Abstract

An energy scale of the superconducting condensate, which we call the effective Fermi temperature TF, can be derived from the magnetic field penetration depth λ determined by muon spin relaxation (μSR) measurements. We classify various superconductors in the crossover from Bose–Einstein (BE) to BCS condensation, based on the ratio Tc/TF. The phase diagram of high- Tc cuprate superconductors, as a function of carrier doping, can be understood in the context of this BE–BCS crossover, if we identify the ‘pseudo-gap’ as the signature of pair formation in the normal state. In particular, the universal linear relation between Tc and ns/m* (superconducting carrier density/effective mass), found in the underdoped region, comes from a general feature expected in the BE condensation of pre-formed pairs. The optimal Tc occurs around the doping concentration at which the condensate energy scale TF becomes comparable to the energy scale hωB of the pair-mediating interaction. A surprising decrease of ns/m* with increasing carrier doping was found in the overdoped Tl 2201 system. This behavior suggests that evolution to the BCS region does not occur in a simple way, but rather is associated with a possible microscopic separation between superconducting and residual normal metallic phases.

During the past several years, we have performed measurements of the magnetic field penetration depth λ of high- Tc cuprate and other superconducting systems using the muon spin relaxation (μSR) technique [1–5].

Abstract

An analytically solvable Ansatz set of Migdal-type self-consistent equations is proposed for the coupling between spin and charge degrees of freedom in the strongly correlated electron system described by the t–t′–J(t–J) Hamiltonian. The small parameter validating Migdal's approximation for this problem is found to be 1/ln(U/t) (when U»t), where U and t are on-site Coulomb repulsion energy and the bare electron hopping integral of the basic Hubbard Hamiltonian (t′ = J =4t2/U). The analytical results, obtained for electron concentrations close to half-filling, demonstrate strong enhancement of the quasi-particle mass, accompanied by a depletion of the particle density of states at the Fermi-level (EF). The spectral density is pushed away from EF into a broad range of energies (of order t) and possesses a sawtooth form. Theoretical predictions for the optical conductivity are derived, which could provide a qualitative explanation of the mid-infrared anomaly observed experimentally in high-Tc cuprates. The variation in quasi-particle energy over the Brillouin zone is found to be of order J only, in good accord with previous theoretical work and the dispersion observed in recent ARPES experiments in 2212 high- Tc compounds.

Introduction and summary of our previous work

It is by now generally accepted that strong Coulomb correlations should play an essential role in the physics of the high-temperature superconductors (HTS). A great number of experimental results obtained since the discovery [1] of the HTSs raise strong doubts about the applicability of ‘classical’ BCS theory to the description of the superconductivity phenomena in these compounds.

Abstract

The spectral function and momentum distribution for holes in an antiferromagnet are calculated on the basis of the t–J model in a slave-fermion representation. The self-consistent Born approximation for a two-time Green function is used to study the dependences on temperature and doping (δ) of the self-energy operator. The numerical calculations show weak dependences on concentration and temperature of the spectral function (quasi-particle hole spectrum) while the momentum distribution function changes dramatically with increasing temperature for T>Td with Td⋍Jδ.

Introduction

The problem of hole motion in an antiferromagnetic (AF) background has attracted much attention in recent years. That is mainly due to the hope of elucidating the nature of the carriers involved in high-Tc superconductivity in copper oxides. It is believed that the essential features of the problem are described by the t–J model with a Hamiltonian written as

Here 〈ij〉 indicates nearest-neighbor pairs, c+iσ = c+iσ (1 − ni − σ) are the electron operators with the constraint of no double occupancy. The properties of a single hole doped in the Néel spin background have been analyzed intensively with various numerical and analytical methods. Among them are exact diagonalization of small clusters [1] and variational calculations [2]. A rather transparent description within a ‘string’ picture has been developed by several authors [3]. A perturbative approach to the problem was proposed by Schmitt–Rink, Varma and Ruckenstein [4] and developed further by Kane, Lee and Read [5] and Martinez and Horsch [6].

Abstract

Two-polaron states on a square lattice are studied in the presence of on-site repulsion U and inter-site attraction V. In the limit of infinite U the exact critical value Vcr for bipolaron formation is obtained as a function of the attraction radius R. The results are compared with the continuum limit of the same model. It is shown that if R≃(2–3) lattice constants then Vcr is of the order of the characteristic phonon frequency in the high-temperature superconductors.

