An important issue—how to calculate the finite temperature properties of the lanthanide metals and their compounds—. General characteristics of the thermal expansion of rare earth aluminates and gallates, as well as the observed.
INTRODUCTION
This is due to the fact that SIC is only essential for localized states, which are the 4f-states. However, there have recently been applications of the quasi-particle self-consistent GW method to localized 4f systems (Chantis et al., 2007).
SALIENT PHYSICAL PROPERTIES
Lattice parameters
For the elementary lanthanides, it is the radius of the atomic sphere, S, that appears instead of the lattice parameter, where S is defined as V¼4=3pS3 with the unit cell volume. Pressure studies have been able to unravel much of the physics of the rare earths.
Magnetic properties and magnetic order
The disturbing potential is simply the scalar product of the magnetic moment,m, and the magnetic field,^ B, experienced by the atom:. If this model is correct, it is only necessary to know the values of the quantum numbers J, L and S to calculate the magnetic moment of the lanthanides.
Fermi surfaces
Large mass improvements were measured, especially for an elemental metal, and understanding the Fermi surface required an approach well beyond the Standard Model (Temmerman et al., 1993). Gd metal is the most thoroughly studied of all lanthanides and one of the few lanthanides studied with the de Haas van Alphen technique (Schirber et al., 1976; Mattocks and Young, 1977).
BAND STRUCTURE METHODS
- Local spin density approximation
- OP scheme
- Local density approximation þ Hubbard U
- Self-interaction-corrected local spin density approximation
- Local self-interaction-corrected local spin density approximation
- The GW method
- Dynamical mean field theory
A second advantage of the SIC-LSD scheme is that localized f-states with different symmetry can be accounted for. Further technical details of the current numerical implementation can be found in Temmerman et al.
VALENCE AND VALENCE TRANSITIONS 1 Determining valence
Valence of elemental lanthanides
The energy differences seen in Figure 4 are large and positive at the beginning of the lanthanide series, indicating a preference for the trivalent state, but towards the middle of the series they drop sharply and eventually become negative for Eu. To understand these anomalous changes in energies, in Figure 5, we show the corresponding energy differences as a function of the change in the number of itinerant f electrons across the lanthanide series, and also for the sulfides.
Valence of pnictides and chalcogenides
The reason is that the true magnetic structures of the lanthanide compounds are not always known. The contribution of the empty spheres to the total spin magnetic moments is not shown.
Valence of ytterbium compounds
The difference between the present electronic structure and the picture provided by Degiorgi et al may be due to the LSD approximation, since the position of the narrow f14 band in theory is solely determined by the LSD potential (no correlation correction). LDAþU calculations on YbN (Larson et al., 2007) include a positive correlation shift of the unoccupied f states leading to an ideal trivalent Yb ion in agreement with Degiorgi et al.
Valence transitions
The results reported in Table 4 show that all the observed pressure transitions in the cerium pnictides and chalcogenides are indeed reproduced (Svane et al., 1998). In CeS the first transition occurs to the B1 phase with delocalized f electrons (Svane et al., 1999), i.e. the theory predicts an isostructural phase transition in CeS similar to CeP. For Yb chalcogenides, the application of pressure leads to a pronounced softening of the pV curve (Jayaraman et al., 1974; Werner et al., 1981; Syassen, 1986).
In Figure 15, the experimental pV-curve (Syassen, 1986) of YbS is compared with the theoretical curves obtained by SIC-LSD for the two valence scenarios f13 and f14 (Temmerman et al., 1999). In the YbAl3 intermetallic compound, pressure causes a continuous increase in valence (Kumar et al., 2008), which is in good agreement with the calculated depletion rate of the 14th f-band in SIC-LSD calculations.
Valence of lanthanide oxides
Calculations based on the LSD approximation tend to give a reliable picture of the electronic structure of CeO2 (Koelling et al., 1983), resulting in the tetravalent ground state configuration, with no localized f electrons. Delocalized f states are allowed to participate in band formation and they occur as part of the tails of the O-p states. In Figure 20A, we show the evolution along the unoccupied f-band sequoia series (hatched area showing the f-band width) and the edge of the non-f-conducting band relative to the top of the O p bands (at zero energy) .
Regarding experimental data, in Figure 20B, one notices a considerable difference between optical (Prokofiev et al., 1996) and conductivity (Lal and Gaur, 1988) measurements of the energy gaps. The horizontal axis in the band structure diagram refers to the wave vector along the symmetry directions indicated by the high symmetry points of A,G, M, K,G, in the BZ of the A-type lattice.
LOCAL SPIN AND ORBITAL MAGNETIC MOMENTS
Hund’s rules
In the description of the Standard Model, section 2.2, it is assumed that the electrons are distributed among the available f-states according to Hund's rules. A good basic theory of electronic structure is one in which Hund's rules fall outside the theory rather than having to be included empirically. As mentioned, in order to apply the SIC to the f-states, it is first necessary to decide which (and how many) states to apply it to based on total energy considerations.
Apparently, it is energetically preferable to set up the spins parallel to each other of the individual states than to have them anti-parallel, and thus Hund's first rule is satisfied. In the heavy lanthanides, similar behavior is displayed, but the lowest total energy is when the orbital moment is a maximum and parallel to the spin moment, also completely in accordance with Hund's rules.
