12.1 INTRODUCTION: STATES OF MANY-PARTICLE SYSTEMS

Transcription

1 Chapte 1 2 : Many- Paticle Systems Chapte 12: Many-Paticle Systems Intoduction: States of Many-Paticle Systems Systems Divisible into Independent Subsystems Distinguishable subsystems Indistinguishable subsystems Quantum Inteactions in Many-Paticle Systems Quantum States as Non-Inteacting Subsystems Femions: Femi-Diac Statistics Bosons: Bose-Einstein Statistics The Dilute Paticle Limit: Boltzmann Statistics Femi and Bose Gases of Elementay Paticles The density of states The chemical potential The wo function The themal and caloic equations of state The entopy; sufaces of constant entopy INTRODUCTION: STATES OF MANY-PARTICLE SYSTEMS To compute the fundamental equation of a quantized system we need to evaluate its patition function Z( ) = n e - E n 12.1 whee n labels the n th eigenvecto of the appopiate Hamiltonian. Howeve, it should be obvious that the patition function of a typical macoscopic system cannot be found by any pactical means. Since the Hamiltonian is a function of some vaiables it is pactically impossible to find even one of its eigenvectos, and cetainly impossible to find all of them. The statistical fomulation of themodynamics has theoetical value whethe o not its cental equations can be solved. It maes the connection between micoscopic and macoscopic behavio and often povides useful qualitative insights into the souces of macoscopic behavio. But its pactical value is limited. Thee is, nonetheless, a lage class of impotant poblems that have been solved with statistical themodynamics. With vey few exceptions, the solutions ae based on some vaiant of a single method of appoach. As we shall show below, equation 12.1 can be solved when the many-paticle system is a composite of non-inteacting o wealy inteacting subsystems that ae individually small and simple enough that thei behavio page 186

2 is mathematically tactable. This is the case when the paticles themselves inteact wealy, as they do, fo example, in an ideal gas. It is also the case when the paticles inteact stongly, but thei collective motion can be analyzed in tems of nomal modes o quasipaticles that behave mathematically as if they wee discete paticles that inteact wealy with one anothe. To solve a poblem in statistical themodynamics one hence begins by finding a way to descibe the system in tems of paticles o quasipaticles that inteact wealy with one anothe, and evaluates an appopiate patition function to establish the fundamental equation. When it is impossible to find a desciption of this type that is sufficiently accuate fo the poblem at hand, one evaluates the patition function of a simple system that is as nealy lie the system of inteest as possible. The fundamental equation of the system of inteest is then appoximated by expessing its diffeence fom the simple system as a petubation and evaluating the effect of that petubation on the fundamental equation. When the petubation is small, the fundamental equation of the system of inteest can be found as accuately as desied. The themodynamics of solids, which is the poblem of geatest inteest to us, offes seveal examples of this pocedue that will be discussed in some detail at a late point. A cystalline solid is modeled as an odeed aangement of ion coes, which contain the atomic nuclei and coe electons, in a sea of valence electons. The ion coes ae in constant motion due to themal vibations. To find the states of the valence electons we neglect the motions of the ion coes by placing the ions at thei equilibium positions on the cystal lattice, and neglect the specific inteactions between the electons by gatheing all the electons except the electon of inteest into a single chage distibution. Even with these appoximations we cannot diectly teat the state of a valence electon on a given atom since the valence states on adjacent atoms ae not othogonal. To define electon states whose mutual inteaction is wea we athe let the electons be in Bloch states that spead though the cystal and incopoate its peiodicity. The Bloch states can be calculated, at least appoximately, and they ae gatheed into bands of one-electon states that fom the band stuctue of the cystal. The static enegy of the cystal lattice is then found by summing the enegies of the individual electons and the enegy of inteaction of the ion coes. The next significant contibution to the enegy is the enegy of lattice vibations in the solid. Again, we cannot teat the vibational states of individual atoms, since the motions of the individual atoms ae stongly coupled togethe. Howeve, if the atom displacements ae small it is possible to descibe the vibational state in tems of its nomal modes, which ae essentially distotional waves (elastic waves o sound waves) that tavel though the cystal. These waves can be teated mathematically as quasipaticles (phonons) that inteact wealy with one anothe. Thei enegy states can be found and summed to detemine the vibational enegy of the cystal. The fee enegy calculated by assuming independent electon and phonon states is not always sufficiently accuate to teat a phenomenon of inteest. In this case we usually do not abandon the model, which would e-immese us in an unsolvable poblem, but athe teat the inaccuacies of the model by calculating electon-electon, electonphonon and phonon-phonon inteactions as petubations of the enegy. This lets us page 187

