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.

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.

Niagara-Welland Power Company Limited fonds, 1833, 1852, 1904-1906, 1909, n.d.

The Welland Power and Supply Canal Company Limited, established in 1893 and incorporated in 1894 with a capital stock of $500,000. The aim of the company was to harness the natural water supply of the Niagara and Welland Rivers. In 1898 the Canadian Electrical News published a report by Henry Symons, QC outlining the main project of the company. This project involves the construction of a canal from the Welland River to the brow of the mountain at Thorold, a distance of 8 miles; the construction at Thorold of a power house, and from Thorold to Lake Ontario, a raceway by which to carry water into the lake. The estimate for the machinery to generate 100,000 horse power is £125,000; for transmission line to Toronto at a voltage of 10,000….The total estimate therefore amounts to £2,452,162, or roughly speaking, $12,000,000. Source: Canadian Electrical News, August 1898, p. 172. In 1899 the company officers petitioned the federal government desiring a name change to the Niagara-Welland Power Company Limited. Officers of the company were Harry Symons, President; Charles A. Hesson, Vice-President; and M.R. O’Loughlin, James B. Sheehan, James S. Haydon, Frederick K. Foster, directors; John S. Campbell, secretary-treasurer. The company’s head offices were located in St. Catharines, with a New York (City) office on Broad Street. In 1905 and 1909 the company petitioned the federal government for additional time to construct its works, which was granted. The company had until May 16, 1915 to complete construction. John S. Campbell (1860-1950) was a graduate of the University of Toronto and Osgoode Hall. During his university years John began his military career first in "K" Company, Queens Own rifles and then later as Commanding Officer of the 19th Lincoln Regiment, from 1906 to 1910. Upon his return to St. Catharines John Campbell served as secretary in the St. Catharines Garrison Club, a social club for military men begun in 1899. After being called to the Bar, he became a partner in the firm of Campbell and McCarron and was appointed to the bench in 1916, serving until retirement in 1934. Judge Campbell served as an alderman for several terms and was the mayor of St. Catharines in 1908 and 1909. He also served as the first chairman of the St. Catharines Public Utilities in 1914. John S. Campbell was married to Elizabeth Oille, daughter of Jerome B. and Charlotte (St. John) Oille. The family home "Cruachan" was located at 32 Church St.

Brown Bess Musket

This original Brown Bess musket is an East India Pattern musket (3rd model) as it has a 39 inch barrel, a brass side plate on the opposite side of the lock and has no tail to it. The tang on the butt plate is also short. The lock mechanism reveals that this musket has been converted to a percussion cap musket, which became popular in the 1820s. It was assembled at the Tower of London and is engraved with a crown, "GR" and an "X", on the top of the barrel. There is also a ".W. Reg.t" marking on the side of the barrel. This rare musket is typical of the muskets carried by British and Canadian soldiers during the War of 1812 and the Rebellion of 1837.

Ann Eliza Hepburne Rooth fonds, 1837-1897

Ann Eliza Hepburne was born in Chippawa, Ontario, in 1821, to William Hepburne and Susan Shannon. In 1842, she married William Anthony Rooth in St. James Cathedral in Toronto. They continued to live in different parts of the Niagara region, including Drummondville, Welland and Port Colborne. William was the editor and proprietor of the Drummondville Reporter, as well as an accountant and insurance agent, and later worked for the Customs Service in Port Colborne. He died in 1878, and Eliza in 1899. Both are buried in Drummond Hill Cemetery in Niagara Falls, Ontario.

Friendship Album

The album includes an illustration of a dog howling, by one of Eliza's friends. There is also a painting of roses. There is a poem signed Denison, that reads:" In truth it is not every book That's suited to the mind; In some forever you may look and no amusement find. But seldom does an album fail To please both grave and gay; It teams with many a merry tale and many a mournful day. Then reader know, whoever thou be Wise, witty, gay or sad; It's like the world in some degree Made up of good and bad". Another poem of eight verses is signed A. McNab. A poem and illustration are included by George Coventry. The illustration shows a man (presumed to be Coventry) looking at a book while others fish and work. Another poem initialed W.A.R. is thought to be by William Anthony Rooth, it is called "To Caroline". Also included in the album is a note from a Major who stayed at Eliza's home Christmas Day 1837. The note thanks Eliza for caring for him while he was ill at her home. A poem by Eliza's brother-in-law, Oliver T. Macklem is also included in the album. An illustration of two birds by Benjamin, Eliza's son, he was ten years old at the time. There is a poem written by L. D. Raymond with an attached newspaper clipping from Welland. The clipping is from L. D. Raymond's 79th birthday and is also a poem, "To the Old Barrister". There is a page of soldier autographs from 1866, those who fought in the Battle of Ridgeway during the Fenian Raid. (http://www.wikitree.com/wiki/Space:Ann_Eliza_Hepburne_Rooth%27s_1837_Album)

Photograph - Hunting Group with Percy Band

A group of men with rifles and hunting dogs. The photograph is dark and cloudy making it difficult to see details. Percy Band is sitting in front.