Phonon dispersion curves and atomic mean square displacement for several fcc and bcc materials

The atomic mean square displacement (MSD) and the phonon dispersion curves (PDC's) of a number of face-centred cubic (fcc) and body-centred cubic (bcc) materials have been calclllated from the quasiharmonic (QH) theory, the lowest order (A2 ) perturbation theory (PT) and a recently proposed Green's function (GF) method by Shukla and Hiibschle. The latter method includes certain anharmonic effects to all orders of anharmonicity. In order to determine the effect of the range of the interatomic interaction upon the anharmonic contributions to the MSD we have carried out our calculations for a Lennard-Jones (L-J) solid in the nearest-neighbour (NN) and next-nearest neighbour (NNN) approximations. These results can be presented in dimensionless units but if the NN and NNN results are to be compared with each other they must be converted to that of a real solid. When this is done for Xe, the QH MSD for the NN and NNN approximations are found to differ from each other by about 2%. For the A2 and GF results this difference amounts to 8% and 7% respectively. For the NN case we have also compared our PT results, which have been calculated exactly, with PT results calculated using a frequency-shift approximation. We conclude that this frequency-shift approximation is a poor approximation. We have calculated the MSD of five alkali metals, five bcc transition metals and seven fcc transition metals. The model potentials we have used include the Morse, modified Morse, and Rydberg potentials. In general the results obtained from the Green's function method are in the best agreement with experiment. However, this improvement is mostly qualitative and the values of MSD calculated from the Green's function method are not in much better agreement with the experimental data than those calculated from the QH theory. We have calculated the phonon dispersion curves (PDC's) of Na and Cu, using the 4 parameter modified Morse potential. In the case of Na, our results for the PDC's are in poor agreement with experiment. In the case of eu, the agreement between the tlleory and experiment is much better and in addition the results for the PDC's calclliated from the GF method are in better agreement with experiment that those obtained from the QH theory.

Calculation of nondifferential properties for atomic ground states /

A new method for sampling the exact (within the nodal error) ground state distribution and nondiflPerential properties of multielectron systems is developed and applied to firstrow atoms. Calculated properties are the distribution moments and the electronic density at the nucleus (the 6 operator). For this purpose, new simple trial functions are developed and optimized. First, using Hydrogen as a test case, we demonstrate the accuracy of our algorithm and its sensitivity to error in the trial function. Applications to first row atoms are then described. We obtain results which are more satisfactory than the ones obtained previously using Monte Carlo methods, despite the relative crudeness of our trial functions. Also, a comparison is made with results of highly accurate post-Hartree Fock calculations, thereby illuminating the nodal error in our estimates. Taking into account the CPU time spent, our results, particularly for the 8 operator, have a relatively large variance. Several ways of improving the eflSciency together with some extensions of the algorithm are suggested.

Sample preparation and heavy metal determination by atomic spectrometry /

Microwave digestions of mercury in Standards Reference Material (SRM) coal samples with nitric acid and hydrogen peroxide in quartz vessels were compared with Teflon® vessel digestion by using flow injection cold vapor atomic absorption spectrometry. Teflon® vessels gave poor reproducibiUty and tended to deliver high values, while the digestion results from quartz vessel show good agreement with certificate values and better standard deviations. Trace level elements (Ag, Ba, Cd, Cr, Co, Cu, Fe, Mg, Mn, Mo, Pb, Sn, Ti, V and Zn) in used oil and residual oil samples were determined by inductively coupled plasma-optical emission spectrometry. Different microwave digestion programs were developed for each sample and most of the results are in good agreement with certified values. The disagreement with values for Ag was due to the precipitation of Ag in sample; while Sn, V and Zn values had good recoveries from the spike test, which suggests that these certified values might need to be reconsidered. Gold, silver, copper, cadmium, cobalt, nickel and zinc were determined by continuous hydride generation inductively coupled plasma-optical emission spectrometry. The performance of two sample introduction systems: MSIS™ and gas-liquid separator were compared. Under the respective optimum conditions, MSIS^"^ showed better sensitivity and lower detection limits for Ag, Cd, Cu, Co and similar values for Au, Ni and Zn to those for the gas-liquid separator.

