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A review of past and current work with mixed crystals of the KH2PO4 family is presented. It concentrates on solid solutions where the cations have been mixed in large proportions to achieve properties different from those of the parent crystals. A large part of the review is devoted to mixed crystals of RbH2PO4 and NH4H2PO4, or RADP, which is a model system for structural glasses that has recently received considerable attention.

NMR spectra of 75As and 39K have been measured in unannealed and annealed crystals of KH2PO4 and KH2AsO4 in the paraelectric phase. The As paraelectric A spectrum disappears, while the B-spectrum intensity increases substantially upon approaching Tc. The K satellite linewidth shows characteristic minima at ±8° from a+45° where the lines in the ferroelectric phase cross. Together with spin-echo data, this is evidence of quasistatic inhomogeneous broadening.

Combining dielectric, Brillouin, and neutron measurements, two spectral components related to freezing are identified in single crystals of ${\mathrm{Rb}}_{0.38}$${(\mathrm{N}{\mathrm{D}}_{4})}_{0.62}$${\mathrm{D}}_{2}$P${\mathrm{O}}_{4}$. The first corresponds to the incommensurate correlations that develop upon cooling and saturate near 80 K. These seem to be mostly static. A second, long-wave dynamic component continues to evolve markedly down to \AA{} 30 K, developing a broad distribution of slow relaxations. The former may provide the hierarchy of constraints leading to the latter.

Inelastic neutron scattering, combined with Brillouin and elastic small-angle neutron scattering, were used to test universality in the spectral dimension d\ifmmode \tilde{}\else \~{}\fi{}. The fracton density of states, obtained here on a new fractal silica aerogel and over a large frequency interval, now gives d\ifmmode \tilde{}\else \~{}\fi{}\ensuremath{\simeq}1.8. This suggests that d\ifmmode \tilde{}\else \~{}\fi{} is a sensitive function of the fractal structure.

Incoherent inelastic neutron spectroscopy combined with Brillouin scattering data are used to estimate the absolute density of states of fractal silica aerogels over the entire frequency range. The results indicate an extended fracton regime, and allow the thermal properties to be calculated in agreement with experiments.

Small angle neutron scattering results obtained on several series of silica aerogels are presented. It is shown that basic catalysis can produce materials that are fractal over at least two orders of magnitude in length. These have bulk densities lower than ∼ 0.02 g/cm3, corresponding to porosities of 99% or larger. In this case, the constituent particles, whose gyration radii are larger than 10 A, have a fractal surface that cannot be smoothened by oxidation. Densification at elevated temperatures mostly increases the particle size. The fractal structure is destroyed when the particle dimensions equal the correlation length at the gel-glass transformation.

Silica aerogels form mutually self-similar fractal structures for a broad range of densities and preparation conditions. This allows to scale dynamical results obtained on longitudinal acoustic waves up to the onset of the phonon to fracton crossover. An “effective” value of the fracton dimension\(\bar \bar d = 1.3 \pm 0.1\) is found, and various possible interpretations of that value are discussed.

X-ray scattering techniques have been used to study the structures of two crystals of Rb1−x(NH4)xH2PO4 withx≃0.2, near to the boundary between ferroelectric and “structural-glass” behavior at low temperature. Below about 83K, both crystals develop shortrange incommensurate correlations with a wavevectorq≃0.13 a* which are presumably related to the glass properties. On cooling below 70 K, the crystal with the slightly larger NH4+ concentrationx, develops the ferroelectric structure in a small fraction of the crystal, while the bulk of the crystal retains the tetragonal structure. The other crystal transforms almost wholly to the ferroelectric phase. The transition to the ferroelectric structure shows considerable hysteresis on heating and cooling, and is spread over about 20 K. The transition is certainly of first order, and the spread in temperature may arise from concentration fluctuations. These results and the structure of the incommensurate modulations are compared with the predictions of a theoretical model for this system.

Temperature- (T) and frequency- (\ensuremath{\nu}) dependent dielectric losses on ${\mathrm{Rb}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$(${\mathrm{NH}}_{4}$${)}_{\mathrm{x}}$${\mathrm{H}}_{2}$PO $_{4}$ single crystals (RADP), and their deuterated isomorphs (D-RADP), can be collapsed to single curves with a scaling variable E(T,\ensuremath{\nu}). This applies for T sufficiently high that equilibrium is established within the sample-thermalization time. If E(T,\ensuremath{\nu}) follows the Vogel-Fulcher law, the scaling will depend only on the freezing temperature ${T}_{0}$ and on the attempt frequency ${\ensuremath{\nu}}_{0}$. For RADP, values in excellent agreement with a study covering a very broad frequency range are thus obtained, while significantly different values apply to D-RADP.

