Bell’s Inequalities express constraints on the correlations of three random, binary variables and they can be applied to the interpretation of quantum mechanics. In an analysis of the theory behind these inequalities, this paper suggests a geometrical interpretation of Bell’s Inequality in the form of a tetrahe- dron in “correlation space” to which correlation measurements are restricted. Using correlated photons produced through spontaneous parametric downconversion, we were able to experimentally demonstrate a set of measurements lying outside of this tetrahedron, negating the binary nature of hidden-variable theories.

Bell's Inequality, hidden variables, EPR paradox, correlation space

[1] A. Einstein, B. Podolsky, and N. Rosen, Can Quantum Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev. 47, 777-780 (1935).
[2] J.S. Bell, On the Einstein Podolsky Rosen Paradox, Phys. (N.Y.) 1, 195-200 (1964).
[3] J.F. Clauser,M.A. Horne, and A. Shimony, R.A. Holt, Proposed Experiment to Test Local Hidden-Variable Theories, Phys. Rev. Lett. 23 (15), 880-884 (1969).
[4] A. Aspect, P. Grangier, and G. Roger, Experimental Tests of Realistic Local Theories via Bell’s Theorem, Phys. Rev. Lett. 47 (7), 460-463 (1981).
[5] D. Dehlinger and M.W. Mitchell, Entangled Photons, Nonlocality, and Bell Inequalities in the Under-graduate Laboratory, Am. J. Phys. 70 (9), 903-910 (2002).
[6] M. Born and A. Einstein, The Born-Einstein Letters: The Correspondence Between Albert Einstein and Max and Hedwig Born, 1920-1955 (Walker, New York, 1971), pp. 157-160.
[7] J.S. Bell, On the Problem of Hidden Variables in Quantum Mechanics, Rev. Mod. Phys. 38 (3), 447-452 (1966).
[8] P.G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, New High-Intensity Source of Polarization-Entangled Photon Pairs, Phys. Rev. Lett. 75 (24), 4337-4341 (1995).

In this paper, we study the role of each of the four chemical components of a catalyst system used in growing carbon nanotubes (CNTs). Our goal is to be able to grow desirable carbon nanotubes by chemical vapor deposition (CVD), which is believed to be the most practical growth method for CNTs. The catalyst used in our process is known for inducing Y-junction CNTs. To the best of our knowledge, there has not been a thorough investigation on the importance of each chemical component used during the CNT growth. To identify the impact that each component has on CNT growth, we prepared nine samples by either excluding or doubling the amount of each component (iron nitrate, aluminum oxide, and molybdenum). We used one of three different solvents including methanol, DI water, or ethanol for the catalyst solutions. We found that 1) all three catalyst components are needed in effective growth of CNTs; 2) molybdenum has a significant role in CNT growth and y-junctions in our system; 3) the solvent has noteworthy effect on the degree of CNT crystallinity of the nanotubes and 4) the growth of multiple samples at the same time may cause interaction from one sample to the next.

Y-Junction, Iron Nitrate Nonahydrate, Aluminum Oxide, and Molybdenum Acetylacetonate

1. C. Oncel, Y. Yurum. Carbon Nanotube Synthesis via the Catalytic CVD Method: A Review on the Effect of Reaction Parameters. Fullerenes, Nanotubes, and Carbon Nanostructures, 14 (2006): 17–37.
2. B. Duong, Y. Peng, M. Ellis, S.Seraphin, H. Xin. Combined Raman Spectroscopy and SEM Analysis of Chemical Vapor Deposition Grown Carbon Nanotubes. Microscopy Society of America and the Microbeam Analysis Society annual conference. (2008) Poster.
3. J. Kong, H. T. Soh, A. M. Cassell, C. F. Quate, H. Dai. Synthesis of individual singlewalled carbon nanotubes on patterned siliconwafers. Nature 395. (1998) : 878 -881.
4. A. R. Harutyunyan, B. K. Pradhan, U. J. Kim, G.Chen, and P. C. Eklund. CVD Synthesis of Single Wall Carbon Nanotubes under “Soft” Conditions. Nano Letters 2.5 (2002): 525-530.
5. Y. C. Choi, W. Choi. Synthesis of Y-junction single-walled carbon nanotubes. Carbon. 43 (2005): 2737-41.
6. A. Jorio, M. A. Pimenta, A. G. Souza Filho, R. Saito, G. Dresselhaus and M. S. Dresselhaus. Characterizing carbon nanotube samples with resonance Raman scattering New Journal of Physics 5 (2003) 139.1–139.17
7. S.Costa, E. Borowiak-Pale, M. Krusznynska, A. Bachmatiuk, R. J. Kalenczuk. Characterization of carbon nanotubes by Raman spectroscopy. Mater Sci-Poland 26.2 (2008): 433-441.
8. G. P. Huffman, N. Shah, Y. Wang, F. E. Huggins. Catalytic Dehydrogenation of Hydrocarbons: Alternative, One Step Process to Produce Pure Hydrogen and Carbon Nanotube Byproducts. Prepr. Pap.-Am. Chem. Soc., Div. Fuel Chem 49.2 (2004): 731-732.
9. J. Kong, A. M. Cassell, H. Dai. Chemical vapor deposition of methane of single-walled carbon nanotubes. Chem. Phys. Lett. 292 (1998): 567-74.
10. S.P. Chai, S.H.S. Zein, and A.R. Mohamed, A Review On Carbon Nanotubes Production Via Catalytic Methane Decomposition. In: 1st National Postgraduate Colloquim NAPCOL (2004).
11.S. Curtarolo, N. Awasthi, W. Setyawan, A. Jiang, K. Bolton, T. Tokune, and A. R. Harutyunyan. Influence of Mo on the Fe:Mo:C nano-catalyst thermodynamics for single-walled carbon nanotube growth, Phys. Rev. B 78.5 (2008).
12. H.J. Dai, A.G. Rinzler, P. Nikolaev, A. Thess, D.T. Colbert, R.E. Smalley. Single-wall nanotubes produced by metal-catalyzed disproportionation of carbon monoxide. Chem. Phys. Lett. 260 (1996): 471-475

