Condensed Matter Physics

Physics

1

SIMPLE LIQUIDS

One could well argue that it is more difficult to treat theoretically a liquid than a gas with a random distribution of molecules or a crystal with a regular structure. Nevertheless, its description is a good place to begin discussing condensed matter physics since the liquid state is no stranger to us and yet has strong molecular correlations. Theoretical techniques developed for a liquid are often applicable to other condensed matter systems.

1.1. Pair distribution function

The liquid state is similar to the solid state in having a free surface, low compressibility, and high density. In fact, its density is only about 5% lower than that of a solid, although there are exceptional cases such as water. However, its fluidity and molecular order are unique. Molecules in liquids can change position continuously, fill out a vessel with a formation of a meniscus, and do not have the long-range order of solids but only short-range order. Both liquids and gases are called fluids but these two are clearly distinguishable except near the critical point.

Since the molecular distribution of a liquid differs from that of a gas or a crystal, its study is important. This study can be performed effectively using two experimental methods: X-ray or electromagnetic wave diffraction and neutron scattering. The former method probes a static distribution that is quite different from those of gases and crystals. The latter provides information on the dynamic correlation of molecules, which is also distinctive. We shall discuss the former in this section and the latter in the next section.

X-ray scattering from a liquid depends not only on the molecular distribution but also on the molecular species. Therefore, we consider only a simple liquid (without any complex molecular structure), homogeneous and in equilibrium [1].

X-ray studies probe the radial distribution function g(r), which is connected with the pair distribution function ρ2(r) such that

The pair distribution function ρ2(r) represents the probability of finding two molecules at a relative distance r. For homogeneous and isotropic systems, this probability depends only on the magnitude of r and approaches n² as r → ∞, where n is the number density. Hence, the radial distribution function g(r) has the asymptotic property:

g(r) → 1,     (r → ∞).

Its deviation from this asymptotic value represents the molecular correlations. For a given distance r, ng(r)4πr² dr is the number of molecules in the spherical shell of width dr at r.

The pair distribution function satisfies the normalization:

where V is the volume and N is the number of molecules in the liquid. The average 〈…〉 is taken if this total number is fluctuating. In such cases, ρ2(r) is defined in a grand ensemble where 〈…〉 represents a grand-ensemble average.

From Eqs. (1.1) and (1.2), we learn that the radial distribution function satisfies

For large 〈N〉, the right-hand side represents the square average fluctuations of the total number N about its average 〈N〉. One can show statistically [2] that

where I(0) is proportional to the intensity of scattered radiation at zero scattering angle and p is the pressure. Note that the quantity

is the isothermal compressibility. Light scattering will be discussed shortly.

Generally, g(r) stays 0 near the origin at r = 0 due to the strong short-range repulsion of molecules, reaches a maximum at a point where nearest-neighbor molecules gather, and then shows weaker maxima and gradually approaches its asymptotic value of 1. Its schematic curve is illustrated in Fig. 1.1.

FIG. 1.1. Radial distribution function of a simple liquid.

The structure factor S(q) is a quantity that is related to g(r) through the Fourier transform:

As a function of q, the graph of S(q) resembles that of g(r) in that it is small for small q, approaches 1 for q → ∞, and wiggles in between. However, S(q) for small q corresponds to g(r) at large r. In particular,

The structure factor S(q) can be expressed in a series in q² starting from the above value with the coefficients given by moments of [g(r) − 1] if the sine function in the integrand of Eq. (1.6) is expanded. The pair distribution function or the structure factor can be determined theoretically on the basis of certain approximations or numerical calculations. Since the determination is one of the major theoretical tasks in the theory of liquids and statistical mechanics, many techniques have been developed, and its detailed discussion must be sought elsewhere, although some theoretical methods are outlined in Appendix for convenience’s sake. In what follows, we present an approximate but very useful approach.

Ornstein and Zernike [3] assumed that the total correlation function

consists of a direct correlation function C(r) and an indirect correlation function via intermediates molecules in such a way that

Fourier transformation of this equation yields

is the Fourier transform of the direct correlation function.

can be expanded in a Taylor series. If C(r) is spherically symmetric, its Fourier transform can be expressed to order q² such that

where

The notation 〈…〉 here represents an average. Using Eq. (1.10) for q = 0 we arrive at

where

From Eq. (1.13) we find that the Ornstein and Zernike equation yields

Hence, the molecular correlation is short-ranged as long as ξ stays finite.