3 etain a pictue of the solid as a collection of paticles in single electon and phonon states, which is almost essential fo physical visualization. The basis of this whole pocedue is the possibility of modeling the system as a collection of individual subsystems (paticles o quasipaticles) that inteact wealy and computing the patition function fo the system of paticles. The best method of division into independent (wealy inteacting) subsystems depends on the natue of the system and may also depend on the phenomenon of inteest. In the following sections we assume that an appopiate division has been made and discuss the computation of the fundamental equation. In computing the fundamental equation we must eep in mind that thee ae thee distinct types of "inteactions" that influence the behavio of systems in quantum mechanics. The fist ae the physical inteactions that detemine the foces between paticles. These fix the fom of the potential, V({q}), in the Hamiltonian of the system. The second type of inteaction elates to the distinguishability of the paticles, and deives fom the fact that the state of the system is unchanged by an intechange of the coodinates and momenta of identical paticles. This inteaction is the souce of the Gibbs facto, which we aleady discussed in connection with the statistical themodynamics of classical systems. The thid type of inteaction is a puely quantum phenomenon that deives fom the symmety of the quantum state, and detemines whethe two identical paticles can exist in the same quantum state. This inteaction is esponsible fo the Pauli Exclusion Pinciple, which assets that no two paticles that have half-integal spins (such as electons) can be in the same one-paticle quantum state in the same system. When the quantum inteaction can be ignoed, as it can be when the paticles ae distibuted ove so many individual states that the puely statistical pobability that two will be found in the same state is negligible, we say that the paticles ae independent. In the following we shall fist conside systems of independent paticles and then teat the quantum inteaction. As we shall see, including the quantum inteaction leads to thee distinct types of quantum statistics, which ae named Femi- Diac statistics, Bose-Einstein statistics, and Boltzmann statistics SYSTEMS DIVISIBLE INTO INDEPENDENT SUBSYSTEMS Conside a system at equilibium in contact with a themal esevoi acoss a igid impemeable wall. The system is hence epesented by a canonical ensemble. Let the system be divisible into N wealy inteacting subsystems that ae independent in the quantum sense and also have canonical distibutions. If the inteaction enegy is negligible the enegy of the n th state of the composite system is just the sum E n = E n 12.2 page 188

4 whee E n is the enegy of the th subsystem when the composite system is in its n th state. The fundamental equation of the composite system depends on whethe the subsystems ae distinguishable. We conside the two cases in tun Distinguishable subsystems If the subsystems ae distinguishable the canonical patition function is Z( ) = n e - E n = n e - E n = n e - En 12.3 Since a state of the composite system is a union of states of the subsystems, and since the subsystem can be in any state consistent with its own bounday conditions, thee is one value of the enegy of the system, E n, fo each possible choice of the enegies of the subsystems, that is, fo each possible choice of the set {E n }. It follows that the summation can be intechanged with the poduct in equation 12.1, giving the esult Z( ) = n e - En = Z ( ) 12.4 whee Z ( ) = n e - En 12.5 is the patition function fo the th subsystem taen by itself. It follows fom equation 12.4 that when the subsystems ae distinguishable the Helmholtz fee enegy is additive: F = - T ln[z( )] = - T ln[z ( )] = F 12.6 whee F is the Helmholtz fee enegy of the th subsystem. The enegy and entopy ae also additive. The intenal enegy is page 189