The temperature-dependence of the upper critical field [1,2], the resistivity and Seebeck coefficient [3], and the universal correlation between the critical temperature and the hole content [4] in the p-type oxide superconductors unambiguously support the validity of the local pair conception for these compounds at low doping 0.06≤n≤0.12. Phonons are the most natural candidates for the bosonic field whose interaction with the carriers (polarons) results in the effective interpolaron attraction. However, there are several arguments against the phonon pairing mechanism. One of them is that the phonon-mediated attraction between polarons is much weaker than the shortrange Coulomb repulsion, hence creation of local pairs is inhibited. The typical estimates for the on-site copper, copper–oxygen, and inter-site copper Coulomb potentials are Udd≃10 eV, Upd≃1 eV and U′dd≃Q.l eV correspondingly [5, 6]. Since the typical phonon frequency ω is of the order of 0.1 eV or less, ω » U/dd, Upd and ω≃U′dd and therefore the existence of the local pair in which the polarons are localized on the same lattice site or on the nearest neighbours is impossible.

Abstract

The spectrum of collective (pair) excitations in the ground state of a dilute twodimensional (2D) attractive Fermi-gas is studied within the functional integral formalism. The linearized equations for the fluctuations about the non-trivial saddle point are analyzed for all coupling regimes, which are characterized by the ratio ε0/εF, where ε0 is the two-fermion binding energy and εF is the Fermi energy. The approximation takes into account propagation of the fluctuations and their interaction with the condensate. In the strong-coupling, or ‘;Bosegas’, regime (ε0/εF> 1) the spectrum is continuous and has the Bogolubov form, but in the weak-coupling limit (ε0/εF«1) there are two types of excitations (different from the two-fermion scattering states): (i) long-wavelength sound-like excitations with the cut-off at momentum qc≃ l/ξ0 (where ξ0 is the Cooper pair size), and (ii) pair excitations with q≃ (8mμ)½, where μ is the chemical potential and m is the fermion mass. The crossover between weakand strong-coupling behavior of the excitation spectrum is found to occur at the value of the coupling parameter ε0/εF≃¼.

Introduction

Since the discovery of high-Tc superconductivity there has been growing interest in studying 2D Fermi gases with attractive interaction, especially in the crossover regime, when the pair size ξ0 is of the order of the interparticle distance. The importance of such a model, which can be regarded as a semiphenomenological model of a 2D superconductor, is highlighted by experimental evidence that the high- Tc superconductors, most of which have a layered structure, have a short coherence length, so that kFξ0≃1, where kF is the Fermi momentum.

Abstract

We report on experimental evidence for different electronic phases in the superconductor YBa2Cu4O8: simple metallic Cu–O chains and highly correlated CuO2 planes. YBa2Cu4O8 is a genuine untwinned compound; hence, we were able to determine the anisotropy of the resistivity along the a and b directions. Along the b direction (chain and plane conduction channels), a normal metallic temperature behavior of the chain dominates. For the a direction (only the plane conduction channel), the resistivity reveals unconventional behavior with a kink at 160 K, becoming linear at higher temperatures. Further, we present results of the dynamical (optical) conductivity of the CuO2 plane as a function of frequency and temperature. The frequency-dependence of the optical conductivity is consistent with a model of ferromagnetic polarons in an antiferromagnetic matrix. This is further confirmed by a depolarization experiment, which is sensitive to crystal regions with different spin polarizations. The temperature behavior of the thermal occupation of the ground state is in agreement with a real-space condensation.

Introduction

From the beginning [1] the high-Tc superconductors (HTSC) have been very challenging systems. Regarding their solid state chemistry, the defect structure with its wide variability in stoichiometry gives rise to homogeneity problems: the samples may show a chemical phase separation of different oxygenated regions. So, proper preparation of homogeneous samples is crucial. Concerning the physics, the normal as well as the superconducting state show unusual features. In addition, as we will show at least for YBa2Cu3O7 (123) or YBa2Cu4O8 (124), one has to take into account the coexistence of different physical phases.

Abstract

Using picosecond pulses we excite a large number of carriers in YBa2Cu3O7−δ and drive the system through an insulator-to-metal transition. As we do this, we study the carrier interaction with the apical O and the 340 cm−1 planar O buckling vibrations by counting the number of non-equilibrium phonons that the carriers generate as they relax towards equilibrium. Counting is done directly by photoexcited anti-Stokes Raman scattering (PEARS). Carrier transport is found to be thermally activated; presumably the carriers relax by hopping between localized states. The in-plane and chain activation energies are found to be very different, suggesting that different carriers are involved: the chain carriers form polarons, which are strongly coupled to apical O vibrations, while the planar holes, which have a smaller activation energy, are suggested to interact less strongly with the lattice, especially in the metallic state.