In this study we look at the SALT-II model of Type IA supernova analysis, which determines the distance moduli based on the known absolute standard candle magnitude of the Type IA supernovae. We take a look at the determination of the shape and color parameter coefficients, α and β respectively, in the SALTII model with the intrinsic error that is determined from the data. Using the SNANA software package provided for the analysis of Type IA supernovae, we use a standard Monte Carlo simulation to generate data with known parameters to use as a tool for analyzing the trends in the model based on certain assumptions about the intrinsic error. In order to find the best standard candle model, we try to minimize the residuals on the Hubble diagram by calculating the correct shape and color parameter coefficients. We can estimate the magnitude of the intrinsic errors required to obtain results with χ2/degree of freedom = 1. We can use the simulation to estimate the amount of color smearing as indicated by the data for our model. We find that the color smearing model works as a general estimate of the color smearing, and that we are able to use the RMS distribution in the variables as one method of estimating the correct intrinsic errors needed by the data to obtain the correct results for α and β. We then apply the resultant intrinsic error matrix to the real data and show our results.

Sloan Digital Sky Survey, SALT-II, Supernova Analysis

Gunn, J. E. et al. 1998 AJ, 116, 3040
Gunn, J. E. et al. 2006, AJ, 131, 2332
Guy, J. et al. 2007, A&A, 466, 11
Frieman, J. A. et al. 2008, AJ, 135, 338
Fukugita, M. et al. 1996, AJ, 111, 1748
Hayden, B. et al. 2010, ApJS, published (arXiv:astroph/ 1001.3428)
Ivezi´c, Z. et al. 2004, Astronomische Nachrichten, 325, 583
Kessler, R. et al. 2009a, ApJS, accepted for publication (arXiv:astro-ph/0908.4274)
Kessler, R. et al. 2009b, PASP, accepted for publication (arXiv:astrop-ph/0908.4280)
Kessler, R. et al. 2009c, ApJ, accepted for publication (arXiv:astro-ph/1001.0738v2)
Lupton, R. et al. 2001, in ASP Conf. Ser. 238: Astronomical Data Analysis Software and Systems X, ed. F. R. Harnden, Jr., F. A. Primini, & H. E. Payne, 269-+
Press, W. H. et. al. 1988, Numerical Recipes in C, (Cambridge University Press, New York, NY)
Sako, M. et al. 2008, AJ, 135, 348

In order to determine the first four energy levels of our anharmonic potential, we will compute the first four eigenvalues of the anharmonic oscillator potential with a quartic term using Heun polynomials and Maple software packages. We will then compare them to the results obtained from the conventional perturbation method, treating the quartic term as perturbation up to the third order. Through this it can be shown that generally the two methods agree well with each other when the perturbing potential is weak. Nevertheless, the perturbation results will start to deviate from those of the exact solutions at stronger perturbation potentials and higher excited states.

Perturbation Method, Anharmonic Potential, Anharmonic Oscillator

1. C. M. Bender and T. T. Wu, Phys. Rev., 184, 1231-1260, (1969).
2. A. Decarreau, M. C. Dumont-Lepage, P. Maroni, A. Robert, and A. Ronveaux, "Formes Canoniques de Equations confluentes de l'equation de Heun." Annales de la Societe Scientifique de Bruxelles, Vol. I-II. (1978): 53-78. A. Ronveaux, ed. “Heun's Differential Equations”. Oxford, England: Oxford University Press, (1995). S. Y. Slavyanov and W. Lay, “Special Functions, A Unified Theory Based on Singularities”. Oxford, England: Oxford Mathematical Monographs, (2000).
3. D. J. Griffiths, “Introduction to Quantum Mechanics” (Upper Saddle River, NJ, Pearson Preston Hall, 2005, second edition) Chapter 6, and the references cited therein, especially Footnote 5 on P. 256.
4. ibid, Chapter 2, Section 2.3.
5. J. Koch, C. Schuck, and B. Wacker, "Excited states of the anharmonic oscillator problems: variational method", Journal of Undergraduate Research in Physics 21, (2008). http://www.jurp.org/
6. E. R. Hedgahl, T. L. Johnson III, S. E. Schnell, and A. R. Ward, "Systematic convergence in applying the variational method to the anharmonic oscillator potentials”, Journal of Undergraduate Research in Physics 21, (2008). http://www.jurp.org/