A treatment of atomic mean square displacement in higher order perturbation theory

A general derivation of the anharmonic coefficients for a periodic lattice invoking the special case of the central force interaction is presented. All of the contributions to mean square displacement (MSD) to order 14 perturbation theory are enumerated. A direct correspondance is found between the high temperature limit MSD and high temperature limit free energy contributions up to and including 0(14). This correspondance follows from the detailed derivation of some of the contributions to MSD. Numerical results are obtained for all the MSD contributions to 0(14) using the Lennard-Jones potential for the lattice constants and temperatures for which the Monte Carlo results were calculated by Heiser, Shukla and Cowley. The Peierls approximation is also employed in order to simplify the numerical evaluation of the MSD contributions. The numerical results indicate the convergence of the perturbation expansion up to 75% of the melting temperature of the solid (TM) for the exact calculation; however, a better agreement with the Monte Carlo results is not obtained when the total of all 14 contributions is added to the 12 perturbation theory results. Using Peierls approximation the expansion converges up to 45% of TM• The MSD contributions arising in the Green's function method of Shukla and Hubschle are derived and enumerated up to and including 0(18). The total MSD from these selected contributions is in excellent agreement with their results at all temperatures. Theoretical values of the recoilless fraction for krypton are calculated from the MSD contributions for both the Lennard-Jones and Aziz potentials. The agreement with experimental values is quite good.

Continuous hydride generation in the presence of L- cysteine for the determination of arsenic, bismuth, antimony and tin in steels by inductively coupled plasma atomic emission spectrometry

Arsenic, bismuth, germanium, antimony and tin were simultaneously determined by continuous hydride generation and inductively coupled plasma-atomic emission spectrometry . I Hydrides were introduced into four different types of gas-liquid separators. Two of the gas-liquid separators were available in-house. A third was developed for this project and a fourth was based on a design used by CET AC. The best signal intensity was achieved by the type II frit-based gas-liquid separator, but the modified Cetac design gave promise for the future, due to low relative standard deviation. A method was developed for the determination of arsenic, bismuth, antimony and tin in low-alloy steels. Four standard reference materials from NIST were dissolved in 10 mL aqua regia without heat. Good agreement was obtained between experimental values and certified values for arsenic, bismuth, antimony and tin. The method was developed to provide the analyst with the opportunity to determine the analytes by using simple aqueous standards to prepare calibration lines. Within the limits of the samples analyzed, the method developed is independent of matrix.

Investigation into the determination of arsenic by direct current plasma atomic emission spectrometer

Improvements have been made on the currently available hydride generator system manufactured by SpectraMetrics Incorporated, because the system was found to be unsatisfactory with respect to the following: 1. the drying agent, anhydrous calcium chloride, 2. the special sample tube, 3. the direction of argon flow through the Buchner funnel when it came to dealing with real sample, that is, with reference only to aqueous extracts of soil samples. Changes that were made on the system included the replacement of anhydrous calcium chloride with anhydrous calcium sulphate and the replacement of the special sample tube with a modified one made from silica. Re-directing the flow of argon through the top of the Buchner funnel appeared to make the system compatible with aqueous extracts of soil samples. The interferences from 1000 ~g/mL of nickel(II) , cobalt(II), iron(III), copper(II) have been eliminated with the aid of 1.4 M hydrochloric acid and 1% (weight/volume) L-cystine. Greater than 90% recovery of 0.3 ~g/mL arsenic signal was achieved in each case. Furthermore, 103% of arsenic signal was accomplished in the presence of 1000 ~g/mL cadmium with 5 M Hel. tVhen each of the interferents was present in solution at 1000 ppm, a recovery of 85% was achieved by using 5 M hydrochloric acid and 3% (weight/volume) L-cystine. Without L-cystine and when 1.4 M hydrochloric acid was used, the recoveries were 0% (Ni), 0% (Co), 88% (Fe), 15% (Cu), 18% (Cd). Similarly, a solution containing 1000 ppm of each interferent gave a zero percent recovery of arsenic. The reduction of trivalent and pentavalent arsenic at a pH less than one has also been investigated and shown to be quantitative if peak areas are measured. The reproducibility determination of a 0.3 Vg/mL standard arsenic solution by hydride generation shows a relative standard deviation of 3.4%. The detection limits with and without Porapak Q have been found to be 0.6 ng/mL and 1.0 ng/mL, respectively.