Brillouin scattering measurements have been performed on a series of fractal silica aerogels under applied uniaxial pressure. The entire nonlinear elastic behaviour scales with the aerogel density using the single scaling exponent that characterizes linear elasticity. Further, the anomalous nonlinear elasticity of the aerogels extrapolate to that of fused silica, strongly suggesting a common origin.

A microscopic model for the mixed crystal Rb1−x(NH4)xH2PO4 is introduced which takes into account the acid protons as well as the NH4 groups. Using Monte Carlo simulation, the topology of the experimentally determined phase diagram is reproduced rather well, including the onset of freezing into a glasslike state.

The various methods to measure the density of states by neutron inelastic scattering are discussed. Purely incoherent scattering data on silica aerogels have been obtained by taking the difference spectra between protonated and identical deuterated samples. The results so far cover the particles modes and the upper part of the fracton regime. Complemented with Brillouin scattering information, these measurements provide an overall picture of the absolute density of states, from which low temperature thermal properties can be calculated, in remarkable agreement with experiments. 1 INTRODUCTION The theory of vibrational excitations in fractal structures predicts the existence of "fractons" 111. Those vibrational modes are localized. Also, in contrast to long-wavelength acoustic phonons, which obey a linear dispersion law, o = v q , where w is the frequency, q is the wavevector, -and v is the constant sound velocity, fractons are expected to exhibit dispersion according to o C C ~ ~ I ~ 11.21. Here D is the fractal (Hausdorff) dimension, and k is a characteristic length for the fracton. Finally, the density of states (DOS) of fractons Z(w) was predicted to be proportional to wa -1 /I/. The dimensions obey the inequality d 2 D > z, with d the Euclidean dimension. As an example, for 3-d percolation clusters and scalar waves the values are D = 2.5 and a = 413. The above predictions are at strong variance with the well-known behavior of acoustic phonons, a@ the differences should be observable experimentally. Silica aerogels are ideally suited for the experimental study of fracton dynamics. They are porous solid materials. Large pieces can be prepated, so that their elastic properties can be measured by static as well as by ultrasonic techniques. The optical transparency of aerogels in a broad range of densities and preparation conditions allows the study of their vibrational dynamics by Brillouin and Raman light scattering. (I) Supported by Neutron Scattering Project IIKW, Natuurkunde, University of Antwerp, 8-2610 Wilrijk, Belgium (2) Laboratoire mixte CNRS-Commissariat a I'Energie Atomique. (3) Laboratory associated to the Centre National de la Recherche Scientifique (CNRS). No 1129. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989424 Structural studies /3,4/ have demonstrated that silica aerogels are made of elementary particles, of typical size a, with more or less defined surfaces. Those particles assemble to form fractal networks, up to an average cluster size 5. At length scales above (, the material is an homogeneous assembly of such clusters. Corresponding to these different length scales, three distinct vibrational regimes are expected, namely, with increasing frequency : phonons, fractons, and particle modes. A crossover from phonons to fractons at oco,12n -1 GHz has been demonstrated by Brillouin scattering experiments 151. This technique also allowed the determination of the dispersion curve for fractons 16,7/, and gave indication for the occurrence of localization at the phonon-fracton crossover. Evidence of an extended fracton domain, ranging approximately from 40 peV to 1 meV, was obtained from very-low frequency Raman scattering IS/. It is obviously of interest to measure directly the DOS of these excitations. 2 EXPERIMENTAL APPROACHES There are several ways to measure a vibrational DOS by inelastic neutron scattering. One basic idea is that the DOS is reflected in the incoherently scattered neutrons. Taking only the one phonon terms in a multiphonon expansion, the expression for the double differential incoherent cross section of a Bravais lattice

Accurate birefringence measurements on monodomain SrTi${\mathrm{O}}_{3}$ platelets give important information about the fluctuations associated with the second-order structural phase transition. Above the transition, the monodomain character is announced by a birefringence tail caused by anisotropic fluctuations. Because of these, a distinct cusp is observed at the transition temperature ${T}_{a}$. Below ${T}_{a}$, the difference $〈\ensuremath{\phi}_{c}^{}{}_{}{}^{2}〉\ensuremath{-}〈\ensuremath{\phi}_{a}^{}{}_{}{}^{2}〉$ between mean-square rotations ${\ensuremath{\phi}}_{c}$ associated with the order parameter and orthogonal rotations ${\ensuremath{\phi}}_{a}$ is obtained.

