Sistemas de Fermi Ideais - Termodinâmica e Comportamento

8 Ideal Fermi Systems 81 Thermodynamic behavior of an ideal Fermi gas According to Sections 61 and 62 we obtain for an ideal Fermi gas fracPVkT equiv ln mathcalQ sumvarepsilon ln 1z ebeta varepsilon 1 and N equiv sumvarepsilon langle nvarepsilon rangle sumvarepsilon frac1z1 ebeta varepsilon 1 2 where beta 1kT and z expmukT Unlike the Bose case the parameter z in the Fermi case can take on unrestricted values 0 le z infty Moreover in view of the Pauli exclusion principle the question of a large number of particles occupying a single energy state does not even arise in this case hence there is no phenomenon like BoseEinstein condensation here Nevertheless at sufficiently low temperatures Fermi gas displays its own brand of quantal behavior a detailed study of which is of great physical interest If we replace summations over varepsilon by corresponding integrations equations 1 and 2 in the case of a nonrelativistic gas become fracPkT fracglambda3 f52 z 3 and fracNV fracglambda3 f32 z 4 where g is a weight factor arising from the internal structure of the particles eg spin lambda is the mean thermal wavelength of the particles lambda h2 pi m kT12 5 while f u z are FermiDirac functions defined by see Appendix E f uz frac1Gamma u int0infty fracx u 1 dxz1 ex 1 z fracz22 u fracz33 u cdots 6 232 Chapter 8 Ideal Fermi Systems Eliminating z between equations 3 and 4 we obtain the equation of state of the Fermi gas The internal energy U of the Fermi gas is given by U equiv left fracpartialpartial beta ln mathcalQ rightzV kT2 left fracpartialpartial T ln mathcalQ rightzV frac32 kT fracgVlambda3 f52z frac32 N k T fracf52zf32z 7 thus quite generally this system satisfies the relationship P frac23 UV 8 The specific heat CV of the gas can be obtained by differentiating 7 with respect to T keeping N and V constant and making use of the relationship frac1z left fracpartial zpartial T right u frac32T fracf32zf12z 9 which follows from equation 4 and the recurrence formula E6 in Appendix E The final result is fracCVNk frac154 fracf52zf32z frac94 fracf32zf12z 10 For the Helmholtz free energy of the gas we get A equiv N mu PV NkT left ln z fracf52zf32z right 11 and for the entropy S equiv fracU AT Nk left frac52 fracf52zf32z ln z right 12 In order to determine the various properties of the Fermi gas in terms of the particle density n NV and the temperature T we need to know the functional dependence of the parameter z on n and T this information is formally contained in the implicit relationship 4 For detailed studies one is sometimes obliged to make use of numerical evaluation of the functions f u z for physical understanding however the various limiting forms of these functions serve the purpose well see Appendix E Now if the density of the gas is very low andor its temperature very high then the situation might correspond to f32z fracn lambda3g fracnh3g 2 pi m k T32 ll 1 13 81 Thermodynamic behavior of an ideal Fermi gas 233 we then speak of the gas as being nondegenerate and therefore equivalent to a classical ideal gas discussed in Section 35 In view of expansion 6 this implies that z ll 1 and hence f u z simeq z Expressions for the various thermodynamic properties of the gas then become P N k T V quad U frac32 N k T quad CV frac32 N k 14 A N k T left ln left fracn lambda3g right 1 right 15 and S Nk left frac52 ln left fracn lambda3g right right 16 If the parameter z is small in comparison with unity but not very small then we should make a fuller use of series 6 in order to eliminate z between equations 3 and 4 The procedure is just the same as in the corresponding Bose case that is we first invert the series appearing in 4 to obtain an expansion for z in powers of n lambda3 g and then substitute this expansion into the series appearing in 3 The equation of state then takes the form of the virial expansion fracP VN k T suml1infty 1l1 al left fraclambda3g v rightl1 17 where u 1n while the coefficients al are the same as quoted in 7114 but alternate in sign compared to the Bose case For the specific heat in particular we obtain CV frac32 N k suml1infty 1l1 frac5 3 l2 al left fraclambda3g v rightl1 frac32 N k left 1 00884 left fraclambda3g v right 00066 left fraclambda3g v right2 00004 left fraclambda3g v right3 cdots right 18 Thus at finite temperatures the specific heat of the gas is smaller than its limiting value frac32 N k As will be seen in the sequel the specific heat of the ideal Fermi gas decreases monotonically as the temperature of the gas falls see Figure 82 later in the section and compare it with the corresponding Figure 74 for the ideal Bose gas If the density n and the temperature T are such that the parameter n lambda3 g is of order unity the foregoing expansions cannot be of much use In that case one may have to make recourse to numerical calculation However if n lambda3 