On the equation of state and atomic mean square displacement of crystals

We have presented a Green's function method for the calculation of the atomic mean square displacement (MSD) for an anharmonic Hamil toni an . This method effectively sums a whole class of anharmonic contributions to MSD in the perturbation expansion in the high temperature limit. Using this formalism we have calculated the MSD for a nearest neighbour fcc Lennard Jones solid. The results show an improvement over the lowest order perturbation theory results, the difference with Monte Carlo calculations at temperatures close to melting is reduced from 11% to 3%. We also calculated the MSD for the Alkali metals Nat K/ Cs where a sixth neighbour interaction potential derived from the pseudopotential theory was employed in the calculations. The MSD by this method increases by 2.5% to 3.5% over the respective perturbation theory results. The MSD was calculated for Aluminum where different pseudopotential functions and a phenomenological Morse potential were used. The results show that the pseudopotentials provide better agreement with experimental data than the Morse potential. An excellent agreement with experiment over the whole temperature range is achieved with the Harrison modified point-ion pseudopotential with Hubbard-Sham screening function. We have calculated the thermodynamic properties of solid Kr by minimizing the total energy consisting of static and vibrational components, employing different schemes: The quasiharmonic theory (QH), ).2 and).4 perturbation theory, all terms up to 0 ().4) of the improved self consistent phonon theory (ISC), the ring diagrams up to o ().4) (RING), the iteration scheme (ITER) derived from the Greens's function method and a scheme consisting of ITER plus the remaining contributions of 0 ().4) which are not included in ITER which we call E(FULL). We have calculated the lattice constant, the volume expansion, the isothermal and adiabatic bulk modulus, the specific heat at constant volume and at constant pressure, and the Gruneisen parameter from two different potential functions: Lennard-Jones and Aziz. The Aziz potential gives generally a better agreement with experimental data than the LJ potential for the QH, ).2, ).4 and E(FULL) schemes. When only a partial sum of the).4 diagrams is used in the calculations (e.g. RING and ISC) the LJ results are in better agreement with experiment. The iteration scheme brings a definitive improvement over the).2 PT for both potentials.