Phenomena of coherent resonant propagation can be considered as resulting from the cooperative interaction of a certain number of excited two-level systems. It is shown that these phenomena can be characterized by a specific "maximum cooperation number" and by the associated "cooperation time." These are defined for the superradiant state, but their meaning and usefulness can be extended to other situations. The alternative description of of superradiance as a spontaneous or as a stimulated effect is also discussed and it is shown that with the help of the new concepts, the Dicke quantum perturbative treatment can be reconciled with the semiclassical theories.

For the description of an assembly of two-level atoms, atomic coherent states can be defined which have properties analogous to those of the field coherent states. The analogy is not fortuitous, but is shown to be related to the group contraction of exponential operators based on the angular momentum algebra to exponential operators based on the harmonic-oscillator algebra. The derivation of the properties of the atomic coherent states is made easier by the use of a powerful disentangling theorem for exponential angular momentum operators. A complete labeling of the atomic states is developed and many of their properties are studied. In particular it is shown that the atomic coherent states are the quantum analogs of classical dipoles, and that they can be produced by classical fields.

The vibrational excitations of well-characterized silica aerogels have been studied with Brillouin, Raman, and inelastic neutron scattering. Fractons are observed over a frequency range directly related to the fractal structure of these materials. The fractal dimension D, the spectral dimension d s , a localization exponent d φ , as well as the density of states N(ω), have been determined in these experiments. The overall behavior is in remarkable agreement with theoretical expectations.

In these lectures, the structure and vibrations of random media that are self-similar at various length scales are described. Small-angle scattering demonstrates that silica aerogels are excellent fractals. Brillouin, Raman, and inelastic neutron scattering are used to investigate their vibrational dynamics. The existence of new localized collective excitations, called fractons, is clearly revealed. Values are obtained for the fractal dimension D, the spectral dimension $$\bar \bar d$$ , an “internal length” exponent σ, as well as for the density of vibrational states N(ω). The picture that emerges is in remarkable agreement with fraction theory.

Systems crossing over from homogeneity to self-similarity at a correlation length ξ are expected to exhibit rather peculiar elastic properties.1 Phonons of well-defined frequency ω and wavevector q can propagate in such media if their wavelength λ = 2π/q is larger than ξ. In contrast, a crossover to fractons, the vibrational excitations characteristic of fractal structures,2,3 should occur as qξ increases towards 1.

Small-angle neutron scattering from silica aerogels reveals that they form mutually self-similar volume fractals for a wide range of densities and preparations. The fractal dimension is D=2.40\ifmmode\pm\else\textpm\fi{}0.03. The fractal behavior extends down to near-molecular sizes. Indication for a volume-to-surface fractal crossover is also obtained. These results apply to neutrally prepared or acid-catalyzed materials. Base-catalyzed gels have widely different properties.

The main theoretical predictions on vibrational excitations of fractal networks are first summarized. The application of the fracton theory to glasses and other amorphous materials is then critically discussed. The early experimental evidence for fractons obtained on demonstrated fractal systems is also surveyed. Finally, the extensive observations performed recently on silica aerogels are reviewed.

Brillouin scattering on porous silica aerogels reveals the onset of the phonon-to-fracton crossover in both the wave-vector and density dependence of the spectra. The results are consistent with a fractal description, giving both the fractal dimension D and the spectral dimension d\ifmmode \tilde{}\else \~{}\fi{}. In the hypothesis that the samples form single infinite clusters, the numerical values suggest that scalar elasticity is significant in these materials.

The observation and the theoretical explanation of phenomena which accompany near-resonant pulse propagation in atomic vapors are described. These phenomena bridge the gap between linear dispersion theory, which applies in the far wings of lines, and strictly resonant effects characteristic of self-induced transparency. The results indicate that very low propagation velocities and strong pulse reshaping are not unique to transparency situations.

Results of depolarized Raman spectroscopy on fractal silica aerogels are presented at frequencies from 0.3 to 50 ${\mathrm{cm}}^{\ensuremath{-}1}$. Three distinct vibration regimes are observed. In the fracton range, a power-law dependence of the Raman susceptibility on the frequency is found. It allows a determination of the superlocalization exponent ${d}_{\ensuremath{\varphi}}$. Alternatively, with use of the "resistance" expression for the exponent ${d}_{\ensuremath{\varphi}}$, a spectral dimension $\stackrel{-}{\overline{d}}$ in fair agreement with previous measurements results.

