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.

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.

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.

Flow injection analysis (FIA) determination of lead by hydride generation in banded coral skeletons /

A flow injection hydride generation direct current plasma atomic emission spectrometric (FI-HG-DCP-AES) method was developed for the determination of lead at ng.ml-l level. Potassium ferricyanide (K3Fe(CN)6) was used along with sodium tetrahydroborate(III) (NaBH4) to produce plumbane (PbH4) in an acid medium. The design of a gas-liquid separator (hydride generator) was tested and the parameters of the flow injection system were optimized to achieve a good detection limit and sample throughput. The technique developed gave a detection limit of 0.7 ng.ml-l(3ob). The precision at 20 ng.ml"* level was 1.6 % RSD with 1 1 measurements (n=l 1). Volume of sample loop was 500 |J.l. A sample throughput of 120 h"^ was achieved. The transition elements, Fe(II), FeOH), Cd(n), Co(II), Mn(n), Ni(II) and Zn(n) do not interfere in this method but 1 mg,l'l Cu(II) will suppress 50 % of the signal from a sample containing 20 ng.ml'l Pb. This method was successfully applied to determine lead in a calcium carbonate (CaC03) matrix of banded coral skeletons from Si-Chang Island in Thailand.

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.

Application of [Lambda] to the fourth perturbation theory in calculating the equation of state of rare gas solids and fcc metals

We have calculated the thermodynamic properties of monatomic fcc crystals from the high temperature limit of the Helmholtz free energy. This equation of state included the static and vibrational energy components. The latter contribution was calculated to order A4 of perturbation theory, for a range of crystal volumes, in which a nearest neighbour central force model was used. We have calculated the lattice constant, the coefficient of volume expansion, the specific heat at constant volume and at constant pressure, the adiabatic and the isothermal bulk modulus, and the Gruneisen parameter, for two of the rare gas solids, Xe and Kr, and for the fcc metals Cu, Ag, Au, Al, and Pb. The LennardJones and the Morse potential were each used to represent the atomic interactions for the rare gas solids, and only the Morse potential was used for the fcc metals. The thermodynamic properties obtained from the A4 equation of state with the Lennard-Jones potential, seem to be in reasonable agreement with experiment for temperatures up to about threequarters of the melting temperature. However, for the higher temperatures, the results are less than satisfactory. For Xe and Kr, the thermodynamic properties calculated from the A2 equation of state with the Morse potential, are qualitatively similar to the A 2 results obtained with the Lennard-Jones potential, however, the properties obtained from the A4 equation of state are in good agreement with experiment, since the contribution from the A4 terms seem to be small. The lattice contribution to the thermal properties of the fcc metals was calculated from the A4 equation of state, and these results produced a slight improvement over the properties calculated from the A2 equation of state. In order to compare the calculated specific heats and bulk moduli results with experiment~ the electronic contribution to thermal properties was taken into account~ by using the free electron model. We found that the results varied significantly with the value chosen for the number of free electrons per atom.

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.

