Molecular dynamics calculation of mean square displacement in alkali metals and rare gas solids and comparison with lattice dynamics

Molec ul ar dynamics calculations of the mean sq ua re displacement have been carried out for the alkali metals Na, K and Cs and for an fcc nearest neighbour Lennard-Jones model applicable to rare gas solids. The computations for the alkalis were done for several temperatures for temperature vol ume a swell as for the the ze r 0 pressure ze ro zero pressure volume corresponding to each temperature. In the fcc case, results were obtained for a wide range of both the temperature and density. Lattice dynamics calculations of the harmonic and the lowe s t order anharmonic (cubic and quartic) contributions to the mean square displacement were performed for the same potential models as in the molecular dynamics calculations. The Brillouin zone sums arising in the harmonic and the quartic terms were computed for very large numbers of points in q-space, and were extrapolated to obtain results ful converged with respect to the number of points in the Brillouin zone.An excellent agreement between the lattice dynamics results was observed molecular dynamics and in the case of all the alkali metals, e~ept for the zero pressure case of CSt where the difference is about 15 % near the melting temperature. It was concluded that for the alkalis, the lowest order perturbation theory works well even at temperat ures close to the melting temperat ure. For the fcc nearest neighbour model it was found that the number of particles (256) used for the molecular dynamics calculations, produces a result which is somewhere between 10 and 20 % smaller than the value converged with respect to the number of particles. However, the general temperature dependence of the mean square displacement is the same in molecular dynamics and lattice dynamics for all temperatures at the highest densities examined, while at higher volumes and high temperatures the results diverge. This indicates the importance of the higher order (eg. ~* ) perturbation theory contributions in these cases.

Irrigation scheduling for Sovereign Coronation grapevine based upon evapotranspiration calculations and crop coefficients /

Several irrigation treatments were evaluated on Sovereign Coronation table grapes at two sites over a 3-year period in the cool humid Niagara Peninsula of Ontario. Trials were conducted in the Hippie (Beamsville, ON) and the Lambert Vineyards (Niagara-on-the-Lake, ON) in 2003 to 2005 with the objective of assessing the usefulness of the modified Penman-Monteith equation to accurately schedule vine irrigation needs. Data (relative humidity, windspeed, solar radiation, and temperature) required to precisely calculate evapotranspiration (ETq) were downloaded from the Ontario Weather Network. One of two ETq values (either 100 or 150%) were used in combination with one of two crop coefficients (Kc; either fixed at 0.75 or 0.2 to 0.8 based upon increasing canopy volume) to calculate the amount of irrigation water required. Five irrigation treatments were: un irrigated control; (lOOET) X Kc =0.75; 150ET X Kc =0.75; lOOET X Kc =0.2-0.8; 150ET X Kc =0.2-0.8. Transpiration, water potential (v|/), and soil moisture data were collected each growing seasons. Yield component data was collected and berries from each treatment were analyzed for soluble solids (Brix), pH, titratable acidity (TA), anthocyanins, methyl anthranilate (MA), and total volatile esters (TVE). Irrigation showed a substantial positive effect on transpiration rate and soil moisture; the control treatment showed consistently lower transpiration and soil moisture over the 3 seasons. Transpiration appeared accurately reflect Sovereign Coronation grapevines water status. Soil moisture also accurately reflected level of irrigation. Moreover, irrigation showed impact of leaf \|/, which was more negative throughout the 3 seasons for vines that were not irrigated. Irrigation had a substantial positive effect on yield (kg/vine) and its various components (clusters/vine, cluster weight, and berries/cluster) in 2003 and 2005. Berry weights were higher under the irrigated treatments at both sites. Berry weight consistently appeared to be the main factor leading to these increased yields, as inconsistent responses were noted for some yield variables. Soluble solids was highest under the ET150 and ET100 treatments both with Kc at 0.75. Both pH and TA were highest under control treatments in 2003 and 2004, but highest under irrigated treatments in 2005. Anthocyanins and phenols were highest under the control treatments in 2003 and 2004, but highest under irrigated treatments in 2005. MA and TVE were highest under the ET150 treatments. Vine and soil water status measurements (soil moisture, leaf \|/, and transpiration) confirmed that irrigation was required for the summers of 2003 and 2005 due to dry weather in those years. They also partially supported the hypothesis that the Penman-Monteith equation is useful for calculating vineyard water needs. Both ET treatments gave clear evidence that irrigation could be effective in reducing water stress and for improving vine performance, yield and fruit composition. Use of properly scheduled irrigation was beneficial for Sovereign Coronation table grapes in the Niagara region. Findings herein should give growers some strong guidehnes on when, how and how much to irrigate their vineyards.

