Bse which is a significantly S k ln higher temperature! Als, even at this higher temperature, the prbability f having  particles (rather than ) in the grund state is ½  far larger than fr distinguishable particles. Cnclusin. It takes a higher temperature t frce the same amunt f energy int a Bse system, and even then, there s a greater ppulatin in the grund state. Density f states effect. Smething that isn t addressed in this simple illustratin is a higher energy level generally has a greater degeneracy, but this effects distinguishable particles and bsns equally. Pulling back a bit and summarizing: here s the cmpetitin between the prbability f being in a
4 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, Okay, let s get quantitative particular state (higher prbability fr lwer energy states) and the number f states with a particular energy. he higher the energy, the less prbable a particular state, but the mre individual states available. Fr Distinguishable particle, nt nly are there mre states available fr individual particles at higher energies, but there are even mre ways f chsing which particles will be in which f the ccupied states. Fr distinguishable particles then, it is preferable t have E energy accunted fr by having particles in different, medium energy states. Fr Bsns n the ther hand, yu re a little mre likely t have E energy accunted fr by having fewer particles in higher energy states leaving mre particle back in the grund state. 3 Regimes: 0, very lw/mderate, high (classical) We re interested in what fractin f the particles in ur system are in the grund state. In the Cndensatin regime, it s startlingly large. Grund State he average ccupancy f the grund state is n ( µ ). Since e there is nly ne grund state (degeneracy f ), the number f particles with this lwest energy is simply n ( µ ). e (Prep fr Pr. 66) Questin: Fr that matter, what wuld be the average ccupancy f ne f the st states? n ( µ ) e Fr a spin0 particle, hw many st states are there? lk in p(r n) space. 3. S the number in the first energy level is 3 3n ( µ ) e n µ What s m during cndensatin, i.e. Lw? We see it in n ( µ ). e At lw, we knw that where is generally n rder f 0 3. Lking at ( µ ), can get quite large nly if e ( µ ) the denminatr gets quite small, i.e., e appraches. S, ( µ ) expanding e ( µ ) arund gives e + ( µ ). te: at first blush, yu might think what the heck are yu ding with a aylr series, is huge, but fr physical
5 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, reasns we knw that the expnent must be tiny fr t be huge, s we can find what µ des that fr us. k + µ µ k (Prep fr Pr. 66) S, µ w, as drps and grws, µ clearly appraches frm belw. S it appraches the grund state energy frm belw. S, what s? Energy Fr an rder f magnitude calculatin, imagine a simple cube f h dimensins L. he smallest mmentum available is p x, L ditt fr the y and z cmpnents f mmentum. S the smallest h h h 3 energy available is h m + +. L L L 8mL 3 34 ( Js) his is n rder f 7 8 ( kg) J, m really small! A crrespnding temperature wuld be 08 K! te: fr a spin 0 particle, there is nly ne grund state. (Prep fr Pr. 66) Questin: Fr that matter, what wuld be the energy f the st state? 6 h h h h m L L L 8mL k Since is quite small, µ means that µ is negative fr elevated temperatures // when is small. Only when temperature drps and the ppulatin f the grund state grws appreciably will µ apprach the very small psitive value f. Cndensatin, dependence n. S, arund what temperature d the particles begin cndensing int the lwest energy state? ns + + sµ µ s µ e e e states states states Prep. Fr HW 74: Yu ll use Excel t d this sum fr the first ~ 00 terms. te that the degeneracy, n and energy structure are bth given in the previus prblem. w / Here, I ve explicitly separated ff the term fr the grund state ppulatin frm all the ther terms fr the states.
6 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, dnw/ µ n s. w/ s µ µ e e d e states his last step f curse is in preparatin fr replacing the sum with an integral, but we can nly get away with that if <<, r in ther wrds << k. he energy steps between states are n rder f which is pretty darned small, s we can get away with this fr mst temperatures. dns. w / g ( ) d µ µ d e e 3/ πm nspin d µ π h e Lk at the cmpetitin in the integrand. vs.. µ e While the lw energy states are by far the mst ppular (accrding t the secnd factr particles really want t be in thse states), they are als the fewest (accrding t the first factr there just aren t that many f these lw energy states), in the end, states with energy much less than k dn t really cntribute t the sum. hat s gd because, it allws us t play a little lse with the lwer end f the integratin withut significantly affecting the result. he tw apprximatins are that and µ are much smaller than the energy where the integrand becmes significant, i.e.,, µ 0. n n spin spin π πm h 3/ πmk π h 0 3/ 3/ 0 πmk πmk nspin.35 nspin.6 π h h 3 3 he integral was Γ ζ.6... Limit f Apprximatin: As we ve already seen at very lw temperatures, µ grws increasingly negative with increasing. S the apprximatin µ 0 is nly gd fr small and mderate temperatures. At high temperatures we can t neglect the chemical ptential. hreshld: Clearly, the apprximatin breaks dwn by the time it predicts that the number f particles in states exceeds the ttal number f particles in the system. S we can define a threshld by e ( ) d x dx x e 3/
7 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, / / m k h n v h mk n c spin Q c spin π π te: this is rughly when the vlume per particle is the quantum vlume when wavefunctins must just verlap. his can be rephrased in terms f the grund state energy: 3/. 0 k c, s it can be cnsiderably mre than the grund state energy (which itself is quite small and is abut between states). With this definitin in hand, we can rewrite the number f particles as C 3/ Rughly speaking, the ppulatin f states ges like: Determine m frm cndensate ppulatin: Fr that matter, fr these mderate temperatures, / / / 3 / C k k C e e µ µ + 3/ ln C k µ c
8 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, Assuming that is quite large, k µ 3/ C his has the same lw behavir as we d already predicted. Clearly, this gets illbehaved as appraches c. m at > c Abve c, he ppulatin f the grund state is negligible, s 3/ πm nspin d µ π h is the e 0 defining equatin fr µ. One can use this t verify the basic frm in Figure Real Wrld Examples It s readily bserved in liquid 4 He. It takes sme ding, but it s als bserved in Rubidium gas.
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