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Chapter 13 Ideal Fermi Chapter 13 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: k T µ,βµ 1, B � � which defines the degenerate Fermi gas. In this limit, the quantum mechanical nature of the system becomes especially important, and the system has little to do with the classical ideal gas. Since this chapter is devoted to fermions, we shall omit in the following the subscript ( ) that we used for the fermionic statistical quantities in the previous chapter. − 13.1 Equation of state Consider a gas ofN non-interacting fermions, e.g., electrons, whose one-particle wave- functionsϕ r(�r) are plane-waves. In this case, a complete set of quantum numbersr is given, for instance, by the three cartesian components of the wave vector �k and thez spin projectionm s of an electron: r (k , k , k , m ). ≡ x y z s Spin-independent Hamiltonians. We will consider only spin independent Hamiltonian operator of the type ˆ 3 H= �k ck† ck + d r V(r)c r†cr , �k � where thefirst and the second terms are respectively the kinetic and th potential energy. The summation over the statesr (whenever it has to be performed) can then be reduced to the summation over states with different wavevectork(p=¯hk): ... (2s + 1) ..., ⇒ r � �k where the summation over the spin quantum numberm s = s, s+1, . , s has been taken into account by the prefactor (2s + 1). − − 159 160 CHAPTER 13. IDEAL FERMI GAS Wavefunctions in a box. We as- sume that the electrons are in a vol- ume defined by a cube with sidesL x, Ly,L z and volumeV=L xLyLz. For the one-particle wavefunction 1 ik r r k =ψ k(r) = e · � | � √V we use periodicity condition, here along thex-direction, at the cube’s walls, eikxx =e ikxx+ikxLx , which is then translated into a condition for the allowedk-values: ikxLx i2πnx 2π e =e , kx = nx , nx Z. Lx ∈ Analogously for they- and for thez direction. Summation over wavevectors. Each state has ink-space an average volume of (2π)3 (2π)3 Δk= = . (13.1) LxLyLz V For largeV we can then replace the sum over all quantum number by →∞ r 1 � V (2s + 1) d3k = (2s + 1) d3k → Δk (2π)3 r � � � V = (2s + 1) d3p , (13.2) h3 � wherek= p/¯h has been used. 1 1 The factor h ( h3N forN particles) introduced “ad hoc” in clas- sical statistical physics in Sect. 8.2 appears naturally in quan- tum statistical physics. It is a direct consequence of the fact that particles correspond to wavefunctions. 13.1.1 Grand canonical potential We consider now the expression (12.36) for the fermionic grand canonical potentialΩ(T, V, µ) that we derived in Sect. 12.5, β(�r µ) Ω(T, V, µ) = k T ln 1 +e − − . − B r � � � 13.1. EQUATION OF STATE 161 Using the substitution (13.2) and d3k=4π k2dk we write the grand canonical potential as � V �∞ 2 β¯h2k2/(2m) − βΩ(T, V, µ) = (2s + 1) 3 4π dk k ln 1 +ze , (13.3) − (2π) 0 � � � where we used the usual expressions ¯h2k2 z=e βµ,� � = r → k 2m for the fugacityz and for the one-particle dispersion and used an explicit expression for the one-particle energies for free electrons� k. Dimensionless variables. Expression (13.3) is transformed further by introducing with β 2m 3/2 x=¯hk , k2dk= x2dx 2m β¯h2 � � � a dimensionless variablex. One obtains 3/2 4V m ∞ 2 x2 βΩ(T, V, µ) = (2s + 1) 2 x dx ln 1 +ze − . − √π 2πβ¯h 0 � � � � � De Broglie wavelengths. By making use of the definition of the thermal de Broglie wavelengthλ, 2πβ¯h2 λ= , � m we then get (2s + 1) 4V ∞ 2 x2 − βΩ(T, V, µ) = 3 dx x ln 1 +ze . (13.4) − λ √π 0 � � � Term by term integration. We use the Taylor expansion of the logarithm, ∞ yn ln(1 +y) = ( 1)n+1 , − n n=1 � in order to evaluate the integral ∞ n ∞ 2 x2 n+1 z ∞ 2 nx2 x dx ln 1 +ze − = ( 1) dx x e− − n 0 n=1 0 � � � � � ∞ n n+1 z d ∞ nx2 = ( 1) dx e− − n −dn n=1 0 � � � � ∞ zn d 1 1 = ( 1)n+1 √π − n −dn 2 √n n=1 � � � 162 CHAPTER 13. IDEAL FERMI GAS term by term. The result is ∞ n ∞ 2 x2 √π n+1 z x dx ln 1 +ze − = ( 1) . 