Introduction

One may ask a very simple question about charge carrier transport in a cuprate superconductor: are the carriers moving through the crystal as in an extended band, or are they hopping from site to site? Given that the structure of these materials is composed of two distinct parts – the CuO2 planes plus charge reservoirs – one may perhaps also wonder whether the behaviour of carriers in the two parts is different. In order to answer such questions, we need a microscopic probe, which, it is to be hoped, might tell us something about both the type of interaction between the carriers and the symmetry and location of the interaction.

Abstract

The BSC theory is extended to strong electron–phonon coupling for λ > 1. In this limit carriers are charged 2e bosons (singlet and triplet inter-site bipolarons). The Anderson localisation of the bosons resulting from disorder is also considered. Several non Fermi-liquid features of copper-based high-Tc oxides, in particular the spin gap in NMR and neutron scattering, the temperature dependent Hall effect, linear resistivity and divergent Hc2(T) are explained.

The strong-coupling extension of the BCS theory

The electron–phonon coupling constant λ in the BCS theory is the ratio of the characteristic interaction energy V = 2Ep of carriers with a bosonic field, for instance of phonons, which is responsible for the coupling to their kinetic energy EF, λ ⋍ V/(2EF). At the point λ ⋍ 1 the characteristic potential energy due to the local lattice deformation exceeds the kinetic energy. This is a condition of small-polaron formation which has been known for a long time as a solution for a single electron on a lattice coupled with lattice vibrations. So long as λ > 1 the kinetic energy remains smaller than the interaction energy and a self-consistent treatment of a many-electron system strongly coupled with phonons is possible with the ‘1/λ’ expansion technique [1]. This possibility results from the fact, which has been known for a long time, that there is an exact solution for a single electron in the strong-coupling limit λ rarr; infin. Following Lang and Firsov (1962) one can apply the canonical transformation exp (S1) to diagonalise the single-electron Fröhlich Hamiltonian (under the ‘Fröhlich Hamiltonian’ we assume that any electron–phonon interaction occurs with its matrix element depending on the phonon momentum).

Abstract

In small-polaron models the hopping amplitude for a carrier from a site to a neighboring site is reduced due to ‘dressing’ by a background degree of freedom. Electron–hole symmetry is broken if this reduction is different for a carrier in a singly occupied site and one in a doubly occupied site. Assuming that the reduction is smaller in the latter case, the implication is that a gradual ‘undressing’ of the carriers takes place as the system is doped and the carrier concentration increases. A similar ‘undressing’ will occur at fixed (low) carrier concentration as the temperature is lowered, if the carriers pair below a critical temperature and as a result the ‘local’ carrier concentration increases (and the system becomes a superconductor). In both cases the ‘undressing’ can be seen in a transfer of spectral weight in the frequency-dependent conductivity from high frequencies (corresponding to non-diagonal transitions) to low frequencies (corresponding to diagonal transitions), as the carrier concentration increases or the temperature is lowered repectively. This experimental signature of electron–hole asymmetric polaronic superconductors as well as several others have been seen in high-temperature superconducting oxides. Other experimental signatures predicted by electron–hole-asymmetric polaron models remain to be tested.

The physics of high-Tc oxides

From the beginning of the high-Tc era there have been indications that small polarons may play an important role in the physics of these materials [1–10]. Among the workers that have not completely abandoned the Fermi liquid framework for this problem most would agree that the physics of the normal state may be described in terms of heavily dressed quasi-particles [11].

Abstract

Recent experimental findings concerning the temperature-dependence of the penetration depth and strong empirical evidence for a universal 3D X–Y critical behaviour in several classes of high- Tc superconductors suggest that Bose condensation of ‘weakly charged’ bosons can be a driving mechanism for the phase transition in extreme type II superconductors. We explore the occurence of analogous behaviour in a simple model of local electron pairs, which is that of hard-core charged bosons on a lattice. We examine the electromagnetic properties of the model in the superfiuid phase as a function of boson concentration and density–density interaction. In the low-density limit, with the use of the exact scattering length, we present new results for the ground state characteristics. The relevance of the obtained results to interpretation of experimental data on high-Tc oxides is discussed.

