50 Production of Low and Ultralow Temperatures; Third Law 375 At ultralow temperatures of this order of magnitude the phenomenon of Bose-Einstein condensation comes into play (see Sect. 53.3). In this case the third law only plays a part at even lower temperatures where quantization of the translational energy would become noticeable, i.e., at temperatures k B T < ∼ 2 2M eff R 2 , where R describes the size scale of the sample and M eff may be the mass of a Na atom in the normal state above the Bose condensation (or in the condensed phase: an effective mass). Anyhow, we should remind ourselves that at these low temperatures and the corresponding low particle numbers (some 10 4 to 10 5 instead of 10 23 ) one should work with the microcanonical ensemble, not the canonical or grand canonical ones. However one should also remember that the basic temperature definition M · v 2 s =3kT is also valid for microcanonical ensembles. 51 General Statistical Physics (Formal Completion; the Statistical Operator; Trace Formalism) Is it really necessary to diagonalize the Hamilton operator ˆ H of the system ( ˆ H → E j ), if one “just” wants to calculate the partition function Z(T )= j e − E j k B T of the system and obtain the thermodynamic potentials or thermal expecta- tion values ˆ A T = j e − E j k B T Z(T ) ψ j | ˆ Aψ j ? The answer to this rhetorical question is of course negative. Instead of di- agonalizing the Hamiltonian we can make use of the so-called trace formalism. This approach is based on the definition: trace ˆ A := j ψ j | ˆ Aψ j , which is valid for every complete orthonormal 1 basis. It is easy to show that the expression on the r.h.s. of this equation, the sum of the diagonal elements of the matrix A i,j := ψ i | ˆ Aψ j , is invariant with respect to a base change. It therefore follows, for example, that Z(T )=tracee −β ˆ H , where e −β ˆ H is the operator, which has in the base of the eigenfunctions of ˆ H amatrix representation with diagonal elements e −βE j . (In another base it can also be defined by the power series ∞ n=0 (−β) n n! ˆ H n .) 1 Orthonormality is not even necessary. On the other hand, operators for which the trace exists, belong to a class of their own (trace class). 378 51 General Statistical Physics (Statistical Operator; Trace Formalism) In addition there is a Hermitian density operator ˆ, also called state oper- ator or statistical operator, whose eigenvalues are just the probabilities p j (ψ j are the corresponding eigenstates). For example, one can write the relation S k B = − j p j ln p j abstractly as S k B = −trace{ˆ ln ˆ} , or even more abstractly: S k B = −ln ˆ ˆ , just as one also uses, instead of the formula ˆ A := j p j ψ j | ˆ Aψ j , the more abstract formula ˆ A ˆ := traceˆ ˆ A. However, one must realize that for this additionally gained freedom of avoid- ing diagonalizing the Hamiltonian there is the penalty of more complicated calculations 2 . For example, it is easy to calculate the partition function if one has already diagonalized ˆ H, whereas without diagonalization of it, calcula- tion of ˆ := e −β ˆ H tracee −β ˆ H becomes very difficult. Indeed, the trace of a matrix product involves a double sum, e.g., traceˆ ˆ A = j,k j,k A k,j . 2 Atypeof“conservation law for effort” holds here. 52 Ideal Bose and Fermi Gases In the following section we shall consider identical particles, such as elemen- tary particles and compound particles, as well as quasi-particles which are similar to light quanta (photons, electromagnetic waves): e.g., sound quanta (phonons, elastic waves) and spin-wave quanta (magnons). These particles or quasi-particles are either a) fermions (particles or quasi-particles with spin s =1/2, 3/2, in units of ), such as electrons, protons, neutrons and He 3 ,aswellasquarks,from which nucleons are formed (nucleons are compound particles made up of three quarks), or b) bosons (particles or quasi-particles with spin s =0, 1, 2, in units of ), such as, for example, the pion, which is an elementary particle of rest mass m 0 ≈ 273 MeV/c 2 consisting of two quarks; or a He 4 particle; or one of the above-named quasi-particles which all possess zero rest mass and, as a result, vanishing chemical potential. Fermionic quasi-particles also exist in solids. For example, in polar semi- conductors there are the so-called polarons, which are electrons accompanied by an attached phonon cloud. This fermionic quasi-particle possesses a non- negligible rest mass. If these particles or quasi-particles do not interact with each other (or only interact weakly), the energy levels are given by: E n 1 ,n 2 , = n max n 1 =0 n max n f =0 (n 1 ε 1 + n 2 ε 2 + ) , and the number of particles is: N n 1 ,n 2 , = n max n 1 =0 n max n f =0 (n 1 + n 2 + ) . There are thus n 1 particles or quasi-particles in single-particle states of en- ergy ε 1 ,etc.Forfermions,n max = 1, while for bosons, n max = ∞.These statements