Sttistic Physics. Soutions Sheet 5. Exercise. HS 04 Prof. Mnfred Sigrist Ide fermionic quntum gs in hrmonic trp In this exercise we study the fermionic spiness ide gs confined in three-dimension hrmonic potenti nd compre it with the cssic cse see Exercise Sheet 3. The eigenenergies of the gs re given by ε = ~ωx + y + z, where = x, y, z, with i {0,,,...}, bes the sttes nd the zero point energy ε0 = 3 ~ω/ ws omitted. The occuption number corresponding to stte is given by n. Consider the high-temperture, ow-density imit z. Derive the grnd cnonic prtition function Zf of this system nd compute the grnd potenti Ωf. Show tht Ωf f4 z, where the function fs z is defined s fs z = = z. s 3 Soution. We begin with the gener definition of the grnd cnonic prtition function within the occuption number formism chpter 3 of the ecture notes nd find Y βε n Y = ze + ze βε. S. Zf = n In order to compute the grnd potenti Ω = /β og Z, we use the series expnsion og + x = x for < x. S. = This expnsion is ppicbe to the ogrithm of the prtition function in S. if ze βε it is wys positive in the high-temperture, ow density imit z. We find z β~ω z og Zf = og + ze = e βε = e =0 = = P 3 if β 0 = z β~ω 3 z = = P e β~ω = = z if β β~ω3 f4 z if β 0 =, f z if β!3 βε S.3 nd obtined both the high nd the ow temperture imits in either cse z must be given. Aterntivey, for the high temperture imit, with the hep of the Euer-Mcurin formu see, e.g., Exercise Sheet 4, we cn pproximte the sum over the oscitor modes by n integr nd find in eding order!3 z β~ω e og Zf = S.4 =0 = Z 3 Z 3 z z d e β~ω = d e β~ω 0 0 = = z 4 = f4 z S.5 = β~ω3 β~ω3 =
Either wy, the high temperture expnsion resuts in the grnd potenti Ω f = β β ω 3 f4z. S.6 b Ccute the prtice number N nd the intern energy U s function of N. In order to get U in terms of N insted of deing with the chemic potenti, introduce the prmeter 3 ω N /3 ρ 4 k B T nd rete it to z using the high-temperture, ow-density expnsion of N up to Oz. Interpret the condition ρ. Finy, expnd U up to second order in ρ, reting it to N. Soution. First, we compute the intern energy of the system, U f = β Ω f β, z S.7 where the derivtive hs to be tken t constnt fugcity z = e βµ. Strting from S.6 we find U f = 3 β β ω 3 f4z, which shows tht the intern energy is proportion to the grnd potenti, U f = 3 Ω f. The verge prtice number cn be computed in simir wy, We hve N f = z z og Z f. N f = z z where we used z f4z = f3z. z β ω 3 f4z = β ω 3 f3z, S.8 S.9 S.0 S. In order to rete the intern energy to the prtice number, we strt with the high-temperture, owdensity expnsion of the tot prtice number, N f = β ω f3z z z. S. 3 β ω 3 8 Rewriting this eqution using the prmeter ρ eds to ρ = z z 8. S.3 The condition z therefore impies ρ. Soving this eqution for z, we obtin z = 4 ± 4 ρ. Choosing the reevnt soution nd expnding + x + x x we find 8 z = ρ + ρ 8. Expnding in ρ ows us to de with the prtice number insted of the chemic potenti. S.4 To interpret the condition ρ we first note tht for this system, the Fermi energy foows ɛ F = 3 ω mx, whie the number of occupied sttes is proportion to 3 mx. The chrcteristic energy sce is thus given by 3 ωn /3. Therefore, this condition requires tht the chrcteristic energy sce is much smer thn
the therm energy high-temperture imit. This mens tht we consider tempertures t which the verge occuption of the sttes is much smer thn one ow-density imit. We write the intern energy up to second order in ρ s U = 3 β β ω f4z = 3 z z 3 β β ω 3 6 = 3 ρ + ρ β β ω 3 6 = 3 3 N + N β ω 3 ω = 3N + N, S.5 β 6 6 where we recover the equiprtition w in eding order nd the positive first order quntum corrections N ω/ 3 distinguishing the fermions from the cssic ide gs. c Compute the het cpcity C. Which quntity hs to be fixed in order to do this? Soution. Since our system does not hve voume s thermodynmic vribe we hve to compute the specific het C N by fixing the number of prtices. Hence, s strting point we use the expression S.5 for the inner energy, where we cn keep N fixed: U C N = = 3Nk B 3 ω T N 8 N. S.6 d Compute the isotherm compressibiity κ T. Soution. By definition κ T = v N, N µ T S.7 where v = N. Therefore, κ T = v z N µ N = v z T N βz f 3z = vβ fz β ω 3 z f 3z N 3 ω 8 N. S.8 e Interpret your resuts for U, C, nd κ T by compring them with the corresponding resuts for the cssic Botzmnn gs. How do the quntum corrections infuence the fermionic system? Soution. In summry we hve found up to first order in ρ: 3 ω U = 3N + N, S.9 6 C N = 3Nk B 3 ω 8 N, S.0 κ T = N 3 ω 8 N, S. These resuts s function of temperture re potted in Fig. ; ech for the cssic nd the fermionic cse. Note tht our expnsions up to second order in ρ re stricy speking ony vid for ρ, whie the pots extend to rger vues of ρ to emphsize the trends. We see tht 3