This short-rangeness of molecular correlation has been found experimentally to be the case except near a critical temperature. Let us examine this point from a somewhat different angle by expanding the free energy density F1(r) about its equilibrium value 〈F1〉. We adopt the Landau expansion:

where a and b are expansion coefficients. The constant a should vanish at the critical point because

The pair distribution function is related to the correlation function of local density fluctuations. The density–density correlation function is given by

where n(r) represents the density at r. One can express the density fluctuation n(r) − n in terms of its Fourier transform nq:

The Fourier transform nq must satisfy

because n(r) − n is real. From Eqs. (1.18) and (1.19) we learn that

The free energy of the entire liquid with volume V is then expressed as

Therefore, we adopt the following as the probability of having |nq|²:

to arrive at the average:

Using this result in Eq. (1.20) we obtain for b ≠ 0 an expression that is equivalent to Eq. (1.15):

In accordance with Eq. (1.17), a = 0 at the critical temperature so that the pair distribution function becomes long-ranged:

Near the critical temperature Tc the parameter a may be assumed to vary such that

We find then

That is, the isothermal compressibility diverges. As we shall discuss later, the exponent of this divergence in real liquids is slightly larger than 1. Correspondingly, the correlation function is frequently expressed as [4]

where η is a small correction. This divergence of κT is due to the long-range molecular correlation that also causes I(0) to diverge. Strong light scattering at Tc, i.e., critical opalescence, has long been observed.

The structure factor is determined by the intensity of scattered radiation in accordance with Eq. (1.13). That is, the Ornstein–Zernike theory predicts that 1/S(q) is expected to be proportional to q².

The structure factor can be determined by light or X-ray scattering. This depends on the interference of scattered waves from intra- and interatomic electrons in liquids. The former is given by the so-called atomic structure factor, which is denoted by f. The latter is determined by the pair distribution function. The combination of these two factors yields the following expression for the scattered intensity:

where N is the total number of molecules, I0 is the intensity of incident radiation and I(s) is given by

Here

represents the difference between the wave vectors k0 and k of the incident and scattered radiation respectively. If no frequency change takes place upon scattering, the magnitude of s can be expressed as

where λ is the wavelength of the radiation. In short, J(s) is determined by S(q) and

Hence, light scattering can directly determine the correlation function. Equations (1.27) and (1.29) show that the reciprocal scattered intensity is proportional to q² for large q as in the Ornstein–Zernike theory but deviations take place for small q. The solid line in Fig. 1.2 shows the q² proportionality. This proportionality has been observed [5]. The dotted lines extend the linear variation to q = 0 where the ordinate is a constant which is proportional to kTnκT. The dashed curves represent the deviation that depends on the exponent η. This is a small parameter close to 0.05.

Once the pair distribution function is determined, thermodynamic functions can be generated. For instance, the equation of state of a classical system is given exactly by [1,2]

where φ(r) is a molecular potential. This equation is called the virial equation of state. From Eqs. (1.5) and (1.6) we find another exact relation:

FIG. 1.2. Inverse relative scattered intensity near a critical point Tc. The dotted lines extrapolate the linear variation and the dashed lines represent actual behaviors.

Equation (1.29) shows that the left-hand side of the above equation is equal to the intensity I(0) in the incident direction in agreement with Eq. (1.7). Thus, the q² plot in Fig. 1.2 should yield the constant on the right-hand side. At the same time, we learn that the higher the temperature the higher the value at q = 0 in Fig. 1.2.

It is often convenient to introduce a coupling constant λ to the interacting part H1 of the Hamiltonian so that the strength of the interaction can be varied. The total Hamiltonian is then of the following form:

One can show that the equation of state is given by

where the pair distribution function is now dependent on λ and p0 is the pressure corresponding to H0.

The radial distribution function of a liquid depends on the nature of the molecules, temperature, and density. Generally, its first peak is reduced when the temperature is increased. However, liquid helium shows the opposite temperature dependence below the λ point.

FIG. 1.3. Radial distribution function of three liquids. Solid curve, liquid ⁴He at 1.94 K; dashed curve, Ne at 35.05 K; dashed-dotted curve, Ar at 163 K. (From Raveché and Mountain [6])

The quantity 4πr²ng(r) yields the number of molecules in the spherical shell of unit width at a distance r from the origin. Approximating the first peak by a Gaussian distribution, it has been found that the number of nearest-neighbor atoms in liquid argon is around 10.6 at 84.4 K, indicating that the liquid has a local structure that is somewhere between those of the fcc and bcc lattices.