are fundamental to quantum mechanics (viz Pauli’s exclusion principle). In addition, if there is no particle interaction (i.e., in a dilute 380 52 Ideal Bose and Fermi Gases Bose or Fermi gas), the partition function can be factorized as follows: Z tot (β,μ)=Z 1 (β,μ) ·Z 2 (β,μ) · ·Z f (β,μ) · , where β and μ are reciprocal temperature and chemical potential which de- termine the mean values of energy and particle number respectively. It is therefore sufficient to calculate the partition function for a single factor, i.e., a single one-particle level. For example, Z f (β,μ):= n max n f =0 e −β·(ε f −μ)n f . For fermions the sum consists of only two terms (n max = 1); a convergence problem does not arise here. For bosons on the other hand, since n max is without an upper limit 1 , there is an infinite geometric series, which converges if μ<ε f 2 . In both cases it then follows that Z f (β,μ)= 1 ± e −β(ε f −μ) ±1 . (52.1) For the ±-terms we have a plus sign for fermions and a minus sign for bosons. Only a simple calculation is now required to determine ln Z f and the expec- tation value n f T := d d(βμ) ln Z f . One then obtains the fundamental expression n f T,μ = 1 e β(ε f −μ) ± 1 , (52.2) where the plus and minus signs refer to fermions and bosons respectively. For a given temperature T = 1 k B β and average particle number N, the chemical potential μ is determined from the auxiliary condition: N = j n j T,μ , 1 An analogy from everyday life: With regard to the problem of buying a dress, French women are essentially fermionically inclined, because no two French women would buy the same dress, irrespective of the cost. On the other hand, German women are bosonically inclined, since they would all buy the same dress, provided it is the least expensive. 2 The limiting case μ → 0 − is treated below in the subsection on Bose-Einstein condensation. 52 Ideal Bose and Fermi Gases 381 as long as for all j we have ε j <μ. (In the boson case, generally μ ≤ 0, if the lowest single-particle energy is zero. As already mentioned, the case μ =0is treated below in the chapter on Bose-Einstein condensation.) The classical result of Boltzmann statistics is obtained in (52.2) when the exponential term dominates the denominator, i.e., formally by replacing the term ±1 by zero. It is often stated that different statistics are required for fermions as opposed to bosons or classical particles, but this is not really the case, since the derivation of (52.2) is made entirely within the framework of the grand canonical Boltzmann-Gibbs statistics, and everything is derived together until the difference between fermions and bosons is finally expressed by the value of n max prescribed by the Pauli principle (see above), which depends on the fact that the spin angular momentum (in units of )isan integer for bosons (→ n max = ∞)andahalf-integer for fermions (→ n max = 1). In this respect the (non-classical) property of spin is crucial. (We have already seen in quantum mechanics how the Pauli principle is responsible for atomic structure and the periodic table of elements.) It is important to make this clear in school and undergraduate university physics and not to disguise the difficulties in the theory. 3 3 e.g., one should mention that spin with all its unusual properties is a consequence of relativistic quantum theory and that one does not even expect a graduate physicist to be able to understand it fully. 53 Applications I: Fermions, Bosons, Condensation Phenomena In the following sections we shall consider several applications of phenomeno- logical thermodynamics and statistical physics; firstly the Sommerfeld theory of electrons in metals as an important application of the Fermi gas formal- ism, see [40]. Actually, we are not dealing here with a dilute Fermi gas, as prescribed by the above introduction, but at best with a Fermi liquid,since the particle separations are as small as in a typical liquid metal. However, the essential aspect of the formalism of the previous chapter – which is that interactions between particles can be neglected – is still valid to a good ap- proximation, because electrons avoid each other due to the Pauli principle. As a result, Coulomb interactions are normally relatively unimportant, as long as the possibility of avoidance is not prevented, e.g., in a transverse direction or in d = 1 dimension or by a magnetic field. 