Thermodynamic properties and Debye-Waller factor of fee materials

We have calculated the equation of state and the various thermodynamic properties of monatomic fcc crystals by minimizing the Helmholtz free energy derived in the high temperature limit for the quasiharmonic theory, QH, and the lowest-order (cubic and quartic), 'A2, anharmonic terms of the perturbation theory, PT. The total energy in each case is obtained by adding the static energy. The calculation of the thermal properties was carried out for a nearest-neighbour central-force model of the fcc lattice by means of the appropriate thermodynamic relations. We have calculated the lattice constant, the thermal expansion, the coefficient of volume expansion, the specific heat at constant volume and at constant pressure, the isothermal and adiabatic bulk moduli, and the Griineisen parameter, for the rare-gas solids Kr and Xe, and gold. Morse potential and modified Morse potential were each used to represent the atomic interaction for the three fcc materials. For most of the calculated thermodynamic properties from the QH theory, the results for Kr and Xe with the modified Morse potential show an improvement over the results for the Morse potential when compared with the experimental data. However, the results of the 'A 2 equation of state with the modified Morse potential are in good agreement with experiment only in the case of the specific heat at constant volume and at constant pressure. For Au we have calculated the lattice contribution from the QH and 'A 2 PT and the electronic contribution to the thermal properties. The electronic contribution was taken into account by using the free electron model. The results of the thermodynamic properties calculated with the modified Morse potential were similar to those obtained with the Morse potential. U sing the minimized equation of state we also calculated the Mossbauer recoilless fraction for Kr and Xe and the Debye-Waller factor (DWF) for Pb, AI, eu, Ag, and Au. The Mossbauer recoilless fraction was obtained for the above two potentials and Lennard-Jones potential. The L-J potential gives the best agreement with experiment for Kr. No experimental data exists for Xe. At low temperature the calculated DWF results for Pb, AI, and eu show a good agreement with experimental values, but at high temperature the experimental DWF results increase very rapidly. For Ag the computed values were below the expected results at all temperatures. The DWF results of the modified Morse potential for Pb, AI, eu and Ag were slightly better than those of the Morse potential. In the case of Au the calculated values were in poor agreement with experimental results. We have calculated the quasiharmonic phonon dispersion curves for Kr, Xe, eu, Ag, and Au. The calculated and experimental results of the frequencies agree quite well for all the materials except for Au where the longitudinal modes show serious discrepancies with the experimental results. In addition, the two lowest-order anharmonic contributions to the phonon frequency were derived using the Green's function method. The A 2 phonon dispersion curves have been calculated only for eu, and the results were similar to those of the QH dispersion curves. Finally, an expression for the Griineisen parameter "( has been derived from the anharmonic frequencies, and calculated for these materials. The "( results are comparable with those obtained from the thermodynamic definition.

Landscape geochemical investigations in the vicinity of a lead deposit near Snertingdal, southern Norway

Landscape geochemical investigations were conducted upon portions of a natural uniform landscape in southern Norway. This consisted of sampling both soil profile samples and spruce tree twigs for the analysis of twelve chemical elements. These elements were cobalt, copper, nickel, lead, zinc, manganese, magnesium, iron, calcium, sodium, potassium and aluminum which were determined by atomic absorption analysis on standardized extraction techniques for both organic and inorganic materials. Two "landscape traverses" were chosen for a comparative study of the effects of varying landscape parameters upon the trace element distribution patterns throughout the landscape traverses. The object of this study was to test this method of investigation and the concept of an ideal uniform landscape under Norwegian conditions. A "control traverse" was established to represent uniform landscape conditions typical of the study area and was used to determine "normal" or average trace element distribution patterns. A "signal traverse" was selected nearby over an area of lead mineralization where the depth to bedrock is very small. The signal traverse provided an area of similar landscape conditions to those of the control traverse with significant differences in the bedrock configuration and composition. This study was also to determine the effect of the bedrock mineralization upon the distribution patterns of the twelve chemical elements within the major components of the two landscape traverses (i.e. soil profiles and tree branches). The lead distribution within the soils of the signal traverse showed localized accumulations of lead within the overburden with maximum values occurring within the organic A horizon of soil profile #10. Above average concentrations of lead were common within the signal traverse, however, the other elements studied were not significantly different from those averages determined throughout the soils of the control traverse. The spruce twig samples did not have corresponding accumulations of lead near the soil lead anomaly. This is attributable to the very localized nature of the lead dispersion pattern within the soils. This approach to the study of the geochemistry of a natural landscape was effective in establishing: a) average or "normal" trace element distribution patterns b) local variations in the landscape morphology and c) the effect of unusually high lead concentrations upon the geochemistry of the landscape (i.e. within the soil profiles and tree branches). This type of study provides the basis for further more intensive studies and serves only as a first approximation of the behaviour of elements within a natural landscape.