There are quantum effects on the first peak of g(r). Figure 1.3 compares the radial distribution functions of three liquids [6]. The solid, dashed, and dashed-dotted curves represent respectively liquid ⁴He at 1.94 K, n = 0.0245 Å−3; Ne at 35.05 K, n = 0.0317 Å−3; and Ar at 163 K, n = 0.08 Å−3. The abscissa is a reduced distance r/a, where a = 2.556 Å for He, 2.786 Å for Ne, and 3.405 Å for Ar. Note that the first peak of liquid helium is lower and appears at larger r in comparison with the cases of argon and neon.

1.2. Dynamic structure factor

The effectiveness of probing condensed matter by X-ray or neutron scattering depends on the energy and momentum that are transferred into the system. With energies of order 10 keV, X-ray and Mössbauer γ-ray experiments probe only spatial correlations. Neutrons can have energies in the range 1−100 meV and wavelengths in the range 1−10 Å. The wavelength λ in angstroms of a neutron and its energy ε in electronvolts are related to each other by

On the other hand, the correlation time of a liquid is of the order of 10−13 s and the correlation length is of the order of 10−8 cm. Therefore, it is possible to investigate both spatial and temporal correlations because neutrons can spend time comparable to the correlation time over the correlation length in a liquid. Neutron scattering provides information concerning diffusion and vibration of molecules, but since the thermal energy kT is 0.86T × 10−4 eV, the observation of energy gain of more than 0.1 eV is difficult. For such a case, energy-loss techniques have been developed.

The space-time correlation function introduced by van Hove [7] can be used effectively to describe neutron scattering. This function measures the correlation of the local density at time 0 with that at time t through the definition

where n is the average number density,

is the local number density at time t, and 〈…〉 is a statistical average.

The same function can be defined in a somewhat more dynamical way by following the motions of molecules. Let us denote the position of the ith molecule at time 0 by ri(0). Let r be the point where the correlation is measured, and denote by r′ the sum

ri(0) + r = r′.

A molecule j may be at this point at time t:

rj(t) = r′.

By the time t, the ith molecule may have moved out of the position ri(0), and can even be at r′. In this case j = i, and Gs(r, t) yields the probability of finding a particle at position r after time t when it was at the origin at time 0. Or, if j ≠ i, another molecule j can be at r′.

The corresponding G(r, t) can be introduced such that it represents the probability of finding any particle at r at time t when a particle is at origin at time 0. According to van Hove, the space-time correlation function is defined by [7]

where

is the position operator of the jth molecule in the Heisenberg representation, H is the Hamiltonian of the system without the neutrons, and 〈…〉 represents a statistical average. Note that the molecules i and j are not necessarily different. If they are the same, the notation Gs(r, t) is used to represent a self-correlation function. Hence,

Especially,

As can be shown easily, the correlation function satisfies

For instant correlations in homogeneous liquids, the relation

holds, where g(r) is the radial distribution function, which depends only on the distance r. This instantaneous correlation function can be probed by X-ray scattering.

If there is no thermal motion and if n(r, t) = n, G(r, t) is reduced to n. This is the limit that is expected in the long-time limit:

Note that in the same limit,

The Fourier transform of the deviation of G(r, t) from its limiting value n is called the dynamic structure factor. This is defined by

The Fourier transform Ss(q, ω) of Gs(r, t) can also be defined. These two Fourier transforms are related to the differential cross-sections for coherent and incoherent scattering respectively such that

Here, b is the strength of the Fermi pseudopotential Vi of a given nucleus i:

such that q = k0 − kare the averages of bi over all the nuclei. The differential cross-section for coherent scattering by N molecules is then given by Eq. (2.11), and that for incoherent scattering cross-section is given by Eq. (2.12).

The dynamic structure factor is reduced to the static structure factor upon integration over ω:

In addition, it satisfies the sum rule:

and is related to the energy-loss cross-section S(−q, ω) by

This relation can be understood by taking the ratio of the cross-section σ(k0 → k) and its inverse σ(k → k0) in consideration of the detailed balancing condition. Note in this respect that

Equation (2.15) represents the first moment of the dynamic structure factor. There are of course many other moments.

Equation (2.16) generates a relation:

That is, the real and imaginary parts of the correlation function are related to each other through [8]

Neutron or laser light scattering has been applied to classical and quantum liquids, polymers, magnetic, and many other condensed systems [9]. Some of the applications will be addressed in later chapters. We shall discuss in the remainder of this section the especially important role the dynamic structure factor plays for critical light scattering.