The central problem of Quantum Optics (laser theory, super-radiance, resonant propagation, etc.) is the description of the interaction between N atoms and an electromagnetic field confined in a cavity of finite volume. A suitable model Hamiltonian for this problem is the following one (ℏ = 1) $$\begin{array}{*{20}{l}} {{\rm{H = }}\sum\limits_{\rm{k}} {{{\rm{\omega }}_{\rm{k}}}} {\rm{a}}_{\rm{k}}^{\rm{ + }}{{\rm{a}}_{\rm{k}}}{\rm{ + }}\frac{{{{\rm{\omega }}_{\rm{o}}}}}{{\rm{2}}}\sum\limits_{{\rm{i = 1}}}^{\rm{N}} {{{\rm{S}}_{{\rm{3}}\left( {\rm{i}} \right)}}} }\\ {{\rm{ + }}\sum\limits_{{\rm{k,i}}} {{{\rm{g}}_{\rm{k}}}\left( {{{\rm{a}}_{\rm{k}}}{\rm{S}}_{\rm{i}}^{\rm{ + }}{{\rm{e}}^{{\rm{i \bullet }}{{{\rm{}}}_{\rm{i}}}}}{\rm{ + a}}_{\rm{k}}^{\rm{ + }}{\rm{S}}_{\rm{i}}^{\rm{ - }}{{\rm{e}}^{{\rm{ - i \bullet }}{{{\rm{}}}_{\rm{i}}}}}} \right)} {\rm{,}}} \end{array}$$ where ak, a k + are Bose operators describing the k-th field mode and S i ± ,S3i are Pauli operators describing the atom located at position x i as a two-level system.

High frequency sound is observed in lithium diborate glass, Li$_2$O--2B$_2$O$_3$, using Brillouin scattering of light and x-rays. The sound attenuation exhibits a non-trivial dependence on the wavevector, with a remarkably rapid increase towards a Ioffe-Regel crossover as the frequency approaches the boson peak from below. An analysis of literature results reveals the near coincidence of the boson-peak frequency with a Ioffe-Regel limit for sound in {\em all} sufficiently strong glasses. We conjecture that this behavior, specific to glassy materials, must be quite universal among them., 4 pages, 4 figures, revised version

The spectra recently published by Tominaga and Urabe are shown to be in good qualitative agreement with the current coupled-mode models of the KDP transition. The need for a careful quantitative analysis of this type of data over a broad frequency range is discussed.

Results obtained on mixed crystals of Rb1−x(NH4)xH2PO4 are reported. These polar materials of the KH2PO4 family are characterized by a high-temperature disordered network of O-H-O bonds. On cooling, different orderings of these bonds correspond to the onset of a ferroelectric polarization parallel to the tetragonal c-axis in RbH2PO4 (x=O), or of antiferroelectric moments perpendicular to it in NH4H2PO4 (x=1). In the mixed crystals of interest, these transitions are suppressed by random NH4 substitutions. The NH+ 4 ions form H-bonds with nearby PO4 tetrahedra, occupying O-sites and frustrating the low-temperature acid-proton order. Glass behavior results over a broad range of concentration 0.22 < x ≲ 0.8.1,2

p‐Type conductivity and stored electron charge densities as large as 1.3 × 10−7 coul/cm2 have been observed in high‐resistivity CdS crystals. These properties were studied by photovoltaic, photopolarization, and thermo‐electric power measurements.

The growth, structure, and dynamics of RbH2PO4–NH4H2PO4 mixed crystals are reviewed. In the intermediate concentration range, the built-in frustration of these materials prevents the ordinary ferroelectric or antiferroelectric transitions, whereby the disordered acid-proton sublattice freezes into a glass. This has been observed with a number of experimental techniques, including dielectric dispersion, optical birefringence, X-ray diffraction and scattering, Raman and Brillouin scattering, NMR, and more recently, specific-heat and neutron scattering. The variety of techniques applicable to this system has allowed the observation of polarization freezing over about 17 orders of magnitude in frequency. This freezing is well described phenomenologically by the Vogel-Fulcher law, while other laws proposed for spinglasses are less successful. Recent hypersonic-relaxation measurements point to the existence of a frozen polarization component at temperatures already much above Vogel-Fulcher freezing, while neutron diffused scattering reveals that the tendency to form superstructures is not static.