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.

Square Root Finding In Graphs

Abstract: Root and root finding are concepts familiar to most branches of mathematics. In graph theory, H is a square root of G and G is the square of H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Graph square is a basic operation with a number of results about its properties in the literature. We study the characterization and recognition problems of graph powers. There are algorithmic and computational approaches to answer the decision problem of whether a given graph is a certain power of any graph. There are polynomial time algorithms to solve this problem for square of graphs with girth at least six while the NP-completeness is proven for square of graphs with girth at most four. The girth-parameterized problem of root fining has been open in the case of square of graphs with girth five. We settle the conjecture that recognition of square of graphs with girth 5 is NP-complete. This result is providing the complete dichotomy theorem for square root finding problem.

Interactions of onion (Allium cepa) and yellow wax bean (Phaseolus vulgaris) in monoculture and intercropping with weeds, Chenopodium album and Amaranthus hybridus

Intercropping systems are seen as advantageous as they can provide higher crop yield and diversity along with fewer issues related to pests and weeds than monocultures. However, plant interactions in intercropped crop species and between crops and weeds in these systems are still not well understood. The main objective of this study was to investigate interactions between onion (Allium cepa) and yellow wax bean (Phaseolus vulgaris) in monocultures and intercropping with and without the presence of a weed species, either Chenopodium album or Amaranthus hybridus. Another objective of this study was to compare morphological traits of C. album from two different populations (conventional vs. organic farms). Using a factorial randomized block design, both crop species were planted either in monoculture or intercropped with or without the presence of one of the two weeds. The results showed that intercropping onion with yellow wax bean increased the growth of onion but decreased the growth of yellow wax bean when compared to monocultures. The relative yield total (RYT) value was 1.3. Individual aboveground dry weight of both weed species under intercropping was reduced about 5 times when compared to the control. The poor growth of weeds in intercropping might suggest that crop diversification can help resist weed infestations. A common garden experiment indicated that C. album plants from the conventional farm had larger leaf area and were taller than those from the organic farm. This might be associated with specific evolutionary adaptation of weeds to different farming practices. These findings contribute to the fundamental knowledge of crop-crop interactions, crop-weed competition and adaptation of weeds to various conditions. They provide insights for the management of diversified cropping systems and integrated weed management as practices in sustainable agriculture.

Place Card - Girl in Blue Dress on Yellow Chest

A place card with an illustration of a girl in a blue dress sitting on a yellow chest.

Place Card - Girl in Red Dress on Yellow Chest

A place card with an illustration of a girl in a red dress sitting on a yellow chest.

License no. 144 of season 1872/73 made out to S.D. Woodruff for 36 square miles in berth no. 192

License no. 144 of season 1872/73 made out to S.D. Woodruff for 36 square miles in berth no. 192. This document is slightly torn and stained along the right hand side. This does not affect the text, April 7, 1873.

License no. 145 of season 1872/73 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198

License no. 145 of season 1872/73 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198, April 7, 1873.

License no. 67 of season 1873/74 made out to S.D. Woodruff for 36 square miles in berth no. 192

License no. 67 of season 1873/74 made out to S.D. Woodruff for 36 square miles in berth no. 192, June 13, 1873

License no. 68 of season 1873/74 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198

License no. 68 of season 1873/74 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198, June 13, 1873.

License no. 11 of season 1874/75 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198

License no. 11 of season 1874/75 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198, May 20, 1874.

License no. 5 of season 1875/76 made out to S.D. Woodruff for 36 square miles in berth no. 192

License no. 5 of season 1875/76 made out to S.D. Woodruff for 36 square miles in berth no. 192, June 1, 1875.

License no. 2 of season 1886/87 made out to S.D. Woodruff for 36 square miles in berth no. 192

License no. 2 of season 1886/87 made out to S.D. Woodruff for 36 square miles in berth no. 192, May 15, 1876.

License no. 3 of season 1886/87 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198

License no. 3 of season 1886/87 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198, May 15, 1876