g gg 1 the functions involved can be expressed as asymptotic expansions in powers of ln z1 we then speak of the gas as being degenerate As n lambda3 g o infty our functions assume a closed form with the result that the expressions for the various thermodynamic quantities pertaining to the system become highly simplified we then speak of the gas as being completely degenerate For simplicity we first discuss the main features of the system in a state of complete degeneracy In the limit T 0 which implies nλ3g the mean occupation numbers of the singleparticle state εp become nε 1eεμkT 1 1 for ε μ0 0 for ε μ0 where μ0 is the chemical potential of the system at T 0 The function nε is thus a step function that stays constant at the highest value 1 right from ε 0 to ε μ0 and then suddenly drops to the lowest value 0 see the dotted line in Figure 81 Thus at T 0 all singleparticle states up to ε μ0 are completely filled with one particle per state in accordance with the Pauli principle while all singleparticle states with ε μ0 are empty The limiting energy μ0 is generally referred to as the Fermi energy of the system and is denoted by the symbol εF the corresponding value of the singleparticle momentum is referred to as the Fermi momentum and is denoted by the symbol pF The defining equation for these parameters is 0εF aεdε N where aε denotes the density of states of the system and is given by the general expression aε gVh34πp2 dpdε We readily obtain N 4πgV3h3 pF3 which gives pF 3N4πgV13 h FIGURE 81 Fermi distribution at low temperatures with x εkT and ξ μkT The rectangle denotes the limiting distribution as T 0 in that case the Fermi function is unity for ε μ0 and zero for ε μ0 accordingly in the nonrelativistic case εF 3N4πgV23 h22m 6π2 ng23 ħ22m The groundstate or zeropoint energy of the system is then given by E0 4πgVh3 0pF p22m2 p2 dp 2πgV5mh3 pF5 which gives E0N 3pF210m 35 εF The groundstate pressure of the system is in turn given by P0 23 E0V 25 nεF Substituting for εF the foregoing expression takes the form P0 6π2 g23 ħ25m n53 n53 The zeropoint motion seen here is clearly a quantum effect arising because of the Pauli principle according to which even at T 0 K the particles constituting the system cannot settle down into a single energy state as we had in the Bose case and are therefore spread over a requisite number of lowest available energy states As a result the Fermi system even at absolute zero is quite live For a discussion of properties such as the specific heat and the entropy of the system we must extend our study to finite temperatures If we decide to restrict ourselves to low temperatures then deviations from the groundstate results will not be too large accordingly an analysis based on the asymptotic expansions of the functions fν z would be quite appropriate However before we do that it seems useful to carry out a physical assessment of the situation with the help of the expression nε 1eεμkT 1 The situation corresponding to T 0 is summarized in equation 19 and is shown as a step function in Figure 81 Deviations from this when T is finite but still much smaller than the characteristic temperature μ0k will be significant only for those values of ε for which the magnitude of the quantity ε μkT is of order unity for otherwise the exponential term in 29 will not be much different from its groundstate value namely e see the solid curve in Figure 81 We therefore conclude that the thermal excitation of the particles occurs only in a narrow energy range that is located around the energy value ε μ0 and has a width OkT The fraction of the particles that are thermally excited is therefore OkTεF the bulk of the system remaining uninfluenced by the rise in temperature¹ This is the most characteristic feature of a degenerate Fermi system and is essentially responsible for both qualitative and quantitative differences between the physical behavior of this system and that of a corresponding classical system To conclude the argument we observe that since the thermal energy per excited particle is OkT the thermal energy of the whole system will be ONk² T²εF accordingly the specific heat of the system will be ONk kTεF Thus the lowtemperature specific heat of a Fermi system differs from the classical value 32 Nk by a factor that not only reduces it considerably in magnitude but also makes it temperaturedependent varying as T¹ It will be seen repeatedly that the firstpower dependence of CV on T is a typical feature of Fermi systems at low temperatures For an analytical study of the Fermi gas at finite but low temperatures we observe that the value of z which was infinitely large at absolute zero is now finite though still large in comparison with unity The functions fν z can therefore be expressed as asymptotic expansions in powers of ln z1 see Sommerfelds lemma E17 in Appendix E For the values of ν we are presently interested in namely 