5 E = - {ln[z( )]} = - {ln[z ( )]} = E 12.7 while the entopy is S = - F T = - F T = S 12.8 whee the entopy of the th subsystem is S = - {ln[z ( )]} + ln[z ( )] 12.9 The gand canonical patition function of the composite system can be found in the same way. The patition function, Z(,µ), is Z(,µ) = N n e - [E n(n)-µn] whee E n (N) is the enegy of the n th state of the system when it contains a total of N paticles. If the system is a composite of independent subsystems then each of its states is the union of one state of each subsystem. When the system is in its n th state with paticle numbe N the th subsystem has the paticle numbe N and enegy E n (N ), whee N = N E n (N) = E n (N ) The paticle numbes in the individual subsystems ange ove all possible values of N, and the enegies, E n (N ), ange ove all possible values of E consistent with N. It follows that Z(,µ) = N n e - [E n (N )-µn ] = Z (,µ) Hence the wo function of a composite system of distinguishable subsystems is page 190

6 (T,V,µ) = (T,V,µ) whee (T,V,µ) is the wo function of the th subsystem Indistinguishable subsystems When the subsystems ae indistinguishable the state of the system is invaiant to the intechange of the coodinates and momenta of any two of them. The n th state of the total system has enegy E n = E but is unchanged by an intechange of the states of subsystems (i) and (j). Since thee ae N! diffeent ways of intechanging the enegies of the N subsystems, each value of E n can be composed of the enegies of N identical subsystems in N! diffeent ways. When the subsystems ae distinguishable the N! states ae distinct, but when the subsystems ae indistinguishable they ae edundant examples of the same state. It follows that the canonical patition function fo the system is simply (N!) -1 times that when the subsystems ae distinguishable. Moeove, since the subsystems ae the same thei patition functions ae equal. The canonical patition function fo the composite system is, theefoe, Z( ) = 1 N! Z ( ) = 1 N! [z( )] N whee z( ) is the patition function of the subsystem. In many cases the subsystems ae not all the same, but can be gatheed into goups of subsystems that ae identical. Let a system contain N subsystems, and let these be divisible into sets of identical subsystems, the i th of which has N i membes. The state of the system is unchanged if the coodinates and momenta of a subsystem ae intechanged with those of any identical subsystem. It follows that the patition function fo the system can be witten 1 Z( ) = (N i )! [z i ( )] N i whee z i ( ) is the canonical patition function fo a subsystem of the i th type. When the system contains indistinguishable subsystems the expected value of the enegy is still a simple sum, since it follows fom equation that page 191

7 E = E = N i E i whee E i is the expected value of the enegy of a subsystem of type (i). Howeve, the Helmholtz fee enegy and the entopy contain additional tems that aise fom the indistinguishability of the paticles (these ae often called the entopy of mixing in mateials science texts). The fundamental equation of a system that contains inds of subsystems, N i of which ae of type (i), is, fom equation 12.17, F = - T ln[z( )] = - N i T ln[z i ( )] + T ln[(n i )!] = - N i T ln[z i ( )] + N i N i T ln e whee e is the base of the natual logaithm and we have used Steling's Appoximation, ln(n!) «N ln N e to obtain the thid fom of the left-hand side. Since the appoximation is almost exact when N > 10, we have used the equality in equation Equation can be ewitten F = N i F i + N i N i T ln e whee F i is the Helmholtz fee enegy of a subsystem of type (i) by itself. The entopy of the system is S = ( E - F) = N i S i - N i N i ln e whee S i is the entopy of a subsystem of the i th type taen in isolation. Note that these elations do not violate the additivity of the classical themodynamic functions. The themodynamic quantities that ae added in classical themodynamics ae those of subsystems that ae sepaated o divided by patitions. The subsystems occupy diffeent pats of space and ae, theefoe, distinguishable. As shown in the pevious subsection the themodynamic quantities of distinguishable subsystems page 192

