We perform a consistent low-energy reduction of the three-band Hubbard model ${\mathit{H}}_{3\mathit{b}}$ to an effective one-band model ${\mathit{H}}_{1\mathit{b}}$, using as an intermediate step a renormalized effective spin-fermion model ${\mathit{H}}_{\mathrm{sf}}$ in which ${\mathrm{Cu}}^{+}$ and ${\mathrm{Cu}}^{3+}$ configurations are contained as virtual states. We map ${\mathit{PH}}_{\mathrm{sf}}$P into ${\mathit{H}}_{1\mathit{b}}$ using two projectors P onto different Zhang-Rice singlets: (1) orthogonal singlets constructed using O Wanner functions, (2) nonorthogonal singlets. In case 1 (2) the mapping is almost exact (exact as shown before by Zhang) if direct O-O hopping and the ${\mathrm{Cu}}^{3+}$ (${\mathrm{Cu}}^{+}$) configuration can be neglected.

We show that from measurements of the reflectivity of a uniaxial medium taken at a finite incidence angle with s- and p-polarized light it is possible to determine the dielectric function both parallel and perpendicular to the optical axis. When applied to layered compounds with its surface parallel to the layers, this technique allows for an accurate determination of the loss function perpendicular to the layers. This is demonstrated for the example of c-axis oriented thin films of the high-${\mathit{T}}_{\mathit{c}}$ superconductor ${\mathrm{Tl}}_2$${\mathrm{Ba}}_2$${\mathrm{Ca}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{10}$, on which we carried out polarized reflectivity measurements at 45° incidence angle above and below ${\mathit{T}}_{\mathit{c}}$.

Exact expression for the isotope-shift exponent and the pressure coefficient of the transition temperature are derived from the BCS gap equation for a density of states (DOS) with a van Hove singularity (VHS). The variations of these quantities with the shift of the Fermi level from the VHS and with ${\mathit{T}}_{\mathit{c}}$ are discussed. The superconducting gap-to-${\mathit{T}}_{\mathit{c}}$ ratio (2${\mathrm{\Delta }}_0$/${\mathit{T}}_{\mathit{c}}$) is also studied. The value of 2${\mathrm{\Delta }}_0$/${\mathit{T}}_{\mathit{c}}$ lies in the range from 3.53 to 4 for the DOS with a VHS at the Fermi level. A comparison of theoretical predictions with experimental results of high-${\mathit{T}}_{\mathit{c}}$ oxide systems are made. A qualitative good agreement is obtained for the pressure coefficient of ${\mathit{T}}_{\mathit{c}}$.

The homologous series ${\mathrm{HgBa}}_2$${\mathrm{Ca}}_{\mathit{n}\mathrm{-}1}$${\mathrm{Cu}}_{\mathit{n}}$${\mathrm{O}}_{2\mathit{n}+2+\mathrm{\delta }}$ (n=1,2,3,4) are the newest and most promising of the layered high-${\mathit{T}}_{\mathit{c}}$ superconducting materials, as demonstrated in setting the highest ${\mathit{T}}_{\mathit{c}}$ among the known superconducting materials. The first of its series, ${\mathrm{HgBa}}_2$${\mathrm{CuO}}_{4+\mathrm{\delta }}$, contains only one Cu-O plane per unit cell and is superconducting with ${\mathit{T}}_{\mathit{c}}$ about 94 K. The superconducting transition temperature as a function of pressure, ${\mathit{T}}_{\mathit{c}}$(P), was measured to 4.5 GPa. ${\mathit{T}}_{\mathit{c}}$ was determined from ac susceptibility measurements and pressure was determined near ${\mathit{T}}_{\mathit{c}}$ from shifts in ruby fluorescence peaks. ${\mathit{T}}_{\mathit{c}}$ increases monotonically with ${\mathit{dT}}_{\mathit{c}}$(P)/dP≊1.8 K/GPa at ambient pressure. The volume as a function of pressure, V(P), was measured to 10 GPa. The lattice parameters were determined by energy-dispersive x-ray-diffraction methods and pressure was determined from shifts in ruby fluorescence peaks. The volume decreases monotonically producing an isothermal bulk modulus ${\mathit{B}}_{\mathit{V}}$=104±17 GPa at ambient pressure. Both results were combined to produce ${\mathit{T}}_{\mathit{c}}$(V), which is more readily compared to theoretical models.

ac-susceptibility (χ’,χ’’) measurements as a function of temperature and ac field amplitude have been carried out on cylindrical samples of the 110-K phase of the Bi-Sr-Ca-Cu-O system. The two single-phase samples used for studies were prepared under different experimental conditions with an aim to understand the regimes of validity of Bean’s and Kim’s critical-state models. Material-dependent parameters such as critical current density as a function of temperature and effective volume fraction of grains, were estimated using these models. Improvements in the critical-current density and intergrain coupling strength were found when a sample was subjected to intermediate cold pressing between sintering. Field variation of intergranular matrix susceptibility of the sintered sample could be fitted well to Kim’s model whereas the data on the press sintered sample were in agreement with Bean’s model.