52 32 and 12 we have to the first approximation f52 z 815π12 ln z52 1 5π²8 ln z2 f32 z 43π12 ln z32 1 π²8 ln z2 and f12 z 2π12 ln z12 1 π²24 ln z2 Substituting 31 into 4 we obtain NV 4πg3 2mh²32 kT ln z32 1 π²8 ln z2 ¹We therefore speak of the totality of the energy levels filled at T 0 as the Fermi sea and the small fraction of the particles that are excited near the top when T 0 as a mist above the sea Physically speaking the origin of this behavior again lies in the Pauli exclusion principle according to which a fermion of energy ε cannot absorb a quantum of thermal excitation εT if the energy level ε εT is already filled Since εT OkT only those fermions that occupy energy levels near the top level εF up to a depth OkT can be thermally excited to go over to the unfilled energy levels In the zeroth approximation this gives kTlnz μ 3N4πgV23 h²2m which is identical to the groundstate result μ0 εF see equation 24 In the next approximation we obtain kTlnz μ εF 1 π²12 kTεF² Substituting 30 and 31 into 7 we obtain UN 35 kTlnz 1 π²2 lnz2 with the help of 35 this becomes UN 35 εF 1 5π²12 kTεF² The pressure of the gas is then given by P 23 UV 25 nεF 1 5π²12 kTεF² As expected the main terms of equations 37 and 38 are identical to the groundstate results 26 and 27 From the temperaturedependent part of 37 we obtain for the lowtemperature specific heat of the gas CVNk π²2 kTεF Thus for T TF where TF εFk is the Fermi temperature of the system the specific heat varies as the first power of temperature moreover in magnitude it is considerably smaller than the classical value 32 Nk The overall variation of CV with T is shown in Figure 82 The Helmholtz free energy of the system follows directly from equations 35 and 38 AN μ PVN 35 εF 1 5π²12 kTεF² 238 Chapter 8 Ideal Fermi Systems TTF 0 0 05 10 15 1 2 3 Cv Nk FIGURE 82 The specific heat of an ideal Fermi gas the dotted line depicts the linear behavior at low temperatures which gives S Nk π2 2 kT εF 41 Thus as T 0S 0 in accordance with the third law of thermodynamics 82 Magnetic behavior of an ideal Fermi gas We now turn our attention to studying the equilibrium state of a gas of noninteracting fermions in the presence of an external magnetic field B The main problem here is to determine the net magnetic moment M acquired by the gas as a function of B and T and then calculate the susceptibility χT The answer naturally depends on the intrinsic mag netic moment µ of the particles and the corresponding multiplicity factor 2J 1 see for instance the treatment given in Section 39 According to the Boltzmannian treatment one obtains a positive susceptibility χT which at high temperatures obeys the Curie law χ T1 at low temperatures one obtains a state of magnetic saturation However if we treat the problem on the basis of Fermi statistics we obtain significantly different results especially at low temperatures In particular since the Fermi gas is pretty live even at absolute zero no magnetic sat uration ever results we rather obtain a limiting susceptibility χ0 which is independent of temperature but is dependent on the density of the gas Studies along these lines were first made by Pauli in 1927 when he suggested that the conduction electrons in alkali metals be regarded as a highly degenerate Fermi gas these studies enabled him to explain the physics behind the feeble and temperatureindependent character of the paramagnetism of metals Accordingly this phenomenon is referred to as Pauli paramagnetism in contrast to the classical Langevin paramagnetism In quantum statistics we encounter yet another effect which is totally absent in clas sical statistics This is diamagnetic in character and arises from the quantization of the orbits of charged particles in the presence of an external magnetic field or one may say from the quantization of the kinetic energy of charged particles associated with their motion perpendicular to the direction of the field The existence of this effect was first established by Landau 1930 so we refer to it as Landau diamagnetism This leads to an additional susceptibility χT which though negative in sign is somewhat similar to the paramagnetic susceptibility in that it obeys Curies law at high temperatures and tends to a temperatureindependent but densitydependent limiting value as T 0 In general the magnetic behavior of a Fermi gas is determined jointly by the intrinsic magnetic moment of the particles and the quantization of their orbits If the spinorbit interaction is negligible the resultant behavior is given by a simple addition of the two effects 82A Pauli paramagnetism The energy of a particle in the presence of an external magnetic field B is given by ε p²2m μ B where μ is the intrinsic magnetic moment of the particle and m its mass For simplicity we assume that the particle spin is 12 the vector μ will then be either parallel to the vector B or antiparallel We thus have