8 ae additive. The subsystems that ae teated hee ae indistinguishable, and must, theefoe, fill the same volume. The patition function of a single paticle, z i ( ), inceases with V, and, hence, F i contains an additive tem of the fom ln(v). Hence F can be witten, F = N i f i - N i Tln(V) + N i N i T ln e = N i f i + N i N i T ln ev whee f i = - Tln[z i ( )/V] is the fee enegy pe unit volume of the i th subsystem. Equation shows that the fee enegy is additive. Combining two identical systems doubles both N i and V, hence doubling F QUANTUM INTERACTIONS IN MANY-PARTICLE SYSTEMS If a quantized system contains N paticles (subsystems) that have wea physical inteactions with one anothe then the Hamiltonian, H, fo the system can be witten H = H + V In this equation H is the Hamiltonian of the th paticle and is a function of the coodinates and momenta of that paticle only. The opeato, V, epesents the inteaction between the paticles. In most cases of inteest to us it is a function of the coodinates of the paticles, but is independent of thei momenta. The paticles have a wea physical inteaction if the inteaction enegy, V, is small in compaison to the sum of the one-paticle enegies, that is, if fo any eigenstate of H, V = n V n << E = n H n When the inequality holds the states of the composite system ae, to the fist ode, states in which each paticle is in an eigenstate of its own Hamiltonian. These ae quantum states, n, such that H n = E n n whee E n is the enegy of the nth state of the th paticle taen by itself. page 193

9 The Hamiltonian of the th paticle, H, opeates only on the coodinates and momenta of that paticle. It follows that if we neglect the inteaction, V, a quantum state that is a simple poduct of the eigenstates of the individual paticles is an eigenstate of the Hamiltonian of the system. If H = n then H n = H n = ' H ' n = ' E ' n n = E n n In deiving equation we have used the identity H ' n = E ' n n ' which follows fom the fact that the opeato H ' opeates only on the coodinates of the paticle labeled ' and leads to the esult H ' and have used the definition n = E ' n n E n = ' E ' n fo the enegy of the n th eigenstate of H when V is negligible. Equation suggests that all possible enegy states of the composite system can be constucted by foming odeed multiples of the enegy eigenstates of the single paticles. This is tue if the paticles ae distinguishable by vitue of being physically diffeent o spatially sepaate fom one anothe. Howeve, it cannot be tue when the paticles ae indistinguishable since equation does not have the pope symmety to be a composite state of indistinguishable paticles. If two paticles ae indistinguishable then an intechange of thei coodinates and momenta cannot change any physical popety of the system. To see how this esticts the symmety of the system let n' be the state that is obtained fom n by intechanging page 194

10 the coodinates and momenta of indistinguishable paticles. Let R be any dynamical vaiable of inteest. Indistinguishability equies that n' R n' = n R n which can hold fo all possible choices of R only if n' = ± n If the uppe sign holds the state is said to be symmetic to an intechange of indistinguishable paticles; if the lowe sign holds it is antisymmetic. The geneal esult is that the quantum state of a many-paticle system is eithe symmetic o antisymmetic to the intechange of indistinguishable paticles. Paticles whose states ae symmetic ae called bosons; those whose states ae antisymmetic ae called femions. It can be shown in geneal that the symmety of the quantum state is detemined by the spin of the paticle. The total spin is always a multiple of (1/2). If the spin is an odd multiple of (1/2) the paticle is a femion and the quantum state is antisymmetic. If the spin is zeo o an even multiple of (1/2) the paticle is a boson and the quantum state is symmetic. It follows that electons ae femions. As we shall see below, phonons, which ae the quasipaticles associated with lattice vibations, ae bosons. The simple poduct function, equation 12.27, is neithe symmetic no antisymmetic, and hence cannot epesent the state of a collection of indistinguishable paticles. To see this, conside a two paticle system in which one paticle has state 1 and the othe state 2. Intechanging the paticles changes the poduct state n = 1 2 to the state n' = 2 1. But n' is neithe identical to n no its negative. Since n n = 1 the ba associated with the et n is n = 2 1. Hence the inne poduct between n' and n is n' n = = since 1 and 2 ae othogonal. The allowable states ae the symmetic combinations n = 1 2 { } fo femions and page 195