We have studied the linewidth of the ${\mathrm{\Gamma }}_{3\mathrm{-}}$${\mathrm{\Gamma }}_2$ (0.5 meV) crystal-field transition of ${\mathrm{Ho}}^{3+}$ in polycrystalline samples of ${\mathrm{Ho}}_{\mathit{x}}$${\mathrm{Y}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Ba}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_7$ (x=0.1 and 0.2) by high-resolution, inelastic neutron scattering. The linewidth increases with temperature, but shows an anomaly at, or slightly above, the superconducting transition temperature ${\mathit{T}}_{\mathit{c}}$. To investigate the connection with superconductivity we have applied the theory of Becker, Fulde, and Keller (BFK) to calculate the normal-state linewidth based on the assumption that the primary relaxation mechanism is the s-f interaction with a normal Fermi liquid on the ${\mathrm{CuO}}_2$ planes. The calculations indicate that there is a reduction in damping associated with the superconducting state, but that at low temperatures, where the relaxation behavior should reflect the symmetry of the gap function, the linewidth increases more rapidly with temperature than might be expected. We have also applied the BFK theory to the damping of crystal-field transitions of ${\mathrm{Tm}}^{3+}$ in ${\mathrm{Tm}}_{\mathit{x}}$${\mathrm{Y}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Ba}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{y}}$, measured recently by Osborn and Goremychkin. From the measured linewidths above ${\mathit{T}}_{\mathit{c}}$ we derive reduced s-f coupling constants ρ’=0.014 for ${\mathrm{Ho}}^{3+}$, and ρ’=0.043 for ${\mathrm{Tm}}^{3+}$. We discuss possible reasons for why these two values differ.

Longitudinal and transverse wave velocities, six kinds of elastic parameters, and dilational and shear internal frictions have been simultaneously measured as functions of temperature between room temperature and 77 K in the high-${\mathit{T}}_{\mathit{c}}$ superconductor ${\mathrm{GdBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$. An abrupt increase from 117 K in bulk modulus and Lamè parameter suggests that the order parameter has a much stronger coupling with volume-nonpreserving distortions than with volume-preserving distortions. The 227 and 120 K peaks in dilational friction might be related to structural change and second-order phase transition due to variation of potential energy, respectively. For shear friction, the 214 and 106 K peaks could be attributed to barium atom relaxation with an apparent activation energy of 0.62 eV, and lattice instability, respectively.

The ESR, magnetic susceptibility, and resistivity measurements of ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4+\mathrm{\delta }}$ doped with Mn are performed in the broad regions of temperature, Mn and Sr concentrations, respectively. The temperature and Mn concentration dependence of the ESR signal can be understood according to a model of the collective motion of the magnetic moments of Mn and Cu ions with the ESR linewidth determined mainly by the spin relaxation ${\mathrm{\Gamma }}_{\mathrm{\sigma }\mathit{L}}$ of the total magnetic moment of Cu ions to the lattice and by their susceptibilities (bottleneck effect). We found that ${\mathrm{\Gamma }}_{\mathrm{\sigma }\mathit{L}}$ is proportional to temperature ${\mathrm{\Gamma }}_{\mathrm{\sigma }\mathit{L}}$≊BT in a broad region of temperatures with B increasing with Sr doping. At low temperatures T<${\mathit{T}}_{\min }$, with ${\mathit{T}}_{\min }$∼20–110 K depending both on Mn and Sr concentration, the ESR line experiences a sharp broadening which we attribute to the partial opening of the bottleneck. It reveals an inhomogeneous contribution to the Mn ESR linewidth and the relaxation rate ${\mathrm{\Gamma }}_{\mathit{s}\mathrm{\sigma }}$ of the Mn ions to the antiferromagnetic fluctuations in the ${\mathrm{CuO}}_2$ layers, which were hidden in the bottleneck regime. We found ${\mathrm{\Gamma }}_{\mathit{s}\mathrm{\sigma }}$∼${\mathrm{\Gamma }}^{\mathrm{\alpha }}$, with α≊2 for T<${\mathit{T}}_{\min }$, and give some speculations on the origin of this broadening.