two groups of particles in the gas i those having μ parallel to B with ε p²2m μB and ii those having μ antiparallel to B with ε p²2m μB At absolute zero all energy levels up to the Fermi level εF will be filled while all levels beyond εF will be empty Accordingly the kinetic energy of the particles in the first group will range between 0 and εF μB while the kinetic energy of the particles in the second group will range between 0 and εF μB The respective numbers of occupied energy levels and hence of particles in the two groups will therefore be N 4πV3h³ 2mεF μB32 and N 4πV3h³ 2mεF μB32 The net magnetic moment acquired by the gas is then given by M μN N 4πμV2m323h³ εF μB32 εF μB32 We thus obtain for the lowfield susceptibility per unit volume of the gas χ₀ Lim B 0 MVB 4πμ²2m32 εF12h³ Making use of formula 8124 with g 2 the foregoing result may be written as χ₀ 32 nμ²εF For comparison the corresponding hightemperature result is given by equation 3926 with g 2 and J 12 χ nμ²kT We note that χ₀χ OkTεF To obtain an expression for χ that holds for all T we proceed as follows Denoting the number of particles with momentum p and magnetic moment parallel or antiparallel to the field by the symbol np or np the total energy of the gas can be written as En p p²2m μB np p²2m μB np p np np p²2m μBN N where N and N denote the total number of particles in the two groups respectively The partition function of the system is then given by QN np np expβEn where the primed summation is subject to the conditions np np 0 or 1 and p np p np N N N To evaluate the sum in 9 we first fix an arbitrary value of the number N which automatically fixes the value of N as well and sum over all np and np that conform to the fixed values of the numbers N and N as well as to condition 10 Next we sum over all possible values of N namely from N 0 to N N We thus have QN N0N eβμB2NN np exp β p p²2m np np exp β p p²2m np here the summation is subject to the restriction p np N while is subject to the restriction p np N N Now let Q0N denote the partition function of an ideal Fermi gas of N spinless particles of mass m then obviously Q0 N np exp β p p2 2m np expβA0 N 13 where A0 N is the free energy of this fictitious system Equation 12 can then be written as QN eβ μ B N N 0N e2β μ BN Q0N Q0N N 14 which gives 1N ln QN β μ B 1N ln N 0N exp 2 β μ B N βA0N βA0 N N 15 As before the logarithm of the sum N may be replaced by the logarithm of the largest term in the sum the error committed in doing so would be negligible in comparison with the term retained Now the value N of N which corresponds to the largest term in the sum can be ascertained by setting the differential coefficient of the general term with respect to N equal to zero this gives 2 μ B A0 N N N overlineN A0 N N N N overlineN 0 that is μ0 overlineN μ0 N overlineN 2 μ B 16 where μ0 N is the chemical potential of the fictitious system of N spinless fermions The foregoing equation contains the general solution being sought To obtain an explicit expression for χ we introduce a dimensionless parameter r defined by M μ N N μ 2 N N μ N r 0 r 1 17 equation 16 then becomes μ0 1 r2 N μ0 1 r2 N 2 μ B 18 If the magnetic field B vanishes so does r which corresponds to a completely random orientation of the elementary moments For small B r would also be small so we may carry out a Taylor expansion of the left side of 18 about r 0 Retaining only the first term of the expansion we obtain r 2 μ B μ0 x N x x 12 19 The lowfield susceptibility per unit volume of the system is then given by χ M V B μ N r V B 2 n μ2 μ0 x N x x 12 20 which is the desired result valid for all T For T 0 the chemical potential of the fictitious system can be obtained from equation 8134 with g 1 μ0 x N 3 x N 4 π V 23 h2 2m which gives μ0 x N x x 12 243 3 3 N 4 π V23 h2 2 m 21 On the other hand the Fermi energy of the actual system is given by the same equation 8134 with g 2 εF 3N 8 π V 23 h2 2m 22 Making use of equations 21 and 22 we obtain from 20 χ0 2 n μ2 4 3 εF 3 2 n μ2 εF 23 in complete agreement with our earlier result 6 For finite but low temperatures one has to use equation 8135 instead of 8134 The final result turns out to be χ χ0 1 π2 12 k T εF2 24 On the other hand for T the chemical potential of the fictitious system follows directly from equation 814 with g 1 and f32z z with the result μ0 x N k T ln x N λ3 V which gives μ0 xN x x 12 2 k T 25 Equation 20 then gives χ n μ2 k T 26 in complete agreement with our earlier result 7 For large but finite temperatures one has to take f32z z z2 232 The final result then turns out to be χ χ 1 n λ3 252 27 the correction term here is proportional to TF T32 and tends to zero as T 82B Landau diamagnetism We now study the magnetism arising from the quantization of the orbital motion of charged particles in the presence of an external magnetic field In a uniform field intensity B directed along the zaxis a charged