Introduction

Some recent experimental results concerning temperature-dependence of the London penetration depth and empirical evidence for a universal 3D X–Y critical behaviour in several families of high- Tc superconductors indicate that Bose condensation can be a driving mechanism for the phase transition in these extreme type II superconductors [1–3]. Motivated by these experimentally established universal trends, we examine the electromagnetic properties of the effective pseudo-spin model of the local pair superconductor, equivalent to a hard-core charged Bose gas on a lattice, for the case of arbitrary electron concentration n. From linear response theory, the current response is obtained and expressions for the paramagnetic and diamagnetic kernels are evaluated in the RPA (random phase approximation) and SWA (spin-wave approximation).

With the advent of high-temperature superconductors, research on polarons and bipolarons has gained renewed attention. It appears that carriers in some high-temperature superconductors are strongly correlated both in the normal and in the superconducting state. In the strong-coupling limit the Fermi-liquid ground state may be destroyed by the formation of small polarons and bipolarons. Experimental and theoretical analysis of such particles is a central issue for current research in superconductivity.

This book contains a series of authoritative articles on the most advanced research on polarons and bipolarons. They were invited for presentation during a workshop on ‘Polarons and Bipolarons in High-Tc Superconductors and Related Materials’, which was held at the Interdisciplinary Research Centre (IRC) in Superconductivity, Cambridge, UK, between 7th and 9th April 1994. Over 50 participants from ten countries took part in this workshop, representing the major research centres currently working in this field. The workshop was held in honour of Sir Nevill Mott in recognition of his important contributions to the physics of polarons.

The editors are grateful to all contributors for their overwhelmingly positive response to the idea of publishing this book. We are indebted to Ken Diffey and Margaret Hilton who ensured the smooth running of the workshop, to all of the referees and to William Beere for their help editing the manuscripts. We also thank Simon Capelin of Cambridge University Press for his encouragement and help throughout the publication of this book.

Abstract

We review a series of exact, analytical and numerical results obtained on the adiabatic Holstein–Hubbard model, many of which are new and non-trivial. We study next the role of the quantum lattice fluctuations that were initially neglected. The possibility of having high-Tc bipolaronic superconductors is analysed on the basis of these results.

We suggest that models that involve only electron–phonon coupling are very unlikely to produce bipolaronic superconductivity, which is prevented by the spatial ordering of the bipolarons associated with a lattice instability. This is due to a very large effective mass of the bipolarons that can be related to the large Peierls–Nabarro energy barrier required to move these bipolarons through the lattice in the adiabatic limit.

We conjecture that high-Tc superconductivity originates specifically from the exceptionally well-balanced competition between electron–phonon coupling and electron–electron repulsion. In the restricted region in which, within the mean field approach, the energy of a bipolaron is close to those of two polarons, a new type of electron pairing occurs by formation of pairs of polarons in the spin singlet state. Such a polaron pair, called a spin resonant bipolaron (SRB), is not the standard bipolaron (both could exist in the adiabatic case). Its Peierls–Nabarro energy barrier can be sharply depressed, almost to zero. As a result of quantum lattice fluctuations, its tunnelling energy is sharply enhanced. Then, superconductivity could persist for a relatively large electron–phonon coupling, with an unusually large critical temperature as a Bose condensate of SRBs before becoming, at larger coupling and in any case, an insulating spin–Peierls polaronic phase.

Abstract

The near infrared and optical absorption of several high-Tc superconductors is investigated using the KBr powder method. The spectra show broad NIR absorption peaks. Examples of absorption spectra obtained using transmission data of thin film work for Ba1−xKxBiO3 (BKB) and single-crystal measurements for Bi2Sr2Ca1−xYxCu2O8+δ (BSCC) are given for comparison. They confirm the results of the KBr spectra. An example of similar investigations on NiO also shows good agreement over the whole spectral range under investigation. These facts provide some evidence that the line profiles deduced by the KBr technique may indicate a correct absorption line shape. A possible explanation of the NIR line profile is discussed.

Introduction

The absorption of various high- Tc systems measured using Kubelka-Munk [1, 2] or standard KBr powder techniques [3–7] is known to possess broad peaks in the near infrared spectral range. Similar features have earlier been discussed in connection with a small-polaron absorption mechanism in the non-stoichiometric compounds of systems like TiO2−x [8], WO3−x [9], NbO2.5−x [10,11] and the ternary compounds NbxW1−xOy [12]. For the superconductors of the system YBa2Cu3O7−δ, La2CuO4+δ and Bi2Sr2CaCu2O8+δ [3–5] it has been shown that the NIR absorption cross-sections increase approximately linearly with decreasing temperature down to the superconducting transition temperatures (Tc). Below Tc the slope decreases to smaller values. Dewing et al. [3, 4] have explained this effect as evidence for Bose–Einstein condensation of small bipolarons for YBa2Cu3O7.