${\mathrm{Bi}}_{2+\mathit{x}}$${\mathrm{Sr}}_{2\mathrm{-}\mathit{x}}$${\mathrm{CuO}}_{\mathit{y}}$ (0.10≤x≤0.40) polycrystalline samples have been studied by x-ray diffraction, resistivity and Hall-coefficient measurements, x-ray and ultraviolet photoemission spectroscopy, and inverse-photoemission spectroscopy (IPES). The lattice constant of the c axis decreases and that of the a(b) axis increases as a function of x. The results of Hall-coefficient measurements indicate that hole concentration is reduced as a function of x, which is consistent with the results of oxygen-concentration measurements. We find no energy shifts of the core levels, valence bands, and conduction bands for the samples, which is completely different from the results for the ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{Ca}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Y}}_{\mathit{x}}$${\mathrm{Cu}}_2$${\mathrm{O}}_{\mathit{y}}$ system. The results of IPES and resistivity measurements show that the metal-semiconductor transition occurs at about x=0.35. The states created by hole doping move to the Fermi level. The change in the electronic states is discussed.

It is known that superconductivity in ${\mathrm{TlSrLaCuO}}_5$ is due to the overlap of empty Tl 6s band with the antibonding Cu 3${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ band at the Fermi level. This motivated us to investigate the effects of isovalent-cation substitution in superconducting ${\mathrm{TlSrLaCuO}}_5$ and semiconducting ${\mathrm{TlBaLaCuO}}_5$. We have succeeded in inducing superconductivity in ${\mathrm{TlBaLaCuO}}_5$ by introducing ${\mathrm{Ca}}^{2+}$ at the Ba sites.

We have measured and calculated the electron-positron momentum distribution of ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_4$ samples for Sr concentrations of 0, 0.1, 0.13, and 0.2. Measured distributions were obtained at room temperature with high statistical precision, greater than 4×${10}^8$ events, in the Lawrence Livermore National Laboratory positron-annihilation angular correlation spectrometer on single-crystal samples fabricated using the traveling solvent floating zone technique. Corresponding theoretical momentum-density calculations were performed using the linear muffin-tin-orbital method. The momentum distribution of all samples contained features derived from the overlap of the positron distribution with the valence electrons. In addition, discontinuities typical of a Fermi surface are seen in the doped samples. The form and position of these features are in general agreement with the Fermi surface and overall momentum distributions as predicted by band theory. However, the evolution of the Fermi surface with doping differed significantly from expectations based on single electron band theories.

We have carried out high-resolution dilatometry experiments on a series of high-quality single crystals of ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_4$ with various Sr content on both the under- and overdoped side of the ${\mathit{T}}_{\mathit{c}}$(x) phase diagram. A simple in situ technique was applied to prevent the usual twinning of the samples in the low-temperature orthorhombic (LTO) phase, which allows us to study the full anisotropy of the thermal expansion of the LTO phase. From the anomalies in the linear expansivities Δ${\mathrm{\alpha }}_{\mathit{i}}$ (i=a,b,c) at ${\mathit{T}}_{\mathit{c}}$ we deduce all three uniaxial stress (${\mathit{dT}}_{\mathit{c}}$/${\mathit{dp}}_{\mathit{i}}$) and strain (${\mathit{dT}}_{\mathit{c}}$/d${\mathrm{\varepsilon }}_{\mathit{i}}$) dependences, which turn out to be large and almost cancel for hydrostatic pressure.

We report detailed experimental results on the magnetic susceptibility of high-quality ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ single crystals for the applied magnetic field H parallel, ${\mathrm{\chi }}_{\mathit{a}\mathit{b}}$, and perpendicular, ${\mathrm{\chi }}_{\mathit{c}}$, to the c direction. The data were obtained from 5 to 300 K, and with ${\mathrm{\mu }}_0$H≤0.6 T, for which ${\mathrm{\chi }}_{\mathit{a}\mathit{b}}$ and ${\mathrm{\chi }}_{\mathit{c}}$ above the superconducting transition correspond to the so-called weak magnetic-field limit, even for reduced temperatures, ɛ≡(T-${\mathit{T}}_{\mathit{C}\mathit{O}}$)/${\mathit{T}}_{\mathit{C}\mathit{O}}$, of the order of ɛ=2×${10}^{\mathrm{-}3}$, where ${\mathit{T}}_{\mathit{C}\mathit{O}}$ is the mean-field-like normal-superconducting transition temperature. For T>${\mathit{T}}_{\mathit{C}\mathit{O}}$, ${\mathrm{\chi }}_{\mathit{c}}$ is found to be temperature independent and its average value for the samples measured here is (2.7±0.5)×${10}^{\mathrm{-}5}$. In contrast, ${\mathrm{\chi }}_{\mathit{a}\mathit{b}}$(T) is found to be temperature independent, and equal to (5.2±1)×${10}^{\mathrm{-}5}$, only above T-${\mathit{T}}_{\mathit{C}\mathit{O}}$≊25 K and up to the highest temperature studied here (300 K). Between ${\mathit{T}}_{\mathit{C}\mathit{O}}$ and T-${\mathit{T}}_{\mathit{C}\mathit{O}}$≊25 K, ${\mathrm{\chi }}_{\mathit{a}\mathit{b}}$(T) is appreciably rounded. The excess diamagnetism above but near ${\mathit{T}}_{\mathit{C}\mathit{O}}$, for H parallel to the c axis, Δ${\mathrm{\chi }}_{\mathit{a}\mathit{b}}$, is analyzed in terms of the existing approaches for independent Gaussian fluctuations of the superconducting-order-parameter amplitude in layered materials and also compared with the paraconductivity measured in the same samples.