particle would follow a helical path whose axis is parallel to the zaxis and whose projection on the xyplane is a circle Motion along the zdirection has a constant linear velocity uz while that in the xyplane has a constant angular velocity eB mc the latter arises from the Lorentz force e u B c experienced by the particle Quantummechanically the energy associated with the circular motion is quantized in units of eℏB mc The energy associated with the linear motion along the zaxis is also quantized but in view of the smallness of the energy intervals this may be taken as a continuous variable We thus have for the total energy of the particle2 ε e ℏ B mc j 12 pz2 2 m j 012 28 Now these quantized energy levels are degenerate because they result from a coalescing together of an almost continuous set of zerofield levels A little reflection shows that all those levels for which the value of the quantity px2 py2 2 m lay between e ℏ B j mc and e ℏ B j 1 mc now coalesce together into a single level characterized by the quantum number j The number of these levels is given by 1 h2 dx dy d px d py Lx Ly h2 π 2 m e ℏ B m c j 1 j Lx Ly e B h c 29 2 See for instance Goldman et al 1960 Problem 63 71 Thermodynamic behavior of an ideal Bose gas We obtained in Sections 61 and 62 the following formulae for an ideal Bose gas fracPVkT equiv ln mathcalQ sumvarepsilon ln 1 zebeta varepsilon 1 and N equiv sumvarepsilon langle nvarepsilon rangle sumvarepsilon frac1z1 ebeta varepsilon 1 2 where beta 1kT while z is the fugacity of the gas which is related to the chemical potential mu through the formula z equiv expmukT 3 as noted earlier zebeta varepsilon for all varepsilon is less than unity In view of the fact that for large V the spectrum of the singleparticle states is almost a continuous one the summations on the right sides of equations 1 and 2 may be replaced by integrations In doing so we make use of the asymptotic expression 247 for the nonrelativistic density of states avarepsilon in the neighborhood of a given energy varepsilon namely2 avarepsilond varepsilon 2 pi V h3 2m32 varepsilon12 dvarepsilon 4 We however note that by substituting this expression into our integrals we are inadvertently giving a weight zero to the energy level varepsilon 0 This is wrong because in a quantummechanical treatment we must give a statistical weight unity to each nondegenerate singleparticle state in the system It is therefore advisable to take this particular state out of the sum in question before carrying out the integration for a rigorous justification of this unusual step see Appendix F We thus obtain fracPkT frac2 pih3 2m32 int0infty varepsilon12 ln 1 zebeta varepsilon dvarepsilon frac1V ln 1 z 5 and fracNV frac2 pih3 2m32 int0infty fracvarepsilon12 dvarepsilonz1 ebeta varepsilon 1 frac1V fracz1 z 6 of course the lower limit of these integrals can still be taken as 0 because the state varepsilon 0 is not going to contribute toward them anyway Before proceeding further a word about the relative importance of the last terms in equations 5 and 6 For z ll 1 which corresponds to situations not far removed from the classical limit each of these terms is of order 1N and therefore negligible However as z increases and assumes values close to unity the term z 1 z V in 6 which is identically equal to N0 V N0 being the number of particles in the ground state varepsilon 0 can well become a significant fraction of the quantity N V this accumulation of a macroscopic fraction of the particles into a single state varepsilon 0 leads to the phenomenon of BoseEinstein condensation Nevertheless since z 1 z N0 and hence z N0 N0 1 the term leftV1 ln 1 z right in 5 is equal to V1 ln N0 1 which is at most ON1 ln N this term is therefore negligible for all values of z and hence may be dropped altogether We now obtain from equations 5 and 6 on substituting beta varepsilon x fracPkT frac2 pi 2mkT32h3 int0infty x12 ln 1 zex dx frac1lambda3 g52z 7 and fracN N0V frac2 pi 2mkT32h3 int0infty fracx12 dxz1 ex 1 frac1lambda3 g32z 8 where lambda h 2 pi mkT12 9 while g uz are BoseEinstein functions defined by see Appendix D g uz frac1Gamma u int0infty fracx u 1 dxz1 ex 1 z fracz22 u fracz33 u cdots 10 note that to write 7 in terms of the function g52z we first carried out an integration by parts Equations 7 and 8 are our basic results on elimination of z they would give us the equation of state of the system The internal energy of this system is given by U equiv left fracpartialpartial beta ln mathcalQ rightzV kT2 left fracpartialpartial T left fracPVkT right rightzV kT2 V g52z left fracddT left frac1lambda3 right right frac32 kT fracVlambda3 g52 z quad 11 here use has been made of equation 7 and of the fact that lambda propto T12 Thus quite generally our system satisfies the relationship P frac23 U V 12 For small values of z we can make use of expansion 10 at the same time we can neglect