Lattice contribution

The result is that Factor III of 2.2.6. given above imposes further symmetry restrictions on the 32 point groups and we obtain a total of 231 space groups. We do not intend to delve further into this aspect of lattice contributions to crystal structure of solids, and the factors which cause them to vary in form. It is sufficient to know that they exist. Having covered the essential parts of lattice structure, we will elucidate how one goes about determining the structure for a given solid. 

The EFG parameters Vzz and described by (4.42a) and (4.42b) do not represent the actual EFG felt by the Mossbauer nucleus. Instead, the electron shell of the Mossbauer atom will be distorted by electrostatic interaction with the noncubic distribution of the external charges, such that the EFG becomes amplified. This phenomenon has been treated by Stemheimer [54—58], who introduced an anti-shielding factor (1 —y 00) for computation of the so-called lattice contribution to the EFG, which arises from (point) charges located on the atoms surrounding the Mossbauer atom in a crystal lattice (or a molecule). In this approach,the actual lattice contribution is given by... 

For iron compounds, (yzz)iat and iat are amplified by approximately (1 —yc ) 10 as compared to the point charge contributions 14z and rj obtained from (4.42a) and (4.42b). Nevertheless, the lattice contribution is usually least significant for most iron compounds because it is superseded by a strong EFG from the valence electrons details will be found in Chap. 5. 

W(VI) compounds with 5cP electronic configuration, the EFG is reasonably weU described by the lattice contribution only. The regular octahedral WOs structure yields zero EFG in agreement with experiment, while the deformed tetrahedral W04 strucmre results in significant quadrupole coupling constants of either sign, e Q > 0 and also <0, depending on the distortion of the O4 tetrahedron. 

Correspondingly, replacement of the phenol (Fig. 33) by m-cresol (with the methyl oriented partially along a, Fig. 35) is associated with an increase of only about 0.65 A in the length of the c axis. The methyl substituent is perfectly well accommodated within the expanded lattice, contributing additional interactions of dispersion without an apparent distortion of the previous ones. All intermolecular distances remain well within range of characteristic van der Waals values 49). 

Usually in a metal, this contribution is much larger than the lattice contribution due to the strong difference in the velocity of heat carriers. Again we wish to point out that there are two phenomena which produce the scattering of the electrons as heat carriers ... 

The Mossbauer spectral isomer shifts of both spin states in [Fe(HB(3,5-(CH3)2pz)3)2] show the expected decrease with increasing pressure as the s-electron density at the iron-57 nucleus increases. In contrast, the quadrupole splitting for the high-spin state is almost independent of pressure whereas that of the low-spin state, which is dominated by the lattice contribution to... 

Since the lattice contribution to the heat capacity varies with T3, the total heat capacity at low temperatures (typically T< 10 K or lower) to a first approximation is given by... 

In Fe2+, the situation is more complicated. Here the six 3d-electrons dominate the magnitude of the electric field gradient, though in a way that is determined by the lattice symmetry. If a Fe2+ ion comes in a more asymmetric environment, the quadrupole splitting decreases in general, because the lattice contribution to the electric field gradient is smaller than the electronic contribution, and has the opposite sign. 

To subtract the cation Mg + from its lattice position in the crystal and to bring it to the surface, we must work against the static potentials (coulombic plus repulsive plus dispersive) at the Mg site. In terms of energy, this work corresponds to half the lattice contribution of Mg + (in the Mg site of interest—i.e.. Ml or M2 see section 5.2) to the bulk static energy of the phase (see also section 1.12) ... 

As in chapter 8, we will refer to contributions from the electron density centered on the nucleus as central contributions, and to the remainder as peripheral contributions. In the spectroscopic literature, the latter are commonly referred to as lattice contributions, a term we will avoid as it conflicts with the common definition of the lattice as a mathematical concept. 

Fig. 3. C/T versus for UAI2 and PuAlj up to 20 K. Also shown is the model lattice contribution for O = 340 K. (Trainor et al. )...