Abstract

The small polaron has proved useful in understanding the transport properties of such low-mobility solids as oxides, glasses, and amorphous semiconductors. Polarons and bipolarons are of interest in high-Tc superconductors. I will first briefly review the basic mechanism for the Hall effect found in the localized regime where transport is due to multi-phonon-assisted transitions between localized small polaron states. The temperature-dependence of the Hall mobility will be reviewed for the non-adiabatic, adiabatic and three- and four site cases. I will then indicate how the magnetic phase factors in the localized regime give the conventional magnetic Lorentz force in a description of polaron band motion or of purely electronic bands of narrow width. This narrow-band regime is more relevant to the normal state of high- Tc materials in which carrier motion is itinerant. I will then survey experimental evidence for the Hall effect due to small polarons and in the narrow-band regime for several materials and conclude with an example of the Hall effect in the normal state of the cuprate superconductors taken from David Emin.

The basic mechanism of the Hall effect in the localized regime

The model used is a straightforward two-dimensional generalization of the molecular crystal model of Holstein [1]. (This case admits only a small-polaron and free-particle solution and no large-polaron solution.) Briefly, the model consists of a site occupied by diatomic molecules with fixed centres of gravity and orientation but variable internuclear separation so that each acts like an Einstein oscillator with fixed frequency, ω0. The oscillators are subject to weak coupling giving rise to dispersion of the vibrational frequencies.

Abstract

We describe the influence of local magnetization on electron localization in concentrated and diluted magnetic semiconductors. This includes a review of transport and optical evidence such as the magnetic-field-induced insulator metal transition in concentrated systems, i.e. Eu chalcogenides and Gd3-xvxS4 (where v = vacancy). It also includes a brief review of salient experimental evidence for polaron formation in the diluted magnetic semiconductor Cd1-xMnxTeiln. In addition, static and dynamic photo-induced magnetic measurements in ZnTe/Cd1-xMnxSe heterostructures are presented and their relevance to high-Tc materials discussed.

Introduction

This paper deals with the effects of local exchange due to interaction of carrier spins with the ionic spins in magnetic semiconductors. We are specifically concerned with the 4f shell of Gd3+ and Eu2+ (S) and the 3d shell of Mn2+ (S), which contribute to the magnetic character of the concentrated and diluted systems respectively. Towards this end, the body of the paper begins with a brief review of the relevant exchange mechanisms leading to polaron formation. In the concentrated systems, this leads to giant magneto-resistive effects in the Eu chalcogenides [1] and very distinctive luminescence [2]. It alsoleads to the spectacular magnetic-field-induced metal-insulator phase transition in the Gd3-xvxS4 compounds [3–5]. In the dilute system, polarons account for novel effects in spin-flip Raman scattering [6, 7] and low-temperature transport properties [8, 9], and for the best examples to date of photo-induced magnetism. The latter properties have been studied both by magneto-optical [10] and by very sensitive magnetic techniques [11, 12]. Furthermore, new femtosecond dynamic studies have given information about the formation and decay of the polarons themselves [10,12].

Abstract

There exists a well-established empirical trend, namely that the best superconductors are among the bad conductors, in which electrons are essentially localized on atomic orbitals. Just this electronic structure is requisite for the metal-insulator transition predicted by N. Mott. I show in this paper that the coexistence of these two remarkable phenomena within the same set of materials is not accidental.

Introduction

It was understood long ago that there exists some relationship between superconductivity and Bose–Einstein condensation. According to Schafroth et al. to get the Bose particles the electrons should be bound somehow into quasimolecules (pairs) [1], and ‘the only obstacle’ to achieving an explanation of superconductivity was the nature of these quasi-molecules. The original belief was that they should have an atomic size to maintain Bose statistics when their concentration is high (of the order of one per unit cell), and the problem of how to overcome Coulomb repulsion seemed insurmountable. However, this had nothing to do with local pairs in the case of the superconductors known in the middle of the 1950s. Being good metals, these superconductors above Tc = 1−10 K display almost-free-electron behaviour with the Fermi energy not less than 5 eV, and it was clear that the superconducting transition here concerns only a very thin shell around the Fermi surface. BCS theory [2] was addressed precisely to these superconductors. It was shown that, at T = 0, the normal state is unstable with respect to formation of Cooper pairs [3], when in the vicinity of the Fermi surface there exists an arbitrary weak attraction between electrons.