Molecular mechanics calculations of tin-bis (triphenylphosphine oxide) complexes

The capability of molecular mechanics for modeling the wide distribution of bond angles and bond lengths characteristic of coordination complexes was investigatecl. This was the preliminary step for future modeling of solvent extraction. Several tin-phosphine oxide COrnI)le:){es were selected as the test groUl) for t.he d,esired range of geometry they eX!libi ted as \-vell as the ligands they cOD.tained r Wllich were c\f interest in connection with solvation. A variety of adjustments were made to Allinger's M:M2 force·-field ill order to inl.prove its performance in the treatment of these systems. A set of u,nique force constants was introduced for' those terms representing the metal ligand bond lengths, bond angles, and, torsion angles. These were significantly smaller than trad.itionallY used. with organic compounds. The ~1orse poteIlt.ial energ'Y function was incorporated for the M-X l')ond lE~ngths and the cosine harmonic potential erlerg-y function was invoked for the MOP bond angle. These functions were found to accomodate the wide distribution of observed values better than the traditional harmonic approximations~ Crystal packing influences on the MOP angle were explored thr"ollgh ttle inclusion of the isolated molecule withil1 a shell cc)ntaini11g tl1e nearest neigl1'bors duri.rlg energy rninimization experiments~ This was found to further improve the fit of the MOP angle.

Molecular mechanics calculations on some phosphine oxide metal complexes in aqueous and organic solvents

Molecular mechanics calculations were done on tetrahedral phosphine oxide zinc complexes in simulated water, benzene and hexane phases using the DREIDING II force field in the BIOGRAF molecular modeling program. The SUN workstation computer (SUN_ 4c, with SPARK station 1 processor) was used for the calculations. Experimental structural information used in the parameterization was obtained from the September 1989 version of the Cambridge Structural Database. 2 Steric and solvation energies were calculated for complexes of the type ZnCl2 (RlO)2' The calculations were done with and without inclusion of electrostatic interactions. More reliable simulation results were obtained without inclusion of charges. In the simulated gas phase, the steric energies increase regularly with number of carbons in the alkyl group, whereas they go through a maximum when solvent shells are included in the calculation. Simulated distribution ratios vary with chain length and type of chain branching and the complexes are found to be more favourable for extraction by benzene than by hexane, in accord with experimental data. Also, in line with what would be expected for a favorable extraction, calculations without electrostatics predict that the complexes are better solvated by the organic solvents than by water.

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.

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.

Polytypism and Silicon carbide : a solid state nuclear magnetic resonance study

A survey of predominantly industrial silicon carbide has been carried out using Magic Angle Spinning nuclear magnetic resonance (MAS nmr); a solid state technique. Three silicon carbide polytypes were studied; 3C, 6H, and 15R. The 13C and 29 Si MAS nmr spectra of the bulk SiC sample was identified on the basis of silicon (carbon) site type in the d iff ere n t pol Y t Y pes • Out to 5.00 A fro mac en t r a lsi 1 i con (0 r carbon) atom four types of sites were characterized using symmetry based calculations. This method of polytype analysis was also considered, in the prelminary stages, for applications with other polytypic material; CdBr 2 , CdI 2 , and PbI 2 " In an attempt to understand the minor components of silicon carbide, such as its surface, some samples were hydrofluoric acid washed and heated to extreme temperatures. Basically, an HF removable species which absorbs at -110 ppm (Si0 2 ) in the 29 Si MAS nmr spectrum is found in silicon carbide after heating. Other unidentified peaks observed at short recycle delays in some 29 Si MAS nmr spectra are considered to be impurities that may be within the lattice. These components comprise less than 5% of the observable silicon. A Tl study was carried out for 29 Si nuclei in a 3C ii polytype sample, using the Driven Equilibrium Single-Pulse Observation of T1 (DESPOT) technique. It appears as though there are a number of nuclei that have the same chemical shift but different T1 relaxation times. The T1 values range from 30 seconds to 11 minutes. Caution has to be kept when interpreting these results because this is the first time that DESPOT has been used for solid samples and it is not likely in full working order. MAS nmr indicates that the 13C and 29 Si ~sotropic chemical shifts of silicon carbide appear to have a reciprocal type of relationship_ Single crystal nmr analysis of a 6H sample is accordance with this finding when only the resultant isotropic shift is considered. However, single crystal nmr also shows that the actual response of the silicon and carbon nuclear environment to the applied magnetic field at various angles is not at all reciprocal. Such results show that much more single crystal nmr work is required to determine the actual behavior of the local magnetic environment of the SiC nuclei.