11 n = 1 2 { } fo bosons. These ae nomalized and ae, espectively, antisymmetic and symmetic to an intechange of paticle states. Genealizing this esult, the enegy eigenstates fo a system of N wealy inteacting femions can be witten in the compact notation n = 1 N! P (-1) P n whee the sum ove P is taen ove all possible pemutations of the one-paticle states of the indistinguishable paticles, and P in the exponent of (-1) indicates the ode of the pemutation. A pemutation is made by pemuting the ode of the states n in the poduct function, since it is undestood that the fist state in the poduct function gives the state of the fist paticle, etc. The facto (1/ N! ) is included to nomalize the summation, which includes N! tems, each of which is individually nomalized. An intechange of identical paticles is ealized by pemuting the ode of states in the lead tem; all othe tems in the summation ae changed accodingly. The state, n', that esults fom an intechange of identical paticles is n' = (-1) P n whee P is the ode of the pemutation associated with the paticle intechanges (P = 1 if two paticles ae intechanged, P = 2 if thee o fou paticles ae intechanged in pais, etc.). The state is hence antisymmetic to an intechange of single-paticle states, as it should be. The symmety of the state of a system of indistinguishable femions leads to the Pauli Exclusion Pinciple, which assets that no two indistinguishable femions can have the same one-paticle state. If any two of the states n in the poduct of equation ae the same then the pemutation that intechanges them epoduces the same tem with the opposite sign. It follows that the tems in the summation ove P cancel in pais, and the summation is zeo. Hence if a system of many paticles contains indistinguishable femions, they must have distinct one-paticle states (the Pauli Exclusion Pinciple). The symmetic state of a system of N bosons in one-paticle states taes the simila fom n = 1 N! P n page 196

12 in which the pemutations ae simply added. The ight-hand side of equation is symmetic to an intechange of one-paticle states. The function does not vanish when two paticles have the same state. Hence if a system of many paticles contains indistinguishable bosons, an abitay numbe of them can have the same one-paticle state QUANTUM STATES AS NON-INTERACTING SUBSYSTEMS To evaluate the patition function fo a system of indistinguishable quantized paticles we need to subdivide it into wealy inteacting subsystems that ae simple enough to be analyzed. Howeve, the paticles themselves cannot usually be thought of as wealy inteacting since the quantum inteaction is stong. Each paticle ecognizes whethe thee is anothe in the same quantum state even if the physical inteaction is negligible. To avoid this poblem we focus ou attention on the quantum states. The th single-paticle state has the popeties of an open subsystem. It has a vaiable numbe of paticles, N, which can only be zeo o one when the paticles ae femions, and a definable enegy E = N whee is the single-paticle enegy in the th state. When the system is at equilibium each state has an expected numbe of paticles, N, that is detemined by the equilibium distibution of paticles ove the available one-paticle states, o, equivalently, by its equilibium with an envionment that includes othe single-paticle states that donate paticles to it o eceive paticles fom it. Adopting this pespective, let each state of the system be epesented by a gand canonical ensemble whose membes include all possible numbes of paticles in that state. The gand canonical patition function fo the th state is Z (,µ) = N e - ( -µ)n whee the summation is ove all allowed values of N, the numbe of paticles in quantum state. The wo function fo the th state is (T,V,µ) = - T ln[z(,µ)] and the expected numbe of paticles in the th state is page 197

13 N = - µ Since the subsystems ae distinguishable (they epesent distinct quantum states), thei wo functions sum. It follows that the wo function fo the system is (T,V,µ) = (T,V,µ) The most common poblem in statistical themodynamics equies a solution fo the Helmholtz fee enegy of a system of N paticles at tempeatue, T. Given equation 12.44, the poblem can be solved by ecognizing that the themodynamic state of a macoscopic system of N paticles with a given value of T is identical to that of an open system whose chemical potential, µ, is such that the expected numbe of paticles, N, is equal to N. It follows that µ is a solution to the equation N = N = - µ The Helmholtz fee enegy is, then, F(T,V,N) = (T,V,µ) + µn The detailed fom of these equations depends on whethe the paticles ae femions o bosons, and whethe the density of paticles is lage o small compaed to the density of states. We shall wo out the thee impotant cases in tun FERMIONS: FERMI-DIRAC STATISTICS If the paticles of inteest ae femions then the gand canonical patition function of a given quantum state contains only the two tems that coespond to N = 0 and N = 1. Hence, fom equation 12.41, Z (,µ) = N e - ( -µ)n = 1 + e - ( -µ) This summation conveges iespective of the sign of µ. The wo function of the th state is (T,V,µ) = - T ln[ 1 + e - ( -µ) ] and the expected numbe of paticles in the th state is page 198