53.1 Electrons in Metals (Sommerfeld Formalism) a) The internal energy U(T,V,N)ofsuchanelectronsystemcanbewritten U(T,V,N)= ∞ 0 dε · ε ·g(ε) ·n(ε) T,μ , (53.1) where g(ε) is the single-particle density; furthermore, dεg(ε)isequalto the number of single-particle energies ε f with values in the interval dε (i.e., this quantity is ∝ V ). b) Similarly, for the number of particles N: N = ∞ 0 dεg(ε) ·n(ε) T,μ . (53.2) The value of the chemical potential at T = 0 is usually referred to as the Fermi energy ε F , i.e. μ(T =0)=ε F . Depending on whether we are dealing at T =0withanon-relativistic electron gas or an ultrarelativistic electron gas (ε F m e c 2 or m e c 2 ,wherem e ≈ 0.5MeV/c 2 is the electron mass), 384 53 Applications I: Fermions, Bosons, Condensation Phenomena we have from the Bernoulli pressure formula either p = 2U 3V or p = U 3V . For electrons in metals, typically ε F = O(5) eV, so that at room temperature we are dealing with the non-relativistic case 1 . An approximation attributable to Sommerfeld, [40], will now be described. Firstly, n(ε) T,μ for T =0 is given by a step function, i.e. n(ε) T →0,μ =0 for ε>ε F and = 1 for all ε<ε F (neglecting exponentially small errors). Furthermore we can write the inte- grals (53.1) and (53.2), again neglecting exponentially small errors, in the form ∞ 0 dε dF dε ·n(ε) T,μ , where F (ε) are stem functions, F (ε)= ε 0 dεf(ε) , of the factors f(ε):=ε ·g(ε)andf(ε):=g(ε) appearing in the integrands of equations (53.1) and (53.2). Compared to n(ε) T,μ , a function whose negative slope behaves in the vicinity of ε = μ as a (slightly smoothed) Dirac δ function: − d dε n(ε) T,μ ≡ 1 4k B T · cosh ε−μ 2k B T 2 (≈ δ(ε −μ)) , the functions F (ε), including their derivatives, can be regarded at ε ≈ μ as approximately constant. On partial integration one then obtains 2 : ∞ 0 dε dF dε ·n(ε) T,μ = ∞ 0 dεF (ε) · − d dε n(ε) T,μ , 1 The next chapter considers ultrarelativistic applications. 2 The contributions which have been integrated out disappear, since F (0) = 0, whereas n(∞) T,μ =0. 53.1 Electrons in Metals (Sommerfeld Formalism) 385 where one inserts for F (ε) the Taylor expansion F (ε)=F(μ)+(ε − μ) · F (μ)+ (ε − μ) 2 2 ·F (μ)+ . On integration, the second, odd term gives zero (again neglecting exponen- tially small terms O(e −βε F )). The third gives π 2 6 (k B T ) 2 ·F , so that, for example, from (53.2) the result N = μ 0 dεg(ε)+ π 2 (k B T ) 2 6 g (μ)+ follows, where as usual the terms denoted by dots are negligible. The integral gives μ 0 dεg(ε)=N +(μ(T ) −ε F ) · g(ε F )+ , so that: μ(T )=ε F − π 2 (k B T ) 2 6 · g (ε F ) g(ε F ) + . From (53.1) we also obtain U(T,V,N)= μ 0 dεεg(ε)+ π 2 (k B T ) 2 6 · [εg(ε)] |ε=μ + . Inserting the result for μ(T ), after a short calculation we thus obtain (with U 0 := ε F 0 dεε · g(ε)): U(T,V,N)=U 0 + π 2 (k B T ) 2 6 g(ε F )+ . By differentiating U with respect to T it follows that electrons in a metal give a contribution to the heat capacity: C V = ∂U ∂T = γk B T, which is linear in T and where the coefficient γ is proportional to the density of states at the Fermi energy ε F : γ ∝ g(ε F ) . 386 53 Applications I: Fermions, Bosons, Condensation Phenomena For free electrons g(ε) ∝ ε 1 2 , so that g (ε F ) g(ε F ) = 1 2ε 2 F , and μ(T )=ε F · 1 − π 2 (k B T ) 2 12(ε F ) 2 + , with negligible terms +. . . . This corresponds to quite a small reduction in μ(T ) with increasing temperature T . This is quite small because at room temperature the ratio k B T ε F 2 itself is only of the order of (10 −2 ) 2 ,asε F ≈ 3 eVcorresponds to a temperature of 3·10 4 K (i.e., hundred times larger than room temperature). In some metals such as Ni and Pd, (k B T ) 2 g g is indeed of the same order of magnitude, but in these metals g (ε F )has a negative sign – in contrast to the case of free electrons – so that at room temperature μ(T ) is here slightly larger than at T =0.(However,insome compounds γ is larger by several orders of magnitude than usual (even at zero temperature) so that one speaks of “heavy fermions” in these compounds.) Electrons in conventional metals behave at room temperature (or generally for (k B T ) 2 ε 2 F ) as a so-called degenerate Fermi gas. The essential results of the previous paragraphs, apart from factors of the order of O(1), can be obtained by adopting the following simplified picture: Only the small fraction k B T ε F of electrons with energies around ε = μ ≈ ε F are at all thermally active. Thus, multiplying the classical result for the heat capacity C V = 3 2 Nk B by k B T ε F , we obtain the above linear dependence of the heat capacity on temperature for electrons in a metal, apart from factors of the order of O(1). In particular g(ε F ) can be approximated by N ε F . In order to calculate the zero-point energy U 0 one must be somewhat more careful. Indeed, U 0 = ε F 0 dεε · g(ε) . . a class of their own (trace class). 378 51 General Statistical Physics (Statistical Operator; Trace Formalism) In addition there is a Hermitian density operator ˆ, also called state oper- ator. section we shall consider identical particles, such as elemen- tary particles and compound particles, as well as quasi-particles which are similar to light quanta (photons, electromagnetic waves):. metals as an important application of the Fermi gas formal- ism, see [40]. Actually, we are not dealing here with a dilute Fermi gas, as prescribed by the above introduction, but at best with a