Reduced local energy calculations on X¹Sigmaâ ½gHâ and 1¹S He /|nGerald F. Thomas. -- 260 St. Catharines [Ont.] : Dept. of Chemistry, Brock University,

The one-electron reduced local energy function, t ~ , is introduced and has the property < tL)=(~>. It is suggested that the accuracy of SL reflects the local accuracy of an approximate wavefunction. We establish that <~~>~ <~2,> and present a bound formula, E~ , which is such that where Ew is Weinstein's lower bound formula to the ground state. The nature of the bound is not guaranteed but for sufficiently accurate wavefunctions it will yield a lower bound. ,-+ 1'S I I Applications to X LW Hz. and ne are presented.

An anharmonic contribution to the Helmholtz free energy O(lambda 6)

The anharmonic contributions of order A6 to the Helmholtz free energy for a crystal in which every atom is on a site of inversion symmetry, have been evaluated The cor~esponding diagrams in the various orders of the perturbation theory have been presented The validity of the expressions given is for high temperatures. Numerical calculations for the diagrams which contribute to the free energy have been worked out for a nearest-n~ighbour central-force model of a facecentered cubic lattice in the high-temperature limit and in the leading term and the Ludwig approximations. The accuracy of the Ludwig approximation in evaluating the Brillouin-zone sums has been investigated. Expansion for all diagrams in the high-temperature limit has been carried out The contribution to the specific heat involves a linear as well as cubic term~ We have applied Lennard-Jones, Morse and Exponential 6 types of potentials. A comparison between the contribution to the free energy of order A6 to that of order A4 has been made.

A29 NMR spin-lattice relaxation study of paramagnetics -doped orthosilictes

A ~si MAS NMR study of spin-lattice relaxation behaviour in paramagnetic-doped crystalline silicates was undertaken, using synthetic magnesium orthosilicate (forsterite) and synthetic zinc orthosilicate (willemite) doped with 0.1% to 20% of Co(II), Ni(II), or CU(II), as experimental systems. All of the samples studied exhibited a longitudinal magnetization return to the Boltzmann distribution of nuclear spin states which followed a stretched-exponential function of time: Y=exp [- (tjTn) n], Olattice relaxation time and paramagnetic dopant ion concentration, with Tni[M2+]i=Tnj[M2+]j for a given dopant and mineral. There are many cases where this correlation is not apparent, however, and this is attributed to the structural, phase, and ion distribution complexities inherent in many of these systems.

Tintype of Girl Peering through Lattice [n.d.]

A young African American girl peers through a lattice window in an unknown photographer's studio in this small black and white tintype. The girl in the photo is unidentified. There are slight scratches on the surface of the tintype, which is cut in a hexagonal shape. This tintype was in the possession of Iris Sloman Bell, of St. Catharines, Ontario. The Sloman - Bell families have relatives who are descended from Black slaves from the United States."Tintypes were the invention of Prof. Hamilton Smith of Ohio. They begin as thin sheets of iron, covered with a layer of black paint. This serves as the base for the same iodized collodion coating and silver nitrate bath used in the ambrotype process. First made in 1856, millions were produced well into the twentieth century. When tintypes were finished in the same sorts of mats and cases used for ambrotypes, it can be almost impossible to distinguish which process was used without removing the image to examine the substrate." Source: American Museum of Photography http://www.photographymuseum.com/primer.html

Calculations Regarding S.D. Woodruff's Property on Ontario Street

Calculations regarding S.D. Woodruff’s property on Ontario Street. 1 page of rough penciled calculations, n.d.

Calculations regarding quantities needed for cutting and ditching

Calculations regarding quantities needed for cutting and ditching (1 page, handwritten), March 19, 1884.

Page of scribbled notes and calculations, 1855

Page of scribbled notes and calculations, 1855.

Sketch of the bridge over Hurst’s Point and calculations about the bridge over the pond at Allenburgh

Sketch of the bridge over Hurst’s Point and calculations about the bridge over the pond at Allenburgh, n.d.