14 N = - µ e - ( -µ) = 1 + e - ( -µ) 1 = 1 + e ( -µ) Equation is nown as the Femi-Diac distibution function. The wo function fo the system is (T,V,µ) = = - T ln[ 1 + e - ( -µ) ] whee, if the system contains N paticles, µ is fixed by the elation N = N = - µ = N = e ( -µ) The entopy of a system of femions is S = - T = ( -µ) 1 + e ( -µ) + ln[ 1 + e- ( -µ) ] The expected value of the enegy can be evaluated diectly fom the Femi-Diac distibution E = N = 1 + e ( -µ) If the system is isolated then its enegy is fixed at the value E. The equation E = E is then an equation fo the tempeatue, T. It follows fom equations that as it should. = E - TS - µ N page 199

15 12.6 BOSONS: BOSE-EINSTEIN STATISTICS If the paticles in a system ae bosons the gand canonical patition function fo the th state contains tems fo all possible numbes of paticles. The patition function can still be computed in closed fom. Fom equation the patition function is Z (,µ) = N =0 e - ( -µ)n = (x) N N = whee x = e -β(ε µ) Since the patition function is eal, the seies must convege fo all values of. It follows that µ must be less than the least value of (µ < 0 if min( ) = 0), so that x 1 and x N 0 as N. In this case (1-x) (x) N N =0 = fom which 1 Z (,µ) = 1 - e - ( µ)n and (T,V,µ) = - T ln[z (,µ)] = T ln[ 1 - e - ( -µ)n ] The expected numbe of paticles in the th state is, theefoe, N = - µ e - ( -µ) = 1 - e - ( -µ) 1 = e ( µ) - 1 Equation is called the Bose-Einstein Distibution. The fundamental equation fo the system can be witten page 200

16 (T,V,µ) = (T,µ) = T ln[ 1 - e - ( -µ) ] The expected value of the total numbe of paticles is N = N = 1 e ( µ) - 1 If the system has a fixed numbe of paticles, N, the equation N = N detemines the chemical potential, µ. The intenal enegy can be evaluated fom the Bose-Einstein distibution function E = N = e ( µ) - 1 If the system has fixed enegy, E, as well as fixed paticle numbe then the equation E = E detemines the tempeatue, T THE DILUTE PARTICLE LIMIT: BOLTZMANN STATISTICS The wo function fo the th state of a quantized system of many paticles can be expessed in the compact fom (T,µ) = - (±T) ln[ 1 ± e (µ- ) ] whee the positive sign applies when the paticles ae femions and the negative sign applies when they ae bosons. When (µ- ) is lage and negative, that is, when eithe the chemical potential, µ, is lage and negative o the minimum value of > µ and is lage, equation appoaches the same limit fo both paticle types: (T,V,µ) = - Te (µ- ) The expected numbe of paticles in the th state is then N = - µ = e (µ- ) When equations and hold the system is said to obey Boltzmann statistics. Equation is called the Boltzmann distibution. page 201

17 The Boltzmann limit is, in fact, pecisely the condition in which the quantum coelations between the paticles can be ignoed. When (µ- ) is lage and negative it follows fom equation that N << and thee is a vanishing lielihood that moe than one paticle is in the same state. In this case it is statistically ielevant whethe the natue of the paticles pemits moe than one in the same state. When the system contains N paticles the chemical potential can be evaluated fom the condition N = N = e µ e - = e µ z( ) whee z( ) = e is the canonical patition function fo a single paticle. It follows that µ = - T ln[z( )] + T ln(n) The chemical potential, µ, is much less than zeo when the facto z( ) N T >> To see the meaning of the condition note that the tems in z( ) fo which the facto is small contibute appoximately 1 to the patition function while those fo which the facto is lage contibute almost nothing. It follows that the patition function can be egaded as a measue of the numbe of available states, the numbe of states that can have a easonable pobability of occupation. Hence the condition is just the condition that the numbe of available states geatly exceeds the numbe of paticles. The wo function of a system of paticles that obey Boltzmann statistics has a simple fom. It follows fom equations and that page 202