N0 in comparison with N An elimination of z between equations 7 and 8 can then be carried out by first inverting the series in 8 to obtain an expansion for z in powers of n lambda3 and then substituting this expansion into the series appearing in 7 The equation of state thereby takes the form of the virial expansion fracPVNkT suml1infty al left fraclambda3v rightl1 13 where u equiv 1n is the volume per particle the coefficients al which are referred to as the virial coefficients of the system turn out to be a1 1 a2 frac14 sqrt2 017678 a3 left frac29 sqrt3 frac18 right 000330 a4 left frac332 frac532 sqrt2 frac12 sqrt6 right 000011 14 and so on For the specific heat of the gas we obtain fracCVNk equiv frac1Nk left fracpartial Upartial T rightNV frac32 left fracpartialpartial T left fracPVNk right right u frac32 suml1infty frac5 3l2 al left fraclambda3 u rightl1 frac32 left 1 00884 left fraclambda3 u right 00066 left fraclambda3 u right2 00004 left fraclambda3 u right3 cdots right quad 15 As T o infty and hence lambda o 0 both the pressure and the specific heat of the gas approach their classical values namely nkT and frac32 Nk respectively We also note that at finite but large temperatures the specific heat of the gas is larger than its limiting value in other words the CV Tcurve has a negative slope at high temperatures On the other hand as T o 0 the specific heat must go to zero Consequently it must pass through a maximum somewhere As seen later this maximum is in the nature of a cusp that appears at a critical temperature Tc the derivative of the specific heat is found to be discontinuous at this temperature see Figure 74 later in this section As the temperature of the system falls and the value of the parameter lambda3 u grows expansions such as 13 and 15 do not remain useful We then have to work with formulae 7 8 and 11 as such The precise value of z is now obtained from equation 8 which may be rewritten as Ne V frac2 pi mkT32h3 g32 z 16 where Ne is the number of particles in the excited states ε 0 of course unless z gets extremely close to unity Ne N³ It is obvious that for 0 z 1 the function g32z increases monotonically with z and is bounded its largest value being g321 1 12³2 13³2 ζ 32 2612 17 see equation D5 in Appendix D Hence for all z of interest g32z ζ 32 18 Consequently for given V and T the total equilibrium number of particles in all the excited states taken together is also bounded that is Ne V 2πmkT³2 h³ ζ 32 19 Now so long as the actual number of particles in the system is less than this limiting value everything is well and good practically all the particles in the system are distributed over the excited states and the precise value of z is determined by equation 16 with Ne N However if the actual number of particles exceeds this limiting value then it is natural that the excited states will receive as many of them as they can hold namely Ne V 2πmkT³2 h³ ζ 32 20 while the rest will be pushed en masse into the ground state ε 0 whose capacity under all circumstances is essentially unlimited N0 N V 2πmkT³2 h³ ζ 32 21 The precise value of z is now determined by the formula z N0 N0 1 1 1 N0 22 which for all practical purposes is unity This curious phenomenon of a macroscopically large number of particles accumulating in a single quantum state ε 0 is generally referred to as the phenomenon of BoseEinstein condensation In a certain sense this phenomenon is akin to the familiar process of a vapor condensing into the liquid state which takes place in the ordinary physical space Conceptually however the two processes are very different Firstly the phenomenon of BoseEinstein condensation is purely ³Remember that the largest value z can have in principle is unity In fact as T 0 z N0 N0 1 N N 1 which is very nearly unity but certainly on the right side of it of quantum origin occurring even in the absence of intermolecular forces secondly it takes place at best in the momentum space and not in the coordinate space⁴ The condition for the onset of BoseEinstein condensation is N VT³2 2πmk³2 h³ ζ 32 23 or if we hold N and V constant and vary T T Tc h² 2πmk N V ζ 32 ²³ 24⁵ here Tc denotes a characteristic temperature that depends on the particle mass m and the particle density N V in the system Accordingly for T Tc the system may be looked on as a mixture of two phases i a normal phase consisting of Ne N T Tc³2 particles distributed over the excited states ε 0 and ii a condensed phase consisting of N0 N Ne particles accumulated in the ground state ε 0 Figure 71 shows the manner in which the complementary fractions Ne N and N0 N vary with T For T Tc we have the normal phase alone the number of particles in the ground state namely z 1 z is O1 which is completely negligible in comparison with the total number N Clearly the situation is singular at T Tc For later reference we note that at T Tc from below the condensate fraction