4 − n5/2 0 n=1 � � � � Grand canonical potential. Defining ∞ n n+1 z 4 ∞ 2 x2 f (z) = ( 1) = dx x ln 1 +ze − (13.5) 5/2 − n5/2 √π n=1 0 � � � � we obtain 2s+1 βΩ(T, V, µ) = V f (z) (13.6) − λ3 5/2 for the grand canonical potential for an ideal Fermi gas. Pressure. Our result (13.6) reduces withΩ= PV to − P s π h2 2 +1 2 ¯ µ/(kB T) = 3 f5/2(z) ,λ= , z=e , (13.7) kBT λ �kBT m which yields the pressureP=P(T, µ). Density. With PV Ω= k BT ln = PV = ln − Z − kBT Z wefind, compare Eq. ( 10.14), ∂ ∂ P Nˆ =z ln =Vz (13.8) � � ∂z Z ∂z k T � �T,V � B �T,V for the number of particlesN. The densityn(T, µ) = Nˆ /V is then given by � � Nˆ 2s+1 n= � � = f (z) . (13.9) V λ3(T) 3/2 where we have defined d ∞ zn f (z) =z f (z) = ( 1)n+1 (13.10) 3/2 dz 5/2 − n3/2 n=1 � Thermal equation of state. As a matter of principle one could solve (13.9) for the fugacityz=z(T, n), which could then be used to substitute the fugacity in ( 13.7) forT andn, yielding such the thermal equation of state. This procedure can however not be performed in closed form. 13.2. CLASSICAL LIMIT 163 Rewriting the particle density. The functionf 3/2(z) entering the expression (13.9) for the particle density may be cast into a different form. Helping is here the result (13.7) forP(T, µ): P 2s+1 λ3 ln PV = 3 f5/2(z), f5/2(z) = Z , = ln , kBT λ 2s+1 V kBT Z which leads to d d λ3 ln f (z) =z f (z) =z Z 3/2 dz 5/2 dz 2s+1 V � � With d d dβ 1 d lnz = = ,β= (13.11) dz dβ dz µz dβ µ � � andλ= (2πβ¯h2)/m we thenfind � d 3λ3 d µV (2s + 1)f (z) = λ3 ln = ln +λ 3 ln . 3/2 dβ Z 2β Z dβ Z � � For the particle density (13.9) wefinally obtain 2s+1 3 d n= f (z), µnV= PV+ ln , nV=N, (13.12) λ3(T) 3/2 2 dβ Z where we have used ln /β=PV. Z Caloric equation of state. The expression for µnV= µN in ( 13.12) leads with d β(�r µ) U= ln +µ Nˆ , = e− − . − dβ Z � � Z r � to the caloric equation of state 3 U= PV . (13.13) 2 The equationU=3PV/2 is also valid for the classical ideal gas, as discussed in Sect. 8.2, but it is not anymore valid for relativistic fermions. 13.2 Classical limit Starting from the general formulas (13.7) forP(T, µ) and ( 13.9) forn(T, µ), wefirst investigate the classical limit (i.e. the non-degenerate Fermi gas), which corresponds, as discussed in Chap. 11, to nλ3 1 , z=e βµ 1. � � 164 CHAPTER 13. IDEAL FERMI GAS Under this condition, the Fermi-Dirac distribution function reduces to the Maxwell- Boltzmann distribution function: 1 β�r ˆnr = ze− . 1 β�r � � z− e + 1 ≈ Expansion in the fugacity. For a small fugacityz we may retain in the series expansion forf 5/2(z) andf 3/2(z), compare (13.5) and (13.10), thefirst terms: z2 3 z f5/2(z) z 5/2 βPλ (2s + 1)z 1 5/2 ≈ − 2 ≈ − 2 (13.14) z2 3 z f3/2(z) z 3/2 nλ (2s + 1)z �1 3/2 � ≈ − 2 ≈ − 2 where we have used (13.7) and (13.9) respectively. � � High-temperature limit. The expression fornλ 3 in (13.14) reduces in lowest approxi- mation to nλ3 z(0) , nλ3 1. (13.15) ≈ 2s+1 � The number of particlesnλ 3 in the volume spanned by the Broglie wavelengthλ 1/ √T is hence small are small. This is the case at elevated temperatures. ∼ Fugacity expansion. Expanding in the fugacityz = exp(βµ) 3 (0) nλ 1 z (1) (0) (0) 3/2 z , z z 1 +z 2− , ≈ 2s+1 1 z2 3/2 ≈ 1 z (0)2 3/2 ≈ − − − − z(0) � � where 1/(1 �x)�� 1+� x forx 1 was used. The equation of state ( 13.14) for the − ≈ 3 � 2 5/2 pressure, namelyβPλ (2s + 1)(z z 2− ), is then ≈ − 3 (0) (0) 3/2 5/2 (0) 2 βPλ (2s + 1) z 1 +z 2− 2 − (z ) ≈ 3 − 3 5/2 nλ =nλ 1 +� 2− � , � � 2s+1 � � when 1 1 2 1 1 3 5 = 5 5 = 5 √2 − √2 √2 − √2 √2 is used. Altogether we thenfind nλ3 PV= Nˆ k BT 1 + . (13.16) � � 4√2(2s + 1) � � In this expression, thefirst term corresponds to the equation of state for the classical ideal gas, while the second term is thefirst quantum mechanical correction. 13.3. DEGENERATED FERMI GAS 165 13.3 Degenerated Fermi gas In the low temperature limit,T 0, the Fermi distribution function behaves like a step function: → 1 T 0 0 if� k > µ → nk = β(� µ) e k + 1 −−−→ 1 if� k < µ − � i.e., lim nk =θ(µ � k).
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