U kbt 3N 0 8 6 4 CN kbt 3N...0 0.9 0 4 6 8 B ΚT kbt N 0.35 0.30 0.5 0.0 0.5 0.8 4 6 8 0.0 3 4 5 6 7 8 9 0 k B T Figure : Thermodynmics of the fermionic gs dhsed, bue compred to the cssic gs soid, bck. Note tht these quntities re computed within the high-temperture, ow-density pproximtion nd re therefore not exct resuts. Sti, they cn be used to observe trends. We set N ω 3 = 00. B In first order in ρ the resuts for the cssic Botzmnn gs in hrmonic trp re recovered cf. Exercise Sheet 3. Due to quntum corrections, the intern energy U for fermions is higher thn the cssic ide gs. This cn be understood by tking quntum sttistics into ccount. Fermions re not owed to occupy the sme stte mutipy Pui. Lowering the temperture, the system tends to occupy ow energy sttes with growing probbiity. Whie the cssic system does not cre bout mutipe occupncies, in the fermionic system the doube occupncy is forbidden nd occuption of ow-energy sttes is thus reduced, incresing the inner energy U f compred to the cssic gs. 4
Exercise. Sommerfed expnsion nd density of sttes Consider thermy equiibrted system of non-intercting fermions with singe prtice sttes beed by the quntum numbers nd corresponding energies ε. Work in the grnd cnonic ensembe nd write the prtice nd energy densities in the form n = fε = dε gεfε, 5 u = ε fε = dε εgεfε, 6 where fε is the Fermi-Dirc distribution function. Wht is gε? Soution. In therm equiibrium, the occuption probbiity of prticur stte with energy ε is given by the Fermi-Dirc distribution fε = e ε µ/ +, nd we find from the definition of n nd u n = fε = dεfε δε ε = dε gεfε, u = ε fε = dε εfε δε ε = dε εgεfε, with gε = δε ε = ωε, nd ωε s in the ecture notes. Tht is, gε is the density of energy eves divided by the voume. S. S.3 S.4 S.5 b The bove expressions for n nd u re of the form dε Hεfε. 7 For tempertures T kb which is typicy the cse for mets, Hε is sowy vrying in the region where df dε 0 significnty nd the Sommerfed expnsion µ dε Hεfε = dε Hε+ π 6 H µ+ 7π4 360 4 H kb T 6 µ+o 8 µ becomes hndy. Mke use of this expnsion up to O k B T µ to expnd n nd u in T. Hint: Use in sef-consistent wy tht µ T in eding order in T nd expnd µ dε Hε εf dε Hε + µ H. 9 For reference on the Sommerfed expnsion see, e.g., Ashcroft, N. W. nd Mermin N. D., Soid Stte Physics, Hot, Rinehrt nd Winston, 976. 5
Soution. From the Sommerfed expnsion we obtin n = u = µ µ dε gε + π 6 kbt g ε + O dε εgε + π 6 kbt [µg ε + gµ] + O 4 kbt S.6 µ 4 kbt. S.7 As we know tht im T 0 µ =, the remining integrs cn be expnded round the chemic potenti see hint: εf ] n dε gε + [µ g + π 6 kbt g, S.8 εf [ ] u dε gε + µ g + π 6 kbt g + π 6 kbt g. S.9 For this to be sef-consistent, we need to check tht the chemic potenti µ = + αt + OT 3, which we see next. µ c Find the chemic potenti µ nd the specific het c v t constnt density n. Soution. As the density is constnt nd we hve t T = 0 tht εf n = dε gεfε = dε gε, S.30 it foows, from Eq. S.8, tht µ = π 6 kbt g g. S.3 Simiiry, u = u 0 + π 6 kbt g, S.3 nd therefore u c v = T n = π 3 k BT g. S.33 From this we see the importnt resuts tht i the chemic potenti vries with respect to temperture in dependence of g ε nd ii tht het mesurments re too to ccess the density of sttes t the Fermi energy. d Determine gε for the cse of free Fermi gs nd ccute its chemic potenti nd specific het from the previous resuts. Compre your resut for the specific het with the one for cssic gs. Soution. For free fermions, the be corresponds to the momentum of the prtices k nd the energies ε re given by the dispersion retion ε k = k m. With the hep of the step function Θ, the density of sttes cn be written s cf. ecture notes S.34 gε = 3/ m ε / Θε, S.35 4π or terntivey gε = 3 / n ε Θε. S.36 6
Using this resut we obtin s for the ow-temperture, high-density imit in the ecture [ ] µ = ε f π kbt, S.37 c v = π kbt nk B. For cssic ide gs Mxwe-Botzmnn distribution we find S.38 c v = 3 nkb, S.39 which mens tht in the fermionic cse the specific het is surpressed by fctor π. S.40 The origin of this surpression ies in the Fermi-Dirc distribution nd the Pui principe. For ow tempertures note tht this cn esiy be sever hunderd Kevin for eectrons in mets ony frction of the fermions, nmey the ones round the Fermi energy, get thermy excited nd contribute to the het cpcity, wheres for cssic gs the prtices cn contribute. e For the free Fermi gs g > 0. This does not need to be the true in more compex systems such s soids cf., e.g., semiconductors. Wht re the consequences of g 0? Soution. The sign of the density of sttes determines whether the chemic potenti increses, decreses or stys constnt with respect to the temperture. A negtive sign woud ed to n increse in the chemic potenti by incresing temperture. 7