Abstract

The observed characteristics of mid infrared (MIR) spectra in doped semiconductors are discussed. These characteristics were explained by Reik and coworkers on the basis of hopping motion of small polarons from a localized site to a neighbouring localized site. The success and limitations of this model are pointed out. Emin, on the other hand, showed the importance of large polarons for the conductivity. The recently observed features of MIR spectra in high-Tc cuprates are then summarized. The low-frequency peak in many cuprates with frequency 0.1−0.2 eV has been ascribed by many investigators to polaronic origin. We have undertaken in the present work numerical studies of polaronic conductivity in the two-site and four-site cluster model by diagonalization of the dynamical matrix. Broadening of the phonon spectra due to damping has been taken into account by considering a small but finite phonon lifetime. For intermediate and strong coupling, a number of peaks in the optical conductivity appear due to bound states with different numbers of phonons. We have also studied the importance of Hubbard U by calculating the optical conductivity as a function of U with two electrons in a two-site model. The experimental results of MIR spectra for the cuprates can be better understood on the basis of the present calculations.

Introduction

It was Landau who first introduced the idea of polarons for explaining the F centres in NaCl as due to self-trapping of electrons [1]. Polarons are the quasiparticles formed by the accompanying self-consistent polarization field and are generated due to the dynamical electron–phonon interaction. As a consequence there is extra scattering of the charge carriers, phonon energies are renormalized and the charge carriers are heavy [2].

Abstract

A review is presented of a recent theory of strong two-band electron selftrapping in a semiconductor, for which hybridization of the related electron state with the band states is essential and gives rise to new features of both electron and atomic dynamics. Pressure-induced phenomena in such materials predicted in the theory are discussed.

Introduction

As demonstrated in many works (see, e.g. [1–9]), the type of self-trapping of quasi-particles, e.g., electrons or holes in semiconductors, depends on properties of the materials. Whatever the origin of self-trapping, in most papers [1–4, 9] generally important contributions come from states of a single energy band, e.g. of the conduction band for electrons. Hence, single-band self-trapping has largely been considered in most works, which holds true, insofar as the characteristic self-trapping energy WST (< 0) is substantially less in magnitude than the interband, or mobility, gap width E(0)g, WST|<<E(0)g. However, there are realistic semiconducting materials in which two-band self-trapping occurs, in the sense that contributions from states of both conduction and valence bands are important and hybridization of the states in the gap gives rise to new effects [5–8]. For instance, a single-band self-trapping energy for a single electron in a harmonic atomic lattice WST= W1≃ – W∂Q2d/(2ka20) may be comparable in magnitude to E(0)g/2, as self-trapping occurs at a soft ‘defect’ exhibiting a small effective atomic spring constant k<<k0, for typical values Qd≈3–5 eV, E(0)g ≈ 1–3 eV, and K0 ≈ 30–50 eV Å2 (hole self-trapping can be treated in a similar way, with trivial substitutions of conduction band states for valence band ones and vice versa).

Condensed matter physics by its very nature deals with systems with a large number of degrees of freedom, i.e. systems for which a statistical description is essential. In this chapter, we will present a rather comprehensive review of the fundamentals of thermodynamics and statistical mechanics. Much of this chapter, especially the parts dealing with homogeneous fluids (ideal and interacting gases and liquids), should be familiar to everyone. They are included here mostly to establish a basis for discussing more complicated ordered systems. As we saw in the preceding chapter, a great deal of useful and experimentally accessible information is contained in correlation functions such as the density-density correlation function Cnn(x,x′). This chapter will define these functions in a statistical mechanical context and develop a method of calculating them using functional differentiation with respect to spatially varying external fields. Functional differentiation will allow us to calculate position-dependent correlation and response functions by a simple generalization of the familiar technique used to calculate the magnetic susceptibility by differentiating the free energy with respect to the magnetic field. This is a very powerful tool that will be used throughout this book. It is presented here first in a familiar context that should make it easy to grasp.

After reviewing the properties of homogeneous fluids, we will introduce order parameters in Sec. 3.5 and show how they modify both thermodynamics and statistical mechanics.