18 (T,V,µ) = (T,V,µ) = - N T If the system is fluid we have, by definition, = - N T (T,V,µ) = - PV Equations show that a system of N wealy inteacting Boltzmann paticles obeys the themal equation of state of an ideal gas: PV = NT When a system of Boltzmann paticles has fixed values of N and T its fundamental function is the Helmholtz fee enegy. The Helmholtz fee enegy is easily evaluated fom equations and 12.70, and is F(T,V,N) = + µn = - NT ln[z( )] + NT ln N e But if the paticles ae independent it is also possible to evaluate the Helmholtz fee enegy diectly fom the patition function. Fom section , Z( ) = 1 N! [z( )] N and the Helmholtz fee enegy is F(T,V,N) = - T ln[z( )] = - NT ln[z( )] + NT ln N e This esult epoduces equation and shows that Boltzmann paticles obey the statistics of indistinguishable independent subsystems. The Boltzmann limit ensues that the paticles inteact wealy in the quantum sense as well as in the physical sense. Boltzmann statistics ae almost always sufficient to teat systems of wealy inteacting atoms o molecules, and ae hence adequate fo most eal gases. Boltzmann statistics geneally fail fo systems of elementay paticles such as electons, phonons o photons because the quantum inteaction is stong. Boltzmann statistics also usually fail fo condensed phases. Since the atoms in a page 203

19 condensed phase inteact stongly thei states ae most conveniently descibed as collections of quasipaticles that have a stong quantum inteaction FERMI AND BOSE GASES OF ELEMENTARY PARTICLES Conside a system that can be descibed as a collection of quantized elementay paticles that have a wea inteaction with one anothe. Thee ae seveal impotant geneal elations that goven the themodynamics of such systems which have the same fom to within a change of sign fo both femions and bosons. The geneal elations ae pesented in compact fom by Landau and Lifshitz (Statistical Physics, Section 55). The deivations ae epeated hee to include some of the impotant steps that ae assumed in thei pesentation The density of states The enegy of an elementay paticle in a fee state is just its inetic enegy = p 2 2m Since a fee paticle can always be teated by classical statistics (it does not inteact) its inetic states ae distibuted ove phase space with the density dw(q,p) = h -3 d 3 pdv Howeve, an elementay paticle also has a spin, s, and each of its inetic states coesponds to g = (2s+1) degeneate states that diffe in the oientation (s z ) of the spin. In the case of an electon s = 1/2 and g=2. We nomally descibe the two independent electon states as spin-up and spin-down. The total numbe of fee-electon states pe unit volume of phase space is, hence, dw(q,p) = g h 3 d3 pdv This equation can be ewitten in tems of the enegy,, by ecognizing that since the behavio of a fee paticle depends only on the magnitude, p, of the momentum vecto, p, and is independent of its diection, the volume element, d 3 p, in momentum space can be taen to be the diffeential volume between shells of adius p and p+dp. Hence page 204

20 dw(q,p) = dw(,v) = g h 3 4πp2 dpdv = g h 3 2π(2m)3/2 1/2 d dv = 2πg 2m 2 d dv h 2 The numbe of states that have enegies between and +d in the volume, V, is whee ( ) is the density of states The chemical potential 3 dw( ) = ( ) d = 2πVg 2m 2 d Now conside a gas of identical, almost-fee paticles in a containe of fixed volume, V, at tempeatue, T. Even though the paticles have at most a wea physical inteaction the quantum inteaction equies that they obey quantum statistics. The expected numbe of paticles in a state that has enegy,, is h N( ) = e ( -µ) ± whee the uppe sign petains to femions and the lowe sign to bosons. It follows that the expected numbe of paticles of the gas that have enegies between and +d is dn = N( ) ( )d 3 2 = 2πVg 2m h 2 d e ( -µ) ± The total numbe of paticles in the gas is obtained by integating equation ove all possible values of. Since the inetic enegy of a fee paticle anges fom 0 to, 3 N = 2πVg 2m h d e ( -µ) ± 1 3 = 2πVg 2mT h zdz e (z- µ) ± page 205