The so-called mean-field-like region will correspond to, approximately, ${10}^{\mathrm{-}2}$≲ɛ≲0.15. Beyond the lower ɛ limit, closer to ${\mathit{T}}_{\mathit{C}\mathit{O}}$, both the paraconductivity and Δ${\mathrm{\chi }}_{\mathit{a}\mathit{b}}$(ɛ) separate from the mean-field region behavior. Although so close to the transition we cannot exclude the influence of small stoichiometric inhomogeneities, these data for ɛ≲${10}^{\mathrm{-}2}$ may be explained on the grounds of the three-dimensional XY model, with a critical exponent of x=-1/3 for the paraconductivity and x=-2/3 for the fluctuation-induced diamagnetism. A scenario compatible with the experimental results is characterized by the values of the correlation length amplitude in the ab plane and in the c direction of, respectively, ${\xi }_{\mathit{a}\mathit{b}}$(0)=(1.1±0.2) nm and ${\xi }_{\mathit{c}}$(0)=(0.12±0.02) nm, two Josephson coupled superconducting planes in the unit-cell length, and a complex order parameter with two real components, i.e., conventional ${}_{}{}^1{}_{}{}^{}$${\mathit{s}}_0$-wave pairing or one-complex-component unconventional pairing.

Results of electronic structure calculations for the hole density in the two-dimensional ${\mathrm{CuO}}_2$ planes are presented in this paper for the three bismuth cuprate superconductors ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{CuO}}_6$, ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{CaCu}}_2$${\mathrm{O}}_8$, and ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{Ca}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{10}$. The hole density in these compounds arises due to electron transfer from the ${\mathrm{CuO}}_2$ planes to the Bi-O layers. Our calculations show that ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{CuO}}_6$ is so highly overdoped that it is really a metal and at the limit of superconductivity. Also ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{CaCu}}_2$${\mathrm{O}}_8$ has a hole density which is higher than is required for a maximum value of ${\mathit{T}}_{\mathit{c}}$. On the other hand, the hole density in the three-layer compound ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{Ca}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{10}$ is nearly optimum, which indicates that apart from the number of layers, an optimum value of the hole density is important for maximizing ${\mathit{T}}_{\mathit{c}}$. The hole densities in the two types of ${\mathrm{CuO}}_2$ layers in this compound are nearly equal although the central layer has a slightly lower value. As the number of ${\mathrm{CuO}}_2$ planes increases in this family of superconductors the total amount of hole carrier density per chemical formula increases due to an enhanced electron transfer to the Bi-O layers. However, this increase is small so that the hole density per ${\mathrm{CuO}}_2$ drops rapidly as the number of ${\mathrm{CuO}}_2$ planes increases.

In this paper, we find that the strong correlation that exists in the high-${\mathit{T}}_{\mathit{c}}$ cuprate oxides can lead to both elastic and inelastic angle-resolved photoemission spectroscopies. In the elastic process the correlation only produces a normalizing factor to the matrix element of the photon-electron interaction; this term accounts for the peak, satisfies the sum rule, and has a superconducting gap. In the inelastic process the correlation can cause the photoelectrons to scatter inelastically with the charge and spin fluctuations and contributes a large background to the spectroscopy. In particular, the background does not satisfy the sum rule, so that the whole intensity is of non-sum-rule type. It also has an energy threshold in the superconducting state that results in a dip in the spectroscopy. Our theoretical predictions agree with the large background, non-sum-rule, and dip observed in the experiments.