The heavy lanthanides
We will now discuss the results of the calculation of the magnetic moments of the heavy lanthanides. Thus, current theory appears to overestimate these valence contributions to the ordered moment of the lanthanides. Surprisingly, by omitting the spin and orbital momenta of the valence states, we obtain an excellent agreement between theory and experiment.
This may indicate that the experiment only provides the contribution of the localized states to the spin and orbital magnetic moments. 5) and the experimental moment derived from the Curie-Weiss behavior of the magnetic susceptibility in the paramagnetic phases (Jensen and Mackintosh, 1991).
The light lanthanides
This shows that in the light lanthanides, the f states are as localized as for the heavy lanthanides. Now let us consider the effect of the crystal environment on the magnetic moment of the lanthanides. In the last two columns of Table 10, we see that the SIC-LSD theory gives values slightly smaller than, but very close to, these numbers.
It is not at all surprising that the moments of the valence electrons can vary so greatly between different crystal structures. Such differences in the way the valence electrons interact with the f-states will undoubtedly lead to strong variations in the valence band moments.
SPECTROSCOPY
Hubbard-I approach to lanthanide photoemission spectra
This is due to the different atomic multiplet characteristics, which serve as a fingerprint of the configuration the lanthanide ion adopts in the solid environment. Not included in the description based on Eq. 37) are the matrix elements between the outgoing photoelectron and the orbital of the hole left behind. The position of the corresponding levels in the reference atom calculation is marked in the figure.
For the high-energy shoulder, there is a 1.5 eV difference in position, which could be due to the neglect of the non-f electrons in the theory. The two experiments differ slightly in the relative intensities of the divalent and trivalent parts.
Relativistic theory of resonant X-ray scattering
The final expression for the resonant part of the scattering amplitude in Bragg diffraction is In the LMTO implementation, the resonance matrix element mð Þ þLtit jkð Þql can be written as This is defined as the difference divided by the sum of the cross section for the moment in one direction and the moment in the opposite direction.
Calculations of the asymmetry ratio for the heavy lanthanide metals have been performed by Brown et al. It is clear that feature A in the scattering corresponds to the peak in the 5d minority spin states.
FINITE TEMPERATURE PHASE DIAGRAMS 1 Thermal fluctuations
Spin fluctuations: DLM picture
An expression for S(2)(q), involving scattering quantities obtained from the electronic structure of the paramagnetic state, can be found in Staunton et al. The band improves the sensitivity to the nest vector for all volumes . The behavior of the paramagnetic spin sensitivity,w, is determined by the direct correlation function,S(2)[Eq.
We can also calculate estimates of the magnetic ordering vectors of all the heavy lanthanides from our susceptibility calculations for gadolinium at the appropriate lattice parameters. Overall, the physical picture that emerges from the magnetic ordering phase diagram, Figure 38, unambiguously links the lattice parameters of the heavy.
Valence fluctuations
Our results have confirmed the critical role that the c/a ratio plays in determining the magnetic order types of the heavy lanthanides and how this is linked to the Fermi surface of the paramagnetic phase. We note that the underestimation of the volume of both the a and g phases is mainly due to the KKR l convergence problem. The free energy of the system is obtained by adding the relevant entropy contributions to the total energy [see Eq.
In addition, the calculated slope of the phase separation line is slightly too steep compared to the experimental phase diagram. If we examine in more detail the discontinuity across the phase separation line shown in Figure 44, we find the magnitude of the discontinuities for the different contributions to the Gibbs free energy.
DYNAMICAL FLUCTUATIONS: THE ‘ALLOY ANALOGY’ AND THE LANDAU THEORY OF PHASE TRANSITIONS
Indeed, when suitably generalized, it can be used to calculate the coefficients in the Landau expansion of the free energy (Gyorffy et al., 1989), thereby turning the phenomenological theory into a material-specific, quantitative first-principles theory. Thus, the first principles Landau Theory alluded to above can be considered a robust and efficient theory of the phase diagrams. A particularly general shortcoming of the 'alloy analogy' approximation is that it cannot describe quantum fluctuations, such as the zero-point fluctuations of spin waves.
It is clear that the full complexity of the scattering events caused by such "targets" is not described by the language of static mean field theory of the "alloy analogy". On the other hand, if TK is Tc, static calculations miss important aspects of the physics.
CONCLUSIONS
ALUMINATES
- Formation and preparation .1 Phase formation and stability
- Crystal structure
- Solid solutions based on RAlO 3 perovskite-type compounds
Some members of the RAlO3 (R¼ LaEu) series are stable over a wide temperature range and melt congruently at temperatures from 2263 to 2438 K (Table 1). Another explanation of the LT structural behavior of PrAlO3 was given by Moussa et al. The temperature dependences of the lattice parameters obtained during cooling and heating (Fig. 4) clearly reflect all three anomalies and 205 K) described in the literature.
The thermal behavior of the GdAlO3 structure was investigated in situ in the temperature range 14–1170 K using synchrotron powder diffraction (Vasylechko et al., 2003a, 2007a). The temperature behavior of the DyAlO3 structure in the range of 2981170 K was reported by Vasylechko et al. The lattice parameters of the orthorhombic structure of HoAlO3 were reported by Toropov et al., 1969; Dernier and Maines, 1971; and Shishido et al., 1995a.
Lattice parameters and cell volume of the TABLE 18 Structural parameters of LuAlO3(S.G.Pbnm) at different temperatures.