21 whee z = ( ). If the gas is contained in an open system of volume V equation is an equation fo the expected numbe of paticles fo given values of µ and T: N(µ,V,T). If the gas is in an isothemal system with volume V and paticle numbe N equation is an equation fo the chemical potential, µ, as a function of N, V and T: µ(n,v,t) The wo function It follows fom equations and that the wo function of a quantum gas of elementay paticles is (T,V,µ) = - ±T 0 ln[ 1 ± e (µ- ) ] N( ) ( )d = - ±2πVTg 2m h ln[ 1 ± e (µ- ) ] d The integal can be cast into an altenate fom though integation by pats: /2 e ( -µ) ± 1 d ln[ 1 ± e (µ- ) ] d = 2 3 ln[ 3/2 1 ± e (µ- ) ] 0 ± = ± /2 e ( -µ) ± 1 d since the baceted tem on the ight-hand side vanishes at both limits. Substituting into equation 12.87, 3 (T,V,µ) = - 4πVg 3 2m h This equation can be witten in the altenate fom 3/2 e ( -µ) ± 1 d (T,V,µ) = - 4πVT5/2 g 3 2m h z 3/2 e z- µ ± 1 dz = VT 5/2 Ï µ T page 206

22 whee Ï[ µ] is a function of the poduct ( µ) only The themal and caloic equations of state The themal equation of state of a gas of elementay paticles can be obtained diectly fom equation Using the elation we have = - PV P(T,V,µ) = 4πT5/2 g 3 2m h z 3/2 e z- µ ± 1 dz To obtain the themal equation of state, P = P(N,T,V), we must solve equation to evaluate µ(n,t,v). This is, in geneal, a difficult mathematical poblem. We can, howeve, show that the themal equation of state educes to the ideal gas equation in the Boltzmann limit, as it should. In the Boltzmann limit, µ << 0. Then z 3/2 0 e z- µ ± 1 d «0 z 3/2 e µ-z d = e µ 0 z 3/2 e -z dz = 3 4 π e µ whee the definite integal is evaluated fom standad integal tables. Theefoe, in the Boltzmann limit P = gt 2πmT 2 e µ h 2 The paticle numbe is, fom equation N = 2πVg 2mT h zdz e (z- µ) ± 1 = 2πVg 2mT h 2 2 e µ 0 3 = 2πVg 2mT 2 e µ 1 2 h 2 3 z e -z π dz page 207

23 Compaing the two equations, = gv 2πmT 2 e µ h 2 3 PV = NT which shows that the equation of state fo a gas of fee quantum paticles educes to that fo an ideal gas as the chemical potential appoaches the Boltzmann limit. The caloic equation of state is found though the same pocedue. The intenal enegy of the gas can be calculated diectly fom the distibution, dn, E(T,V,µ) = 0 N( ) ( )d = 2πVg 2m h /2 d 0 e ( -µ) ± 1 = (T,V,µ) which, again, must be solved in conjunction with equation to find the caloic equation of state, E(T,V,N). Howeve, since = -PV equation poves the impotant geneal theoem: fo all gases of nealy fee paticles the intenal enegy obeys the simple elation E = 3 2 PV In the case of a Boltzmann gas equation shows that PV = NT, hence E = 3 2 NT which is the appopiate caloic equation of state. Equation holds in geneal fo wealy inteacting gases. Equation is esticted to cases in which the quantum inteaction is negligible The entopy; sufaces of constant entopy Seveal othe impotant elations that goven the behavio of systems of elementay paticles can be infeed diectly fom equation The entopy is page 208

24 S = - T = 5 2 VT3/2 Ï( µ ) - VT 3/2 ( µ) ( µ) [Ï( µ)] and the paticle numbe is = VT 3/2 5 2 Ï( µ) - ( µ) ( µ) [Ï( µ)] N = - µ = VT 3/2 ( µ) [Ï( µ)] Hence, S = Nƒ( µ) whee the function ƒ depends on the poduct ( µ) only. It follows fom equations that fo an adiabatic pocess in a gas of elementay paticles with fixed paticle numbe, N µ T = const VT 3/2 = const PT -5/2 = const PV 5/3 = const whee the values of the constants depend on the quotient (S/N). In the case of a dilute gas of classical paticles equations hold, and ae usually deived as a consequence of the paticula values of the isometic and isobaic specific heats, C V and C P. The theoem stated above shows that these elations hold in geneal fo gases of elementay paticles that have a wea physical inteaction, even when the specific heats ae not constant and have no simple elation to one anothe. page 209