We report on experimental evidence of different electronic phases in the superconductor ${\mathrm{YBa}}_2$${\mathrm{Cu}}_4$${\mathrm{O}}_8$: simple metallic CuO chains and highly correlated ${\mathrm{CuO}}_2$ planes. ${\mathrm{YBa}}_2$${\mathrm{Cu}}_4$${\mathrm{O}}_8$ 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 plane conduction channel), the resistivity reveals an unconventional behavior with a kink at 160 K, becoming linear at higher temperatures. Further, we present results of the dynamical (optical) conductivity of the ${\mathrm{CuO}}_2$ 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.

Using symmetry arguments we show that for a particle-hole symmetric system that has antiferromagnetic long-range order at half-filling, and does not phase separate, the chemical potential will move across the gap when the dopants are changed from holes to electrons.

Temperature-dependent far-infrared reflectivity of rare-earth substituted ceramic R${\mathrm{Ba}}_2$${\mathrm{Cu}}_4$${\mathrm{O}}_8$ with R=Y, Er, Dy, and Gd is reported. The same number of modes are observed in the rare-earth substituted compounds as in previously reported ${\mathrm{YBa}}_2$${\mathrm{Cu}}_4$${\mathrm{O}}_8$ and, except for the Y–rare-earth mode, the shifts in the mode frequencies are small. The temperature-dependent behavior of the phonons and the electronic background in all the samples is also similar to earlier reports of ${\mathrm{YBa}}_2$${\mathrm{Cu}}_4$${\mathrm{O}}_8$. Modes near 310 and 500 ${\mathrm{cm}}^{\mathrm{-}1}$ soften significantly with decreasing temperature beginning 30 or 40 K above the zero-resistance ${\mathit{T}}_{\mathit{c}}$. Other modes display little sensitivity to temperature. There is an enhancement in the reflectivity in the superconducting state relative to the normal state for frequencies less than 600 ${\mathrm{cm}}^{\mathrm{-}1}$ that grows in magnitude rapidly below ${\mathit{T}}_{\mathit{c}}$. The spectral range of the enhancement is independent of temperature, rare-earth substitution, and ${\mathit{T}}_{\mathit{c}}$, a behavior that cannot be described by weak-coupling BCS theory.

In this article we report on thermally excited flux creep and the critical transport current density ${\mathit{j}}_{\mathit{c}}$ in high-quality epitaxial ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{CaCu}}_2$${\mathrm{O}}_{8+\mathrm{\delta }}$ thin films. Both dissipative mechanisms are governed by the highly anisotropic behavior of this compound which was investigated by means of the angular dependence of the magnetoresistivity (γ≥150). The activation energy U for thermally excited flux creep was evaluated with respect to temperature, magnetic field, and applied current density j. U(T,B,j) is essentially increasing linearly with falling temperature, power-law dependent on the field (U∝${\mathit{B}}^{\mathrm{-}\mathrm{\alpha }}$ with α≊0.5), and almost independent of current density for j≤${10}^5$ A/${\mathrm{cm}}^2$. These experimental results are consistent with the concept of plastic flux creep. The critical current density exhibits high absolute values [${\mathit{j}}_{\mathit{c}}$(77 K, B=0)=4×${10}^5$ A/${\mathrm{cm}}^2$] and was measured for magnetic fields in the configuration B∥c up to 10 T and also with respect to various Θ angles between the c axis and field direction. The decrease of ${\mathit{j}}_{\mathit{c}}$ with increasing B was found to be significantly reduced in comparison to single crystals and prior results on thin films. A further enhancement in the ${\mathit{j}}_{\mathit{c}}$(B,T) behavior could not be achieved by chemical doping through partial substitution of copper by zinc.

We present a symmetry analysis of the fully ordered state of solid ${\mathrm{C}}_{60}$ assuming that the angle ${\mathrm{\Gamma }}_0$ by which the ${\mathrm{C}}_{60}$ molecules are rotated about their respective 〈111〉 axes is arbitrary. We find that in the general case the transition is driven by two order parameters: (1) a ${\mathrm{\Gamma }}_2^{+}$(1) scalar order parameter at the Γ point and (2) a six-component order parameter belonging to the ${\mathit{X}}_5^{+}$(6) irreducible representations of Fm3m; however, in the very particular case that ${\mathrm{\Gamma }}_0$=[120°+${\cos }^{\mathrm{-}1}$(1/4)]/2 or ${\cos }^{\mathrm{-}1}$(1/4)/2, the ${\mathrm{\Gamma }}_2^{+}$(1) scalar order parameter vanishes identically. The implications of this analysis for the order-disorder transition in pristine ${\mathrm{C}}_{60}$ and ${\mathrm{K}}_{1.3}$${\mathrm{C}}_{60}$ are discussed.