vanishes as follows N0 N 1 T Tc³2 32 Tc T Tc 25 A knowledge of the variation of z with T is also of interest here It is however simpler to consider the variation of z with v λ³ the latter being proportional to T³2 For 0 v λ³ 2612¹ which corresponds to 0 T Tc the parameter z 1 see equation 22 For v λ³ 2612¹ z 1 and is determined by the relationship g32z λ³ v 2612 26⁶ ⁴Of course the repercussions of this phenomenon in the coordinate space are no less curious It prepares the stage for the onset of superfluidity a quantum manifestation discussed in Section 76 ⁵For a rigorous discussion of the onset of BoseEinstein condensation see Landsberg 1954b where an attempt has also been made to coordinate much of the previously published work on this topic For a more recent study see Greenspoon and Pathria 1974 Pathria 1983 and Appendix F ⁶An equivalent relationship is g32z g321 Tc T³2 1 see equation 8 For v λ³ 1 we have g32z 1 and hence z 1 Under these circumstances g32z z see equation 10 Therefore in this region z v λ³¹ in agreement with the classical case⁷ Figure 72 shows the variation of z with v λ³ Next we examine the P Tdiagram of this system that is the variation of P with T keeping v fixed Now for T Tc the pressure is given by equation 7 with z replaced by unity PT kT λ³ ζ 52 27 which is proportional to T⁵² and is independent of v implying infinite compressibility At the transition point the value of the pressure is PTc 2πm h²³2 kTc⁵² ζ 52 28 ⁷Equation 6212 gives for an ideal classical gas ln Q zV λ³ Accordingly N z ln Q z zV λ³ with the result that z λ³ v FIGURE 71 Fractions of the normal phase and the condensed phase in an ideal Bose gas as a function of the temperature parameter T Tc FIGURE 72 The fugacity of an ideal Bose gas as a function of v λ³ with the help of 24 this can be written as PTc ζ52ζ32NV kTc 05134NV kTc Thus the pressure exerted by the particles of an ideal Bose gas at the transition temperature Tc is about onehalf of that exerted by the particles of an equivalent Boltzmannian gas8 For T Tc the pressure is given by P NV kT g52zg32z where zT is determined by the implicit relationship g32z λ3ν NV h32πmkT32 Unless T is very high the pressure P cannot be expressed in any simpler terms of course for T Tc the virial expansion 13 can be used As T the pressure approaches the classical value NkTV All these features are shown in Figure 73 The transition line in the figure portrays equation 27 The actual PTcurve follows this line from T 0 up to T Tc and thereafter departs tending asymptotically to the classical limit It may be pointed out that the region to the right of the transition line belongs to the normal phase alone the line itself belongs to the mixed phase while the region to the left is inaccessible to the system In view of the direct relationship between the internal energy of the gas and its pressure see equation 12 Figure 73 depicts equally well the variation of U with T of course with ν fixed Its slope should therefore be a measure of the specific heat CV T of the gas We readily observe that the specific heat is vanishingly small at low temperatures and rises with T until it reaches a maximum at T Tc thereafter it decreases tending asymptotically to the constant classical value Analytically for T Tc we obtain see equations 15 and 27 CVNk 32 VN ζ52 ddT Tλ3 154 ζ52 vλ3 8 Actually for all T Tc we can write PT PTc TTc52 05134NekTV We infer that while particles in the condensed phase do not exert any pressure at all particles in the excited states are about half as effective as in the Boltzmannian case FIGURE 73 The pressure and the internal energy of an ideal Bose gas as a function of the temperature parameter TTc which is proportional to T32 At T Tc we have CVTcNk 154 ζ52ζ32 1925 which is significantly higher than the classical value 15 For T Tc we obtain an implicit formula First of all CVNk T 32 T g52zg32zv see equations 11 and 26 To carry out the differentiation we need to know zTv this can be obtained from equation 26 with the help of the recurrence relation D10 in Appendix D On one hand since g32z T32 T g32zv 32T g32z on the other z z g32z g12z Combining these two results we obtain 1z zTv 32T g32zg12z Equation 33 now gives CVNk 154 g52zg32z 94 g32zg12z the value of z as a function of T is again to be determined from equation 26 In the limit z 1 the second term in 37 vanishes because of the divergence of g12z while the first term gives exactly the result appearing in 32 The specific heat is therefore continuous at the transition point Its derivative is however discontinuous the magnitude of the discontinuity being CVTTTc0 CVTTTc0 27Nk16π Tc ζ322 3665 NkTc see Problem 76 For T Tc the specific heat decreases steadily toward the limiting value CVNkz0 154 94 32 Figure 74 shows all these features of the CV Trelationship It may be noted that it was the similarity of this curve with the experimental one for liquid