Investigations into the determinations of hydride-forming elements

Analytical methods for the determination of trace amounts of germanium, tin and arsenic were established using hydride generation coupled with direct current plasma atomic emission spectrometry. A continuous gas flowing batch system for the hydride generation was investigated and was applied to the determination of germanium(Ge), tin(Sn), antimony(Sb) and lead(Pb) (Preliminary results suggest that it is also applicable to arsenic)As) ). With this system, the reproducibility of signals was improved and the determination was speeded up, compared with the conventional batch type hydride generation system. Each determination was complete within one minute. Interferences from a number of transition metal ions, especially from Pd(II), Pt(IV), Ni(II), Cu(II), Co(II), and Fe(II, III), have proven to be very serious under normal conditions, in the determination of germanium, tin, and arsenic. These interference effects were eliminated or significantly reduced in the presence of L-cystine or L-cysteine. Thus, a 10-1000 fold excess of Ni(II), Cu(II), Co(II), Fe(II), Pt(IV), Pd(II), etc. can be tolerated without interference, In the presence of L-cystine or L-cysteine, compared with absence of interference reducing agent. The methods for the determination of Ge, Sn, and As were examined by the analyses of standard reference materials. Interference effects from the sample matrix, for example, in transition metal-rich samples, copper, iron and steel, were eliminated by L-cystine (for As and Sn) and by LI cysteine (for Ge). The analysis of a number of standard reference materials gave excellent results of As and Sn contents in agreement with the certified values, showing there was no systematic interference. The detection limits for both germanium and tin were 20 pg ml- I . Preliminary studies were carried out for the determination of antimony and lead. Antimony was found to react with NaBH4, remaInIng from the previous determinations, giving an analytical signal. A reversed injection manner, i.e., injection of the NaBH4 solution prior to the analyte solution was used to avoid uncertainty caused by residual NaBH4 present and to ensure that an excess of NaB H4 was available. A solution of 0.4% L-cysteine was found to reduce the interference from selected transition metal ions, Co(II), Cu(II), Ni(II) and Pt(IV). Hydrochloric acid - hydrogen peroxide, nitric acid - ammonium persulphate, and potassium dichromate malic acid reaction systems for lead hydride generation were compared. The potassium dichromate - malic acid reaction medium proved to be the best with respect to reproducibility and minimal interference. Cu(II), Ni(II), and Fe(II) caused strong interference In lead determinations, which was not reduced by L-cysteine or Lcystine. Sodium citrate, ascorbic acid, dithizone, thiosemicarbazide and penicillamine reduced interferences to some extent. Further interference reduction studies were carried out uSIng a number of amino acids, glycine, alanine, valine, leucine and histidine, as possible interference reducing agents in the determination of germanium. From glycine, alanine, valine to leucine, the interference reduction effect in germanium determinations decreased. Histidine II was found to be very promising In the reduction of interference. In fact, histidine proved more efficient than L-cystine and L-cysteine In the reduction of interference from Ni(II) in the determination of germanium. Signal enhancement by easily ionized elements (EIEs), usually regarded as an interference effect in analysis by DCP-AES, was studied and successfully applied to advantage in improving the sensitivity and detection limit in the determination of As, Ge, Sn, Sb, and Pb. The effect of alkali and alkaline-earth elements on the determination of the above five hydride forming elements was studied. With the appropriate EIE, a signal enhancement of 40-115% was achieved. Linear calibration and good reproducibility were also obtained in the presence of EIEs. III

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.

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.

Development and characterization of concentric capillary nebulizer used in inductively coupled plasma analysis /

A simple, low-cost concentric capillary nebulizer (CCN) was developed and evaluated for ICP spectrometry. The CCN could be operated at sample uptake rates of 0.050-1.00 ml min'^ and under oscillating and non-oscillating conditions. Aerosol characteristics for the CCN were studied using a laser Fraunhofter diffraction analyzer. Solvent transport efficiencies and transport rates, detection limits, and short- and long-term stabilities were evaluated for the CCN with a modified cyclonic spray chamber at different sample uptake rates. The Mg II (280.2nm)/l\/lg 1(285.2nm) ratio was used for matrix effect studies. Results were compared to those with conventional nebulizers, a cross-flow nebulizer with a Scott-type spray chamber, a GemCone nebulizer with a cyclonic spray chamber, and a Meinhard TR-30-K3 concentric nebulizer with a cyclonic spray chamber. Transport efficiencies of up to 57% were obtained for the CCN. For the elements tested, short- and long-term precisions and detection limits obtained with the CCN at 0.050-0.500 ml min'^ are similar to, or better than, those obtained on the same instrument using the conventional nebulizers (at 1.0 ml min'^). The depressive and enhancement effects of easily ionizable element Na, sulfuric acid, and dodecylamine surfactant on analyte signals with the CCN are similar to, or better than, those obtained with the conventional nebulizers. However, capillary clog was observed when the sample solution with high dissolved solids was nebulized for more than 40 min. The effects of data acquisition and data processing on detection limits were studied using inductively coupled plasma-atomic emission spectrometry. The study examined the effects of different detection limit approaches, the effects of data integration modes, the effects of regression modes, the effects of the standard concentration range and the number of standards, the effects of sample uptake rate, and the effect of Integration time. All the experiments followed the same protocols. Three detection limit approaches were examined, lUPAC method, the residual standard deviation (RSD), and the signal-to-background ratio and relative standard deviation of the background (SBR-RSDB). The study demonstrated that the different approaches, the integration modes, the regression methods, and the sample uptake rates can have an effect on detection limits. The study also showed that the different approaches give different detection limits and some methods (for example, RSD) are susceptible to the quality of calibration curves. Multicomponents spectral fitting (MSF) gave the best results among these three integration modes, peak height, peak area, and MSF. Weighted least squares method showed the ability to obtain better quality calibration curves. Although an effect of the number of standards on detection limits was not observed, multiple standards are recommended because they provide more reliable calibration curves. An increase of sample uptake rate and integration time could improve detection limits. However, an improvement with increased integration time on detection limits was not observed because the auto integration mode was used.