When condensed matter is considered as a continuum in the so-called hydrodynamical regime, the equation of continuity, the equation of motion, and the energy conservation equation relate the density, pressure, and temperature fluctuations with each other. The fluctuating hydrodynamical modes represented by these macroscopic quantities are nonlinearly coupled with each other. In particular, the Fourier transform nq(ω) of the density fluctuations in time and space, which determines S(q, ω) is obtained by solving these equations by linearization, decoupling, and some other approximations. Its standard solution is of the form [10]:

Here,

c is the sound velocity at zero frequency; γ = cp/cv is the specific heat ratio; b is called the longitudinal kinetic viscosity, which depends on the shear and bulk viscosities η and ζ; n is the equilibrium density; m is the mass; and λ is the thermal conductivity.

The roots of the denominator of Eq. (2.19) can be used to identify the processes for the decay of density fluctuation. If the last term with q⁴ is neglected, the three roots are given approximately by

where

The quantity DT is called the thermal diffusivity and Γ is the attenuation coefficient of sound. Equation (2.19) can now be simplified as a sum of three terms:

One can show that the first and second terms are related to the decay of entropy fluctuations and that of pressure respectively [10]. The Rayleigh line corresponds to the nonpropagating entropy fluctuations at constant pressure and the Brillouin lines to sound.

FIG. 1.4. Scattered intensity in the hydrodynamical regime. Rayleigh line, center and Brillouin lines, both sides.

The inverse Fourier transform of Eq. (2.23) yields

On the other hand, S(q, ω) is given by the Fourier transform:

The above form of nq(t) yields

Accordingly, the spectrum consists of three Lorentzian lines, the Rayleigh line at ω = 0 and two Brillouin lines at ω = ± cq, which are also called Stokes (+) and anti-Stokes (–) lines respectively. The spectrum is illustrated schematically in Fig. 1.4.

The Rayleigh line has strength (1 − γ−1) and half-width

This width is around 10⁷ rad/s for laser light. When the two Brillouin lines are combined the strength is γ−1. The half-width is given by

which is typically 10⁹ rad/s. The Brillouin shift

is around 10¹¹−10¹² rad/s.

The total intensities of the Rayleigh line and each of the Brillouin lines are given by

Hence, the total scattered intensity is

The intensity ratio of the Rayleigh and Brillouin lines is called the Landau–Placzek ratio. This is given by

Thus, dynamical scattering provides a wealth of information; the total scattered intensity yields the isothermal compressibility, the intensity ratio of Rayleigh and Brillouin lines, the specific heat ratio, ΔωR the thermal diffusivity, ωB the sound velocity, and ΔωB the sound attenuation.

We have been concerned so far with the hydrodynamical regime. This regime is specified by the requirement that the correlation length ξ is small in comparison with q−1:

However, as the critical temperature is approached the correlation length diverges, causing deviations from this condition. In fact, the critical region is characterized by the opposite condition:

The transport coefficients depend on wave number q and frequency ω in this regime due to the spatial and temporal correlations between the molecules, which are reflected into intense coupling between fluctuating quantities. Thus, deviations from the above Rayleigh linewidth ΔωR take place as qξ increases from its hydrodynamical limit. These deviations have been studied theoretically and experimentally and important progress in the critical regime has been made in several stages [11–13] since the late 1960s. As a result, dynamical scaling theory has been developed.

A phenomenological approach to the critical regime can be made by using the following power-law variations of the relevant quantities in the above hydrodynamical results:

The approximate values of the exponents are a = 0.58, v = 0.67, ψ = 0.5 and γ = 1.25, where ε = T/Tc − 1.

The Rayleigh line is predominantly determined by the heat diffusion mode. According to the critical variations in Eq. (2.34), the Rayleigh linewidth approaches zero as

The exponent γ – ψ ~ v has been found to be around 0.6 for CO2, Xe, and SF6 [12] as theoretically expected. The corresponding decrease in the thermal diffusivity toward the critical point is called the critical slowing down of fluctuations.

The Brillouin shift is determined by the sound velocity c at zero frequency. Since this velocity approaches zero as εα/2 the Brillouin shift is expected to approach zero as

If Γ/cv is dominant, the Brillouin linewidth diverges as

Thus, the Brillouin lines approach the Rayleigh line and predominantly determine the linewidth. Note that from these variations critical exponents can be determined. Note that DT ~ ξ−(γ−a)/v is ξ-dependent and that γ is not the specific heat ratio, although the customary notation has been used.