That the wavefunction is not much changed by solvation is seen in Fig. 4, where the hole population profile is shown for coupling to the lattice displacements and the environment separately and together. The parameters used are to=0.2 eV, a lK=0.2 eV. The lattice contribution can be included in the hole Hamiltonian (Eq. 10) by adding -2a /K to g i from Eqs. 13 and 14. 

Here K, Ki and the Debye temperature d are parameters which can be obtained by fitting (1) and (2) to the thermal expansion in the paramagnetic temperature range. The electronic contribution in (1), ee = K T2, is usually much smaller than the lattice contribution ephom i.e. in most cases it makes no difference when only fphon is taken into account for determining the nonmagnetic contribution to the thermal expansion. 

Fig. 5. Temperature variation of the hexagonal lattice parameters and of the volume of pure gadolinium measured by x-ray powder diffraction (this work). The values have been normalized to 300 K in order to show the relative changes. (The values at 300 K are a = 3.632 0.002 A, c = 5.782 0.002 A.) The lines represent the extrapolation of the lattice contribution from temperatures above Tq assuming a Debye temperature of 184 K (Bodnakov et al. 1998). The lowest part of the figure shows the magnetovolume effect, obtained by subtracting the lattice contribution from the volume expansion.

Fig. 14. Anisotropic thermal expansion of GdCuSn measured by x-ray powder diffraction (Gratz and Lindbaum 1998). The lines represent the extrapolation of the lattice contribution from the paramagnetic range by fitting...

McKean 182> considered the matrix shifts and lattice contributions from a classical electrostatic point of view, using a multipole expansion of the electrostatic energy to represent the vibrating molecule and applied this to the XY4 molecules trapped in noble-gas matrices. Mann and Horrocks 183) discussed the environmental effects on the IR frequencies of polyatomic molecules, using the Buckingham potential 184>, and applied it to HCN in various liquid solvents. Decius, 8S) analyzed the problem of dipolar vibrational coupling in crystals composed of molecules or molecular ions, and applied the derived theory to anisotropic Bravais lattices the case of calcite (which introduces extra complications) is treated separately. Freedman, Shalom and Kimel, 86) discussed the problem of the rotation-translation levels of a tetrahedral molecule in an octahedral cell. 

Also, the heat capacity is affected by the axial ZFS parameter and, in excess of the lattice contribution, it shows a Schottky anomaly as modeled in Fig. 2. In the zero magnetic field the isofield heat capacity Ch collapses to the usual Cp and stays isotropic. 

The lattice contribution for all the crystals CUl is equal to 6R at / > 150 K. This assumption is true for most crystals of hydrocarbons with large molar mass. 

Low-temperature adiabatic heat capacity (Cp) measurements then were carried out on microcrystalline BABI to look for evidence of ordering below 2K 190 The expected anomaly was somewhat broad but readily seen in the right-hand chart of Fig. 25, with a maximum at 2 K the shape was consistent with expectations for a low-dimensional antiferromagnet. After subtraction of molecular lattice contributions by a Debye-type extrapolation from the higher... 

Fig. 25 Adiabatic heat capacity for BABI below 200 K (upper left) and over full temperature range (lower left). Magnetic heat capacity after subtraction of lattice contributions (upper right), and with fitted curves (lower right) using the following models (A) square planar AFM system with /2D/k = —1.6K (B) square planar bilayer AFM system with /2D/fc = — 1.2K and interlayer Jjk = — 1.9K (C) AFM spin pairing with Jjk = —2.8 K D is same model as B, with /2D/k = — 1.4K and interlayer Jjk = —1.3 K (from the magnetic analysis of Fig. 24).

The lattice contribution in the lanthanide fluorides is only known with sufficient accuracy when the f shell of the metal ions is empty (4f°) or completely filled (4f14). In these cases Sexs is zero and the experimental entropy corresponds to Siat. Also in case the f-shell of the metal... 

Fig. 10. The variation of Sexp (o, hexagonal O, monoclinic) and Slat ( ) in the lanthanide trichloride series at 298.15 K the broken line shows the lattice contribution (see text).

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