Application of thermal ion-molecule reactions in tackling spectroscopic inferences in inductively coupled plasma mass spectrometry

Part I: Ultra-trace determination of vanadium in lake sediments: a performance comparison using O2, N20, and NH3 as reaction gases in ICP-DRC-MS Thermal ion-molecule reactions, targeting removal of specific spectroscopic interference problems, have become a powerful tool for method development in quadrupole based inductively coupled plasma mass spectrometry (ICP-MS) applications. A study was conducted to develop an accurate method for the determination of vanadium in lake sediment samples by ICP-MS, coupled with a dynamic reaction cell (DRC), using two differenvchemical resolution strategies: a) direct removal of interfering C10+ and b) vanadium oxidation to VO+. The performance of three reaction gases that are suitable for handling vanadium interference in the dynamic reaction cell was systematically studied and evaluated: ammonia for C10+ removal and oxygen and nitrous oxide for oxidation. Although it was able to produce comparable results for vanadium to those using oxygen and nitrous oxide, NH3 did not completely eliminate a matrix effect, caused by the presence of chloride, and required large scale dilutions (and a concomitant increase in variance) when the sample and/or the digestion medium contained large amounts of chloride. Among the three candidate reaction gases at their optimized Eonditions, creation of VO+ with oxygen gas delivered the best analyte sensitivity and the lowest detection limit (2.7 ng L-1). Vanadium results obtained from fourteen lake sediment samples and a certified reference material (CRM031-040-1), using two different analytelinterference separation strategies, suggested that the vanadium mono-oxidation offers advantageous performance over the conventional method using NH3 for ultra-trace vanadium determination by ICP-DRC-MS and can be readily employed in relevant environmental chemistry applications that deal with ultra-trace contaminants.Part II: Validation of a modified oxidation approach for the quantification of total arsenic and selenium in complex environmental matrices Spectroscopic interference problems of arsenic and selenium in ICP-MS practices were investigated in detail. Preliminary literature review suggested that oxygen could serve as an effective candidate reaction gas for analysis of the two elements in dynamic reaction cell coupled ICP-MS. An accurate method was developed for the determination of As and Se in complex environmental samples, based on a series of modifications on an oxidation approach for As and Se previously reported. Rhodium was used as internal standard in this study to help minimize non-spectral interferences such as instrumental drift. Using an oxygen gas flow slightly higher than 0.5 mL min-I, arsenic is converted to 75 AS160+ ion in an efficient manner whereas a potentially interfering ion, 91Zr+, is completely removed. Instead of using the most abundant Se isotope, 80Se, selenium was determined by a second most abundant isotope, 78Se, in the form of 78Se160. Upon careful selection of oxygen gas flow rate and optimization ofRPq value, previous isobaric threats caused by Zr and Mo were reduced to background levels whereas another potential atomic isobar, 96Ru+, became completely harmless to the new selenium analyte. The new method underwent a strict validation procedure where the recovery of a suitable certified reference material was examined and the obtained sample data were compared with those produced by a credible external laboratory who analyzed the same set of samples using a standardized HG-ICP-AES method. The validation results were satisfactory. The resultant limits of detection for arsenic and selenium were 5 ng L-1 and 60 ng L-1, respectively.