According to dynamical scaling, the half-width ΔωR is a homogeneous function of q and inverse correlation length ξ−1. If the order of homogeneity is s, then

In particular, in the limit qξ → ∞,

Experiment shows that the linewidth is proportional to q³ so that the degree of homogeneity s can be 3. From Eqs. (2.27) and (2.34) we find that indeed

A more specific expression for ΔωR is given by

According to Kawasaki [11], the function is given in the first approximation by

From the behaviors of K(x) for small and large x, we find

We can express Eq. (2.43a) in the form of Eq. (2.27):

with an effective thermal diffusivity given by

where

The quantity 6πηξcan be interpreted as an effective Stokes friction of a sphere of radius ξis q-dependent as a result of strong couplings between fluctuations of relevant quantities in the critical region. These couplings occur because the Fourier transforms of the macroscopic variables such as velocity, energy, and pressure in the equations of motion, continuity, and conservation are all dependent on ξ. These variables determine the propagation modes of acoustic waves, viscous flows, and heat diffusion. The couplings between these modes are enhanced through their ξ dependences as the critical point is approached.

FIG. 1.5. Thermal diffusivity of CO2 as a function of ΔT = T − Tc. Curve a, theoretical background contribution; curve b, the contribution from the critical part of the linewidth based on Kawasaki’s function K(x) of Eq. (2.42); curve c, the sum of the critical and background contributions. The circles are the data. (From Swinney and Henry [12])

Figure 1.5 illustrates DT as a function of ΔT = T − Tc for carbon dioxide on the critical isochore given by Swinney and Henry [12]. The theoretical curves represent: (a) the background contribution, (b) the contribution of the critical part of the linewidth; (c) the sum of the critical and background contributions. The black circles represent the measured values at scattered angle 90°. The large background contribution, which is not included in the above formula, causes ambiguity in analyzing the theoretical result.

We note in Eq. (2.44) that

is a characteristic frequency, its inverse being the relaxation time τ for thermal diffusion. Acoustic modes with frequently larger than τ−1 are expected to show dispersion since they are no longer effectively coupled with the heat-diffusion mode. Such a coupling is important for the Brillouin components, as can be seen from Eqs. (2.22). As the critical point is approached and the characteristic frequency drops to zero, the number of these modes increases and the sound velocity is no longer constant but decreases. Hence, the study of the sound velocity near Tc becomes an important subject. Kawasaki’s result in Eq. (2.42) is based on the replacement of shear viscosity by a frequency- and q-independent constant. Such an approximation is not

FIG. 1.6. Reduced diffusion coefficient [D] as a function of the scaling variable x = qξ. Dashed and solid curves represent respectively Eq. (2.42) and K*(x) of Eq. (2.48).

Hence, theoretical improvements have been attempted [13]. For instance, Burstyn et al. [13] have introduced a new function:

where zη ~ 0.06 is a universal critical exponent and S(x) behaves such that

where a0 and a∞ are constants. They also proposed a reasonable procedure to eliminate the background contribution that is not included in Eqs. (, they determined the dimensionless diffusion coefficient

as a function of x = qξ, as shown in Fig. 1.6. The dashed curve represents Kawasaki’s result K(x), which shows deviations for large x. The solid curve is their approximate result, while the symbols are the data for three scattering angles as indicated. [D] represents K(x) or K*(x).

1.3. Theory of condensation

The condensation of a gas into a liquid is one of the oldest known and most familiar phase transitions. In fact, the important discovery of the existence of a critical temperature was made by Andrews in 1869. Therefore, it is understandable that considerable effort has been made toward its theoretical description. This effort has undoubtedly contributed to the developments of statistical mechanics and many-body theory. However, no completely satisfactory theory of condensation has been constructed as yet. Nevertheless, this phenomenon involves some basic features of interacting particles and serves as a challenging classical subject for condensed matter physics.

The liquid state is realized not only by condensation but also by melting. These transitions into a liquid are generally first-order because the first-order thermodynamic derivatives of the Gibbs free energy are discontinuous. It follows the path of minimum values of the free energy, but a metastable phase with a higher free energy may still occur. A supercooled liquid, which we observe as in the form of icy rain, is a typical example. Its stability contrasts with that of a glass, which is also in a metastable state.

1.3.1. Yang–Lee theory

Yang and Lee [14] developed in 1952 a general theory of condensation for a system of molecules with a hard core. This theory starts with the recognition that only a finite number of hard-sphere molecules can be put in a given volume V. Then, the grand partition function Ξ(T, V, z) must be a finite polynomial:

where QN is the configurational partition function of N molecules:

and M is the maximum number of molecules that can be put in the volume.

The grand partition function can be rewritten such that

The roots zk of the grand partition function must be complex because all the QN are positive. This means that no divergence is expected for

when the fugacity z is changed in accordance with density variations.

However, this situation may change in the thermodynamic limit in which

M → ∞,     V → ∞,     M/V = n = constant.

In order to find what would happen in this important limit, we note that a term with zk must