He4 Figure 75 that prompted F London to suggest in 1938 that the curious phase transition that occurs in liquid He4 at a temperature of about 219 K might be a manifestation of the BoseEinstein condensation taking place in the liquid Indeed if we substitute in 24 data for liquid He4 namely m 665 1024 g and V 276 cm3mole we obtain for Tc a value of about 313 K which is not drastically different from the observed transition temperature of the liquid Moreover the interpretation of the phase transition in liquid He4 as BoseEinstein condensation provides a theoretical basis for the twofluid model of this liquid which was empirically put forward by Tisza 1938ab to explain the physical behavior of the liquid below the transition temperature According to London the N0 particles that occupy a single entropyless state ε 0 could be identified with the superfluid component of the liquid and the Ne particles that occupy the excited states ε 0 with the normal component As required in the FIGURE 74 The specific heat of an ideal Bose gas as a function of the temperature parameter TTc The specific heat of liquid He4 under its own vapor pressure after Keesom and coworkers The isotherms of an ideal Bose gas 72 BoseEinstein condensation in ultracold atomic gases of the atoms in the trap Atoms that are stationary are just off resonance and so rarely absorb a photon Moving atoms are Doppler shifted on resonance to the laser beam that is propagating opposite to the velocity vector of the atom Those atoms preferentially absorb photons from that direction and then reemit in random directions resulting in a net momentum kick opposite to the direction of motion This results in an optical molasses that slows the atoms This cooling method is constrained by the recoil limit in which the atoms have a minimum momentum of the order of the momentum of the photons used to cool the gas This gives a limiting temperature of hf22mc2 k 1 μK where f is the frequency of the spectral line used for cooling and m is the mass of an atom In the next step of the cooling process the lasers are turned off and a spatially varying magnetic field creates an attractive anisotropic harmonic oscillator potential near the center of the magnetic trap Vr 12 m ω12 x2 ω22 y2 ω32 z2 The frequencies of the trap ωα are controlled by the applied magnetic field One can then lower the trap barrier using a resonant transition to remove the highest energy atoms in the trap If the atoms in the vapor are sufficiently coupled to one other then the remaining atoms in the trap are cooled by evaporation If the interactions between the atoms in the gas can be neglected the energy of each atom in the harmonic oscillator potential is εl1l2l3 ħω1 l1 ħω2 l2 ħω3 l3 12 ħω1ω2ω3 where lα 012 are the quantum numbers of the harmonic oscillator If the three frequencies are all the same then the quantum degeneracy of a level with energy εħωl32 is l1l22 see Problem 326 For the general anisotropic case the smoothed density of states as a function of energy suppressing the zero point energy and assuming ε ħωα is given by aε 0 0 0 δε ħω1 l1 ħω2 l2 ħω3 l3 dl1 dl2 dl3 ε2 2 ħω03 where ω0 ω1 ω2 ω313 this assumes a single spin state per atom The thermodynamic potential Π see Appendix H for bosons in the trap is then given by ΠμT kT4 2 ħω03 0 x2 ln1 ex eβμ dx kT4 ħω03 g4z where z expβμ is the fugacity and gνz is defined in Appendix D Volume is not a parameter in the thermodynamic potential since the atoms are confined by the harmonic trap The average number of atoms in the excited states in the trap is NμT ΠμT kTħω03 g3z For fixed N the chemical potential monotonically increases as temperature is lowered until BoseEinstein condensation occurs when μ0 z1 The critical temperature for N trapped atoms is then given by kTcħω0 Nζ313 where ζ3g31 1202 While the spacing of the energy levels is of order ħω0 the critical temperature for condensation is much larger than the energy spacing of the lowest levels for N 1 A typical magnetic trap oscillation frequency f 100Hz For N2 104 as in Cornell and Wiemans original experiment kTcħω0 255 The observed critical temperature was about 170nK Anderson et al 1995 For T Tc the number of atoms in the excited states is NexcitedN ζ3NkTħω03 TTc3 so the fraction of atoms that condense into the ground state of the harmonic oscillator is N0N 1 TTc3 see de Groot Hooyman and ten Seldam 1950 and Bagnato Pritchard and Kleppner 1987 In the thermodynamic limit a nonzero fraction of the atoms occupy the ground state for T Tc By contrast the occupancy of the first excited state is only of order N13 so in the thermodynamic limit the occupancy fraction in each excited state is zero A comparison of the experimentally measured Bosecondensed fraction with equation 8 is shown in Figure 77 72A Detection of the BoseEinstein condensate The linear size of the ground state wavefunction in cartesian direction α is aα ħmωα